SLAC-R-574 18 th Advanced ICFA Beam Dynamics Workshop on Quantum Aspects of Beam Physics Pisin Chen, Editor Stanford Linear Accelerator Center Stanford University Stanford, CA 94309 SLAC-Report-574 April 2003 Prepared for the Department of Energy under contract number DE-AC03-76SF00515 Printed in the United States of America. Available from the National Technical Information Service, U.S. Department of Commerce, 5285 Port Royal Road, Springfield, VA 22161.
Transcript
SLAC-R-574
18th Advanced ICFA Beam Dynamics Workshop on Quantum Aspects of Beam Physics
Pisin Chen, Editor
Stanford Linear Accelerator Center Stanford University Stanford, CA 94309
SLAC-Report-574 April 2003
Prepared for the Department of Energy under contract number DE-AC03-76SF00515
Printed in the United States of America. Available from the National Technical Information Service, U.S. Department of Commerce, 5285 Port Royal Road, Springfield, VA 22161.
Table of Contents
A. Quantum Fluctuations in Beam Dynamics
The Diffractive Quantum Limits of Particle Colliders. . . . . . . . . . . . . . . . . . . . 3C.T. Hill
Quantum Fluctuations in Free-Electron Lasers . . . . . . . . . . . . . . . . . . . . . . . . . . 19C.B. Schroeder, C. Pellegrini, and P. Chen
Quantum Models in Beam Physics and Signal Analysis . . . . . . . . . . . . . . . . . .564M. Manko
Radiative Corrections in Symmetrized Classical Electrodynamics . . . . . . . . 574J.R. Van Meter, A.L. Troha, D.J. Gibson, A.K. Kerman,P. Chen, and F.V. Hartemann
PREFACE
The 18th Advanced ICFA Beam Dynamics Workshop on “Qauntum Aspects ofBeam Physics” was held from October 15 to 20, 2000, in Capri, Italy. This wasthe second workshop under the same title. The first one was held in Monterey,California, in January, 1998. Following the footstep of the first meeting, thesecond one in Capri was again a tremendous success, both scientifically andsocially.
About 70 colleagues from astrophysics, atomic physics, beam physics,condensed matter physics, particle physics, and general relativity gatheredto update and further explore the topics covered in the Monterey workshop.Namely, the following topics were actively discussed:
1. Quantum Fluctuations in Beam Dynamics;2. Photon–Electron Interaction in Beam handling;3. Physics of Condensed Beams;4. Beam Phenomena under Strong Fields;5. Quantum Methodologies in Beam Physics.
In addition, there was a newly introduced subject on6. Astro-Beam Physics and Laboratory Astrophysics.
The nature of the so-called “Unruh radiation,” an analog of the famousHawking radiation when a particle is undergone a violent acceleration, waswarmly discussed (or even debated) during the Monterey Conference. In viewof the rapidly growing interest in laboratory investigations of astrophysicalphenomena, the organizers of the Capri workshop has decided to formallyrecognize these activities with the name “astro-beam physics,” and a dedi-cated working group formed.
The conference started with a series of plenary talks that providedoverviews on the progress of the particular subjects within quantum beamphysics during the past two years, as well as reports on the speakers’ recentimportant findings. While all plenary talks were outstanding, the ones givenin the Astrophysics Session by F. Mirabel, R. Ruffini, and L. Scarsi, wereparticularly memorable due to some warm human elements. The six top-ics listed above were sorted into four Working Groups. The Working Groupchairs not only gave introductory overviews of the topics, but also gave Sum-mary Reports at the end of the conference. Their dedication was the key tothe sustained enthusiasm throughout the week. About 40 wonderful presen-tations were made during the parallel working group sessions that covered a
very wide range of exciting subjects. All the presentations, as well as newrevelations induced from the meeting, are now recorded in this book. Theexcellent program of the conference was due to a large extent to the inputsfrom the Advisory Committee and the planning of the Program Committee.
In addition to the effort of fund-raising from the various Italian scientificagencies, the conference Organizing Committee, led by Dr. Stefania Petracca(University of Benevento), has constructed a series of exciting social programsthroughout the week. On the opening night, there was a piano recital inthe church Certosa di San Giacomo by the famous pianist, Maestro FrancoisJoel Thiollier. On Wednesday, a whole-day excursion to Pompeii was made.During the conference banquet on Thursday night, a local folk song-and-dancegroup made a warm and inspiring performance on the famous Neapolitansongs. All participants were invited to join the performers in making musictogether. Of course, the beautiful scenery of the island of Capri required nopersuasion. Everyone left his/her heart there when we had to return to thereal world after the conference. On behalf of all the participants, I would liketo thank all the Organizing Committee members from various institutions inItaly, as well as the conference secretaries for their hard work.
Based on the continuing success of this workshop series, we are confidentthat the importance of quantum beam physics will continue to grow andwe invite all the colleagues to join in the exploration of quantum effects inlaboratory and astrophysical beam phenomena.
Pisin ChenStanford, California
Section 1
Quantum Fluctuations
in Beam Dynamics
THE DIFFRACTIVE QUANTUM LIMITS
OF PARTICLE COLLIDERS
C T HILL
Fermi National Accelerator Laboratory
PO Box Batavia Illinois
Email hillfnalgov
Quantum Mechanics places limits on achievable transverse beam spot sizes of particle accelerators We estimate this limit for a linear collider to be x hcfEwhere f is the nal focal length E the beam energy and the intrinsic transverse Gaussian width of the electron wavefunction is determined in the phasespace damping rings and for a linear focusing channel is the Gaussian width ofthe transverse groundstate wavefunction A crude estimate in a circular damping
ring yields p
hceB where B is the typical wiggler magnetic eld strength
in this system For the NLC nm and x O nm about two ordersof magnitude smaller than the design goal We can recover an estimate of theclassical result when we include radiative relaxation e ects We also consider aselfreplicating solution in a synchrotron and obtain x
phcfE O nm
We discuss formulation of quantum beam optics relevant to these issues
Introduction
Particle accelerators are designed and built based essentially upon the classical theory of point charges interacting with electromagnetism Neverthelessparticles are described by wavefunctions and diractive limits must existas to how well they can be localized in a given optical apparatus The rstquantum mechanical eects to arise in a potentially limiting way might beexpected to be diractive in nature In this paper we take a rst look atthe problem of estimating the quantum diractive limits of accelerators Webegin with an important system an idealized ee linear collider We areinspired to consider this because the desired goals for eg the NLC beamspot size are ambitious To achieve the desired luminosity requires a nmbeam spot in one transverse dimension the vertical or y direction in the NLCreports We will estimate that this criterion is about two orders of magnitude above the quantum limit Indeed we will describe how to estimate theclassical design result for the beam spot size itself from quantum mechanicsobtaining rough agreement with the NLC specications
What might we expect for a quantum diractive limit Assume we aretrying to hit a target of transverse size x Then we must require transverse
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momentum uncertainty px given by geometry
L L
pxpz
x ∆xL
∆θThe momentum uncertainty and target size must satisfy the Heisenberg
uncertainty relation
xpx h
Thus we obtain the limit
x s
hL
pzr
hcL
Erelativistic
Putting in numbers for the NLC E GeV L m eg nal focallength design goal of the NLC obtain
x nm compared to nm design goal
Are we therefore close to the quantum limitThe above argument contains fallacy px is the momentum uncertainty
at time t when projectile is launched Minimizing pxt implies abroad coherent nearplanewavefront conguration for the particle at launchpx in uncertainty principle is momentum spread at the time we hit thetarget at t Lc This must be large corresponding to the small localizationx required to hit a small target Thus we can evade the limit in principleby requiring a specially prepared wavefunction localization of particle in thetransverse dimension must shrink as the particle approaches the target
This occurs when a transverse wavepacket describing a quantum mechanical particle passes through a lens For our discussion we will use theParaxial Gaussian Transverse WavePacket Approximation Gaussians arenatural choice because they extremalize the Heisenberg uncertainty relationThe previous result will hold in a synchrotron but we obtain a dierent resultfor a linear collider
The linear collider result is conceptually simple The diractive limit onthe beam spot size in the x direction is given by the Rayleigh formula for amassless wave of energyE which has passed through an eective aperature and focused over a focal length f That is
x hcf
E
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where f is the overall nal focal length E the beam energy We emphasizethat the aperature is not a mechanical aperature eg it is not thebeam pipe size Rather is roughly the initial state Gaussian width of thetransverse quantum wavefunction as it enters the linac upstream from thedamping rings where the wavefunction has been prepared we ignore furthersynchrotron radiation downstream Indeed if the initial state were preparedin a linear focusing channel LFC the result would be exact and wouldthen be the true groundstate Gaussian width of the LFC Our simplifyingassumptions for the case of circular damping rings are crude and we expect aresult only to within an order of magnitude precision without further detailedmodeling
The initial state can be considered to be an ensemble of particles each insimple harmonic oscillator SHO transverse wavefunctions where the Gaussian envelope groundstate width is determined by the damping ring wigglersystem This is given by
rhc
eB
where B is the typical magnetic eld in the damping system of order TeslaFor a typical linear collider taking f m E GeV B Teslayields nm and thus x nm as a diracive limit Hencethe NLC beam spot size is safely above the quantum limit by about two ordersof magnitude
f should be viewed as the overall nal focal length of the compound lensFor example if the beam is magnied by a defocusing lens of f followedby a space of a then focused by a nal lens of f the resulting diractivesize becomes f f a and the diractive spot size fE is in principlereducible However the longitudinal depth of focus is then reduced by anamount af
In general the individual particle initial state is an excited SHO transverse wavefunction of average principle quantum number n This increasesthe expected diractive spot size to x pnfE In fact since n hit is easily seen that this result is independent of h and therefore should beequivalent to a classical derivation of the beam spot size n can be crudely estimated from radiation relaxation following the original arguments of Sands
This yields a result of x nm roughly consistent with the NLC designreport calculations for the vertical beam spot
Presently we construct a transverse Gaussian wavepacket with a longitudinal plane wave structure and propagate it through an optical system Gaussians extremalize the Heisenberg uncertainty relation and they
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are also the groundstate solutions in continuous linear focusing channels andeg magnetic lenses wigglers to a reasonable approximation etc and canbe approached by synchrotron radiation relaxation RemarkablyGaussian transverse wavefunctions which solve the quantum Schroedingerequation for propagation through the optical system neglecting synchrotronradiation are controlled entirely by the classical lens matrices of the system While Gaussian optics is a standard formalism in treating electronmicroscopy to our knowledge the behavior of a quantum Gaussian beamin a synchrotron has not been previously formulated and we will indicatethe selfreplicating solution to a synchrotron by an application of lens matrixmethods
Assume for simplicity that there is only one spatial transverse dimensionx and let z be the longitudinal spatial dimension In the KleinGordonequation we include a transverse simple harmonic oscillator SHO potentialterm which is dependent upon z For the analysis we set h c
m Kzx
Then with expiEt pzz and E pz m the KG equationbecomes the transverse Schoedinger equation
i
z
Er Kzx
where
K KE
z has the conventional interpretation with z replacing time The parameterKz is zdependent corresponding to the nite longitudinal structure of thelens system
Let us now postulate a Gaussian form for the wavefunction centered atthe transverse position x carrying a transverse momentum p
exp
Azx x ipx C
In this expression Az is the complex Gaussian kernel x and p are realand C simply parameterizes the overall normalization Hence the Gaussianwavefunction has four real parameters After substituting this anzatz theSchroedinger equation eq yields the following equations of motion for thewidth
iA
z
A
EKz
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and for x and p we obtain the classical Hamilton equations
pz
Kxxz
pE
Note that the centroid x and centroid momentum p motions are decoupledfrom that of the Gaussian kernel Az and vice versa anharmonic eectswould generally couple these quantities Moreover the boundary conditionson Az and of the centroid x and centroid momentum p are independentNote that the last equation is just the zvelocity expressed in terms of themomentum for a particle of mass E Remarkably eq can be seen tobe equivalent to the classical Hamilton equations by identifying Az iP zXz where P z and Xz are generalized complex momenta andpositions which satisfy the eqs
We remark that with particular boundary conditions ReAz can beseen to be equivalent to e where e is the Compton wavelength of theelectron and z is the accelerator lattice function as in the work ofVenturini presented at this meeting Venturinis construction is essentiallyequivalent to ours and makes contact with the parlance of the machine design parameters eg the function His boundary conditions appear to berestricted to a matched wavefunction ie one which lies within the envelope of the classical emittance phase space more extreme cases are allowed inprinciple
Now we impose an initial condition at z L that the particle has beenprepared into a transverse Gaussian wavepacket specied to have a pure realwidth given by
A ReAL ImAL
We assume that the centroid of the initial wavepacket is moving parallel tothe zaxis thus p and x x is initially arbitrary at z L
The wavepacket enters a lens at z z in which Kz K and exitsat z Upon entry of the lens Az is given by the free drift solution ofeq from z L to the lens over the drift distance L
Az A
iALE
where we assume a thin lens zL In the thin lens to a good approximation for small z we have from
eq
A Az izK
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Here we neglected the term izAE in the dierential equation for Azwhich only gives negligible free particle spreading in the lens Moreover theclassical centroid motion of the wavepacket is found from eq
p Kxz x x
Upon exiting the lens the particle propagates again in free space a distance with K Hence we nd
A A
iAE
and
p Kxz x x pE
Note that the classical trajectory of the oaxis particle is de ected backtoward the lens axis x
The focal length f is dened such that xf hence
f E
Kz
The kernel of the wavepacket can now be obtained by solving eqsrecursively to obtain
A
iLE
Ef
f iE LE LEf
Note that the Gaussian kernel has an imaginary part which changes frompositive focusing to negative defocusing upon passage of the geometricalfocal length L f Thus the transverse probability distribution becomes
jj N z exp
x x
f E L ff
and the transverse size of the wavepacket is given by
f E
L ff
In Figure we give a numerical integration of the Schroedinger equation inthe preceding discussion which conrms the validity of our solution Note thatfor nite L the Gaussian width is focused to a minimum at z f fL In the limit L the transverse size of the wavepacket reaches a minimumat the focal point f where the new eective transverse size is
x f
E
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-6
-4
-20
2
4
6
8
10
12
-3 -2 -1 1 2
Figure The numerical evolution of a Gaussian wavefunction with L evolvingthrough a lens E K z and A hence and thegeometrical focal length f a dashed line is xz crossing the axis at the geometrical
focal point b dotted Gaussian width p
ReAz which focuses at fq Also shown are
ReAz solid and ImAz dotdashed note the ImAz receiving a positive kickupon passing through the lens
This is the usual Rayleigh diractive minimum fa if we regard a asan eective aperture size through which the beam has passed and E pzc hc the usual quantum wavelength of the particle
What if the initial prepared wavefunction is not the groundstate of aSHO pure Gaussian but is rather an excited eigenstate of principle quantumnumber n Hence at z L neglecting x and p we assume
L Hnxp exp
x
C
where Hn is the nth Hermite polynomial It is straightforward to obtainthe solution at a focal point
f xn exp
Azx
and thus the arbitrary solution is focused to a Gaussian times a power of xThis gives a focal point spot size
x
pnf
E
Now this result may seem counterintuitive we are starting with a broaderinitial distribution by the factor
pn and we might guess that this would pro
duce a smaller focal point by an amout pn The wavefunction however
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is not smooth in x ie the Hermite polynomial yields a distribution of transverse momentum and the initial state has ears each of typical Gaussianwidth but displaced o the optical axis by
pn These produce the
pn
enhancement of the focal spot Yet another way to see this is to note that onecan make a classical o!axis centroid motion of the groundstate Gaussian bysuperimposing large n states and the Gaussian width will yield the minimalfE result Of course the quantum state of interest to us will typicallyhave a large value of n determined by radiative relaxation We consider thisin the next section
The actual linear acceleration phase is inconsequential to this result Theabove discussion assumed a uniform drift in the longitudinal zdirection ieconstant energy E If the particle is accelerating linearly then E becomeszdependent Ez Ef EzLE It is easily seen that the only eecton our solution is to replace L by L lnEf EE where E is the initialenergy Ef the nal energy and E in the above expressions is everywherereplaced by Ef For the rst NLC we have Ef GeV E GeV The linear acceleration phase is thus equivalent to free drift through aneective distance of L lnEfE km where L km The amountby which a wavepacket of initial size of nm spreads throughout the NLCacceleration phase is about a factor of However this spreading is irrelevantto computing the nal diractive limit as seen in eq where the free driftlength L has completely cancelled from the expression at the classical focalpoint and only the initial quantity together with the local quantities fand E controls the diractive limita
Thus the ultimate diractive limit is controlled by the initial boundaryconditions on the wavefunction size ie by and not by the interveningunitary lens system What in general determines For the NLC the initialwavefunction as well as the initial classical distribution is prepared in thedamping rings Damping rings are essentially a system of magnets arrangedas wigglers which induce synchrotron radiation and cool the classical beambunches of electrons They are designed to produce roughly a four order ofmagnitude reduction in one of the transverse dimension phase space volumesie xpx the transverse emittance As the system cools classically it isalso relaxing quantum mechanically This occurs because the particles in thewiggler chain experience a transverse SHO potential and synchrotron radia
aWe remark that the proper way to view the quantum spreading in the transverse phasespace is to use Wigner functions which depend upon both x and a quantum momentump The Wigner function isocontours deform in a manner that is conformal to the classicalemittance envelope so while the wavefunctions spread in x the Wigner envelopes actuallyshear in x and p and remain contained in the transverse phasespace
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tion pushes highly excited wavefunctions toward the Gaussian groundstate inthis potential However there are also reexcitation transitions which eventually come into equilibrium and a typical average SHO principle quantumnumber is established While this is certainly an oversimplied view of theactual system we will use it as a starting point to estimate
Magnetic Focusing and Damping
We now sumarize the details of the motion of a transverse wavepacket in amagnetic eld This is discussed in detail in the classic work of Sokolov andTernov We will use the more transparent WKB approximation expandingabout the classical radius of motion Hence one should use caution in comparing solutions eg principle quantum numbers refer to dierent thingsFor example large n in the usual framework corresponds to small n butlarge classical radius jz presently
Consider a particle moving in a planar orbit in a uniform magnetic eldaligned in the z direction B zB in a cylindrical coordinate system r zThe vector potential can be chosen as A rB with Ar and Az We examine the transverse motion of a relativistic electron in the plane z For an anzatz of the form p
rei r teiEth the KG equation becomes
E m
r
r
erB
r
note is not the physical angular momentum because it is gauge dependentdue to presence of the vector potential
This has the apparent form of a onedimensional Schroedinger equationwith an eective potential
V r
r
erB
r
The potential has a minimum at
r R p
eB
r
eB
We consider small radial uctuations with around the large orbitalradius R as r R x Expanding
E m
x V R
xV R
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and
V R eB V R eB
Thus the high orbital angular momentum Landau levels are approximateeigenstates of the SHO potential dened by V x
xV R or K
eB E The states are labelled by n where n is a principle SHO quan
tum number the energy eigenvalues of these levels are given by
E m eB n
In the presence of the gauge interaction the physical angular momentum isLz i eRAR Hence the physical angular momentum is
jz
eBR
where we use the explicit solution for R from eq in the latter expressionTherefore to make the classical correspondence we identify the angular momentum with that of an entering beam particle of momentum p to obtainRp j This yields consistency with the familiar expression for theclassical orbital radius and the total energy
R peB
E m eBjz n
Transitions that increase n but decrease jz are allowed hence synchrotronradiation can be excitatory as well as relaxational The fact that the energyis degenerate depending upon the combination jz n is a consequence of thesymmetry in the choice of the classical orbital center Note that the solutionformed with the anzatz ei for large is actually a solution of vanishingphysical momentum it is a zeromode associated with the translational invariance of the center of the particles orbit
The groundstate in the transverse dimension is a Gaussian with A jeBj given by
p
ReA eBhc
For a typical eld strenth of Tesla we obtain nm The spring constant is Oe hence we say that the dipole magnet is weakly focusing forquadrupoles V R eBEa where a denes a gradient hence strong focusing This description applies to wigglers even though the dipole magneteld is alternating in z if the magnitude of the B eld is roughly constant
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It is well known that an equilibrium between deexcitatory and excitatory transitions for a particle in a damping system or synchrotron will beestablished and there will be an equilibrium value of n This is estimatedas follows The typical energy of synchrotron radiated photons is
E
R
E
me
A unit step in a quantum number n or jz produces only a small energy change eBE R The dipole approximation selection rules imply largeallowed changes in jz but only unit steps in n
jz E
ReB
E
me
E
me
n
Bear in mind that we are treating n as the principle quantum number in theWKB focusing channel dened by expanding about R and dipole transitionsinvolving the operator A r will change n by a unit these can be excitatoryThen jzR is essentially the change in longitudinal electron momentumimparted to the photon In transitions though R changes there is no suddentranslation in the transverse position of the electron wavefunction only atransition in motion ie the virtual center of the orbit changes
Over a radiative energy loss time interval the number of emitted photonsis
n E
E m
e
eBE R
me
E
The principle quantum number n undergoes a random walk by roughlypn
hence the equilibrium n is of order pn Using E GeV and B Tesla whence R m we nd n and n
Hence our diractive limit is now increased bypn and we
thus have a beam spot sizepn nm Why is this result so close to
the design goals of the NLC that are obtained by classical physics Indeedwe believe that this result is a quantum derivation of the classical result" Thequantum number n scales as h while our diractive limit scales as x ph hence the product
pnx is independent of h This moreover assures us
that the ultimate quantum limit is of order pn smaller than the minimal
classical analysis We note that the Oide eect may be understood as ablowing up of n in intense nal focus magnets where large transverse energyphotons are radiated A more detailed discussion of synchrotron radiationrelaxation is beyond the scope of the present paper
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Quantum Particle in a Synchrotron
The solution to the Schroedinger equation for passage of a free particle througha lens eq can be completely described by the simple classical opticalmatrix methods If one passes a classical ray moving in the z direction througha lens system the outgoing state of the transverse x canonical variables maybe written as
xpE
out
M
xzpzE
in
where detM
The unimodular matrix M for a compound sequence of lens elements is thecorresponding sequential product of the individual matrices of the elementsFor example a sequence of free propagation distance L followed by defocusing lens focal length f followed by a space a followed by a focusing lensfocal length f followed by free propagation distance yields the result
xpE
a
f af L a aL
f af aL
f
af a
f aLf
xz
pzE
The zero of the matrix element in F f fa implies the systemis net focusing with composite focal length F
The eect of this particular lens system in quantum mechanics eg onthe Gaussian kernel A as dened in eq can be easily derived from theSchroedinger equation
A A af aLf iEaf
af af iAEL a aLf af aLf
The focal length F is whereM and at the focal length we obtain thewidth
F f
aEwhere ReA
This result is the minimal diractive quantum limit for the composite lenssystem and it is again determined by the initial width of the quantum stateb
Of course beyond there is actually no new information in the aboveformula for A than is already present in the lens matrix for the classical ray
bHere we can take a holding f xed to cause F however for a f thelongitudinal dimension z of the focal point depth of focus becomes small as faowing to the aLf and aLf terms inM Eventually the nite longitudinal distributionof the beams becomes problematic we have not looked in detail at optimization of this
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optics If the lens matrix isMij then we see by comparison with eq thegeneral result for the Gaussian kernel
Theorem I Aout MAin iME
M iMAinE
That is if we write the outamplitude as
Aout iEND
then DN
M M
M M
iEAin
From this result we can easily derive the quantum limit on beam size at theclassical focal length The classical focal length occurs where M Atthis point we haveM M
Hence we readily nd
Theorem II ReAf EReA
MReA ImA
x
If A h is pure real then
x M
E
F
E
noting that M F f is a length scale comparable to the focal lengthat the classical focal length eg F fa in our previous compound lensexample see previous footnote
Now if the magnet system is periodic as in a synchrotron we expectquantum states that are approximately periodic solutions in the matrix Periodic solutions must be eigenstates of the matrix M Consider rst themotion within a very thick lens ie a continuous transverse SHO potentialFor an innitesimal displacement in the zdirection the lens matrix is
MSHO
z
KzE
which is unimodular to Oz The eigenvalues of Mh are iz
pKE The stable quantum solutions in the lens are therefore the eigen
vectors
iEA
M M
M M
iEA
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hence we nd
A pKE
Hence the stable solution in the linear focusing channel is indeed the Gaussian groundstate solution in the SHO potential We have simply recoveredthe usual Gaussian groundstate in this limit
Consider now a synchrotron ie periodic magnet lens lattice basedupon the above lens conguration We assume an innite series of alternatingdipoles with spacing a and focal lengths f Replacing L a in thematrix elements Mij of eq gives the lens matrix for the synchrotron
M
a
f af a a
f af af a
f
af a
f a
f af
The condition that we have a periodic solution is
A MA iME
M iMAE
Using detM we nd
A iE M
hM M
M M
i
Stable quantum solutions solutions that are normalizeable Gaussians for all therefore require
TrM
This is of course the familiar stability condition for the classical motion Notethat TrM af which is independent thus when the conditionis met for particular choices of f and a it holds everywhere The stabilitycondition is the usual one f a
The solution for A is
A E
f a f f
a
f
i
a
f
f
where f a and aConsider the special case of a system in which f a We see that the
minimum Gaussian width occurs at a given by
minReA fpE
x
This implies that the minimum achievable beam spot size in a synchrotron isp
fE
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Conclusion
In conclusion we estimate the minimal quantum beam spot size achievable ina linear collider to be given by
x hcf
E
where f is the overall nal focal length E the beam energy and is theinitial transverse size of the wavefunctions prior to acceleration This maybe viewed as a direct transcription of the Heisenberg uncertainty principle is prepared in the synchronous damping rings typically wigglers and peBhc where B is the magnetic eld strength As a crude estimate we
have for an NLCclass machine B Tesla f m E GeV hence nm and x O nm
Radiation damping implies that the initial state wavefunction has anaverage equilibrium principle quantum number n estimated to be n meeBE
or for the above parameters n Then x p
nhcfE is classical and yields x nm roughly consistent with theclassical vertical nal focus beam spot size of the NLC nm This may bea useful way to approach other phenomena such as the Oide eect in whichlarge uctuation in n in strong nal focus magnets can occur broadening thebeam spot size
We have shown that the transverse wavefunction in linear optics is entirely controlled by the classical equations of motion and hence a simplecalculus based upon lens matrices is constructed or else the function canbe employed Several simple theorems are proved and a selfreplicatingsolution apropos a quatum particle in a synchrotron is constructed
In a synchrotron information about the initial is lost and the minimaltransverse beam spot size is x
phcfE which is O nm for typical
high energy synchrotrons
Acknowledgements
I wish to thank W Bardeen D Burke P Chen D Finley J D Jackson LMichelotti R Noble C Quigg A Tollestrup and especially R Raja and RRuth for useful discussions
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References
Physics and Technology of the Next Linear Collider SLAC Report submitted to Snowmass NLC ZDR design ande Physics WorkingGroup June eg pg we refer to y pg
P Chen in Quantum Aspects of Beam Physics ed P Chen WorldScientic
M Sands Physical Review AA Sokolov I M Ternov Radiation from Relativistic Electrons AIP
Translation Series New York Z Huang P Chen R Ruth Phys Rev Lett Z Huang R Ruth Phys Rev Lett Radiation Damp
ing and Quantum Excitation in a Focusing Dominated Storage RingSLACPUB Apr
W Glaser in Handbuch der Physik Vol ed S Flugge SpringerBerlin p
J A Arnaud Beams and Fiber Optics Academic Press New York see also J B Rosenzweig in Quantum Aspects of Beam
Physics ed P Chen World Scientic A Yariv in Quantum
Electronics Wiley New York V Fock in Electromagnetic Diraction and Propagation Problems Chap
Pergamon Oxford R Jagannathan R Simon E C G Sudarshan and N Mukunda
Phys Rev Lett A R Jagannathan Phys Rev A
S A Khan and R Jagannathan Phys Rev E R Ja
gannathan and S A Khan Quantum theory of the optics of chargedparticles in Advances in Imaging and Electron Physics Vol Ed PW Hawkes Academic Press San Diego
L I Schi Quantum Mechanics McGrawHill pg C T Hill The Diractive Quantum limits of Particle Colliders
FERMILABPUBT hepph# submitted to Phys RevD
D A Edwards and M J Syphers An Introduction to the Physics ofHigh Eneregy Accelerators J Wiley $ Sons Inc
H Kogelnik Bell System Tech J A Gerrard and J MBurch Intro to Matrix Methods in Optics Wiley New York
M Venturini in these proceedings R Ruth private communication obtains x of ours for NLC
within the precision of our crude approximations
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QUANTUM FLUCTUATIONS IN FREEELECTRON LASERS
C B SCHROEDER C PELLEGRINI
Department of Physics and Astronomy
University of California Los Angeles CA USA
P CHEN
Stanford Linear Accelerator Center
Stanford University Stanford CA USA
In this paper we present a manyparticle fullyquantized theory of the freeelectronlaser We examine quantum corrections for the highgain singlepass freeelectronlaser operating in the selfamplied spontaneous emission mode It is shown thatquantum eects become signicant when the photon energy becomes comparableto the energy radiated by a single electron at saturation Initiation of the freeelectron laser process from quantum uctuations in the position and momentumof the electrons is considered and the parameter regime for enhanced startup isidentied
Introduction
The freeelectron laser FEL holds great promise as a source of intense coherent radiation FEL operation in the selfamplied spontaneous emissionSASE mode has received much attention recently as a candidate for the nextgeneration light sources for coherent xrays Presently there are major proposals in the United States and Europe to construct a SASE FEL operatingin the xray regime
An FEL amplies coherent radiation by means of a relativistic electronbeam passing through a periodic static magnetic eld undulator The FELprocess can be understood as the scattering of virtual undulator photons bythe electron beam into photons of the radiation eld ie an exchange of photons between the undulator and the radiation with the electrons providing thenecessary momentum In the limit of an ultrarelativistic electron beam theFEL process is analogous to Compton scattering In addition to xray production by conventional FELs there are proposals to replace the undulator witha counterpropagating highintensity optical laser pulse and generate xraysby stimulated Compton scattering of a comparatively lowenergy electronbeam
Previous quantum mechanical treatments of the FEL have beensuccessful in describing the weakeld regime Madey rst described thesmallsignal FEL gain by calculating the quantum mechanical transition rates
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Later Colson et al calculated the relativistic electron wavefunctions usingquantum electrodynamics in the weakeld regime These results have beenderived assuming a small electron recoil due to emission and absorption ofdiscrete photons and therefore limits the applicability of these theories tocorrections in the smallsignal gain or weakeld regime of the FEL An extensive review of solving the Schrodinger equation in the electron beam restframe through perturbation in the electron recoil is presented in the work byDattoli and Renieri
In this paper we present a manyparticle quantum theory of the FELwhich is applicable in the highgain SASE mode of operation We calculatethe evolution of the expectation value of the photon number operator bysolving the Heisenberg equations SASE is initiated by noise in the beam Wetherefore examine quantum uctuations in the position and momentum of theelectrons as an eective source of noise for initiation of the FEL process
FEL Hamiltonian
The NeelectronM mode quantized Hamiltonian operator describing the FELprocess in the frame moving at the mean velocity of the electron beam can bewritten as
H MX
h
aya
NeXj
hpj
MX
hg
ayau
NeXj
eij Hc
The parameter hkrm determines the electron recoil and the parameter
g pe
mcV k
determines the strength of the coupling between the undulator and radiationeld where m me K is the eective mass and K is the normalizedvector potential of the undulator magnetic eld Here kr is the Lorentz contracted resonant wavenumber and k and are given by Eqs and in the Appendix respectively The Hilbert space operators in Eq satisfythe commutation relations i pj ikkrij and a a
y where
j kzj is the phase pj pjh
kr is the normalized to the recoil providedby a photon exchange between the undulator and the resonant radiation eld
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electron axial momentum and a and ay are the photon annihilation and creation operators of the radiation eld
The details of the derivation of the Hermitian Hamiltonian operator describing the FEL process Eq are presented in the Appendix In particularwe have assumed that the undulator eld is much larger than the radiationeld
ayuau
aya
and that the number of undulator photons is very
largeayuau
such that we may treat the undulator eld classically andreplace the undulator creation and annihilation operators with cnumbersThis is wellsatised for conventional undulators with K
The total electron and photon momentum operator commutes with theHamiltonian
NeX
j
kr pj
MX
kaya
A H
and is a constant of motion The emission of photons is balanced by recoil ofthe electrons
Heisenberg Equations
To study the highgain FEL we will use the Heisenberg picture and evolve thequantum mechanical operators The time evolution of the operators is givenby the Heisenberg equations
djdt
ihj H
kkr
pj
dpjdt
ihpj H i
MX
kkrg
ayaue
ij aaue
ij
dadt
iha H ia igau
NeXj
eij
It is convenient to dene operators representing the observables of the collective motion of the electron beam Consider a bunching operator
b
Ne
NeXj
eij
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and a collective momentum operator
Ne
NeXj
eij pj
which is the normalized axial momentum averaged over phase The Heisenberg equations for the time evolution of the collective operators are
dbdt
ihb H i
kkr
b ikkr
ddt
ih H i
kkr
ikkr
Ne
NeXj
eij pj
iMX
gkkr
Ne
NeXj
hauae
ijj auayeijj
i
dadt
ia igauNeb
The above Heisenberg equations are valid for a general multimode lasereld We will now examine the case of singlemode operation in detail For asingle laser mode dropping the mode label the Heisenberg equations forthe collective operators can be combined to yield
d
dt i
k
kr
d
dt i
ig
k
krauauNe
a
igk
kraua
yNeXj
eij igk
krau
NeXj
eij pj
If we assume that the electron beam is initially unbunched such thatPNe
j
eij
Ne then we can neglect the rst term on the righthandside of Eq We may also neglect the term quadratic in the normalizedmomentum ie the second term on the righthandside of Eq providedaya
Ne cNu where is the radiation wavelength Nu is the num
ber of undulator periods and c is the Compton wavelength By neglectingthese terms we are limiting our analysis to a regime before saturation occursin the FEL process In this regime the Heisenberg equations can be combinedto yield the following linear equation for the evolution of the annihilationoperator
d
d i
Nsat
d
d i
i
a
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where Nu is the renormalized number of undulator periods kr kkr is the renormalized frequency detunning and
Nsat mec
h
is the number of photons emitted per electron at saturation As we will showthe parameter Nsat is a critical parameter for characterizing the quantumeects Here is the dimensionless FEL parameter
eNeK
meV u
The linear evolution equation for the annihilation operator Eq hasthe general solution
a
Xj
cjeij
fNe N
sat b fN
e N
sat fa
where the coecients are
f Nsat
ei
Nsat
ei
Nsat
ei
f ei
ei
ei
f
ei
ei
ei
and j satises the dispersion relation
Nsat
Nsat
Nsat
Nsat
In the classical limit ie limNsat Eq reduces to the characteristiccubic equation of classical FEL theory The time evolution of the number of
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laser photons is given by the expectation value of the number operator aya
aya
jfj
aya
NeNsat
jfj
byb
jfjN
sat
y
f fNsat
by
f fN
sat
yb
Here we have assumed that initially there are no correlations between theelectrons and photons Note that the rst term on the righthandside ofEq is the contribution due to stimulated emission while the last fourterms in the brackets are due to spontaneous emission A solution of thisform Eq was originally studied by Bonifacio and Casagrande althoughin their work the noncommutativity of the collective operators was neglected
Stimulated Radiation
In the highgain regime near resonance the gain due tostimulated radiation can be expressed using Eqs and as
aya
aya jfj AeLLgekkr
k
where A is the gain coecient Lg is the power gain length and k is the FELradiation bandwidth The dependence of A Lg and k on the parameterNsat is shown in Fig This gure shows the deviation from the classicallypredicted values as the parameter Nsat approaches unity and the FEL processmoves from the classical to the quantum regime In particular the gureshows the increase in the power gain length as Nsat is reduced This reductionin gain is due to the strong recoil of the electrons in the parameter regimewhere Nsat The strong recoil moves the electrons o resonance after theemission of a photon thereby decreasing the probability of emitting additionalphotons and reducing the gain
To lowestorder in the parameter Nsat the gain coecient power gain
length and FEL radiation bandwidth satisfy
A
Nsat
Lg
pku
Nsat
k kr
s p
Nsat
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satN
classLLg
classAA
classσσ k
Figure The power gain length Lg dashed line the gain coecient A solid line andthe FEL radiation bandwidth k dotted line normalized to the classically predicted valuesLclass
pku Aclass and class kr
p as functions of the
parameter Nsat
In the limitNsat these quantities reduce to the classical onedimensionalresults
Spontaneous Radiation
In the SASE mode of operation the spontaneous emission emitted in the undulator is coherently amplied by the FEL process To evaluate the spontaneousemission it is necessary to know the initial expectation values of the collectiveoperators in Eq and therefore the initial wave function of the electronbeam
The conjugate operators pj and j satisfy the Heisenberg uncertaintyrelation for any state If we assume that the initial position and momentumfor each electron is described by a minimumuncertainty wavepacket thenthe variances in position and momentum for each electron are described by
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the equality
p
where the variance operators are j hji and p pj hpji If the initial state of the electron beam is described by the productof Ne minimumuncertainty wavepackets satisfying Eq than the initialexpectation values of the collective operators in Eq arebyb
Ne
h ehii
ehi
Ne
NeXi
eihii
Ne
NeXj
eihji
by
yb
Neehi
Ne
NeXj
hpji
ehi
Ne
NeXi
eihii
Ne
NeXj
eihji
y
Ne
hp
ehii
Ne
NeXj
hpji
ehi
Ne
NeXi
eihii
Ne
NeXj
eihji
The center of the quantum wavepacket can be interpreted as the classicalposition for each electron such that the classical bunching parameter is
bc
Ne
NeXj
expi hji
For an initially random longitudinal distribution of electron wavepacket centroids in phase ie classical shot noise Nejbcj With Eqs and assuming the electron beam is initially monoenergtic hpji theexpectation value of the photon number operator Eq can be expressedas
ayaaya
jfj Nsat
h ehi Nejbcj
i jfj
ehi Nejbcj
f f f f
Nsat
hp
ehi Nejbcji jfj
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With no initial classical bunching jbcj eg an ideal crystalline beamand no initial input radiation
aya
Eq gives the number of
laser photons radiated starting from only quantum uctuations in the electronbeam This is the minimum spontaneous radiation produced by any beampassing through the undulator
If we consider resonant radiation in the highgain regime then the expectation value for the number of photons generated by the coherent amplicationof the spontaneous radiation is
aya
AeLLgNsat
Nsat
Nsat
ehi
p
N
sat
Nejbcj
Nsat
Nsat
ehi
In the classical limit ie
p
and limNsat Eq
reduces to haya
expLLclassNemec
jbcj the wellknownresult for a SASE FEL in the onedimensional highgain regime Assuming
and Nsat ie the nearclassical regime Eq can berewritten as
aya AeLLgNsat
Nejbcj
Nejbcj
Nsat
p
N
sat
The rst term in the brackets of Eq represents the classical bunchingwhile the remaining terms represent the eective bunching due to quantumuctuations in the position and momentum of the electrons
If we assume that the Gaussian wavepacket for each electron is weaklyinteracting with the radiation eld then the expectation of the variance inphase will evolve in time as
t
hkrt
m
If we further assume the wavepacket is localized such that the variance isminimized over the length of the undulator ie the nearclassical regime wend the initial value which minimizes the variance in phase to be
NuNsat With these assumptions the expectation value for the numberof photons Eq can be expressed as
aya
AeLLgNsat
Nejbcj
Nsat
Nejbcj
Nu Nu
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Log 10
(γ)
Log10 (λ[nm])
LCLS-SLAC
TESLA-DESY
Compton Backscatter
Classical Regime(Nsat >>1)
Quantum Regime
Nsat = 1
Figure Radiation wavelength and electron beam energy parameter space for proposedxray FEL sources Linac Coherent Light Source SLAC TESLA FEL DESY and xray production by Compton backscattering of an optical laser pulse Dashed line satisesNsat for and indicates the transition from the classical to the quantumregime
For a conventional FEL designed to reach saturation Nu Thereforeone may expect startup from quantum uctuations if N
sat Nejbcj Forthe case of an electron beam with classical shot noise Nejbcj Eq indicates that the relative increase in startup due to the eective bunchingproduced by the quantum uctuations in the electron beam will be of theorder N
sat
Discussion
As evident from Secs and quantum eects manifest themselves in theSASE FEL when the parameter Nsat approaches unity In other words whenthe photon energy is comparable to the energy of the radiation emitted perelectron at saturation h mec
Figure shows the radiation wavelengthand beam energy parameter space for several proposed xray sources The
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dashed line in the gure satises parameters such that Nsat for the FELparameter and roughly indicates the transition between the classical and quantum regimes for that choice of FEL parameter The operatingpoints for the two major xray SASE FEL proposals are plotted in the parameter space as well as for xray production by Compton backscattering ofan optical laser pulse with laser wavelength laser m and laser intensityIlaser Wcm As the gure shows the conventional FEL sourcesare in the classical regime For the case of xray production by Comptonbackscattering of an optical laser pulse one must include the eects of theelectron recoil on the dynamics In this case we can expect deviations fromthe predictions of classical theory and possibly enhanced startup due to thequantum uctuations in the electron beam
The photon number expectation value Eq can be solved explicitlyusing Eqs The solution for Nsat assuming an electron beamwith initial bunching due to classical shot noise Nejbcj and no initialradiation
aya
is shown in Fig The inset in the gure shows
the initial enhancement of the radiation due to the eective bunching fromquantum uctuations in the electron beam compared to what is predicted byclassical theory The larger plot shows that for a suciently long undulatorthe photon intensity is reduced from what is predicted by classical theory dueto the increased power gain length caused by the strong electron recoil in thisparameter regime Nsat
Summary
In this paper we have presented a manyparticle fullyquantized matter andradiation elds theory of the freeelectron laser The Heisenberg equationswere solved for a singlemode radiation eld in the regime before saturationThe stimulated amplication of the radiation was computed in the highgainregime For FEL parameters satisfying Nsat the gain was shown todecrease compared to the classically predicted value owing to the strongelectron recoil The initiation of spontaneous radiation due to quantum uctuations in the position and momentum of the electron beam was examinedThe minimum spontaneous radiation emitted by the beam passing throughthe undulator was calculated In the nearclassical regime the eective bunching of the beam was shown to increase by a factor of N
sat from the classicalshot noise value due to initial quantum uctuations
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<a† a
>Nsa
t-1
4πρNu
Figure The solid curve is the amplication of the spontaneous radiation intensity
ayaN
sat versus the undulator length Nu for Nsat The dotted curve is
the result derived from the classical equations of motion ie Nsat The inset showsthe plot for short undulator lengths
Acknowledgments
One of the authors CBS would like to thank Andrew Charman for manyenlightening discussions We also thank HD Nuhn for useful conversations
This work was supported by the U S Department of Energy under Contract No DEFGER
Appendix
In this appendix we derive the Hermitian Hamiltonian operator for the FELprocess Eq The classical Hamiltonian describing the mattereld inter
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action can be written as
H
NeXj
rm
ec
c Pj e A
where P is the canonical momentum A is the eld vector potential Ne is thenumber of electrons in the beam me is the mass of the electron e is thecharge of the electron and c is the speed of light We will assume that theelectron beam is suciently dilute to make spacecharge eects negligible
The Coulomb gauger A is chosen and we assume the radiation eldspropagate along the z direction The vector potential may be decomposedsuch that A AL Au where Au is the vector potential describing theundulator and AL is the vector potential describing the laser eld which mayconsist of M modes The vector potentials of the laser and undulator may bewritten as
AL MX
sch
V
aee
ikzt aeeikzt
Au
sch
uV
auee
ikuz aueeikuz
where kc is the frequency of the mode and u kuc is the undulatorfrequency Here a and au are the complex amplitudes of the elds whichare contained in the volume V For deniteness both the laser and undulatorelds are assumed to be circularly polarized such that e x iy
p
We will assume that the electron beam is initially injected along the zaxis Since the transverse canonical momentum Pj is a constant of motionP A and the Hamiltonian Eq can be written as
H
NeXj
rmc pjc
eAL Au Au AL
e A
L
where pj is the axial electron momentum and m is the renormalized eectivemass
m me
p K me
Here K is the undulator parameter K ej Aujmec and
p K is
the Lorentz factor associated with the quiver motion of the electrons due tothe transverse undulator magnetic eld
We will assume that the number of undulator photons is much greaterthan the number of laser photons j Auj j ALj With this assumption we
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may neglect the term quadratic with respect to the laser vector potentialSubstituting the vector potentials into the Hamiltonian yields
H mcNeXj
pjmc
mc
MX
hg
aa
ue
ikkuzjt cc
where the strength of the coupling between the undulator and radiation eldis determined by the parameter
g e
mVpu
In the laboratory frame k ku and the relativistic electron beam ismoving in the zdirection ie along the direction of propagation of the radiation elds We dene the resonant frequency r krc with respect to theenergy of the electron beam such that
kr kuk
where k k is the Lorentz factor owing to the axial velocity of the
electron beam k and the total beam energy is mec mec
k mckConsider a Lorentz transformation to a frame moving with velocity
f kr
ku kr ku
kr
in the zdirection with respect to the laboratory frame ie a transform to theframe where the electron beam is at rest In this frame the phase becomes
ku k zj t kzj t
where the primes indicate the coordinates in the frame moving at f and
k f ku k f kr
k kr
kr
f k f kuf c krc
k kr
kr
Here kr pkukr krk is the Lorentz contracted resonant wavenumber
In this frame the electron motion is nonrelativistic and the interactionHamiltonian becomes
H
NeXj
pjm
MX
hg
aa
u
NeXj
eikz
jt cc
A
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Here we have assumed that emcjALjjAuj The conditionemcjALjjAuj hgjaujjajmc may be expressed as jajNe Nsat
where Nsat and are given by Eqs and respectively Thiscondition will always be wellsatised provided since the FEL processsaturates at jaj NeNsat
In addition we can consider a canonical transformation to remove thetime dependency in the Hamiltonian Using an actionangle generating function the Hamiltonian becomes
H
MX
h
aa aa
NeXj
pjm
MX
hgaa
ue
ij cc
where the phase is j kzj In the main body of this paper we will drop
the prime on the time parameter with the understanding that we are workingin the moving frame where the motion of the electrons is nonrelativistic
We will construct a quantum Hamiltonian operator from the classicalHamiltonian through the Dirac prescription for quantization the quantumHamiltonian is assumed to have the form of the classical Hamiltonian according to the correspondence principle and the canonical dynamical variablesare associated with Hilbert space operators The Hermitian Hamiltonian describing the FEL process is
H
MX
h
aya
NeXj
hpj
MX
hg
ayau
NeXj
eij Hc
Here hkrm determines the strength of the electron recoil and pj pjh
kr is the electron axial momentum normalized to the recoil provided by aphoton exchange between the undulator and the resonant laser The operatorssatisfy the commutation relations
i pj
i
kkrij
a ay
for the phase and momentum operators and the nonHermitian photon annihilation a and creation ay operators Note that these commutation relationsare satised for all time since the Hamiltonian is Hermitian and therefore thetimeevolution operator is unitary We assume that the number of undulator
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photons is very largeayuau
and therefore we treat the undulator eldclassically
In this work it is also assumed that the wave functions of the electronsno not overlap and they may be treated as distinguishable particles ie thenumber of available states is much larger than the number of electrons andtherefore there will be no degeneracy in the electron wavefunctions This willbe valid provided the phase space volume of the electrons is suciently diluteie k
Ne
c where k and are the longitudinal and transverse emit
tance of the electron beam respectively and c is the Compton wavelength
References
Linac Coherent Light Source LCLS Design Study Report The LCLSDesign Study Group Stanford Linear Accelerator Center SLAC ReportNo SLACR
J Rossbach et al Nucl Instrum Methods A F Glotin JM Ortega R Prazeres G Devanz and O Marcouill!e
Phys Rev Lett J C Gallardo R C Fernow RPalmer and C Pellegrini IEEE J Quantum Electron
J M J Madey J Appl Phys P Bosco W B Colson R A Freeman IEEE J Quantum Electron
QE W Becker and J K McIver Phys Rev A G Dattoli and A Renieri Laser Handbook Vol eds W B Colson
C Pellegrini and A Renieri Elsevier Sci See for example The Physics of Free Electron Lasers E L Saldin E
A Schneidmiller and M Yurkov SpringerVerlag R Bonifacio and R Casagrande Nucl Instrum and Methods A
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QUANTUM EQUATION OF ELECTRON MOTION
K.-J. KIM
Argonne National Laboratory, 9700 S. Cass Ave., Argonne, IL 60439 USA, and
The University of Chicago, 5801 South Ellis Ave., Chicago, IL 60637 USA
The quantum mechanical equation of motion of an electron in nonrelativistic approximation is derived using a certain operator reduction technique well known in the discussion of quantum synchrotron radiation. The result reproduces that obtained previously by Moniz and Sharp, which was shown to be free of unphysical behavior.
1 Introduction
The equation of motion of an electron has been studied by many authors since the original work of Abraham [1] and Lorenz [2]. The effects of the electromagnetic self-force have been computed in nonrelativistic approximation, including terms linear in the electron coordinate. In the point particle limit, the equation exhibits unphysical behavior such as runaway and preacceleration. The topic has been reviewed extensively in the literature [3-7].
Two important results were reported in 1977 by Moniz and Sharp [8]. First, the problem of runaway and acausal behavior in the classical equation of motion was shown to disappear if the electron radius a is taken to be larger than re, the classical electron radius. Thus, the problem in the point particle limit, or more generally in the case a < re, was attributed to the fact that the mechanical mass m0 must be negative in this case so that the observed mass m = m0 + δm is positive, where δm is the electromagnetic mass. Second, the Heisenberg operator equation of motion for a nonrelativistic electron coupled to a quantized electromagnetic field was derived. The quantum mechanical equation in the linear approximation was shown to be free from the problems of runaway and acausal behavior even in the point particle limit. The result is in accord with the classical argument in the above since an electron in quantum mechanics should behave as an extended object of about the size of the Compton wavelength λ. Note that the Compton wavelength is 137 times larger than the classical electron radius.
However, the derivation of the quantum mechanical equation of motion by Moniz and Sharp is difficult to follow since it involves elaborate manipulation of operator orderings in infinite series. The difficulty may be one of the reasons why their very interesting results have not received serious attention.
The purpose of this paper is to present another, hopefully more transparent, derivation of the quantum mechanical equation of motion. The calculation involves a technique to reduce products of two operators at different times that was employed by Baier and Katkov in their study of quantum behavior of synchrotron radiation [9]. It turns out that the electron’s quantum equation of motion obtained
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in this way reproduces precisely the equation derived by Moniz and Sharp. A shorter account of the derivation was given previously in a review paper on the electron’s equation of motion [7].
In Section 2, we discuss the classical equation of motion as an introduction to the topic and also present a particular approach to compute the self-force, which can be generalized easily to the case of quantum mechanical calculation. In Section 3, the products of physical quantities occurring in Section 2 are interpreted as symmetric products of Heisenberg operators. It is noted that terms that are apparently nonlinear in classical mechanics could, in fact, contain linear terms due to the noncommuting properties of operators. Identification of the linear terms is facilitated by employing an operator reduction technique developed by Baier and Katkov mentioned above. Appendix A contains some details of the operator algebra used in Section 3.
2 Classical Equation
We start by deriving the equation of motion of an electron in nonrelativistic classical mechanics. The electron is assumed to be a charge distribution centered on a trajectory y(t). The charge density at x is given by
( ) ,(t) - f yx (1)
where e is the electron charge, and f(x) is a spherically symmetric function, normalized so that
∫ = .1xd)(f 3x (2)
The nonrelativistic equation of motion is determined by the Lorentz force law:
∫ ×+−+= xd))((t)c
1)()((t)(fe)t(m 3
ext0 xByxEyxFy &&& . (3)
Here m0 is the bare mass, the dot represents the time derivative, and Fext is the external force. The integral term in Eq. (3) is the self-force, in which E and B are, respectively, the electric and magnet fields (in Gaussian units).
The self-force due to the magnetic field is of the second and higher order in the electron coordinate, and therefore neglected. To calculate the self-force due to the electric field, we follow the recent derivation by Low [10] since it can be easily extended to the quantum mechanical case. We work with the potentials A and φ in the Coulomb gauge. The scalar potential is given by
∫ ′=φ′−
−′xde)t,( 3))t((f
xxyx
x . (4)
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The vector potential is obtained by solving the wave equation
( ) ( ).(t)f)t()t(c
e4=
tc
12
22
2
2 yxyy A −
∇∇∇⋅−π
∇−
∂∂
⊥⊥&& (5)
Note that the RHS is the transverse part of the current [3,6]. Next, we compute E = -∇φ - ( )tc∂∂A/ , and insert the results into Eq. (3). The
electric force arising from the scalar potential vanishes due to the spherical symmetry of f(x):
( ) ( ) .0)t(f1
)t(fxdxde 322 =−′′−
∇−′∫ yxxx
yx- (6)
To obtain the vector potential, it is convenient to introduce the momentum representation:
kd)(f
~)t,(
~e
)2(
1
)(f
)t,(A 3i3
π=
∫ ⋅
k
kAx
x xk . (7)
The solution of Eq. (5) with the retarded boundary condition is
( )
′⋅−′′−′π= ∫∞
′⋅− )t()t()(f~
ek
)tt(kcsintde4t),(
~ t
-
)t(i ykk
kykkA
2yk && , (8)
where k=k . Noting that the function )(f~
k depends only on k because of the
spherical symmetry
,)k(f~
)(f~
=k (9)
the last factor in the integrand of Eq. (8) can be simplified as
.)t(3
2)t(
k)t(
2yyk
ky &&& →⋅− (10)
In this way, the self-force is found to be
∫∫∞
τττππ−=
0
233
2ext0 kccos),t(d)k(f
~kd
)2(
4e
3
2)t()t(m YFy&& , (11)
where
).-t(ee)τ,t( )t(i)t(i τ= τ−•• yY ykyk & (12)
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Equations (11) and (12) will be the starting point of the quantum derivation in Section 3. In classical mechanics, Eq. (12) can be further simplified by noting that the exponential factors in Eq. (12) can be replaced by 1 in linear approximation, i.e.,
Y (t,τ) = y& (t-τ). (13)
Equations (11) and (13) lead to:
.c
1t
dt
d
-
) f()f( dde
3
2)t()t(m
2
2332
ext0
′−−
′′
′−= ∫ xxyxx
xxxxFy&& (14)
This is equivalent to the well-known classical equation of motion in the power series derived by Lorentz [2,3,6].
For the case of a spherical shell of radius a,
( )
ka
kasin)k(f
~,
a4
a)(f
2=
π
−δ=
xx . (15)
Equations (11) and (13), or Eq. (14) become
[ ])t()c/a2t(ca3
e)t()t(m
2
2
ext0 yyFy &&&& −−+= . (16)
This differential-difference equation was derived by Sommerfeld [11] and also by Page [12].
Equation (16) becomes, in the limit a→0,
,)t(c3
e2)t(m)t()t(m
3
2
ext0 yyFy &&&&&&& +δ−= (17)
where
.ac2
e
3
4m
2
2
=δ (18)
Equation (17) is known as the Abraham-Lorentz equation. It is the simplest form of the equation of motion, taking into account the electromagnetic self-force in a nonrelativistic linear approximation and in the point particle limit.
The difficulties of the Abraham-Lorentz equation exhibiting runaway behavior and preacceleration have been extensively reviewed [3-7]. The difficulties may be traced to the fact that the electromagnetic mass δm could become arbritrarily large. If δm becomes larger than the “observed” mass m = m 0 + δm, then the “bare” mass m0 must be negative. A particle with a negative mass at rest may spontaneously
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radiate electromagnetic energy and pick up speed without limit as a result of radiation reaction, since the positive electromagnetic energy can be balanced by the negative kinetic energy. Such a phenomenon cannot occur for a positive m0 due to the energy conservation.
The above interpretation of the runaway behavior and preacceleration was explicitly demonstrated using the Sommerfeld-Page equation by Moniz and Sharp [8]. They studied the analytic properties of the frequency-domain solution of the inhomogeneous equation and showed that the motion is indeed causal and finite as long as δm < m, i.e.,
.r3
2a e> (19)
Here re is the classical electron radius associated with the observed mass m = m0 + δm:
.cm
er
2
2
e = (20)
In quantum mechanics, it is reasonable to expect that the effective size of an electron is of the order of the Compton wavelength
.r137mc
λ e≈= h (21)
Since λ >> re, there is some hope that the electron’s equation of motion may become finite in quantum mechanics.
Before taking up the quantum mechanical calculation in the next section, we write down the expression of electromagnetic mass δm, given by Eq. (18) in the case of the spherical shell model, for a general symmetric change distribution as follows:
.)k(f~
dkc
e
3
π4
-
)(f)(fdd
c
e
3
2δm2
02
233
2
2
∫∫∞
=′
′′=
xx
xxxx (22)
This expression again diverges linearly in the point particle limit .1)k(f~
→
3 Quantum Mechanical Calculation
The Heisenberg equation of motion in quantum mechanics is similar to that in classical mechanics except that it is necessary to pay close attention to the ordering of the products of noncommuting operators. In particular, terms that are higher order in classical mechanics may not be neglected in quantum mechanics. Thus,
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consider the product yp, where y and p are, respectively, the position and momentum operators of an electron:
,2
i)pyyp(
2
1
]p,y[2
1p,y
2
1yp
h++=
+= + (23)
where ,+ is an anticommutator, [,] is a commutator, and ħ is the Planck constant. The product yp, which is apparently of the second order in electron variables and thus neglected in classical calculation, may not be neglected in quantum mechanics. The correct thing to do is to neglect the Hermitian combination yp + py, but not the non-Hermitian product yp.
Moniz and Sharp dealt with these complications by carefully manipulating each term in the power series of operators [8]. The calculation was subtle and laborious and therefore difficult to follow.
Quantum mechanical analysis of the electron’s equation of motion has been carried out recently again based on a different but more transparent approach [7]. Remarkably, the results were the same as that obtained by Moniz and Sharp. In the following, we summarize the calculation of the later approach.
The equation for a Heisenberg operator is the same as in the corresponding classical equation, except a proper account must be provided for the noncommuting nature of the operators. In the present case, one starts from an equation identical to Eq. (11) in Section 2, but the operator Y is given by the symmetrized version of the classical expression, Eq. (12), as follows:
( )
( )( ) .][eeee4
1
,e,e4
1t,
i11
ii
1ii
112
12
+⋅−⋅⋅−⋅
++⋅−⋅
−→++=
=τ
kkyy
yY
1ykyikykyk
ykyk
&&
&
(24)
Here y2 = y(t), y1 = y(t-τ), + is again the anticommutator, and [k → – k]+ indicates terms obtained by changing the sign of k and taking the Hermitian conjugate.
Noting that p = P - eA/c = m0 y& is the kinetic momentum operator (P =
canonical momentum), we obtain
λ−=⋅−⋅ kyy kyk cee 1yi
1i 11 && , (25)
where λ = h /m0c is the Compton wavelength for the bare mass m0. In the classical derivation in Section 3, we replaced the factor exp (ik⋅y1) by 1 since it would at most contribute to nonlinear terms in Eq. (24). Such a procedure is not justified in quantum mechanics, as is clear from Eq. (25).
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Similarly, the factor exp (ik⋅y2) ⋅ exp (-ik⋅y1) in Eq. (24) cannot simply be replaced by 1. This product is reduced by a technique used by Baier and Katkov [9] in their calculation of quantum synchrotron radiation. First noting that exp (iτH/ħ), where H is the Hamitonian, is a time translation operator,
The derivation is given in Appendix A. Using these results, Eq. (24) becomes
( ) .] [2
c)(i1
m2
kiexp
2
112
0
2
−→+
λ−−⋅+
τ−= +kkk
yyykY 1&h (29)
Noting that terms containing odd powers in k vanish after d3k-integration and
,k3
1k 22
x = one finally obtains (keeping only terms linear in y)
( )
τλτ=τ 2kccos)-t(t,
2yY & ( )
τλτ−λ− 2
kcsin)-(t(t)6
ck 22
yy . (30)
The linearized electron’s equation of motion in nonrelativistic quantum mechanics is then given by inserting Eq. (30) into Eq. (11). Expanding the operator y(t-τ) in Eq. (30) in a Taylor series around τ = 0 and inserting it in Eq. (11), an equation involving a sum of derivatives of y(t) is obtained. It can be shown that the coefficients of these derivatives are exactly those derived in Moniz and Sharp [8]:
m0 y(t) = Fext(t) - 2n
2n
0nn
nn
2
2
dt)t(d
c!n
A)1(
c
e
3
2+
+∞
=∑
− y , (31)
where
nc
n B)1n(
1)2n(3
1A
λ∂∂
+λ+
λ∂∂
+λ
+= , (32)
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and
( ) .)(f)(fdd
4k2
k21n
)!k2n(
!nB
k22x
k21n
k2
0kn
xxxxxx ′∇′−′
×
λ−
−++
=
′+−
∞
=
∫
∑ (33)
The coefficient of )t(y&& in Eq. (31) is the quantum mechanical self-mass δm and is given by
δm = .16
1c
e
3
202
2
Ω
∂λ∂λ+
∂λ∂λ+
(34)
Here
,dk)4/k1(
)k(f~
P2
o22
2
0 ∫∞
λ−π=Ω (35)
where P denotes the principal value integration. Note first that in the limit λ → 0,
Ωo = ∫∞
πo
2dk)k(f
~P
2 , (36)
and the theory reproduces the classical result, as it should. In general, the quantum mechanical self-mass given by Eq. (34) is finite.
