Post on 27-Mar-2015
transcript
F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 1
Università di Urbino
Italy
F. VetranoUniversità di Urbino & INFN Firenze, Italy
Atom interferometers for gravitational wave
detection: a look at a “simple” configuration
F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 2
Università di Urbino
ItalyPerformance and Sensitivity
Frequency response: phase difference at the output when the input is a “unity amplitude” GW
Noise spectrum:power spectral density of phase fluctuations read at the output
Sensitivity:the smallest amplitude wave that can be detected at a fixed S.N.R. (usually 1)
)( I )( F )( output input
Frequency Response
dt) n(t n(t) T
1 lim )(corr
T/2
T/2-Tnn,
de )(corr )(P-
inn,n
1 )(P ),(h~
N
SnGW
Hz
1 Ωh
~ h
~ss
F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 3
Università di Urbino
Italy
The Ingredients of Sensitivity - 1
As an example look at the performance of an optical interferometer(a Michelson with suitable technical solutions when a plane GW with “+”polarization is impinging on it along a direction perpendicular to its arms):
Frequency Response:
h c
Ltcos
c
L sinc
4 L
tc 2
; c
Ltcos
c
L sinch
c
L2t
Input (GW)output(phase difference)
Frequency Response
Geometrical TermProbe Term
Configuration Term
Geometrical Term: Scale factor related to the dimension of the detector (the length of Michelson arms, and their angular relation)
Probe Term: the Physics for detection (interference of optical beam)
Configuration Term: the geometrical arrangement of components of the detector (refraction, reflection and recombination of the same beam on suspended mirrors in an orthogonal – arms Michelson)
F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 4
Università di Urbino
Italy
The Ingredients of Sensitivity - 2Because of the discrete nature of light and/or atomic beams, we have a unavoidable limit in reading the interferometer output: the Shot Noise. We adopt the “Shot Noise limited Sensitivity” as a first criterium for comparing performances.
Noise spectrum (Shot Noise only):Assuming poissonian distribution we have:
Standard Deviation fluctuations at the output
Power Spectral Density
N N N S 1
N D S P 2 1
Correlation
The minimal detectable signal amplitude at S.N.R. = 1 is supplied by (η2 is a “efficiency of the process”)
where η=η2/η1 is a “efficiency number” (we put η=1 from now on) and for a Michelson interferometer
) ( N dt)n(t n(t) T
1 lim )(corr 2
1
T/2
T/2-Tnn,
(Shot Noise is a white noise)
(η1 is a kind of “reading” efficiency)
N N 1 2 )F( N
1 )(h
~
c
Ltcos
c
L sinc
4 L )( F
F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 5
Università di Urbino
Italy
Why we hope in Atom Interferometry ?
)(G N L
)(h~
Shot Noise limited sensitivity - Matter Waves versus Optical Waves:a naive approach
Probe Term: 8
2P
L
POW
MW 10v
c
cmc
f
vm
2
max gain for fast – not
relativistic atoms
Shot Noise 2
Atom
Photon 10N
N
min loss for 100 W laser and the max value found in literature for Atom flow
(~ 10 )
18
Six order of magnitude at our disposal assuming the same order of magnitude for geometrical term. Are we able to use this resource? And what about the configuration term G(Ω)?
F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 6
Università di Urbino
Italy
Towards the evaluation of the S.N. limited sensitivity
Source T T
g, 0
e, k
g, 0
g, 0
e, k
Detection
The absorption (emission) of momenta modifies both internal and external states
We use the ABCD formalism, applied to a wave packet represented in a gaussian basis (e.g. Hérmite-Gauss basis).
