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Quantum Effects in BECs and FELs
Nicola Piovella, Dipartimento di Fisica and INFN-Milano
Rodolfo Bonifacio, INFN-Milano
Luca Volpe (PhD student), Dipartimento di Fisica-Milano
Mary Cola (Post Doc), Dipartimento di Fisica-Milano
Gordon R. M. Robb, University of Strathclyde, Glasgow, Scotland.
work supported by INFN (QFEL project)
Outline
1. Introductory concepts
2. Classical FEL-CARL Model
3. Quantum FEL-CARL Model
4. Propagation Effects
5. Quantum SASE regime
Free Electron Laser (FEL) 22 w
r
Collective Atomic Recoli Laser (CARL)
Pump beam p
Probe beam p
R. Bonifacio et al, Opt. Comm. 115, 505 (1995)
Both FEL and CARL are examples of collective recoil lasing
Cold atoms
Pump field
Backscattered field(probe)
CARL
FEL
“wiggler” magnet(period w)
Electron beam
EM radiation w /<< w N
S N
S N
S N
S N
S N
S
At first sight, CARL and FEL look very different…
~p
electrons
EM pump, ’w
(wiggler)
BackscatteredEM field’ ’w
Connection between CARL and FEL can be seen
more easily by transforming to a frame (’)
moving with electrons
Cold atoms
Pumplaser
Backscatteredfield
Connection between FEL and CARL is now clear
FEL
CARL
~p
Collective Recoil Lasing = Optical gain + bunching
In FEL and CARL particles self-organize to form compact bunches ~ which radiate coherently.
N
j
i jeN
b1
1 bunching factor b (0<|b|<1):
Exponential growth of the emitted radiation:
Both FEL and CARL are described using the same ‘classical’ equations, but different independent
variables.
AieNz
A
z
A
ccAez
N
j
i
ij
j
j
11
2
2
1
.).(
FEL:
;)( tzkk w
CARL:;2kz
N
NA photons2||
gL
zz
cL
tzz 0
1
v
;4w
gL ;gw
c LL
3/13/20 nBk
mcw
3/2
3/13/1
a
L nP
)/(1 czztz recrec
m
krec
22
CARL-FEL instability animation
Animation shows evolution of electron/atom positions in the dynamic pendulum potential together with the probe field intensity.
01
z
A
)cos(||2)( AV
Linear Theory (classical)
2 1 0
Maximum gain at =0
runaway solution
See figure (a)
)()()( 0 CARLFEL
k
mc
rec
pr
zieA
zeA 32||
We now describe electrons/atoms as QM wavepackets, rather than classical particles.
Procedure :
Describe N particle system as a Q.M. ensemble
Write Schrodinger equation for macroscopic wavefunction
),( z
Quantum model of FEL/CARL
Include propagation using a multiple-scaling approach
),,( 1zz
Canonical Quantization
p
Hp
H
ccAep i ..
..2
2
ccAeip
H i
Quantization (with classical field A) :
ipp ˆ ip ]ˆ,ˆ[ HH ˆ
Hz
i ˆ
so
Aiezdzd
dA
ccezAiz
i
i
i
2
0
2
2
2
),(
..)(2
1
so
)()( 0 FEL
k
mc
k
pp z
R. Bonifacio, N. Piovella, G.R.M.Robb and M.Cola, Optics Comm, 252, 381 (2005)
Quantum FEL Propagation model
Here describes spatial evolution of on scale of and describes spatial evolution of A and on scale of cooperation length, Lc >>
1z
1 ( ) /r cz z v t L
We have introduced propagation into the model, sodifferent parts of the electron beam can feel different fields :
So far we have neglected slippage, so all sections of the e-beamevolve identically (steady-state regime) if they are the same initially.
Aiezzdz
A
z
A
ccezzAi
z
i
i
2
0
21
1
12
2
),,(
.].),([2
where
4cL
Quantum Dynamics
Only discrete changes of momentum are possible : pz= n (k) , n=0,±1,..