Moniz and Sharp evaluated Eq. (34) as a function of the electron radius a and found that it is maximum around a ~ λ with the value
.137
mm
c
em
2
2
a
=α=λ
≈δλ≈
(37)
Thus the self-mass in quantum mechanics is much less than m. Therefore, the mechanical mass m0 is about the same as the observed mass m, which is clearly positive. We may thus anticipate that the quantum mechanical equation of motion will not exhibit unphysical behavior such as runaway or preacceleration.
Whether the quantum mechanical equation of motion exhibits runaway or acausal behavior can explicitly be studied by writing Eq. (31) in a Green’s function form. Such an analysis in the point particle limit was carried out by Moniz and Sharp [8]. It is found that the motion is indeed causal with no runaways if
.75.1re λ< (38)
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Thus inequality is clearly satisfied since re/λ ≈ 1/137.
4 Acknowledgments
This work was carried out while the author was collaborating with Andy Sessler on a review article on the electron’s equation of motion [7]. His encouragement is graciously acknowledged. The work is supported by the U.S. Department of Energy, Office of Basic Energy Sciences, under Contract No. W-31-109-ENG-38.
References
1. Abraham, M., Theorie der Elektrizitat, Vol. II: Elektromagnetische Theorie der Strahlung, Leipzig, Teubner (1905).
2. Lorentz, H.A., The Theory of Electrons, Leipzig: Teubner (1909) (2nd edition, 1916). 3. Panofsky, W.K.H., and Phillips, M., Classical Electricity and Magnetism, Reading, MA:
edition 1990). 5. Pearle, P., Classical Electron Models, in Electromagnetism: Path to Research, D.
Teplitz, Ed., New York: Plenum, 1982. 6. Jackson, J.D., Classical Electrodynamics, 3rd Edition, New York: Wiley, 1999. 7. Kim, K.-J. and Sessler, A. M., The Equation of Motion of an Electron, Proceedings of
the 8th Workshop on Advanced Acceleration Concepts, Baltimore, MD, AIP Conference Proceedings 472, 3-18 (1998).
8. Moniz, E.J. and Sharp, D.H., Radiation reaction in nonrelativistic quantum electro-dynamics, Phys. Review D 15, 2850 (1977).
Preprint MIT- CTP-2522 (1997). 11. Sommerfeld, A., Simplified deduction of the field and the forces of an electron moving
in any given way, Akad. van Wetensch. te Amsterdam 13 (1904) (English translation 7, 346 (1905)).
12. Page, L., Phys. Rev. 11, 376 (1918). 13. Dragt, A.J., and Finn, J.M., J. Math. Phys. 17, 2215 (1976). 14. Cahn, R.N., and Jackson, J.D., unpublished LBL note.
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APPENDIX A
Given two operators A and B, we write
)s,t(X)BA(stA eee =+ . (A.1)
According to the Campbell-Baker-Hausdorff theorem [13], X can be expressed as a linear combination of commutators involving A and B. Assuming that B is small, we evaluate X for the first order in B. Thus, we consider X in the form
The coefficients Cm can be determined easily by using a technique due to R.N. Cahn and J.D. Jackson [14]: expand both sides of Eq. (A.1) and equate the coefficients of those terms of the form APB, i.e., terms where B occurs in the final position. This leads to
)ACACs(ts
1e)1e(e 2
21
A)ts(sAtA K+++
+−=−
+. (A.3)
In the special case t = –s, this becomes
.)e1(AACACs sA1221
−− −=+++ K (A.4)
Thus we determine
!)1m()s(C 1mm +−−= + (A.5)
Inserting this into Eq. (A.2), the operator X(–s,s) is determined up to terms linear in B.
The result can be written in a more compact form by using the identities:
τAs
0
sA1 edτ)e1(A −−− ∫=− , (A.6)
[ ] K+τ−+−=− ]B,A[,A!2
)(]B,A[τBeBe
2τAτA . (A.7)
Equation (A.5) can be shown to be equivalent to the following identity:
)s,s(X)BA(ssA eee −+− = , (A.8)
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where
∫ +τ=− ττ−s
o
2AA .)B(OBeed)s,s(X (A.9)
Let us now apply Eq. (A.9) to the case
,its hτ−→−=
HA→ , (A.10)
.)t(km
kpB
o
τ−⋅−=⋅
−→ y&hh
We obtain
,eee X)ykH(iH
i
=−
τ−
τ&h
hh (A.11)
( ) .yinorderhigherofterms)t()t(ie)t(ediXHi
0
Hi+τ−−⋅=τ−⋅τ=
τ−τ τ
−
∫ yykyk hh &
We have therefore proved Eq. (28) to the first order in y.
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The equilbrium phase distribution of stored colliding electron beams is studiedfrom the viewpoint of Vlasov-Fokker-Planck (VFP) theory. Numerical integrationof the VFP system in one degree of freedom revealed a nearly Gaussian equilibriumwith non-diagonal covariance matrix. This result is reproduced approximately inan analytic theory based on linearization of the beam-beam force. Analysis ofan integral equation for the equilbrium distribution, without linearization, estab-lishes the existence of a unique equilbrium at sufficiently small current. The roleof damping and quantum noise is clarified through a new representation of thepropagator of the linear Fokker-Planck equation with harmonic force.
1 Introduction
The competition of damping and noise from synchrotron radiation in quantaresults in a unique equilibrium state of a stored electron beam of sufficientlysmall current 1,2. Although very familiar, this is a noteworthy example of amacroscopic effect of quantal processes. At higher current, more complicatedquasi-equilbrium states have been observed 3 and simulated 4. These stateshave been called “sawtooth” modes; they are nearly periodic with very longperiod, comparable to the damping time. Again, the existence of such modesdepends on the presence of damping and quantum noise.
An elegant and natural mathematical framework for study of equilibriumand non-equilibrium phase space distributions is based on the VFP equation,which is to say the ordinary Vlasov equation for self-consistent multi-particlemotion, augmented with Fokker-Planck terms to account for damping andnoise. Recently, stable, long-term numerical integration of the VFP equationfor longitudinal motion has been achieved 4. Calculations 4 with a realisticwake field gave good agreement with several aspects of observations at theSLAC damping rings 3.
capri: submitted to World Scientific on May 7, 2001 1
Two counter-rotating stored beams, undergoing beam-beam collisions,can be treated by coupled VFP equations for the distribution functions of thetwo beams. This approach was formulated for the Vlasov part of the systemby Chao and Ruth 5, in a model with one degree of freedom. In this paper weadopt their model, extended to include Fokker-Planck terms. Generalizationsto more realistic models are certainly possible, and are on the agenda forfuture work.
The first order of business in the beam-beam problem is to determine theequilibrium distribution, understood as a distribution that is periodic in themachine azimuth s. Later, one will want to study stability of the equilib-rium. If it is stable at small current, as expected, then the threshold currentfor instability and the character of trans-threshold, time-dependent motionare of interest. Surprisingly, the question of existence and character of anequilibrium state is rarely mentioned in the extensive beam-beam literature,although a few authors do recognize it as an open problem 6,7,8. A first step 5
is to note that when the beam-beam force is linearized, the transverse motionfor one beam has the familiar Courant-Snyder description, but with a “dy-namic beta function”. Consequently, there should be a conserved action , andthe equilibrium distribution function in action-angle coordinates should be afunction of action alone. This argument alone does not solve the equilibriumproblem, however, since the dynamic beta function and the correspondingaction are nonlinear functionals of the charge distribution of the opposingbeam. Determination of the charge densities of the two beams so as to beself-consistent in the steady state is thus a remaining nonlinear problem. Cer-tain aspects of this problem were treated by Furman, Ng, and Chao 6, andby Chao 7. Another approach 8,9 is to average the beam-beam force over aturn, neglecting its almost impulsive character. This leads to coupled integralequations of Haıssinski type 9, but seems unnecessarily rough as a physicalmodel.
We have found it better to avoid both action-angle coordinates and aver-aging of the force. Also, we do not linearize the beam-beam force, except foran approximate analytic treatment which models in a simple way the impor-tant aspects of the full problem. We treat the full VFP system by numericaltime-domain integration, and also by a nonlinear integral equation for theequilibrium distributions. We must defer to a longer report the comparisonof our results to those of Furman et al. 6,7.
capri: submitted to World Scientific on May 7, 2001 2
2 Definitions and Equations
We treat vertical betatron motion with normalized phase-space variables (q, p)defined in terms of the vertical lattice function β(s) and emittance ε as
q = y(βε)−1/2 , p = (βy′ − β′y/2)(βε)−1/2 , (1)
where y is the vertical displacement and the prime denotes d/ds. The Hamil-tonian is H = (p2 + q2)/2 and the independent “time” variable of Hamilton’sequations is the phase advance θ =
∫ s
0 du/β(u). The two beams may have dis-similar optics and intensities; we distinguish their properties by superscripts(1), (2).
The Chao-Ruth model is intended to represent vertical motion of beamswith width Lx much greater than height Ly. We calculate the force as thoughit came from uniform planes of charge normal to the y-axis, which is to sayfrom a charge density of the form ρ(x, y, z) = σλ(y), where
∫ ∞−∞ λ(y)dy = 1
and σ is the total charge per unit area in (x, z)-space accounting for chargeat all y. To get the electric field E(y) we apply Gauss’s Law to a semi-infinitecylinder running along the y-axis to y = +∞, with a face perpendicular tothe axis at y. We then do the same for a cylinder running from y to −∞, andeliminate E(∞) = −E(−∞) between the two resulting equations to find E(y).An analogous calculation of H(y) by Ampere’s Law shows that the magneticforce is precisely equal to the electric force for an ultrarelativistic particle.The full Lorentz force on an ultrarelativistic particle is in the y-direction andhas the following value in m.k.s. units:
e(E + v × B)y =eσ
ε0
∫ ∞
−∞sgn(y − y′)λ(y′)dy′ , (2)
where sgn(x) is 1 for x > 0 and −1 for x < 0. Now suppose that the beam haswidth Lx and length Lz, and bears a charge ±eN . We approximate the forceit exerts on a particle in the other beam by (2) with σ = ±eN/LxLz. Thisforce acts only during the transit time of the particle through the oncomingbeam. During that time the particle moves a distance ∆s = Lz/2 in the labframe, so that the force as a function of s with IP at s = 0 is
±e2Nh(s)ε0LxLz
∫ ∞
−∞sgn(y − y′)λ(y′, s)dy′ , (3)
where h(s) is 1 for 0 < s (mod C) < Lz/2 and 0 otherwise, where C is thecircumference of the reference orbit. Since the transit time is tiny comparedto a betatron period, it seems reasonable to concentrate this force at s = 0.To do that we replace 2h(s)/Lz, made up of step functions of unit integral,
capri: submitted to World Scientific on May 7, 2001 3
with δC(s) =∑
n δ(s − nC), the periodic delta function of period C. Theresulting force is smaller by a factor of 2 than that of Chao and Ruth 5, butagrees with later papers of Chao 7 and Forest 10.
Knowing the force we can find the kick in transverse momentum py, andtranslate that into the kick of y′ = py/P , where P is the total momentum.For relativistic beams of opposite charge, dy′/ds for beam (1) has the valuegiven in Eq.(4) of Ref.[5], reduced by a factor of 2.
We now turn to VFP theory with two phase-space distribution functionsf (i)(q, p, θ) , i = 1, 2, normalized to unit integral. The corresponding particleposition densities λ(i)(q, θ) are obtained by integrating over p. In general, θ isdefined differently for the two beams, but we do not distinguish the θ’s with asuperscript. Just remember that each equation and each function has its ownindependent variable. Translating our results in terms of (y, y′, s) to (q, p, θ),we find the coupled VFP equations for beams of opposite charge as follows:
∂f (1)
∂θ+ p
∂f (1)
∂q−
[
q + (2π)3/2ξ(1)∑
n
δ(θ − 2πν(1)n) ·
·∫ ∞
−∞sgn(q − q′)
∫ ∞
−∞f (2)(q′, p′, θ)dq′dp′
]∂f (1)
∂p
= 2α(1) ∂
∂p
[
pf (1) +∂f (1)
∂p
]
, (and 1 ↔ 2) . (4)
The beam-beam parameter is ξ(1) = N (2)β∗(1)re/((2π)1/2γ(1)σ(1)y L
(2)x ). Here
β∗ is the beta function at the IP, re = e2/(4πε0mc2) is the classical electronradius, γ is the Lorentz factor, and σy = (β∗ε)1/2 is the bunch height. Theright hand side of (4) is the Fokker-Planck contribution, with damping con-stant α(1) = 1/(2πν(1)n
(1)d ), where nd is the number of turns in a damping
time. Our phase space coordinates have been defined so that the dampingand diffusion constants are equal.
Equation (4) has only a formal significance, since the θ- dependent fac-tors multiplying the delta function actually change discontinuously at the IPwhere the delta function acts. Consequently, we cannot say how to evaluatethose factors without further analysis. Actually, the correct change of the dis-tribution function at the IP is easy to see. Let f (1)(q, p, 0−) and f (1)(q, p, 0+)represent the distributions just before and just after θ = 0 (mod 2πν(1)).Then by the usual argument from probability conservation 4 the distribu-tion is changed by the inverse of the kick map; i.e., by the Perron-Frobeniusoperator for that map:
f (1)(q, p, 0+) = f (1)(q, p− F (q, 0−), 0−) , (5)
capri: submitted to World Scientific on May 7, 2001 4
where
F (q, 0−) = −(2π)3/2ξ(1)∫
sgn(q − q′)f (2)(q′, p′, 0−)dq′dp′ . (6)
For propagation of the distribution function between IP kicks, we havein (4) a linear Fokker-Planck equation with harmonic force. The propaga-tor or fundamental solution of that equation is known 11, namely a functionK(z, z′, θ) , z = (q, p) such that for any initial distribution f(z, 0) the solutionat time θ is
f(z, θ) =∫
K(z, z′, θ)f(z′, 0)dz′ . (7)
There are several equivalent representations of K. The following form, derivedfrom a probabilistic argument, is especially appealing:
K(z, z′, θ) =1
2π(det Σ)1/2exp
[−(z − eAθz′
)T Σ−1(z − eAθz′
)/2
],
Σ = I − eAθeAT θ . (8)
Here T denotes transposition and eAθ is the transfer matrix for the single-particle harmonic motion with damping. With damping constant α we have
eAθ = e−αθ
(cosΩθ + (α/Ω) sin Ωθ (1/Ω) sinΩθ
−(1/Ω) sinΩθ cosΩθ − (α/Ω) sin Ωθ
)
, (9)
Ω = (1 − α2)1/2 , det eAθ = e−2αθ . (10)
Let K denote the operator corresponding to the kernel K(z, z′, θ). The actionof K has a simple expression in Fourier space. Writing h for the Fouriertransform of h, we have
Kh(v) = exp[−vT eAθΣeAT θv/2
]h(eAT θv) . (11)
3 Numerical Integration of the VFP Equation
The kick map (5) followed by the action of K gives the complete propagationof the distribution function over one turn, and thus specifies the meaningof the delta function in the VFP equation. For numerical work it is highlyinefficient to use K, however. Instead, we shall follow the method of [4],based on operator splitting. We write ∂f/∂θ = LV (f) + LFP (f), where LV
and LFP are the operators associated with the Vlasov and Fokker-Planckterms, respectively. We make a θ step under LV alone followed by a θ stepunder LFP alone, and so on. It turns out that in this problem the step for
capri: submitted to World Scientific on May 7, 2001 5
LFP can be a full turn, owing to the small value of the damping constant α.For LV we apply the Perron-Frobenius (PF) operator for the map T = RK,where K is the beam-beam kick and R the phase-space rotation through angle2πν. The PF operator is discretized on a grid in phase space, being definedat off-grid points by local polynomial interpolation 4. The kick is calculatedat grid points from values of the distribution on grid points. Then f(T−1(z))is computed for grid points z by interpolation to give an update of f at gridpoints. For LFP we use a divided-difference discretization and a simple Eulerstep 4 with ∆θ = 2πν. In a typical run we use a 201 × 201 grid, and thecalculation takes 6.5 hours for 30000 turns on a 400 MHz work station. Thealgorithm conserves charge to one part in 105 (or 104 at high current) overseveral damping times, and reproduces the known solution for zero current.
We show results for parameters suggested by PEP-II design values;namely, ν(1) = 0.6342 , ν(2) = 0.6387 , n(1)
d = 5014 , n(2)d = 8579 , ξ(1)/ξ(2) =
1.11, where beam (1) is in the high energy ring (9 GeV) and beam (2) inthe low energy ring (3.1 GeV). Keeping the ratio of beam-beam parame-ters at the stated value, we increase ξ(1) in steps, starting with a smallvalue such as 0.01. The initial distribution for each beam is the Gaussianf0(z) = exp(−zT z/2)/2π , z = (q, p), which is the solution for zero current.Figure 1 shows the normalized r.m.s. bunch size σ(1)
q for beam (1), just be-fore the IP. It undergoes rapid oscillations in a region of transient behaviorextending to about 150 turns, and then decreases slowly, reaching a steadystate at about 2 damping times. Figure 2 is a contour plot of − log f (1), wheref (1) is the final distribution at ξ(1) = 0.028. The solid lines are curves givenby − log f (1)(z) = c, with c = 2, 3, 4, 5, 6. At smaller c the contours appear tobe nearly elliptical, indicating a nearly Gaussian behavior.
To test the deviation from a Gaussian we compute the covariance matrixM of the final f , and look at the contour plot of − log g, where g is theGaussian with the same covariance, namely
g(z) =exp(−zTM−1z/2)
2π(detM)1/2. (12)
The dotted curves in Figure 2 represent − log g(z) = c, for c equal to 3 and6. They lie close to the corresponding solid curves, with more deviation atc = 6. Figure 3 shows a graph of the force in the equilibrium state.
The threshold of instability of the equilibrium state is somewhere betweenξ(1) = 0.0280 and ξ(1) = 0.0373. Figure 4 shows σq at the latter value. Thefast oscillations at large time are reminiscent of what was found in longitudinalsingle beam motion with wake field 4.
capri: submitted to World Scientific on May 7, 2001 6
Figure 1. Normalized dimensionless beam
size σ(1)q vs. turn number, ξ(1) = 0.0280.
−3 −2 −1 0 1 2 3
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
Figure 2. A contour plot of − log f(1),where f(1) is the equilibrium distributionfor beam (1) just before the IP. The dot-ted curves are for the Gaussian with thesame covariance.
Figure 3. Beam-beam force F (q) in theequilibrium state.
Figure 4. σq above the threshold of insta-bility, at ξ(1) = 0.0373.
4 Equilibrium with Linearized Force, without Radiation
For a first step in an analytical discussion we take two beams with equalproperties and turn off the synchrotron radiation by putting α(i) = 0 in (4).We linearize the beam-beam force as a function of q, but do not linearize theVlasov equation in its dependence on f . By (6) the Taylor expansion of theforce is
F (q) = −(2π)3/2ξ
[(∫ 0
−∞−
∫ ∞
0
)λ(q′)dq′ + 2λ(0)q + O(q2)
]
. (13)
capri: submitted to World Scientific on May 7, 2001 7
Motivated by the numerical results, we seek an equilibrium having the generalGaussian form (12) just before the beam-beam kick. Integrating this formover p to get λ we find λ(q) = exp(−q2/(2m11))/
√2πm11, hence F (q) =
−4πξq/√m11 + · · ·. We define η = 4πξ/
√m11, and note that the condition
of equilibrium, which is to say periodicity, is g((RK)−1z) = g(z), where Ris the one-turn transfer matrix for the betatron motion, and K is the matrixtransformation representing the kick:
R =(
cos 2πν sin 2πν− sin 2πν cos 2πν
)
, K =(
1 0−η 1
)
. (14)
In other words, with T = RK (hence detT = 1), the equilibrium condition is
TMT T = M , (15)
where M must be symmetric and positive definite. For any matrix T withunit determinant, (15) has infinitely many symmetric solutions M , since thesystem regarded as three linear equations for three unknowns m11 , m22 ,m12
has zero determinant. Provided that t12 = sin 2πν = 0, all solutions can beexpressed in terms of one parameter, m11. In fact,
m22 = − t21t12
m11 , m12 = m21 =t22 − t11
2t12m11 . (16)
Also, the T that we actually have depends on M only through m11, so that(16) gives the general solution of (15). From the definition of T = RK wefind
m22 = m11 + 4πξ cot 2πν√m11 , m12 = 2πξ
√m11 . (17)
Remarkably, the nonlinear equation (15) for M has been solved by linearmeans.
We have yet to impose the condition that M be positive definite, whichis equivalent to the conditions m11 > 0 , detM > 0 taken together. It is easyto check that detM > 0 if and only if
ξ <
√m11
2π
[1
| sin 2πν| + cot 2πν]
. (18)
Through (17) we have an infinite family of Gaussian equilibriaparametrized by m11 > 0, provided that sin 2πν = 0 and ξ satisfies (18). Ac-tually, in the present case without radiation the Gaussian is irrelevant: onlythe invariance of the quadratic form zTMz under T was essential to the argu-ment. Let us look for an equilibrium of the form f(z) = Φ(zTM−1z)/N (M),whereM is positive definite, Φ is an arbitrary positive function on the positivereal line, and N is a normalization constant chosen to make
∫f(z)dz = 1.
capri: submitted to World Scientific on May 7, 2001 8
Calculating the latter integral by a change of variable that diagonalizes Mthen scales by eigenvalues, we find N (M) = π
√detM
∫ ∞0 Φ(x)dx. Further-
more, integration of f(0, p) over p gives
λ(0) =2
∫ ∞0 Φ(x2)dx
π√m11
∫ ∞0 Φ(x)dx
, (19)
the same dependence on m11 as in the Gaussian case. It is now clear that thesolution is again given by (17), if we replace ξ by κξ, where
κ =2√
2/π∫ ∞0 Φ(x2)dx
∫ ∞0 Φ(x)dx
. (20)
All we require of Φ is that the integrals in (20) exist.An interesting exercise, which we leave to the reader, is to explore the
behavior of M as ξ approaches the limit (18). The ellipses become long andthin, since only one eigenvalue vanishes. In the limit sin 2πν → 0 the boundon ξ depends on the sign of tan 2πν near the limit. It expands to infinity fortan 2πν > 0 but shrinks to zero for tan 2πν < 0.
The angles of inclination of axes of the ellipse depend only on the tune.The eigenvectors of M−1 are v1 = (cosπν, − sinπν), v2 = (sinπν, cosπν).Then just before the IP, one axis of the ellipse is displaced by an angle −πνwith respect to the q-axis. The beam-beam kick rotates this axis by +2πν,to compensate the lattice motion which rotates the ellipse by −2πν. This isequivalent to saying that the kick reflects the distribution in the q-axis.
5 Equilibrium with Linearized Force, Including Radiation
To include radiation we can follow the plan of the previous section, exceptfor replacing the rotation R by propagation according to the linear Fokker-Planck equation. The latter is handled most easily in Fourier space, by meansof formula (11). The presence of the exponential factor makes clear that wewill be restricted to Gaussian distributions in this case. Again we assume thatthe equilibrium distribution has the form (12) just before the IP. Translatingthe equilibrium condition into Fourier space and applying ( 11), we find thatM must satisfy
M − TMT T = eAθΣeAT θ , T = eAθK , θ = 2πν . (21)
Here we have 3 inhomogeneous equations for m11, m22, m12 (since 2 of the 4equations are equivalent by symmetry). Multiplying on the right by (T T )−1
and recalling that detT = e−2αθ, we see that it is easy to eliminate m12 and
capri: submitted to World Scientific on May 7, 2001 9
m22. First suppose that t12 = 0, which is to say sin Ωθ = 0. Defining γ = e2αθ
If t12 = 0, the equation for m11 is linear, and we get the complete M inexplicit form:
m11 =±σ11
2 sinhαθ,
m12 =±1
2 sinhαθ[σ21 ∓ e−αθη m11
],
m22 =±1
2 sinhαθ[η σ21 + σ22 ∓ 2η coshαθ m12
]. (24)
The sign is to agree with the sign of cosΩθ = ±1.Now t11 and t21 are linear in η, and t22 and t12 are independent of η,
where η = 4πξ/√m11. It follows that (23), after multiplication by x =
√m11,
is a cubic equation for x. To lowest order in the damping constant α thecoefficients of the cubic simplify. Cancelling an overall factor of α, we get thelowest order form of the equation, independent of α:
Of course, this makes sense only if sin θ = 0, so that the expansion is usefulonly if sin Ωθ = sin θ + O(α2) is not too close to 0. With that restriction,(25) provides a good model of the exact polynomial (23). For typical valuesof α (around 10−5 in our numerical examples) we find that the roots of thetwo cubics agree to about 3 digits over a grid in the (ξ, ν) parameter space,including values of ν fairly close to 1/2, say ν = 0.505.
For zero current the equation (25) reduces to x3 − x = 0. The root x = 1is the correct solution for zero current, corresponding to the unperturbedGaussian. The roots x = −1, 0 are unphysical. To lowest order in thecurrent parameter ξ, the roots of (25) are
x± = ±1 − πξ(cot θ + cos 2θ/θ) , x0 = −2πξ(cot θ − cos 2θ/θ) . (26)
capri: submitted to World Scientific on May 7, 2001 10
Now x0 may be positive or negative, but even if positive it corresponds to anunphysical solution, since the corresponding lowest order form of M is notpositive definite.
Similarly, for non-zero current and the exact solution of equation (23) wefind one positive root which corresponds to a positive definite M , one negativeroot that is clearly unphysical, and one root near zero that can be eitherpositive or negative, but is unphysical even if positive since the correspondingM is not positive definite. This result was seen in a numerical explorationover a fine grid in (ξ, ν) space. This outcome is quite satisfactory: as expectedfrom experience in the single-beam problem, inclusion of radiation reduces theinfinite family of Vlasov solutions to a single solution of Gaussian type.
With radiation we have found no analog of the constraint (18). Never-theless, the beam can get larger than the beam pipe at large current or innear-resonant conditions (sin Ωθ small). This is seen in the small-α form ofM for sin Ωθ = 0. By (24) we find
m11 = 1 + O(α) , m12 = − η
2αθ+ O(1) , m22 =
η2
2(αθ)2+ O(1/α) . (27)
Since m22 = O(α−2), the ellipses are long and thin in the p direction. Al-though the equilibrium exists mathematically even at a resonance, it may beunrealizable in the machine.
Figure 5 compares the result of the present linearized model with numer-ical integration of the VFP equation for equal beams with ν = 0.6364, ξ =0.0266, nd = 5000. The angle of tilt of the near-elliptical curves is givenquite well by the linearized theory, but in other respects the agreement issomewhat rough. The covariance matrix of the VFP solution is m11 =0.8297, m22 = 1.016, m12 = 0.1075, whereas that of the linearized modelis m11 = 0.9095, m22 = 1.185, m12 = 0.1593. It is interesting that the Gaus-sian determined by the covariance matrix of the VFP solution gives a betterfit to the VFP solution than the Gaussian from the theory with linearizedforce.
6 Integral Equation for the Equilibrium state
We write coupled integral equations for the equilibrium distributions f (1), f (2)
just after the IP. The difference in convention compared to the above discus-sion (after rather than before) arises from a technical point in the analysis.The equations are
capri: submitted to World Scientific on May 7, 2001 11
−3 −2 −1 0 1 2 3
−2.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
Figure 5. Some contours of equal probability for a VFP solution (solid line) and for themodel with linearized force (dashed line)
g(i)(q, p+ (2π)3/2ξ(i)∫
sgn(q − q′)g(j)(z′)dz′) = f (i)(z) , i = j
g(i)(z) =∫K(i)(z, z′, 2πν(i))f (i)(z′)dz′ , (28)
with∫f (i)(z)dz = 1 . (29)
It is essential that the normalization constraint (29) be regarded as part ofthe definition of the mathematical system.
The physical meaning of (28) should be obvious: we start just after theIP, propagate for one turn by the linear Fokker-Planck operator, then applythe beam-beam kick, and require that the result be equal to what we startedwith.
It is possible to analyze these equations without any approximations, us-ing methods of functional analysis. By applying the implicit function theoremin an appropriate Banach space, one can show that there is a solution, uniquein that space, at sufficiently small ξ(i). The proof will be published elsewhere.In accord with the linearized theory with radiation, a non-resonance conditionis not required. The method of proof is readily generalized to the beam-beamproblem with two degrees of freedom.
The system (28), (29) is analogous to the Haıssinski equation for longitu-dinal motion with wake field, but certainly quite different in form. It lives on
capri: submitted to World Scientific on May 7, 2001 12
phase space, and it depends on both the damping and diffusion constants (theratio of the two being buried in our choice of variables). The principal depen-dence is on the ratio, however, as the discussion of the previous section andnumerical VFP solutions indicate. The Haıssinski equation lives on q-space,and it depends only on the ratio of damping and diffusion constants.
7 Conclusion
We have found a fairly complete depiction of the equilibrium state in a beam-beam interaction model with one degree of freedom. It remains to work outrelations to the dynamic beta function description, and to extend the theoryto 2 or 3 degrees of freedom. In higher dimensions we expect the equilibria tobe generally similar to what we have found, but time-dependent phenomena athigh current should be much richer than in 1 degree of freedom. Preliminaryresults on high-current motion suggest that many interesting results can beexpected from numerical integration of the VFP equation in the time domain.
8 Acknowledgments
We enjoyed many helpful discussions with Yunhai Cai, Alex Chao, MathiasVogt, Sam Heifets, and Ron Ruth. Our work was supported in part by Depart-ment of Energy Contracts DE-AC03-76SF00515 and DE-FG03-99ER41104.
References
1. M. Sands, The Physics of Electron Storage Rings - An Introduction,SLAC-121 (1970).
2. J. Haıssinski, Nuovo Cimento 18 B, 72 (1973).3. B. V. Podobedov, Saw-Tooth Instability Studies at the Stanford Lin-
4. R. Warnock and J. Ellison, SLAC-PUB-8404, published in Proc. 2ndICFA Advanced Accelerator Workshop on the Physics of High BrightnessBeams, UCLA, November 9-12, 1999 (World Scientific, Singapore, 2001),and SLAC-PUB-8494, to be published in Proc. 2000 European ParticleAccelerator Conf..
5. A. Chao and R. D. Ruth, Particle Accelerators 16, 201-216 (1985).6. M. A. Furman, K.-Y. Ng, and A. W. Chao, SSC-174 (1988)7. A. Chao, SSCL-346 (1991)8. P. Zenkevich and K. Yokoya, Particle Accelerators 40, 229-241 (1997).
capri: submitted to World Scientific on May 7, 2001 13
9. S. Heifets, private communication.10. E. Forest, Particle Accelerators 21, 133-156 (1987).11. S. Chandrasekhar, Rev. Mod. Phys 15, 1-91 (1943), reprinted in Selected
Papers on Noise and Stochastic Processes, N. Wax, Ed. (Dover, NewYork, 1954).
capri: submitted to World Scientific on May 7, 2001 14
POSSIBLE QUANTUM MECHANICAL EFFECTON BEAM ECHO∗
ALEX CHAO AND BOAZ NASH
Stanford Linear Accelerator Center, Stanford Unversity, CA 94309
The echo effect in charged particle beams provides a link between macroscopicmeasurable beam parameters and microscopic phase space motion of the beam.Since quantum mechanics dictates a granularization of the phase space, it influenceshow the phase space behaves microscopically, and thus potentially affect how theecho effect behaves macroscopically. In this study, we propose to examine thepossible measurable macroscopic effects of quantum mechanics on beams throughits echo effect.
Figure 1 shows a schematic for the classical echo effect. We considertransverse motion in 1-D giving a 2-D phase space with coordinates (q, p).The motion is taken to be that of a harmonic oscillator with frequency ωwith a perturbative non-linear term. Thus, a particle of mass m will follow acircular trajectory (in scaled (q, p/mω) phase space) with a frequency weaklydependent on initial amplitude. At t = 0 the beam is subjected to a dipolekick resulting in a translation of the phase-space distribution. Subsequently,the beam distribution filaments because of the non-linearity and will fill upan annular region. This filamentation results in a diminishing of the beamcentroid signal until it is essentially zero. After a sufficient amount of time τ ,the beam is subjected to a second, quadrupole kick which causes a squeezing ofphase space. Subsequently, the squeezed filamented phase space continues toevolve in some sense undoing the filamentation process such that at a time 2τthe phase space becomes bunched again, yielding a non-zero centroid signal.This beam centroid signal at time 2τ is called the echo.
The mechanism of the echo effect depends on a detailed microscopic cor-relation of phase space dynamics over a long time period of 2τ . This propertyhas led to the application of using the echo as a sensitive measure to detectslow diffusion of particle motion in phase space 1,?. The question being askedhere is whether quantum mechanical requirement on phase space would alsoaffect the echo signal over the long time 2τ .
The simplest example of quantum mechnical effect on beam observablescan be found by replacing the point-particles by wavepackets in an equilibriumbeam distribution in phase space with rms size σ. If all wavepackets are
∗WORK SUPPORTED BY DEPARTMENT OF ENERGY CONTRACT DE–AC03–76SF00515.
t=τ t
t
t=0
P/mω
P/mω
q q
t=τ t=2τ
t=2τ
P/mω P/mωP/mω P/mω
q q qq
(Dipole Kick)
(Filamentation) (Filamentation)
(Quadruple Kick) (Echo)
Beam Centroid
12-20008578A1
Figure 1. Schematic for the classical echo effect.
displaced quantum mechanical ground states, then the beam emittance isfound to be
ε =mωσ2, classicalmωσ2 + 1
2 h, quantum mechanical (1)
This is a quantum mechanical contribution to beam emittance. However,the magnitude of the quantum mechanical contribution is too microscopic tobe measured (unless one has an extremely cold beam). It is hoped that aquantum mechanical effect on echo could be more macroscopic.
In obtaining Eq.(1), we have assumed that particles do not interact withone another. This excludes any space charge Columb interactions. It also ig-nores the effect of the exclusion principle. The latter is particularly importantwhen the beam is extremely cold 3.
It is possible that the centroid motion would not be affected by quantummechanical considerations when one ignores the exclusion principle. However,as demonstrated by Eq.(1), quantum mechanics does affect the beam’s secondmoments. It is therefore possible that the first quantum mechanical effecton echo occurs to the higher order echo when the beam is first kicked by aquadrupole kick, followed by a sextupole kick, and the echo signal is to appearby observing the second moment of the beam.
We would like to model the foregoing classical narrative of the echo effectusing quantum mechanical wavepackets instead of classical point-particles.Thus, we must translate each component of such an echo experiment intoquantum mechanical language. Because transverse motion is non-relativistic,we can use the Schrodinger equation to describe the propagation of single-particle wavepackets 7,?. The picture of an evolving distribution in phasespace gives way to an evolving Wigner function 5,? from which macroscopicparameters such as beam centroid position or beam emittance can be derived.
There are two “active” elements to the echo experiment described above.The beam is first given a dipole kick, or displaced, and then given a quadrupolekick, or squeezed. These two transformations can be described via two uni-tary operators, namely, the displacement operator and the squeeze operator 4.In the absence of non-linearities, the effect of the displacement and squeezeoperators on the time dependence of the quantum state of each particle canbe obtained by standard quantum mechanical treatements. However, in orderto model the filamentation process in the beam, we need to treat the caseincluding non-linearity, ideally, by generalizing the linear analysis. We knowthat 〈q〉 and 〈p〉 will follow the classical non-linear equations of motion andperform the classical filamentation. What happens to the shapes of the wave-functions over time, however, is not readilly apparent, and constitutes themain focus of the subsequent study.
In classical mechanics one replaces the detailed granularity of individualparticles in phase space with a time dependent phase space density functionin order to talk about macroscopic beam properties. Examples of such beamproperties are emittance and beam centroid position. One calculates suchquantities in quantum mechanics using the Wigner function.
Given the initial wavefunctions ψj(q, 0) for all particles j = 1, 2, ...N , wecan compute ψj(q, t). We propose to write the answer as ψ(q, t; q0, p0) where(q0, p0) are the initial centroid coordinates of an injected particle. We thencompute the statistical Wigner function as
W (q, p, t) =1
2πh
∫ ∞
−∞dp0
∫ ∞
−∞dq0 ψ0(q0, p0)
×∫ ∞
−∞dy ψ∗(q +
y
2, t; q0, p0)ψ(q − y
2, t; q0, p0)e−ipy/h (2)
For the case of the harmonic oscillator, it turns out that there is a short-cut to obtaining the time-dependent Wigner function. One can show that theWigner function as defined above satisfies
W (q, p, t) = W (q, p, 0) (3)
where
q = q cosωt+p
mωsinωt (4)
p = p cosωt−mωq sinωt . (5)
Thus, the Wigner function simply rotates with frequency ω.a
As an example, we calculate the Wigner function corresponding to a Gaus-sian distribution of “squeezed state”4 particles. At t = 0 we let
ψ(q, 0; q0, p0) =(b2
π
) 14
e−b2
2 (q−q0)2+
ip0h (q−q0) , (6)
where b is real positive number. For b = β =√
mωh we have a squeezed state.
Because of Eq. (3), we just use the t = 0 wave-functions to compute theWigner function. First, we compute∫ ∞
−∞dy ψ∗(q +
y
2, t; q0, p0)ψ(q − y
2, t; q0, p0)e−ipy/h = 2e−b2(q−q0)
2− β4
b2(
p+p0mω )2
(7)To compute the t = 0 Wigner function, we need to multiply by the classicaldistribution and integrate over q0 and p0. For our example, we choose acentered Gaussian distribution
ψ0(q0, p0) =1
2πσqσpmωe− q2
02σ2
q− p2
02σ2
pm2ω2. (8)
Computing the t = 0 Wigner function, and putting in the time-evolution, theresult is
W (q, p, t) =1
πh√
(2σ2qb
2 + 1)(2σ2p
β4
b2 + 1)e− b2
2σ2qb2+1
q2−
(β4
b2
)2σ2
p
(β4
b2
)+1
p2
m2ω2
(9)
which is a rotating bi-Gaussian. The Wigner function for the displaced Gaus-sian classical distribution which occurs in the discussion of the echo effect canbe similarly computed.
For the case b = β, the Wigner function reduces to
W (q, p, t) =1
πh√
(2σ2qβ
2 + 1)(2σ2pβ
2 + 1)e− β2
2σ2qβ2+1
q2− β2
2σ2pβ2+1
p2
m2ω2 (10)
aThe obvious generalization for the non-linear case where q and p would simply follow theclassical motion is, unfortunately, not generally true. See 6 and references therein, but notetheir slightly different definition of the Wigner function in comparing our Eq. (3).
Taking the limit in which σq = σp = σ, the Wigner function gains rotationalsymmetry in (q, p
mω ) so that the time-dependence goes away:
W (q, p, t) =1
πh(1 + 2σ2β2)e− β2
1+2σ2β2 (q2+ p2
m2ω2 ) (11)
One can show that the quantum emittance given by Eq.(1) follows from thisequation.
A generalization of the calculation is needed when a small non-linearityis added.
We thank P. Chen, A. Dragt, A. Kabel, M. Venturini for many usefuldiscussions.
References
1. G. Stupakov, SSCL Report 579 (1992).2. L.K. Spenzouris, J.-F. Ostigy, P.L. Colestock, Phys. Rev. Lett. 76, 620
(1996).3. A. Kabel, this conference.4. L. Mandel and E. Wolf, “Optical Coherence and Quantum Optics”, Cam-
bridge, 1995.5. A. Dragt, p.655-657, Proc. Quantum Aspects of Beam Physics, World
Scientific, 1999.6. T. Curtright and C. Zachos, U. Miami report TH/1/98, ANL report
ANL-HEP-PR-98-132 (1998).7. Z.R. Huang, P. Chen, and R. Ruth, Phys. Rev. Lett. 74, 1759 (1995).8. M. Venturini, this conference.
Old and new cooling methods are discussed in reference to e ion and beams
PACS numbers Dh Fv Eg
Introduction
Dierent cooling methods were suggested to decrease the emittances ofcharged particle beams in storage rings Among them methods based onsynchrotron radiation damping electron cooling laser cooling in traps
and in storage rings ionization cooling cooing of ions through theinelastic intrabeam scattering and stochastic cooling The majorityof cooling methods are based on a friction of particles in external electromag netic elds or in media when the Liouvilles theorem does not work Onlystochastic method of cooling is not based on a friction It consists in theindividual observation of particles and the action of external control eldsintroduced in the storage ring for the correction of particle trajectories
The friction is determined by the next processes the spontaneous in coherent emission of the electromagnetic radiation in external elds producedby bending magnets undulatorswigglers laser beams etc ionization andexcitation of atoms of a target at rest installed on the orbit of the storagering transfer of the kinetic energy from particles of a being cooled beamto particles of a co propagating cold beam of e e or ions in the processof the elastic scattering excitation of being cooled ions and emission ofphotons by these ions through the inelastic intrabeam scattering e
pair production by photons of a laser beam in elds of being cooled ions A friction originating from a media or in the process of emission scat
tering of photons by charged particles in external elds leads under deniteconditions to a damping of amplitudes of both betatron and phase oscillationsof these particles In this case particles of a beam lose theirs momentumAt that the friction force is parallel to the particle velocity and therefore themomentum losses include both the transverse and longitudinal ones Longitu dinal momentum losses are compensated by a RF accelerating system of the
bessonov submitted to World Scientic on February
storage ring Meanwhile it tends to a certain equilibrium This is because therate of the momentum loss of the particle is higherless than the equilibriumone when the particle momentum is higherless than the equilibrium oneThe transverse vertical momentum of particles disappears irreversibly Thetransverse radial and longitudinal particle oscillations are dispersion coupledthrough energy losses Their damping rates can be corrected or redistributedthrough this coupling by insertion devicestargets undulators laser beamswedge shaped material targets having in general case non homogeneous eldstrengths or density in the radial direction placed at straight sections of stor age rings to introduce an additional friction of particles and by a correctionof the ring lattice parameters Such a way the ordinary compression of phase space density for a given ensemble of particles takes place The enhancedcompression of phase space density of beams based on selective interaction ofparticles of the beam and a target is possible as well
In this paper ordinary and enhanced cooling methods based on a frictionare discussed in reference to e ion and muon storage rings
Ordinary threedimensional radiative cooling of particle
beams in storage rings by laser beams
In the ordinary three dimensional cooling the particle beams are cooled un der conditions when all particles interact with the external elds or mediaindependent of their energy and amplitudes of betatron oscillations Insertiondevices in this case overlap the particle beam and are motionless The dier ence in rates of energy losses of particles of the beam having maximum andminimum energies is not high That is why the cooling time of the particlebeam both in the longitudinal and transverse planes is high and equal to thetime interval at which particle energy losses in the target are equal to aboutthe two fold initial energy of the particle
In the case of laser cooling particles lose their energy mainly in the pro cess of backward Compton or backward Rayleigh scattering of photons Theelectronic or nuclear ion transitions and broadband laser beam have to beused when ions are cooled The physics of the three dimensional laser coolingis similar to the synchrotron radiation damping The dierence is in new re gions where the emission of photons takes place lattice parameters of theseregions and in spectral distribution of scattered photons
In the ordinary three dimensional laser cooling of not fully stripped ionbeams the average cross section of the photo ion interactions fretrL is larger than the Compton Thompson cross sectionT re cm by about a factor of trreL
bessonov submitted to World Scientic on February
which is large about for many cases of the practical interest Inthe previous expressions value f is the oscillator strength re emc theclassical electron radius tr the transition wavelength of the atom Lthe line width of a laser beam When photo ion scattering takes place indispersion free straight section then the damping times of horizontal x andvertical y betatron oscillations are equal The damping times of betatronand phase oscillations of ions are
x y
D
P
where P fNintloss is the average power of the electromagnetic radiationemitted scattered by the ion f the frequency of the ion beam revolution inthe storage ring Nint IsatlintDchL D the number of ioninteractions with the laser beam photons per one ion laser beam collision lintthe length of the interaction region of the laser and ion beams D ILIsatthe saturation parameter Isat gg gh
tr
cL thesaturation intensity gg the degeneracy factor of the state loss htr hL the average energy of scattered photons and thepersonal and central relative energies of ions in the beam tr ctr
The expression in Eq is specic to the assumption that the spectralintensity of the laser beam I x y is constant inside its bandwidth and thearea of the laser beam occupied by the being cooled ion beama Moreoverwe assume that the length of the ion decay ldec csp is much less than thelength of the dispersion free straight section where the spontaneousdecay time sp nat gfre
trgc is the probability of
the spontaneous photon emission of the ion or the natural linewidthThe quantum nature of the laser photon scattering provides excitation of
betatron and phase oscillations of ions The relative rms energy spread ofthe ion beam at equilibrium is given by
p DhtrMc
In the present case where the interaction takes place in a dispersion freestraight section the excitation of both the horizontal and vertical betatron
aThe dispersion coupling leads to a redistribution of the longitudinal and radial dampingtimes when the radial gradient of the laser beam intensity Ix is introduced Experiments conrm this observation The same idea is in the scheme of the ionizing coolingof muon beams
bessonov submitted to World Scientic on February
motions is due to the fact that the propagation direction of emitted pho tons is not exactly parallel to the vector of the ion momentum The rmsequilibrium horizontal ion beam emittance is found to be
x
htrMc
x
In x is the average horizontal beta function in the interactionregion The equilibrium vertical emittance is obtained by replacing x by y The rms transverse size of the ion beam at the waist x
px x
The three dimensional laser cooling of ion beams can be realized by amonochromatic laser beam with accelerating elds of radio frequency cavitiesas well In this case every ion interacts with the laser beam only a part of timewhen it passes the ion resonance energy in the process of phase oscillationsThat is why it has greater damping time at the same saturation parameteras in the case the broadband laser and the same cooling conguration atthat the power of the broadband laser is higher The cooling of a bunchednon relativistic beam of Mg ions kinetic energy keV was observedrst in the storage ring ASTRID in the longitudinal plane
A three dimensional cooling of electron and proton beams based on thebackward Compton scattering of laser photons in the dispersion free straightsections of the storage rings can be used In this case we can usethe expressions if we replace the values T IsatD IL andaccept D The method is identical to that suggested in papers
where magnetic wigglers were used instead of laser beamsb
Enhanced cooling of particle beams
The damping time and the emittance of particle beams can be shortenedsignicantly by using a selective interaction of particles and laser beams Forthis purpose we have to choose such targets which interact with particleshaving denite energies or amplitudes of betatron oscillations and do notinteract with another particles of the same beam For example target caninteract with particles of the energy higher then minimum energy of the beamand do not interact with particles of minimum and lesser energy In this casethe rate of the energy loss of particles is not increased but the dierence inthe rates of losses of particles of the beam having maximum and minimumenergies will be increased essentially and all particles will be gathered at the
bElectromagnetic waves are objects which belong to the type of undulatorswigglers
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minimum energy in a short time equal to the time interval at which a particleloses the energy equal to the initial energy spread of the particle beam Belowwe will consider one and two dimensional enhanced cooling schemes
The selectivity of interaction can be achieved in dierent ways Amongthem note methods based on a resonant and energy threshold interactionsinteraction of ions with a monochromatic and broadband laser beam havingsharp frequency edges e pair production partial overlapping in the radialdirection of a being cooled particle beam and a target
Onedimensional laser cooling of ion beams in storage rings
A typical version of one dimensional laser cooling of ion beams is based on theresonant interaction of unbunched ion beam and homogeneousmonochromaticlaser beam overlapping the ion one in the transverse direction The initialfrequency of the laser beam in this version of cooling is chosen so that photonsinteract rst with the most high energy ions Then the frequency is scanningfrequency chirp in the high frequency direction and ions of a lower energybegin to interact with the laser beam and decrease their energy The scanningof the laser frequency is stopped when all ions are gathered at the minimumenergy of ions in the beam The rate of scanning must correspond to thecondition r drdt Pmax where now Pmax is determined by the previousequation for P if we substitute in the expression for the saturation intensityL nat tr and accept maximum cross section ofthe photon ion interaction max g
trg The resonance ion energy is
r Mcr where r trLtrL
The cooling time and the energy spread of the cooled beam are determinedby the average power of scattered radiation and either the average energy ofscattered photons or the natural line width
inP
max
htrMc
nat
The considered method is one of possible one dimensional ion coolingmethodsc The rst similar method was used for cooling of non relativisticion beams Relativistic version of such method was developed in
One dimensional laser cooling of bunched ion beams by monochromaticlaser beam is possible with accelerating elds of radiofrequency cavities The broadband laser beam with a sharp low frequency edge can be used as
cThe laser frequency can be constant Ions in the direction of given resonance energy canbe accelerated by eddy electric elds or by phase displacement mechanism
bessonov submitted to World Scientic on February
well In this case the edge frequency must have such a value that only ionswith energies above the equilibrium one can be excited
Twodimensional cooling of particle beams in storage rings
One dimensional laser cooling is highly ecient in the longitudinal directionbut rather dicult in the transverse one unless a special longitudinal radialcoupling mechanism is applied synchro betatron resonance dispersioncoupling Moreover this resonance cooling can be applied only to compli cated ions In papers a three dimensional radiative ion cooling methodis proposed and considered above Nevertheless the quest for new more e cient enhanced three dimensional cooling methods remains vital for cooling ofelectrons protons muons and both not fully stripped and fully stripped highcurrent ion beams Such two dimensional and two stage method of coolingbased on alternative targets and selective interactions was suggested in thepaper In this paper the internal target T at the rst stage and externaltarget T of constant thickness at the second one are moved in the transversedirections to the central part of the vacuum chamber in turn The RF accel erating system of the storage ring is switched o The edges of targets musthave the form of a at sharp boundaries
In this method the next observation was used The velocity of a particleinstantaneous orbit x depends on the distance xT x between the targetand the instantaneous orbit and on the amplitude of particle oscillations AWhen the instantaneous orbit of a particle enters the target at the depthhigher than the amplitude of the particle oscillations then its velocity x inis maximum one by the value and negative In the general case particlesdo not interact with the target every turn That is why the velocity x x in W where W is the probability of a particle to cross the targetIt is determined by the inverse ratio of the period of betatron oscillationsof a particle to a part of the period which is determined by conditions the deviation of the particle from the instantaneous orbit is greater than thedistance between the orbit and the target the deviation is directed to thetarget The valueW where arccos arccos xT xA labels correspond to rst and second targets
The change of the amplitudes of betatron oscillations of particles aftera change of their instantaneous orbit by the value x is determined by the
equation Ax x
or Ax x
A where x
isthe particle deviation from the instantaneous orbit averaged through the rangeof phases of betatron oscillations where the particle cross the target
The value x
Asinc where sinc sin signs
bessonov submitted to World Scientic on February
and are related to the rst and second stages of cooling Thus thecooling processes at the rst and second stages are determined by the systemof equations
A
x sinc
xt
x in
This system of equations permits to investigate the main processes con nected with cooling of particle beams
At the rst stage of the two dimensional cooling an internal target Tmust be located at the orbit position xT At the initial step the tar get overlaps only a part of the particle beam so that particles with largestinitial amplitudes of betatron oscillations interact with the target The in teraction takes place only at the moment when the particle deviation causedby betatron oscillations is directed toward the target and has approximatelythe maximum initial amplitude In this case immediately after the interactionand loss of energy the position of a particle and the direction of its momentumremain the same but the instantaneous orbit is displaced inward in the direc tion of the targetd The radial coordinate of the instantaneous orbit and theamplitude of betatron oscillations are decreased to the same value owing tothe dispersion coupling After every interaction the position of the instanta neous orbit approach the target more and more and the amplitude of betatronoscillations is coming smaller until it will reach some small value when theinstantaneous orbit will reach the edge of the target Up to this moment theinstantaneous orbit go in the direction to the target but the particle depthof dipping do not move forward deeper into the target At the moment whenthe instantaneous orbit of the particle enter the target the amplitude of theparticle betatron oscillations is much decreased The instantaneous orbit willcontinue its movement in the target with constant velocity x in if the targetis homogeneous one has constant thickness and when its depth of dipping inthe target is greater then the amplitude of particle betatron oscillations
The particle beam has a set of amplitudes of betatron oscillations andinstantaneous orbits To cool the beam we must move the target in thedirection to the particle beam or move instantaneous orbits of the particlebeam in the direction to the targete at some velocity vT until the targetwill reach the instantaneous orbit having maximum energies Then the targetmust be removed or the particle beam must be returned to the initial position
dWe neglect scattering of particles in the target for the time of the enhanced coolingeA kick decreasing of the value of the magnetic eld in bending magnets of the storagering a phase displacement or eddy electric elds can be used for this purpose
bessonov submitted to World Scientic on February
for a short time As the result all particles will have small amplitudes ofbetatron oscillations and increased energy spread
At the second stage of cooling an external target T is moving with avelocity vT jvT j j xinj from outside of the working region of thestorage ring in the direction of a being cooled particle beam or instantaneousorbits of the particle beam are moving in the direction of the target Theinstantaneous orbits of particles will go in the same direction with a velocityj x j j xinj after the moment of their rst interaction with the target Inthis case the target will start to interact rst with particles having the largestamplitudes of betatron oscillations and the highest energies Then it willinteract with particles of lesser amplitudes and energies When the targetwill pass through the instantaneous orbit of particles having minimum initialenergies then it must be removed to the initial position
Let for the simplicity the initial spread of positions of instantaneousorbits x is much greater then the spread of the amplitudes of betatronoscillations xb of the beam In this case high energy particles rst andthen particles with smaller energies will interact with the moving target untilthe target will reach the instantaneous orbit with the least energy Thenthe target must be removed or the particle beam instantaneous orbits mustbe returned to the initial position for a short time The particles of thebeam having maximum energy and zero amplitudes of betatron oscillationswill interact with the target during the time t
xvT For thistime the instantaneous orbits of particles will pass the distance j x injt
k b At that particles having minimum energy and zero amplitudesof betatron oscillations will stay at rest Hence it follows that the spread ofinstantaneous orbits of these particles will be compressed to the value xf x k At that if the amplitudes of betatron oscillations are not zerothen according to they will be increased
Discussion
The dynamics of positions of instantaneous orbits and amplitudes of beta tron oscillations of particles strongly depends on the target velocity when theinstantaneous orbits are deepened into the target on the depth less then theamplitude of particle betatron oscillations see The moving target beginto interact with particles of the beam located at dierent instantaneous orbitsat dierent moments of time and that is why can compress or decompress thespread of these orbits and change the spread of betatron oscillations Thesefeatures of interaction of moving target can be used for developing of newschemes of enhanced three dimensional cooling
At the rst stage of cooling there is the signicant decrease of amplitudesof betatron oscillations transverse cooling and at the same time a greater
bessonov submitted to World Scientic on February
increase of the spread of instantaneous orbits longitudinal heating If thedegree of transverse cooling is dened by the coecient of compressionCtr p jkjjkj then the increase of the spread of the instantaneous orbits of
the beam decompression will be Dl Ctr Here and below we introduced
the relative velocities of targets k vT xinAt the second stage of cooling there is a signicant decrease of the spread
of instantaneous orbits of particles dened by the compression coecientCl xxf and at the same time lesser value of increase of am plitudes of betatron oscillations The degree of the transverse heating de compression can be about the square root of the degree of the longitudinalcooling Dtr
pCl In this case the degree of selectivity of interaction
of a target with instantaneous orbits must be high It can be realized easierif we will locate the target in the straight section of the storage ring with alow beta and high dispersion function
When the interaction of the particle beam and the target has the reso nance or threshold nature then we can use transverse cooling of particles atthe rst stage of the two dimensional method of cooling and then to use onedimensional method of longitudinal cooling at the second one
At the second stage contrary to the rst one the degree of longitudi nal cooling can be much greater then the degree of heating in the transverseplane That is why in the case of non resonant interaction we can use theemittance exchange between longitudinal and transverse phase spaces us ing the synchro betatron resonance and such a way to have enhanced two dimensional cooling of the particle beam based on the second stage of coolingonly In this case the rst stage of cooling can be omitted and the second onecan be repeated many times
On cooling of electron and ion beams in linear accelerators
Electron beams can be cooled in linear accelerators if external elds un dulators electromagnetic waves producing radiation friction forces will bedistributed along the axes of these accelerators The physics of cooling ofelectron beams under conditions of linear acceleration is similar to one instorage rings where external elds are created in the dispersion free straightsections of the rings The expected emittances of cooled beams are small inboth transverse directions and relatively high in the longitudinal one
First the eect of undulatorwiggler radiation damping on the transversebeam emittance was studied by ATing and PSprangle for linear acceleratorsbased on inverse free electron lasers In the same eect applied to thecase of the radio frequency linear accelerators is considered In and later in
bessonov submitted to World Scientic on February
the cooling of electron beams in linear accelerators was investigated wherea laser beam was used instead of a wiggler General formulas in this casesare similar Some peculiarities are in the hardness of the emitted radiationwhich determines the energy spread and transverse emittance of being cooledbeams The hardness of the backward scattered laser radiation is more highthen undulatorwiggler radiation That is why laser beams for damping canbe used at small MeV electron energies A strong focusing ofelectron and laser beams at the interaction point is necessary in this caseDamping wigglers can be used at high energies GeV and in thelimits of more long distances along the axis of the accelerator about somekilometers
The analogies with the enhanced particle cooling in storage rings arepossible in linear accelerators as well when bending magnets will be used fordispersion separation of particles and selective cooling Monochromatizationof ion beams can be realized by broadband lasers with sharp frequency edgelocated at the exit of the linear accelerator
Conclusion
In this review we have presented dierent methods of cooling of relativisticparticle beams in a single particle approximation We hope that the devel opment and adoption of these methods will lead to the next generation ofstorage rings for colliders of dierent particles e muons ions new lightsources in optical to X ray and ray regions ion fusion sources ofgravitational radiation in IR and more hard regions and so on
References
DBohm and LFoldy Phys Rev v MSands Phys Rev v p KWRobinson Phys Rev v No p AHoman
RLittle JMPeterson et al Proc VI Int Conf High Energy AccelCambridge Mass p
AAKolomensky and ANLebedev Theory of Cyclic AcceleratorsNorth Holland Publ Co MSands The physics of electron stor age rings an Introduction SLAC Report Nov unpub lished HBruk Accelerateurs Circulaires de Particules Press Univer sitaires de France HWiedemann Particle Accelerator Physics I! II Springer Verlag New York
bessonov submitted to World Scientic on February
GIBudker Atomnaya Energia YaADerbenevANSkrinskii Sov Phys Rev v p
DJWineland and HDehmelt Bull Am Phys Soc TWH"ansch and ALShawlov Opt Commun v p
PJChannel JApplPhys v p LDSelvoRBonifacio WBarletta Optics Communications v p
SShr"oder et al Phys Rev Lett v No p JSHangst MKristensen JSNielsen et al Phys Rev Lett v
JSHangst KBerg Sorensen PSJessen et al Proc IEEEPart Accel Conf San Francisco May NY v p
ONeil G Phys Rev AShoch Nucl Instr Methv p
AAKolomensky Atomnaya Energia v No p YuMAdo Atomnaya Energia v No p ANSkrinskii Uspekhi Fiz Nauk v No p a CRubbia Nucl Instr Meth A b M JMiesner
RGrimm MGrieser et al Phys Rev Lett v No p RGrimm UEisenbarth MGrieser et al Advanced ICFA Beam Dy
namics Workshop on Quantum Aspects of Beam Physics World Scien tic Ed Pisin Chen Monterey California USA p
ILauer UEisenbarth MGrieser et al Phys Rev Lett v No p
S van der Meer CERN Internal Report CERNISR PO EGBessonov FWilleke Preprint DESY HERA December EGBessonov physics Phys Rev STAB presented EGBessonov a Preprint FIAN No b Proc of the Internat
Linear Accel Conf LINAC Tsukuba KEK August Vol pp c Journal of Russian Laser Research No p d Proc of the Internat Conf EPAC London June July Vol pp
EGBessonov Proc of the th Int Free Electron Laser ConferenceFEL Nucl Instr Meth vA pp
EGBessonov and Kwang Je Kim Phys Rev Lett vol No p
EGBessonov K JKim Proc of the Part Accel Conf andInt Conf on High Energy Accel p Proc th European ParticleAccel Conference Sitges Barcelona June v p
DVNeuer Nucl Instr Meth vA p JSHangst et al Phys Rev Lett v No p
bessonov submitted to World Scientic on February
K JKim Proc of the th WS on Advanced Accel Concepts LakeTahoe CA AIP Conf Proc p LBNL
Zh Huang RDRuth Phys Rev Lett v No p AHutton Particle accelerators v p RChasman GKGreen Proc th All Union Conference on charged
particle accelerators Dubna Oct v p Moscow Nauka
LEmery Proc Workshop on th Generation Light Sources Febr p EdMCornacchia and HWinick SSRL
EGBessonov Preprint FIAN No Moscow Proc VI All UnionConf on charged particle accelerators Dubna Oct DubnaJINR p Trends in Physics Proc IV EPS General ConfYork UK Bristol p
WPetrich et al Phys Rev A v No p EBonderup Proc th Accelerator School Ed STurner CERN
v p Geneva DHabs et al Proc of the Workshop on Electron Cooling and New
Techniques Lengardo Padowa Italy World Scientic p EGBessonov Bulletin of the American Physical Society Vol No
May p HOkamoto AMSessler and DM"ohl Phys Rev Lett
TKihara HOkamoto YIwashita et al Phys Rev E v No p
EGBessonov K J Kim FWilleke Physics ACTing PASprangle Particle Accelerators v p NSDikanskii AAMikhailichenko Preprint BINP Novosibirsk
Proc VI All Union Particle Accel Conf v p DubnaD
PSprangle EEsarey in High Brightness Beams for Advanced AccelApplicat AIP New York p Phys Fluids B p
VTelnov Phys Rev Lett p EGBessonov Advanced ICFA Beam Dynamics Workshop on Quantum
Aspects of Beam Physics World Scientic Ed Pisin Chen MontereyCalifornia USA p
bessonov submitted to World Scientic on February
ON THE DARWIN LAGRANGIAN
EG BESSONOV
Lebedev Physical Institute AS Leninsky prospect
Moscow Russia
Email bessonovsgilpimsksu
In this paper we explore some consequences of the retardation eects of Maxwellselectrodynamics to a system of charged particles The specic cases of three interacting particles are considered in the framework of classical electrodynamicsWe show that the solutions of the equations of motion dened by the DarwinLagrangian in some cases contradict to the energy conservation law
PACS numbers De Fh c ib
Introduction
Darwin Lagrangian for interacting particles is an approximate one rst derived by Darwin in This Lagrangian is considered to be correct tothe order of c inclusive To this order we can eliminate the radiationmodes from the theory and describe the interaction of charged particles in pureactionatadistance terms Although the Darwin Lagrangian has had its mostcelebrated application in the quantummechanical context of the Breit interaction it has uses in the purely classical domain In this paper we exploresome consequences of the retardation eects of Maxwells electrodynamics toa system of charged particles The specic cases of three interacting particlesare considered in the framework of classical electrodynamics
Below we will present the detailed and typical derivation of the DarwinLagrangian and Hamiltonian for a system of charged particles to correct somemisprints made by some authors Then we will show that the solutions ofthe equations of motion dened by the Darwin Lagrangian in some casescontradict to the energy conservation law
Derivation of the Darwin Lagrangian and consequences
The Lagrangian for a particle of a charge ea in the external eld of an anotherparticle of a charge eb is
La maca eab
eacAb va
qabpkd submitted to World Scientic on February
where ma is the mass of the particle a c the light velocity a p a
relativistic factor of the particle a a jvacj va the vector of a velocity of
the particle a b and Ab the scalar and vector retarded potentials producedby the particle b
The scalar and vector potentials of the eld produced by the charge b atthe position of the charge a can be expressed in terms of the coordinates andvelocities of the particle b for b to the terms of order vbc
and for Ab toterms vbc
b ebRab
Ab ebvb vb RabRabR
ab
cRab
where Rab jRabj Rab Ra Rb Ra and Rb are the radiusvectors of theparticles a b respectively vb jvbj vb the velocityvector of the particle b
Substituting these expressions in we obtain the Lagrangian La for
the particle a for a xed motion of the other particles b The Lagrangian ofthe total system of particles is
L Lp Lint
where the Lagrangian of the system of free particles Lp and the Lagrangianof the interaction of particles Lint are
Lp X
a
maca
X
a
mac X
a
mac
X
a
mac
Lint X
ab
eaebRab
X
ab
eaeb Rab
ab X
ab
eaeb R
ab
a Rabb Rab
The motion of a particle a is described by the equation dPadt L Ra
where P Lva is the canonical momentum of the particle This equationaccording to can be presented in the form see Appendix
dpadt
X
ab
eaebR
ab
abRab X
ab
eaebR
ab
Rabab
X
ab
eaeb R
ab
bRab
X
ab
eaeb R
ab
Rabb
Rab X
ab
eaeb c
b