Single interferometer with M.Z. geometry and light-field beam-splitters
F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 7
Università di Urbino
Italy
Suppose the Hamiltonian quadratic at most:
ic massrelativistM ; ~
q)t(gM-p(t)fqq 2
M
2
pq
M2
pp
2
qp H
BBBB
Determine the ABCD Matrices - 1
Evolution (via the Ehrenfest theorem) through Hamilton’s equations:
)t()t(
)t()t()t(
)t(g
)t(f
M
pq
)t(
dq
dH
M
1dp
dH
M
pq
dt
d
F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 8
Università di Urbino
Italy
Determine the ABCD Matrices - 2
The integral of Hamilton’s equations is:
A perturbative expansion leads to:
dt')'t(expτ dt'δ(t')γ(t')
β(t')α(t')expτ
)t,t(D)t,t(C
)t,t(B)t,t(A
dt'g(t')
f(t')
)t,'t(D)t,'t(C
)t,'t(B)t,'t(A
)t,t(
)t,t(
)t,t(
)t,t(
M
)t(p)t(q
)t,t(D)t,t(C
)t,t(B)t,t(A
M
)t(p)t(q
t
ot
t
otoo
oo
ot
t oo
oo
o
o
o
oo
o
oo
oo
.....''dt)''t()'t('dt'dt)'t( 10
01
)t,t(D)t,t(C
)t,t(B)t,t(A t
ot
t
ot
't
otoo
oo
time ordering operator
F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 9
Università di Urbino
Italy
where:
)Y,X,q,p, w.p.(t)t,t(iS
exp )Y,X,q,p,t.(p.w 1111121cl
22222
1
1
2121
2121
2
2
21
21
1
1
2121
2121
2
2
Y
X
DC
BA
Y
X
M
pq
DC
BA
M
pq
Under paraxial approximation, the evolution of the gaussian wave packet is determined by the classical action Scl and by the use of the ABCD matrices:
Evolution of a gaussian wave packetunder ABCD description
(X/Y is the complex radius of curvature for the gaussian w.p.)
F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 10
Università di Urbino
Italy
The Beam Splitter influenceStandard 1st order perturbation approach for weak dipole interaction
ttt theorem
The B.S. (neglecting possible dispersive properties) introduces a multiplicative amplitude Qbs and a phase factor simply related to the laser beam quantitiesω*, k*, Φ*
where q* = qcl(tA), qcl being the central position of the incoming atomic w.p., withrespect to the laser source, and tA = central time of e.m. pulse (used as an atom beam splitter).
qk t exp Qbs
F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 11
Università di Urbino
Italy
Phase shift for a sequence of pairs of homologous paths - 1
kβ1
kβ2kβ3 kβi
kβN
kα1 kα2kα3
kαikαN
t1 t2 t3 ti tN tD
Mβ1 Mβ2 Mβ3 Mβi MβN
Mα1 Mα2 Mα3 Mαi MαN
β1
β2
β3
βi
βN βD
α1 α2 α3 αi
αN
α D
t
q
F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 12
Università di Urbino
Italy
Phase shift for a sequence of pairs of homologous paths - 2
From previous results:
α Dα Dβ Dβ D
N
1jα jβ jjα jβ jα jα jβ jβ j
N
1jj1jj1j
qq pqq p1
tωωqkqkt,tSt,tS1
w.p. propagation
Phases imprinted by the B.S. on the atom wavesSplitting at the exit of the interferometer
Space integration around the mid (exit) point, equal masses on both the paths and identical starting points q1α = q1β lead to simplified expression
where all qj are evaluated by using ABCD matrices.
N
1jα jβ jjα jβ j
α jβ j
α jβ j tωω2
qqkk
F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 13
Università di Urbino
Italy
• Choose a system of coordinates
• Calculate ABCD matrices in presence of GW at the 1st order in the strain amplitude h
• Apply ΔΦ expression (previous slide) to the settled interferometer
• Use ABCD law to substitute all qj in Δφ expression
• Fully simplify
• Print ΔΦ
• End
Note : the job should be worked in the frequency space (Fourier transform)
The Machine
F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 14
Università di Urbino
ItalyHow about coordinates ? - 1
q)t(gM-p(t)fqq 2
M
2
pq
M2
pp
2
qp H
BBBB
Coordinates (and GW) are in the Hamiltonian:
Starting from usual Lagrangian function (signature +,-,-,-)
where gμν is the metric tensor, in the weak field approximation
the first order expansion leads
to the Hamiltonian function :
To be compared with previous general expression.