pz kn=1n=0n=-1
is momentum eigenstate corresponding to eigenvalue ( )n kine
2| |n nc p
n
inn ezzczz ),(),,( 11
probability to find a particle with p=n(ħk)
2,0
Aiccz
A
z
A
cAAccin
z
c
nnn
nnnn
*1
1
1*
1
2
2
classical limit is recovered for
many momentum states occupied,
both with n>0 and n<0
1
-15 -10 -5 0 5 100.00
0.05
0.10
0.15
(b)
n
p n
0 10 20 30 40 5010-9
10-7
10-5
10-3
10-1
101
=10, no propagation
(a)
z
|A|2
steady-state evolution:
01z
A
Quantum limit for
iezczcz )()(),( 10
Only TWO momentum states involved : n=0 and n= - 1
n=0
n=-1
Dynamics are those of a 2-level system coupled to an optical field,described byMaxwell-Bloch equations
1
0 100 2000
2
4
6
8
10
z
|A|2
0 100 200
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
z
<p>
1.0
01z
A
0 1 2 3 4 50
2
4
6
8
10
N(
)/N
/2
-20 -15 -10 -5 0 5 10 150.00
0.05
0.10
0.15
pn
n
0 1 2 3 4 50.0
0.5
1.0
1.5
2.0
N(
)/N
/2
-5 -4 -3 -2 -1 0 1 2 3 4 5
0.1
0.2
0.3
0.4
0.5
0.6
pn
n
Bunching and density gratingCLASSICAL REGIME >>1 QUANTUM REGIME <1
2| ( ) | 2| ( ) |
( ) inn
n
c e
zieA Quantum Linear Theory
014
12
2
-10 -5 0 5 10 150.0
0.2
0.4
0.6
0.8
1.0
(a) (b) (c) (d) (e) (f)
(f)(e)
(d)
(c)
(b)(a)
|Im|
Classicallimit
Quantum regime for <1
max at
2
1
)(
)(2/)( 0
CARL
FELkmc
recp
r
width
012 1
QUANTUM CARL HAS BEEN OBSERVED WITH BECs IN SUPERRADIANT REGIME (MIT, LENS)
When the light escapes rapidly from the sample of length L,we see a sequential Super-Radiant (SR) scattering, with atoms recoiling by 2ħk, each time emitting a SR pulse
KA
L
cK
Aicczd
dA
cAAccin
zd
dc
nnn
nnnn
*1
1*
1
2
2
damping of radiation
k2n=-2
n=0
n=-1
0 250 5000.000
0.001
0.002
z
|A|2
0 250 500
-4
-2
0
z
<p>
BECLASER
k2
)(sec|| 222 gNzhNA
SEQUENTIAL SUPERRADIANT SCATTERING
Superradiant Rayleigh Scattering in a BEC(Ketterle, MIT 1991)
for K>>1 and
K
• Production of an elongated 87Rb BEC in a magnetic trap
• Laser pulse during first expansion of the condensate
• Absorption imaging of the momentum components of the cloud
Experimental values:
= 13 GHzw = 750 mP = 13 mW
laser beam kw,
BEC
absorption imaging
trap
g
Experimental evidence of quantum CARL at LENS
2p k
L.Fallani et al, PRA 71 (2005) 033612
The experiment
pump light
n=0(p=0)
n=-1(p=2ħk)
n=-2(p=4ħk)
Temporal evolution of the population in the first three atomic momentum states during the application of the light pulse.
Particles at the trailing edge of the beam never receive radiation from particles behind them: they just radiatein a SUPERRADIANT PULSE or SPIKE which propagates forward.
if Lb << Lc the SR pulse remains small (weak SR).
if Lb >> Lc the weak SR pulse gets amplified (strong SR) as it propagates forward through beam with no saturation.
The SR pulse is a self-similar solution of the propagation equation.
PROPAGATION EFFECTS IN FELs : SUPERRADIANT INSTABILITY
1
i
z
Ae
z
A
Strong SR (Lb=30 Lc) from a coherent seed
SR in the classical model:
c1 L
vtzz
R. Bonifacio, B.W. McNeil, and P. Pierini PRA 40, 4467 (1989)
Ingredients of Self Amplified Spontaneous Emission (SASE)
i) Start up from noiseii) Propagation effects (slippage)iii) SR instability
The electron bunch behaves as if each cooperation length would radiate independently a SR spike which is amplified propagating on the other electrons without saturating. Spiky time structure and spectrum.
SASE is the basic method for producing coherent X-ray radiation in a FEL
CLASSICAL SASE
CLASSICAL SASE
Example from DESY (Hamburg) for the SASE-FEL experiment
Time profile with many randomspikes (approximately L/Lc)
Broad and noisy spectrum atshort wavelengths (X-FEL)
SASE : NUMERICAL SIMULATIONS
cLL 30
CLASSICAL REGIME: 5 QUANTUM REGIME: 1.0
Classical behaviour : both n<0 and n>0 occupied
CLASSICAL REGIME: 5 QUANTUM REGIME: 1.0
SASE: average momentum distribution
Quantum behaviour : sequential SR decay, only n<0
0.1 1/ 10 0.2 1/ 5
0.3 1/ 3.3 0.4 1/ 2.5
Quantum SASE:Spectral purification and multiple line spectrum
• In the quantum regime the gain bandwidth decreases as line narrowing.
• Spectrum with multiple lines. When the width of each line becomes larger or equal to the line separation, continuous spectrum, i.e., classical limit. This happens when
3/ 24
3/ 24 1 0.4
CLASSICAL SASEneeds:GeV Linac (Km)Long undulator (100 m)High cost (109 $)yields:Broad and chaotic spectrum
FEL IN SASE REGIME IS ONE OF THE BEST CANDIDATE FOR AN X-RAY SOURCE (=1Ǻ)
QUANTUM SASEneeds:MeV Linac (m)Laser undulator (~1m)lower cost (106 $)yields:quasi monocromatic spectrum
CONCLUSIONS
• Classical FEL/CARL model- classical motion of electrons/atoms- continuous momenta
• Quantum FEL/CARL model- QM matter wave in a self consistent field- discrete momentum state and line spectrum
• Quantum model with propagation- new regime of SASE with quantum ”purification’’ - appearance of multiple narrow lines