Rab
Rab
bRab
R
ab
qabpkd submitted to World Scientic on February
where pa maava is the kinetic momentum of the particle aThe Hamiltonian of a system of charges in the same approximation must
be done by the general rule for calculating H from L H va Pa L According to the value see Appendix
H Hp H int
where
Hp X
a
maca
X
a
pm
ac pac
X
a
mac X
a
pa ma
X
a
pacm
a
H int X
ab
eaebRab
X
ab
eaeb cmambRab
papb X
ab
eaeb cmambR
ab
pa Rabpb Rab
The constant valueP
amac in can be omitted Here we would like
to note that contrary to expressions presented in the last two items inthe term H int of the equation has the positive sign and the momentumpa maava includes factor of the particle TheHamiltonian expressed through the canonical momentum has the form
where the ordinary momentum pa is replaced by the canonical one Pa and thesigns of the last two terms are changed a When the particles are movingin the external electromagnetic eld then the term
Pa ea eaPa Amac
eaj Aj mac
is included in the Hamiltonian where and A are the external
scalar and vector potentials In the term eajAj mac
is omittedThe Lagrangian does not depend on time That is why the Hamilto
nian is the energy of the system When particles are moving along the axis x then the Hamiltonian of the
system of particles is described by the expression
H X
a
pm
ac pac
X
ab
eaebRab
papb
cmamb
aIn the Hamiltonian includes small letters for momentum pa mava that is pa in isthe kinetic momentum It dier from because of its derivation is based on erroneousconnection of small corrections to Lagrangian and Hamiltonian If L LL then withoutany approximation H HH where H
Pab
va PaL H P
abva PaL
Pa Pa Pa Pa Lva Pa Lva is the extra term to the canonical
conjugate momentum In this connection was used but the termP
abva Pa was
omitted In our case this term dier from zero as L depends on velocity At the same timeif we will start from the denition H vaLva L then we will receive
qabpkd submitted to World Scientic on February
where i pi are the xcomponents of the particle relative velocity and kineticmomentum respectively
The xcomponent of the force applied to the particle a from the particleb according to in this case is dpadt eaebR
ab
b eaeb bcRab This
force corresponds to the electric eld strength Eb rb c Abtproduced by the particle b and determined by the equations As was tobe expected in the case of the uniform movement of the particle b b the case mb ma the electric eld strength produced by the particle b inthe direction of its movement is b times less then in the state of restb
The dynamics of three particles
Next we consider the dynamics of three particles a b d according to the Darwin Lagrangian and Hamiltonian see Fig Let particles a b have chargesea eb e masses ma mb m and velocities va vb v cThe particle d is located at the position x at rest vd its charge andmass are qM
xavavbb
Fig A scheme of two particle interaction
eb ea
In this case the Hamiltonian is the energy of the system which accordingto can be presented in the form
H Mc mc Mc mc e
R
eq
R
where is the initial relativistic factor of the particles a b corresponding tothe limit R R jRaj the distance between the particle a and the originof the coordinate system
bIn accordance with the fact that electric and magnetic elds dened by potentials andexpressions take into account the retardation
qabpkd submitted to World Scientic on February
From the equation we can receive the dependence between the distanceR and the factor of the particles a b In this case the next solutions existWhen q e
then
R e eq
mc
When q e then according to the particle is moving with
the constant momentum velocity and energyFrom these solutions it follows the next conclusions When q e
then the turning point exist at which p v
and According to the minimal distance between particle a and theorigin of the coordinate system
Rmin e eq
mc e eq
T
where T is the initial kinetic energy of the particle aAccording to the value Rmin ra where ra emac
is theclassical radius of the particle a In conformity with the energy conservationlaw the potential energy eU eeq Rmin of two particles at the turningpoint is equal to the initial kinetic energy of the particles T Retardationdoes not lead to any results which contradict to common sense The termin the electric eld strength and in the force which is determined bythe acceleration of the opposite particle will compensate the decrease of therepulsive forces corresponding to the uniformly moving particles
When q e then according to the particles a b aremoving uniformly v v In this case particles can reachthe distance R x which is not reachable for them under the conditionof the same energy expense T and a nonrelativistic bringing closer of theparticles This conclusion is valid in the arbitrary relativistic case as in thiscase there is no emission of the electromagnetic radiation It contradicts tocommon sense as the particles can be stopped at any position Rs to give backthe kinetic energy T in the form of heat and so on and moreover contraryto the energy conservation law they will produce an extra energy eURs under the process of slow moving aside of these particles by extraneous forcesunder conditions of repulsive electromagnetic forces
When e q e then the particles a b will be brought closerunder the condition of an acceleration by attractive forces and fall in towardeach other At the same time under such value of charge q of the particle d
qabpkd submitted to World Scientic on February
in the nonrelativistic case the particles a b will repel each other such a waythat the position R x will not be reachable for them if the same energyexpense T will be used for slow bringing closer of the particles
When q e then the particles a b will be brought closerunder the condition of an acceleration by attractive forces After a stop theparticles will not experience any force
When q e then the particles will be brought closer under thecondition of an acceleration by attractive forces and fall in toward eachother At that there is no necessity in the initial acceleration of particles tothe energy mc They can be released at some nite distance with zerovelocity
In the cases the velocities of particles can reach the values compared with the light velocity when Darwin Lagrangian does not valid Nevertheless in these cases obviously the process of fall in and contradiction withthe energy conservation law will take place as well In the cases thesum of the kinetic energies of particles before a stop by extraneous forces willbe higher than the initial ones T T Moreover the potential energy ofthe particles after the stop will be positive eURs
In the case the unphysical situation will appear when particles willbe stopped at the distance Rs re and the total energy of the system atthis position will be negative eURs Mc mac
This result isthe known fact for a system of two particles of the opposite sign which isbeyond of the present consideration In this case the contradiction with theenergy conservation law will take place as well as the attractive forces in theprocess of bringing particles closer will be higher then the attractive forces inthe process of slow moving aside of these particles
Conclusion
In the framework of classical electrodynamics there are many open or perpetual problems such as the problem of the particle stability the problemof the selfenergy and momentum of particles the nature of the particlesmass the problem of the runaway solutions There is a spectrum of opinionsconcerning the importance and the ways of a nding of the answers on thesequestions Unfortunately the eorts of the majority of authors are directedto avoid similar questions but not to solve them see eg In additionthey base themselves on the laws of conservation of energy and momentumostensibly following from the electrodynamics in the most general case andpresenting electrodynamics as the consistent theory In such stating the arising questions by their opinion do not have a physical subject of principle and
qabpkd submitted to World Scientic on February
the diculties in their solution are on the whole in the eld of the mathematicians In reality the corresponding equations for the energy and momentumincluding the system of particles and elds does not describe the laws of conservation of energy and momentum These equations are not treated correctway in the existing textbooks The electrodynamics of particles and elds isnot consistent theory
The result presented in this paper is the reminiscence of the nonconsistency of the classical MaxwellLorentz electrodynamics It can be considered as a new open question of the classical electrodynamics The existenceof the received solution is a genuine eect of electrodynamics of point particleswith retardation
Appendix
The canonical momentum of the particle a is
Pa L
va pa pa
where
pa X
ba
eaeb c
bRab
RabRab
b
R
ab
The time derivative of the canonical momentum is
dPadt
d
dt
L
va pa Fa
where pa dpadt Fa dpadt or
Fa X
ba
eaeb R
ab
Raba b b a bRabb bRab a b
X
b a
eaeb R
ab
RabbRab a bRab
X
ba
eaeb c
b
Rab
RabRab
b
R
ab
The directional derivative of the Lagrangian is
L
Ra
X
ab
eaeb Rab
R
ab
ab
X
ab
eaeb R
ab
aRabb bRab
a
qabpkd submitted to World Scientic on February
X
ab
eaeb R
ab
RabRabaRab
b
From the equation of motion dPadt L Ra and equations it follows the equation
The value vk Pk and the Hamiltonian H vaLva L are equal
va Pa X
ab
eaeb
abRab
a Rabb Rab
P
ab
maca
a
H X
a
va Pa L X
a
pm
ac pac
X
ab
eaebRab
cpapb
pm
ac cpa
pm
bc cpb
cRabpaRabpb
R
ab
pm
ac cpa
pmbc cpb
In the approximation c the Hamiltonian leads to
References
CGDarwin Phil Mag Landau L D and E M Lifshitz The Classical Theory of Fields rd
Reversed English edition Pergamon Oksford and AddisonWesley Reading Mass
J D Jackson Classical Electrodynamics John Wiley Sons SColeman JH Van Vleck Phys Rev J De Luca Phys Rev Letters VMehra and J De Luca Phys Rev E No E G Bessonov Preprint FIAN No Moscow
httpxxxlanlgovabsphysics D Ivanenko A Sokolov Klassicheskaya teoriya polya Gostechizdat M
L the Classische Feldtheorie AkademieVerlag Berlin M A Markov Uspekhi Fiz Nauk v p R P Feynman Lectures on Physics Mainly Electromagnetism and
Matter AddisonWesley London EGBessonov Photon Old Problems in Light of New Ideas Book
series Contemporary Fundamental Physics Nova Science Publishers p
qabpkd submitted to World Scientic on February
Section 2
Photon-Electron Interaction
and Monitoring
in Beam Production, Cooling
Coherent Atom Optics with BoseEinstein Condensates
K Bongs S Burger S Dettmer K Sengstock and W Ertmer
Institut fur Quantenoptik Universitat Hannover
D Hannover Germany
The coherent manipulation of BoseEinstein Condensates by far blue detuneddipole potentials is discussed Classical atom optical elements like mirrors andwaveguides are realised and the dynamics of coherent matter waves interactingwith these elements is investigated The local manipulation of the phase of thecondensate wavefunction by temporally applied dipole potentials represents a powerful tool for the design of matter waves We use this method in particular for thecreation of dark solitons in BoseEinstein condensates and study their dynamics
Introduction
The experimental realisation of BoseEinstein condensation BEC in weaklyinteracting atomic gases has stimulated many new physical elds inatomic molecular and optical physics BoseEinstein condensates in atomicgases are unique quantum mechanical systems in several respects The dilutenature in combination with the low temperature and the purity of these systems allows to observe and to compare phenomena known from other elds likesuperuidity vortices or spin domains and their theoretical treatmentfrom rst principles In addition completely new areas such as nonlinearatom optics are developing One of the most interesting future prospects forBoseEinstein condensates is their application as a source of coherent matterwaves eg in atom optics and atom interferometry This oers a signicant advance similar to the introduction of lasers in light optics Moreoverthe macroscopic size of a BoseEinstein condensate opens the exciting possibility to manipulate the quantum mechanical wavefunction itself which allowsfor the creation of fundamental excitations and facilitates the application ofspecically tailored wavefunctions
An important aspect for future applications is the coherent manipulationof matter waves by external potentials Laser induced dipole potentials haveproven to be extremely feasible in this respect as they allow for variable spatialgeometries as well as for fast and exible timing In this article we willdiscuss mirrors and waveguides for matter waves based on dipole potentialsThe method of phase manipulation of the condensate wave function by a localapplication of pulsed dipole potentials to parts of the condensate is used tocreate and to study dark solitons in BoseEinstein condensates
We will concentrate on the case of far detuned light elds where sponta
neous emission processes can totally be neglected during the atomlight interaction This allows to treat the corresponding dipole potentials as conservativeand thus coherent potentials
The characteristics of the atomlight interaction can be divided into thefollowing two regimes depending on the duration of the applied light elds
Spatial regimeThe spatial regime is analogous to classical light optics but with the rolesof matter and light interchanged ie here a matter wave moves inthe presence of xed light eld potentials The atomlight interaction isrepresented by the dipole force the atoms experience in these light eldsWe will discuss two examples for this regime namely an atom mirrorand a matter waveguide formed by a light sheet and a doughnutshapedLaguerreGaussian light tube respectively
Temporal regimeThe temporal regime corresponds to the case of a pulsed light eld interacting with a matter wave which is practically xed during the pulse timeIn the case of spatially homogeneous light intensities the interaction results in a pure phase shift of the matter wave function The application oflight elds with several homogeneous regions of dierent intensities leadsto the imprint of spatially varying phases This regime oers the excitingpossibility of phase engineering of matter waves leading to holographicmatter wave design In contrast the application of light elds with intensity gradients is characterised by a sudden momentum transfer ontoparts of the wavefunction in a kicklike interaction We will discusshere the creation of dark soliton states as an example of fundamentalexcitations in BoseEinstein condensates in this regime
Mirrors for BEC
There is still a demand for ecient mirrors with purely specular reection inatom optics Compared to light optics several aspects lead to a more complexsituation in atom optics Due to the dierent dispersion relation matter wavesare often subject to velocity dependent ie dispersive processes Decoherencedue to spontaneous processes has to be considered and in the high densityregime nonlinear eects induced by the interparticle interaction appearVarious dierent concepts have been studied in recent years The reectionfrom evanescent waves and periodic magnetic surfaces has reacheda very high standard with the advantage that atoms with relatively high kinetic
energy can be reected On the other hand in both methods surface roughnesshinders a purely specular reection
We have studied the reection of atoms from at light elds In thiscase the mirror potential can be realised arbitrarily at but on the otherhand the gradient of the light eld potential is weaker leading to a slightlysofter mirror compared to evanescent waves In particular we have studiedin detail the coherent reection of BoseEinstein condensates released frommagnetic traps on light elds formed by far blue detuned light Here weconcentrate on the description of the two dierent regimes for this kind of atommirror ie soft and hard mirror and on a more intuitive explanation ofthe observed self interference structures of condensate wavefunctions after thereection Note that soft mirrors which also circumvent the problem ofsurface roughness were recently realised by soft gradient magnetic elds andlaser Raman reection
Figure Experimental setup for the reection of BoseEinstein condensates from a repulsivedipole potential light sheet
In the experimental realisation a BoseEinstein condensate of Rb atomstypically containing atoms is released from a magnetic trap falls underthe inuence of gravity and is reected by a blue detuned light sheet The distance between the magnetic trap and the light eld was varied between andm Fig In contrast to single particle ie classical atom optics withthermal atomic ensembles the macroscopic wavefunction of a BoseEinsteincondensate leads to macroscopic interference structures in the reected ensemble see Fig Thus a quantum mechanical treatment is essential Onevery important point concerning the initial evolution of the condensate is thenonlinearity induced by the interparticle interaction It is essential for thepreparation of the wavefunction during the rst few ms of time of ight but ithas no direct inuence on the reection process which can be treated as linear
Figure Darkeld images for BoseEinstein condensates bouncing o a light sheet positioned m below the magnetic trap Each image was taken with a new condensate andwith an additional time delay of ms
In the following we will concentrate on these aspectsThe evolution of the condensate wavefunction can be summarised as followsinto several steps
During the rst few ms after switching o the magnetic trapping potential the meaneld energy stored in the Bose condensed ensemble istransformed into kinetic energy For our experimental parameters morethan of the meaneld energy is transformed within the rst msof time of ight It is only during this time that the nonlinear term inthe GrossPitaevskii equation governing the evolution of the condensatewavefunction gives a signicant contribution This nonlinear preparationstage is nevertheless essential for the appearance of macroscopic interference structures in the reected atomic cloud
During the following evolution under the inuence of gravity the wavepacketfurther accelerates and matter wave dispersion in free space leads to acorrelation between position and wavevector Therefore the resultingwavepacket is mainly determined by the velocity spread induced by thenonlinearity during the initial evolution period
The interaction of the atoms with the mirror potential strongly dependson the height of the dipole potential with respect to the kinetic energyof the atoms when hitting the light sheet The following two regimes canbe identied
Hard mirrorIn the case of a relatively high dipole potential the dynamics ofthe atomic cloud can be described to a good approximation as thereection from an innitely high potential step ie an ideal hard
mirror Thus the eect of this mirror on the matter wavepacketonly consists in a reversal of the zcomponent of the wavevectorsie a reversal of the vertical motion
Dispersive mirrorIf the dipole potential is only slightly stronger than the kinetic energy of the atoms the zcomponent of the wavevectors is still reversed but the time delay for this process is now wavenumber dependent This results in a dispersive reection changing the initialcorrelation between position and the wavevector of the wavepacketIn this regime it is possible to prepare a wide range of positionwavevector correlations also allowing for the observation of macroscopic interference structures during the further evolution as shownin Fig
Note that the interaction between the atoms can be neglected during thereection process even if there is some compression of the cloud in thevertical direction Due to an additional expansion of the cloud in thedirection of observation the ensemble stays that dilute that the meaneld contribution to its energy can be neglected This is conrmed bynumerical simulations of the GrossPitaevskii equation
After the interaction with the mirror the wavepacket evolves again under the inuence of gravity The interaction of the wavepacket with thelinearly rising gravitational potential corresponds to a dispersive reection This leads to purely quantum mechanical self interference structures occurring in the matter wavepacket around the classical turningpoint These structures strongly depend on the wavepacket preparationand become large scaled for the choice of a soft mirror
The occurrence of interference structures can be explained in a more intuitive classical picture as follows Even an ensemble consisting of classicalparticles can lead to a doublepeaked structure after bouncing o an idealhard mirror due to the crossing of the trajectories of particles with dierentinitial velocities An important condition for the occurrence of these structuresis an initally small spatial extent and a relatively large velocity spread of theensemble
Fig shows trajectories of classical particles following Newtons law ofmotion which all start at the same initial height above the mirror at the sametime but with dierent velocities After reection and passing the upper turning point there are two regions in which the trajectories of particles withdierent initial velocities overlap leading to a doublepeaked density distribution This is even more pronounced when being convoluted with the correct
Figure Classical trajectories of pointlike noninteracting particles being reected from ahard mirror with dierent initial vertical velocities The dotted line corresponds to particleswith zero initial velocity The dashed vertical line indicates a time at which a splitting ofthe ensemble occurs due to the crossing of two velocity classes of trajectories
Figure Absorption images of a thermal ie noncondensed atomic ensemble being reected from a hard mirror m below the magnetic trap After reaching the upper turningpoint a splitting of the cloud can be observed
initial density distribution This behaviour is clearly visible in the experimentally observed reection of a thermal ensemble as shown in Fig For coherentquantum mechanical wavepackets the crossing of trajectories of particles withdierent velocities just corresponds to interference structures
In our case of a BoseEinstein condensate the additional velocity spreaddue to the conversion of meaneld energy into kinetic energy is essential forthe velocity distribution prior to the reection and thus for the splitting of theensemble and the interference structures According to numerical simulationsa condensate being reected from a hard mirror will only show interferenceswith high spatial frequencies the fringes being too small to be imaged byour detection system The use of a more dispersive soft mirror allowsfor the preparation of the wavepacket such that the macroscopic interference
structures of Fig occur
Finally let us emphasise that the observed interference contrast corresponds to the contrast found in numerical simulations based on the GrossPitaevskii equation giving evidence of the coherence of the reection process The estimation of spontaneous processes for the reection of Rb atomsfrom the nm light elds gives values below per bounce so that thisdecoherence process can be neglected As the light eld can easily be createdby a fast modulation of a regular Gaussian shaped light beam the atness ofthe mirror in two dimensions is guaranteed We could observe no additionaltransverse heating within our detection resolution mrad for the transversespreading after one bounce
In conclusion the discussed at far blue detuned light eld represents auseful tool for the reection of slow atoms As the hardness can be variedwithin certain limits the dierent regimes of dispersive and nondispersivereection allow for fundamental studies eg of phase shifts and coherenceproperties of the reected ensembles
Loading BECs into a de Broglie waveguide
The use of optical bres for the guiding of photons is a well developed technique with many applications ranging from high speed data communication tofundamental physics Correspondingly waveguides for atomic matter wavespromise to be an extremely useful tool in various atom optical applicationsfor a review see eg Guiding of atoms has been demonstrated eg in experiments using hollow bres or freely propagating light beams An additional purpose of waveguides for matter waves is to hold atomic ensembles against gravity thus allowing long interaction and observation timesfor earth based systems Of particular interest are new interferometer designswith an unprecedented sensitivity eg long arm sagnac interferometers usingguided atoms
With the availability of coherent matter waves the demand for coherentguides for matter waves is even higher Congurations similar to those wellestablished in linear and nonlinear light optics with laser light in bres can bea basis forthe new regimes of coherent linear and nonlinear matter wave optics
The operation of single mode guiding in reach within the nearest futurewill allow for fascinating aspects like pure D congurations for coherent matter waves controlled collisions in strongly conned D systems as well as thestudy of fundamental excitations like solitons in D or quasi D regimes
As a class of waveguides the dipole potentials of tubeshaped light eldsagain have the advantage of roughness free manipulation tools well suited
for exible geometries and for fast timing An additional advantage of dipoletraps eg compared to magnetic traps is the possibility of trapping atomsindependently of their magnetic substate
We use the repulsive dipole potential of a blue detuned LaguerreGaussiandoughnut laser beam propagating in vacuum as a waveguide and conningthe atoms in a region of low laser intensity Light scattering which leads todecoherence is suppressed for atoms in the transverse ground state by morethan three orders of magnitude in comparison with the scattering rate in theintensity maximum With the use of doughnut beams of dierent orders different potential geometries were realised In earlier experiments the loadingof a BoseEinstein condensate into a TEM
waveguide was studied In this
paper we report on experiments using a LaguerreGaussian beam of rst order TEM
to hold and guide the atoms The radial intensity distributionIr in a doughnut mode of lth order TEM
l is given by
Ir lPrllrl
er
r
where P and r are the laser power and the beam waist respectively A rstorder doughnut mode thus results in a radially symmetric harmonic potentialWith a power of P W at nm and a beam waist of r m the resulting dipole potential at the focal plane has a maximum value of KkBwith a transverse oscillation frequency of kHz for Rb atoms In the exper
Figure Schematic setup for the guiding of BoseEinstein condensates within a doughnutmode light eld
iment described here the BoseEinstein condensates were nonadiabaticallytransferred into the waveguide potential with a loading rate of up to Thewaveguide was aligned with the long condensate axis see Fig with a slighttilt allowing for gravitational acceleration Due to the nonadiabatic transferseveral transverse modes of the conning potential were occupied leading toheating of the axial degree of freedom The evolution of the BEC inside thedoughnut mode is governed by gravity expansion of the atomic ensemble due
to the initial meaneld energy and also by these heating eects Fig showsthe density distribution of the atomic ensemble for dierent times after thetransfer into the doughnut waveguide
Fig shows rst results on the broadening of the atomic density distribution within the waveguide for two dierent radial alignments of the waveguideIn a simple model the nonadiabatic loading leads to an alignment dependentenergy in the transverse degrees of freedom Due to the nonlinear interparticleinteraction this energy is also transferred to the axial degree of freedom nallyleading to a constant axial expansion rate corresponding to the temperatureof the ensemble Heating eects due to beam uctuations are supposed to givea minor contribution to the system energy as they are highly suppressed byadiabatic following processes in the low frequency range From the experimental data axial temperatures of approximately K and K can bededuced from the constant expansion corresponding to energies much largerthan the meaneld energy of the original condensate nK The datagives evidence of two time scales in the axial expansion due to a transfer ofthe initial radial energy to axial energy which occurs with a time delay on theorder of to ms
In order to form an analogue to an optical laser cavity for an atom laserit is possible to introduce two additional dipole potential mirrors closing thewaveguide see Fig As has been shown in the previous section the reectivity of the mirrors is velocity dependent Therefore when applying Braggpulses transferring an appropriate momentum to the condensate it should bepossible to couple out parts of the condensate A main advantage of this schemewould be the directed output of coherent matter waves into the doughnut waveguide Note that the interparticle interactions play an important role in theBragg outcoupling process when the mean free path between atomatom collisions becomes smaller than the size of the condensate Fig shows the caseof a beamsplitter realised with Bragg scattering in freee space for atomicensembles with the mean free path above and below their size In Fig athe BoseEinstein condensate has been released from the magnetic trap msprior to the Bragg pulse whereas in Fig b the Bragg pulse was applied tothe trapped ensemble with the trapping elds switched o immediately afterthe pulse avoiding any inuence on the further evolution The densities werebelow cm in a and cm in b corresponding to a meanfree path of m and m respectively The time of ight used toseparate the dierent velocity components was ms in a and ms in band the condensate size was m The velocity of sound in both cases wassmaller than the relative velocity of mms of the two Bragg componentsThe nearly perfect splitting is clearly visible in the case where the mean free
path of the particles during the splitting process is suciently large so thatno collisional processes inuence the splitting In the case where the mean freepath is much smaller than the condensate size during the separation enhancedcollisions of identical bosons lead to a collisional cloud A suppression of collisions in high density samples seems feasible for a relative velocity below thevelocity of sound in the atomic ensemble In this case phonons are excited forBragg scattering and superuid suppression of collisions occurs if hypernechanging Raman transitions are used to couple the atoms to a moving state
The fundamental atom optical elements discussed so far ie atom mirrorsand matter waveguides are classical elements in the sense that they arebased on a conservative potential directly inuencing the density distributionof the atomic ensemble Recently holographic methods have been developed
which mainly eect the phase of an atom or the collective phase of a Bosecondensed ensemble The following section describes the application of theatom holographic technique of phase imprinting which allows for engineeringof a variety of excited states in coherent matter waves eg to convert theground state of a BoseEinstein condensate into the state of a dark soliton
Phase engineering of matter waves
creation of dark solitons in BoseEinstein condensates
As the harmonic oscillator archetypically describes oscillation processes solitons are nowadays the paradigm of nonlinear waves Dark solitons as an important class of macroscopically excited Bose condensed states are of particularinterest within the new eld of nonlinear atom optics to explore nonlinearproperties of matter wavesSolitonlike solutions of the GrossPitaevskii equation are closely related to similar solutions in nonlinear optics describing the propagation of light pulses inoptical bres Here bright soliton solutions correspond to short pulses wherethe dispersion of the pulse is compensated by the selfphase modulation ie theshape of the pulse does heuristically not change Similarly optical dark solitons correspond to intensity minima within a broad light pulse In the case of nonlinear matter waves bright solitons are expected only for anattractive interparticle interaction swave scattering length a whereasdark solitons also called kink states are expected to exist for repulsive interactions a Recent theoretical studies discuss the dynamics and stabilityof dark solitons as well as concepts for their creation Conceptually solitons as particlelike objects provide a link of BEC physics touid mechanics nonlinear optics and fundamental particle physics
Dark solitons in matter waves are characterised by a local density minimumand a sharp phase gradient of the wave function at the position of the minimumFig a Due to the balance between the repulsive interparticle interactiontrying to reduce the minimum and the phase gradient trying to enhance it theshape of the soliton does not change The macroscopic wave function of a darksoliton in a cylindrical harmonic trap forms a plane of minimum density DSplane perpendicular to the symmetry axis of the conning potential Thusthe corresponding density distribution shows a minimum at the DS plane witha width of the order of the local correlation length A dark soliton in ahomogeneous D BEC of density n is described by the wave function see
and references therein
kx pn
i vk
cs
s vk
cs tanh
x xkl
s vk
cs
with the position xk and velocity vk of the DS plane the correlation lengthl an
and the speed of sound cs panhm where m is the
mass of the atomFor T in D dark solitons are stable In this case only solitons with
zero velocity in the trap center do not move otherwise they oscillate alongthe trap axis However in D at nite temperature dark solitons exhibitthermodynamic and dynamical instabilities The interaction of the soliton withthe thermal cloud causes dissipation which accelerates the soliton Ultimatelyit reaches the speed of sound and disappears The dynamical instabilityoriginates from the transfer of the axial soliton energy to the radial degreesof freedom and leads to the undulation of the DS plane and ultimately tothe destruction of the soliton This instability is essentially suppressed forsolitons in cigarshaped traps with a strong radial connement such as inour experiment
As can be seen from equation the local phase of the dark soliton wavefunction varies only in the vicinity of the DS plane x xk and is constantin the outer regions with a phase dierence k between the parts left andright to the DS plane see Fig a To generate dark solitons experimentally we apply the method of phase imprinting which also allows one tocreate vortices and other textures in BoseEinstein condensates A homogeneous potential Upi generated again by the dipole potential of a far detunedlaser beam is applied to one half of the condensate wave function Fig bThe potential is pulsed on for a time tp such that the wave function locallyaquires an additional phase factor ei pi with pi Upi tph Thepulse duration tp is chosen to be short compared to the correlation time of the
condensate tc h where is the chemical potential This ensures that theeect of the light pulse corresponds approximately to a change of the phase ofthe BEC whereas changes of the density during this time can be neglectedNote however that at larger times due to the imprinted phase pi Fig bone expects an adjustment of the phase and of the density distribution in thecondensate This will lead to the formation of a dark soliton with k piin general and also to additional structures Fig shows simulations of theD GrossPitaevskii equation for the dynamics of the condensate wavefunctionafter the phase imprinting at t ms After ms the imprinted phase stepleads to the formation of a density minimum and a maximum both travelling inopposite directions These two structures take away part of the initial phasegradient pi Due to matter wave dispersion and the repulsive interparticleinteraction the density maximummoves with the speed of sound and broadenswhereas the density minimum travels with smaller velocity and preserves itsshape This density minimum thus corresponds to a moving dark soliton
In these experiments we produce BoseEinstein condensates of Rb every s containing typically atoms in the F mF state withless than of the atoms being in the thermal cloud The fundamentalfrequencies of our static magnetic trap are x Hz and Hz along the axial and radial directions respectively The condensates arecigarshaped with the long axis xaxis oriented horizontally
For the phase imprinting potential Upi we use a blue detuned far oresonantlaser eld nm of intensity I Wmm pulsed for a time tp sresulting in a phase shift pi on the order of
Spontaneous processes canbe totally neglectedA high quality optical system is used to image an intensity prole onto theBEC nearly corresponding to a step function with a width of the edge lesmaller than m The corresponding potential gradient leads to a force transferring momentum locally to the wave function and supporting the creation ofa density minimum at the position of the DS plane for the dark soliton Notethat the velocity of the soliton also depends on leAfter applying the dipole potential we let the atoms evolve within the magnetictrap for a variable time tev We then release the BEC from the trap switchedo within s and take an absorption image of the density distributionafter a time of ight tTOF ms reducing the density in order to get a goodsignaltonoise ratio in the images
In a series of measurements we have studied the creation and successivedynamics of dark solitons as a function of the evolution time and the imprintedphase Figures a and b show density proles of the atomic clouds for
two dierent evolution times in the magnetic trap tev The phase imprintingpotential Upi has been applied to the lower part of the condensate with x and the potential strength in this measurement was estimated to correspondto a phase shift of
For short evolution times Fig a the density prole of the condensateshows a pronounced minimum contrast about After a time of typicallytev ms a second minimum appears Both minima contrast about each travel in opposite directions and in general with dierent velocities alongthe long condensate axis Fig b
One of the most important results is that both structures move with velocities which are smaller than the speed of sound cs mms for our experimental parameters and depend on the applied phase shift pi Thereforethe observed structures are dierent from sound waves in a BoseEinstein condensate as rst observed at MIT We identify the minimum moving slowlyin the negative x direction to be the DS plane of a dark soliton Figure shows the experimentally observed evolution of this dark soliton With dierent parameter sets for the imprinted phase pi which is determined by theproduct of laser intensity and imprinting time and the width of the imprintedphase step le the velocity of the dark soliton could be varied experimentallybetween vk mms Fig and vk mms
In addition to the dark soliton the dipole potential creates a density wavewhich consumes part of the imprinted phase pi and which travels in thepositive x direction with a velocity close to cs After opening the trap acomplex dynamics results in the appearence of a second minimum behind thedensity wave see Fig
All our experimental observations agree very well with theoretical investigations and numerical simulations of the D GrossPitaevskii equation performed by A Sanpera M Lewenstein and GV Shlyapnikov
The experimental results also show clear signature of the presence of dissipation in the dynamics originating from the interaction of the dark solitonwith the thermal cloud We observe a decrease of the contrast of the darksoliton by on a time scale of ms This is in contradiction with anondissipative dynamics where the contrast should even increase for a darksoliton moving away from the trap center The decrease of the contrastcan therefore only be explained by the presence of dissipation decreasing theenergy of the dark soliton As the lifetime of the dark soliton is sensitive tothe ensemble temperature the studies of dissipative dynamics of dark solitonswill oer a possibility for thermometry of BoseEinstein condensates in thecondition where the thermal cloud is not discernible
Parallel to our work dark solitons in nearly spherical BoseEinstein con
densates were observed at NIST
Interaction of several solitons in BoseEinstein condensates
The study of dark solitons as particlelike objects can be extended to severalkinks within a BoseEinstein condensate In this case the characteristics oftheir interaction can provide a direct link of the physics of nonlinear matterwaves not only to nonlinear optics but also to fundamental particle physicsespecially to eld theory where the concept of particles and antiparticlescan directly be applied to dark solitons
Recent theoretical studies controversially discuss the dynamics of two darksolitons in BoseEinstein condensed ensembles In this case two dierent interaction regimes depending on the relative orientation of the two phasegradients can be studied see Fig
Two kinks ie two dark solitons with the same phase gradient lead toan eective interaction which is attractive whereas the interaction between akink and an antikink with opposite phase gradients is eectively repulsive
We present here rst experimental results on the interaction of two counterpropagating dark solitons within a BoseEinstein condensate To generatethem experimentally we apply again the method of phase imprinting as discussed in the previous section The homogeneous potential Upi of the fardetuned laser beam nm is now applied to the middle of the condensate wave function Fig generating two dark solitons with opposite phasegradients ie a kink and an antikink After a variable evolution timetev within the magnetic trap we release the condensate and take an absorptionimage of the density distribution after a time of ight tTOF ms
Figure shows the evolution of two dark solitons ie the experimentallyobserved positions of the two density minima during the rst few ms in themagnetic trap after the phase imprinting The dark solitons approach eachother with approximately constant velocity and after about to ms theyoverlap in the central region of the condensate see also Fig a and b
For longer evolution times up to about ms we observed quite dierentsituations The absorption image in Fig c shows one single minimum ofthe density distribution after ms which still is at the same position as it wasafter only ms of evolution time within the magnetic trap
The fact that our experimental results uctuate reproducibly indicates avery critical dependence of the soliton interaction on the specic experimentalparameters We currently investigate in detail the successive dynamics of acrossing or reection of dark solitons as well as possible bound states of two
solitons in the presence of dissipation
Conclusions
We have discussed the coherent manipulation of matter waves by far bluedetuned dipole potentials We presented classical elements like mirrors andwaveguides based on spatially distributed light elds as well as innovative phasemanipulation methods realised with pulsed dipole potentials
The complex dynamics of BoseEinstein condensates falling under gravityand bouncing o a mirror formed by a far detuned sheet of light has beenstudied After reection the atomic density prole develops splitting andinterference structures A comparison with the behaviour of bouncing thermalclouds allows to identify quantum features specic for condensates
BoseEinstein condensates of Rb have been loaded to a linear waveguidefor atomic de Broglie waves The waveguide is created by the optical dipoleforce of a far oresonant blue detuned LaguerreGaussian laser beam of highorder and the transport and broadening of the atomic cloud inside the waveguide has been studied
The method of phase imprinting was used to create dark soliton statesin BoseEinstein condensates as fundamental excitations in nonlinear matterwaves Investigations of the dynamics of dark solitons in BoseEinstein condensates were presented First experiments concerning the interaction of twodark solitons were discussed
Acknowledgements
Part of the work presented here has been done in fruitful and stimulating cooperation with !L Dobrek M Gajda M Lewenstein K Rz"a#zewski A Sanpera and GV Shlyapnikov We also acknowledge important support in dierent stages of the experiment by G Birkl D Hellweg M Kottke M Kovacevand T Rinkle
This work is supported by SFB of the Deutsche Forschungsgemeinschaft
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Rolston S L and Phillips W D Science Bloch I Hansch T W and Esslinger T Phys Rev
Lett Adams C S Sigel M and Mlynek J Phys Rep
Kasevich M A Weiss D and Chu S Opt Lett
$ Aminoff C G Steane A M Bouyer P Desbiolles P
Dalibard J and CohenTannoudji C Phys Rev Lett
Szriftgiser P Gu%eryOdelin D Arndt M and DalibardJ Phys Rev Lett $ Landragin A Courtois JYLabeyrie G Vansteenkiste N Westbrook C I and AspectA Phys Rev Lett
Christ M Scholz A Schiffer M Deutschmann R andErtmer W Opt Comm
Roach T M Abele H Boshier M G Grossman H L
Zetie K P and Hinds E A Phys Rev Lett Sidorov A et al Quant Semiclass Opt Bongs K Burger S Birkl G Sengstock K Ertmer
W Rza zewski K Sanpera A and Lewenstein M Phys Rev Lett
Burger S Bongs K Sengstock K and Ertmer W inMartellucci S editor Int School of Quant Electr th
Course Erice Sicily Kluwer Academic Publishers in print Kohl M et al Proceedings of the spring meeting of the
Deutsche Physikalische Gesellschaft Bonn Dowling J P and GeaBanacloche J Adv At Mol Opt
Phys Renn M J Montgomery D Vdovin O Anderson D Z
Wieman C E and Cornell E A Phys Rev Lett
Renn M J Donley E A Cornell E A Wieman C E andAnderson D Z Phys Rev A R
Ito H Nakata T Sakaki K Ohtsu M Lee K I and JheW Phys Rev Lett
Wokurka G Keupp J Sengstock K and Ertmer W Procs EQEC QFG
Schiffer M Rauner M Kuppens S Zinner M Sengstock
K andW Ertmer W Appl Phys B Kuppens S Rauner M Schiffer M Sengstock K and Ert
mer W Phys Rev A StamperKurn D M Chikkatur A P Gorlitz A Inouye
S Gupta S Pritchard D E and Ketterle W PhysRev Lett
Dobrek L Gajda M Lewenstein M Sengstock K Birkl
G and Ertmer W Phys Rev A R Kivshar Y S and LutherDavies B Physics Reports
& Ruprecht P A Holland M J Burnett K and Edwards
M Phys Rev A Zhang W Walls D F and Sanders B C
Phys Rev Lett $ Reinhardt W P and Clark C W J Phys B L$ Jackson A D Kavoulakis G M andPethick C J Phys Rev A
Busch T and Anglin J condmat Muryshev A E van Linden van den Heuvell H B and
Shlyapnikov G V Phys Rev A R Fedichev P O Muryshev A E and Shlyapnikov G V
Phys Rev A
Dum R Cirac J I Lewenstein M and Zoller P Phys Rev Lett
Scott T F Ballagh R J and Burnett K J Phys B L
In our case the dark solitons move with a velocity of about cs andthey are already dynamically stable at a smaller radial connement thanin the case of standing dark solitons
The imprinted phase is estimated from measured laser parameters andtables of atomic data In addition the onset contrast for darksoliton creation was measured to correspond to in agreementwith the theoretical results
Andrews M R Kurn D M Miesner HJ Durfee
D S Townsend C G Inouye S and Ketterle W Phys Rev Lett
Burger S Bongs K Dettmer S Ertmer W Sengstock
K Sanpera A Shlyapnikov G V and Lewenstein M Phys Rev Lett
Denschlag J Simsarian J E Feder D L Clark C W
Collins L A Cubizolles J Deng L Hagley E W Helmer
son K Reinhardt W P Rolston S L Schneider B I andPhillips W D Science
Sanpera A et al Proceedings of the spring meeting of theDeutsche Physikalische Gesellschaft Bonn
Busch T private communication Shlyapnikov GV private communication
Figure Atoms from a BEC loaded to a doughnut waveguide with dierent evolution timesinside the waveguide The atomic ensemble is accelerated to the left by gravity due to aslight tilt of the light eld with respect to the horizontal plane Broadening of the cloudoccurs due to heating eects and a release of meaneld energy
Figure Longitudinal expansion of BoseEinstein condensates loaded into a doughnut waveguide with two dierent radial alignments The squares triangles are data points for aslight relatively large misalignment between the centre of the doughnut mode and the BECin the magnetic trap The lines are just added to guide the eye
Figure Concept for continuous outcoupling of coherent matter waves from a BoseEinsteincondensate in a dipole trap which simultaneously acts as a matter waveguide
Figure Bragg scattering of free BoseEinstein condensates with a transfer eciency ofapproximately The Bragg pulses were applied to a sample with a mean free path abovea and below b the sample size
Figure a Density and phase distribution of a dark soliton state with k Thedensity minimum has a width of l b Phase imprinting potential Upi and associatedphase distribution
Figure Phase distribution a and density distribution b of the condensate wave function after the phase imprinting obtained numerically from Dsimulations of the GrossPiteavskii equation for dierent evolution times in the magnetic trap The dark soliton isindicated by an arrow
Figure Absorption images of BoseEinstein condensates with kinkwise structures propagating in the direction of the long condensate axis for two dierent evolution times tev in the magnetic trap a tev s and b tev ms The dark soliton is marked byan arrow N and tTOF ms
Figure Position of the experimentally observed density minimum corresponding to thedark soliton versus evolution time in the magnetic trap
Figure Density and phase distribution of two dark solitons in a BEC a with oppositephase gradients kinkantikink and b with the same phase gradient kinkkink
Figure Position of the experimentally observed density minima corresponding to the twodark solitons versus evolution time in the magnetic trap
Figure Absorption images of BoseEinstein condensates with kinkantikink structurespropagating in the direction of the long condensate axis for dierent evolution times tev in the magnetic trap a tev s b tev ms and c tev ms N
and tTOF ms
THE WEAKLY-INTERACTING PHOTON GAS IN 2D:BEC, SUPERFLUIDITY, AND QUANTIZED VORTICES
RAYMOND Y. CHIAO, COLIN MCCORMICK, AND JANDIR M. HICKMANN*DEPT. OF PHYSICS, UNIV. OF CALIFORNIA, BERKELEY, CA 94720-7300,
USA
The conditions under which a Bose-Einstein condensate (BEC) of photons can formis discussed. In analogy with the recently observed atomic BECs, a photonic BECcan occur when photons weakly interact with each other while inside a harmonictrap. Photon-photon collisions can happen when photons interact with each othervia atoms excited off resonance by a few Doppler widths. At the macroscopiclevel, this leads to the atoms forming a Kerr nonlinear optical medium. Thephotons, along with the atomic medium, are placed inside an optical resonatorwhich forms a harmonic trapping potential in two transverse dimensions for thephotons. For this paraxial geometry, the effective mass of the photons for theirtransverse, two-dimensional dynamics is hω/c2, where ω is the frequency of thelight. The scattering length of S-wave photon-photon collisions is of the orderof a few nanometers. This is in agreement with an earlier experiment, in whichhead-on photon-photon collisions mediated by rubidium atoms in a vapor cell wasbeen observed. Multiple photon-photon collisions inside the resonator lead to theformation of a “photon fluid,” which is closely connected with the formation of thephotonic BEC. The signature of BEC would be the formation of a central spike inthe transverse momentum distribution of the photons, on top of a broad pedestal(the depletion) of uncondensed photons. Further implications of this photonic BECstate are the superfluidity of the photon fluid, when produced by laser light in apure quantum state, and the appearance of quantized vortices in this superfluid.
1 Introduction
Photons are known to be bosons. Therefore the question naturally arises: Canphotons undergo Bose-Einstein condensation (BEC), just like other bosons?If not, why not? This question has recently become an especially interestingone in light of the fact that cold, dilute, weakly interacting bosonic atomshave been observed to form Bose-Einstein condensates.1 Since the interactionbetween atoms and photons is reciprocal one, it is natural to ask whetherphotons can undergo BEC due to the presence of atoms, just as atoms canundergo BEC due to the presence of photons.
Conventional wisdom tells us that photons cannot undergo BEC. This lineof reasoning begins with the observation that in a blackbody cavity, photonnumber is not conserved, due to the possibility of photon emission and ab-sorption (i.e., creation and annihilation) by the atoms at the cavity walls, andtherefore that the photon chemical potential vanishes. Thus in the Planckblackbody radiation problem, the average photon occupation number for a
given mode of frequency ω is given by Bose-Einstein distribution
n(ω) = (exp[hω − µ/kT ] − 1)−1, (1)
but with the photon chemical potential µ being identically zero. This isconnected with the fact that the rest mass of the photon is also identicallyzero. In contrast, in an atomic BEC, due to the finite rest mass and finitechemical potential of atoms, there is a conservation of the atoms in the system.As a result of the vanishing of the photon chemical potential, there does notexist a finite, nonvanishing critical temperature for the condensation of thephotons into a BEC in the case of blackbody radiation.
However, it is well known that a BEC-like phenomenon does indeed occurin a laser, where above threshold there occurs a macroscopic occupation of asingle mode of the laser cavity by many identical photons. This suggests thatthe bosonic nature of photons indeed plays an important role in this BEC-likephenomenon. Nevertheless, laser action does not produce a BEC, since theinverted atoms of the total system are far away from thermodynamic equilib-rium, whereas BEC occurs in thermodynamic equilibrium. Furthermore, incontrast to the case of atomic BECs, where BEC occurs due to the stimulatedatom-atom scattering, which is number conserving, the fundamental interac-tion responsible for laser action is stimulated emission of photons, which isobviously not a number conserving process, since there is an amplification ofthe photon number due to the gain of the laser.
This leads us to search for photonic BECs in other physical settings moresimilar to that of the atomic BECs. Here we shall examine a special casein which photon particle number is conserved, namely, in stimulated photon-photon scattering mediated by virtual atomic excitations. When light is tunednear an atomic resonance, the effective photon-photon collision cross sectioncan be quite large. However, if the light frequency is detuned with respectto the resonance by a few Doppler widths, the medium becomes essentiallytransparent, so that photons are not created or annihilated by the atoms. Inthis kind of photon-photon scattering, photon number is conserved, resultingin a nonzero chemical potential for the photons in the system.
The result of stimulated photon-photon scattering is similar to that ofstimulated atom-atom scattering in the atomic BECs. This leads to a macro-scopic occupation of the ground single-particle state (i.e., the zero transversemomentum cavity mode) by most of the photons, with the remainder of theuncondensed photons occupying higher-lying states (i.e., higher transversemomenta modes), which form the depletion. Therefore there should result acentral spike in the transverse momentum distribution at zero transverse mo-
mentum, which represents the photonic BEC, on top of a broad but sparselypopulated pedestal of uncondensed photons, in complete analogy with theatomic BEC.
Experimentally, an optical cavity can be designed to behave like a har-monic trap for photons in the two transverse dimensions of the cavity, bymeans of curved mirrors (or a lens) placed inside the cavity. Such a cavity pos-sesses many transverse Hermite-Gaussian modes, which are formally identicalto energy eigenstates of a two-dimensional isotropic simple harmonic oscilla-tor. These correspond to the single-particle states of harmonically trappedatomic BECs. When the cavity is filled with laser light, and the photons areallowed to weakly interact with other photons via the atoms inside this cavity,there should result a macroscopic condensation of the photons into the lowesttransverse mode of the cavity, in addition to a depletion consisting of un-condensed photons in the higher transverse modes. This is the experimentalsignature for a photonic BEC.
Since photons are one of the gauge bosons of the standard model, it isimportant to determine if BEC can indeed occur for this elementary particle.Also, we hope to make an experimental connection with the illustrious the-oretical work on the weakly-interacting Bose gas performed by Bogoliubov,2
Lieb and Liniger,3 Yang and Yang,4 Mottelson,5 and many others.
2 Nonlinear optics of transverse mode interactions
The Gross-Pitaevskii equation has been highly successful in describing thebehavior of the atomic BECs. Thus we expect a similar equation to describephotonic BEC. Indeed, a nonlinear optical analysis of the photonic systemshows that this is the case.
The light inside a cavity is governed by Maxwell’s equations in the parax-ial and slowly-varying envelope approximations. In the presence of a nonlin-ear medium inside the cavity, these equations reduce to a driven nonlinearSchrodinger-like equation, otherwise known as the Lugiato-Lefever equation,6
for the slowly-varying envelope of the transverse field E inside the cavity,driven by an input laser field Ed
i∂E∂t
= − c
2kn0∇2
⊥E − (∆ω)E − 6ωn0n2 | E |2 E − iγ(E − Ed),
where k = n0ω/c is the optical wavenumber of the carrier field of frequencyω, n0 is the linear refractive index of the medium, n2 is the Kerr coefficient,∆ω = ω − ωc is the detuning of the driving field from the cavity resonance,and γ is the cavity damping rate due to the finite reflectivity of the cavity
mirrors. The slowly-varying electric field envelope E in the two transversedimensions of the cavity becomes the complex order parameter ψ after weapply appropriate scaling transformations; the above equation then takes onthe dimensionless form
i∂ψ
∂t= −1
2∇2
⊥ψ − ∆ψ − η | ψ |2 ψ − iΓ(ψ − ψd). (2)
The scaling is based on a typical beam size, i.e., the transverse coordinatescale x0. the time scale is given by the time of flight through the Rayleighrange of a diffraction beam of this size, t0 = kn0x
20. Thus t becomes the
dimensionless time t/t0; the electric field strength becomes the dimensionlesselectric field strength ψ = E/E0, where E0 is the characteristic electric fieldstrength (c/6ω|n2|kx2
0)1/2 required to accumulate a 2π nonlinear phase shift
in the time t0. (Nonlinear effects become comparable to diffraction effectswhen E ≈ E0). The transverse Laplacian is scaled by x0, the dimensionlessdetuning is ∆ = (∆ω)t0, the sign of the nonlinearity is η (η = +1 for theself-focussing, and η = −1 for the self-defocussing Kerr nonlinearity), andΓ = γt0.
The above equation is formally identical to the Gross-Pitaevskii equation,apart from the inhomogeneous term ψd which represents the external drivinglaser. Hence one expects the same solutions and therefore equivalent physicalbehaviors of the photonic BEC as compared to the atomic BEC, when theexternal laser drive is turned off. We have indeed observed this to be case innumerical simulations of this equation (see below).7
A harmonic trap for the photons can be implemented in a number ofways, for example, by letting the cavity mirrors become slightly curved, or bydisplacing mirrors of fixed curvature into a near-confocal configuration,8 or byplacing a lens inside a ring resonator.9 Due to the curvature of these opticalelements, there results many closely spaced transverse modes. The transversemode eigenfrequencies, in the absence of photon-photon interactions, are iden-tical to those of a 2D isotropic simple harmonic oscillator in nonrelativisticquantum mechanics, and the transverse mode eigenfunctions are identical tothe Hermite-Gaussian wavefunctions of this quantum problem. This allows usto map the transverse mode problem to the harmonic oscillator problem, andthus to view the cavity as an effective transverse harmonic trapping potentialfor photons in the two transverse dimensions of the cavity.
One can add to this harmonic trap nonlinear interactions among the pho-tons, which lead to a four-wave mixing of the various transverse modes insidethe cavity mediated by the atomic Kerr nonlinearity. In this system, nonlinearoptics predicts the formation a central spike in the transverse light intensity
distribution, in which the lowest, on-axis transverse mode will be become themost intense one, on top of a pedestal of weakly excited off-axis transversemodes (which we shall call the “depletion”), due to mode competition. Thishas already been observed in numerical simulations of the Lugiato-Lefeverequation.10 The formation of the central spike happens for both signs of thenonlinearity; here we focus on the self-defocusing sign, since this correspondsto the repulsive sign of interactions for stable atomic BECs.1
Although this central spike was found in a purely classical-field analysis,this does not mean that this phenomenon is a purely classical one. In fact,we shall argue here that the physical significance of this central spike is thatit represents a macroscopic occupation of the lowest single-particle state (i.e.,the lowest transverse mode) by the photons due to the Bose-Einstein con-densation, arising from the four-wave interactions (i.e., collisions) among thephotons. This formation of a central-spike on top of a depletion has not yetbeen observed experimentally.
When the transverse mode density is very high, one can understand theformation of the central spike from the classical point of view of four-wave mix-ing of plane waves, in which two strong plane waves which converge towardsthe central axis of the cavity from the input pump laser beam amplify twoweak plane waves which propagate collinearly along the central axis of the cav-ity through a stimulated four-wave mixing process. Notice that this processconserves photon number. Although all four interacting waves have the nearlythe same frequency, due to cross-phase modulation the phase-matching of thefour waves can occur through the phenomenon of weak-wave retardation,11,12
since the two weak waves along the central axis possess shorter wavevec-tor lengths for the self-defocusing sign of the Kerr nonlinearity than the twostrong off-axis waves, even when all four waves are degenerate in frequency.Exponential growth of the central spike at the expense of the off-axis wavesthus arises from the stimulated emission of the four-wave mixing process. Thiscan happen in the two transverse dimensions of the nonlinear cavity, implyingthe possibility of BEC in two dimensions, completely analogous to the predic-tion of atomic BECs in two dimensions.13 After many four-wave interactions,numerical simulations show that the system comes into an equilibrium wherethere exists a central spike consisting of a strongly excited on-axis mode, ontop of a pedestal of weakly excited off-axis modes, i.e., the remnant of theinput laser modes.10
Now we present an alternative viewpoint from quantum field theory. Fromthis point of view, we shall see that the meaning of the central spike is thatBEC is occurring in this weakly interacting Bose gas composed of photons.Furthermore, we shall see that we should expect the production of nonclassi-
cal, squeezed states of light due to four-wave mixing.The second-quantized Hamiltonian governing the interactions among the
photons in a cavity is given by14
H = Hfree +Hint
Hfree =∑
p
ε(p)a†pap
Hint =12
∑κpq
V (κ)a†p+κa†q−κapaq , (3)
where the operators a†p and ap are creation and annihilation operators, re-spectively, for photons with transverse momentum p, which satisfy the Bosecommutation relations
[ap, a†q] = δpq and [ap, aq] = [a†p, a
†q] = 0 . (4)
The first term Hfree in the Hamiltonian represents the energy of the freephoton system. The second term Hint arises from four-wave mixing throughthe Kerr nonlinearity. This quartic term in a’s and a†’s represents the pairwiseannihilation of two particles, here photons, of momenta p and q, and thepairwise creation of two particles with momenta p+κ and q−κ. In other words,this term represents a scattering process with a momentum transfer κ betweena pair of particles with initial momenta p and q, along with the assignment ofan energy V (κ) to this scattering process. Momentum is obviously conservedin this process. It is well known that the quartic term is responsible for theproduction of squeezed states of light,15 and that with a exponentiation leadsto the unitary squeezing operator.16
Here we have neglected the harmonic trapping potential, so that inter-actions are occurring between plane-wave states of the photons, which corre-spond to the transverse modes in a flat-mirror cavity system.
We have observed the scattering process corresponding to this quarticterm in a photon-photon scattering experiment in a rubidium vapor cell.17
Back-to-back 90 photon-photon S-wave scattering from two head-on collidingbeams of light mediated by atoms was observed by means of coincidencedetection. In particular, the momentum conservation implied in the abovescattering term was experimentally confirmed.
3 Experiments
We have performed previous work on high-Finesse nonlinear Fabry Perot cav-ities and observed the formation of a nonlinear mode in a cylindrical form ofthis cavity.19 Because of the narrow input bandwidth and the cylindrical cur-vature of the mirrors, in that system both the longitudinal and one transversedegree of freedom were “frozen out,” making the system one-dimensional.We are now studying the two-dimensional system, with spherical rather thancylindrical mirror curvature, and using a ring resonator configuration insteadof a two-mirror Fabry-Perot configuration, in order to produce traveling waves,rather than standing waves, inside the cavity. In this way, we avoid creatingthe undesirable closely-spaced inhomogeneities in the atomic nonlinearity dueto the nodes and antinodes of the standing wave.18
In our ongoing experiments, we inject a narrow-bandwidth (≤ 1 MHz),high-power (≈ 1 W) laser beam from a titanium sapphire laser system into ahigh-Finesse nonlinear ring cavity in a nearly confocal configuration. The Kerrnonlinearity is provided by a rubidium vapor cell with anti-reflection-coatedwindows placed inside the cavity. The choice of a ring resonator rather thana standing-wave configuration is to eliminate the inhomogenous saturation ef-fects in the nonlinear medium.18 The cavity is 1 meter long and has a finesseof around 300, giving it a cavity linewidth of 1 MHz, comparable to the band-width of the input laser. For the cavity mirrors we use piezoelectric-actuated,remotely-controlled motorized mirror mounts with a resolution better than30 nm and stability better than 10 nm. The laser is tuned to a cavity reso-nance, but detuned slightly by a Doppler width or so to the red side of therubidium D2 line at 780 nm. Thus most of the input light will be transmittedinto the cavity but will suffer negligible absorption by the atomic vapor. Thenonlinearity will be tunable in magnitude and sign both by the laser detuningand by the temperature of the cell, which controls the vapor density, and issufficiently large to provide multiple four-wave interactions within a cavityring-down time. Using a CCD camera, we plan to measure the transverseprofile in the far field region i.e., the intensity versus momentum distribution,of the light transmitted by the cavity, to observe the predicted central spikeon top of the depletion. However, we do not have any results to report as ofyet.
To measure the squeezing of the output light from the cavity, we will use aspatial filter to select the central spike from the pedestal. The central spike isthen recombined at a beam splitter with a portion of the original laser beamfunctioning as a local oscillator. Two balanced detectors are placed at thetwo output ports of the beam splitter, and their photodetection currents are
subtracted by means of a differential amplifier and measured as a functionof the phase shift of the local oscillator beam at the beam splitter. Thisbalanced homodyne technique has been used in the past for detection of asqueezed state of light,15,20 and we plan to utilize this standard technique formeasuring the amount of squeezing in the central spike and in the pedestal ofthe output light from our nonlinear cavity.
In analogy with atomic BECs, there should be a critical “temperature” atwhich the macroscopic occupation of the lowest transverse momentum modeoccurs. To observe the phase transition in which the central spike suddenlydisappears as the noise in the input is increased beyond a certain criticalpoint, we plan to pass the input laser light through a pair of crossed AOMs21
to deflect and modulate the pump beam before it enters the nonlinear cavity.The AOMs are modulated with Gaussian radio-frequency noise, so that theoutput from these AOMs consists of deflected light with a Gaussian trans-verse momentum profile, which can be thought of as a Maxwell-Boltzmanndistribution in the transverse momenta of the photons incident on the non-linear cavity. The width of this distribution function can be characterized byan effective temperature. We plan to measure the ratio of the central spikeintensity relative to the pedestal as a function of this effective temperature.If the photonic BEC picture is a good one, we expect to observe a phasetransition in which the central spike will suddenly disappear as a function ofincreasing effective temperature, when it reaches a critical temperature whichcorresponds to the BEC transition temperature.