νμμν xxg mc L
1 ; η g μνμνμνμν
M2
pppp2pppp
M
1H ji
ij
oio
ijij
iooo
o
F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 15
Università di Urbino
ItalyHow about coordinates ? - 2
Finally:
The matrices α,β,γ,δ are fully determined by the metric
(as usual greek indexes run from 0 to 3; latin indexes from 1 to 3)
In the following we assume for simplicity f = g = 0 and GWs with “+” polarization, propagating along the z axis (j = 3).
ijijij
ij
ijoi
ii
jij
i2oo
fqc
qg2qqc
dxdxdxdxds2
F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 16
Università di Urbino
ItalyFermi Coordinates - 1
Metric essentially rectangular (near a line), with connection vanishing along the line, and series expansion:
Laboratory Reference Frame:
where h is the amplitude of the “+” polarized GW.We assume z = 0 as the plane of the interferometer and we develop our calculations on this plane.
)1 than less modulus(with number realε
dx dx x 0xxε R ηds νμ3βααβμνμν
2
lightspeed of ; c2, 1m, n
c
dzdtxxh
2
1dxdxηds
2
nmm n
νμμν
2
F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 17
Università di Urbino
ItalyFermi Coordinates - 2
It is easy to obtain: α = δ = 0; β = 1; γ = Ω² h/2, which leads to the following expressions for A,B,C,D matrices:
(for a single Fourier component)
F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 18
Università di Urbino
ItalyFermi Coordinates - 3
And finally we write the I/O relation through the response function:
where all the quantities are expressed in the FC system and ћ is the reducedPlanck constant.
The index 1 refers to the first interaction between atoms and photons beams.
4321ΩΤsiniΩΤcos2
2ΩΤ/
2ΩΤ/sin1 q1k2T
2
Ωh2Ω
2
2ΩΤ/
2ΩΤ/sinΩΤcos
ΩΤ
ΩΤsinΩΤ2sini
ΩΤ
ΩΤcosΩΤ2cos2
2ΩΤ/
2ΩΤ/sinΩΤsin
2
1k1p
M
1 1k2TΩΩ h ΩΔ
F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 19
Università di Urbino
ItalyEinstein Coordinates - 1
In this system the “mirrors” are free falling in the field ofthe GW, and the metric is
Hence α = δ = γ = 0; β = h η, where η is the minkowskian matrix, and hthe amplitude of the “+” polarized GW; we deduce immediately the ABCD matrices:
dzh)dy1(h)dx1(dtc ds 2 2 2 2 22
1e eiΩ
Ωhtt,tt B
0,tt ; C1,ttD,tt A
121 tti Ω t i1212
121212
F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 20
Università di Urbino
Italy
Einstein Coordinates - 2We cannot use the same k for every atoms/photons interaction; from the metric for a null geodesic we have
By inserting these kj values in the general expression for Δφ, we obtain
where all the quantities are expressed in the EC system. But the transformation matrix S from FC to EC behaves as S = 1 + 0(h); so the two expressions for Δφ in the two systems of coordinates are identical (as expected from the gauge invariance property of Δφ).
j t i
1j
j1j
e 2
h kk
δ k k k 2
t, h1c
dt
dxv
4321ΩΤsiniΩΤcos2
2ΩΤ/
2ΩΤ/sin1 q1k2T
2
Ωh2Ω
2
2ΩΤ/
2ΩΤ/sinΩΤcos
ΩΤ
ΩΤsinΩΤ2sini
ΩΤ
ΩΤcosΩΤ2cos2
2ΩΤ/
2ΩΤ/sinΩΤsin
2
1k1p
M
1 1k2TΩΩ h ΩΔ
F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 21
Università di Urbino
Italy
FC: fiducial observer: the laser device is free falling; a tidal force actson the atoms; the interaction points move and imprinted phases changeaccordingly.
EC: Atoms are free falling; no forces on them; the space between interactionpoints shows a variable index of refraction; the imprinted phases changeaccordingly.
Two different descriptions; same (physical) result, obviously.
Descriptions and Result
h(t)dΩf 2
2
)t(h1n
A.S.
L.B. L.B.
A.S.