4 The Photonic BEC Analogy
Nonlinear optics theory explains the mean-field transverse mode dynamicsof the cavity light, but the above quantum field approach suggests anotherpotentially fruitful way of understanding these phenomena. The similaritybetween the nonlinear Schrodinger or Lugiato-Lefever equation (Eq. (2)) andthe mean-field Gross-Pitaevksii (GP) equation for atomic BECs
ih∂ψ
∂t= − h2
2m∇2ψ + V (x)ψ +
4πh2
ma | ψ |2 ψ (5)
has been noted.7 In the limit of ψd = 0 (the absence of a driving input field)the similarity becomes even more pronounced, and Eq. (2) becomes formallyidentical to the GP equation with a damping term (a “lossy” BEC) usedto describe atomic BECs, where the BEC slowly decays due to three-bodycollisions.1 Experimentally, we can achieve the limit of ψd = 0 by suddenly
shutting off the input laser, and observing the output light as it slowly leaksout of the nonlinear cavity on a time scale set by the cavity ring-down time.Because of this formal equivalence, we can gain insight into the nonlinearoptics problem by reasoning in analogy with the atomic BEC problem.
In accordance with this analogy, the function ψ plays the role of the orderparameter in Eq. (2). It is a 2D complex function, and thus the correct analogto the nonlinear cavity problem is a 2D BEC, which has recently been shownto be possible for atomic BECs in a 2D trapping potential.13 The eigenmodesof ψ in the absence of the nonlinearity (scaled Hermite-Gaussian transversemomentum modes) are the same as the eigenstates of the 2D simple harmonicoscillator potential, so that a cavity with curved optical elements, in thisanalogy, plays the role of the harmonic trapping potential. Equation (2) istwo-dimensional because the input laser bandwidth is so much smaller thanthe free spectral range, that the longitudinal degree of freedom has been“frozen out” and only one longitudinal mode is occupied. This reasoningleads us to the appropriate cavity analog of an effective mass for the photonfor its transverse dynamics. Consider the energy-momentum relation for a freephoton in the cavity. Rewriting the longitudinal momentum of the photonplong in terms of its frequency ω, we obtain in the paraxial approximation
E2 = (cp⊥)2 + (cplong)2
≈ (cp⊥)2 + (hωn0)2
= (cp⊥)2 +m2effc
4
where meffc2 = hωn0 with n0 ≈ 1 being the linear refractive index at ω. This
is the relativistic energy-momentum relation for a two-dimensional particlewith an effective mass of meff . Thus we identify this as the effective mass ofthe photon. For λ = 780 nm we have meff ≈ 1.6 eV/c2.
Equating the nonlinear terms in Eqs. (2) and (5) we find the followinganalog of the scattering length a:
a = − 3hω3
2n0c2n2. (6)
For our experimental conditions n0 ≈ 1 and n2 ≈ 6 × 10−6 cm3/erg, givingus a ≈ −1.5 nm. This value is tunable in both magnitude and sign in therange −10 nm ≤ a ≤ +10 nm. The limits are set by requiring the detuningto be outside the Doppler width (∆ ≈ 350) MHz to prevent absorption andthe temperature (which controls the vapor density) to be less than around100C for experimental reasons. The above analysis leads to a particle pic-ture, in which the physical meaning of the scattering length a is that it is
the effective hard-sphere radius of the colliding particles (photons) with aneffective mass meff . This implies that there should exist effectively elastichard-sphere collisions between pairs of photons, and this has been verifiedin coincidence-counting experiments which demonstrate the existence of elas-tic photon-photon collisions mediated by rubidium atoms mentioned above.17
This picture is equivalent to a four-wave mixing picture, since they both leadto the same equation (Eq. (2)) which describes the dynamics of the nonlinearcavity. It should be mentioned that this hard-sphere picture also makes anexperimental connection with the past theoretical work on the hard-sphereBose gas problem by Lee and Yang, and by others.25
In the particle picture, the condition for the formation of the centralspike in the transverse mode profile is that there must be a sufficient numberof hard-sphere collisions within a cavity ring-down time, so that the systemwill come into equilibrium with itself and produce a “photon fluid”.14 In thewave picture, the equivalent condition is that there is a sufficient numberof four-wave interactions so that the transverse mode intensity distributioncomes into equilibrium with itself. Consider the case when the cavity lengthL is 1 meter and the cavity mirrors have reflectivities of R = 0.99 for a cavityfinesse F = 300. The optical nonlinearity is provided by rubidium vapor at80oC, corresponding to a number density of 1012 rubidium atoms per cubiccentimeter. A circularly-polarized CW laser beam is incident on rubidiumatoms, and is detuned by around 600 MHz to the red side of a closed two-level transition, for example, the |F = 2,mF = +2〉 → |F ′ = 3,mF ′ = +3〉transition of the 87Rb D2 line. Thus the Kerr coefficient is that of a puretwo-level atomic system excited well off resonance (i.e., with a detuning muchlarger than the absorption linewidth)12
n2 = πNatom µ4/h3∆3 ≈ 6 × 10−6 cm3/erg (7)
where Natom is the atomic number density of the atomic vapor, µ is thematrix element of the two-level atomic system, and ∆ is the detuning ofthe laser frequency from the atomic resonance frequency. In intensity units,n2I ≈ 5 × 10−8 cm2/Watt. Thus the ∆ ≈ 600 MHz detuning of the laserfrom the atomic resonance used in the above example is considerably largerthan the Doppler width of 340 MHz of the rubidium vapor, and the residualabsorption arising from the tails of the nearby resonance line gives rise to aloss which is much less than the loss arising from the mirror transmissions,so that the medium can be considered as being essentially transparent. Anintracavity intensity of 4 W/cm2 would result in ∆n = |n2|E0
2 ≈ 2 × 10−7.For this intensity, the photon number inside the cavity (mostly in the BEC)is N0 ≈ 8 × 1010. The cavity ring-down time τcav = 2FL/c ≈ 2 µs is much
longer than the mean photon-photon collision time τcoll = (2ωn2|E0|2)−1 ≈ 1ns. Since there would be approximately two thousand hard-sphere photon-photon collisions within a cavity ring-down time, this many-body systemshould quickly come into equilibrium with itself. Therefore a central spikecorresponding to BEC should indeed form in the transverse photon momen-tum distribution.
The other parameter requiring an analogy is temperature. When theinput laser beam is in the TEM00 mode and in a coherent state of this mode,it is in a pure quantum state. If the laser beam is well-coupled into thecavity, the spread in transverse momentum is zero, and thus the temperatureis zero. We can “heat” the incident light by passing it through crossed AOMsand randomly modulating its deflection angle, which will artificially createa spread of random momentum. With random Gaussian noise fed to theAOMs, the resulting transverse momentum distribution can be characterizedby a Maxwell-Boltzmann distribution and hence by an effective temperature.We can control the effective temperature by the amplitude of the Gaussiannoise.
In the BEC picture is it easy to understand the transverse momentumdistribution of the cavity light. The condensation of the system into its low-est energy state, and the presence of a pedestal of uncondensed photons, the“depletion,” is a consequence of the nonlinear interaction term in Eq. (2).The sub-Poissonian statistics of the output light can also be understood inthis picture, since the bosons leaking out of a BEC are predicted to be pro-duced in pairs (see Eq. (3)), and thus are correlated with each other. Thisleads to a squeezed state of the light, which has already been observed in anonlinear resonator without transverse degrees of freedom.20 The BEC pictureis useful because it predicts some phenomena more easily than the classicalnonlinear optics theory. In particular, by “heating” the input light to the cav-ity as discussed above, there should be a phase transition at some particulareffective critical temperature. This BEC-like phase transition will take thesystem out of its “condensed” phase of highly-collimated output and into an“uncondensed” phase in which the output beam has a much larger spread oftransverse momentum with no central spike. Observing this transition andmeasuring its effective temperature is one of our future experimental goals.
Using the nonlinear Fabry-Perot cavity,19 we have observed spatial soli-ton formation in a standing-wave, cavity-bound nonlinear medium. Spatialsoliton formation leads to previously unobserved “soliton resonances” in thetransmission spectrum of the cavity. In terms of the interacting Bose gaspicture, this observation indicates that the photon gas can significantly inter-act with itself, i.e., it can “thermalize” and become a “photon fluid,” before
leaving the cavity.By means of the photon-photon scattering experiment,17 we have directly
measured the interaction time in photon-photon collisions, the microscopicprocess underlying the collective, many-body photon effects we are studyinghere. The interaction time is critical to the interpretation of the experimentsin terms of an interacting Bose gas: a long delay on the interaction time wouldinvalidate the interacting Bose gas picture. This time scale has been measuredto be sub-nanosecond, and is in good agreement with a many-body theory ofthe scattering process. Thus the interaction is much faster than other timescales in the cavity experiments, and the interacting Bose gas picture is thusvalidated.
5 The Bogoliubov problem
Here we re-interpret the above results in terms of one particular quantummany-body problem, the one first solved by Bogoliubov.2,22 Suppose that onehas a zero-temperature system of bosons which are interacting with each otherrepulsively, e.g., a dilute system of small, bosonic hard spheres. In order tomake the problem tractable theoretically, let us assume that these interactionsare weak. In the case of light, the interactions between the photons are infact always weak, so that this assumption is a good one. However, theseinteractions are nonvanishing, as demonstrated by our photon-photon collisionexperiment.17 We start with the Bogoliubov Hamiltonian
H = Hfree +Hint
Hfree =∑
p
ε(p)a†pap
Hint =12
∑κpq
V (κ)a†p+κa†q−κapaq , (8)
where the operators a†p and ap are creation and annihilation operators, re-spectively, for bosons with momentum p, which satisfy the Bose commutationrelations
[ap, a†q] = δpq and [ap, aq] = [a†p, a
†q] = 0 . (9)
The first term Hfree in the Hamiltonian represents the energy of the freeboson system, and the second term Hint represents the energy of the inter-actions between the bosons arising from the pairwise potential energy V (κ),which is the Fourier transform of the potential energy V (r2 −r1) in configura-tion space of a pair of bosons located at r2 and r1. Here the assumption that
the interactions are weak means that the second term in the Hamiltonian ismuch smaller than the first, i.e., |V (κ)| |ε(κ)|.
6 The Bogoliubov dispersion relation for the photon fluid
In an ideal Bose gas at absolute zero temperature, there exists a Bose conden-sate consisting of a macroscopic number N0 >> 1 of particles occupying thezero-momentum state. This feature should survive in the case of the weakly-interacting Bose gas, since as the interaction vanishes, one should recover theBose condensate state. Hence following Bogoliubov, we shall assume here thateven in the presence of interactions, N0 will remain a macroscopic number inthe photon fluid.24 This macroscopic number will be determined by the in-tensity of the incident laser beam which excites the cavity system, and turnsout to be a very large number compared to unity. For the ground state wavefunction Ψ0(N0) with N0 particles in the Bose condensate in the p = 0 state,the zero-momentum operators a0 and a†0 operating on the ground state obeythe relations
a0 |Ψ0(N0)〉 =√N0 |Ψ0(N0 − 1)〉
a†0 |Ψ0(N0)〉 =√N0 + 1 |Ψ0(N0 + 1)〉 . (10)
Since N0 1, we shall neglect the difference between the factors√N0 + 1
and√N0. Thus one can replace all occurrences of a0 and a†0 by the c-number√
N0, so that to a good approximation [a0, a†0] ≈ 0. However, the number of
particles in the system is then no longer exactly conserved, as can be seen byexamination of the term in the Hamiltonian∑
κ
V (κ)a†κa†−κa0a0 ≈ N0
∑κ
V (κ)a†κa†−κ , (11)
which represents the creation of a pair of particles, i.e., photons, with trans-verse momenta κ and −κ out of nothing.
However, whenever the system is open one, i.e., whenever it is connectedto an external reservoir of particles which allows the total particle numberinside the system (i.e., the cavity) to fluctuate around some constant averagevalue, then the total number of particles need only be conserved on the aver-age. Formally, one standard way to compensate for the lack of exact particlenumber conservation is to use the Lagrange multiplier method and subtracta chemical potential term µNop from the Hamiltonian (just as in statisticalmechanics when one goes from the canonical ensemble to the grand canonicalensemble)23 H → H ′ = H − µNop, where Nop =
∑p a
†pap is the total number
operator, and µ represents the chemical potential, i.e., the average energy for
adding a particle to the open system described by H. In the present context,we are considering the case of a cavity with low, but finite, transmissivitymirrors which allow photons to enter and leave the cavity. This permits arealistic physical implementation of the external reservoir, since the cavitymirrors allow the total particle number inside the cavity to fluctuate due toparticle exchange with the beams outside the cavity. However, the photonsremain trapped inside the cavity long enough so that a condition of thermalequilibrium is achieved after very many collisions, which is indeed the casefor the experimental numbers. This leads to the formation of a photon fluidinside the cavity.26
Assuming that there is little depletion of the Bose condensate due to theinteractions (i.e., N ≈ N0 1), the ground state energy is given by
E0 ≈ 12N2
0V (0) ≈ 12N2V (0), (12)
or a chemical potential of µ ≈ NV (0) ≈ N0V (0).This implies that the effective chemical potential of a photon, i.e., the
energy for adding a photon to the photon fluid, is given by the number ofphotons in the Bose condensate times the repulsive pairwise interaction en-ergy between photons with zero relative momentum. It should be remarkedthat the fact that the chemical potential is nonvanishing here makes the ther-modynamics of this two-dimensional photon system quite different from theusual three-dimensional, Planck blackbody photon system, for which µ = 0.It should also be remarked that the conventional wisdom which tells us thatBose-Einstein condensation and superfluidity are impossible in 2D bosonicsystems, does not apply here. To the contrary, we believe that superfluidityof the topological, 2D Kosterlitz-Thouless kind (with algebraic decay of longrange order) is possible for the photon fluid.27 Furthermoe, recent studies ofthe weakly-interacting Bose gas problem in 2D harmonic traps show that atrue BEC can form in such traps.13
Following Bogoliubov, we now introduce the following canonical transfor-mation:
ακ = uκaκ + vκa†−κ
α†κ = uκa
†κ + vκa−κ . (13)
Here uκ and vκ are two real c-numbers which satisfy the condition u2κ−v2
κ = 1,to insure that the Bose commutation relations are preserved for the creationand annihilation operators for the quasi-particles, α†
κ and ακ, viz.,
[ακ, α†κ′ ] = δκ,κ′ and [ακ, ακ′ ] = [α†
κ, α†κ′ ] = 0 . (14)
The final result is that the diagonalized Hamiltonian describes a collectionof noninteracting simple harmonic oscillators, i.e., quasi-particles, or elemen-tary excitations, of the photon fluid, with the energy-momentum relation, orBogoliubov dispersion relation,
ω(κ) ≈[κ2N0V (κ)
m+
κ4
4m2
]1/2
. (15)
For small κ this dispersion relation is linear in κ, indicating that the na-ture of the elementary excitations here is that of phonons, i.e., sound wavespropagating inside the photon fluid at the sound speed
vs = limκ→0
ω(κ)κ
=(N0V (0)m
)1/2
=( µm
)1/2
. (16)
Thus in the above analysis, we have shown that all the approximationsinvolved in the Bogoliubov theory are valid ones for the case of the 2D pho-ton fluid inside a nonlinear cavity. Hence the Bogoliubov dispersion relationshould indeed apply to this fluid; in particular, there should exist sound wavesin the photon fluid. As additional evidence for the existence of these soundwaves, we have recently found that the same Bogoliubov dispersion relationemerges from a classical nonlinear optical analysis of this problem in the limitof a vanishing cavity decay rate.19 The velocity of sound found by the classicalanalysis is identical to the one found in Eq. (16) for the velocity of phononsin the photon fluid, provided that one identifies the energy density of thelight inside the cavity with the number of photons in the Bose condensate asE20 = 8πN0hω/Vcav, where Vcav, the cavity volume, is also the quantization
volume for the electromagnetic field, and provided that one makes use of theknown proportionality between the Kerr coefficient n2 and the photon-photoninteraction potential V (0) = 8π(hω)2n2/Vcav.28
7 Conclusions
We suggest that the above Bogoliubov dispersion relation implies that thephoton fluid formed after many photon-photon collisions is actually a photonsuperfluid. This means that a superfluid state of light should exist. Althoughthe exact definition of superfluidity is presently still under discussion, in par-ticular, whether the atomic BECs are superfluids or not,29 one indication ofthe existence of a photon superfluid would be that there is a critical transi-tion from a dissipationless state of superflow, i.e., a laminar flow of the photonfluid below a certain Landau-like critical velocity past an obstacle,30 into a
turbulent state of flow, accompanied by energy dissipation associated withthe shedding of a von-Karman street of quantized vortices past this obstacle,above this critical velocity.31
This conclusion has recently been reinforced by the observation in nu-merical simulations of a vortex-shedding phenomenon above the Landau-likecritical velocity in the Lugiato-Lefever equation.7 In these simulations, thegeneration of quantized vortices was observed to occur past a cylindrical ob-stacle placed inside a nonlinear Fabry-Perot cavity, after the input drive laserfield was turned off. The typical vortex core size is given by the light wave-length divided by the square root of the nonlinear index change. Thus thevortex core size should be around a few hundred microns, so that this dra-matic nonlinear optical phenomenon should be readily observable in the cavityexperiments.
In summary, we argue that the similarities between the above phenomenaand those of a true BEC are not accidental: a BEC of photons can indeedform when photons weakly interact with each other through four-wave mix-ing inside a cavity. Accompanying the formation of a “photon fluid” arisingfrom many photon-photon collisions, is the formation of a superfluid corre-sponding to the pure state of the input light field. That this photon fluid isa superfluid could be verified by the observation of two predicted features:the first being the appearance of sound waves in this superfluid in accordancewith the Bogoliubov dispersion relation, and the second being the appearanceof a vortex-shedding phenomenon when the Landau-like critical velocity isexceeded, and the driving laser field is suddenly turned off.
8 Acknowledgments
*J.M.H. is on leave from the Department of Physics, Universidade Federal deAlagoas, Maceio - AL - Brazil with support from CNPq. This work was alsosupported by the ONR and the NSF.
References
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2. N. Bogoliubov, J. Phys. (U.S.S.R.) 11, 23 (1947).3. E. H. Lieb and W. Liniger, Phys. Rev. 130, 1605 (1963); H. B. Thacker,
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(2001).8. We thank W. D. Phillips of NIST for suggesting this to us.9. We thank C. Ropers for suggesting this to us.
10. A. M. Dunlop, E. M. Wright, and W. J. Firth, Opt. Comm. 147, 393(1998).
11. R. Y. Chiao, P. L. Kelley, and E. Garmire, Phys. Rev. Lett. 17, 1158(1966).
12. R. W. Boyd, Nonlinear optics, (Academic Press, Boston, 1992), and thereferences therein.
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14. R. Y. Chiao and J. Boyce, Phys. Rev. A 60, 4114 (1999); R. Y. Chiao,Opt. Comm. 179, 157 (2000).
15. R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley,Phys. Rev. Lett. 55, 2409 (1985).
16. R. Y. Chiao and T. F. Jordan, Phys. Lett. A 132, 77 (1988); H. J. Kimbleand D. F. Walls, J. Opt. Soc. Am. B 4, 1450 (1987).
17. M. W. Mitchell, C. J. Hancox, and R. Y. Chiao, Phys. Rev. A 62, 043819(2000).
18. We thank J. C. Garrison for suggesting this to us.19. J. Boyce, J. P. Torres, and R. Y. Chiao, Opt. Lett. 24, 1850 (1999);J.
Boyce, “Transverse effects in nonlinear optical cavities,” Ph. D. Thesis,U. C. Berkeley, 1999 (unpublished).
20. M. Shirasaki and H. A. Haus, J. Opt. Soc. Am. B7, 30 (1990).21. R. Onofrio, D.S. Durfee, C. Raman, M. Koehl, C.E. Kuklewicz and W.
Ketterle, Phys. Rev. Lett. 84, 810 (2000).22. D. Pines, The Many-Body Problem (Benjamin, New York, 1961).23. N. M. Hugenholtz and D. Pines, Phys. Rev. 116, 489 (1959); G. W. Goble
and D. H. Kobe, Phys. Rev. A 10, 851 (1974).24. Since we have assumed a zero-temperature Bose gas, following Bogoli-
ubov we start this calculation with the ground state of the system inthe macroscopically occupied zero-momentum number state |N0, p = 0〉.However, it is also possible to derive the same dispersion relation starting
from a system in a coherent state |α, p = 0〉. We thank D. Thouless forsuggesting this to us.
25. K. Huang, Statistical Mechanics, 2nd ed. (Wiley, New York, 1987).26. Here we consider a pure quantum state of the system produced by a pure
coherent state of the incident laser beam. The entropy and temperatureof the resulting system inside the cavity is zero. In this case, BEC in twodimensions should occur (V. N. Popov, Functional Integrals in QuantumField Theory and Statistical Physics (Reidel, Dordrecht, 1983)).
27. J. M. Kosterlitz and D. J. Thouless, J. Phys. C 6, 1181 (1973).28. R. Y. Chiao, I. H. Deutsch, J. C. Garrison, and E. W. Wright in Frontiers
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distinguishes the onset of superfluid turbulence from the onset of normalhydrodynamic turbulence.
Various aspects of photon-electron interactions in the beam physics are discussed,in particular, application of lasers for beam production, cooling, acceleration, mon-itoring, diagnostics and especially for obtaining high energy γγ, γe colliding beams(photon colliders). Condense beams are also discussed shortly.
1 Introduction
This is an introductory talk to a subject of a working group at the workshopon quantum aspects of beam physics. Instead of giving general overview of twoquite different themes given in the title, I would like to focus on one strategicdirection: high energy photon colliders based on linear colliders which includealmost all items.
Linear colliders consist of the following parts: an injector, a cooling sec-tion, an accelerator and an interaction region. All these sections use technolo-gies based on electron-photon interactions.
1.1 Injector
Indeed, polarized electrons for linear colliders are produced by photo-guns,Fig.1a. The polarization degree about 85 % has been achieved. Polarizationis very important for many experiments. Small normalized emittances inphoto-guns open also the way to X-ray free electron lasers.
Polarized positrons can be produced using a two-step scheme, Fig.1. Atfirst, electrons pass through an undulator producing polarized photons withthe energy of several tens of MeV, and then these photons hit some targetand produce e+e− pairs.1 Most higher energy electrons and positrons havehigh degree of polarization. Instead of undulator on can use a laser (similarphysics).2 Polarization degree 50–80 % is expected.
1.2 Cooling
Electrons are cooled usually in damping rings where emission of photons inthe bending magnets and wigglers (undulators) leads to decrease of beam
laser
(polarized)photo−cathode
e
e
e e+ −,(polarized)
e E ~ 50 GeVγ
a)
e e,+ −
(polarized)b)
Injector
1) Photo−gun
2) Two−step
+ − production
laser
Undulator
of polarized e, e
e
E ~5 GeV
eE ~150 GeV
Figure 1. Electron and positron injectors
emittances. Additional decrease of emittances can be achieved using laserscooling,3,4,5 see Fig.2, where instead of wigglers lasers are used. The equilib-rium normalized emittance in proportional to βλC/λ, where λ is the periodof undulator, λC is the Compton wavelength of the electron and β is thebeta-function in the cooling region. a So, from first sight, usual wigglers withseveral cm periods can provide emittances smaller than in the case of lasers.This is correct, however, in damping rings emittances are determined not onlyby wigglers, but rather by the radiation in the bending magnets, intra-beamscattering and the space charge tune shift. In contrast, the laser cooling canbe done for one pass (linear laser cooling) where all effects enumerated aboveare not important.
Radiative laser cooling is very promising way for increasing γγ luminosityat photon colliders where collisions effects are not important and in currentprojects6 the luminosity is determined by the geometric luminosity of electronbeams. The main problems here is very high required laser powers, peak andaverage. Multiple use of the same laser flash is a possible solution.5
Application of laser cooling for storage rings is doubtful, because in Comp-ton scattering electrons lose large fraction of their energy and are simplyknocked out from the beam. It may be practical only for low energy stor-age rings with energies 10–100 MeV.7 Ultimate emittance in this case will bedetermined by intra-beam scattering and space charge effects.
aThe β-function in damping rings is two order of magnitude larger than can be obtainedin the local place of the laser cooling.
Cooling
cooling at storage rings
cooling at linear colliders
3) Optical stochastic cooling
E 0 E ε n ~ ε n,0 EE 0
laser
A
2) Radiative laser
1) Radiative lasere
e
e
(electrons,ions)
laser
Figure 2. Laser (and optical) cooling.
Another approach for storage ring is an Optical stochastic cooling, Fig.2.8
In this scheme, particles radiate in the pick-up undulator, this wave carryingthe spatial information about positions of particles particles in the beam isamplified in an optical amplifier and then cools stochastically the beam inthe undulator kicker. The optical stochastic cooling can provide cooling instorage rings faster then the usual radiative cooling and allows to reach smalleremittances. Recently, Mikhailichenko suggested further improvement of theoptical stochastic, “holographic cooling”,9 where instead of the spontaneousradiation in the pick-up undulator he suggests to use the stimulated radiationwhich is much more intensive.
1.3 Acceleration
Acceleration of particles is achieved in interactions of particles with RF fieldwhich presents a coherent state of photons in an accelerating cavity.
It worth to note that the energy gain proportional to the strength of thefield is achieved due to interference of the RF field and the field radiated bythe bunch in the cavity (without RF field). This is just because the final fieldenergy is equal
∫(E0 + Erad)2dV ∝ E2
0 + 2E0Erad + E2rad. The second term
depends linearly on the RF field and also on the field radiated by the beam.In free space the radiation is only due to acceleration in the external field,therefore Erad ∝ E0 and the net energy gain is proportional to E2
0 .Laser field can be be much stronger than the RF field and using lasers
for obtaining higher accelerating gradients look promising. Such experimentsare underway. The acceleration proportional to laser field can be obtained
Acceleration
1) Laser acceleration in semi−free space
laser
laser
E ∆ energy ~E(in free space energy is 0 or ~ E2)∆
laser
2) Laser accel. by a
acceleration3) Ponderomotive
e
laser bunch energy∆ ~ dEdx
2
4) Laser slicing technique
traveling lasers focus
(see Chattopadhyuy talk)
Figure 3. Acceleration of beams.
in two kind of devices shown in Fig.3. In the first case the beam intersectssome diaphragm-mirror system, in the second case small open cavities feededby the laser with a traveling focus can used10 (at this workshop all ideas arewelcomed).
THe third kind of the laser acceleration in free space is due to ponderomo-tive forces in the nonuniform laser field, Fig.3(3).11 This method has demon-strated experimentally.
Such systems are developed by enthusiasts and may be laser acceleratorswill find some area of applications. As for high energy linear colliders withthe laser acceleration, perspectives are not so clear, first of all because thebeam-beam effects in e+e− collisions dictate that the average beam powershould be at the level of 10 MW or higher. The corresponding laser powershould be at least one order of magnitude higher, which is unrealistic.
There are many interesting application of lasers for manipulation withelectron beams, for example, cut of a very short part of the electron bunchwhich then generates a short X-ray flash. The laser slicing technique is dis-cussed the talk by S. Chattopadhyay at this workshop.
1.4 Interaction region
Interaction regions at colliders present many opportunities for laser applica-tions, Fig. 4. The most exciting is a High Energy Photon Collider (γγ, γe)
Interaction region
( )γ γ e
1) Photon collider
e
laser
laser
3)
dd Ω
σ λ= f ( )
Beam size measurement
a) laser "wire"e
laser
b) Shintake monitor
eI e
2) Measurement of polarization
γγ,γ e
γ e γγ e
e
eE ~ E , L ~L
4) Backgroundsγ e e e+e− γ + field e+e− γ γ hadrons . . .
Figure 4. γe interactions at an the interaction region.
based on a linear collider with conversion of electrons to high energy photonsusing Compton scattering of laser light off high energy electrons.12,13 Thisoption has been included in Conceptual designs of all linear colliders projects(NLC,17JLC,18 and TESLA19) and recently in the TESLA Technical DesignReport.6 There are good chances that at least one linear collider project willbe approved. Photon colliders are discussed in more details in the next section.
Another well know application of lasers is the measurement of the electronpolarization. The circular polarization is measured using the difference in thecross sections for forward and backward circular laser photons. The linearpolarization is measured by azimuthal asymmetry (up-down) in scaatteringof linearly polarized laser photons.
Lasers are used successfully for measurement of electron beam sizes,Fig.4c, using the Compton scattering on a laser wire20 or on a diffractivepattern produced by two interfering laser beams.21 The second method al-
lows to measure beam transverse sizes more then one order smaller than thelaser wavelength.
Beside many useful applications, electron-photon and photon-photon col-lisions cause also sources of various backgrounds, Fig.4d, which are harmfulin most case, but also can be used for diagnostics of the beam collisions.For example, one can measure beam sizes at the IP using the deflection ofsoft secondary electrons (positrons) in the field of opposite electron (positron)bunch22.
The rest of my talk is devoted to the project of the Photon Collider atTESLA. The Technical Design Report of TESLA is just published.6 At theend, some phenomena in dense beams are discussed mostly in view of possiblelow emittance beams at linear colliders.
2 Photon Colliders
The unique feature of the e+e− Linear Colliders (LC) with the energy fromhundreds GeV to several TeV is the possibility to construct on its basis aPhoton Linear Collider using the process of the Compton backscattering oflaser light off the high energy electrons.12,13,14,15
0
.
electronbunch
C (e). γe
αγ(e)
laser
IP
b
Figure 5. Scheme of γγ, γe collider.
An electron beam of energy E0 travels towards the interaction point (IP)and at a distance b ∼ 1 − 5 mm from it intersects the focus of a laser beam,Fig.5. After scattering the photons have energies close to E0 and followthe electron directions to the IP, with a small additional angular spread of∼ mc2/E0 < 10−5. There they collide with a similar pulse of high energyphotons or electrons. Using a laser with a flash energy of several Jules onecan “convert” almost all electrons to high energy photons. The spot size ofthe photon beam at the IP will be close to that of the electrons and therefore
the luminosity of the γγ and γe collisions will be similar to the “geometric”luminosity of the basic e−e− beams.
The maximum energy of the scattered photons is
ωm =x
x+ 1E0; x ≈ 4E0ω0
m2c4 15.3
[E0
TeV
] [ ω0
eV
], (1)
where E0 is the electron beam energy and ω0 the energy of the laser photon.For example, for E0 = 250 GeV, ω0 = 1.17 eV, i.e. λ = 1.06 µm (Nd:glasslaser), we obtain x = 4.5 and ωm = 0.82E0.
The high energy photon spectrum becomes more peaked for increasingvalues of x. It turns out that the value x ≈ 4.8 is the optimum choice forphoton colliders, because for x > 4.8 the produced high energy photons createQED e+e− pairs in collision with the laser photons, and the γγ luminositywill be reduced.13,15,16 Hence, the maximum c.m.s. energy in γγ collisions isabout 80% (and 90% in γe collisions) of that in e+e− collisions.
A typical luminosity distribution in γγ collisions is characterized by ahigh energy peak and a low energy part, see section 3. The peak has a widthat half maximum of about 15%. The photons in the peak can have a highdegree of circular polarization.
2.1 The Physics
Physics in e+e− and γγ, γe collisions is quite similar because the same particlescan be produced. However, reactions are different and can give complemen-tary information. Some phenomena can best be studied at photon collidersdue to better accuracy (larger cross-sections) or larger accessible masses (asingle resonance (in γγ and γe) or a pair of light and heavy particles (inγe). Detail consideration of physics program at photon colliders can be foundelsewhere.23,6
2.2 The Photon Collider at TESLA
Recently the TESLA team has published the Technical Design Report.6 ThePhoton Collider is included in the project, though all technical aspects, es-pecially the laser system, should be developed in more detail in the next 2–3years.
The resulting parameters of the photon collider at TESLA for the energyof electron beams 2E0 = 200, 500 and 800 GeV are presented in Table 1. Forcomparison the e+e− luminosity at TESLA is also included. It is assumedthat the electron beams have 85% longitudinal polarization and that the laser
photons have 100% circular polarization. The laser wave length is 1.06 µmfor all energies.
Table 1. Parameters of the photon collider at TESLA.
The maximum energies in γγ and γe collisions given in Table 1 are some-what lower than follow from Eq.1 due to the nonlinear effects in Comptonscattering.24 This lead to decrease of the effective value of x by a factor of(1+ ξ2), where the parameter ξ2 = 0.15, 0.3, 0.4 for 2E0 = 200, 500, 800 GeV,respectively. One can see that for the same energy
Lγγ(z > 0.8zm) ≈ 13Le+e− . (2)
Note that cross sections in γγ collisions are typically by one order of magnitudehigher than in e+e− collisions. The relation (2) is valid only for the beamparameters considered. A more universal relation is (for k2 = 0.4)
Lγγ(z > 0.8zm) ≈ 0.09Lee(geom). (3)
Simultaneously with γγ collisions there are also γe collisions with compa-rable luminosity, so one can study both types of collisions simultaneously.
The normalized γγ and γe luminosity spectra for 2E0 = 500 GeV areshown in Fig. 6.25 The γγ luminosity spectrum is decomposed into two parts
Figure 6. γγ and γe luminosity spectra at TESLA(500). Solid line for total helicity of thetwo photons 0 and dotted line for total helicity 2. Lower curves with cut on longitudinalmomentum.
with the total helicity of the two photons 0 and 2. We see that in the highenergy part of the luminosity spectra the photons have a high degree of po-larization. The low energy luminosity is produced mainly by photons aftermultiple Compton scattering and beamstrahlung. These events have a largeboost and can be easily distinguished from the central high energy events.Fig. 6(left) also shows the same spectrum with an additional cut on the longi-tudinal momentum of the produced system, which suppresses the low energyluminosity to a low level.
The γγ luminosity distributions, including all their polarization charac-teristics, can be measured using processes γγ → l+l−, where l = e or µ. Theγe luminosity can be measured using the process γe → γe.
2.3 Luminosity limitations due to beam collisions effects at TESLAenergies
Beam collision effects in e+e− and γγ, γe collisions are different. In particular,in γγ collisions there are no beamstrahlung or beam instabilities, there is onlyone effect: the coherent e+e− pair production by the photon in the field ofthe opposite electron beam. Therefore, it was of interest to study limitationsof the luminosity at the TESLA photon collider due to beam collision effects.The simulation25 was done for the TESLA beams and the horizontal size ofthe electron beams was varied. Fig. 7 shows the dependence of the γγ (solid
curves) and the γe (dashed curves) luminosities on the horizontal beam size forseveral energies. One can see that all curves for the γγ luminosity follow their
Figure 7. Dependence of γγ and γe luminosities in the high energy peak on the horizontalbeam size for TESLA at various energies.
natural behavior: L ∝ 1/σx. Note that while in e+e− collisions σx ≈ 500 nm,in γγ collisions the attainable σx with the planned injector (damping ring)is about 100 nm. In γe collisions the luminosity at small σx is lower thanfollows from the geometric scaling due to beamstrahlung and displacement ofthe electron beam during the beam collision.
So, we can conclude that for γγ collisions at TESLA one can use beamswith a horizontal beam size down to 10 nm which is much smaller than thatin e+e− collisions. As a result, having beams with very smaller emittances,the γγ luminosity in the high energy peak can be, in principle, several timeshigher than the e+e− luminosity.
Production of the polarized electron beams with emittances lower thanthose possible with damping rings is a challenging problem. There is onemethod, laser cooling3,4,5 which allows, in principle, the required emittancesto be reached. However this method requires a laser power one order ofmagnitude higher than is needed for e → γ conversion. This is not excluded,may be for the second generation of photon colliders.
2.4 Photon collider factories
At current TESLA parameters the rate of the neutral Higgs boson h0 in γγcollisions will be higher by a factor of 1–10 for the Higgs mass Mh = 120–250
GeV.25 The rate of charged scalars (charged Higgs bosons or supersymmetricparticles expected in some theories) is higher by a factor of 8 not far from thethreshold. The rate of WW pairs will be also higher approximately by thesame factor. The list of examples could be continued. If methods to decreaseelectron beam emittances were found (the laser cooling or something else)then γγ factories of some particles would be possible.
3 Lasers-optics
The photon collider at TESLA requires the laser with wavelength about 1µm, the flash energy 5 J, duration 1.5 ps and the repetition rate 14 kHz. Theaverage power for two beams should be about 140 kW. All parameters arereasonable for exception of the repetition rate (average power). To overcomethis problem each laser bunch in the scheme considered for TESLA, see Fig.8,is used for the e→ γ conversion many times (12 in the current design) 25,6. Thelaser pulse is sent to the interaction region where it is trapped in an opticalstorage ring. This can be done using Pockels cells (P), thin film polarizers(TFP) and 1/4-wavelength plates (λ/4). The same laser can be used for thewhole range of the TESLA energies.
e e
∆ T(loop)=
1 221
P
λ/4
in
∆T(loop)=
2 psλ/4
337 ns
337 ns
out
TFP
Figure 8. Optical trap (storage ring)
4 Condensed Beams
Theoretically, using some cooling techniques one can decrease beam emit-tances to very small values. What are limits?
4.1 Space charge
The first simple limit is due to the beam space charge when repulsion forcesare larger than focusing forces provided by quadrupoles. This gives the limit
√εnxεny >
rpNβ
γ2σz, (4)
where N is the number of particles in the beam, rp is the classical radiusof particles (rp = e2/mpc
2), β is the beta-function of the focusing channel,γ = E/mc2, σz - bunch length.
4.2 Crystalline beams
The second effect is connected with repulsion of individual particles. If thebeam is focused in all directions, the ground state of such a system representsa crystalline structure. Such crystalline beams can be formed when kineticenergy of particles in the beam rest system in all directions is much smallerthan the two particle Coulomb potential energy e2/a. Such systems havebeen studied by numerical methods and experimentally (not successfully yet).In one of the latest paper on this subject26 the conditions for forming thecrystalline beams in damping rings have been formulated and it was shownthat at one of the existing facilities such beams can be obtained. This subjectis discussed detail in the talk by J.Wei at this workshop.
4.3 Quantum limits
The Heisenberg uncertainty principle can be written (for γ 1) as
εni ≥ λ−c
2, (5)
where λ− = h/mc is the Compton wave length and εni are the normalizedemittances:
εnx = γεx, εny = γεy, εnz = σzσγ . (6)
These inequalities are valid always for a single particle and for bosonbeams. For fermions, such as protons or electrons there is an additional re-striction posed by Fermi-Dirac statistics: each quantum state can be occupiedby no more than one fermion. The maximum density in the 6D phase spacefor fermions with spin 1/2 is dN/(dPdr) = 2/(2πh)3. For Gaussian beamswith the maximum density in the center this condition corresponds to
εnxεnyεnz >Nλ−3
c
2. (7)
For emittances considered now for linear electron colliders the left part of (7)is of the order of 10−9−10−8 cm3, while the limit is equal 5 ·10−22 cm3, whichis much smaller.
For bosons special techniques allow to reach quantum condensate states,see talks by R. Chiao, W.Ertmer and T.Hirose in these proceedings. Thisis very exciting field, with interesting physics and many nice experimentaltechniques.
Acknowledgments
I would like to thank P.Chen for inviting me to chair the Working Group andpatient waiting of this written contribution (this time has overlapped withthe editor work on the TESLA TDR). Many thank to Stefania Petracca fororganization of the nice workshop in the beautiful place.
References
1. V.E.Balakin and A.A.Mikhailichenko, Preprint INP 79-85, Novosibirsk,1979.
2. T. Hirose et al., Nucl. Instr. &Meth. A 455 (2000) 15.3. V. Telnov, SLAC-PUB-7337, Phys. Rev. Lett., 78 (1997) 4757, erratum
tum aspects of beam physics, Monteray, USA, 1998, World Scientific,p.173-189, e-print hep-ex/9805002.
5. V. Telnov, Nucl. Instr. and Meth. A 455 (2000) 80, hep-ex/0001028.6. Technical Design Report, DESY 2001-011, ECFA-97-182.7. Z. Huang and R. Ruth, Phys. Rev. Lett. 80 (1998) 976.8. A.A. Mikhailichenko, M.S. Zolotorev,Phys. Rev. Lett. 71 (1993) 4146.9. A. Mikhailichenko, CLNS 01/1726
13. I. Ginzburg, G. Kotkin, V. Serbo, V. Telnov,Nucl.Instr. & Meth. 205(1983) 47.
14. I. Ginzburg, G. Kotkin, S. Panfil, V. Serbo, V. Telnov, Nucl.Instr.&Meth.219 (1984) 5.
15. V. Telnov, Nucl. Instr. &Meth. A 294 (1990) 72.16. V. Telnov, Nucl. Instr. &Meth. A 355 (1995) 3.17. Zeroth-Order Design Report for the Next Linear Collider LBNL-PUB-
5424, SLAC Report 474, May 1996.18. JLC Design Study, KEK-REP-97-1, April 1997.19. R.Brinkmann et al., Nucl. Instr. and Meth. A 406 (1998) 13, hep-
ex/9707017.20. Sakai, N. Sasao, S. Araki, Y. Higashi, T. Okugi, T. Taniguchi, J. Urakawa,
25. V.I. Telnov, Proc. Intern. Workshop on High-Energy Photon Colliders,Hamburg, Germany, 14-17 Jun 2000. Submitted to Nucl. Instr. Meth.A, hep-ex/0010033.
26. J. Wei, H. Okamoto, A. Sessler, Phys. Rev. Letters, 80 (1998) 2606.
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PHOTON-ELECTRON INTERACTIONS IN BEAM PRODUCTION, COOLING AND MONITORING
F.V. HARTEMANN
Institute for Laser Science and Applications,
Lawrence Livermore National Laboratory, Livermore, CA 94550
This paper is a summary of the various presentations given in Group B, at the workshop on “Quantum Aspects of Beam Physics”, held on the island of Capri, in Italy, during October 2000. Two main themes emerged from our discussions: emittance, cooling, and condensates, on the one hand; radiation, on the other hand. Fundamental questions about the quantum limit of emittance, the behavior of fermionic beams at extremely low temperatures, and radiative coupling under novel physical conditions, are being pushed to the forefront of accelerator physics because of rapid developments of concomitant technologies; these questions are addressed here, in the challenging and informal format characterizing this workshop. Of particular interest, our debate on the extension of the concept of Bose-Einstein condensation to photonic states is noteworthy, as well as novel ideas to produce short wavelength radiation, with numerous exciting applications ranging from the high energy physics frontier and the gamma-gamma collider, to ultrafast material science, molecular biology, and x-ray protein crystallography.
1 Introduction
With rapid progresses in technology, such as the development of tabletop terawatt lasers using chirped-pulse amplification (CPA), high-gradient rf photoinjectors, and sophisticated laser cooling techniques, as exemplified by Bose-Einstein condensation (BEC), fundamental questions about the quantum limit of emittance, the behavior of fermionic beams at extremely low temperatures, and radiative coupling under novel physical conditions, are being pushed to the forefront of accelerator physics; these questions are addressed here, in the challenging and informal format that characterized this workshop.
The goal of our working group on photon-electron interactions in beam production, cooling and monitoring, was thus to examine various novel developments in the aforementioned multi-disciplinary area of research, with special emphasis on the following subjects: BEC; the emittance and cooling of fermionic beams; and radiative coupling, including Compton scattering, free-electron lasers (FELs), synchrotron radiation and cooling, photo-emission in high-gradient rf photoinjectors, and harmonic bunching. Finally, the exact relation between Bose-Einstein condensation and photonic states, such as squeezed states, laser coherent states, and mode-locked states, was also one of the focal points of our debates, as well as discussions regarding the role of quantum mechanics in these questions, in contrast with classical and semi-classical models which can prove useful in certain situations.
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Two main themes emerged from our discussions: emittance, cooling, and condensates, on the one hand; radiation, on the other hand. While a number of papers clearly belonged to the overlap of both sets, I have divided this paper within these two main sections, without prejudice.
2 Emittance, Cooling, and Condensates
Before giving a more detailed summary of each presentation given on the subject of emittance, cooling, and condensates, I would like to briefly present the salient questions discussed during our round table on BEC, and so-called “crystalline” beams.
It was first pointed out that BEC is a purely quantum effect, while crystalline beams rely on the balance between the Coulomb repulsive force and external forces, including focusing and cooling; crystalline beams can be described by classical or semi-classical models, whereas BEC must be treated in a fully quantum mechanical fashion. It is also interesting to note that an emerging concept during this workshop, beyond the various working groups, was the distinction between wave-mechanical and pure quantum effects: for example, BEC is a true quantum effect, as it relies on spin, whereas a number of questions pertaining to beam dynamics and radiation can be adequately treated with models ranging from classical and semi-classical electrodynamics, to wave-mechanical models. A good example of the former type of theory is Compton scattering, in the small recoil limit, where it coincides with Thomson’s classical theory in the non-relativistic limit, and where photon numbers can be obtained by scaling the entire calculation with Planck’s constant, while the latter approach can be successfully applied to beam diffraction and other effects; again, Planck’s constant appears only as a scale in such cases.
R. Chiao, of UC Berkeley, proposed that 4-wave mixing could be used to produce a photonic state similar to BEC; in particular, an insightful analogy with mode-locking of the axial mode in a laser cavity was proposed, where the transverse mode structure in a photonic BEC would consist of a coherently phased superposition of transverse eigenmodes resulting in a “super-mode”, or BEC mode. It was also noted that BEC implies the existence of a temperature below which the quantum wave-functions of the bosons in the condensate start to coherently overlap; within this conceptual framework, it is clear that a pure photonic coherent state, such as that approached by a laser, has zero temperature; in contrast, the proposed 4-wave mixing scheme uses a large set of modes, to which an equivalent temperature can be ascribed; BEC then corresponds to a condition where the transverse coherence of the compound state exceeds its thermal fluctuations.
Furthermore, a novel scheme using light-by-light scattering, mediated by free electrons instead of atoms, and using a second laser to stimulate the process, was speculated to potentially lead to photonic BEC. It was also pointed out that the phase-dependent gain characterizing free-electron laser could be used to lead to squeezed states. Finally, an intense discussion on the comparison between
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squeezing and BEC led to the temporary conclusion that the conceptual foundations of this fascinating area of physics may still require some elucidation.
2.1 Quantum Ground State and Emittance of a Fermionic Particle Beam in a Circular Accelerator
This paper was presented by A. Kabel, of the Stanford Linear Accelerator Center (SLAC); the main thrust of this work being to estimate the minimum phase space volume occupied by an ensemble of fermions subjected to focusing forces and rf acceleration. The question that spurred these calculations was whether the current technology even remotely approached this quantum limit or not; this is an important question, as it has immediate repercussions in terms of the development of new techniques to further improve beam quality at accelerator facilities. In a somewhat predictable and anti-climactic fashion, the answer is a clear no, and a lot of work is still required to push the beam quality at current facilities toward this ultimate quantum limit.
As this is intended as a summary, only the salient features of the calculations are given here; for more details, we refer the interested reader to Kabel’s paper in these proceedings. We also note that finite-temperature effects are included in this calculation. The Hamiltonian of the system, using rest-frame coordinates, is given by
( )22 2 2 2
2 ,2 2 2 2 2
= + + −µ + κ −ρ + κ +ΦH yx zz x y
pp p x yxp (1.1)
where , and x y z correspond to the radial, transverse, and longitudinal coordinates,
respectively. The radius of curvature of the ring is given by ρ and the focusing
strengths are modeled by ( ), , ;κx y z t finally, the rf potential is ( ), .Φ z t The first
task is to diagonalize the dispersion away, using the following canonical transformation:
1 1 2 1
1 1 2 1
2 1 2 2
2 1 2 2
0 0
0 0.
0 0
0 0
α α β β = γ γ δ δ
p p
q q
p p
q q
(1.2)
with the result that
2 2 22
1 21
.2 2=
ω= + +µ∑H i i i
i
p qq p (1.3)
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This very simple Hamiltonian clearly shows coupled, quantized harmonic oscillators, from which the dynamics of the system can be determined; the fermionic nature of the electron beam is taken into account by anti-symmetrizing the resulting eigen-wavefunctions. The beam emittance is then defined as the ensemble-averaged, single-particle expectation value of the action, .J For a properly scaled harmonic oscillator problem, the action is identical to the energy level: ;=J n therefore, we find
( ) ( )
!,1 1
ξ Ωε = = = =ω ω + ω +
di Fi i
i i i
En Nd
N d d (1.4)
where ξF is the Fermi energy of the system, while 3=d is the dimensionality of the problem; furthermore, the product emittance can also be determined:
( ) ( ) ( ) ( )1 2 32 3
1
2 8 3! , , , .
1 9 4=
π ε = ε = ε = π ε = π ε = π + ∏
ddd
ii
Nd N N Nd
(1.5)
Considering an actual ring, the rest-frame frequencies can be approximated by the tune parameters: / .ω ≈ ω ≈ βγνx y L The longitudinal momentum is quantized in
units of 2 / ,π γL which yields 2 / 2 ,ω = π γl m L and we find
2 2
53
7200.
16807
ξ πε = =ω γ ν
Fx
x
Nq
Lm (1.6)
For example, in the case of a ring with 2 m,= πL 10,γ = 1010 ,=N and 100 ,ν = π
we obtain an emittance of approximately 2 Compton wavelengths; needless to say, this is well below any experimentally realized emittance; for a practical comparison of the quantum emittance limit with that obtained on current machines, a good figure of merit is the ratio of the quantum emittance, *,ε to the actual emittance, :ε
on the HERA electron ring, we find 10* / 2 10 ,−ε ε ×; while for protons we have 10* / 3 10 ;−ε ε ×; for PEPII, we have 10* / 9 10 ,ε ε ×; for electrons, and
9* / 6 10ε ε ×; for positrons. In conclusion, for fermionic beams, the current machine performance is very remote from the minimal emittance due to quantum fluctuations in the ring; in addition, the collective behavior of the fermions can be evaluated both for zero and non-zero temperatures; finally, as a possible consequence of the deviation from a pure Boltzmannian distribution, one may expect deviations from the standard IBS theory for very dense bunches subjected to
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strong focusing, including scattering from and to oscillator eigenstates, limited to unoccupied states by Pauli’s exclusion principle.
2.2 Radiative Laser Cooling of Electron Beams
This presentation was given by my co-chair, V. Telnov, form BINP; its main focus is the systematic study of various radiative cooling schemes aimed at decreasing the emittance of high-energy electron or positron beams, by means of isotropic radiative recoil, followed by anisotropic, axial re-acceleration, thus decreasing the transverse beam emittance. One of the most promising methods for this type of cooling mechanism involves high-energy electrons, in the 5 GeV range, which are subjected to high-intensity laser pulses to induce radiation. These lasers pulses can be considered as electromagnetic micro-wigglers, in close analogy with FEL theory and technology.
A number of key questions addressed in this talk include the minimal emittance that can be attained using this technique, the laser pulse energy required for efficient cooling, the influence of the beam energy spread, and that induced by the laser scattering, spin depolarization effects, ponderomotive forces and scattering, as well as radiation damage issues for the laser.
One of the salient equations to be considered is that governing the scaling of the interaction point (IP) spot size with the beam energy, ,γ normalized emittance,
,εn and axial spread, :σz
/ .σ = ε σ γIP n z (1.7)
An estimate of the ultimate emittance can be given by considering the fact that in the electron bunch frame, the laser photon energy is close to 0 ,γωh where 0ω is the initial laser frequency; furthermore, in that frame, the dipole scattering is almost isotropic. At equilibrium, the electron temperature matches the photon temperature, and we have ,γ≈eT T which yields 2
0 0/ ;⊥ ≈ γp m m on the other hand, we have, by
definition, 0/ / ,⊥ γ ≡ ε γβnp m c and we find, for the minimal emittance,
0 0
,λ Β Βε ≈ =λ λ
hCn m c
(1.8)
where Β is the beta-function of the IP-focused electron beam. More detailed calculations yield
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( ) [ ][ ] ( )[ ]
( )
8* 3 3
0 0
*
0
7 10 mm31 1 cm rad ,
5 m
31 1.5 ,
5
xCnx x
Cny y
−× Βλπε ≈ + ξ Β + ξ ⋅λ λ µ
λπε ≈ + ξ Βλ
;
(1.9)
and where the parameter ξ is defined as ( )2
0 0 0/ ,ξ = ωheB m c and controls
synchrotron radiation effects, while the first terms in parenthesis, in Eq. (1.9), correspond to Compton scattering. A comparison with typical parameters on current machines, such as TESLA or ILC, indicates that laser cooling could reduce the beam emittance by up to 3 orders of magnitude; furthermore, when space-charge effects are taken into account, an improvement of up to 10 orders of magnitude could be reached.
Finally, in terms of the various electron-photon interactions involved in beam production, cooling and monitoring, a number of configurations were discussed: photoelectron guns, which can produce low emittance, spin-polarized beams; production schemes for spin-polarized positrons, including the generation of polarized γ-rays using Compton scattering, which then mediate the creation of preferentially spin-polarized pairs in a target; laser radiation cooling and optical stochastic cooling, where beam fluctuations are amplified and fed back with the proper phase shift to decrease the transverse momentum of the beam; and laser acceleration by ponderomotive forces, or by a traveling laser focus in an open structure.
2.3 Bose-Einstein Condensation of Ortho-Positronium
The motivation behind this work, presented by T. Hirose, of Tokyo University, is to design an experiment aimed at extremely detailed studies of the properties of leptonic atoms, which offers the potential to probe the electroweak interaction with unprecedented accuracy. The lifetime of para-Positronium ( PsP ) is 0.125 ns, as it
decays via a 2-photon process; this precludes cooling to the desired BEC transition with current techniques; by comparison, the 3-photon decay of ortho-Ps ( Ps⊥ ) yields a lifetime of 142 ns, which makes it possible to reach BEC: the low mass of Ps⊥ makes the BEC condition, 2.162,ρ = λ >Dn accessible with temperatures of
the order of o1 K, in sharp contrast with the extremely low temperatures required with ordinary atoms. In the BEC condition, n is the density of the bosons, and
Ps/ 3λ/ = hD c m kT is the De Broglie wavelength of the Ps atoms. Another
felicitous aspect of Ps⊥ is the fact that there exists a transition near 500 nm, with a relatively short upper state lifetime of a few ns, which allows for 28 transitions
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down to the one-photon recoil limit within 200 ns; however, no magnetic field can be used to trap the atoms, because of the Zeeman effect which induces a transition from Ps⊥ to Ps .P
It is hoped that the aforementioned one-photon recoil limit, which can be reached quickly using lasers tuned near the 500 nm transition, will be sufficient to approach the BEC transition; more sophisiticated cooling schemes are also under consideration, and we refer the interested reader to the paper on this subject published in these proceedings.
2.4 The Role of Quantum Mechanics in Neutrino Factories
This work was presented by A. Sessler, of Lawrence Berkeley National Laboratory (LBNL); the spirit of the talk was centered on the theme of purely quantum mechanical effects, as they arise in the design and operation of neutrino factories. As mentioned in the introduction to this section, it should be noted that an emerging concept during this workshop, beyond the various working groups, was the distinction between wave-mechanical and pure quantum effects; within this context, A. Sessler’s talk was both timely and important.
One of the first remarks concerns the sharp contrast between e-, e+, p and p
colliders, which, save for the creation of anti-particles, can be designed almost entirely following classical considerations, and neutrino factories, where quantum effects are ubiquitous: π production, µ production, µ cooling, etc.
There are a number of important reasons to operate a muon collider: first and foremost, the synchrotron radiation losses scale as 4 ,−m which confers an enormous
advantage to µ over e-, as / 206.77;−µ =e
m m one also maintains the lepton
advantage, as the available center-of-mass (CM) energy is not partitioned between various internal components; the Higgs production cross-section scales as 2 ;m finally, due to the low beam-beam radiation, high resolution beams can be delivered at the IP, with a predicted 5/ 3 10 .−∆ ×;p p Finally, it is estimated that a 3 TeV collider complex could be sited at an existing high-energy physics laboratory, and built in stages, including a neutrino storage ring, a Higgs factory or intermediate energy collider, and a high energy collider.
The main components of a µµ collider start with a 15GeV/c proton accelerator,
with a production of 221.5 10× p/year; a π production target surrounded by capture solenoid magnets is followed by a π-decay channel, where µ are produced at a rate of 211.5 10× µ/year; a µ ionization cooling channel is then used to decrease the transverse momentum of the beam by isotropically reducing the momentum and anisotropically re-accelerating the beam; µ with 100 MeV/c are thus produced, which are subsequently accelerated to 2 TeV for the production of Higgs, t t, WW, etc.
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One of the key difficulties with µ is their rapid decay: 62.19703 10 s;−µτ = ×
this problem requires that the µ be quickly accelerated near the speed of light so that the lab frame decay time is boosted to much larger values. In turn, this requirement translates into the necessity to have efficient and rapid cooling of the µ beam produced by π decay. Evidently, these considerations are based on purely quantum mechanical effects. The pion decay reaction, ,µπ→ µ+ ν is isotropic in the π rest
frame, where the lifetime is 82.9 10 s.−πτ ×; Furthermore, the µ are spin-polarized,
with their spin aligned with their momentum in the π frame; this means that there is a strong energy-spin correlation for the µ in the lab frame.
Finally, the cooling of the µ beam presents special challenges; at it propagates through a material, the beam emittance εn is modified by both energy loss and multiple scattering:
2
22 2
1.
2µ
µ
θ ε ε γβ= − + σ εβ n n
rn
ddEd
dz dz E dz (1.10)
The minimum equilibrium emittance, * ,εn is determined by setting Eq. (1.10) equal to zero; we obtain
( )2
* 02
0.014,
2 µµ
βε
β ;n
R
dEm cL
dz
(1.11)
where 0β is the betatron function. Equation (1.11) can then be used to compare
different target materials to decrease the µ momentum before re-acceleration in an rf
cell; the figure of merit is ( )/ ,µRL dE dz and it is found that liquid Hydrogen (LH)
provides the best number with / 0.29 MeV/cm,µ =dE dz and a range of 8.9 m.=RL
This is the current option retained in design studies, and computer simulations have shown a reduction of εx from 1400 mm.mrad down to 800 mm.mrad over a distance of 20 m, without a serious degradation of the energy spread; this distance is short compared to 658.653 m.µτ =c
2.5 Extremely Low Emittance, Low Current RFQ, with Ion-Trap Injector
This paper was given by A. Ogata, of Hiroshima University; the basic idea is first produce an ion crystal, which is then accelerated by an RFQ, thus producing a low charge beam with extremely low emittance. Linear ion crystals could be produced at
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mK temperatures, thus enabling ion implantation on very small semiconductors or nano-structures; radiation damage studies on DNA and extremely small biological samples could also be envisioned.
7Li+ was identified as a good candidate for cooling and acceleration: there is a closed 2-level transition with a short upper state lifetime for efficient laser cooling, and the charge to mass ratio is sufficiently large to allow for quick acceleration in the RFQ. The energy goal is 1 MeV/nucleon at the output of the RFQ, and it is estimated that the current must be limited to a single ion per rf bucket to avoid coherent synchrotron radiation heating, which would disrupt the Coulomb ion crystal produced after the cooling section. The decay time from the S1 to the S0 levels of 7Li+ is 50 s, with a transition wavelength of 548 nm, with a linewidth of 2 3.7 MHz;π× however, as the fraction of metastable Li is 310 ,−< some improvement will be required.
The temperature that can be reached with laser cooling can be estimated as follows:
2 2
21 , 1 ,
4 2cos sin
f fT T
k k⊥γ γ = + = + θ θ P
h h (1.12)
where γ is the natural width of the transition, and 1/ 3f = for isotropic scattering.
The radial size of the spot is comparable to the laser wavelength, ,σ ≈ λ while the divergence can be estimated as ’ / ,thv vσ ; with the thermal velocity given by
/ ;th iv kT m; we can then estimate the normalized emittance:
’/ / .n thv c v cε = βε = σσ = λ A laser incidence angle, / 20,θ = π yields
59.3 K, and 2.51 mK,T T⊥= µ =P for 7Li+; the minimum transverse emittance is
therefore * 153.12 10 m.n−
⊥ε = × Here, space-charge effects have been neglected, as the crystalline ion beam is extremely tenuous. The previous number can be compared with the quantum, or De Broglie, emittance,
( )16/ 4 8.61 10 / MeV m;DB DB E−ε = λ π = × we see that the previous number
approaches the quantum emittance at the proposed operating energy of 1 MeV. Finally, a number of other important points must be considered: first and foremost, the question of the mechanical tolerances in the RFQ, as they can quickly induce transverse kicks much greater than the transverse momentum corresponding to the minimum emittance; next, the vacuum must be extremely good to minimize scattering; finally, the injection of the cooled ion crystal into the RFQ is very challenging.
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3 Radiation
In this section, we briefly discuss a number of radiation mechanisms made possible by recent advances in the field of tabletop terawatt lasers, low emittance rf photoinjectors, and femtosecond synchronization between laser and electron beams. Spontaneous and stimulated Compton scattering are two important radiation coupling mechanisms that are considered for such projects as the proposed γ−γ collider and the Linac Coherent Light Source (LCLS), as well as to develop compact, tunable x-ray sources with numerous applications ranging from ultrafast material science to biomedical imaging and x-ray protein crystallography.
3.1 Free-Electron Laser Theory Using a Two-Times Green Function Formalism
H. Takahashi, from Brookhaven National Laboratory (BNL), presented a talk on theoretical models aimed at handling the FEL interaction in the limit of ultra-high intensities, in the nonlinear regime. The two-time Green function formalism yields a quantum description of the FEL interaction, capable of handling the radiative recoil problem at high energies, such as those envisioned for the γ−γ collider; the lowest cutoff of the theory agrees with the classical FEL dispersion relation, while the aforementioned recoil can be treated using a higher-order pertubative approach. In addition, a quantum description of plasma electrons in a strong electromagnetic field can complement the two-time Green function formalism to yield a detailed theory of stimulated Compton scattering in extremely high electromagnetic fields. This formalism uses a classical description of the unperturbed system, and quantum perturbations, in analogy with the notion of a classical trajectory as the average of the quantum position and momentum operators.
The Hamiltonian for the FEL satisfies the second quantization, and a dispersion relation can be derived at low laser intensities:
( )( ) ( )
( ) ( )0
2222 2 2
22 2
12222 2 22
3
/ 2
2 / 2
,24
p
b
p ce pb
k k
q mq c
qv q m
q kq kq k v
k m
−
=±
γω ω − − γ ω− − γ
ω ω ω++ = ω− + − + γ γγ ∑
h
h
h
(1.13)
where 204 /p e n mω = π is the non-relativistic plasma frequency, with 0n the
unperturbed density of the beam, and where 0k is the wiggler wavenumber, and
ceω is the electron cyclotron frequency in the applied guide field, while bv is the
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beam velocity. This relation is only slightly different from the one derived from classical electrodynamics:
( ) ( ) ( )122 2 2 2
2 202 2 2 20 05 2 3
0
11 3 ,
2p p ce p
b D
q kq c qk v q k
k
− ω ω ω ω+ ω − − = ω− − + + λ γ γ γ
(1.14) where the Debye wavelength is given by 2 2
0/ 4 ,D T n eλ = π with T the electron beam temperature. In the case when the beam temperature is zero, Eqs. (1.13) and (1.14) coincide.
In the case of a FEL pumped by a high intensity laser, a number of effects can occur, including self-amplification of spontaneous emission (SASE), multi-photon effects in the high intensity laser field, radiative recoil, and squeezing. For a quantum description, the coherent field must be properly described, and previous work done by H. Takahashi on the quantum description of plasma electrons in high-intensity electromagnetic fields could be useful; in this case, the wavefunction of particle i is expressed as
( ) ( )0
, exp ’ ’ ,2
t
i i i i ii
it C i k t dt
m
ψ = ⋅ − + ⋅
∫r k r q A (1.15)
where , , and i i iC mk are respectively the normalization constant, canonical momentum, and mass of the particle. The electromagnetic fields are represented by a superposition of transverse modes, namely,
( ) ( ) ( )ˆ ˆcos sin .xs s s ys s ss
t xA t yA t = ω +θ + ω +θ ∑A (1.16)
The total Hamiltonian of the plasma is then expressed as
( )( ) ( ) ( )
( )
2
, , ,
1,
2 2is i s Fi
ii s t ij s ti s s t i j i j s ti
k t EV V
m ≠
+ ⋅ − = + − + −
∑ ∑ ∑ ∑ ∑q A
r r r rH
(1.17) where FE is the Fermi potential of the fermion, and ( )ij s tV −r r is the Coulomb
interaction potential between particles i and .j The relevance of this formalism to the quantum FEL problem at high intensities can then be summarized by formally separating the vector potential as follows:
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( ) , / 1.n
C n Q Q Cn
= + δ δ∑A A A A A =C (1.18)
Here, the classical vector potential corresponds to a quantum average, and a perturbation expansion is used to study the higher-order nonlinear effects within the quantum formalism.
3.2 Inverse Free-Electron Laser Microbuncher
This work was presented by X. Wang, from BNL; the main thrust of this experimental research program is to build upon the recent inverse FEL (IFEL) results obtain at BNL, to produce extremely short relativistic electron bunches, with a high harmonic content, excellent transverse emittance, and low energy spread. Such microbunches could be used for a number of exciting applications, including laser-plasma acceleration, and coherent harmonic generation of femtosecond uv, vuv, and soft x-ray pulses.
The aforementioned experimental results at the STELLA facility at BNL have clearly demonstrated IFEL bunching and energy modulation in a controlled experiment using an IFEL pre-buncher, and buncher, using two magnetostatic wigglers and properly phased, high-intensity CO2 laser pulses. For a linearly polarized wiggler, it is well-known that there are strong odd-harmonic components; in particular, the 3rd harmonic can be used for efficient bunching. Nonlinear simulations indicate a good potential for strong harmonic IFEL bunching and high harmonic coherent radiation; as modest powers are required, a vuv IFEL might be possible.
The motivation beyond a harmonic IFEL is five-fold: first, this represents a natural extension of the rf linac technology to laser acceleration; second, a planar undulator is almost perfect for bunching; third, a good, efficient micro-buncher is important for the electron beam injector in a laser-driven accelerator, and a high harmonic IFEL for femtosecond x-ray production; fourth, the IFEL is unique because it induces both energy modulation and bunch compression; finally, the transverse emittance can be preserved during the harmonic bunching and compression. Simulation of a single-stage harmonic IFEL buncher show a bunched output with 1% relative energy spread and a phase compression to 0.02 rad; a two-stage IFEL harmonic buncher can produce a phase compression to 0.05 rad, with 95% of the charge, and a very small energy spread.
3.3 Three-Dimensional Theory of Emittance in Compton Scattering
This work is presented in some detail in a paper published in these proceedings.
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3.4 Photocathode RF Gun and Electron Emission
This second paper presented by H. Wang, of BNL, was focused on the questions of quantum efficiency, the Schottky effect, and the promptness of photo-emission in a high-gradient, low emittance, S-band rf gun. Electron photo-emission is a quantum effect; high-field phenomena, including the Schottky effects are not well understood at the present. The Schottky effect can be well studied in rf photoinjectors, by varying the relative phase between the photocathode laser pulse and the accelerating rf fields in the gun. In particular, the so-called enhancement factor, due to surface irregularities, impurities, and dielectric inclusions remains a problem that can be adequately approached using such studies; furthermore, the question of laser-induced explosive emission, or laser-gated field emission, which may be one of the mechanisms at work, can also be studied in detail.
In a high-field rf gun, a number of electron emission mechanisms compete: field emission, photo-emission, thermionic emission, and secondary emission due to electrons or ions, to name a few. The field emission, which leads to dark current in the gun, is described by the Fowler-Nordheim current density,
( ) ( ) ( ) ( ) ( )
133 2
2
4 2ˆ, , exp sin ,
ˆ ˆˆ 3 , , , ,8eme F kT kT
j T F v yheF d F y d F yh r y
− − φ π π φ = φ φπ φ
(1.19)
where φ corresponds to work function, T is the temperature, and F is a phenomenological enhancement factor. For thermionic emission, the Fowler-Nordheim theory yields
( ) ( ) ( )2
3
4, exp ,em e kT e
j TkTh
π − φ− ∆φ φ =
(1.20)
and for photo-emission, the quantum efficiency η is proportional to
2
0
.4
e Eh
βη∝ ν − φ+ πε (1.21)
Here, E is the rf field applied to the cathode, while β is an important parameter corresponding to field enhancement. In an rf gun, if the photocathode laser energy is constant, the charge extracted from the gun should be simply follow the space-charge limit given by the local rf field applied; instead, the high field bends the surface bands of the metal, and yields a phase-dependent charge, involving the Schottky effect, where the work function for photo-emission becomes field
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dependent. This type of effect can be studied in great detail with an rf gun; in particular, β can be determined and compared with theory.
Acknowledgments
First and foremost, it is a pleasure to acknowledge the great organizational skills and scientific creativity of P. Chen and S. Petracca, who made this workshop possible, in a wonderful setting. I would also like to acknowledge the invaluable help of my co-chair, V. Telnov. Last but not least, I would like to thank D.T. Santa Maria for some extremely valuable interactions.
FREE ELECTRON LASER THEORY USING TWO TIMES GREEN FUNCTION
FORMALISM
HIROSHI TAKAHASHI
Brookhaven Natioanal Laboratory Upton New York, 11973
In this paper, we present a quatum theory for free electron laser obtained by firstly using the Two time’s Green Function method developed by Matsubara for solid physics theory. The dispersion relation for the laser photon obtained is limited to the case of low intensity of the laser due to the decoupling the correlation function in low order. For the analysis of the self-amplified emission (SASE), the high intensity laser radiation which strongly affect the trajectory of the free electron is involved, the use of the classical approximation for laser can formulate the laser radiation with multiple frequency. To get the quantum effects in the high intensity laser, use of the perturbation theory, and the expansion methods of state function using the coherent, squeeze and super-radiant states have discussed. 1. Introduction The classical formalism for an electro-magnetic field has been used to formulate a theory for the free electron laser Because the electron beam used for the free electron laser was not very well refined in the past, its formulation by using quantum mechanical formalism was not required. However due to the recent refinements of the electron particle beam used for the free electron laser, it is necessary to formulate a theory and to it give a good foundation by taking the quantum effect into account the Twenty years ago, I derived a quantum theory based on the two times Green function formalism derived by Matsubara for solid physics theory[1]. The dispersion relation is compared with the classical theory obtained by Kwan[2] .
When a relativistic electron moving in a electro-magnetic field and the magnetic field B , The Hamiltonian for such a system may be written as H = ∑i ( c αi.(pi -eA/c) + βi mc2 ) +Hr + H ee +Hs (1.1) where A is the sum of the vector potential As of the B-field and the vector potential Ar of the electromagnetic wave; A =As +Ar, m and c are the mass of the electrons and the speed of light, respectively, while α and β are the Dirac matrix operators, the suffix stands for the ith particle. H r is the Hamiltonian for the electromagnetic wave. Hee is the Hamiltonian for the interaction between electrons , and Hs is the Hamiltonian for the Static magnetic field. The last Hamiltonian, which is the constant of motion, is not
incorporated in the dynamic of the system; it can thus be neglected in the following derivation. The Hamiltonian Hee and Hr are given by Hee = ∑ i>j e
2 /rij, (1.2) Hr = ∑ qλ h ωq b
+q,λ b q,λ (1.3)
where b+
q,λ and b q,λ are the photon creation and annihilation operator, respectively, for photon momentum h q and polarization state λ. . The photon energy is given by h ωq = hcq. The unperturbed system is described by an electron moving in the B-field and the free radiation. Hamiltonian for the unperturbed system is H 0 = H e + H r (1.4) where He = ∑i ( c αi(pi -eAsi /c) + β i mc2 ) (1.5) The interaction Hamiltonian is expressed as H int = - ∑i e αiA ri+_Hee (1.6) For simplicity we consider the B-field to be periodic with spacing d in the direction, and having no spacial dependence in the transverse (x-y) plane . Hence it will be described by B0⊥ = B0(excos K0z + eysinK0z) (1.7) where K
0 = 2 π/ d.
We note that He is invariant under all transformations in the transverse plane and discrete
translations in the z direction corresponding ti the periodicity of the field. This results in the
Bloch type solution
ψ k (r ) = e ikr u k (z) (1.8) where uk(z) has the same periodicity as the static magnetic field. It is important to note that the function is a four component spinor u k (z) = ∑ K=+-K0 CKk,e ik.z (1.9)
where C is a free electron spinor with an energy eigenvalue
Ek2 = h2c2k2 +m2 c4
(1.10)
To derive the dispersion relation of the free electron laser, the Hamiltonian of the relativistic
electron moving in a electron-magnetic field and he magnetic field B may be written as The second
quantization formalism for the Hamiltonian for the unpertubed system can be obtained eq,(2.4) , as
follows ; as
H0 = ∑ kσ T kσ a+
kσ a kσ + ∑ qλhωq b +q,λ b q,λ (1.11) where a+
kσ is the electron creation operator for the electron momentum hk and spin σ, akσ is the corresponding annihilation operator, and T kσ, is its energy . The interaction Hamiltonian can be expressed by
k3σ2(t)ak4σ2 (t)ak2σ1(t) (1.12) The first term is due to the electron creation-photon interaction with M k1k2q,σ1σ2λ = -e √[4 πc2 h /2ωqλv] (2π)2(δ(k1x-k2x +qx ) δ(k1y-k2y+qy ) N d∑ K=-K0
K0 (δ(k1z -k2z +qz, K) F (k 1 k 2 , σ 1 σ 2 ) (1.13)
In equation (1.25), N the number of spacing, is derived from the 2. The Green function of the photons The propagation of the photon in this system can be obtained by solving for Green’s function of the photon. The one photon retarded Green function can be written as G qλ(t-t’) = < A qλ(t); A qλ
+(t’)>> = -i θ(t-t’)<[ A qλ(t); A qλ+(t’)]> (2.1)
And where <> denote the statistical average. Where [C,D] = CD –DC, θ (t) =1 for t>0, and =0 for t<0, ( 2.2) A qλ = bqλ +b+
qλ (2.3) and where <,> denotes the statistical average . To obtain the equation of motion for Green’s function , we consider the equations of the motion for the creation and annihilation operators of the photons and electrons from the
Hamiltonian defined in eqs. (1.11) and (1.12): ih ∂Aqλ(t)/∂t = [Aqλ(t), H ] = hωqλ(bqλ(t)-b+qλ(t)) = hωqλB qλ(t) (2.4) where H =H0 +H int (2.5)
ih ∂Bqλ(t)/∂t = [Bqλ(t), H] = hωqλAqλ(t) + ∑k1k2,σ1σ2 M k1k2q,σ1,σ2λ a+k1σ1(t)ak2σ2(t) (2.6) ih ∂ak1σ1(t)/∂t = [ak1σ1(t) , H] = Tk1σ1a k1σ1(t)- ∑k2q2;σ1λ M k1k2q,σ1σ2λ aq,λ(t)Aq,λ(t) +∑ k2k3k4,σ2V k1k3k4k2,σ1,σ2σ2σ1(t) a+k3,σ2 (t)ak4,σ2(t)ak2,σ1(t) (2.7)
ih ∂a+k2σ2(t)/∂t = [a+k2σ2(t) , H]= - Tk2σ2a+k2σ2(t) +∑k1q;σ1λ1 M k1k2q,σ1σ2λ a+k1σ1(t)Aqλ(t) +∑ k1k3k4,σ2V k1k3k4k2,σ1σ2σ2σ1(k) a+k3σ2 (t)a+k2σ1(t)ak4σ2(t) (2.8) The equation of motion for the Green function Gqλ(t-t’) is obtained from -h2 (∂2(t)/∂t2 + ωqλ
2) Gqλ(t-t’) = 2 ωqλh2δ(t-t’) + ωqλh ∑k1k2,σ1σ2 M k1k2q,σ1σ2λ Gk1σ1k2σ1;qλ(t-t’), (2.9) where Gk1σ1k2σ2;qλ(t-t’)= << a+k1σ1 (t)ak2σ2(t); Aqλ+(t’)>> (2.10) To solve eq.(2.9), the equation of motion for the Green’s function defined in eq.(2.10) is obtained by
differentiating it with respect to time t as follows:
ih ∂Gk1σ1k2σ;qλ (t-t’)/∂t = (-Tk1σ1+ T2σk2) Gk1σ1k2σ;qλ(t-t’) +∑ k3q1,σ3λ1 M k2k3q1,σ2σ3λ1 << a+k1σ1 (t)a+k2σ2 (t) Aq1,λ(t); Aq,λ
+(t’) >> -∑ k3q1,σ3λ1 M k3k1q1,σ3σ13λ1 << a+k3σ3 (t)ak2σ2(t) Aq1λ(t); Aqλ+(t’) >> +∑ k3k4k5,σ3k V k5k3k4k1,σ1σ3σ3σ1(k) << a+k5σ1(t)a+k3σ3(t)ak4σ3(t)ak2σ2 (t); Aqλ+(t’) >> -∑ k3k4k5,σ3k Vk2k3k4k1σ2σ3σ3σ2(k) <<a+k1σ1(t)a+k3σ3(t)ak4σ3(t)ak5σ2(t); Aqλ+(t’) >> (2.11) By successive differentiation of the Green’s function with respect to t, hierarchy of equations is obtained. In order to close the hierarchy equation, we approximate the higher order Green’s function by expressing it in terms of the low order Green’s function. The
de-coupling of the higher order Green’s function is carried out as follows: << a+k1σ1(t)a+k3σ3(t)Aq1λ(t); Aqλ+(t’) >>≅ δk1,k3δ σ1,σ3n k1, σ1δ q1;q δ λ1,λ<< Aq1λ(t); A+
qλ(t’) >> (2.12) where n k1, σ1=< a+k1σ1(t)ak1σ1(t)> is the density of electrons with momentum h k1 and spin σ1
<< a+k5σ1 (t)a+k3σ3 (t)ak4σ3 (t)ak2σ2(t);A
+qλ (t’) >> ~
δk5,k2δ σ1,σ2 n k2, σ1<< a+k3σ3 (t)a+k4σ3 (t); A
+qλ (t’) >>
-δk5,k4δ σ1,σ3 n k4, σ1<< a+k3σ3 (t)a+k2σ2 (t); A
+qλ (t’) >>
-δk3,k2δ σ3,σ2 n k1, σ1<< a+k3σ3 (t)a+k2σ2 (t); A
+qλ(t’) >> (2.13)
The first term on the r. h.s. of eq.(2.13) contribute to the direct Coulomb interaction; the second and third terms contribute to the exchange Coulomb interaction. The Fourier components of the equation of motions for the Green’s function (2.13) G k1σ1k2σ;qλ(ω) is thus obtained as (hω+Tk1σ1 -Tk2σ2) G k1σ1k2σ;qλ(ω) = ∑k 4πe2/k2[nk2σ2 H (k1,k2,k, σ1 )- nk1σ1 H (k1,k2,k, σ2) ], x∑k3,k4 H* (k3, k4,-k, σ3 ) G k3σ3k4σ3;qλ(ω) +(nk2σ2- nk1σ1) Mk2k1q;σ2σ1λ Gqλ (ω). (2.14) Solving eq.(2.14) and substituting it into eq. (2.9), we obtain the equation of motion for photon Green’s function: [[ h2 (ω2-ωqλ
p ω2p /4γ3 ∑k=+-k0(((q+K)/K)2[(ω-(q+K).Vb)2 - 1/4( h(q+K)/mγ)2 +ω2
p /γ] -1 (2.16) This dispersion relation is slightly different from the one obtained by Kwan et al using the classical formalism : [ω2-q2c 2 -ω2
p/γ] = [1/2ω2
p ω2ce /γ5(q+K0)
2/K0)2 [(ω-(q+K0 ).Vb)
2-ω2p /γ3 ( 1+3(q+K0)
2)λ 2D] -1 (2.17)
Where λ2D
=T/4π n0
e2
and T I s the temperature of electron beam.
Although I formulated the quantum mechanical theory for the free electron laser, it was limited to a low intensity laser because of the approximation used to de-couple the higher-order correlated function of Eq.(2.12). Hence, it could not be applied for the large intensity amplification of the laser, such as the self amplification emission (SASE), which produces a high-intensity laser from the small noise signal or input signal. 3. High Intensity Laser
To formulate a theory for a high intensity laser, the higher correlated function should not be de-coupled as in eq. (2.12 ). Further, the equation of the motion of the high-order correlated Green function should be obtained by differentiating it the same way as in Eq. (2. 11 ). This differentiation creates a correlated function of a much higher order. By successively differentiating them, a correlated function which is of a higher order than the previous one can be derived. Since the higher correlated function becomes smaller as the order increases, by de-coupling the higher order’s functions as products of the lower correlation ones, similar to eq. (2.12) successive simultaneous equations can be closed. However, it is difficult to solve the simultaneous equations analytically; therefore a numerical method using a computer might be required, although it poses the problem how to integrate these continuous functions. To obtain the closed form of the correlation function, the use of the classical description for the EM field greatly simplifies the formula. When the laser intensity is high, and the number of photons associated with this laser intensity becomes large, their photons are not correlated, such that their phases are randomly distributed, and the EM field can be treated as classical one. Although the photons are not correlated each other, the high -intensity laser affects to the trajectory of electrons, the laser produced from the emission from the electrons is very much affected by this high-intensity laser. In analyzing electron plasma under such a high -intensity laser field, I derived the Green’s function of
electrons by treating the EM wave as the classical one. In this formalism without a lengthy integration of the simultaneous equation, as used in the above, the intense laser field with multiple of the frequency are simply calculated from the formula of the electron’s correlated function,. However, the highly correlated photon is not taken into ccount this classical formula, and the delicacy of the coherence due to quantum effects is also totally discarded. When the EM is treated as classical field A(t), the wave function of the electron can be expressed as
The total Hamiltonian H is expressed as H = ∑s (1/(2m ) (k(i) + q(i) A(t) )2 - EF(i) ) +1/2 ∑s,t V
(i,i) (rs –r t)) + ∑s,t
(i≠j) V (i j) (rs –r t)) (3.2)
where Vs are the Coulomb interaction Hamiltonians. In this Hamitonian, Hamitonian of electron is expressed by 1/(2m ) (k(i) + q(i) A(t) )2 which includes the classical vector field A(t) The density operator of the i-th particle is expressed in the second quantization formalism as ρ (i)(r,t) = ψ* (i) (r,t) ψ(i)(r,t)= ∑k1,k2 a+k1
(i)exp( ik (i)1 r) ak2 (i)exp( ik (i) 2 r ) (3.3)
The Fourier transformation of the density correlation function operator <ρ (i) (r,t), ρ (i) (r’,t)(r,t)> can be expressed by ∫∫ dr dr’<ρ (i) (r,t) , ρ (i) (r’,t) > exp( ik (i) (r-r’) =∑k1,k2σ1σ2 <a+k1σ1
(i)(t) ak1σ1 (i)(t)a+k2+kσ2
(j) (t’)ak2+kσ 2 (j) (t’) > (3.4)
Fourier Transformation of Green’s functions Gk1σ1
(i,j)(k, t-t’) = ∑k2,σ2 <<a+k1σ1(i)(t) ak1σ1
(i) (t)a+k2+kσ2(j) (t’)ak2 σ2
(j) (t’) >> (3.5) and the vector potential A(t) is sum of many modes as A(t)= ∑s [ Axscos ( ωst+ θs)+ Ays sin( ωst+ θs) ] (3.6) then, Fourier Transformation of Green’s functions Gk1σ1
(i)( k, E) (3.8) 4. Perturbation Theory One way to save partially the quantum effect on the laser is to use the perturbation method by treating a vector potential Aem associated with EM field as sum of a classical field Ac and the Ap. By treating Ap as the operator-expressed creation and annihilation of the photons, the quantum effects on the high-intensity laser can be analyzed as in the first paper, where the vector potential A was composed of the wiggler field Aw and the photon fields. By adding the classical vector potential Ac to the wiggler vector potential,Aw , we can deal with high intensity laser in a similar way as adopted the first paper. 5. Coherent , Squeeze, and Super-Radiant Theories Another way to deal with this problem is to use the coherent state description which gives the sound foundation of the classical formula. The coherent state is defined as α> = exp (-α2/2)∑n=0 to inf αn/√n! n> ( 5.1) Here n> =1/√n! (a+)n ϕ0>, ϕ0> = vacume state> ( 5.2) is the eigen state of th number operator N=a+ a containing light quanta of ( k, η). For our consideration it is useful to split a+ and a into a sum of Hermitian opertors i.e. a = ( u+ip) /√(2h) a+ = ( u-ip) /√(2h) (5.3) Coherent state α > is the eigen state of the non-Hermitian annihilation operator a with the complex eigen value α =( u+ip) /√(2h) If this complex eigen value α , which labels the coherent states runs over the whole
complex plane, the coherent states becomes over-complete for the Hilbert space. Such an over-complete sets of the coherent states can not be used for our consideration because they are linearly dependent. However, Bargmann al. and Perelomov proved that subset of the over complete sub-state form a complete set. This subset is given by α > : α = √π( l +im) ; l = 0,+-1, +-2,; m= = 0,+-1, +-2,…… ( 5.4) This fact was originally stated by von Neumann, without proof. Therefore these states are called von Neumann lattice coherent state(VNLCS). Using the VNLCS state, Toyoda et al [4] provide the sound foundation of the classical formalism for the high intensity photon field. However, in their formulation, classical approximation is applied to the radiation field, assuming that the electron state is not subjected to such an approximation. To use the coherent state for electron, the formalism developed by Ohnuki et al [5] should be applied which describe the coherent state of fermion particle (ξ)n > ≅ exp(-∑j=1
nξj aj+)0> ( 5.5 )
where ξs are Grassmann number instead of the complex number α for boson field . This discretization introduces the uncertainty principle of quantum formulation, the delicacy of the quantum effect on the theory such as the super radiant state and the squeeze states which is prominent in the quantum theory of the laser can be studied as discussed as the paper on . The one of advantage of using these coherent state description is a many modes created by the EM field can be formulated without difficulty. Further more another transition associated with the super-radiant and between squeezed state can be formulated by including it in the eigen values α, The squeezed states which might be created by the free electron laser is described by the u or p in the Eq.(5.3) As mentioned in the state can be expanded by including the these squeeze or super –radiant state, we can formulate the free electron theory which has effect of the these states. This coherent state is over complete state and the some dicretalization is needed to reduces the over-completeness to the conventional completeness. Here the uncertainty of the quantum description is comes in.
Reference
[1] Hiroshi Takahashi “ Theory of the free elctron laser Physica 123 C (1984) 225-237 North-Holland Amsterdam [2] T. Kwan., J.M.Dawson and A.T. Lin. Phys. Rev. 16A (1977) [3]Hirshi Takahashi “Interaction between a Plasma and a Strong Elctromagnetic Wave “ Physica 98C (1980) 313-324
[4] Tadashi Toyoda and Karl Widermuth “ Charged Schroedinger particle in a c-number radiation field” Phy. Rev. 10D, p2391 (1980)
[5] Yosio Ohnuki and Taro Kashiwa “Coherent State of Fermi Operators and the Path Integral” Prog. Theo. Physics, 60. 548, (1978)
capri-wang3.DOC submitted to World Scientific 6/29/01 : 11:28 AM 1/10
HARMONIC INVERSE FREE ELECTRON LASER MICRO-BUNCHER
S. POTTORF
Department of Physics & Astronomy, State University of New York at Stony Brook Stony Brook, NY 11794
X.J. WANG
Brookhaven Accelerator Test Facility, NSLS, Brookhaven National Laboratory Upton, NY 11973, USA E-mail: [email protected]
The Inverse Free Electron (IFEL) and its applications are first briefly reviewed. The concept of harmonic IFEL was proposed for electron beam micro-bunching. A tightly micro-bunched electron beam could be used either as a laser accelerator injector or for femto-second X-ray generation by ultra-high harmonic radiation. For a planar undulator, the electron beam is strongly coupled to both the fundamental and odd harmonic if the undulator parameter is greater than 1, K > 1. The 1-D equation of motion for the IFEL was first extended to the third harmonic. 1-D simulations were performed for both fundamental only IFEL and harmonic IFEL. Two configurations of harmonic IFEL were considered. First, a single undulator with both fundamental and third harmonic IFEL presented simultaneously was studied. Second, a configuration employing a fundamental IFEL followed by a third harmonic IFEL. Better and efficient bunching was achieved for both harmonic IFEL compared to the fundamental IFEL only.
1 Introduction
The concept of the IFEL was proposed by Palmer [1] shortly after the free-electron laser (FEL) was re-introduced by Madey in 1971 [2]. The basic physics of the IFEL is very much similar to the FEL interaction. The undulator magnet introduces a small transverse velocity in the direction parallel to the electric vector of the electromagnetic wave. The interaction between the wave and electrons leads to an exchange of energy between the electron beam and wave. In the IFEL, the energy exchange leads to a loss of energy from the electromagnetic wave while the electromagnetic wave gains energy in the FEL. Extensive studies were carried out at Brookhaven National Laboratory [3,4] to explore the possibility of IFEL based electron accelerator. IFEL acceleration of an electron beam was observed at the Brookhaven Accelerator Test Facility (ATF) using a 10.6µm CO2 laser [5]. The IFEL was also investigated as a UV or X-ray source with techniques such as laser seeded single-pass FEL [6] and high-gain harmonic radiation generation [7]. In the high-gain harmonic generation (HGHG) FEL [6], the interaction of the electron beam through the IFEL effect with a seed laser in the first undulator (modulator) leads to energy modulation of the electron beam. Energy modulation
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is transformed into spatial modulation (bunching) in the second HGHG magnet (dispersion section), and a micro-bunched electron beam generates coherent radiation and amplification in the last section of the HGHG undulator (radiator). More than seven orders of magnitude gain was reported at second harmonic of seed laser at the ATF HGHG experiment [6]. The main attractions of HGHG compared to the Self Amplified Spontaneous Emission (SASE) [8] are better longitudinal coherence and use of a shorter undulator and it suffers from the availability of seed laser. In the scheme of high harmonic radiation, IFEL interaction of the electron beam with a high power laser (GW) leads to strong micro-bunching of the electron beam. Large harmonic contents of micro-bunched electron beam could produce coherent radiation through the second undulator in high harmonic radiation [7]. The IFEL effect was used to slice the femto-second electron beam out of pico-second long electron bunches in the ALS storage ring using a femto-second high power laser [9,10]. The IFEL also plays an important role in a proposed optical stochastic cooling technique [11]. IFEL micro-bunching with harmonic contents of the IFEL were experimentally observed at the ATF [12]. Furthermore, IFEL micro-bunched electron beams were captured and accelerated in a staged laser acceleration experiment [13]. The IFEL as an injector for laser accelerators has many merits. Since the IFEL interaction and bunching occur in vacuum, it only slightly degrades the quality of the electron beam. The same laser can be used both as a buncher and as an accelerator of an electron beam. The synchronization between electron micro-bunches and acceleration field is built in. In both high harmonic generation and injection for laser accelerations, single stage IFEL bunching requires a high power laser, and efficiency is low. In the traditional RF linacs, those problems were overcome by using multi-stage harmonic bunchers. We are proposing a similar technique that could also be used as an IFEL. In a planar undulator with large undulator parameter, K>1, the electron motion in the undulator contains a fast longitudinal component, k∆z ≈-ξsin(2k0z), where ξ= K2/2(1+K2). This fast longitudinal motion leads to strong spontaneous emission at odd harmonics, n=1,3,5… [14]. Similarly, the electron beam is also strongly coupled to odd harmonics in the IFEL interaction for planar undulators. Odd harmonic buncher cavities were employed in RF linacs because they generate a square wave form, therefore the harmonic IFEL can also be explored to improve electron beam micro-bunching. In the following section, the derivation of 1-D IFEL equations (pendulum equations) for the fundamental frequency is reviewed. We then re-derive the equations of motion for two cases. First, we consider a third harmonic laser seed in the IFEL, and in the second case, the IFEL laser is composed of fundamental and third harmonics. In section III, we describe the simulations used to solve the 1-D IFEL equations of motion, then we discuss the energy-phase space and energy and phase distribution results from the simulations.
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2 1-D Harmonic IFEL Theory
Assuming an electron beam co-propagates with a laser along the undulator axis in the z direction, only longitudinal motion of the electrons will be considered. Following the basic formalism used in reference 3, the relativistic Hamiltonian is given by,
( )[ ] Φ+=Φ++−= emcecmceH 221
4222 γAP (1)
where ApP e+= is the canonical momentum with vp mγ= being the kinetic
momentum and A the vector potential. The scalar potential Φ in the Hamiltonian is set to zero since space charge effects are not considered. The vector potential for a planar undulator is,
( )zkA www sin2xA = (2)
where wwk λπ2= with wλ is the undulator period. The magnetic field is,
)cos(2ˆ zkk wwww yAB =×∇= . (3)
For 1<<zx pp , the motion due to the undulator alone is given by Hamilton’s
canonical momentum
( )zkeA wwx sin2xp −= (4a)
0=yp (4b)
( )[ ]
−+−−= zkKKmc wz 2cos12
11ˆ 22
2γγxp (4c)
where K is the undulator parameter,
mc
eBK ww
πλ
2= . (5)
The longitudinal motion of the electron can be written as,
zzz o ′+= (6)
where
+−=2
2
2
11
γK
ctzo (7)
and its time derivative is,
+−=2
2
2
11
γK
cdt
dzo . (8)
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The small oscillations, z′ , are found by subtracting dtdzo from dtdz ,
( )[ ]zzkcK
dt
zdow ′+=
′2cos
2 2
2
γ. (9)
Expanding z′ in powers of 21 γ , to the lowest order z ′ is
( )[ ]zzkk
cKz ow
w
′+=′ 2sin4 2
2
γ. (10)
The vector potential including the laser field is now given as
Using Maxwell’s equations, t∂∂−= AE , gives us the electric field of the laser
( )LLLL tzkE φω +−= cos2xE (12)
where LL AE ω≈ , LLk λπ2= , Lλ , ω are the laser amplitude, wavelength,
and frequency. Note that we use a constant vector potential and initial phase LA
and Lφ in the following equations.
The energy equation can be obtained from the time derivative of the Hamiltonian,
tHdtdH ∂∂= , to give
Lxcm
e
dt
dEp •=
22γγ
. (13)
Substituting xp and LE into equation (13) we have
( ) ( )LLwL tzkzk
mc
eKE
dt
d φωγ
γ +−−= cossin2 . (14)
Eq. (14) can be rewritten as
( ) ( )[ ]zkmc
eKE
dt
dwLL
L 2sinsin −+−+−= φψφψγ
γ. (15)
Where ( ) tzkk wL ωψ −+= is the electron phase. For the longitudinal
position described by Eqs. (6,7,8), the phase ψ can be written, if the energy
exchange is small for a single undulator period,
ψψψ ′+= o (16)
where
+−=2
2
2
1
γψ K
kkct Lwo (17)
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and
( )oww
L zkK
k
k2sin
4 2
2
γψ =′ (18)
are the mean phase and small phase oscillations. The phase equation is then given by taking the time derivative of Eq. (17),
−=
2
2
1γγψ N
wo ck
dt
d (19)
where 2
1 22 K
k
k
w
LN
+=γ is the resonance energy.
Substituting Eq. (16) into the energy equation, (15), we have
( ) ( )
( ) ( ) ( ) ( )zkzk
mc
eKE
dt
d
wLowLo
LoLoL
2sincos2cossin
sincoscossin
−′+−−′+−
′++′+−=
ψφψψφψ
ψφψψφψγ
γ (20)
If γ and LE change slowly (in our case LE is constant), then we can average
Eq.(20) with respect to the mean position, oz , over half a wiggler period,
( ) ( )
( ) ( )
( ) ( )
( ) ( )
−+−
−+−
++
+−=
∫
∫
∫
∫
π
π
π
π
θ
θθθξφψ
θθθξφψ
θθξφψ
θθξφψπγ
γ
2
0
2
0
2
0
2
0
sinsincos
sincossin
sinsincos
sincossin2
1
d
d
d
dmc
eKE
dt
d
Lo
Lo
Lo
LoL
(21)
where 2
2
4γξ
w
L
k
Kk= , and ow zk2=θ , using the integral representation of the
Bessel function given as,
( ) ( )∫ ΘΘ−Θ=π
π
2
0
sincos21
dmuuJm (22)
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where Κ,2,1,0=m . Since sine functions average out to zero, we have
( ) ( ) ( )[ ]ξξφψγ
γγθ
10sin JJmc
eKE
dt
d
dt
dLo
L −+−=→ . (23)
Making the independent variable to be z , using the approximation, cdtdz ≈ , the 1-D IFEL equation is,
( ) ( ) ( )[ ]ξξφψγ
γ102
sin JJmc
eKE
dz
dLo
L −+−= (24a)
−=
2
2
1γγψ N
wo k
dz
d. (24b)
For a 3rd harmonic IFEL, the vector potential of the laser is
( )[ ]33 sin2ˆ φω +−= tzknA LxA (25)
where 3=n here. The electric field is then
( )[ ]33 3sin23ˆ φω +−= tzkE LL xE (26)
where 33 AE ω≈ . Following the same procedure used in obtaining Eq. (24), we
have,
( ) ( ) ( )[ ]ξξφψγ
γ333sin
32132
3 JJmc
eKE
dz
do −+−= (27a)
−=
2
2
1γγψ N
wo k
dz
d. (27b)
For an IFEL that uses a laser seed that includes up to the third harmonic gives equations of motion in 1-D as
( ) ( ) ( )[ ]( ) ( ) ( )[ ]
−++−+
−=ξξφψ
ξξφψγ
γ333sin3
sin
2133
1011
2 JJE
JJE
mc
eK
dz
d
o
o (28a)
−=
2
2
1γγψ N
wo k
dz
d (28b)
3 1-D Simulation for harmonic IFEL and Discussion
Table 1 lists the parameters used for our studies. The undulator parameter and beam energy are those used in the Stella experiment, so we used a Stella IFEL code to check our 1-D model for the fundamental laser seed only. IFEL Equations
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(24,27,28) were solved using the Euler method of integration. The equations were
converted to dimensionless units by introducing a unit length oz ,
2mc
eEz L
o = (29)
where ( ) 21rooL IE εµ= and API Lr = , where rI is the irradiance, LP is
laser power, and A is the cross-section of the laser. For the third harmonic only,
33 LEE = . For the third and fundamental harmonics, LE is defined as,
31 EEEL += (30)
and 313 EE = so that ( ) LEE 431 = . The initial distribution of electrons is
generated in such way that it is uniform in phase and Gaussian centered about the resonant energy (Fig.1). For a fixed laser power, the drift distance after the undulator was varied (20 cm to 1 meter) to optimize the bunching.
Table 1. Parameters for harmonic IFEL simulation.
value description
wλ 3.3 cm Undulator period
Nγ 85.76 beam energy
Lλ 10.6 Fundamental IFEL laser wavelength
Lw 26.4 cm Undulator length
K 1.93 Undulator parameters
PL 150 MW Laser power
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0.6 0.4 0.2 0 0.2 0.4 0.60.02
0.015
0.01
0.005
0
0.005
0.01
0.015
0.02
ran
0.0015 0.001 5 104
0 5 104
0.001 0.00150
20
40
60Initial Energy Distribution
f3
int3
0.6 0.4 0.2 0 0.2 0.4 0.60
20
40Initial Phase Distribution
f4
int4
Figure 1. Electron Beam Initial distribution, phase space (top), energy spectrum (middle), bunch distribution (bottom).
Figure 2. Electron Beam distributions after the fundamental IFEL (left) or third harmonic IFEL (right) using a single undulator. Phase space (top), energy spectrum (middle), bunch distribution (bottom).
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Figure 3. Electron beam distribution for 1st and 3rd harmonic IFEL (left). 1st harmonic IFEL preceding a 3rd harmonic IFEL (right). Phase space (top), energy spectrum (middle), bunch distribution (bottom).
Two configurations of the IFEL harmonic buncher were considered. In the first configuration, only one undulator is employed. In the second configuration, two undulators were employed. For the first configuration, single frequency interactions are shown in figure 2 for the fundamental (left) and third harmonic (right) IFEL. Also, in the first configuration, a combined frequency interaction of fundamental and third harmonics is shown in figure 3 (left). For the second configuration, single frequency interactions are used for each stage of the two undulators. Figure 3 (right) shows the beam distribution after using a fundamental IFEL for the first stage followed by a third harmonic IFEL for the second stage. The third harmonic laser power is about 10% of the total laser power for both configurations. Comparing fig.2 and 3 we see that the combined harmonic IFEL and the staged IFEL improves the bunching by about one order of magnitude. 1-D simulations have shown significant improvement of electron beam micro-bunching using a harmonic IFEL. We are now planning a third harmonic IFEL experiment at the BNL ATF using a 1 µm photocathode RF gun driving laser. The second stage of the experiment will demonstrate harmonic IFEL bunching using both 1 µm and its third harmonic in a single stage harmonic IFEL. We are also planning to inject a harmonic IFEL bunched electron beam into an IFEL accelerator to reduce the energy spread, then use the accelerated beam to study high gain ultra-high harmonic FEL.
0.6 0.4 0.2 0 0.2 0.4 0.60.02
0.01
0
0.01
0.02
i2
i2
0.02 0.015 0.01 0.005 0 0.005 0.01 0.015 0.020
50
100Exit Energy Distribution
E2hist
Eint
0.6 0.4 0.2 0 0.2 0.4 0.60
100
200Exit Phase Distribution
hist2
lower2 upper2
int2
0.6 0.4 0.2 0 0.2 0.4 0.60.015
0.01
0.005
0
0.005
0.01
0.015
i2
i2
0.015 0.01 0.005 0 0.005 0.01 0.0150
10
20Exit Energy Distribution
E2hist
Eint
0.6 0.4 0.2 0 0.2 0.4 0.60
50
100
150Exit Phase Distribution
hist2
lower2 upper2
int2
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4 Acknowledgements
We would like to thank the ATF and Dr. I. Ben-Zvi for providing fund to the Stony Brook Research Foundation to support this study. The discussions with D.C. Quimby, W. D. Kimura of STI, and A. van Steenbergen are gratefully acknowledged. We would like also to thank D.C. Quimby and STI for making their IFEL computer program available for this study. This work is supported by US DOE under contract No. DE-AC02-98CH10886.
References
1. R. Palmer, J. Appl. Phys. 43, 3014(1972). 2. John M. Madey, J. Appl. Phys. 42, 1906 (1971). 3. E. D. Courant, C. Pellegrini, W. Zakowicz, Phys. Rev. A 32, 5, 2813 (1985). 4. A. Fisher, J. Gallardo, J. Sandweiss and A. van Steenbergen, Nucl. Instrum.
Methods Phys. Res. Sect. A 341 ABS 111(1994). 5. A. van Steenbergen, J. Gallardo, J. Sandweiss, J.-M Fang, M. Babzien, X. Qiu,
J. Skaritka and X.J. Wang, Phys. Rev. Lett. 77, 2690 (1996). 6. L.H. Yu et al, Science, 289, 932-936 (2000). 7. V.V. Goloviznin and P.W. van Amersfoot, Phys. Rev. E 55, 6002(1997). 8. R. Bonifacio, C. Pellegrini and L.M. Narducci, Opt. Commun. 50, 373 (1984). 9. R. W. Schoenlein et al, Science, 287, 2237-2240 (2000). 10. A.A. Zholents and M.S. Zolotorev, Phys. Rev. Lett. 76, 912 (1996). 11. A. Mikhalichenko and M. Zolotorev, Phys. Rev. Lett. 71, 4146 (1993). 12. Y. Liu, X.J. Wang, D. Cline, M. Babzien, J.M. Fang, J. Gallardo, K. Kusche, I.
Pogorelsky, J. Skaritka, and A. van Steenbergen, Phys. Rev. Lett. 80, 20, 4418 (1998).
13. W.D. Kimura et al, “STELLA Experiment: Design and Model Predictions”, BNL-66297 (1998).
14. W. B. Colson, IEEE J. Quantum Electron. QE-17, 1417 (1981).
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TERAWATT LASER PULSE MULTIPLEXING FOR COMPTON SCATTERING X-RAY SOURCES
D. U. L. YU, a D. NEWSHAM, a J. ZENG, a A. SMIRNOV, a F. V. HARTEMANN,b A. L. TROHA, b,c A. LE FOLL,b,d D. J. GIBSON, b,c AND H. A. BALDIS b,c
aDULY Research, Inc., Rancho Palos Verdes, CA 90275 bInstitute for Laser Science & Applications, Lawrence Livermore National Laboratory,
Livermore, CA 94550 cDepartment of Applied Science, University of California, Davis CA 95616
dPermanent Address: Ecole Polytechnique, 91128 Palaiseau, France
1 Introduction
Remarkable advances in ultra-short pulse laser technology based on chirped-pulse amplification (CPA), 1-4 and the recent development of high-brightness, relativistic electron sources 5-7 allow the design of novel, compact, monochromatic, tunable, femtosecond x-ray sources using Compton scattering. 8-21 Such new light sources are expected to have a major impact in a number of important fields of research, including the study of fast structural dynamics, 22-25 advanced biomedical imaging, 26 and x-ray protein crystallography; 27 however, the quality of both the electron and laser beams is of paramount importance in achieving the peak and average x-ray spectral brightness required for such applications, and the average brightness of the source is also a crucial parameter.
In this paper, novel optical methods aimed at enhancing the average brightness of Compton x-ray sources are proposed, and a proof-of-principle experimental realization of such a multiplexing laser system is described. Our goal is to recycle the laser light produce by a CPA system, in order to take advantage of the fact that, in Compton scattering, the energy of the scattered light is extracted from the electron beam kinetic energy rather than from the drive laser pulse energy. This can easily be seen by considering the 4-momentum radiated away:
23 ,( )µ
µ µ µτ =d G u a a (1)
where we have used electron units, and where µ µτ=u d x is the electron 4-velocity
along its world-line ( );µ τx a d uµ µτ= is the corresponding 4-acceleration. In
electron units, mass is measured in units of 0,m charge in units of ,e length in units
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of the classical electron radius 15
02.818 10 m,r −= × and time in units of
0/ .r c
Equation (1) shows that the radiated energy-momentum scales quadratically with the electron acceleration, but also depends linearly upon the electron 4-velocity,
( )1, .uµ = γ β This, in turn, shows that the relativistic Doppler effect stretches the
dipole radiation pattern of the transversally accelerated electron in the direction of the electron beam; the resulting recoil slows down the electrons, thereby extracting their kinetic energy. Therefore, the drive laser pulse energy is not utilized in the interaction, and can be recycled for further scattering with another electron bunch.
With this in mind, our approach is to multiplex the drive laser pulse, either actively, using a ring where the pulse is trapped, conditioned, circulated, and can be re-amplified, or passively, where the pulse is circulated into a confocal mirror system; in this case, use of a periscope to rotate the polarization of the input pulse and a broadband thin film polarizer (TFP) allows for up to 20 passes at the focal interaction point. For our experimental demonstration, we have used a 7-pass confocal system, which has resulted in 14 pulses at the interaction point.
This paper is organized as follows: in Sec. 2, we briefly describe the overall Compton scattering system, including the rf linac and the laser system, and we present three-dimensional computer simulations of a Compton x-ray source using an X-band plane-wave transformer linac and a tabletop terawatt laser; in Sec. 3, the passive ring system and the corresponding experimental results are discussed, as well as the active ring system developed by DULY Research, Inc.; 28 finally conclusions are drawn in Sec. 4.
2 RF Linac and Laser System for Compton Scattering and Computer Simulations
The overall schematic of the system is given in Fig. 1. Because the interaction involves sub-picosecond laser and electron pulses, synchronization is key for such a
Compton scattering source. While a number of options have been proposed, the use
Figure 1 Schematic of Compton scattering x-ray source.
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of an rf photoinjector to produce high-quality, low emittance, relativistic electron bunches, with excellent synchronization with an external laser appears to be the most promising: in such a device, the electron are produced by an incoming laser pulse via the photo-electric effects, and accelerated by strong (100 MV/m) rf fields in a linac structure.
We have performed preliminary computer simulations of the x-rays produced by Compton scattering a tabletop, terawatt laser pulse off a relativistic electron beam produced by an X-band plane-wave transformer (PWT) linac, currently being developed by DULY Research, Inc. We first briefly describe the theoretical background underlying the three-dimensional computer Compton scattering code developed by F.V. Hartemann and A. Le Foll; we then present the architecture of the code, and computer simulations where the electron beamline is modeled using the PARMELA-SUPERFISH codes.
Our first task is to demonstrate the Hartemann-Le Foll (HLF) theorem: in the linear regime, where the 4-potential amplitude satisfies the condition 0/ 1,!eA m c and in the absence of radiative corrections, 20,21 where the frequency cutoff is
20 / ,ω ! hm c as measured in the electron frame, the spectral photon number
density scattered by an electron interacting with an arbitrary electromagnetic field distribution in vacuum is given by the momentum space distribution of the incident vector potential at the Doppler-shifted frequency:
( )
( )( ) ( )
2
2
300 04 2
00
11 , exp .
2
sx
s
s sss
d N k
d d
i d
µ
ω Ω
α= × + ⋅ ω − ⋅ − ⋅ κ γγ ωπ ∫! % uk
k u A k k k k x k3s
(2)
Here, ( ) ( )ˆ, 1,µ = ω = ωss s sk nk is the 4-wavenumber of the wave scattered in the
observation direction ˆ ,n at the frequency ;sω 20/ 2 1/137.036α = ε !e hc is the
Figure 2 Compton scattering interaction geometry.
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fine structure constant; ( )00 0,µ = γu u is the electron initial 4-velocity; ( )0
00,µ =x x
is its initial 4-position, and we have introduced the scattered light-cone variable,
0 0 0 .= − = − ⋅µµκ γ ωs
s s su k u k The term ( ) 01+ κ ⋅ / sk u is to be considered as an
operator acting on the Fourier transform of the spatial components of the 4-potential, ( ), ,µ =A V A
( ) ( ) ( )4
44
1exp ,
2
ν νµ ν µ ν ν=
π∫ !%A k A x ik x d k (3)
while the term ( )0exp ⋅ik x is shown to give rise to the coherence factor. 29,30
The HLF Theorem is then applied to the specific case of Compton scattering 15-
21 in a three-dimensional Gaussian-elliptical focus. 31,32 The effects of the electron beam phase space topology are also included, in the form of energy spread and emittance. 33-35
In our analysis, charge is measured in units of e, mass in units of 0 ,m length is
normalized to a reference wavelength, 10 ,−k while time is measured in units of the
corresponding frequency, ( ) 110 0 .
−−ω = ck Neglecting radiative corrections, 20,21,36 the
electron motion is governed by the Lorentz force equation
( )/ .νµ µ ν ν µτ = − ∂ −∂du d A A u Here, /µ µ= τu dx d is the electron 4-velocity along
its world-line, ( );µ τx τ is the electron proper time; µA is the 4-potential from
which the electromagnetic field derives; finally, ( ),µ∂ = −∂ ∇t is the 4-gradient
operator. 37,38 The electromagnetic field distribution considered here corresponds to a vacuum interaction; therefore, the 4-potential satisfies the wave equation,
0 ,νµ ν µ µ = ∂ ∂ = !A A and can be expressed as a superposition of plane waves, as
described in Eq. (3). Furthermore, we choose to work in the Lorentz gauge, where 0.µ
µ∂ =A Here, the symbol ( )0 0,µ = 0 represents the null 4-vector.
Introducing the maximum amplitude of the 4-potential, A, we can linearize the Lorentz force equation, provided that 1,!A a condition which is typically satisfied in most experimental situations: for example, in the case of an ultrahigh intensity laser focus, this condition translates into a maximum intensity below 1017 W/cm2 for visible wavelengths. We then write 0 1 2 ...,µ µ µ µ= + +u u u u where ,µ ∝
n nu A and the
Lorentz force equation yields, to first order,
( )1
0 .µ νµ ν ν µ− ∂ −∂
τ!
duA A u
d (4)
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To solve Eq. (4), we Fourier transform the first-order 4-velocity perturbation into momentum space and use the orthogonality of complex exponentials. Fourier transforming this result back into space-time, we finally obtain
( ) ( ) ( ) ( )4
00 44
0
1exp .
2
νν νν ν
µ µ ν µ ν µ ννν
+ −
π ∫ !
%%!
A k uu x u d k A k k ik x
k u (5)
The linearization procedure used here is manifestly covariant.
We now consider the second half of the demonstration of the HLF Theorem. The electromagnetic radiation scattered by the accelerated charge is described by the number of photons radiated per unit frequency, per unit solid angle, which is determined by Fourier transforming the electron trajectory into momentum space: 39
( ) ( ) ( )
2 2
2ˆ exp .
4
+∞µ µµ−∞
αω = × τ − τ τ ω Ω π ∫ us
x ss
s
d N kik x d
d dn (6)
Here, we have used the fact that n is a unit vector to simplify the double cross product. The spatial component of the electron 4-velocity is replaced by the linearized solution given in Eq. (5); with this, and now using the Coulomb gauge, we have
( )
( )( ) ( ) ( ) ( )
2
2
4032
1exp .
4
sx
s
ss
s
d N k
d d
d d k k k i k k x
µ
+∞ µµ µ µ µ µ−∞
ω Ω
α = × τ + ⋅ − τ ω κ π∫ ∫ ! % %k
k A u A4
(7)
The electron 4-position is now approximated by ( ) 0 0 ;µ µ µτ + τ!x x u this corresponds
to the lowest-order convective term due to the ballistic component of the electron motion, and excludes harmonic production mechanisms; 16-21 this approximation is valid for high Doppler-shift scattering, where the transverse oscillation scale is given by 0 0 0/ .λ γ λ!A Using the light-cone variables, sand ,κ κ we find that
( )
( ) ( )4
2
2
40 06
exp exp .64
sx
s
ss s
s
d N k
d d
d k i k k x d i
µ
+∞µµ µ µ −∞
ω Ω
α = × + ⋅ − τ κ − κ τ κπ ω ∫ ∫!
% %kk A u A
(8)
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The integral over proper time yields a δ-function: ( )exp sd i+∞
−∞τ κ − κ τ ∫
( )2 .s= πδ κ − κ We now perform the change of variable 14 4/ ,
−µ ν µ ν= ∂ ∂d k q k d q
where we have introduced the 4-vector ( ) ( )0, ,q k u kµν µ µ = κ = k k
( )0 0 , ,= γ ω− ⋅u k k which allows us to perform the integral over the δ-function; we
then find that the Doppler condition derives from the equality of
0 0 0 ,µµκ = − = γ ω − ⋅s
s s su k u k and 0 0 0 :µµκ = − = γ ω− ⋅u k u k this yields the well-
known Compton scattering relation ( ) ( )0 0 0 0 ˆ/ ,ω = γ ω− ⋅ γ − ⋅s nu k u in the limit
where recoil is negligible. Finally, using the Jacobian of the transform, 1 1
0/ ,− −
ν µ∂ ∂ = γq k we obtain the HLF Theorem.
The HLF Theorem is now applied to the case of a three-dimensional laser focus. The transverse laser profile is specified at the focal plane, and propagated using the method discussed in Ref. 31, where the vector potential derives from a generating function: ;= ∇×A G in this manner, the Coulomb gauge condition,
0,∇ ⋅ =A is automatically satisfied. For a linearly polarized Gaussian-elliptical
focus, with focal waists 0 0and ,x yw w and a monochromatic wave at the central
frequency 0 1,ω = with a Gaussian envelope of duration 0 ,ω ∆ = ∆φt the 4-potential is represented in momentum-space by
( ) ( )
( ) ( )
22200
0 0 0
2 2 2
1exp
2 2 2 2
ˆ ˆ .
y yx xx y
z x y z x
w k tw kA k A w w t
k k k i k k
µ ν
∆ ω− π = ∆ − − −
×δ − ω − − − + x z
%
(9)
Here, we recognize the -spectrum,⊥k the frequency spectrum, the propagator,
( ),µµδ k k and the curl operator, as expressed in momentum space.
The scattered radiation can now be determined by using the HLF Theorem; to obtain an analytical result than can be further exploited to include the phase space topology of the electron beam interacting with the laser pulse using the method outlined above, the paraxial propagator formalism 31,32 is used: the phase function
( )2 2 2 ,δ − ω − −z x yk k k is replaced by ( ) ( )2 20 0/ 2 / 2 . δ −ω+ + z x yk k k k k The
accuracy of the paraxial approximation has been studied in detail, 31,32 and found to be extremely good over a wide range of parameters; however, in the case of Compton scattering, the following conditions must also be satisfied:
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( )2 2 2 20 0 , 0 , . κ > ∆φ κ − κ − x y z s x yw u u With this proviso, the integrals over the
transverse wavenumber components converge, and can be performed analytically: 41 we use the well-known integral of the exponential of a complex, second-order polynomial,
2 2
2 2 22 2 2
2 22 2
( 2 ) ( 2 )
1 ( ) ( 2 )( ) ( 2 ) arctan2
2 24,
ax bx c i px qx r
p p q pr b p abq a ra b ac aq bq cp ia a pa p
e e dx
e ea p
+∞ − + + + +
−∞
− − − +− − − + − + + π=+
∫ (10)
and the fully three-dimensional x-ray spectral brightness is now obtained as a function of the electron initial position and velocity, 0 0and ,x u as well as the laser
parameters, and the scattered 4-wavenumber, ( )ˆ, .µ = ω ωss sk n Frequencies are
normalized to the laser pulse central wavelength, 0 0 1,= ω =k and axial positions
are measured from the laser focal plane, lying at 0.=z The complete result is quite complex, and was tracked analytically using Mathematica.42
In order to fully exploit the results derived here, we have developed a three-dimensional code describing the radiation scattered by a distribution of eN point
charges having the same charge-to-mass ratio as electrons. The 6 -dimensionaleN phase space is generated by randomly loading the particles in prescribed statistical distribution, or can be the output of an electron beam optics code, such as PARMELA. The 6 -dimensionaleN phase space can be modeled using Gaussian
distributions: in that case, a given particle, numbered 1 ,≤ ≤ ei N is assigned a
random position, ( )0 0, ,i ix u in phase space; its charge is then scaled as
2 2 20 0 00 0 0
22 200 000 0
exp
,
i i ii
yi yxi x zi z
x y z
x x y y z zq
x y z
u uu u u u
u u u
− − −= − − − ∆ ∆ ∆ − − − − − − ∆ ∆ ∆
(11)
to reflect the probability distribution. Here, 0 0and x y now correspond to transverse
spatial offsets, 0z represents timing jitter, and ∆ ∆x y are the electron bunch waist
size, while ∆z is the bunch duration; 0u corresponds to the electron beam
incidence, while ∆ xu and ∆ yu are related to the beam horizontal and vertical
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emittance; finally, ∆ zu represents energy spread. Note that far from head-on
collisions, one needs to perform the appropriate projections to correctly relate the quantities discussed above to the conventional parameters describing the electron
beam phase space. The code keeps track of the total charge, 1
,=∑ eN
iiq and after
performing the incoherent summation over all particles, the charge is re-scaled to the desired value.
In running the code initially, it was determined that good statistical convergence is obtained for 43 10 ;≥ ×eN generally, we have used 50,000 particles in the results presented here. Two different types of output data files are created by the code: spectral brightness measured at a prescribed scattering angle, or angular maps at a specified x-ray frequency; for angular maps, the code also integrates the flux over the map, in a small (typically 1 eV) x-ray photon energy interval. This last result is important for x-ray protein crystallography and other applications.
Figure 3 Left: transverse electron bunch distribution at focus. Right: corresponding transverse momentum distribution.
The case of a realistic beam, simulated with PARMELA, is presented in Figs. 3 to 6. The laser parameters are as follows: 0 800 nm,λ = 0 50 m,w = µ 200 fs,∆ =t and a pulse energy of 50 mJ. The electron bunch energy is 25.1 MeV, its charge is 0.5 nC, its duration is 0.65 ps, the relative energy spread is 0/ 0.38%,∆γ γ = the
beam normalized emittance is 1 -mm mrad,π ⋅ and the focal spot size matches the laser focal distribution. A repetition rate of 1 kHz is used to scale the average spectral brightness of the source; the maximum brightness compares well with that produced by bend magnets on synchrotron beamlines. The angle of incidence is 180o, and the spectra are observed on-axis; the angular maps are obtained at the spectral maximum. One-dimensional, cold beam results are shown for comparison, and clearly demonstrate the importance of the theoretical model developed here, as the warm, three-dimensional brightness is seen to be considerably smaller than that predicted by a simple one-dimensional theory.
-0.016 -0.008 0.000 0.008 0.0160
50
100
150
200
250
T ransverse Position, x (cm ) Transverse M om entum , px /m 0c
-0.12 -0.06 0.00 0.06 0.120
2
4
6
8
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We also note that the laser-driven Compton source can produce much shorter x-ray flashes than those currently generated at synchrotrons: sub-100 fs pulses will be readily produced, in contrast with FWHM in the 35-100 ps range at the Advanced Light Source (ALS) and 170 ps at the Advanced Photon Source (APS). 43
Figure 4 Left: axial electron bunch distribution at focus. Right: corresponding axial momentum distribution.
3 Laser Ring Experiments
The CPA laser system used to drive our experiment consists of a short pulse oscillator (90 nm optical bandwidth, 10 fs pulses), using mirror-controlled dispersion, and a 1 mJ, 1 kHz, CPA system, with a minimum output pulse duration below 40 fs. The compressor in the amplifier can be de-tuned to yield pulses with a
38.02 38.04 38.06 38.08 38.100
5
10
15
20
25
Axial Position, z (cm)
49.4 49.7 50.0 50.3 50.60
2
4
6
Axial Momentum, pz/m0c
0.0E+00
5.0E+07
1.0E+08
1.5E+08
2.0E+08
15.0 15.2 15.4 15.6 15.8 16.0
X-Ray Energy (keV)
Ave
rage
On-
Axi
s B
right
ness
(s.
u.)
Figure 5 Average, on axis x-ray
brightness 2 2( / / / 0.1% / ).xN mm mrad bw s
Figure 6 X-ray angular distribution, with a 1 eV filter at 15.6 keV.
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duration > 200 fs. We now briefly describe the experimental setup, which is illustrated in Fig. 7. The compressed output of the Spitfire is nominally > 0.7 mJ, 55 fs, at 1 kHz. The pulse is s-polarized and reflects off the TFP, with a reflection coefficient, R = 75%. The pulse travels in the ring and is focused 7 times in the 7-pass confocal mirror setup. The current radius of curvature is 1 m, but can easily be changed. The pulse exits the 7-pass setup, collimated again, and its rotation is switched to p by the periscope; the pulse now transits through the TFP with a transmission coefficient, T > 99%. The p-polarized pulse travels again through the 7-pass, now having produced 14 high-intensity foci, and is rotated back to s-polarized by the periscope; it is then reflected off the TFP for diagnostics. We also note that an electro-optic (EO) shutter can be inserted in the ring: in this case, the shutter would be off after the first trip, to allow for re-injection through the TFP; after the second pass through the confocal system, once the polarization has been switched back to s by the periscope, the EO can be turned on, so that after the EO, the pulse is back to p-polarized and trapped in the ring. For each additional turn in the ring, 7 high-intensity foci are produced. When the EO is shut off, the pulse is reflected out on the TFP, and can be captured for diagnostics.
In the absence of the EO modulator, because we use mostly reflective optics, dispersion is small and B-integral effects are negligible. The main limitation is then the peak intensity on the various optics; however, in the fully passive configuration, the system is both very simple to align and quite robust. If the EO modulator is inserted, the peak intensity is limited to 1 GW/cm2 at the EO switch; furthermore, for high pulse energies, stretching would be required before the EO switch, as well as compression after, in order to limit the B-integral effects: for a 0.1 mJ, 50 fs pulse, with a beam radius near 5 mm, propagation through 2 cm of KDP resulted in a temporal broadening to 400 fs, due to nonlinear self-phase modulation, as illustrated in Fig. 9 using a single-shot auto-correlator.
The main issues for the passive ring are the following: first good spatial overlap of the individual foci is required; this as been measured using a wire scan at low intensity, as shown in Fig. 8; second the temporal spacing of the pulses must be carefully adjusted, so that interaction with a train of electron bunches can be properly synchronized; third the laser pulse quality must be maintained over the ring. Because high-damage threshold reflective optics are used in the system, most of the problems of dispersion, and spatial and spectral degradation are easily managed; the synchronization can be ensured by carefully controlling the distance between the various mirrors in the ring; furthermore, a small delay line can be inserted to guarantee that the re-circulation delay matches the electron bunch train format, and corresponds to a sub-harmonic of the rf system used to accelerate the photoelectrons, as illustrated in Fig. 1. Note that a 1 ps pulse has very little bandwidth and will not disperse; the 55 fs pulse is much better for measuring roupe velocity dispersion (GVD) effects, as well as nonlinear B-integral effects. The intensity is maintained below the damage threshold of the EO shutter.
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Figure 9 Left: single-shot auto-correlation trace of Spitfire output pulse. Right: same, after propagating through 2 cm kDP.
With this experimental setup, we were able to produce pulse trains of 14 individual pulses, as illustrated in Fig. 10. The measurement technique consists in using a pellicle near the location of the overlapping foci, to reflect part of the signal
55 fs
0
0.2
0.4
0.6
0.8
1
-150 -50 50 150
Position (um)
Inte
nsity
(a.
u.)
Figure 7 Schematic of laser multiplexer with integrated 7-pass setup.
Figure 8 Wire scan of 7 overlapping foci.
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to a fast avalanche photodiode. This measurement is performed at low power to avoid breakdown and damage to the pellicle, and the decrease of the pulse intensity observed in Fig. 10 corresponds to the de-Qing of the optical resonator by the pellicle, which couples a significant fraction of the laser pulse out of the ring. Without the pellicle, the losses for 7 passes where measured with an energy meter, and found to be less than 4%; some additional losses are due to the quarter-wave plate and re-injection optics, but the pulse amplitude can be maintained near 75% of its original value after 14 passes. Furthermore, as the ring is passive, we have confirmed that the auto-correlation traces of the input and output (after 14 passes, the pulse is automatically switched out by the TFP) pulses are nearly identical, as well as their spectra. The simplicity of the technique, and its relative ease of implementation make it a good candidate to improve the overall repetition rate of a Compton scattering x-ray source by a factor of 20. Of course, for this technique to be useful, one needs to match the drive laser pulse format with that of the electron beam. We believe that this should be feasible because, the typical duration of the rf pulse powering the linac is of the order of a few µs; as compared to the duration of the laser pulse trains, which is only a few tens of ns. In fact, one should minimize the rf pulse duration, to alleviate cooling problems, and the corollary frequency stabilization of the linac; increase the overall system efficiency; and minimize the degradation of the rf accelerating structure due to dark current at high accelerating gradient, near 100 MeV/m.
We have also performed preliminary experiments designed to test the validity of the active ring concept. As mentioned above, one of the key difficulties in that case is the large value of the B-integral, when operating at high peak powers. One possible way to alleviate this problem is to use a stretched pulse, in which case the nonlinear effects become negligible, and only GVD effects are to be taken into account. In such a scheme, the pulse would be stretched to propagate through the EO shutter, and recompressed before being focused in the ring. The peak fluence on the EO shutter is also reduced in this approach, and the only difficulties are linked
Figure 10 Double laser pulse train measured near the overlapping foci with a pellicle; coaxial cable dispersion is visible.
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with the fact that the compressor efficiency is only 50%, which implies that the pulse must be re-amplified during each re-circulation; in that case, gain narrowing should also be considered carefully.
Our initial work with an active ring has shown that we could obtain up to 10 re-circulation passes, which now yields a total number of 70 passes at the focus. However, because of the aforementioned B-integral problems, our demonstration was performed at low power, in contrast with the passive ring results presented in this paper. Further work in this direction is planned.
4 Conclusions
In conclusion, we have presented detailed three-dimensional simulations of a Compton scattering x-ray source using an X-band PWT under development at DULY Research, Inc., and a tabletop terawatt laser, which could use laser pulse multiplexing to achieve the high repetitions rates required by some applications. We have also discussed experiments pertaining to demonstrate such laser pulse multiplexing, at peak powers relevant for Compton scattering. Our main results include the successful re-circulation of 14 lasers pulses in a passive ring, with energies up to 1 mJ, and a typical duration of 55 fs; since the laser amplifier has a 1 kHz repetition rate, these experiments have demonstrated an effective laser repetition rate near 15 kHz, at the mJ level. The spectral content and auto-correlation traces of the pulses remained nearly unchanged in the passive ring system, and good focal overlap was obtained.
References
[1] M.D. Perry and G. Mourou, Science 264 (5161), 917-924 (1994). [2] G.A. Mourou, C.P.J. Barty, and M.D. Perry, Phys. Today 51 (1), 22-28 (1998). [3] C.P.J. Barty, et al., Opt. Lett. 21, 668 (1996). [4] D.P. Umstadter, C. Barty, M. Perry, and G.A. Mourou, Opt. Phot. News 9 (1), 41 (1998). [5] Advanced Accelerator Concepts, 8th Workshop, edited by W. Lawson, C. Bellamy, and D.F. Brosius, American Institute of Physics, Conference Proceedings No. 472, Woodbury, NY, 1999. [6] D. Yu et al., Proc. Particle Accelerator Conference 1999, 2003 (1999). [7] S.G. Biedron, et al., Proc. Particle Accelerator Conference 1999, 2024 (1999). [8] R.W. Schoenlein, et al., Science 274 (5285), 236-238 (1996). [9] W.P. Leemans, et al., Phys. Rev. Lett. 77, 4182 (1996). [10] W.P. Leemans, et al., IEEE J. Quantum Electron. QE11, 1925 (1997). [11] C. Bula, et al., Phys. Rev. Lett. 76, 3116 (1996). [12] D. Burke, et al., Phys. Rev. Lett. 79, 1626 (1997). [13] C. Bamber, et al., Phys. Rev. D60 (9), 092004/1-43 (1999).
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[14] V.N. Litvinenko, et al., Phys. Rev. Lett. 78, 4569 (1997). [15] W. Greiner and J. Reinhardt, Quantum Electrodynamics, Springer-Verlag, Berlin, 1994, Chap. 3.7. [16] E. Esarey, S.K. Ride, and P. Sprangle, Phys. Rev. E48, 3003-3021 (1993). [17] F.V. Hartemann, et al., Phys. Rev. E54, 2956-2962 (1996). [18] S.K. Ride, E. Esarey, and M. Baine, Phys. Rev. E52, 5425-5442 (1995). [19] E. Esarey, P. Sprangle, and J. Krall, Phys. Rev. E52, 5443-5453 (1995). [20] F.V. Hartemann and A.K. Kerman, Phys. Rev. Lett. 76, 624-627 (1996). [21] F.V. Hartemann, Phys. Plasmas 5, 2037-2047 (1998). [22] C.W. Siders, et al., Science 286 (5443), 1340 (1999). C. Rose-Petruck, et al., Nature 398, 310 (1999). [23] A.H. Chin, et al., Phys. Rev. Lett. 83, 336 (1999). [24] A.M. Lindenberg, et al., Phys. Rev. Lett. 84, 111 (2000). [25] R.A. Robb, Three-Dimensional Biomedical Imaging: Principles and Practice, Wiley-VCH Publishers, New York, NY, 1995. [26] R. Fitzgerald, Phys. Today 53 (7), 23 (2000). [27] J. Drenth, Principles of Protein X-Ray Crystallography, 2nd Ed., Springer-Verlag, New York, NY, 1999. K.E. van Holde, W. Curtis Johnson, and P. Shing Ho, Principles of Physical Biochemistry, Prentice-Hall, Inc., Upper Saddle River, NJ, 1988, Chap. 6. [28] D.U.L. Yu and D. Bullock, US Patent No. 5,701,317 (1997). [29] F.V. Hartemann, Phys. Rev. E61, 972-975 (2000). [30] M.J. Hogan, et al., Phys. Rev. Lett., 81, 4867-4870 (1998). [31] F.V. Hartemann, et al., Phys. Rev. E58, 5001-5012 (1998). [32] B. Quesnel and P. Mora, Phys. Rev. E58, 3719-3732 (1998). [33] M. Reiser, Theory and Design of Charged Particle Beams, John Wiley and Sons, New York, NY, 1994, Chaps. 3 and 6. [34] H. Wiedemann, Particle Accelerator Physics, Vol. 1, 2nd Ed., Springer, New York, NY, 1999, Chaps. 5 and 8. [35] B.E. Carlsten, Nucl. Instrum. Methods Phys. Res. A285, 313-319 (1989). [36] P.A.M. Dirac, Proc. R. Soc. London, Ser. A167, 148 (1938). [37] W. Pauli, Theory of Relativity, Dover, New York, NY, 1958, Secs. 19, 27, 28. [38] R.P. Feynman, R.B. Leighton, and M. Sands, The Feynman Lectures on Physics, Vol. 2, Addison-Wesley, Reading, MA, 1964, Chap. 25. [39] J.D. Jackson, Classical Electrodynamics, 2nd Ed., John Wiley and Sons, New York, NY, 1975, Chap. 14. [40] F.V. Hartemann, et al., accepted for publication in Phys. Rev. E. [41] I.S. Gradshteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, 4th Ed., Academic Press, Orlando, FL, 1980, Eqs. 3.923, 3.924, and 3.323.3. [42] S. Wolfram, The Mathematica Book, 3rd Ed., Wolfram Media, Champaign, IL, 1996. [43] D. Atwood, Soft X-Rays and Extreme Ultraviolet Radiation, Cambridge University Press, Cambridge, U.K., 1999.
We describe a possible accelerator system to provide an ion beam of extremelylow emittance and low current. It can used as a nano beam source for singleion implantation onto semiconductors and for the systematic study of radiationdamage in bio-molecules and semiconductors. The whole system basically has aradio-frequency quadrupole configuration with a Paul ion trap as theinjector. Ionsare confined and cooled in the trap section, and then accelerated to ∼MeV throughan RFQ linac. Design speculation is made assuming acceleration of 7Li+ ions.
1 Introduction
An abbreviation RFQ for “radio-frequency quadrupole” reminds atomic physi-cists of a Paul trap1 and accelerator physicists of a kind of ionlinac2. Bothhave the cross-section as shown in Fig. 1(a). Particular ions are radiallyconfined with an RF potential of quadrupole symmetry generated by fourelectrodes. The transverse dynamics is governed by the Mathieu equationwhile the longitudinal dynamics are somewhat different in two systems. In alinear trap, axial confinement is usually achieved by applying a DC voltage totwo end electrodes which yield a longitudinal potential well3. As to an RFQaccelerator, the tip of each electrode vane has a “modulation” as illustratedin Fig. 1(b) to produce an additional axial electric field for beam acceleration.
This paper proposes a possible ion accelerator combining these two RFQ’s.It is used as a high-energy microbeam source4,5,which may be considered asa crystal-beam source8.A small number of heavy ions such as 7Li+ are laser-cooled in the initial trap region and then transferred into the following RFQlinac. The output energy is ∼1MeV/nucleon. In the present system, trappedsingle-species plasma is quickly cooled with a laser light to the milli-Kelvinrange or even a lower temperature.A Coulomb crystals is then formed as al-ready confirmed experimentally in several laboratories9,10. The correspondingcrystalline structure should be a one-dimensional (1D)“string;” namely, indi-vidual ions are aligned along the axis of thequadrupole symmetry so thatthe beam cross section is minimum.The ions are diagnosed by the fluores-
qabp˙1: submitted to World Scientific on November 5, 2001 1
Figure 1. (a) End view of the RFQ. The dc-biased RF voltages U ± V cos ωt areapplied to the pairs of electrodes facing to each other. The RF voltages of adja-cent electrodes are 180 degrees out of phase. (b) Side view of two counter-facingelectrodes of the RFQ accelerator; ion source section (left) and linac section (right).Ions are first stored and cooled in the trap section between the two end electrodes,pre-accelerated by the acceleration electrode, and injected into the linac section.
cence from excited ions11. The ultracold ions are pre-accelerated by anotherelectrode, and injected into the linac section. A similar idea of using a trapsystem as an ion source has previously been discussed for the production ofhighly charged ion beams while no cooling was considered12.Microbeams andnanobeams have recently been employed for the study of radiation damagein cells and DNA’s4,5. They control precisely the dose to individual cells ormolecular chains by localizing the irradiation onto a tiny area. The radiationeffect is explored by the image processing of visual fields of a microscope.Similar requirements must be met in the study of radiation damage of semi-conductor elements used in rockets and artificial satellites in outer space. Thissystem can also be applied to the implantation of single ions on a semiconduc-tor tip,circuit inspection and mask repairing in semiconductors,where a beamenergy of lower than 100keV is preferred6,7. On the other hand, the radiationdamage experiments require a beam energy of at least several MeV per nu-cleon, so that the ions have the sufficiently long range in the cells. Since highbeam intensity is rather harmful in such applications, 1–10 ions/pulse withlow pulse frequency should be enough. The number of ions must be exactinstead,and their precise positioning is essential. Submicrometer ion beamshave been produced either by combination of the liquid-metal and the fieldemission ion source7 or by the multicusp ion source13. The present system issuperior in two points to these sources. First, it produces a beam with muchsmaller emittance;in other words both the beam size and the beam diver-gence can be reduced. The above-mentioned sources cannot minimize these
qabp˙1: submitted to World Scientific on November 5, 2001 2
two quantities simultaneously.Second, it produces a very low current beamwith the δ-function-like radial distribution; in other words, the beam does nothave any background. The current of the beam from conventional ion sourceshave continuous distributions13. The present paper is organized as follows. Insection 2, an ion species suitable both for cooling and acceleration is selectedand laser properties for cooling is discussed. The conditions of attaining aone-dimensional string Coulomb crystal are then examined.The relation be-tween the beam temperature in the trap and the transverse emittance in thelinac is also explored. In Section 3, we roughly calculate the fundamentalparameters of RFQ linac, checking out the evolution of beam energy and thenecessary number of the cells. How tomaintain the 1D structure during theacceleration is also discussed in this section. Finally, brief discussion is madein Section 4.
2 Ion-Trap Injector
2.1 selection of an ion species
Let us first choose an ion species suitable both for cooling and for acceleration.An ion with a closed two-level transition is preferable for laser cooling. Thelifetime of the upper state must be short. From the viewpoint of acceleration,the charge-to-mass ratio should be as large as possible. It is impractical toaccelerate singly-charged heavy ions to a final energy of several MeV/nucleonwith an RFQ linac alone. The role of the RFQ linac considered below is toraise the energy to ∼1MeV/nucleon,so that the beam can be further acceler-ated with a DTL if necessary.Needless to say, bare nuclei such as protons arenot coolable with lasers. It would be possible to cool H− ions, but no data isavailable. Helium is hard to ionize. Since the next-best choice is lithium, weconsider here 7Li+ ion as an example. Use of double-charged light ions suchas 9Be2+ may be considered, which almost doubles the gradient. It is alsoworthy to explore the applicability of the evaporative cooling, which is effec-tive to bare nuclei such as protons14. Speculations on these ideas are howeverout of the scope of this paper. At least two methods have been reported toproduce Li ions. One is the use of liquid-metal (lithium) ion source7,and theother is stripping of LiH− ions in a gas target. The latter is used for exper-iments of beam cooling in the storage ring TSR15,16.These methods produceboth 1S0 and 3S1 states of this ion. The 1s2s3S1 state of 7Li+ is metastable11,which lies about 60eV above the ground state 1S0 and decays there with arate of 1/50s−1. The 3S1 state is connected to the 1s2p3P2 state through theoptical transition having the wavelength of 548nm, realizable by a dye laser.
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The line width of the upper state is ∼ 2π×3.7MHz, corresponding to the lifetime of 43ns. The initial temperature of the ions depends on the conditionof neutral atoms as well as on the vacuum pressure of the system. If it is afew hundred K17, the approximate cooling time is less than 1ms. Therefore,even if we consider the storage time additionally, the bunch frequency above10Hz will bepossible.The fraction of metastable Li ion is very small, < 10−3,without any trick17.The vast majority of the ions, stable and unaffected bythe laser,may disrupt the cooling process via intrabeam interaction. In theTSR experiments15,18, the fraction of the metastable ions in a beam amountsto more than 20%, owing to the selective acceleration of them17. Because thistechnique is not applicable to our case, we have to seek some other means,such as the excitation of ground state ions, to increase the fraction.
2.2 1D string structure
Beam cross section becomes smallest when all the ions are aligned on thetrap axis as a1–D string. There is no problem to produce such crystals in thePaul trap.In addition, the string structure could be almost maintained duringthe acceleration in the linac section as discussed later. When the number oftrapped ions goes beyond some critical level, however, the crystalline structurein the trap converts from 1D string to 2D zigzag.At higher density, the ionsarrange themselves into a 3D helical pattern19. Suppose an infinitely longcylinder of radius R0 with a line charge density ρ. If we assume a uniformdistribution of particles,the repulsive Coulomb force acting on a particles withchargeeN at the radial position r is
Fint =eNρr/(2πε0R2
0), r < R0,eNρ/(2πε0r), r > R0.
(1)
Requiring the external confining (focusing) force Fconf = −eNrK/4πε0 tobalance with the Coulomb force at r = R0, we obtain R0 = (2ρ/K)1/2. HereK is a constant parameter representing the strength of the focusing field. Theaverage volume of the cylinder per particle is πR2
0eN/ρ = 2πeN/K. Settingthis equal to a volume of 4πr3WS/3 yields
rWS =(
3eN2K
)1/3
, (2)
which is called Wigner–Seitz radius. The 1D string structure is formed, pro-vided that the spacing of adjacent ions is greater than ∼ 1.4rWS
19. The
qabp˙1: submitted to World Scientific on November 5, 2001 4
equation of ion motion in the radial direction of the Paul trap is
mid2u
dt2+ eN(U − V cosωt)
u
r20= 0. (3)
where mi is the ion mass, u = x or y, and U − V cosωt denotes the voltagewith frequency ω applied to the electrodes. Equation (3) can be rewritten inthe form of the Mathieu equation:
d2u
dξ2+ (a− 2q cos 2ξ)
u
r20= 0, (4)
where ωt = 2ξ, a = 4eNU/(r20miω2), and q = 2eNV/(r20miω
2), with r0 beingthe minimum radius of the electrode tips. The values of ω, r0, V and U haveto be selected so that the operating point stays in the stable region in a − qspace.The special case of a = 0 or U = 0 requires 0 < q < qM ∼ 0.9. We canderive from eq.(3), Fconf ∼ −eNV r/r20, or K ∼ 4πε0V /r20, where V is theeffective value of V cosωt or V/
√2. The Wigner–Seitz radius then becomes
rWS =(
3eNr208πε0V
)1/3
. (5)
The size of a linear Paul trap9 can be on the order of mm. Assuming r0=1mmand V = 100V, we have rWS = 2.78µm for N = 1. Though this estimate isvalid only for an infinitely long trap, the recent experiment showed that it alsoexplains the phase transition in a short trap fairlywell10. The length of a stringcrystal could be longer than that of the region in which the laser intersectsthe trap axis9. The laser focal radius of 10µm(corresponding to the Raileighlength of ∼ 50πµm if the laser wavelength is∼ 500nm) and the intersectingangle of π/20 results in the string length longer than60µm. The numberof ions on the string can thus be over 10 with the parameters r0 = 1mm,V = 100V and N = 1. A DC voltage to attain the axial confinement hasbeen rather small in the hitherto experiments; a few volt. This is because thefield must be much smaller than the Coulomb field of the ion string, otherwiseindividual ions execute energy oscillation independently in the potential valley,which destroys the string structure. The measure of the Coulomb field of thestring is (1/4πε0)[eN/d2], where d is the distance between neighboring ions.Assuming d = 2rWS ∼ 5.56µm and N = 1, we have 46.5V/m.
2.3 Attainable Temperature and Emittance
The radial size σ of a string crystal in the Paul trap can be reduced to aroundthe laser wavelength λL by laser cooling9. The thermal transverse velocity
qabp˙1: submitted to World Scientific on November 5, 2001 5
of the ions is approximately given by vth = (kT/mi)1/2 where k denotesthe Boltzman constant. When the linear string is accelerated to the non-relativistic velocity v without any heating,the radial divergence is simply σ′ =vth/v. The normalized emittance of the beam is then estimated from
εN = βε = (v/c)σσ′ = vthλ/c, (6)
where c is the speed of light. The laser is directed to the trapping region ata finite angle θ from the trap axis in order to enhance the transverse coolingefficiency. The theoretical Doppler limits of the laser cooling for axial andradial motions are givenby20,
T‖ = (1 + f‖/ cos2 θ)hΓ/(4k), (7)
T⊥ = (1 + 2f⊥/ sin2 θ)hΓ/(2k), (8)
respectively, where Γ denotes the natural line width and f‖ = f⊥ = 1/3 forisotropic scattering. Assuming θ = π/20, we have T‖ = 59.3µK and T⊥ =2.51mK, or v‖ = 0.264m·s−1 and v⊥ = 1.71m·s−1 for 7Li+ ions. With the laserwavelength λL = 548nm, eq. (6) gives εNperp = 3.12×10−15m. Spectral widthof the laser has to be below the value corresponding to the temperature aimedat: ∆ffwhm < kT/πh. The Doppler limit of 59.3µ Krequires ∆f < 2.47MHz.
3 Linac Section
3.1 parameters of the RFQ linac
For the purpose of accelerating a continuous beam directly from an ion sourceat high transmission efficiency, a conventional RFQ linac typically has foursections: i.e. radial matching section, shaper section, buncher section, andaccelerator section21.However, our beam from the trap is already bunched.The precise control of injection timing allows us to surf the ion onto theoptimum RF phase. Furthermore, no transverse matching is necessary.Someor all of the first three sections could then be omitted,which shortens the totallength of the accelerating structure.
Accelerating gradient per nucleon in an RFQ is approximately given bydW
dz=πNeA(z)V2Mβ(z)λ
cosφ(z), (9)
where M is the mass number of the ion (M = 7 in the present example), Vis the RF voltage, λ is the RF wavelength in a free space and A(z) is thedimensionless parameter defined by
A(z) =m2 − 1
m2I0(k(z)r0) + I0(mk(z)r0), (10)
qabp˙1: submitted to World Scientific on November 5, 2001 6
Figure 2. Operating points of the RFQ in V − ω space for three different valuesof r0; (a) r0 =0.5mm,(b) 1mm and (c) 2mm. In each panel, region between twothick dot–dash lines is stable, while the solid line indicates the Kilpatrick limit.Three broken lines show the traces of constant V ω values corresponding to V ω =20 × 1012, 50 × 1012, and 100 × 1012Vs−1. The plausible operating point is markedwith a cross.
where m is the modulation parameter, k(z) = 2π/L(z) = 2π/(β(z)λ) and I0is the modified Bessel function of zeroth order. If k(z)r0 1, we have A ∼ 0.6for m = 2. The maximum possible value of V is empirically evaluated fromthe Kilpatrick’s formula22; i.e.,
ω[Hz] = 10.3E[MV/m]2 exp(−8.5/E[MV/m]). (11)
We can double this electric field in a pulse operation with low repetition,just like in our case.We do so in the following estimate, assuming E = V/r0.To give an explicit picture of the linac, let us first estimate the fundamentalparameters, neglecting their z dependence. We here put a = 0 in eq. (4)for simplicity. Because dW/dz is proportional to ωV , it is convenient tosearch for the optimum operating point in V − ω space. Figure 2 shows thecases of three different values of r0; r0 = 0.5mm, 1mm and 2mm.The twothick dot-dash lines in each panel represent the stability boundaries evaluatedfrom the condition 0 < q < 0.9 while the Kilpatrick limit is indicated by asolid line. Three broken lines show the traces of constant V ω values; V ω =20×1012, 50×1012, and 100×1012Vs−1. Each cross mark shows the plausible
qabp˙1: submitted to World Scientific on November 5, 2001 7
operating point. This figure indicates that the acceleration gradient is almostindependent of the value of r0. Since the accelerating structure becomesmore compact at a higher RF frequency, a smaller r0 is preferable, though itmakes machining more difficult. We here choose r0 = 1.0mm as an example.The operating point of Fig. 2 (b)leads to ω ∼ 2.5GHz or f = 397MHz andV ∼50kV.
Figure 3. Energy evolution (a) and the number of cells (b) in the RFQ. f = 397MHz,V = 50kV, r0 = 1mm, m = 2, φs = 0.
Figure 3 shows the beam energy evolution and the necessary number ofcells in the RFQ with the above parameters, with m = 2. The initial beamenergy is 50keV and the synchronous phase is zero because of the reasonclarified in the next subsection. Ion acceleration to ∼ 7MeV requires 1.5mand ∼ 400 cells.
3.2 injection to the RFQ linac
Ions are forced to execute synchrotron oscillations once entering the linacsection. One may then worry about if the longitudinal time-dependent fielddestroys the 1D structure. Fortunately, we can evade this problem controllingthe number of ions in an RF bucket. For instance, Coulomb interactionsbetween the ions are negligible provided that each bucket contains only asingle ion. No heating effect due to Coulomb collisions then occurs duringacceleration. This ideal situation can readily be realized by properly choosingthe extraction voltage from the trap. Assume that all ions in the trap gainthe same longitudinal velocity v1 at the same timing by a low extraction field.One of the DC electrodes of the trap producing the longitudinal confinementcan be used to give v1. The extracted ions must be further accelerated toa sufficiently high energy, say a few tens of keV, before injected into theRFQ linac. Since no matching procedure is required in the present case, all
qabp˙1: submitted to World Scientific on November 5, 2001 8
we have to do is to put a static accelerating gap in between the trap andlinac section, as shown in Fig. 1. After the pre-acceleration, the inter-particlespacing has become widened from d to (v2/v1)d wherev2 is the ion velocityat the entrance of the linac. Accordingly, it is possible to deliver one ion intoone RF bucket by adjusting the ratio v2/v1. As an example, let us considerthe same RFQ parameters as employed in Fig. 3. Since the distance betweenthe neighboring RF buckets is βλ,v2/v1 must be equal to nβλ/d where n is aninteger. When d = 2rWS with rWS = 2.78µm, v2/v1 ∼ 530n. This equationyields v1 ∼ 2.2 × 103m sec−1 for n = 1, corresponding to 0.178 eV.
4 Discussion
We have some technical problems in the design and construction of this linac.One is to maintain the vacuum at about 10−8 Pa along the whole beamorbit. According to TSR experiments16, the residual gas collisions with themetastable Li ions lead to a loss rate of 1/35s−1 in a vacuum of ∼ 10−8Pa,which is higher than the conversion rate of the ions from the excited to theground states. The small aperture of the linac, r0 = 1.0mm in the presentexample, worsens the conductance and makes the problem more difficult tohandle. The small aperture also reduces the RF coupling between four cavitiesseparated by the vanes. It may thus be necessary either to install some fieldstabilizer for maintaining the equilibrium among the four cavities, or to simplyincrease the aperture size. Accuracy in machining the linac tube directlyaffects the characteristics of the system. A larger aperture size eases therequirements on the machining as well. In summary, we have described alow-current accelerator system consisting of an ion trap injector and an RFQlinac. Laser cooling will realize a transverse emittance in the nm range, andthe injection scheme discussed in the last section will realize negligibly smallenergy width. Although there remain some technical problems to overcome,we believe that the proposed scheme canbe one promising solution for nano-beam production.
Acknowledgements
One of the authors, A. O., acknowledges useful discussions with Dr. Y. Furu-sawa, Dr. T. Hori,Prof. K. Yoshida, Prof. S. Otani and Prof. A. Noda.
References
1. P. K. Ghosh, “Ion Traps,” Clarendon Press (1995).
qabp˙1: submitted to World Scientific on November 5, 2001 9
2. T. P. Wangler, “Principles of RF Linear Accelerators,” John Wiley andSons, Inc.(1998)
3. J. D. Prestage,et al., J. Appl. Phys. 66 (1989) 1013.4. M. Folkard, et al., Int. J. Rad. Biol. 72 (1997) 375, 387.5. G. R. Geard, et al., Nucl. Instr. Meth. B54(1991) 411.6. I. Ohdomari, Oyo Butsuri 64 (1995) 777.7. J. Orloff, Rev. Sci. Instr. 64 (1993) 1105.8. J. Wei, et al., Phys. Rev. Lett. 73 (1994) 3089.9. M. G. Raizen, et al., Phys. Rev. A45 (1992) 6493.
10. M. Drewsen, et al.,Phys. Rev. Lett. 81 (1998) 2878.11. M. H. Prior and R. D. Knight, Opt. Comm. 35 (1980) 54.12. A. Boggia, et al., Proc.PAC 1997, Vancouver (1997) 1619.13. Y. Lee, et al., Rev. Sci. Instr. 71 (2000) 722.14. W. Ketter and N. J. van Druten, Adv. Atom. Mol. Opt. Phys.37 (1997)
181.15. S. Schroder, et al., Phys. Rev. Lett. 64 (1990) 2910.16. W. Petrich, et al., Phys. Rev. 48(1993) 2127.17. J. S. Hangst, ANL/PHY-93/1 (1992).18. S. N. Atutov, et al., Phys. Rev. Lett. 80 (1998) 2129.19. R. W. Hasse and J. P. Schiffer, Ann. Phys. 203 (1990) 419.20. W. M. Itano and D. J. Wineland, Phys. Rev. 25 (1982) 35.21. K. R. Crandall, et al., Proc. 1979 Linear Accelerator Conf., Montauk,
Brookhaven National Laboratory Report BNL-51134 (1979) p205.22. W. D. Kilpatrick, Rev. Sci. Instr. 28 (1957) 824.
qabp˙1: submitted to World Scientific on November 5, 2001 10
Figure 1: Solid lines denote the Bose-Einstein critical density of Ps, H, Rb,as a function of temperature. PsBECoccurs in the left-upper area for each line.
Figure 2: Energy level and decayrates of the low lying states of Psparticipating in the laser cooling.
,
& ! *6(! - *+*) # # : )* C 1 G+ C # ! ! # ,; @ " ,)) C - .<*)
.<*)A*+*) - ,( " ! # ()) B ! # * C # ! # # !
E " 3 9 " ()) # # 1 ! / - .<*)
# ,;
0 1m
Chopper
B uncher
Ion P umpS econdary
Vaccum S ys tem
B eam Monitor
P r imaryVaccum S ys tem
B ending S olenoid Coil
Helmholtz CoilAccelerator
L ead S hield
R I S ource 22Na
Bunched e+
Beam
Slow e+
Beam
Figure 3: Schematic Illustration of slow e beam generator+
Figure 8: Spatial distributions of o-Ps after 220 nsec for laser-on and laser-offobtained by the Monte Carlo simulation: the details of the simulation aredescribed in the text.
; B :2 & & B 8 := C C & C 8 C 7 8 7 + , M $ ! " &- ...' &
*) B :2 & B 8 := 8 C2 B L C 7 / /& ! 0
** 8 := B 8 B & B B :2 & C C83# 8 -&& 0 0 ,2 *> *;;;
*( B :2 & & B 8 := C C & C 8 C 7 8 7 + , M$ ! " & -
*, & C B :2 L : 8 := I C= & B 8 C C L E= 8 C2 -&& 0 0 ,2*> *;;;
*. 8 B & B B :2 8 := & C L : & E -&& 0 0 ,2 (>; *;;;
*+ / 0! " & ()* *;;>*> 8 C2 B L B :2 & C & B '
! " -
**
HIGH ENERGY ELECTRONS AND SYNCHROTRONRADIATION FROM A PHOTONIC BAND-GAP FIBER
ACCELERATOR
RAYMOND Y. CHIAO, DANIEL SOLLI, AND JANDIR M. HICKMANN*
Dept. of Physics, Univ. of California, Berkeley, CA 94720-7300, USA
It should be possible to construct scaled versions of conventional linear accelera-tors using hollow-core photonic band-gap waveguides and pulsed lasers. We haveshown that it is feasible to produce high-energy electron beams from these tabletopsources. In addition, it should be possible to couple a weak, counter-propagatinglight pulse into the waveguide, creating an electromagnetic wiggler. Our calcu-lations show that it is realistically possible to generate a significant amount ofcoherent, hard x-ray radiation from such a device.
The high energy photons characteristic of synchrotron radiation find im-portant uses in many areas of physics, chemistry, and biology. The theoreticaldetails of synchrotron radiation are well known and were originally obtainedby Schott in 1912 and later modernized and expanded by Schwinger in 1949.Since synchrotron theory is present in many sources in the physics literature1,only the important concepts and results will be summarized here as neededto provide background for our new idea.
It is possible to guide light in air or vacuum using a special type of pho-tonic band-gap (PBG) fiber, without any metallic structure.2 These hollow-core PBG waveguides consist of a central channel surrounded by a two-dimensional dielectric photonic crystal.3,4 In contrast to standard opticalfibers which confine light to a central core with total internal reflection,hollow-core PBG waveguides confine electromagnetic energy, within a spe-cific frequency range, by Bragg reflection in the photonic crystal. As a result,the fibers have negligible energy leakage because guided electromagnetic wavesbecome evanescent in the dielectric medium5. The diameter of these struc-tures scales with the wavelength being guided; thus we write the diameter dof the hollow-core as,
d = 2 a = nλ0 (1)
where a is the radius, n is a number (typically n ≈ 10), and λ0 is the wave-length.
The transverse magnetic TM01 mode is the fundamental mode. The axialcomponent of the electric field must satisfy Bessel’s equation with m = 0,
d 2Ez
dρ2+
1ρ
dEz
dρ+ k2
tEz = 0, (2)
where the z-axis is parallel to the fiber, ρ is the radial distance from the z-axis,and kt is the transverse component of k0. The solution of this equation is,
Ez = Ez0 J0(ktρ). (3)
The other two nonzero field components of the TM01 mode can be found fromMaxwell’s equations in cylindrical coordinates. The results are,
Eρ = ikz
k2t
∂Ez
∂ρ, (4)
Bφ = ik0
k2t
∂Ez
∂ρ, (5)
where φ is the azimuthal angular coordinate perpendicular to the fiber axis.6
The boundary conditions require that the z-component of the electricfield must vanish at the surface of the dielectric; thus, kta must be the firstzero of J0. This mode can be excited with a Bessel beam that matches theseboundary conditions. If light, with power Pa , propagates in the fiber, theamplitude of the axial component of its electric field is given by,
Ez0 = Ea sin θ, (6)
where Ea is the amplitude of the total electric field, and θ is the angle betweenk0 and the z-axis. Therefore, the boundary conditions are satisfied if,
sin θ =2.405π n
. (7)
If the mode phase velocity, ω/kz, is closely matched to the vacuum speedof light, the axial electric field can accelerate charged particles to relativisticvelocities. For small θ, we have,
vphase ≈ ω
k0
(
1 +θ2
2
)
(8)
and
Ez0 ≈ Eaθ. (9)
Since the phase velocity has a second order θ dependency, fairly large ac-celerating fields can be obtained with a relatively small effect on the phase
velocity. In any case, the initial velocity of the electrons must be fairly closelymatched with the phase velocity of the mode; thus, the electrons must bepre-accelerated to fairly relativistic velocities. However, as in conventionallinear accelerators such as SLAC, even for small θ there will be potentiallysignificant differences between the phase velocity, the group velocity, and thevelocity of the electrons. For our geometry, we anticipate normal dephasinglengths on the order of a few millimeters. This “walk-off” problem is sig-nificant, but may be solved by lowering the phase velocity of the mode withperiodic corrugations (“loading”) along the fiber. It should be possible to cor-rugate the waveguide during the fabrication process with etching techniquesor ultraviolet radiation.
The accelerating force from infrared and visible radiation can be muchgreater than that of conventional radio-frequency accelerators. By integratingthe Poynting vector over the central channel, it is possible to calculate thepower in the TM01 mode. Using this result, we may write the magnitude ofthe axial component of the electric field in the central channel (in practicalunits) as,
Ez0(MV/cm) ∼= 457
√Pa(MW )n2λ0(µm)
, (10)
where Ez0 is expressed in megavolts/centimeter, and Pa is in megawatts.If this electric field accelerates an electron over a length L, the electron willacquire an energy, E = γmc2 with,
γ ∼= 894(
L(cm)n2λ0(µm)
)√Pa(MW ). (11)
It is possible to create an electromagnetic wiggler by coupling a lightwave of the same wavelength into the waveguide. If an electron beam isaccelerated by the field Ea to highly relativistic energies and subsequentlyallowed to interact with the counter-propagating electromagnetic wave, thepath of the particles will be approximately sinusoidal. Synchrotron radiationwill be radiated along the axis of the waveguide.
The shape of the magnetic field in the TM01 can be determined fromequations (3) and (5),
Bφ ∝ −J1(ktρ). (12)
Thus, Bφ is zero along the fiber axis (this is also evident from a symmetryargument), and it attains a maximum at ktρ ≈ 1.84. From equation (5), it isclear that magnetic and electric fields are out of phase by 90 degrees. Thus,in a plane where the z-component of the electric field is a maximum, the
magnetic field vanishes identically. However, to one side of this null, the signof the magnetic field is such that it focuses the electron beam. Furthermore,the magnitude of the accelerating component of the electric field is a smallfraction of the maximum amplitude of the transverse electric and magneticfields. Therefore, the on-axis electrons experience a significant magnitudeof focusing at the point at which the magnetic force is comparable to theaccelerating electric force, which occurs a small fraction of the wavelengthaway from the null. Hence, stably accelerated electrons are located close tothe maximum of the axial electric field.
In accordance with conventional synchrotron formalism, we calculate thenon-dimensional magnetic strength, K, which can also be written as the nor-malized vector potential a0. Expressed as a function of the system parameters,
K ∼= 0.0186(√
f
n
)√Pa(MW ), (13)
where we have assumed the power of the counter-propagating wave is a frac-tion f of that of the accelerating wave. Undulators are defined to haveK << 1, and wigglers have K >> 1. For pulsed laser systems with peakpowers in the neighborhood of 100 MW , we see that K << 1, characteristicof the undulator regime.
Electromagnetic undulators radiate significant power at the minimumwavelength, λmin ≈ λ0/4γ2, where λ0 is the wavelength of the periodic mag-netic structure. Typical undulators have periodicities on the order of centime-ters; whereas, the periodic magnetic field associated with an electromagneticwave at mid-infrared laser frequencies is on the order of 10 µm. Thus, fromthis type of scheme, one expects wavelengths on the order of 10, 000 timessmaller, for an equivalent electron energy, than that from conventional undu-lators. In terms of the parameters of the system, the minimum wavelength inangstroms is,
λmin(A) ∼= 0.00313
(n4 [λ0(µm)]3
[L(cm)]2
)1
Pa(MW ). (14)
In the undulator limit, the power radiated by an electron at all wave-lengths and into all angles is proportional to the square of the electron energyand to the square of the magnetic field. In terms of the system parameters,the power radiated per electron is,
P (µW ) ∼= 2.01 × 103
(f
n6
)
[L(cm)]2[Pa(MW )]2
[λ0(µm)]4, (15)
where P is expressed in microwatts. If the electron beam is forced to radiateover a length L′, there are N = L′/λ0 undulator periods. From coherence ar-guments, there is a central radiation cone with opening angle θcen
∼= 1/γ√N ,
and relative spectral bandwidth ∆λ/λ ∼= 1/N . The total power radiated bya beam of electrons into this central cone is then, PNe/N , where Ne is thetotal number of electrons in the beam over length L′, and P is the powerradiated per electron. In terms of a beam current, the total power in the cen-tral cone is then independent of L′. As a function of the system parameters,the total power within a relative bandwidth ∆λ/λ and opening angle θcen isapproximately,
P totcen(µW ) ∼= 1.26 × 105
(f
n6
)
[L(cm)]2[Pa(MW )]2
[λ0(µm)]3I(mA), (16)
where I is the electron beam current in milliamperes. The total power ra-diated at all wavelengths and into all angles is given by PNe. Thus, theexpression for the total radiated power in milliwatts is,
P tot(mW ) ∼= 4.19 × 105
(f
n6
)
[L(cm)]2[Pa(MW )]2
[λ0(µm)]4L′(cm)I(mA). (17)
Of course, it is also possible to find the spectral brightness of this ra-diation, however, this calculation requires specific knowledge of the electronbeam size, σ, and divergence, σ′. The phase space volume (or equivalently,the emittance) is proportional to the product of these two parameters. The “undulator condition” requires σ′ 2 << θ2cen; however, if N is large, this condi-tion is not strictly satisfied for realistic electron beams. As a result, numericalmethods must be used to obtain accurate values for the spectral brightness.In this case, the analytic expression can at best only provide a very roughestimate of the spectral brightness, and give useful insights for parameter de-pendencies. According to the standard definition in terms of 0.1% relativebandwidth, the on-axis spectral brightness is roughly,
B
[photons/s
mm2mrad 2(0.1%BW )
]
≈ 5.6 × 104 γ2N 2K2I [mA]
(σ [mm])2, (18)
where we have assumed that the electron beam size is the same along bothaxes, the beam divergence is negligible compared with the central openingangle of radiation, and K << 1.
Equations (14)-(18) show extremely strong dependencies on n and λ0.These parameters determine the strength of the accelerating electric field, aswell as the periodicity and strength of the undulating magnetic field. The
strong dependence on these parameters is a significant motivation for scal-ing down to small wavelengths and cavities. Hence, we propose to use lasersinstead of radio-frequency sources, and hollow core dielectric waveguides forparticle acceleration. These dielectric structures have the important advan-tage over plasma-based devices in that most of the hard-to-control plasmadegrees of freedom are frozen out by the dielectric structure. The dielectricbreakdown strength for typical materials, such as optical fiber glass, increasesdramatically from radio to optical frequencies, so that breakdown should notbe a problem for realistic optical materials and laser power levels, as has al-ready been demonstrated for femtosecond laser pulses propagating throughordinary optical fibers.
In order to construct a tabletop synchrotron radiation source, a pulsedlaser with high peak output power is required. The pulsed TEA CO2 laseris an appropriate source; it generates a 10.6 µm pulse with a pulsewidth ofabout 100 ns and a peak power of approximately 100 MW .7 Photonic band-gap fibers or omnidirectional multilayer-structures designed to guide waves inthe 10 µm wavelength regime have been recently constructed and tested.8 Forthe numerical estimates listed in the table below, we assume n = 10 (d = 10λ),f = 0.1 (Pc = 0.1Pa), I = 0.1 mA (peak current), and σ = 0.1 mm. In prac-tice, there may be constraints on the maximum length L′, which we have notexplicitly considered, due to the finite opening angle of radiation; if intenseionizing radiation is incident on the dielectric walls, the structure may bedamaged. However, the characteristic opening angles are so small that thisshould not be a significant problem for reasonable values of L′.
γλmin
AN
∆λ/λ×100%
θcen
mradP tot
cen
µWP tot
mWB
L = 10L′= 1
84 3.7 943 .11 .39 1.1 .33 1012
L = 10L′= 10
84 3.7 9430 .011 .12 1.1 3.3 1014
L = 100L′= 10
840 .037 9430 .011 .012 110 330 1016
L = 500L′= 100
4200 .0015 94300 .0011 .00077 2600 83000 1019
Table 1: For this table Pa = 100 MW , Ez = 4.3 MV/cm, K = 0.0059; B is given
in photons/s
mm2mrad2(0.1%BW ), and L and L′ are given in centimeters.
Hollow-core waveguides capable of guiding laser wavelengths in the 1 µmrange also exist. Thus, synchrotron radiation may be generated using these
wavelengths in a similar manner as described above. An appropriate lightsource for this application is the pulsed titanium-sapphire laser; which is ca-pable of generating fast pulses with high repetition rates and peak power onthe order of 1 GW . Similar numerical calculations show that it should bepossible to generate synchrotron photons of much shorter wavelengths thanwould be possible from comparable lengths of the 10 µm wavelength fibers.However, experiments at 1 µm wavelengths will be more difficult because evenmore stringent constraints on the phase space volume of the electron beam arenecessary. It is also important to note that interesting synchrotron radiationcan be generated from electrons of relatively modest energies using undulatorswith small periodicities. For example, if λ0 = 1 µm and γ = 10, fundamentalsynchrotron radiation wavelengths of approximately 50 A are possible.
These numerical estimates illustrate that useful high energy electronbeams and high energy, spatially coherent, monochromatic synchrotron ra-diation can be produced in tabletop sources. High energy electron beams areof great interest for research and manufacturing purposes. A compact sourceof coherent, highly directional x-rays and gamma rays would be extremelyimportant for many scientific disciplines. For example, medical imaging andradiation therapy require highly directional photon sources with radiation out-puts at wavelengths ranging from 1 to 0.01 A. This type of synchrotron sourcecould also lead to compact, high energy free-electron lasers if a gain-feedbackmechanism can be devised.
*J.M.H. is on leave from the Department of Physics, Federal University ofAlagoas, Maceio, AL, Brazil with support from CNPq.
References
1. D. Attwood, Soft X-Rays and Extreme Ultraviolet Radiation (CambridgeUniversity Press, 2000), Chapter 5; J. Jackson, Classical Electrodynam-ics, (John Wiley & Sons, 1999), 3rd ed., pp. 671-694.
2. R. F. Cregan, et. al., Nature, 285, 1537-1539, (1999).3. E. Yablonovich and T. J. Gmitter, Phys. Rev. Lett,. 63, 1950-1953
(1989).4. A. J. Ward, Contemporary Physics 40 (2), 117-137 (1999).5. J. M. Hickmann, and R. Y. Chiao, “Fundamental mode of a hollow-core
hexagonal photonic crystal fiber and its application to electron accelera-tion”, in Nonlinear Guided Waves and Their Applications, OSA TechnicalDigest (Optical Society of America, Washington DC, 2001), pp. 73-75.
6. S. Ramo, J. Whinnery, and T. Van Duzer, Fields and Waves in Commu-nication Electronics (John Wiley & Sons, 1993), 3rd ed., Chapter 8.
7. A. E. Siegman, Lasers (University Science Books, Mill Valley, California,1986), pp. 309-310.
8. Y. Fink, D. Ripin, S. Fan, C. Chen, J. Joannopoulos, E. Thomas, J.Lightwave Tech., 17 (11), 2039-2041 (1999).
Section 3
Beam Phenomena under
Strong EM Fields
RECENT RESULTS IN CRYSTAL CHANNELING EXPERIMENTS
Crystals in γ,γ Colliders
E. UGGERHØJ ISA University of Aarhus
Denmark
The very recent indications of Higgs-candidates at CERN have led to a strong interest in new types of facilities, like γ,γ colliders. This again leads to a search for strong high energy gamma-sources. In the present paper it is shown that single crystals are unique radiators due to the strong crystalline fields of 1012v/cm or more. Along axes the radiation emission is enhanced more than two orders of magnitude – a 243 GeV electron loose around 150 GeV in a 0,7 mm thick 110 diamond. This dramatic effect is mostly due to radiation cooling followed by capture to channeling states. The radiation is emitted in a foreward angular cone of around 40 rad. The total energy loss does not follow the classical γ 2-law but turns towards the γ 2/3-law obtained in a full QED calculation.
I Introduction When a parallel beam of high energy electrons and positrons traverses a crystalline target along directions far from major axes and planes the well-known incoherent bremsstrahlung (IB) is emitted. As the incident directions are turned towards planar directions the projecticles start to feel the crystalline structure and coherence in the emitted radiation results in the peak structure of coherent bremsstrahlung (CB). For a review see ref. 1. In CB the peak energies are given by the lattice constants and the incident angle with respect to the planes. The spectra are the same for electrons and positrons. If, however, the incident projectiles are incident nearly parallel to axial and planar directions they become trapped in the potential wells formed by smearing the nuclear charges along axes or planes. These so-called continuum potentials lead to a strong steering effect – the channeling effect. This special motion also gives rise to coherence effects in the emitted radiation – the so-called channeling radiation (CR). In channeling positrons are pushed away from the axial and planar nuclei whereas electrons are focused around the crystalline nuclei. So electrons and positrons “see” different potentials and thereby the emitted radiation spectra are different – in contrast to the CB case.
The main theoretical concepts of channeling were formulated in the classical paper by Lindhard [2]. Today the field is well established and described in many review papers, e.g. ref. 3. A short review is found in my paper from the
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Monterey Conference in Monterey, Jan. 1998. The main aspects of CR was firstly described by Kumakhov and co-workers [4], see also conference proceedings like ref. 5.
Radiation emission from multi GeV electrons and positrons penetrating single crystals has created large interest in recent years. The channeled particles “feel” the electromagnetic fields from the axes and planes which correspond to (1011-1012) volts/cm at distances of (0.05-0.1) Å from crystal axes. Radiation emission from such fields causes a significant recoil, because the emitted radiation may exceed the incoherent bremsstrahlung by more than two orders of magnitude. Although multi-GeV electrons and positrons move along classical trajectories the emission process demands a quantal description because the coherent part of the emitted photon spectrum may extend nearly up to the particle energy. On the other hand at the high energies the transverse excursion of the particles is small during the formation time for the photons so the variation of the axial and planar potentials during this time is small and can be neglected. Therefore the radiation emission process appears as in a constant electromagnetic field. This so-called constant field approximation (CFA) as developed by the Baier group [6] has turned out to be a good approximation for small incident angles to axes and planes. The critical angle for this approximation is given by mc2 . Here U is the depth of the axial/planar potential and m the electron rest mass. For incident angles in the Born approximation is justified and for in CFA is a good approximation. Furthermore, it should be pointed out that probabilities for QED processes in electromagnetic fields from crystal axes/planes are determined by the magnitude of these fields in the partiale rest frame. Hence the quantum parameter is given by = /o where o= v/cm and is 10161043 5-106. This means that crystalline targets are unique for investigating QED-processes in strong fields. It should be pointed out also that because of small incident angles to the crystal axes/planes the projectiles move in these dramatically strong fields over several microns. Recent results for ultrarelativistic particles can be found in ref. 7.
In the present paper the most recent experimental results from the CERN experiment NA-43 are presented – covering: radiation emission, radiation cooling, energy loss, photon multiplicities, pair production and shower formation (see coming issue of NIMB). In the the proceeding from the Monterey Conference, Jan 1998 I already presented some experimental data on these subjects. Below these discussions are referred to as: Mont 98. Last but not least the possibility of using crystals as -sources for photon colliders is discussed.
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II Experimental
The experiment was performed in the North Area of the CERN SPS. The beam H2 is a tertiary one containing electrons, positrons or pions with energies ranging from 35 GeV to 300 GeV. The normal beam divergence is 30 rad, but can be changed.
Beam
e--
BEAMDUMP
Z=0
DC1
B1
Sc1-2 Vac.tube
Vac. Chamber Iwith crystal
on goniometer
Vac. Chamber IIwith crystal
on goniometer
B2
Sc4 Sc6
Tr6B8
40m
DC2
61m 65m 75m
DC3 DC4HeI Sc9
Sc10b
AKS Sc11
77m
DC5 E. M. Calorimeter
HeIIIHeII
81m
DC6
e+
e+/e-- e--
25X0
SSD
γγ
C-188-2A
Figure 1 A schematic drawing of the NA-43 setup. Details are discussed in the text.
The experimental arrangement is shown schematically in Fig. 1. A drift chamber (DC1) was placed 40 m in front of the target and another drift chamber (DC2) was placed immediately (15 cm) behind the target. In this arrangement the influence of multiple scattering on angular resolution is minimum on the incident side. A third drift chamber (DC3) is placed 20 m behind the target. Vacuum tubes, (10-2 torr) are used between the drift chambers. The angular resolution on incident and exit sides are 5 rad or better due to the good position resolution (100 ) of the drift chambers. Scintillation counters (SC1, SC2, SC3) were used to define the useful fraction of the beam. Bend 1 and Bend 2 remove the background from upstream material. By means of Bend 8 the beam was bent away from the photon detectors and into the dump. DC4 is used to define the exit momentum of e+/e-. The radiated energy is determined by a lead-glass array with a resolution of 7% (FWHM) at 150 GeV particle energy. Finally DC5 and DC6 are placed downstream a smaller magnet (Tr6) in front of which a conversion crystal or foil is positioned. These two drift chambers and Tr6 are used as a pair spectrometer to determine the momentum of the e+/e- pair produced in the photon converters. The crystals are mounted in goniometers that are thermally insulated to secure temperature stability. The emitted photon multiplicity can be measured in two ways: 1) by the pair spectrometer for which the minimum photon energy that can be detected is 5 GeV or 2) by a fully depleted solid state detector (SSD) in front of which is mounted a 1 mm thick Pb-foil for conversion. Using the SSD arrangement the minimum photon energy detectable is 0.5 GeV. The SSD was 0.5 mm thick with an area of 600 mm2. From the particle data book it is found that the probability of a photon converting to e+/e-
pairs is 13% in the 1 mm thick lead. For normalization and background subtraction
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we used photon spectra for incidence along random directions in the crystal. Single photon spectra were here assumed. From the lead-glass spectrum certain energy windows for emitted energy have been considered. The conversion probability for photons in this window gave the average photon multiplicity.
0 40 80K [GeV]
1200.15
0.20
0.25
R=
N1/
Nto
t
0 40 80K [GeV]
1200.04
0.06
0.08
0.10
1.5 mm Diamond-Randomwith Pb converterR = (223 ± 2) 10-3
1.5 mm Diamond-Randomwithout Pb converterR = (78 ± 2) 10-3
c
Figure 2 Alignment of a 0.6 mm thick (110) Si crystal using radiation emission from 150 GeV electrons. b) Pulse height spectrum from the 0.5 mm thick fully depleted solid state detector (SSD) placed just behind the Pb-converter for multi-photon detection. c) Photon conversion probability in the 1 mm thick lead and mylor windows (left) and mylor windows alone (right).
From the very good energy resolution of the SSD it is possible to discern between 1, 2, 3,…… minimum ionizing particles (MIP). In Fig. 2b a typical SSD
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spectrum, used for multiplicity measurements, is depicted. Figure 2c shows random conversion probabilities (in %) with and without the 1.0 mm thick Pb-converter. The conversion without the 1.0 mm Pb comes from the mylar windows, air gaps,etc. The difference between the two conversion probabilities (14%) is in a very good agreement with the theoretical value mentioned above. The thickness of the Pb-converter was chosen such that the preshower is linear dependent of the number of incident photons. In Fig. 2a is shown a typical alignment procedure. Planar and axial directions are found as the ones giving maximum radiation emission. From the planar scans a stereogram can be constructed and the axial position is determined. The accuracy of the axial setting is 5 rad or better. The pair production for different planar orientations in the second crystal can be used to analyze the polarization of radiation from the first crystal. In the present setup, the first goniometer holds the 0.7 mm thick diamond crystal. In front of the second crystal, a scintillator, Sc9, is positioned to reject those events where a photon has converted upstream. Immediately downstream of the second crystal, two counters, a solid-state detector (SSD) and a scintillator, Sc11, detect the pairs produced in the crystal. For the pair spectrometer the specific energy of each converting photon is measured, whereas when using the SSD only the total energy of all photons is measured in the calorimeter. The pair spectrometer can also be used to find the direction of the emitted photons. Since the photons are emitted within 1/ which is comparable to the angular resolution of the DCs, the approximate particle direction can be measured at the moment of photon emission. The energy of these photons is measured by the lead-glass array. In this way the angular distribution of the electron beam can be measured in front and behind the first crystal but also the direction of the electron just before emitting the photon inside the crystal can be detected, see below. As mentioned above the upgraded detector has more drift chamber windows and therefore the background increases, which can be serious especially when using thin crystals. Therefore accurate background measurements are important. The random yield for the 0.7 mm diamond crystal equals 0.58% radiation lengths (X0). The additional material in the beamline (mylar foil + DC’s + airgaps) equals 0.96% (X0). Hereby the expected enhancement of the random yield (incl. background) with respect to the background alone for the 0.7 mm diamond is 2.7 – in very good agreement with data. So background subtraction can be performed correctly – even for low-Z crystals with thicknesses below 1 mm. Likewise, the photon multiplicity registered by use of the pair spectrometer is in good agreement with the expected value for a threshold at 4.5 GeV.
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III Radiation Emission and Multiplicities In the following the experimental radiation spectra are plotted as a function of emitted energies. The emitted radiation intensity is normalized to that from an equivalent amorphous target of the same thickness – giving the plotted enhancements. When lines are drawn through data-points it is only to guide the eye. 1. Axial case In figures 3 and 4 are shown photon spectra for 149 GeV electrons and positrons incident on the 0.7 mm thick 110 diamond crystal for which the channeling angle, 1=30 rad. The incident polar angle regions are given below the spectra. Further on two types of multiplicity curves are shown: One type (right column) measured with the solid state detector (SSD) where the lowest energy threshold is 0.5 GeV – and another type (middle column) measured with the pair spectrometer where the lowest photon energy is 4.5 GeV. These detailed multiplicity measurements have been possible only by the upgrade of the NA43 detector. In general the spectra agree with the overall channeling picture for e+, e- i.e. channeled electrons are focused around the nuclei in the atomic rows and emit hard photons, whereas channeled positrons are pushed away from the strong fields and emit mostly soft photons. From figure 4 it is seen that outside the channeling angle region, positron spectra show an increase towards higher energies as compared to electron spectra. This is due to the fact that the positrons due to deceleration in the axial field stay somewhat longer in the strong field than electrons – and radiate somewhat more (larger enhancement). As will be shown below, electrons incident just outside the channeling region will be cooled and thereby captured in the channeling region and emit more hard photons. In the incident angle region from (0-10) rad the photon peak corresponds to nearly 45% of all incident particles. This number is much larger than the number of channeled electrons that are captured in channeling states after surface transmission – which is around 5%. The present picture was also seen in the very first experiments of the present type performed by Belkacem et al.[8]. Later on some of us [9] performed the same type of experiment with much improved angular resolution and on thin crystals. Here it was found that the photon peak for electrons disappeared for thin Si-crystals (100 .) This led to the conclusion that the photon peak was mostly due to multiphoton effects. Such an explanation is dubious because multi-photons would smear out the peak and not give rise to the flat part around 75 GeV. In more detailed spectra even minima are found around 75 GeV photon energies.
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Figure 3 Radiation enhancements for 150 GeV e+/e- (left column) pair spectrometer multiplicities (middle column) and SSD multiplicities (right column) as a function of photon energies for different incident angle regions in units of rad. The open squares refer to electrons whereas filled squares refer to positrons. Note the change of vertical scales. The crystal was a 0.7 mm thick (110) diamond. An explanation for the effect can be found in the book of Baier’s group [6]. Beyond a certain thickness, L0, the electrons penetrating the crystal may be separated into two groups, one giving rise to radiation as from an amorphous substance and one that suffers increased multiple scattering as well as increased energy loss because they come very close to the nuclei. The latter group is
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‘intermixed’ so strongly below a certain transverse energy that it suffices to describe the radiation ‘… by the simulation of a cascade at some intermediate value of for all the particles …’. The thickness L
i
i
0 is determined by ef = 2(L0)/a =
1 where U(
2s
ef) = , being the transverse coordinate and ai
oL
s the screening distance. The observed photon peak becomes most pronounced at crystal thicknesses which in the case of diamond 110 corresponds to about the 0.7 mm used in the experiment. This simple model destribes the behaviour well.
2L
Figure 4 Same as fig. 3 but for increased incident angle regions.
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A similar idea was brought up by some of us [10], where we investigated the influence of multiphoton effects on the spectra. The crystal was divided into slices so thin that in each slice the probability for emission of more than one photon was negligible. Finally the spectra from all slices were folded – but no photon peak was obtained in this way. A photon peak was obtained only when particles with small impact parametres were weighted highly. This situation corresponds to radiation cooling. So the experimental spectra may be reproduced by proper account of radiation cooling and multiple scattering.
00
0.1
0.2
0.3
0.4
0.5
10 20 30INCIDENT ANGLE REGION [µrad]
40 50
Npe
ak/N
tota
l
149 GeV243 GeV
Figure 5 Fraction of incident particles giving rise to radiative energy losses of 60% or more (Npeak) normalized to the total number of incident particles (Ntotal).
In figure 5 we show the fraction of electron events which give rise to energy losses in the interval [0.6-1] of the initial energy for 149 and 243 GeV electrons. Outside the critical angles, Npeak/Ntotal converge for the two data sets, whereas for small angles the value corresponding to 149 GeV exceeds that of 243 GeV significantly. As L0 does not vary much between these energies, the explanation must be found elsewhere and we propose that this is another manifestation of the quantum suppression of the energy loss. From the photon spectra it is clear that electrons incident within the channeling angle 1 = 30 rad emit hard photons in contrast to the positrons. Similar, but much less detailed investigations have been performed before on Ge [8]. Next it should be pointed out that the multiplicities for electrons and positrons are practically the same – apart from incident angles (0-10) rad. So within the channeling region the average emitted photon energies are much higher for electrons than for positrons.
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2 Planar case In an earlier experiment [11] we found for the first time a very pronounced high-energy photon peak when 149 GeV electrons are incident along the (110) planes and
0
10
20
30
40
50
0 1000
0.51
1.52
2.53
3.5
50 100 1500
10
20
30
40
50
60
0 100
Figure 6 Radiation spectrum (a), Photon multiplicity (b) and single photon counting spectrum for 149 GeV electrons incident along the(111) planes in a 0.7 mm thick diamond and at 0.6 mrad to the 110 axis. at 0.3 mrad to the 100 axis in a diamond crystal. In figure 6 is shown the same type of effect incident at ±10 rad to the (111) planes in the 0.7 mm diamond crystal and at 0.6 mrad to the 100 axis.The very pronounced peak at 110 GeV shown in figure 6a is due to this new type of coherent bremsstrahlung emitted when the electrons cross the rows of atoms forming the (111) crystal planes. In the Lindhard theory these incident directions are called: ‘the strings of strings region’[SOS][2]. The photons are expected to be nearly 100% planar polarized – like CB is in many cases. An estimate of the polarization of these photons was obtained recently by NA-43 [17] by comparing the pair production along two perpendicular crystal planes in a second crystal placed 40 m behind the radiator. Here large asymmetries were found – implying a large degree of polarization. In figure 6b the corresponding multiplicity spectrum is shown from which it appears that the electron emits around 1.7 photons each with an energy above ..0.5 GeV in the region of the peak. Finally, in figure 6c the ‘single photon spectrum’, i.e. the energies of the emitted photons is shown and it is proven that the ‘strings of strings’-peak consists of very high energy photons followed by less energetic ones. We emphasize that figure 6c shows a counting spectrum, not a power-spectrum; the effect is thus very strong. Now this effect is being used as a γ-source in the new CERN collaboration – NA59 [12]. Here a second crystal is used to turn planar polarized photons into circular ones. These circularly polarized photons could open new ‘windows’ in high
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energy physics, like measuring the contribution from gluons to the spin of the nucleon. Furher on it should be pointed out that this so-called ‘strings of strings’ (SOS) incident angle region would be an excellent source for high energy gamma-rays. The enhancement is a factor of 10 larger than for normal CB. The effect could also be used in coming photon colliders. III RADIATIVE ENERGY LOSS The energy loss for GeV electrons and positrons is practically all due to radiation emission. In figure 7 is shown the average radiative energy loss of 149 GeV and 243 GeV electrons and 149 GeV positrons incident along the (110) axis. From the curves the channeling effect for positive and negatively charged particles is clear i.e. for incident angles smaller than ψ1 = 30 µrad the positrons are pushed away from the nuclei in the string and thereby radiate less, whereas the electrons are focused around the nuclei and radiate more. The overall radiative energy loss is dramatic i.e. well-aligned electrons lose around half their energy in a 0.7 mm thick crystal where for a comparison the energy loss in an amorphous 0.7 mm foil is less than 1 GeV. So the energy loss is enhanced almost two orders of magnitude. Secondly it should be noticed that this strongly enhanced energy loss is not just found for the rather small channeling angles – it continues far outside the channeling angular region. Here it only decreases rather slowly for increasing incident angles which is due to the strong crystalline fields. For increasing particle energy the potential for distances r a from the axis becomes more and more important as pointed out by Kononets [13]. This also shows up especially in the 243 GeV data. Here the strong increase in energy loss inside the channeling region is smeared and starts already at 2-3
54
1 and not as for the 149 GeV case at 1. This smearing effect is also caused by a strong radiative cooling effect for incident angles outside the channeling region (see below). Here electrons are captured into the channeling potential and thereby have an increased radiation emission. In figure 7b is shown the energy loss curve from 7a – but scaled to the incident energy and critical angle. From this plot it is clear that for increasing particle energy the energy loss from the channeling regions E ½ pv1 is
saturating whereas the contributions from E
2
½ pv 21 are increasing. Again this is
in agreement with the theoretical calculations [13] showing increasing influence from the axial potentials for distances r (3 – 4) a.
Last but not least, the data in figure 7b also show the onset of quantum suppression in energy loss. From a classical description of radiation emission the radiative energy loss follows a 2-dependence for the particle energy. For increasing particle energies this dependence should turn into a 2/3-dependence according to
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0 50
υin [µrad]
100 150
20
40
60
80 Electrons
Positrons
Era
d [G
eV]
a
AVERAGE RADIATED ENERGYscaled to incident energy and the critical angle
00 1 2 3
θin/φc
4 5 6
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Era
d /E
in
Electrons Ein = 149.1 GeV/c
Electrons Ein = 243.1 GeV/cb
Figure 7 Averaged radiative energy loss for 149 GeV e+/e- and 243 GeV electrons traversing the 0.7 mm thick diamond close to the 110 axial direction.
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theoretical calculations [6, 13]. From 7b it is seen that 149 GeV electrons lose on the average 50 GeV (7a) whereas 243 GeV electrons lose around 90 GeV, but a 2-dependence would require an increase by a factor of 2.7. Instead the energy loss increases approximately as 1.3. This deviation from the 2-dependence as predicted by accurate QED-calculations is due to a self-suppression effect which is caused by the strong fields. Such fields deflect the particles so strongly along the formation length lf that the photon is not emitted. This suppression means that the fractional energy loss decreases for increasing particle energy [6, 13]. V RADIATIVE COOLING and CAPTURE The dramatic enhancements of radiation emission from multi-GeV electrons/positrons traversing single crystals have through the years raised the question about radiative cooling – or: Is it possible to reduce the transverse energy of particles by going through a crystal and thereby obtain smaller exit angles out than incident angles in ? When a particle emits a high energy photon its transverse energy E ½γmv + U (r
) decreases and thereby the angle to the crystal axis
also decreases. This so-called radiative cooling will counteract the multiple scattering and the particles might come out from the crystal with a smaller angle to the axis than the incident one.
2
The upgrade of the NA-43 detector and the availability of thin diamond crystals has led to a real breakthrough in the investigations of radiative cooling. In Mont 98 and ref. 14 we were able to demonstrate the very first results on radiative cooling for 149 GeV electrons traversing Si crystals. Positrons on the other hand were “heated” – in good agreement with simple estimates based on changes in transverse energy. – See Mont 98. In figure 8 is shown new cooling data for 243 GeV electrons incident along the 110 axis in the 0.7 mm diamond crystal. Here very strong cooling effects are found. As compared to the 149 GeV data the cooling here is giving negative
already for a radiative energy loss of a few GeV. According to theory [13, 14] the radiative cooling scales as (δ out in
2 2
2/ δL)rad Z3for small and (δ 2/ δL)rad Z2 for >>1. Compared to the multiple scattering (δ 2/ δL)ms Z2 /E2 this means that the net radiative cooling is expected to be much stronger at 243 GeV, as observed.
For incidence within 1 the exit angles are practically all inside the channeling regime showing that nearly all incident particles experience radiative capture to channeling states – a very exceptional situation for negatively charged particles.
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-2000
-1000
0
1000
2000
0 50 100 150
Erad
2 to 20 GeV
20 to 40 GeV
40 to 60 GeV
Cooling plus Multiple Coulumb Scattering243 GeV/c e- on <110>-axis of .7mm Diamond
60 to 80 GeV
∆ [
µrad
2 ]
∆ <θout2 -- θin
2>θin [µrad]
———
Figure 8 Radiation cooling for 243 GeV electrons penetrating the 0.7 mm diamond along directions close to the 110 axis. The curves show the differences between exit ( out) and incident ( in) polar
angles squared i.e. .- in . 2out
2
These new results clearly show that electrons incident on crystals outside the channeling region are cooled and thereby can be captured into the channeling region. Unfortunately for a practical use these cooled electrons have lost a considerable amount of energy – i.e. the total emittance (transverse and longitudinal) is not getting smaller. On the other hand radiative cooling is a real advantage for producing strong γ-sources. For normal channeling radiation the incident angle
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region is very narrow, especially in the GeV region. The cooling effect increases this incident angle region strongly. The very strong cooling effects in diamond is due to the fact that radiation emission is enhanced more than two orders of magnitude and multiple scattering is minimal as compared to other crystals. VI PHOTON EMISSION ANGLES Calculations on radiation emission by the Frankfurt group [15] using the Dirac equation show a rather surprising result i.e. that the hard photons are emitted with
0 20 40 60 80 100 120 140 160
10
0
20
30
40
50
SINGLE PHOTON ENERGY [GeV]
<θ γ
>
[µra
d]
Figure 9 Average angles of emission with respect to the 110 axis in diamond. Single photons emitted by 149 GeV electrons incident within 20 rad to the axis as a function of photon energy. large angles to the crystalline axis – for 50 GeV electrons the photons in the interval 34-50 GeV are emitted with a typical angle of 140 µrad to the 110 axis of Ge. These results were obtained to determine whether or not axial radiation would be applicable for γ-γ physics, e.g. in the hunt for the Higgs boson [16]. Figure 9 shows the average angle of photon emission, , as a function of
the photon energy, E
γ, for 149 GeV electrons incident with an angle 20 rad to the 110 axis. In contrast to the theoretical expectations mentioned above, the average emission angle is 36 rad with about 55% of the photons emitted within 10 rad to the axis. For the 243 GeV data, the corresponding result is = 24 rad
for entry angles 16 rad. We note that in both cases is close to the
respective (
in
in
1) values of 23 and 30 rad. This is also what is expected from the above discussed radiative capture which will bring the electrons into high-lying channeling states.
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These results are very encouraging for the application of channeling/strong field-radiation for γ-γ physics because crystals could be used to produce very narrow γ-ray beams for high energy photon colliders. In the case of the strings-of-strings radiation the emission angles are of the same order as for axial radiation, furthermore, the hard photons are usually followed by only one photon of a few GeV and the photons could be linearly polarized [17]. Conclusion From the presented experimental results it is clear that single crystals – expecially diamonds – are unique for investigations of the influence of strong fields on QED-processes. Crystal fields are 1012V/cm or more and the incident particles move in these strong fields up to 100 m -–in contrast to nucleus-nucleus collisions. Crystals are excellent targets for producing hard gamma rays. The dramatically enhanced radiation emission leads to strong angular cooling for electrons – which again leads to radiative capture to high-lying channeled states. Among the emitted photons from an electron or positron there is normally at least one very hard photon emitted in the first part of the crystal. The strong crystalline fields lead to enormous radiative energy losses. In just 0.7 mm (110) diamond a 150 GeV og 243 GeV electron loose 60% of its total energy which should be compared to the corresponding energy loss of less than 1% in an amorphous foil of the same thickness. The present experiments have shown that for channeled particles the hard photons are emitted in a very narrow angular cone around the axis. So crystals would be unique for future γ,γ-colliders. Here the production of rare particles are more favorable and cleaner than in e+e- colliders. Finally it should be pointed out that the continued demand for higher beam energies and luminosities mean that many beam phenomena involve quantum effects like described above. For colliding beams the Lorenz boost is dramatic. References
b. Coherent Radiation Sources. Ed. A.W. Sáez and H. Überall Springer 1985
2. J. Lindhard. Mat Fys, Medd. Dan. Vid. Selsk. 34 (1965)
3. Channeling and other penetration Phenomena. Ed. E. Uggerhøj. Nuclear Science. Applications 3 (1989)
4. Radiation of Relativistic Light Particles during Interactions with Crystals. Ed M.A. Kumakhov and R. Wedell Spektrum 1991
Error! Unknown switch argument. submitted to World Scientific : 8/2/01 : 10:45 AM
16/Error! Unknown switch argument.
5. Workshop on Relativistic Channeling. Ed. R.A. Corrigan and J.A. Ellison Nato ASI Series. Physics Vol. 165 1986
6. Electromagnetic Processes at high Energies in Oriented Single Crystals. V.N. Baier V.M.Katkov and V.M. Strahkovenko. World Scientific 1998
7. Channeling and other crystal effects at relativistic energy. NIMB 119 (1996) 1-316. Topical issue by H.H. Andersen, R.E. Corrigan and E. Uggerhøj
8. A. Belkacem et al., Phys. Rev. Lett. 53, 2371 (1984) A. Belkacem et al., Phys. Rev. Lett. 58, 1196 (1987)
9. R. Medenwaldt et al., Phys. Lett. B (1990) 517 10. R. Medenwaldt et al., Phys. Rev. Lett. 63 (1989) 2827 11. R. Medenwaldt et al., Phys. Lett. B 281 (1992) 153 12. A. Apyan et al., Proposal to the CERN SPS Committee, CERN/SPSC 98-17,
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)M
UNRUH EFFECT IN STORAGE RINGS a
JON MAGNE LEINAAS
Department of Physics
University of Oslo
PO Box Blindern
N Oslo
Norway
A uniformly accelerated system will get thermally excited due to interactions withthe vacuum uctuations of the quantum elds This is the Unruh eect Also asystem accelerated in a circular orbit will be heated but in this case complicationsarise relative to the linear case An interesting question is in what sense thereal quantum eects for orbital and spin motion of a circulating electron can beviewed as a demonstration of the Unruh eect This question has been studied anddebated I review some of the basic points concerning the relation to the Unruheect and in particular look at how the electron can be viewed as a thermometeror detector that probes thermal and other properties of the vacuum state in theaccelerated frame
Introduction
A uniformly accelerated observer will at least in theory see Minkowskivacuum as a thermally excited state with a temperature determined by theacceleration Thus in the accelerated frame a natural denition of the vacuumstate is the Rindler vacuum which is dierent from the Minkowski vacuumWhen expressed in terms of the Rindler eld quanta Minkowski space is thermally excited A quantum system that is uniformly accelerated will act as adetector or thermometer that probes the temperature of the Minkowski vacuum state in the accelerated Rindler frame When coupled weakly to theradiation eld it will end up in a stationary state with a thermal probabilitydistribution over energy levels
There is a close relation between this eect the Unruh eect and theHawking eect ie the eect that a black hole emits thermal radiationThe accelerated observer in Minkowski space then corresponds to a stationaryobserver with a xed distance from the event horizon of the black hole Thisobserver will detect the local eects of the Hawking temperature
An interesting question is whether the Unruh eect can be seen in anyreal experiment There have been several suggestions but there are obviousproblems with the implementation of such experiments A main problem is
aTalk delivered at the th Advanced ICFA Beam Dynamics Workshop on Quantum Aspectsof Beam Physics Capri October
that the acceleration has to be extremely high to give even a very modesttemperature Thus a temperature of K corresponds to an acceleration of ms A detector indeed has to be very robust to take such anacceleration without being destroyed
Some years ago it was suggested by John Bell and myself that that anelectron could be viewed as such a detector It certainly is robust at this levelof acceleration and in an external magnetic eld the measured occupation of thespin energy levels would give a way to determine the temperature However forlinearly accelerated electrons the obtainable temperature and the time scale forreaching equilibrium for realistic values of the accelerating elds makes it clearthat such eects cannot not be seen in existing particle accelerators Recentlythere have been interesting suggestions of how to obtain much more violentacceleration of electrons by use of laser techniques but even in this case thetime scale for spin excitations is too long There are however suggestions ofother ways to see the Unruh eect in these cases
For electrons circulating in a storage ring the situation is dierent Theacceleration is larger than in linear accelerators and the time available is su cient to reach equilibrium In the discussion with John Bell the question cameup Could the the SokolovTernov eect which predicts a equilibriumpolarization lower than be related to the Unruh eect in the sense thatthe upper spin energy level measured in the rest frame is partly occupied dueto the heating In two papers this question was studied an the answer was aqualied yes See also several later papers some of these included in theproceedings from the ICFA meeting in Monterey two years ago Forcircular motion there are however important complications relative to the caseof linear acceleration One point is that the simple connection to temperatureis correct only for linear acceleration For circular motion the excitations maybe described in terms of an eective temperature but this temperature is notuniquely determined by the acceleration as in the linear case However as amore important point the electron does not act as a simple point detectorThe spin is also aected by oscillations in the orbit and the Thomas precessionmakes the direct coupling to the magnetic eld act dierently from the indirectcoupling mediated through oscillations in the particle orbit The net eect isthat the correct expression for the electron polarization will deviate from onederived by a naive application of the Unruh temperature formula
In this talk I will give a brief review of the connection between the Unruhand the SokolovTernov eect and I will stress some points which I nd interesting concerning this connection First I will review how the Unruh eect inprinciple could be demonstrated as a spin eect for linearly accelerated electrons Then I will discuss the relation between linear acceleration and circular
motion and nally consider the spin eect for circulating electrons I shouldstress that all the basic elements in the discussion of the Unruh eect for electrons in a storage ring have been presented in the papers referred to aboveHowever the discussion of the spin eect I will do in a slightly dierent wayby treating the spin and orbital motion on equal footing
Linear acceleration and the Unruh eect
A pointlike object which is uniformly accelerated ie which has a constantacceleration as measured in the instantaneous inertial rest frame describesa hyperbolic path in space time We may associate a comoving frame withthe motion with the time like unit vector as tangent vector of the trajectoryand the three space like vectors spanning the hyperplane of simultaneity asdened by the moving object The directions of the three orthogonal spacelike vectors are determined only up to a rotation but if a nonrotational frameis chosen FermiWalker transported this degree of freedom is eliminated Thenonrotational frame is also a stationary frame in the sense that the accelerationis xed with respect to the unit vectors
The local frame can be extended in a natural way to an accelerated coordinate system the Rindler coordinate system This is a stationary coordinatesystem in the sense that the metric is independent of the time coordinate Thetransformation from Rindler coordinates x to Cartesian coordinates x tis with the xaxis chosen as the direction of acceleration
x x c
a cosh
a
c t
x
cc
a sinh
a
c y y z z
The Rindler time is the proper time of the trajectory x y z which is assumed to be the trajectory of the pointlike object However anytrajectory x const is equivalent to this in the sense of having a constantproper acceleration although with a value of the acceleration that depends onthe xcoordinate
The coordinate system x is well behaved only in a part of spacetime At nite distance from the object here chosen as the origin thereis a coordinate singularity The two hyperplanes x ct which intersectthere dene event horizons for the accelerated object For points with x ctbehind the future horizon an emitted light signal will not reach the object atany future time A light signal emitted from the object at any past time willnot be able to reach the points with x ct behind the past horizon
The Rindler coordinate system is similar to the Schwarzschild coordinatesystem of an eternal black hole In fact the Rindler coordinate system can be
seen as a limit case of black hole geometry for points at nite distance fromthe horizon with the mass of the black hole tending to innity In this limitthe space time curvature vanishes and only the eect of acceleration due togravitation remains
The Rindler coordinate system can be regarded as constructed from acontinuous sequence of inertial frames the instantaneous rest frames of the accelerated object This implies that the time evolution of a system described inthe accelerated coordinate system can be expressed in terms of the generatorsof the Poincare group which transform between inertial frames for innitesimaly dierent values of For linear acceleration only the time translationand the boost generator in the xdirection are involved and the Hamiltonianwill have the form
H H a
cKx
If H is independent of the situation is stationary in the accelerated frameand the ground state as well as the excited states of the accelerated systemcan be dened as the eigenstates of H Such a physical system has to have anite size to avoid problems related to the coordinate singularity
If a uniformly accelerated system described by a stationary HamiltonianH is coupled to a quantum eld and this eld is in the Minkowski vacuumstate the vacuum uctuations will cause transitions in the accelerated systemThe important point is as pointed out by Unruh that transitions to higherenergy with respect to H is caused by positive frequency uctuations withrespect to the Rindler time The quantum elds in the vacuum state haveonly negative frequency components with respect to Minkowski time t but interms of they have both positive and negative frequency parts
Let us consider the case of an accelerated electron Then H and Kx arerespectively the Dirac Hamiltonian and the boost operator of Dirac theoryfor the inertial rest frame at time If we neglect the uctuations in thetrajectory and simply constrain the particle coordinates to the classical pathx the Hamiltonian H is reduced to a spin Hamiltonian of the form
Hspin
h
with determined by the external magnetic eld and by the radiation eld
e
mcg Bext e
mcg Brad
The primed elds refer to the inertial rest frame The simplest situation
would be to consider an external magnetic eld Bext in the same directionthe xdirection as the accelerating electric eld
The coupling to the radiation eld causes transitions between the spin upand spin down states of the particle in the external magnetic eld Standardrst order perturbation theory gives for the transition probabilities per unittime
eg
mc
Z
expiC id
with referring to transitions updown in spin and with C as the vacuumcorrelation function of the magnetic eld along the accelerated trajectory
CB BB
ha
c
hsinh
a
ci
B Bx iBy
The correlation function C has a periodicity property with respect to shiftsin the imaginary time direction
CB c
ai CB
which makes it easy to solve the integral by closing the integration contouralong the shifted path i ca For the closed contour we nd
exp
c
a
egh
mc
a
c
and the ratio between transitions up and down is
R
expc
a
In an equilibrium situation the ratio between transitions up and downdenes the relative occupation probabilities of the two spin levels The dependence of the energy shows that it has the form of a Boltzmann factorcorresponding to a temperature
kTU ah
c
This is the Unruh temperature which is generally associated with an accelerated system It has exactly the same form as the temperature of a blackhole
kTBH h
c
whith the acceleration a corresponding to the the surface gravity of theblack hole
The spin eect discussed above is a special realization of the Unruh eectThe interesting point is that this eect has a universal character it does notdepend on details of the accelerated system or on its coupling to the radiationeld The temperature eect follows from general features of the vacuumstate and special properties of the accelerated trajectory which lead to thesymmetry properties of the vacuum correlation functions One should alsonote that the limitation to point detectors is not a necessary restriction Forthe case considered here the uctuations in the particle path can be takeninto account The Hamiltonian H will then depend on both spin and orbitalcoordinates and the electron in this sense has to be treated as an extendedsystem If the accelerating elds give rise to a time independent Hamiltonian inthe accelerated frame the probability distribution over the energy levels of H
will still have a thermal form This follows from general symmetry propertiesof the vacuum correlation functions in the Rindler coordinate system andis closely related to PCTinvariance An interesting complication isthat the local temperature of such an extended system would vary over theextension of the system On the other hand such a variation is also presentfor a hot system in a gravitational eld where the variation in temperatureis induced by the redshift which follows from dierences in the gravitationalpotential
A system of linearly accelerated electrons could in principle be used todemonstrate the Unruh eect in the way discussed above Thus the spinpolarization would depend on the spin precession frequency as
P R
R tanh
c
a
and by varying the strength of the external magnetic eld the functional formof P could be demonstrated Unfortunately this is not a realistic situation forreal particle accelerators If we use a value for the electric eld E MVmwe nd a rather low corresponding temperature T K Howeverthe main complication is the long time for reaching equilibrium From theexpression for the transition probabilities we nd a typical time of sIn the lab frame this would be enhanced even further by the time dilatationeect So this is not very promising If much larger accelerations can beobtained the thermalization time would be strongly reduced it varies witha as a but the limited time available in a linear accelerator is neverthelessa serious problem
The limitations present for linear accelerators are not there for circular
-2 -1 1 2 3
-1
-0.5
0.5
1
conventional
thermal
P
g
Figure The equilibrium polarization P as a function of g The curve referred to as conventional represents the conventional result whereas the curve referred to as thermal is based on the use of the relation between temperature and acceleration derived in the caseof linear acceleration
accelerators Thus much higher values for the proper acceleration are obtained mainly due to the relativistic factor for the transformation from thelab frame magnetic eld to the rest frame electric eld A typical accelerationlimited by synchrotron radiation eects is a ms corresponding toa temperature of T K And more importantly for electrons in a storagering the time needed to reach equilibrium is available typically of the order ofminutes to hours
In the magnetic eld of a circular accelerator the electrons will graduallybuild up a transverse polarization due to spin ip radiation This is wellknownfrom calculations by Sokolov and Ternov and later by others and the eecthas been seen in real accelerators The polarization will under ideal conditionsreach the equilibrium value of
In the gure the theoretical curve for the polarization as a function of thegfactor is shown In the same gure also the corresponding curve based onthe simple formula obtained in the case of linear acceleration is shownMuch of the discussion of the Unruh eect in storage rings has been based oncomparison of these curves For some critical remarks see Jackson In thefollowing I will examine this question again There are two important pointsinvolved in understanding the similarity and dierence between the two curvesThe rst is the question of the dierence between linear acceleration and acceleration in a circular orbit The other question concerns the approximation
where we treat the electron as a point detector
Stationary world lines
The world line of a uniformly accelerated particle and the world line of aparticle moving with constant speed in a circular orbit are special cases ofwhat has been referred to as stationary world lines These spacetime curvesare self similar in the sense that there is no geometric dierence between twopoints on the trajectory Such a curve can be generated by a timeindependentPoincare transformation which in general will involve rotation in addition toboostb
H H a
c K J
In the same way as for linear acceleration H can be seen as a time evolutionoperator which jumps between inertial frames the instantaneous rest framesof the particle It therefore generates a full accelerated coordinate system inMinkowski space a system where the particle sits at rest This is a stationarysystem in the sense that the spacetime metric is independent of the timeparameter The operator H can also be interpreted as the Hamiltonian ofa quantum system described in the accelerated frame For the acceleratedparticle H is time independent when the accelerating elds are stationary inthis frame
By making use of the freedom to choose orientation of coordinate axis H
can be brought into the form
H H a
cKx z Jz x Jx
The physical interpretation of the parameters is that they correspond to acceleration and angular velocity of a stationary frame moving with particle asmeasured relative to an inertial rest frame but they also have a geometricinterpretation as curvature torsion and hypertorsion of the world line of theaccelerated particle
The trajectory of uniform linear acceleration corresponds to x z and the circular orbit with constant acceleration and velocity corresponds tox z av with v as the velocity of the particle It is interesting to notethat a continuous interpolation can be made between the two cases by changingz while x This looks as a purely formal interpolation since v has to
bSuch motion has been referred to as group motion and has been examined in the contextof relativistic Born rigid motion
2 4 6 8 10
-10
-7.5
-5
-2.5
2.5
5
7.5
10
x
y ω>a/c
ω<a/c
ω=a/c
Figure Stationary world lines projected on the x yplane for varying values of the angularvelocity with xed proper acceleration a
exceed the speed of light but that is not really the case If the circular motionis not described in the rest frame of the center of the orbit but rather in therest frame of the circulating particle at one of point of the orbit then there is asmooth interpolation between physically realizable trajectories For z acthe trajectories in the x yplane are cycloids ie they are periodic orbitsFor z ac they are nonperiodic see gure The limit case z ac canbe interpreted as the limit of circular motion where the radius tend to innityand the velocity v c while the proper acceleration is xed
In the same way as for linear acceleration all the acceleration dominatedtrajectories z ac have an event horizon This horizon disappears in thelimit z ac and is not present for the rotation dominated trajectoriesz ac ie for circular motion It is interesting to note the similarity withthe situation of a rotating Kerr black hole The black hole is characterized bytwo parameters mass and spin or surface gravity and angular velocity Forxed mass there is an upper limit to the angular velocity for which the eventhorizon exists For angular velocities beyond this one has the unphysicalsituation with a spacetime singularity but no event horizon In addition tothe event horizon a rotating black hole is characterized by the presence of astatic limit a limiting distance from the black hole within which no physicalobject can be stationary with respect to a distant observer The circularmotionin Minkowski space corresponds to a situation with no event horizon but thereis in this case a static limit outside which no object can be stationary in theaccelerated coordinate system The presence of this limit is easy to understand
due to the rotation points with xed space coordinates will at some distancefrom the center of rotation move with a velocity larger than the velocity oflight At such a point a physical body cannot stay xed
As previously mentioned a detector which is uniformly accelerated throughMinkowski vacuum is related to a detector which sits at rest in a stationarycoordinate frame outside a large black hole The black hole can be viewedas a thermodynamic system and in a stationary state the detector will be inthermal equilibrium with the black hole with a locally determined temperature that depends on its position due to the redshift eect This suggeststhat in a similar way a detector following a trajectory with z is relatedto a stationary detector outside a rotating black hole Also a rotating blackhole can be viewed as a thermodynamic system with the angular velocity nowacting as a chemical potential for the conserved angular momentum Howeverin this case we do not expect the occupation probabilities over the detectorsenergy levels to be determined simply by the temperature and angular velocityThis is because the detector will not be coupled to the black hole in a rotationally invariant way In a similar way rotational invariance will be broken in theaccelerated frame for any trajectory with z and there is no reason whythe probability distribution in this case should have a thermal form Only forz ac that will be the case The deviation from thermal form is wellknownfor circular motion and also for the case z ac this has been discussed
However even though there is no exact temperature associated with themotion where z the notion of an eective temperature is meaningful ashas been discussed before The vacuum state in some approximate meaningseems hot in the accelerated frame and in addition there is a relative rotationbetween the stationary detector and the vacuum In the following I will reexamine the case of the circulating electron from this point of view Theintention is to show that such a picture of the vacuum state is relevant and todemonstrate that much of the discrepancy between the polarization curve andthe curve derived from the assumption of thermal excitations is due to the waythe electron works as a detector
Electrons in a storage ring
When we consider electrons in a storage ring under ideal conditions wherethey move in a rotationally symmetric magnetic eld and with the correctiondue to radiation loss neglected they can be described by a timeindependentHamiltonian H of the form previously discussed
H H a
cKx a
vJz
This Hamiltonian refers to an accelerated rotating coordinate system whichrotates with the frequency of electrons moving along a classical referencetrajectory in the magnetic eld The transformation between the acceleratedand the lab frame coordinates can be written as
x x R cos av y sin av
y y cos av x R sin av
z z
t vyc
with R as the bending radius of orbit and as the relativistic gamma factorThe electrons described by the Hamiltonian are not restricted to the refer
ence trajectory x y Also quantum uctuations in the orbital motionare included However we may assume deviation from this orbit both in position and velocity to be small which makes it possible to linearize in thedeviation and to consider a nonrelativistic approximation
The operators H Kx and Jz are Dirac operators in the inertial rest frameof the reference trajectory at time
H c mc e g eh
mci E B
Kx
cxH Hx
Jz xpy ypx
hz
A term for the anomalous magnetic moment g has been introduced inthe expression for H All the coordinates and elds refer to the inertial restframe of the reference trajectory but here and in the following I will omit theprimes on these coordinates and elds p e
cA is the mechanical moment
of the electron and and are the standard Dirac matricesWhen a FoldyWouthuysen transformation is performed and the equations
are linearized in the orbital uctuations the Hamiltonian can be reduced toa nonrelativistic form which involves only spin and vertical oscillations Thecoupling between the horizontal oscillations and the spin can be consideredas a higher order eect when we primarily are interested in the spin degreeof freedom However the coupling to vertical oscillations cannot be neglectedas has been discussed in earlier papers When we take into account only theexternal accelerating elds the resulting electron Hamiltonian gets the simpleform
He p
m
mz
hz
mchpy h
cgzx
where p is the momentum in the zdirection acg is the spin precession
frequency A conning harmonic oscillator potential has been introduced forthe vertical motion In a weak focussing machine this is introduced by agradient in the magnetic eld e
mBr
The coupling to the radiationeld gives an additional term
H eh
mcg B e z Ez
which we treat as a perturbation The elds B and Ez are rest frame eldsThus the transformation to the accelerated frame gives us a fairly simple
and straight forward way to calculate the equilibrium properties of the electronbeam We rst determine the eigenvectors and eigenvalues of He which isa twolevel system coupled to a harmonic oscillator and we then nd theoccupation of the levels by calculating the transition probabilities induced byH
The coupling terms between spin and orbital motion in are normallyquite small If they are treated perturbatively we nd to rst order the following expressions for the eigenvectors of He
jni jni i
rh
mc
g
pn
jn i
g
pn
jn i
jni is the eigenvector of the uncoupled system with n referring to the harmonic oscillator and to the spin levels Note the resonance between spinand orbital motion for The energy levels then are degenerate andimproved expressions close to resonance could be found by use of degenerateperturbation theory but I will not do that here Also note that to higherorder in perturbation theory also higher order resonances for n will bepresent
Let us rst consider the transition matrix elements for nonspin ip transitions It is clear from the form of H that the small terms of will only givesmall contributions to the matrix elements If they are neglected the result is
hnjH jni ie
rh n
mEz
For spin transitions the situation is dierent Even if the coupling terms aresmall they give signicant contributions since they are inuenced by the coupling of the charge to the radiation eld which is much stronger than the spin
coupling The result for spin ips is
hnjH jni eh
mc
gB
g
Ez
The equilibrium populations pn of the energy levels can be found be assuming detailed balance Thus the transition probabilities can be expressedin terms of the electromagnetic elds as for linear acceleration and the relative populations can be determined as the ratio between the probabilities fortransition between the levels one way and the other Thus for xed spin theratio is determined by the matrix element and thereby by the correlationfunction of the rest frame electric eld
R pnpn
CE
CE
CE
Zd exp i hEz Ez i
Note that the ratio R is independent of the state n This implies thatthe excitation spectrum has a thermal form
pn N exp n lnR
with a frequency dependent temperature
eTeff h lnR
The relative population of spin up and down for the same n is determinedby ratios between spin ip transitions It has a similar form
R pnpn
D
D
but now with a composite correlation function
D gCB gCEB CE
where CB is the correlation function of the magnetic elds B and B andCEB is the mixed correlation function of B and Ez
The resonance term of is small except close to the resonance Ifit is neglected the expression for the relative populations of the spinlevels will reproduce the standard result for the equilibrium polarization P
-2 -1 1 2 3 4
-1
-0.5
0.5
1
g
P
indirect
direct total
Figure Evaluated polarization curves based on Eq in the text Total refers to thefull expression and agrees with conventional result displayed in Fig Direct refers tothe curve when uctuations in the orbit are suppressed and indirect refers to the curvewhere only the spin excitations due to uctuations in the orbit are retained
RR earlier shown in Fig It is interesting to see how this is builtup from contributions from the direct coupling between the spin and magneticeld and the indirect one transmitted through uctuations in the orbit Theeect of the uctuations is demonstrated by not including the two terms CEB
and CE in The uctuations in the orbit are then eectively suppressedand only the coupling to the magnetic moment of the electron is retained Theresult is a changed curve P g denoted direct in Fig It indeed has aform very similar to the one derived from the temperature formula denoted thermal in Fig The eective temperature indicated by the curve is howeversomewhat higher than the Unruh temperature TU for the same acceleration
It is instructive also to consider the curve obtained if the contributionsto the spin transitions caused by the direct coupling is suppressed and onlythe indirect one due to uctuations in the orbit are retained That meansincluding only CE in Eq The result is referred to as indirect in FigIt is very similar to the one obtained from the direct coupling to the radiationeld although diering by a shift of units along the gaxis
The relative shift of the curves obtained from the direct and indirect coupling of the spin to the radiation eld is fairly easy to understand The directcoupling of the radiation eld to the spin is rotationally invariant and thecorresponding occupation probabilities have approximately a thermal formin the frame which is nonrotational with respect to the vacuum This is
-1 1 2 3
-1
-0.5
0.5
1
-4 -3 -2 -1
-0.99
-0.98
-0.97
-0.96
-0.95
-0.94
g
P
g
P
AB
A
B
Figure Comparizon of the conventional polarization curve A with the one obtained byreplacing the true correlation functions along the accelerated path with thermal correlationfunctions B
the FermiWalker transported frame where the direct coupling term is proportional to g The vertical uctuations are insensitive to rotations in thex yplane but the coupling between the orbital motion and the spin is timeindependent in the stationary frame which is rotating relative to the FermiWalker transported frame This gives rise to occupation probabilities whichare approximately thermal in the stationary frame The relative rotationbetween these two frames is represented by the shift along the gaxis
Based on this understanding we may conclude that much of the dierencebetween the two curves in Fig is due to the way the electron acts as a detectorSee also the related discussion by Unruh As a nal point I will illustratethis by considering a hypothetical situation where the detector dened by theelectron Hamiltonian and the coupling term is put in contact withan electromagnetic heat bath and where the detector is rotating relative to thethermal state Thus the true correlation functions along the orbit are replacedby thermal correlation functions in the evaluation of the polarization In Figthe correct polarization curve is compared with the modied curve obtained inthis way The temperature of the thermal state as well as the angular velocityare used as tting parameter We note that a good approximation is obtainedfor an eective temperature Teff TU and an angular velocity only slightlydierent from true one ac There are some details though whichare dierent in the two cases and which show that the correct curve is nottruly thermal
Concluding remarks
An interesting observation made by Fulling and others is that the naturaldenition of a vacuum state is related to the choice of the spacetime coordinates In a curved spacetime this makes the notion of a vacuum statehighly nontrivial and physical eects like the Hawking radiation may berelated to the question of what is the correct vacuum state
Even in at space there are nontrivial vacuum eects associated withaccelerated coordinate frames For a linearly accelerated system the Rindlercoordinate system Minkowski vacuum appears as a thermally excited stateAlso for other stationary coordinate systems which include rotation and notonly acceleration Minkowski vacuum appears as an excited state althoughnot characterized by a uniquely dened temperature
It is highly interesting that these vacuum eects that usually are considered to be detectable only under extreme situations can be related to measurable polarization eects of electrons in a storage ring The natural way tosee this connection is to describe the electron beam in the accelerated frameof an orbiting reference particleThe electron can be described as simple quantum mechanical system which includes the vertical motion coupled to the spinand with transitions between the states of this system induced by the radiationeld Since the Minkowski vacuum state appears excited in this frame transitions both up and down in energy are induced and equilibrium is produced asa balance between these two processes
Due to the coupling between spin and orbital motion the electron acts asa nontrivial detector Thus the system is not rotationally invariant and thiscomplicates the detection of the vacuum eects The vacuum can be seen asbeing hot in the accelerated frame not in the stationary frame of the detectorbut rather the nonrotational FermiWalker transported frame In fact up tominor details the correct polarization curve can be reproduced if the rotatingdetector dened by the electron Hamiltonian is excited by an electromagneticheat bath rather than the Minkowski vacuum state
The description of the orbiting electrons in the accelerated frame is natural for the discussion of how the spin eects are related to the Unruh eectBut I would also like to stress that it gives a conceptually simple way to studythe quantum eects of electrons in a storage rings Under the ideal conditionsconsidered here the electron is described as a harmonic oscillator coupled to atwolevel system with transition probabilities determined by correlation functions of the radiation eld It should be of interest to examine this approachfurther also beyond the simplest approximation used here
References
S Fulling Phys Rev D PCW Davies J Phys A WG Unruh Phys Rev D S Hawking Nature Comm Math Phys JS Bell and JM Leinaas Nucl Phys B P Chen and T Tajima Phys Rev Lett R Chiao private communication AA Sokolov and IM Ternov Dokl Akad Nauk SSR
!Sov Phys Dokl " YaS Derbenev and AM Kondratenko Zh Eksp Teor Fiz
!Sov PhysJETP " VN Baier Usp Fiz Nauk !Sov PhysUsp
" JS Bell and JM Leinaas Nucl Phys B WG Unruh Acceleration radiation for orbiting electrons in Quantum
Aspects of Beam Physics Ed Pisin Chen World Scientic DP Barber Unruh eect spin polarization and the Derbenev
Kondratenko formalism in Quantum Aspects of Beam Physics Ed PisinChen World Scientic
JM Leinaas Accelerated electrons and the Unruh eect in QuantumAspects of Beam Physics Ed Pisin Chen World Scientic
W Rindler Am J Phys G Sewell Ann Phys NY RJ Hughes Ann Phys NY JS Bell RJ Hughes and JM Leinaas Z Phys C JD Jackson Rev Mod Phys JD Jackson On eective temperatures and electron spin polarization in
storage rings in Quantum Aspects of Beam Physics Ed Pisin ChenWorld Scientic
JR Letaw Phys Rev D G Salzman and AH Taub Phys Rev
Relativistic Jets in Microquasars
Felix Mirabel
CEADSMSAp CESaclay France IAFECONICET Argentina
The discovery and subsequent study of microquasars lead to major progress in ourunderstanding of the nature of relativistic jets seen elsewhere in the universeand the connection between the accretion onto compact objects and the formation of collimated jets A detailed account of the major progress accomplished untilpresent was published in Annual Review of Astronomy Astrophysics Mirabel Rodrguez Here I review the questions that remain unanswered as wellas the future perspectives that this new eld of research is opening
Major progress accomplished
In relativistic sources located in the Milky Way twosided moving jets can beobserved and therefore can be overcome several of the ambiguities that haddominated the physical interpretation of extraglactic jets where so far onlythe motions of oneside of the jets could be followed From the observation oftwosided moving jets in microquasars the system of equations can be solvedFor the rst time an upper limit of the distance to a microquasar was derivedfrom the proper motions using special relativity constraints see Mirabel Rodrguez
Major progress has also been made in the understanding of the accretionejection phenomenology From multiwavelength observations the connection between accretion ow instabilities observed in the Xrays with theejection of relativistic plasma observed at radio infrared and possibly opticaland Xray wavelengths has been established on a rm basis After the softercomponent of the Xray emitting plasma disappears the inner accretion diskis rapidly reestablished and the plasma that produces the hard Xray component is blown away Mirabel et al in the form of collimated relativisticjets Dhawan Mirabel Rodrguez
The Xray source GRS CastroTirado et al has becomea main target to study accreting black holes of stellar mass However thediscovery and study of other microquasars is important not only to enhancethe statistical sample but also to make progress on the dierent aspects of thediverse phenomenology that each new object of this class has revealed Mirabel Rodrguez
Open questions
What is a microquasar
By this term we usually design stellarmass black holes that mimic on a smallerscale many of the phenomena seen in quasars The question is whether under this concept we should also include neutron star systems with jets Although the physical meaning of the quasarmicroquasar analogy as proposedby Mirabel Rodrguez is valid for black holes because in many casesthe nature of the compact object is unknown we may currently include in thissubclass of stellar sources neutron star binaries that produce relativistic jets
Are all accreting black holes of stellar mass microquasars
All microquasars are accreting compact objects of stellar mass but is thereverse true This question can be reformulated as follows do all Xray blackhole binaries produce jets From the theoretical point of view jets are needed toliberate angular momentum On the other hand observations show that Xrayemitting black holes in addition to possible synchrotron emission associatedto sporadic outbursts always exhibit atspectrum compact counterparts ofsynchrotron emission It has been proposed Fender and demonstratedby high resolution images Dhawan Mirabel Rodrguez that these at spectrum compact radio counterparts are thick collimated jets of AU sizescales Furthermore from multiwavelength observations at radio infraredand Xray wavelengths of the largescale jets in SSW we know thatin some circumstances jets exist without being seen Therefore the answerto this question is that probably all accreting black holes of stellar mass aremicroquasars
Are the Lorentz factors of the bulk motions in the jets from quasars dierentfrom those in microquasars
In other words is the quasarmicroquasar analogy valid The microquasar jetsobserved so far have bulk speeds that are statistically smaler than in quasarsFor instance in quasars are found jets with bulk speeds of up to c butno microquasar with such jets has so far been found Is this due to somefundamental dierence that would invalidate the physical analogy betweenquasars and microquasars or is it rather due to selection eects related to theDoppler favoritism needed in order to measure the proper motion of the ejectain distant quasars Although from the observational point of view this remainsan open question theoretically it is not clear why the bulk motion in jets from
supermassive black holes would be dierent from that in the jets from stellarmass black holes
Could the terminal velocity of the jet be a diagnostic for the nature of thecollapsed object NS versus BH
In all areas of astrophysics there have been increasing evidences that accretionis always related to the production of collimated jets The observed out owspeeds seem to be of the order of the escape velocity from the surface of theaccreting object Objects as diverse as very young stars nuclei of planetarynebulae and accreting white dwarfs have jets with nonrelativistic velocities km s whereas neutron stars and black holes produce jets withrelativistic speeds V c Livio Mirabel Rodrguez If thepresent trends between the escape velocities from the accreting objects andthe velocity of the out ows were conrmed as a strong correlation it wouldimply that gravity is important and dominates over MHD mechanisms Meieret al However it is not clear whether we will be able to discriminatebetween neutron stars and black holes by knowing the terminal velocity of thecollimated outows For instance Sco X which problably contains a neutronstar has lobes moving with an average velocity of c Fomalont Geldzahler Bradshaw Since only a handful of relativistic jet sources have beendiscovered so far in the Galaxy Mirabel Rodrguez we need thediscovery and study of more sources to obtain a statistically signicant numberto answer this question
Why are QPOs in microquasars and not in AGNs
The scale of time of the instabilities in the accretion disk of black holes isproportional to the mass of the black hole QPOs with periods of sec inblack holes of M would correspond to QPOs of day to years inblack holes with masses of and M respectivelly QPOs of sec inmicroquasars correspond to typical uctuations of of the Xray ux AtXrays the companion binary stars are much dimmer than the accretion diskOn the other hand the accretion disk in AGNs is cooler Mirabel Rodrguez and QPOs should be observed in the UV and optical wavelengths atwhich stars mostly radiate and interstellar absorption is important SinceAGNs may be blurred by nuclear starbursts it may be dicult to detect uctuations of from AGNs that are embedded in starburst nuclei
One then may ask about the analogous of the large scale oscillations of the ux with periods of min regularly observed in GRS Greiner Morgan Remillard In black holes with masses in the range
of to M analogous QPOs would correspond to periods in the rangeof to years which are dicult to monitor in human timescales SamsEckart Sunyaev
To image the jet close to the black hole do microquasars oer an advantagewith respect to AGNs
No The Schwarzschild radii of BHs are proportional to their mass and inthese units nearby AGNs can be imaged closer to the BH than microquasarsFor instance M which is in the Virgo cluster at a distance of Mpc andcontains a BH of M has been imaged with a resolution of timesthe Schwarzschild radius Junor Biretta Livio A given angular sizeprojected on the supermassive BH of M is in terms of its Schwarzschildradius smaller than the same distance projected on a BH of M at adistance of kpc from the Sun Therefore relative to supermassive BHs in theLocal Universe the study of stellar mass BHs presents an advantage becauseof the time scales but not as far as the dimensions of length are concerned
What is the connection between accretion and ejection
In GRS large amounts of Xray emmiting plasma L disappear in less than a few seconds and soon after synchrotron jets are formedFender Pooley Eikenberry et al Mirabel et al Recentanalysis shows that the Xray ux that suddenly disappears only correspondsto the softer component keV The formation of the jets starts later atthe time of a spike that consists of a sudden increase in the ux of the softcomponent simultaneous with a decrease of the ux of the harder component keV In the context of current models these observations imply thatthe inner accretion disk rst disappears Belloni et al and when theinner accretion disk is being reestablished a shock is produced which triggersthe blow up of the plasma that was emmiting the hard Xrays in the form ofcollimated jets at relativistic speeds Dhawan Mirabel Rodrguez InGRS an analogous correlation between the sudden disappearanceof the keV measured by BATSE and major ejection events seen in theradio with the VLA had been observed see Mirabel Rodrguez forreferences and the discussion of this issue
What fraction of the inow goes into the jets
The observations described above indicate that the matter and energy emittingin the soft Xrays that suddenly disappears is perhaps advected into the BH
since it is not immediately translated neither in hard Xrays nor in synchrotronjets Because of the uncertainties in estimating the mass of the Xray emittingplasma it is dicult to estimate the fraction of the in ow that goes into thejets On the contrary assuming equipartition the amount of mass and energyof the jets can be calculated Fender Pooley Mirabel et al
Now there are increasing evidences that the power of the jets may be alarge fraction of the accretion power For instance in its very high stateGRS may have a shortterm jet power of erg s Mirabel Rodrguez Fender which is a large fraction of the observedaccretion power This is consistent with the model for a rapidly rotating BHwith a high accretion rate by Meier
What is the mechanism that launches the jets
MHD power which was nicely reviewed by Meier and may play animportant role However it is still unclear the relative importance of therotating thin accretion disk mechanism by Blandford Payne and thatfrom the framedragged accreting matter inside the ergosphere of a rotating BHBlandford Znajek The Xray spectral properties of microquasarswith powerful jets eg GRS GRO J indicate that theseare rotating BHs with spins near maximum Zhang Cui Chen whichis consistent with the Blandford Znajek mechanism On the other handthe Blandford Payne mechanism is consistent with the following two recentobservations the large opening angle of of the jet in M JunorBiretta Livio which seems to be the two dimensional image of a jetwith a magnetic polar eld angle The M jet strongly collimates at Schwarzschild radii rs from the BH collimation continuing out to rs The semicontinuous emanation of the jets in the infrared and radioduring time intervals of the order of minutes as observed in GRS Eikenberry et al Fender Pooley Mirabel et al
Although several numerical simulations have been made there are still nodenitive tests to discriminate the relative role of these two MHD mechanisms
What are the QPOs of maximum x frequency
Because QPOs with maximum x frequency have been observed many timesin some microquasars eg Hz in GRS Morgan Remillard Greiner it is believed that they are related to fundamental propertiesof the BHs such as its mass and spin The problem is that there are morethan dierent alternative explanations in terms of General Relativity see
Mirabel Rodrguez for references The theories should provide theobservational tests that would discriminate among the models proposed
Are the jets discrete plasmons or semicontinuous ows with internal andorexternal shocks
Internal shock models of microquasar jets Bodo Ghisellini KaiserSunyaev Spruit propose that the plasma velocity is smaller than thepattern velocity Outbursts in the core lit up the jets are by shock frontsthat travel along the jets which accelerate the relativistic particules that emitthe synchrotron radiation Internal shock models relax the requirements onthe power of the central engine because much of the energy underlying theoutbursts is stored in the continuous jet An alternative model of the twinmoving radio lobes observed in Sco X Fomalont Geldzahler Bradshaw is that they are intetraction of the energy ow from beams with theinterstellar medium The lobes advance at c but the beam velocity isc While the observations of GRS and SS would be moreconsistent with the internal shock model the moving lobes in Sco X seemto be shocks in working surfaces of an external medium While the observedvelocities of the ejecta in GRS c and SS c havebeen the same over several years whereas in Sco X the speeds for dierentpairs of components at dierent times range between c and c Thissuggests that dierent physical processes may dominate in dierent sources
Jets in microquasars ray burst GRB afterglows and AGN show analogous phenomenologies It is interesting that the internal shock model originallydeveloped for GRBs Meszaros Rees and the external shock model originally proposed for the terminal lobes in AGNs Blandford Rees arenow being applied to microquasars Kaiser Sunyaev Spruit FomalontGeldzahler Bradshaw Conversely the plasmon model originally proposed to interpret the microquasar jets is being proposed in the cannonballmodel of GRBs by Dar De Rujula
Have extragalactic microquasars been identied
Ultraluminous Xray compact sources have been identied in several nearbyspiral galaxies Colbert Mushotzky Makishima et al as wellas in dwarf galaxies Mirioni Pakull These enigmatic sources haveluminosities in the range of Lx erg s and it has been proposedby Colbert Mushotzky that they are BHs of M Makishimaet al propose that some of these superluminous sources are BHs as themicroquasars GRS and GRO J with masses below M
and high disk temperatures because rapid rotation gets the disk closer to theBH hence hotter Unfortunately the VLBA array does not have the sensitivity to image in its plateau state extragalactic supermicroquasars as GRS
Why superluminal microquasars have unbroken power law photon spectra
Grove has shown that the microquasars with superluminal jets havepowerlawgammaray states with an index of Contrary to CygnusX and E the superluminal sources GRO J and GRS have unbroken power laws up to almost MeV with no indicationof Comptonization A similar photon spectrum has been observed by SAX inXTE Frontera et al Since we don t know what is the originof the electrons that produce the gammaray photons it is still unclear whysome BHs would have broken whereas others have unbroken power law photonspectra The sensitivity of INTEGRAL in this domain of energy will certainlyprovide new perspectives on this issue
Will microblazars be found
In most microquasars where the angle between the line of sight and the axisof ejection has been determined large values are found Namelyexcept some exceptions as in the case of Sco X where FomalontGeldzahler Bradshaw the axis of ejection in most cases is close tothe plane of the sky This is consistent with the statistical expectation sincethe probability of nding a source with a given is proportional to sin We then expect to nd as many objects in the range as inthe range However this argument suggests that we shouldeventually detect objects with a small For objects with we expectthe timescales to be shortened by and the ux densities to be boostedby with respect to the values in the rest frame of the condensation Forinstance for motions with v c the timescale will shorten bya factor of the ux densities will be boosted by a factor of andthe photon spectrum of the source will be very hard Then for a galacticsource with relativistic jets and small we expect fast and intense variationsin the observed ux Microblazars may be quite hard to detect in practiceboth because of the low probability of small values and because of the fastdecline in the ux
Could microquasars be unidentied sources of gammarays detected by EGRET
There is the indication that the synchrotron component of the jets in microquasars could reach the Xray domain Marko Falcke Fender Thiswould require the ejection of plasma containing electrons with Lorentz factors as in Blazars In this context the recent observation by Paredes et al of persistent radio jets from LS which is located in the error box ofan EGRET source raises the possibility that microquasars could be persistentsources of gammarays
Could microquasars be sources of eV cosmic rays
Due to opacity by cosmic infrared photons the sources of eV cosmicrays must be within Mpc from the Galaxy AGNs have been proposed aspossible sources of such high energy particles As shown above the bulk andintrinsic Lorentz factors of the electrons in microquasars are comparable tothose in AGNs Since they are much closer and numerous than the later thevery high energy cosmic rays could be produced by shocks in microquasar jets
Are Gammaraybursts extreme microquasars
Gammaray bursts are at cosmological distances and ultrarelativistic bulkmotion and beaming appear as essential ingredients to solve the enormousenergy requirements CastroTirado et al Beaming reduces the energyrelease by the beaming factor f !" where !" is the solid angle ofthe beamed emission Additionally the photon energies can be boosted tohigher values BHs formed by core collapse producing out ows with bulkLorentz factors have been proposed as sources of ray bursts Meszaros Rees Recent studies of gammaray afterglows suggest that they arehighly collimated jets since breaks and a steepening from a power law in time tproportional to t ultimately approaching a slope t have been observedin light curves CastroTirado et al
It is interesting that the power laws that describe the light curves ofthe ejecta in microquasars show similar breaks and steepening of the radio ux density Rodrguez Mirabel In microquasars these breaks andsteepenings have been interpreted Hjellming Johnston as a transitionfrom slow intrinsic expansion to free expansion in two dimensions Besideslinear polarizations of about were recently measured in the optical afterglows providing strong evidence that the afterglow radiation from gammaraybursters is at least in part produced by synchrotron processes Linear polarizations in the range of have also been measured in microquasars at
radio Rodrguez et al Hannikainen et al and optical Scaltritiet al wavelengths
In this context microquasars in our own Galaxy seem to be less extremelocal analogs of the superrelativistic jets associated to the more distant ray bursters which are jets with Lorentz factors orders of magnitude largerTherefore the physical mechanism that launches the jets in ray bursters arelikely to be dierent to the microquasars Furthermore GRBs do not repeatand seem to be related to catastrophic events and have much larger superEddington luminosities Therefore the scaling laws in terms of the black holemass that are valid in the analogy between microquasars and quasars do notseem to apply in the case of ray bursters
Acknowledgements
This work was partially supported by Consejo Nacional de InvestigacionesCientcas y Tecnicas de Argentina
#$ Belloni T Mendez M King AR van der Klis M van Paradijs J Ap J L
#$ Blandford RD Payne DG MNRAS #$ Blandford RD Rees MJ MNRAS #$ Blandford RD Znajek RL MNRAS #$ Bodo G Ghisellini G Ap J L#$ CastroTirado AJ et al Astrophys J Supp Ser #$ CastroTirado AJ et al Science #$ Colbert EJM Mushotzky RF ApJ #$ Dar A De Rujula A astroph#$ Dhawan V Mirabel IF Rodrguez LF ApJ #$ Eikenberry SS Matthews K Morgan EH Remillard RA Nelson RW
Ap J L#$ Fender RP astroph#$ Fender RP Pooley GG MNRAS #$ Fomalont EB Geldzahler BJ Bradshaw CF submitted to
ApJ Letters#$ Frontera L et al Proceedings of the IV Integral meeting Ali
cante Sept #$ Greiner J Morgan EH Remillard RA ApJ L#$ Grove JE ASP Conference Series p#$ Hannikainen DC et al ApJ #$ Hjellming RM Johnston KJ Ap J #$ Junor W Birettta JA Livio M Nature
#$ Kaiser CR Sunyaev R Spruit HC AA #$ Livio M Physics Reports #$ Makishima K et al ApJ #$ Marko S Falcke H Fender R astroph#$ Meier DL Edgington S Godon P Payne DG Lind KR Nature
#$ Meier DL Review in this issue#$ Meszaros P Rees MJ Ap J L#$ Mirabel IF Rodrguez LF Nature #$ Mirabel IF Rodrguez LF ARAA #$ Mirabel IF et al AA L#$ Mirioni L Pakul M Private communication#$ Morgan EH Remillard RA Greiner J ApJ #$ Paredes J M Marti J Ribo M Massi M Science #$ Rodrguez LF et al ApJSupp #$ Rodrguez LF Mirabel IF Astron Astrophys L#$ Sams BJ Eckart A Sunyaev R Nature #$ Scaltriti F et al AA L#$ Zhang NS Cui W Chen W ApJ L
BEPPOSAX AND EUSO : TWO APPROACHES FOR THE EXPLORATION OF THE “ EXTREME UNIVERSE “.
LIVIO SCARSI Istituto di Fisica Cosmica e Informatica / CNR and University of Palermo
BeppoSAX and EUSO, two space missions apparently operating in distant chapters of science , share their interest in the same fundamental field concerning “ extremes “ in astroparticle physics. The processes concerned are those of “ explosions “ involving an energy release up to (10^ 52 – 10^54 ) erg in time spans of seconds or fraction of a second (Gamma Ray Bursts ) occurring at cosmological distances ( z >~ 1 ) or the acceleration / decay of Cosmic Ray particles with energy up to E >1020 eV. In this contribution an overview along these lines is given for BeppoSAX and EUSO.
1 Introduction
Gamma Ray Bursts ( GRBs ) and Cosmic Ray particles with energy E > 1020 eV (EECRs) represent two " extremes " in the energetic of the Universe.
GRBs, localized at cosmological distance, reveal explosive phenomena involving an energy output in the range 1052 to 1054 erg lighting up with a powerful flash the normally gray gamma ray sky in the “tens of keV – MeV” band; the "Burst", lasting from a fraction of a second to seconds, is equivalent to the burning, in this interval of time, of a solar mass or the total energy output of our Galaxy in more than a century. BeppoSAX, the Italian Satellite with a Dutch participation launched in 1996 and today still in operation, has played a key role in tearing out the curtain of mystery about the GRB nature, still remained closed after 30 years from their discovery.
The observation in the Cosmic Ray flux of particles carrying an energy in excess of 1020 eV (or about 10 joules) was first made by John Linsley in 1962 with the giant ground array for Extensive Air Showers operating at Volcano Ranch in New Mexico ( U.S.A.) ; in these last 40 years , despite the great effort made in the field , because of their extremely low flux, only an handful of about 40 events has been reported by the ground based experiments operating around the world. The energy packed in these particles is many orders of magnitude higher of that obtainable by human controlled particle accelerators and it reaches ultra relativistic values in excess of 1011 for the Lorentz factor. The nature of the EECR events and their sources are still unknown: as for the GRB this chapter represents today a major challenge for science. " EUSO - Extreme Universe Space Observatory " is a mission based on the International Space Station, now in the planning phase and aiming to
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start operations in 2007/8 : EUSO ambition are the solution of the EECR mystery and the opening of the High Energy Neutrino Astronomy channel.
I have been from the very beginning and I am today deeply involved with both ventures, BeppoSAX and EUSO. In the following I will try to sketch for both some of the lights and shadows of their “ life “, always exciting and far from trivial. Apparently they seem far apart, but in depth they share the flavor of the research at the frontier and the taste for the far-out in contrast with the look for a more refined description of a chapter already conceptually known. BeppoSAX is now approaching the end of its orbital life time; EUSO, still in its embryonic status, is struggling to become a valid hunter in the sky.
2 BeppoSAX : a Satellite conceived to introduce Italy in the " Big Space Science " of X Ray Astronomy.
2.1 History.
An " Announcement of Opportunity " for a national scientific satellite was issued in 1981 by the Italian " Piano Spaziale Nazionale " which was at the epoch under the responsibility of the National Council of Research (CNR) and it was later transformed in what is today the " Agenzia Spaziale Italiana - ASI “. A Consortium of 4 Italian CNR Institutes (IFCTR in Milano, ITESRE in Bologna, IAS in Frascati and IFCAI in Palermo, all with previous experience in Cosmic Ray Physics with ground based experiments and in X-Gamma Ray Astronomy within the ESRO / ESA European space program), one dutch Group from SRON/Utrecht and a Group from the ESA Space Science Department entered the context. The proposal submitted adopted the name “ SAX “, a straightforward acronym for " Satellite per Astronomia X ".
The mission was intended to operate as a broad band X ray Observatory covering the interval from the fraction of keV to few hundreds keV .
The main scientific objectives addressed: - Broad band spectroscopy in the energy range 0.1 keV to 300 keV, with
imaging property up to 10 keV ; - Variability studies for bright sources on a time scale from milliseconds
to days and months; - Systematic long term source variability through periodic surveys on
selected regions of the sky; - Gamma Ray Burst investigation.
The scientific payload was based on detectors developed by the partners of the
proposing Consortium and consisted of :
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- A set of co aligned NFI (Narrow Field Instruments with field of view of the order of 1°) to cover the wide dynamic range planned, based on Gas scintillation proportional counters (LECS and MECS from 0.1 keV to 10 keV; HPGSPC from 3 keV to 60 keV ) and NaI/CsI crystal scintillators ( PDS from 30keV to 300keV ) , all coaligned ;
- A set of Wide Field Cameras (WFC) covering a field of view of 20 x 20 degrees, with axis at 90° from the common axis of the NFI, surveying the sky in the 2-30 keV range to monitor the variability of celestial sources and searching for transient sources such as GRBs, thus complementing the activity of the other instruments on board.
A schematic view of the scientific payload (in the version actually flown), is given in Fig.1.
Figure 1. BeppoSAX scientific payload
The Spacecraft requested: - Three axis stabilized about 700 Kg on equatorial low altitude (500-
600 Km) orbit; injection utilizing the Shuttle to join a parking
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configuration and then exploiting the Italian developed intermediate stage IRIS for the final step.
Another proposal (OOXA - Osservatorio Orbitante per X Astronomia) was
submitted, in answer to the AO, for a mission devoted primarily to the investigation with fine spectroscopy of the low energy X ray ( < 2 keV ) coronal emission from conventional stars, effect recently discovered by the Nasa satellite “Einstein” . This proposal was strongly supported by Riccardo Giacconi, Eistein P.I., who was considering the possibility of a substitute for the " Coronal Explorer " derived from " Einstein " and unsuccessfully proposed to NASA by his Group.
Following a preliminary phase of feasibility studies carried out by Industry,
with the determinant advice from a Peer Committee composed by eminent scientists (Edoardo Amaldi, Giuseppe Occhialini, Bruno Rossi and Lodewijk Woltjer), SAX was chosen in July 1982 by PSN for the "go ahead", aiming at launch in 1988 .
But ..... The “Challenger” disaster in 1986 and the consequent cancellation by NASA
of any commercial flight involving the Shuttle (like that regarding the combination IRIS / SAX) imposed for SAX first a long stand by and then a reorientation, with the choice of an expendable vehicle (Atlas Centaur) for the injection in orbit. All that introduced cost increases and substantial delays. The program had a new start in 1989, with a goal for flight in 1995 / 6.
The life of the " new SAX " (the mission did not change its denomination) has
never been easy from the beginning. From 1990 on, the Italian Space Agency was troubled by internal turmoils rotating around a hot dispute and divergences about programmatic issues between the ASI President and Remo Ruffini then Chairman of the Board overviewing the Agency activity for Science . SAX, being the major program in Science, has been involved in the crisis and criticized about its management, putting in question, at moments, even the scientific validity of the mission itself. In his quarrel with the ASI Top, Remo found a powerful and influential ally in Riccardo Giacconi who had turned the initial disappointment for the lost competition in the AO outcome in an open and persistent hostility against SAX, not missing occasion of attacking the mission whenever possible. The period 1992-1993 was particularly turbulent. The SAX issue was brought repeatedly to the press national and international, raising occasionally to “ Nature “, beside being object of several inquiries by "ad hoc inspecting Committees both at the national ( ministerial ) and international ( e.g. European Science Foundation ) level.
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SAX survived. The lift-off occurred from Cape Canaveral on April 30, 1996, successfully, with a " nominal " acquisition of the programmed orbital parameters. On May 15, after having verified the " nominal " working condition for all the spacecraft / payload and related ground operational systems, SAX was dedicated to Giuseppe Occhialini, adopting the denomination “ BeppoSAX “, " Beppo " being the nickname for Giuseppe.
In its lifetime BeppoSAX has performed with extraordinary success and it is my
privilege to report in this Conference about the highlights and main scientific results; the magic venture is still in its going.
I wish however to begin with a " special " announcement, possibly only of a
very personal relevance, but I hope of some interest also for the outsiders: thank to Pisin Chen, today is a reconciliation day between Remo Ruffini and myself. The SAX " war " is over: Science interests and mutual respect have won. A picture with both of us, taken in the garden outside, will certify the happening.
2.2 BeppoSAX : 4 years in orbit . Highlights and main results. Gamma Ray Bursts.
The GRB investigation, although representing only a chapter out of the several in which BeppoSAX has brought fundamental contributions, is considered as the major success of BeppoSAX. The discovery of the GRB X-ray afterglow has brought to the identification of the optical counterpart and at the end to the measurement of the z source value and the attribution of the phenomenon to a cosmological event.
Hunting for the identification of the GRB sources has been one of the scientific objectives listed in the original mission planning for BeppoSAX.
The hunt procedure is sketched in Fig 2 a, b. The WFCs are continuously monitoring for transient X ray events a fraction of about 2.5 % of the sky to search for a possible x ray emission in the ( 3 - 30 ) keV interval accompanying the typical low energy gamma emission of a GRB lighted up in the field of view of one of the WFCs. BeppoSAX is expected to register about an event for month. The WFCs can localize the GRB position in "real time" (in practice at the level of the satellite orbital period of 90 minutes) with a precision of few arc minutes, reaching 3’ with a more refined analysis. The WFC finding allows BeppoSAX to point to the target its Narrow Field Instruments: this occur within a time delay of 5 to 10 hours depending on the requirement to send telecommands to the spacecraft at subsequent passages over the BeppoSAX control station at Malindi (Kenya). The NFIs (essentially LECS and MECS) can detect the presence of the eventual X-ray afterglow persisting after the gamma flash, measure its decay curve and perform a spectral analysis; in addition they can refine the source localization down to the arc
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minute and issue the required information for the pointing of ground based large optical and radio telescopes.
Figure 2.
The first event for which it is has been possible to obtain the complete sequence: trigger by the combination of the on board GRBM (Gamma Ray Burst Monitor) warning and direct detection in the WFC FoV, the pointing of the NFI package, observation of the X Ray afterglow followed by the optical observation of the source, occurred on February 28, 1997 for the GRB970228 (GRBs are identified
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by a 3 two figure tag indicating year, month, and day of occurrence). Fig.3 shows the “historical“ X – Ray image of the GRB970228 afterglow, the first ever observed.
Figure 3. BeppoSAX follow-up observations of the region of the Gamma-ray bursts GRB970228. MECS data. The bright source in the fields is the new X-ray source 1SAX J0501.7+1146 associated with the Gamma-ray burst GRB970228 (IAUC 6576).
Spectral analysis of the optical observation placed the source at z = 0.7, giving
therefore the indication of a cosmological distance for the object. Details about GRB 990510 are given in the set of Figures 4 to 7. Fig.4 shows
the X – Ray images as seen at t = ( 8 – 18 ) hours following the gamma burst and at t = ( 35 – 45 ) hours. The optical observations were carried out by the ESO VLT telescope in Chile. The break in the light curves appearing about 1 day after the gamma flash Fig.5) is suggesting that the emission is beamed ( therefore easing the energetic to values around 1052 erg, down from the 1054 erg value corresponding to an isotropic emission. The detailed spectrum given in Fig.6 places the source at z=1.619. The presence of linear polarization at a level P = 1.7 +/- 0.2 is evidentiated in Fig.7.
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Figure 4. GRB990510: a new spectacular “Gamma Bang” detected by BeppoSAX.
Figure 5. GRB990510 optical afterglow
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Figure 6. GRB990510 ESO VLT spectrum (left).
Figure 7. GRB990510 ESO VLT. First detection of linear polarizxarion of the afterglow of a Gamma-ray burst (right).
34 GRB events have been catched by BeppoSAX in the 41 months intercurring
between January 11, 1997 and June 15, 2000; the very first one, occurred on July 20, 1996, was identified only afterwards in the data archives, too late for attempting a pointing of the NFIs.
The average rate of detection by the WFCs of about 10 / year corresponds to the rate expected: The occurrence of a detectable X–ray afterglow has been revealed in 21 cases (+ other 3 ambiguous), the remaining 10 events being accounted for by the impossibility by technical reasons to perform a follow – on.
From the optical observations the z value has been measured in 7 cases (Table 2) proving without ambiguity a cosmological range of distances for the location of the GRB sources (z around 1).
Burst BATSE Trigger
GRBM Trigger
X-Ray Afterglow
TOO After Hours
Optical Transient
Radio Transient
1 960720 5545 Y - 1038 N/A N/A 2 970111 5773 Y Y 16 N N 3 970228 - Y Y 8 Y, Z=0.7 N 4 970402 - Y Y 8 N N? 5 970508 6225 Y Y 6 Y, Z>0.835 Y 6 971214 6533 Y Y 7 Y, Z=3.418 N 7 971227 6546 N Y 14 N N 8 980109 6564 N - N N N 9 980326 6660 Y - N T (SN?) N 10 980329 6665 Y Y 7 T/h Y 11 980425 6707 Y Y 9 SN/h Y 12 980515 - N Y 10 N N 13 980519 6764 Y Y 9 N Y 14 980613 - N Y 10 Y, Z=1.096 N 15 981226 - N Y 10 N Y 16 990123 7343 Y Y 6 Y, Z=1.60 Y 17 990217 - N ? 6.5 N N 18 990510 7560 N Y 8 Y, Z=1.62 N 19 990625 7619 N - N N/A N/A 20 990627 - N Y 8 Y N 21 990704 - Y Y 7.5 ? N 22 990705 - Y ? 11.5 Y N 23 990712 - Y - N Y, Z=0.43 N 24 990806 7701 Y Y 8 N N/A 25 990907 - Y ? 11 N N/A 26 990908 - N - N N N 27 991014 7803 Y Y 13 N N 28 991105 7841 Y N N N 29 000210 - Y Y 7.2 N N 30 000214 - Y Y 12 N N 31 000424 8086 N N N N 32 000528 - Y Y 12 N N 33 000529 - Y Y 7.5 N N 34 000615 - Y 10 N/A N/A 35 000620 - Y - N N/A N/A
Table 1. 35 GRB observations by BeppoSAX WFC (by 16 June 2000).
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GRB Z 970228 0.695 970508 >0.835 971214 3.412 980329 1.096 980613 0.966 990123 1.598 990510 1.619
Table 2. GRB Z values from optical observations.
To summarize, we can try to sketch the GRB situation as emerging today:
- Gamma Ray Burts are very distant ( z in the region of 1 or even greater ) and extremely luminous objects. In their explosive phase (lasting from a fraction of a second to several seconds) they show a gamma emission in the range from the keV to some hundreds keV (occasionally extending to the GeV region). The energy release in gamma has been observed to reach 1054 erg, in the assumption of isotropic emission. A spectacular otburst , lasting for 10 seconds (GRB 990510 ) outshone the brilliance of the entire Universe.
- GRBs show an “ Afterglow “ in the radio – optical - X domain. The duration of the afterglow observed goes from hours to months.
- In 4 out of 5 GRB Xray afterglows a strong redshifted Fe line has been observed in emission. In one case the Fe line has been observed in absorption (at the phase of onset of the GRB).
The implications of these observations are staggering and appear to be
inconsistent with most proposed models, including popular favorites like ipernovaes and merger of binary neutron stars.
I report a suggestion advanced by L.Stella and M.Vietri: the idea is that the GRB originates in a double event. First a Supernova (Hypernova) explosion takes place, by which a massive star implodes to form a collapsed object, while matter is ejected into space. Then, months – years later, with a second much more powerful explosion the GRB is produced by the collapsed star. It is the dense and Fe rich Supernova ejecta which emit the observed radiation characterized by the Fe line when they are illuminated by the flux due to the GRB: the line appears in absorption at the inset, followed by emission in the following relaxation period of the excited ejecta.
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3 EUSO: Using High Energy Cosmic Rays and Neutrinos as Messengers from the Unknown Universe.
(This section is heavily derived from the contribution made by the Author to the International Workshop: “ Observing Ultrahigh Energy Cosmic Rays from Space and Earth “ organized at Metepec , Puebla , Mexico in August 2000 ).
The Cosmic Radiation can be considered the "Particle channel" complementing
the "Electromagnetic Channel" proper of the conventional Astronomy. The mission “Extreme Universe Space Observatory – EUSO“ is devoted to the investigation of the Extreme Energy Cosmic Radiation (EECRs with E > 51019 eV) and the High Energy Cosmic Neutrino Flux, aiming at the exploration of the highest energy processes present and accessible in the Universe. The results obtained will extend our knowledge about the extremes of the physical word and tackle the basic problems still open with a large impact on Fundamental Physics, Astrophysics and Cosmology. A classic presentation of the Cosmic Ray Energy Spectrum is shown in Fig.8; an unconventional view (which I borrowed from a colleague of Karlsruhe, where it was first shown at the Cerimonial organized in honor of Dr. Shatz) is given in Fig.9 to illustrate in an anthropomorphic perspective the features conventionally nominated “knee“ (around 1015 eV) and “ankle“ (above 51018 eV). The remarkable “feminine leg“ in the figure is that of the famous german movie star Marlene Dietrich.
Figure 8. The Cosmic Ray energy spectrum: classical presentation.
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Figure 9. The Cosmic Ray energy spectrum: anthropomorphic presentation.
Today substantial progresses have been made in the knowledge of the nature of
Cosmic Rays of relatively modest energy (those reaching up to the “knee“ at 1014-1015 eV); the Cosmic Radiation on the higher energy side on the other hand presents us with the challenge of understanding its origin and its connection with fundamental problems in Cosmology and Astroparticle Physics.
Focal points are represented by: i) The change in the spectral index at ~5×1018 eV ("Ankle"); this could correspond to:
- a change in the Primary elemental composition connected with a different source or confinement region in space;
- a change in production mechanism in the original sources; - a change in the interaction process in the first collision inducing the
shower in the Atmosphere. ii) Existence of "Cosmic Rays" with energy E>1020 eV: (EECR) (Fig.10). A direct question arising is: what is the maximum Cosmic Ray energy, if there
is any limit? Addressing the theoretical issue concerning the production and propagation of 1020 eV Primary quanta is problematic and it involves processes still little known. The energy loss mechanism related to the interaction of hadronic particles with the 2.7 Kelvin Universal Radiation Background (Greisen-Zatsepin-Kuzmin effect), conditions the mean free path of Cosmic Radiation. This effect limits the distance of the sources of Primary EECRs to less than 50-100 Mpc, a short distance on a cosmological scale, opening the problems related to the nature of the sources and their distribution in the Universe.
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Figure 10. Cosmic Ray energy spectrum: evidence for events above the GZK cutoff value as given from the Akeno experiment; recent data from the HiRes experiment confirm the Akeno picture. The dotted line shows the expectation from an extragalactic source distribution which is uniform in the Universe.
Focusing the attention on the primary sources, two general production
mechanisms have been proposed for the EECRs:
- BOTTOM-UP, with acceleration in rapidly evolving processes occurring in Astrophysical Objects. The scenario involves astrophysical objects such as, e.g. AGNs and AGN radio lobes. The study of these objects is, besides radio observations, a main goal of X-ray and Gamma-ray astrophysics of the late 90`s. An extreme case in this class is represented by the Gamma Ray Bursts, found to be located at cosmological distances. The observation of “direction of arrival and time” coincidences of GRBs and Extreme Energy Neutrinos (E1019 eV) in the EUSO mission could provide a crucial test for the identification of the observed GRBs as EECR sources in spite of their location at distances well above the GZK limit.
- TOP-DOWN Processes. This scenario arises from the cascading of
ultrahigh energy particles from the decay of topological defects. Cosmic Strings would play an essential role for releasing the X-bosons emitting the highest energy quarks and leptons. This process could occur in the nearby Universe. The relics of an early inflationary phase in the history of the Universe may survive to the present as a part of
dark matter and account for those unidentified EECR sources active within the GZK boundary limit. Their decays can give origin to the highest energy cosmic rays, either by emission of hadrons and photons, as through production of EE neutrinos.
From the Astroparticle Physics point of view, the EECRs have energies only a
few decades below the Grand Unification Energy (1024-1025 eV), although still far from the Plank Mass of 1028 eV.
Cosmic Neutrinos, not suffering the GZK effect and being immune from magnetic field deflection or from an appreciable time delay caused by Lorentz factor, are ideal for disentangling source related mechanisms from propagation related effects.The opening of the Neutrino Astronomy channel will allow to probe the extreme boundaries of the Universe. Astronomy at the highest energies must be performed by neutrinos rather than by photons, because the Universe is opaque to photons at these energies.
3.1 Observational problems
The extremely low value for the EECR flux, corresponding to about 1 event per km2 and century at E > 1020 eV, and the extremely low value for the interaction cross section of neutrinos, make these components difficult to observe if not by using a
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detector with exceptionally high values for the effective area and target mass. The integrated exposure ( ~ 2×103 km2 yr sr) available today for the ground based arrays operational over the world is sufficient only to show the "ankle" feature at ~5×1018 eV in the Cosmic Ray energy spectrum and the existence of about ten events exceeding 1020 eV; the limited statistics excludes the possibility of observing significant structures in the energy spectrum at higher energies. Experiments carried out by means of the new generation ground-based observatories, HiRes (fluorescence) and Auger (hybrid), will still be limited by practical difficulties connected to a relatively small collecting area (<104 km2 sr) and by a modest target mass value for neutrino detection.
To overcome these difficulties, a solution is provided by observing from space (Fig.11) the atmosphere UV fluorescence induced by the incoming extraterrestrial radiation, which allows to exploit up to millions km2 sr for the acceptance area and up to 1013 tons as target for neutrino interaction. This is the philosophy of the “AirWatch Programme” and “EUSO” is a space mission developed in the AirWatch framework.
The Earth atmosphere in fact constitutes the ideal detector for the Extreme Energy Cosmic Rays and the companion Cosmic Neutrinos. The EECR particles, interacting with the air nuclei, give rise to propagating Extensive Air Showers (EAS) accompanied by the isotropic emission of UltraViolet fluorescence (300-400 nm) induced in Nitrogen by the secondary charged particles in the EAS as result of a complex relativistic cascade process; an isotropically diffuse optical-UV signal is also emitted following the impact on clouds, land or sea of the Cherenkov beam accompanying the EAS. A Shower corresponding to a Primary with E>1019 eV forms a significant streak of fluorescence light over 10-100 km along its passage in the atmosphere, depending on the nature of the Primary, and on the pitch angle with the vertical.
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Figure 11. Observation of EAS from Space.
Observation of this fluorescence light with a detector at distance from the
shower axis is the best way to control the cascade profile of the EAS. When viewed continuously, the object moves on a straight path with the speed of light. The resulting picture of the event seen by the detector looks like a narrow track in which the recorded amount of light is proportional to the shower size at the various penetration depth in the atmosphere. From a Low Earth Orbit (LEO) space platform, the UV fluorescence induced in atmospheric Nitrogen by the incoming radiation can be monitored and studied. Other phenomena such as meteors, space debris, lightning, atmospheric flashes, can also be observed; the luminescence coming from the EAS produced by the Cosmic Ray quanta can be on the other hand disentangled from the general background exploiting its fast timing characteristic feature.
EUSO observes at Nadir from an orbital height of about 400 km. It is equipped with a wide angle Fresnel optics telescope (60° full FoV) and the focal plane segmentation corresponding to about 1 km2 pixel size on the Earth surface. The area covered on Earth is of about 160000 km2. Exploiting the high speed of the focal plane detector (10 ns class), EUSO is able to reconstruct the inclination of the shower track by the speed of progression of the projected image on the focal surface and to provide the tri-dimensional reconstruction of the EAS axis with a precision of a degree (or better) depending on the inclination. By measuring the EAS front luminosity with the photoelectrons (PE) detected by the MAPTs covering the focal surface, EUSO registers the longitudinal development of the EAS.
3.2 EUSO General Requirements and Main Goals
For a significant observation from a space mission the assumed values are: a) Effective geometrical exposure of (5×104-105) km2 sr considering a duty cycle of 0.1-0.15; b) EAS energy threshold at about 5×1019 eV.
EECR statistics. About 103 events/year (an order of magnitude above those expected by the presently planned ground based experiments) to allow a quantitative energy spectral definition above 1020 eV, together with the evidence of possible anisotropy effects and clustering (if any) for the directions of arrival.
Neutrino events. The expected event rate ranges from several events/year (AGN, GRB source) to several events/day according to the effectiveness of the "topological defects" hypothesis. From the observational point of view, the neutrino induced EAS can be distinguished from background and from other EECR EAS by triggering on horizontal showers initiating deep inside the atmosphere. Moreover neutrinos with energy of about 1015- 1016 eV interacting in the solid earth and emerging upward in the atmosphere create showers which can be detected by EUSO by means of the Cherenkov beamed signal induced in the atmosphere, extending the capability of EUSO to this lower neutrino astronomy energy band. A horizontal tau-neutrino event at energies greater than 1019eV can be identified by a “double bang” structure. Both the initial shower at the interaction, and another, by the -decay, can be seen because of the long enough path-length (~ 1000 [E/1020 eV] km) for -decays observable by EUSO. Tau-neutrinos above 1015 eV, on the other hand, will be observed and identified as Earth-penetrating “upward” showers (by Cherenkov). High flux by the oscillation and the low detection threshold energy for them allow EUSO to make oscillation experiments in space as well as astrophysics of AGN above 1015 eV.
Gamma Ray Bursts. GRBs themselves are detectable via the UV induced fluorescence by the incoming gamma ray plane wave as a glow modulated in intensity with the same light curve of the gammas.
The other optical atmospheric phenomena, like those induced by electrical discharged of meteoroid impacts, represent a very interesting field of research for themselves.
Balloon and micro-sat programs to measure the night sky UV background have been initiated.
3.3 EUSO Schematic Outline
EUSO, originally proposed to ESA for a free-flyer LEO mission [2], has been approved for an “accommodation study” on the ISS International Space Station.
Under the assumption of both a LEO (~ 500 km altitude) free-flyer mission or the ISS accommodation (400 km average altitude), the coverage of the observable
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atmosphere surface at the scale of thousand kilometers across and the measurement of very fast and faint phenomena like those EUSO is interested in, requires:
optical system with large collecting area (because of the faint fluorescence
signal) and wide equivalent field of view covering a sizable half opening angle around the local Nadir (to reach geometrical factor of the order of 106 km2 sr) ,
focal plane detector with high segmentation (single photon counting and high pixelization), high resolving time (~10 ns), contained values for weight and power,
trigger and read-out electronics prompt, simple, efficient, modular, capable to handle hundreds of thousands of channels, and comprehensive of a sophisticated on-board image processor acting as a trigger.
Fig.12 and Fig.13 show an artistic view of ESA “Columbus Orbital Facility”
and of the ISS with EUSO attached at Columbus.
Figure 12. The Columbus Orbital Facility (COF) with the External Payload Facility (EPF) Platforms.
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Figure 13. EUSO at the COF-EPF.
3.4 EUSO Payload: The “Main Telescope”
The EUSO Main telescope is presented schematically in the artistic view of Fig.14. The instrument consists of three main parts: Optics, Focal surface detector, Trigger and Electronics System. An effective synergy between the parts constituting the instrument is of fundamental importance for achieving the EUSO scientific objectives. Optics, detector elements, system and trigger electronics have to be matched and interfaced coherently to obtain a correct response from the instrument. Scientific requirements have been of guidance for the conceptual design of the apparatus and in the choice among various possible technical solutions. The design criteria are based on the following assumptions:
380 km orbit Pixel size at ground: 1 km2 FOV of 30° Event energy threshold 51019 eV
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Figure 14. View of the EUSO Main Telescope.
The observation from space calls for an approach different from that of the
conventional ground based fluorescence experiments. For space application the instrument has to be compact as much as possible, highly efficient, and with a built-in modularity in its detection and electronics parts.
3.4.1 The Optics
The optical system required for EUSO aims at finding the best compromise in the optical design, taking into account the suitability for space application in terms of weight, dimensions and resistance to the strains in launch and orbital conditions.
The optical system views a circle of radius ~220 km on the Earth and resolves 0.80.8 km2 ground pixels: this determines the detector size to be adopted to observe the events. The forgiving resolution requirements of EUSO suggest the consideration of unconventional solutions, identified in the Fresnel lens technology. Fresnel lenses provide large-aperture and wide-field with drastically reduced mass and absorption. The use of a broader range of optical materials (including lightweight polymers) is possible for reducing the overall weight.
The present Fresnel optical camera configuration study (FoV 60°) considers two plastic Fresnel lenses with diameter 2.5 m and iris diaphragm 2.0 m diameter.
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3.4.2 The Focal Surface Detector
Due to the large FOV and large collecting area of the optics, the focal surface detector is constituted by several hundreds of thousands of active sensors (2105 pixels). The detector requirements of low power consumption, low weight, small dimension, fast response time, high quantum efficiency in UV wavelength (300400 nm), single photoelectron sensitivity, limit the field of the possible choices to a very few devices. A suitable off-the-shelf device is the Multi-Anode Photomultiplier Hamamatsu R5900 series. These commercial photomultipliers meet closely the requirements imposed by the project. Pixel size, weight, fast time response and single photoelectron resolution are well adaptable to the EUSO focal surface detector. The organization in “macrocells” of the focal surface (a macrocell is a bidimensional array of nn pixels) offers many advantages as easy planning and implementation, flexibility and redundancy. Moreover, modularity is ideal for space application. The MultiAnode Photomultipliers represent, in this contest, a workable solution.
3.4.3 Trigger and Electronics System
Special attention has been given to the trigger scheme where the implementation of hardware/firmware special functions is foreseen.
The trigger module named OUST (On-board Unit System Trigger) has been studied to provide different levels of triggers such that the physics phenomena in terms of fast, normal and slow in time-scale events can be detected. Particular emphasis has been introduced in the possibility of triggering upward showers (emerging from the earth, “neutrino candidate”) by means of a dedicated trigger logic.
The FIRE (Fluorescence Image Read-out Electronics) system has been designed to obtain an effective reduction of channels and data to read-out, developing a method that reduces the number of the channels without penalizing the performance of the detection system. Rows wired-or and columns wired-or routing connections have been adopted inside every single “macrocell” (nn pixels unit, 100 macrocells constitute the focal surface detector) for diminishing the number of channels to read-out.
3.5 Expected Results
Extensive simulations have been elaborated by O. Catalano at IFCAI/CNR. Fig.15 and Fig.16 report the expected results for EUSO in the ISS version,
compared with those referred to the free-flyer version of the original proposal to ESA: in the two versions the results appear almost identical , with the lower altitude for the ISS compensating the reduced dimensions of the optics for what concerns the “threshold”.
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Figure 15. Differential EECR counting rate: comparison between EUSO on the ISS and the original free-flyer proposal. The dashed zone shows the spectral structure induced by the GZK effect.
Figure 16. Neutrino expectation: the different shadowed areas refer to Topological Defects (TD) and Greisen (by interaction of the Primary (CR).
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4 Acknowledgments
The determinant contribution of M.C. Maccarone is deeply acknowledged.
References
1. See for example: Scarsi, L., “BeppoSAX: Three years of operations. Highlights and significant results”, in Current Topics in Astrofundamental Physics, Kluwer Academic Publishers, 483-497 (2001).
2. Scarsi, L., et al., “EUSO – Extreme Universe Space Observatory”, Proposal for the ESA F2/F3 Mission, January 2000, and references therein (2000) http://www.ifcai.pa.cnr.it/ ~EUSO/docs/EUSOproposal.pdf
3. ESA/ESTEC, “EUSO Accommodation Study Report”, ESA/MSM-GU/2000.462/AP/RDA, December 2000.
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GAMMA RAY BURSTS AND VACUUM POLARIZATIONPROCESS IN ELECTROMAGNETIC BLACK HOLES
REMO J. RUFFINI
Physics Department andInternational Center for Relativistic Astrophysics, University of Rome, I–00185
Roma, Italy
The developments of the elctromgnetic black holes physics and vacuum polarizationprocess are presented in the interpretation of gamma ray bursts.
1 Introduction
I have accepted with great pleasure the invitation to speak on our theory ofGamma ray Bursts by Pisin Chen for the interest of the program of this meet-ing which presents important developments on vacuum polarization effects inhigh energy collisions on which he himself has given so many important con-tributions.An additional factor has also been the beauty of the location ofthe school, in the Capri Island. What to me is also very important is that Ifollow the talks of two pioneers who have opened up some of the most excit-ing fields of Relativistic Astrophysics: I. F. Mirabel who has first pointed theexistence of so called supraluminal behaviour in microquasars galactic sourcesand Livio Scarsi, the ideator and leader of the Beppo Sax Satellite which hasdiscovered the afterglow of Gamma Ray Bursts sources. This last result hasled to an unprecedented collaboration between all fields of space based opti-cal, X and Gamma rays observatories in coincidence with the largest earthbased observatories leading to the clear conclusion that Gamma ray Burstsare indeed of cosmological origin and their energy can be in some cases ≥ 1054
ergs (see Costa 200133). An energy of 1054 ergs/burst can be easily visualizedin its enormity: if we consider the light emitted by a star like our own sunand we multiply such luminosity by the number of stars in our galaxy (1012)and multiply again such numbers by the total number of galaxies in the Uni-verse (109), this is of the order of 1054 ergs/sec. In other words, during theburst the luminosity of a single GRB can equal the energy emission of theentire Universe. It exist the very clear possibility that the interpretation ofboth these phenomena, the Micrroquasars and the GRBs, are deeply rootedin the physics of Black Holes to which I have devoted special attention in mytheoretical research. I will give some motivation for this possibility in thefollowing.
capri-ruffini: submitted to World Scientific on September 18, 2001 1
2 On three happenings related to GRBs
The case of Gamma Ray Bursts is for me personally very intriguing. Therehave been moments in my life which appear to be intertwined with some ofthe relevant events that are leading to the understanding of such most uniquephenomena. Moreover each scientific contribution I have achieved and evenapparently occasional occurrences in my life seemingly disconnected appearto acquire special meaning in reaching the understanding of such phenomena.
The first of such happening was the collaboration during my career,started at Princeton in 1967 with John Archibald Wheeler and the one startedin 1968 at Tbilisi and then in Moscow with Yakov Borisovich Zel’dovich. Boththese scientists were heads and founders of the research in relativistic astro-physics in their countries. But also interesting was the fact that both of themhad a principal role, before adressing their attention to the implications ofEinstein theory of general relativity to astrophysics, in the nuclear arms racerespectively in the USA and Soviet Union. Johnny Wheeler was a leader inthe first tactic H Bomb explosdion on the Bikini Atoll. Ya. B. Zeldovich afterhaving contributed to the defense of his country with the Katiuscia rocketshad developed with Andrej Sakahrov, the soviet A and H bomb. He was alsothe proposer of a most unusual and intolerable project to have an H bombexplode on the far side of the moon to demonstrate at ounce the “maturity”of the nuclear and space technology reached by the Soviet Union in the earlysixties.
The second happening occurred in 1975. Herbert Gursky and myselfhad been invited by the AAAS to organize a session on neutron stars, blackholes and binary X ray sources for their annual meeting in San Francisco.During the preparation of the meeting we heard that some observations madeby the military Vela satellites, conceived in order to monitor the LimitedTest Ban Treaty of 1963 banning atomic bomb explosions, had just beenunclassified. Doubtlessly the unorthodox proposal of Zel’dovich had beenamong the motivations to develop such a grandiose militar monitoring system.We asked Ian B. Strong to report, for the first time in a public meeting, onthese just observed-released gamma ray bursts (GRBs)(Strong 1975)28, SeeFig. 1.
It was clear since the earliest observations that these signals were notcoming either from the Earth or the planetary system. By 1991 a greatimprovement in knowledge of the distribution of the GRBs came with theNASA launch of the Compton Gamma-Ray Observatory which in ten yearsof observations gave beautiful evidence for the perfect isotropy of the angulardistribution of the GRB sources in the sky, see Fig. 2. The sources had to
capri-ruffini: submitted to World Scientific on September 18, 2001 2
Figure 1. One of the first GRBs observed by the Vela satellite. Reproduced from Strong inGursky & Ruffini (1975)28.
either be at cosmological distances or very close to the solar system in ordernot to reflect the anisotropic galactic distribution.
The third happening occurred in 1989 when I was elected President ofthe scientific committee of the Italian Space agency (ASI) and the committeefound itself involved with the scrutiny of the first Italian scientific satellite: theSAX satellite. The project was a collaboration between Italy and Netherland:The total estimated cost was roughly 50 millions US dollars, fairly shared bythe two partners: 25 milions for Italy and 25 for the Netherlands. The satel-lite was supposed to fly in 1985. The program had already been delayed fouryears by the time of our scrutiny started. The costs had correspondingly “sky-rocketed” to almost 250 millions US dollars, “fairly” shared by the partners:225 millions from Italy and 25 from the Netherlands. . . . The real momentof panic came when we learned that the Dutch had run out of money. Theycould not afford to pay for the wide field x-ray cameras, they were supposedto contribute. We decided to intervene offering to pay roughly six millionsUS dollars from the limited budget of our committee in order to avoid anyfurther delay and especially to avoid the loss of one of the crucial instrumentsof the scientific mission. As the delays were augmenting and the expenses ofthe mission were correspondingly “skyrocketing” further, the ambiance soon
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Figure 2. Angular distribution of GRBs in galactic coordinates from the Compton GROsatellite.
deteriorated to inadmissible pressures.Before quitting I insisted on the imper-ative to accomplish the mission no matter what. In these hard days Livio andI were on opposite sides, but we managed to keep the highest personal andscientific esteem for each other.
The satellite finally was launched in 1996 at a cost still today unknown:it is not yet possible to ascertain if it passed and, if so, of how much the onebillion Us dollars mark. Soon after the lunch three of the four gyroscopesfailed apparently due to the improper choice of space qualified instrumentsand the satellite was left without an effective pointing capability.
In spite of all that, thanks also to the determinate action of a num-ber of strongly dedicated and courageous young physicists educated at “LaSapienza” who had joined the Milano based original team, the newly namedBeppo-SAX satellite was able to conclude one of the most successful everscientific missions in Astronomy and Astrophysics. They discovered the after-glows of the GRBs, which in turn have allowed the optical identification of thesources and the determination of their cosmological nature. I am happy today
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to see Livio here again in the presentation of these most beautiful scientificresults. I am also happy to see that also the completion of the wide field xray camera and the many imperatives to conclude the mission have lead to asuccessful epilogue.
Thinking about this past situation with hindsight, I have reached a ratherunorthodox conclusions: if SAX had flown on time in 1985 and possibly withinits planned budget, it would have been a managerial success and quite a sav-ing for the Italian treasury, but very likely not a scientific success. The reasonis that in 1985 neither the Space Telescope nor the very large telescopes likeKECK and VLT, which have been essential to the optical identification of theGRBs and the establishment of their cosmological distance, were functioning.Of course I do not want to make propaganda in favour of wrong doings, butit appears as if a tremendous force directs human actions not only exploit-ing great scientific ideas but making use as well of weakness, mistakes andmismanagement in order to reach a final important scientific goal!
While the expenditure on Beppo-SAX were “skyrocketing,” equally expo-nentially increasing were the numbers of competing theories trying to explainGRBs (ee a partial list in Fig. 3).
The observations of the Beppo-SAX satellite had a very sobering effecton the theoretical developments for GRB models.Almost the totality of theexisting theories, see above partial list, were at once wiped out, not being ableto fit the stringent energetics requirements imposed by the observations.
This led us to the return to the model of GRBs, which we had quietlyadvanced and theoretically developed in all its conceptual complexity start-ing from the work with Thibau Damour in 197530, essentially based on thepossibility of extracting mass-energy from a Black Hole. Such a process isbased on the mass-energy formula for Black Holes I had found in 1971 withDemetrios Christodoulou24. Following this work it had become evident thatBlack Holes far from being energy sinks were in principle the most energeticsource of energy in the Universe. The extraction of energy from Black Holescan in fact surpass, in principle, chemical, nuclear or thermonuclear energysource both in absolute value and intensity. In practice it is now clear thatthis happens in GRBs as predicted by our model.
I will shortly recall the background on which our model was conceived,how the concepts of “alive” versus “dead” black holes was introduced, thekey role of the reversible and irreversible transformations in Black Holes andthe role of quantum vacuum polarization process, and finally indicate somecurrent developments.
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Figure 3. Partial list of theories before the Beppo-SAX, from a talk presented atMGIXMM34.
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3 Early steps in the study of Black Holes
The year 1968 with the discovery of pulsars in 1968 and especially with thediscovery of the pulsar in the Crab Nebula, can be considered the birthdateof relativistic astrophysics. The observation of the period of that pulsar andits slow-down rate not only clearly gave unequivocal evidence for the iden-tification of the first neutron star in the galaxy but also contributed to theunderstanding that the energy source of pulsars is very simply the rotationalenergy of a neutron star.
I was in Princeton in those days initially as a postdoctoral fellow at theuniversity in the group of John Archibald Wheeler, then as a member ofthe Institute for Advanced Study, and finally as an instructor and assistantprofessor at the University. The excitement over the neutron star discoveryboldly led us directly to an as yet unexplored classic paper by Robert JuliusOppenheimer and Snyder “on continued gravitational contraction”5 and thisopened up an entirely new field of research to which I have dedicated theremainder of my life and it is still producing some quite important resultstoday. An “effective potential” technique had been used very successfully byCarl Størmer in the 1930s in studying the trajectories of cosmic rays in theEarth’s magnetic field (Størmer 1934)6. In the fall of 1967 Brandon Cartervisited Princeton and presented his remarkable mathematical work leadingto the separability of the Hamilton-Jacobi equations for the trajectories ofcharged particles in the field of a Kerr-Newmann geometry (Carter 1968)7.This visit had a profound impact on our small group working with JohnWheeler on the physics of gravitational collapse. Indeed it was Johnny whohad the idea of exploiting the analogy between the trajectories of cosmic raysand spacetime trajectories in general relativity, using the Størmer “effectivepotential” technique in order to obtain physical consequences from the setof first order differential equations obtained by Carter. I still remember theexcitement of preparing the 2m × 2m grid plot of the effective potential forparticles around a Kerr black hole which finally appeared later in print (Rees,Ruffini and Wheeler 1973,19748; see Fig. (4). Out of this work came thecelebrated result for the maximum binding energy 1 − 1√
3∼ 42% for coro-
tating orbits and 1 − 53√
3∼ 3.78% for counter-rotating orbits in the Kerr
geometry. We were very pleased to be later associated with Brandon Carterin a “gold medal” award for this work presented by Yevgeny Lifshitz: in thefourth and last edition of volume 2 of the Landau and Lifshitz series (TheClassical Theory of Fields 1975), both Brandon’s work and my own workwith Wheeler were proposed as named exercises for bright students! In ourarticle “Introducing the Black Hole” (Ruffini and Wheeler 1971)9 we first pro-
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Figure 4. “Effective potential” around a Kerr black hole, see Ruffini and Wheeler 1971
posed the famous “uniqueness theorem” stating that black holes can only becharacterized by their mass-energy E, charge Q and angular momentum L.This analogy between a black hole and a very elementary physical system wasimaginatively represented by Johnny in a very unconventional figure in whichTV sets, bread, flowers and other objects lose their characteristic featuresand merge in the process of gravitational collapse into the three fundamentalparameters of a black hole, see Fig. 5. That picture became the object of agreat deal of lighthearted discussion in the physics community. A proof ofthis uniqueness theorem, satisfactory for some case of astrophysical interest,has been obtained after twenty five years of meticulous mathematical work(see e.g., Regge and Wheeler10, Zerilli11,12, Teukolsky13, C.H. Lee14, Chan-drasekhar 15.) However, the proof still presents some outstanding technicaldifficulties in its most general form. Possibly some progress will be reached inthe near future with the help of computer algebraic manipulation techniquesto overcome the extremely difficult mathematical calculations (see e.g., Cru-ciani (1999) cru, Cherubini and Ruffini (2000)17 Bini et al. (2001)18, Bini etal. (2001)19).
It is interesting that this analogy, which appeared at first to be almosttrivial, has revealed itself to be one of the most difficult to be proved requiringan enormous effort, unsurpassed in difficulty both in mathematical physics andrelativistic field theory.
I am profoundly convinced that the solution of this problem from a math-ematical physics point of view will have profound implications for our under-standing of the fundamental laws of physics.
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Figure 5. The black hole uniqueness theorem.
4 From “dead” to “alive” black Holes
We were still under the sobering effects of the pulsar discovery and the veryclear explanation by Tommy Gold and Arrigo Finzi that the rotational energyof the neutron star had to be the energy source of the pulsar phenomenon,when the first meeting of the European Physical Society took place in Florencein 1969. In a stimulating talk Roger Penrose20 advanced the possibility that,much like in the case of pulsars, the rotational energy of black holes could,in principle, be extracted in an analogous way. The first specific example ofsuch an energy extraction process by a gedanken experiment was given us-ing the above-mentioned effective potential technique in Ruffini and Wheeler(1970)21, see Figure (6), and then later by Floyd and Penrose (1971)22. Thereason for showing this figure here is a) to recall the first explicit computationand b) to recall the introduction of the “ergosphere,” the region between thehorizon of a Kerr-Newmann metric and the surface of infinite redshift were theenergy extraction process can occur, and also c) to emphasize how contrived,difficult and also conceptually novel such an energy-extraction mechanism canbe. It is a phenomenon which is not localized at a point but which can occurin an entire region: a global effect which relies essentially on the concept of afield. It can only work, however, for very special parameters and is in generalassociated with a reduction of the rest mass of the particle involved in the pro-cess. It is almost trivial to slow down the rotation of a black hole and increaseits horizon by accretion of counter-rotating particles, but it is extremely diffi-cult to extract the rotational energy from a black hole by a slow-down process,as also clearly pointed out by the example in Fig. (6). The establishment of
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Figure 6. Decay of a particle of rest-plus-kinetic energy E into a particle which is capturedby the black hole with positive energy as judged locally, but negative energy E1 as judgedfrom infinity, together with a particle of rest-plus-kinetic energy E2 > E which escapes toinfinity. The cross-hatched curves give the effective potential (gravitational plus centrifugal)defined by the solution E of Eq.(2) for constant values of pφ and µ. (Figure and captionreproduced from Christodoulou 197023, in turn reproduced before its original publicationin9 with the kind permission of Ruffini and Wheeler.)
this analogy offered us the opportunity to appreciate once more the profounddifference between seemingly similar effects in general relativity and classicalfield theories. In addition to the existence of totally new phenomena, like thedragging of the inertial frames around a rotating black hole for example, wehad the first glimpse of an entirely new field of theoretical physics presentin and implied by the field equations of general relativity. The deep discus-sions of these problems with Demetrios Christodoulou, who was a 17 year oldPrinceton student at the time, my first graduate student, led us to the discov-ery of the existence in black holes physics of both “reversible and irreversibletransformations.”
Indeed it was by analyzing the capture of test particles by a Black Holeendowed with electromagnetic structure, for short an EMBH, that we iden-tified a set of limiting transformations which did not affect the surface areaof an EMBH. These special transformations had to be performed very slowly,with a limiting value of zero kinetic energy on the horizon of the EMBH, seeFig. 7. It became then immediately clear that the total energy of an EMBHcould in principle be expressed as a “rest energy” a “Coulomb energy” anda “rotational energy.” The rest energy is “irreducible,” the other two beingsubmitted to positive and negative variations, corresponding respectively toprocess of addition and extraction of energy.
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Figure 7. Reversing the effect of having added to the black hole one particle (A) by addinganother particle (B) of the same rest mass but opposite angular momentum and charge ina “positive-root negative-energy state.” Addition of B is equivalent to subtraction of B−.Thus the combined effect of the capture of particles A and B is an increase in the mass ofthe black hole given by the vector B−A. This vector vanishes and reversibility is achievedwhen and only when the separation between positive root states and negative root statesis zero, in this case the hyperbolas coalesce to a straight line. Reproduced from24.
While Wheeler was mainly studying the thermodynamical analogy, I ad-dressed with Demetrios the fundamental issue of the energetics of EMBHsusing the tools of reversible and irreversible transformations. We finally ob-tained the general mass-energy formula for black holes (Christodoulou andRuffini 1971)24:
E2 = M2c4 =(
Mirc2 +
Q2
2ρ+
)2
+L2c2
ρ2+
, (1)
S = 4πρ2+ = 4π
(
r2+ +L2
c2M2
)
= 16π(G2
c4
)
M2ir , (2)
with
1ρ4+
(G2
c8
)(Q4 + 4L2c2
) ≤ 1 , (3)
where Mir is the irreducible mass, r+ is the horizon radius, ρ+ is the quasi-spheroidal cylindrical coordinate of the horizon evaluated at the equatorialplane, S is the horizon surface area, and extreme black holes satisfy the equal-ity in eq. (3). The crucial point is that transformations at constant surface
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area of the black hole, namely reversible transformations, can release an en-ergy up to 29% of the mass-energy of an extremal rotating black hole andup to 50% of the mass-energy of an extremely magnetized and charged blackhole. Since my Les Houches lectures “On the energetics of black holes” (B.C.De Witt 1973 )25 I introduced the concepts of “alive” black hoes, endowedof mas-energy ,of rotation and angular momentum and “dead” black holesuniquely characterized by their masses: one of my main research goals hasbeen to identify an astrophysical setting where the extractable mass-energyof the black hole could manifest itself. As we will see in the following, I proposethat this extractable energy of an EMBH is the energy source of gamma-raybursts (GRBs).
5 The paradigm for the identification of the first “black hole”in our galaxy and the development of X-ray astronomy.
The launch of the “Uhuru” satellite by the group directed by Riccardo Giac-coni at American Science and Engineering, dedicated to the first systematicexamination of the universe in X-rays, marked a fundamental leap forward andgenerated a tremendous momentum in the field of relativistic astrophysics.The very fortunate collaboration soon established with simultaneous observa-tions in the optical and radio wavelengths allowed generated high quality dataon binary star systems composed of a normal star being stripped of matterby a compact massive companion star: either a neutron star or a black hole.
The “maximum mass of a neutron star” was the subject of the thesisof Clifford Rhoades, my second graduate student at Princeton. A criteriawas found there to overcome fundamental unknowns about the behavior ofmatter at supranuclear densities by establishing an absolute upper limit to theneutron star mass based only on general relativity, causality and the behaviourof matter at nuclear and subnuclear densities. This work, presented at the1972 Les Houches Summer School (B. and C. de Witt 1973), only appearedafter a prolongued debate (see the reception and publication dates!) (Rhoadesand Ruffini 1974)26.
The three essential components in establishing the paradigm for the iden-tification of the first black hole in Cygnus X1 (Leach and Ruffini 1973)27
were
• the “black hole uniqueness theorem,” implying the axial symmetry of theconfiguration and the absence of regular pulsations from black holes,
• the “effective potential technique,” determining the efficiency of the en-ergy emission in the accretion process, and
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• the “upper limit on the maximum mass of a neutron star,” discriminatingbetween an unmagnetized neutron star and a black hole.
These results were also presented in a widely attended session chaired by JohnWheeler at the 1972 Texas Symposium in New York, extensively reported onby the New York Times. The New York Academy of Sciences which hostedthe symposium had just awarded me their prestigious Cressy Morrison Awardfor my work on neutron stars and black holes. Much to their dismay I neverwrote the paper for the proceedings since it coincided with the one submittedfor publication (Leach and Ruffini 1973)27.
The definition of the paradigm did not come easily but slowly maturedafter innumerable discussions, mainly by phone, both with Riccardo Giacconiand Herb Gursky. I still remember an irate professor of the Physics Depart-ment at Princeton pointing out publicly my outrageous phone bill of $274 forone month, at the time considered scandalous, due to my frequent calls to theSmithsonian, and a much more relaxed and sympathetic attitude about thissituation held by the department chairman, Murph Goldberger. This workwas finally summarized in two books: one with Herbert Gursky (Gursky andRuffini 1975)28, following the 1973 AAAS Annual Meeting in San Francisco,and the second with Riccardo Giacconi (Giacconi and Ruffini 1978)29 follow-ing the 1975 LXV Enrico Fermi Summer School (see also the proceedings ofthe 1973 Solvay Conference).
6 the Heisenberg-Euler critical capacitor and the vacuumpolarization around a macroscopic black hole
In 1975, following the work on the energetics of black holes (Christodoulouand Ruffini 1971)24, we pointed out (Damour and Ruffini, 1975)30 the ex-istence of the vacuum polarization process a’ la Heisenberg-Euler-Schwinger(Heisenberg and Euler 193531, Schwinger 195132) around black holes endowedwith electromagnetic structure (EMBHs). Such a process can only occur forEMBHs of mass smaller then 7.2 · 106M. The basic energetics implicationswere contained in Table 1 of that paper (Damour and Ruffini, 1975)30, whereit was also shown that this process is almost reversible in the sense introducedby Christodoulou and Ruffini (1971)24 and that it extracts the mass energy ofan EMBH very efficiently. We also pointed out that this vacuum polarizationprocess around an EMBH offered a natural mechanism for explaining GRBs,just discovered at the time, and the characteristic energetics of the burst couldbe ≥ 1054 ergs, see Fig. 8.
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Figure 8. Summary of the EMBH vacuum polarization process (see Damour & Ruffini(1975)30 for details).
7 The ergosphere versus the dyadosphere of a black hole
The enormous energy requirements of GRBs evidenced by the Beppo-SAXsatellite, very similar to the ones predicted in Damour & Ruffini (1975)30
have convinced us to return to our earlier work in studying more accuratelythe process of vacuum polarization and the region of pair creation aroundan EMBH. This has led to a) the new concept of the dyadosphere of anEMBH (named for the Greek word dyad for pair) and b) the concept of aplasma-electromagnetic (PEM) pulse and c) the analysis its temporal evolu-tion generating signals with the characteristic features of a GRB.
In our theoretical approach, we claim that through the observations ofGRBs, we are witnessing the formation of an EMBH and therefore followthe process of gravitational collapse in real time. Even more important, thetremendous energies involved in the energetics of these sources have theirorigin in the extractable energy of black holes given in Eqs. (1)–(3) above.
Various models have been proposed in order to extract the rotationalenergy of black holes by processes of relativistic magnetohydrodynamics (see
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e.g., Ruffini and Wilson (1975)35). It should be expected, however, that theseprocesses are relevant over the long time scales characteristic of accretionprocesses.
In the present case of gamma ray bursts a sudden mechanism appears tobe at work on time scales of the order of few seconds or shorter and they arenaturally explained by the vacuum polarization process introduced in Damour& Ruffini (1975)30.
The fundamental new points we have found re-examining our previouswork can be simply summarized, see Preparata, Ruffini and Xue (1998a,b)36
for details:
• The vacuum polarization process can occur in an extended region aroundthe black hole called the dyadosphere, extending from the horizon radiusr+ out to the dyadosphere radius rds. Only black holes with a masslarger than the upper limit of a neutron star and up to a maximum massof 7.2 · 106M can have a dyadosphere.
• The efficiency of transforming the mass-energy of a black hole intoparticle-antiparticle pairs outside the horizon can approach 100%, forblack holes in the above mass range.
• The created pairs are mainly positron-electron pairs and their number ismuch larger than the quantity Q/e one would have naively expected onthe grounds of qualitative considerations. It is actually given by Npairs ∼Qe
rds
h/mc , where m and e are respectively the electron mass and charge.The energy of the pairs and consequently the emission of the associatedelectromagnetic radiation as a function of the black hole mass peaks inthe gamma X-ray region.
Let us now recall the main results on the dyadosphere obtained inPreparata, Ruffini and Xue (1998a,b)36. Although the general considerationspresented by Damour and Ruffini (1975)30 refer to a rotating Kerr-Newmannfield with axial symmetry about the rotation axis, for simplicity, we thereconsidered the case of a nonrotating Reissner-Nordstrom EMBH to illustratethe basic gravitational-electrodynamical process. The dyadosphere then liesbetween the radius
rds =(h
mc
) 12(GM
c2
) 12 (mp
m
) 12(e
qp
) 12(
Q√GM
) 12
(4)
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and the horizon radius
r+ =GM
c2
[
1 +
√
1 − Q2
GM2
]
. (5)
The number density of pairs created in the dyadosphere is
Ne+e− Q−Qc
e
[
1 +(rds − r+)
hmc
]
, (6)
where Qc = 4πr2+m2c3
he . The total energy of pairs, converted from the staticelectric energy, deposited within the dyadosphere is then
Etote+e− =
12Q2
r+
(
1 − r+rds
)(
1 −(r+rds
)2)
. (7)
The analogies between the ergosphere and the dyadosphere are many andextremely attractive:
• Both of them are extended regions around the black hole.
• In both regions the energy of the black hole can be extracted, approachingthe limiting case of reversibility as from Christodoulou & Ruffini (1971)24.
• The electromagnetic energy extraction by the pair creation process in thedyadosphere is much simpler and less contrived than the correspondingprocess of rotational energy extraction from the ergosphere.
8 The EM pulse of an atomic explosion versus the PEM pulseof a black hole
The analysis of the radially resolved evolution of the energy deposited withinthe e+e−-pair and photon plasma fluid created in the dyadosphere of anEMBH is much more complex then we had initially anticipated. The collab-oration with Jim Wilson and his group at Livermore Radiation Laboratoryhas been very important to us. We decided to join forces and propose a newcollaboration with the Livermore group renewing the successful collaborationwith Jim in 1974 (Ruffini and Wilson 1975)35. We proceeded in parallel: inRome with simple almost analytic models to be then validated by the Liver-more codes (Wilson, Salmonson and Mathews 1997,1998)37,38.
For the evolution we assumed the relativistic hydrodynamic equations,for details see Ruffini et al. (1998,1999)39,40. We assumed the plasma fluid
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of e+e−-pairs, photons and baryons to be a simple perfect fluid in the curvedspace-time. The baryon-number and energy-momentum conservation laws are
(nBUµ);µ = (nBU
t),t +1r2
(r2nBUr),r = 0 , (8)
(T σµ );σ = 0 , (9)
and the rate equation:
(ne±Uµ);µ = σv [ne−(T )ne+(T ) − ne−ne+ ] , (10)
where Uµ is the four-velocity of the plasma fluid, nB the proper baryon-number density, ne± are the proper densities of electrons and positrons (e±),σ is the mean pair annihilation-creation cross-section, v is the thermal veloc-ity of the e±, and ne±(T ) are the proper number-densities of the e± at anappropriate equilibrium temperature T . The calculations are continued untilthe plasma fluid expands, cools and the e+e− pairs recombine and the systembecomes optically thin.
The results of the Livermore computer code are compared and contrastedwith three almost analytical models: (i) spherical model: the radial compo-nent of the four-velocity is of the form U(r) = U r
R , where U is the four-velocity at the surface (r = R) of the plasma, similar to a portion of a Fried-mann model, (ii) slab 1: U(r) = Ur = const., an expanding slab with constantwidth D = R in the coordinate frame in which the plasma is moving, (iii)slab 2: an expanding slab with constant width R2−R1 = R in the comovingframe of the plasma. We compute the relativistic Lorentz gamma factor γ ofthe expanding e+e− pair and photon plasma.
Figure (9) shows a comparison of the Lorentz factor of the expanding fluidas a function of radius for all the models. One sees that the one-dimensionalcode (only a few significant points are plotted) matches the expansion patternof a shell of constant coordinate thickness.
In analogy with the notorious electromagnetic radiation EM pulse of cer-tain explosive events, we called this relativistic counterpart of an expandingpair electromagnetic radiation shell a PEM pulse.
In recent work we have computed the interaction of the expanding plasmawith the surrounding baryonic matter (Ruffini, et al. 2000)41, see Fig .10. Wehave also been able to follow the expansion process all the way to the pointwhere the transparency condition is reached and what we have defined the“proper GRB” (P-GRB) is emitted (Bianco, et al. 2001)42, see Fig. 11. Wehave then proceeded to develop the basic work to describe the afterglow ofGRBs (Ruffini, et al. 2001)43. These results of our theoretical model havereached the point where they can be subjected to a direct comparison withthe observational data.
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Figure 9. Lorentz γ as a function of radius. Three models for the expansion pattern of thee+e− pair plasma are compared with the results of the one dimensional hydrodynamic codefor a 1000M black hole with charge Q = 0.1Qmax. The 1-D code has an expansion patternthat strongly resembles that of a shell with constant coordinate thickness. Reproduced fromRuffini, et al. (1999)40.
9 Three new paradigms for the interpretation ofGRBs—obtained after the Capri meeting
Starting from this theoretical background, presented in CAPRI, we havemoved ahead to fit the observational data on the basis of the EMBH model.We have used the GRB 991216 as a prototype, both for its very high ener-
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Figure 10. Lorentz gamma factor γ as a function of radius for the PEM pulse interactingwith the baryonic matter of the remnant (PEMB pulse) for selected values of the baryonicmatter. Reproduced from Ruffini, et al. (2000)41.
getics, which we have estimated in the range of Edya ∼ 9.57 × 1052 ergs, aswell as for the superb data obtained by the Chandra and RXTE satellites.In order to understand the GRB phenomenon, we have found it necessary toformulate three new paradigms in our novel approach:
1. The Relative Space-Time Transformation (RSTT) paradigm (see Ruffini,Bianco, Chardonnet, Fraschetti, Xue (2001a)44).
2. The Interpretation of the Burst Structure (IBS) paradigm (see Ruffini,
capri-ruffini: submitted to World Scientific on September 18, 2001 19
0
2e+054
4e+054
6e+054
8e+054
1e+055
1.2e+055
1.4e+055
0 0.005 0.01 0.015 0.02 0.025
Flu
x (e
rg/s
)
Arrival Time (ta) (s)
Computed Profile
p1*tap2+p3
p4*e(p5*ta)+p6
Figure 11. P-GRB from an EMBH with M = 100M and Q = 0.1Qmax. Reproduced fromBianco, et al. (2001)42.
Bianco, Chardonnet, Fraschetti, Xue (2001b)45).
3. The Multiple-Collapse Time Sequence (MCTS) paradigm (see Ruffini,Bianco, Chardonnet, Fraschetti, Xue (2001c)46).
These results are currently being expanded in a detailed presentation.
10 Conclusions
In view of the above experiences, I have formulated some conclusions whichmay be of more general validity:
• Analogies. The analogies between classical regimes and general relativis-tic regimes have been at times helpful in giving the opportunity to glanceon the enormous richness of the new physical processes contained in Ein-stein’s theory of spacetime structure. In some cases they have allowedus to extend our knowledge and formalize new physical laws, the deriva-tion of Eqs. (1)–(3) is a good example. Such analogies have also provided
capri-ruffini: submitted to World Scientific on September 18, 2001 20
dramatic evidence of the enormous differences in depth and physical com-plexity between classical physics and general relativistic effects. The casesof extracting rotational energy from a neutron star and from a rotatingblack hole are good examples.
• New paradigms and their verifications. The establishment of newparadigms is essential to scientific process and is certainly not easy to do.Such paradigms are important in order to guide a meaningful comparisonbetween theories and observations and much attention should be given totheir development and inner conceptual consistency. Always the majorfactor driving the progress of scientific knowledge is the confrontationof theoretical predictions and the new paradigms of interpretation withobservational data. In recent years the evolution of new technologies haspermitted dramatic improvement in the sensitivity of the observationalapparata. It is very gratifying that in this process of learning the struc-ture of our Universe, the observational data intervene not in a marginalway, but with clear and unequivocal results: they confirm the correcttheories and their paradigms by impressive agreement and they disprovethe wrong ones by equally impressive disagreement.
• The Gamma Ray astrophysics present a new area of research which tran-scends all preceding ones. It implies the verification of general relativisticeffects in totally unexplored and new regimes, it implies as well the ex-trapolation of current knowledge of subnuclear high energy physics intoregimes again unexplored up to now on our planet, it needs technicaldevelopments in instrumentation both from space and from the groundunprecedented both for complexity, accuracy and need of coordination.The understanding of this phenomena requires a totally novel style ofresearch and is extremely promising for promoting the discovery of newfundamental physical laws in Nature.
References
1. E. Fermi, Rend. Acc. Lincei 6, 602 (1928).2. L. Landau, Phys. Zeits. Sov. 1, 285 (1932).3. S. Chandrasekhar, Introduction to the theory of the stellar structure,
(Univ. of Chicago Press, Chicago, 1931).4. J.R. Oppenheimer and G.M. Volkoff, Phys. Rev. 55, 374 (1939).5. J.R. Oppenheimer and H. Snyder, Phys. Rev. 56, 455 (1939).6. C. Størmer, Astrophysica Norvegica 1, 1 (1934).7. B. Carter, Phys. Rev. 174, 1559 (1968).
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