F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 22
Università di Urbino
Italy
The main contributions - 1
ΩΤsiniΩΤcos2
2ΩΤ/
2ΩΤ/sin1 q1k2T
2
Ωh2Ω
2
2ΩΤ/
2ΩΤ/sinΩΤcos
ΩΤ
ΩΤsinΩΤ2sini
ΩΤ
ΩΤcosΩΤ2cos2
2ΩΤ/
2ΩΤ/sinΩΤsin
M
1p 1k2TΩΩ h
ΩΔ
A kind of “clock term”, related to the travel of the beam from the laser to thefirst interaction point, viewed through the A.I. as a read-out. For a discussionabout this term see: S. Dimopoulos et al, Phys.Rev D, 122002 (2008)
We discuss here only the first term, in which we have neglected the smallercontribution k²ћ / 2M* (in next few slides we put G(Ω) = [Ω T …… ]/2)
F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 23
Università di Urbino
Italy
It’s easy to rewrite the phase difference as:
that is:
to be compared with what we wrote in slide 3 (optical Michelson)
The main contributions - 2
h )G( TM
pp
p
2k - 11
1
1
)(h G L
4 - MW
opening angleGeometrical dimension
Geometrical termProbe (matter wave)
Configuration term
F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 24
Università di Urbino
ItalyShot Noise Limited Sensitivity
Considering only the first term of the slide 23, and supposing the A.I.
“shot noise” limited as clarifyed in slide 4
at the level of S.N.R. = 1 we have (with η = 1)
which has the expected form (see slide 5).
Nη
1 ΩΦ Δ Sh.N.
G
1
L 4
N
1 h
~MW
F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 25
Università di Urbino
Italy
100 200 500 1000 2000 50000
0.2
0.4
0.6
0.8
1
1.2
2
2
T2
Tsin
T
Tsin 2 - 1
2
T2
Tsin
2
G
The Configuration Term
lG(Ω)l
Frequency [Hz]
0.001 0.002 0.005 0.01 0.02 0.05 0.10
0.2
0.4
0.6
0.8
1
1.2
ToF 50 s
1 2 5 10 20 500
0.2
0.4
0.6
0.8
1
1.2
ToF 0.1 s
10 20 50 100 200 5000
0.2
0.4
0.6
0.8
1
1.2
ToF 0.01 sToF 1ms
branches) theof bandwidth thedetermines (this L
v2 : poles adiacent between Distance
1 and 0 betweenrapidly more and more oscillates ΩG ; Ω0G Ω2
0
F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 26
Università di Urbino
Italy
The Scale Factor Σ
G
1
N vTp
/2 h
~
tr
G
1
L 4
N
1 h
~MW
Σ = Σ1 Σ2
s J 10 5 -351
We need to have Σ2 as larger as we can, but:
• T is not free (the bandwidth behaves as 1/T)
• vT is the longitudinal dimension L of the A.I. (coherence problem)
• Ptr T/M is the transversal dimension of the A.I. (coherence and handling problems)
F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 27
Università di Urbino
ItalySome sensitivity curves
s10ToF
Js 10 4.3 3
14- 2
s 10ToF
Js 10 4.3 2
12- 2
s 50ToF
Js 10 5.5 15- 2
We represent the first branch only of the sensitivity curves
Let us consider in some detail a specific interesting example (see next slide)
F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 28
Università di Urbino
Italy
NS Binary
Coalescencelrs
SlowPulsars
LMXRBs&
Perturbed“newborn”NS
A rough picture of Sources & Detectors
-24
-22
-20
-18
h [1/sqrt Hz]
f [Hz] - 4 - 2 0 2 4
10 10 10 10 10
Galactic binaries
Coalescence of
massive B
H
IntermediateBH-BH Coalescence
SN corecollapse
msPulsars
1
23
1 LISA2 LIGO – Virgo3 A.I.
NS Binary
Coalescencehrs
F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 29
Università di Urbino
Italy
Numbers
Js 10 1.1 Σ
s; 4.0ToF13 -
2
F [Hz]
h [
1/√
Hz]
m 10 4 L 3
m/s 10 v 4
17- tr 10 2.7 Np
10N 18 m/s Kg 10 2.7 p 26 - tr m/s 16v tr
H
) Lyman H , 10 ( photons UV4 7 -
0.3 0.5 0.7 1 1.5 2
1. 1021
1.5 10212. 1021
3. 1021
5. 10217. 1021
Js 10 1.1 Σ
s; 4.0ToF13 -
2
0.1 1 10 100 1000
3.1023
5.1023
Virgo S.N.-limited Sensitivity
A.I. S.N.-limited Sensitivity
F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 30
Università di Urbino
Italy
Some conclusions
• Comprehensive approach to the problem with (hopefully) reliable calculation of Frequency Response function for atom interferometers
• L and VL frequency disfavoured from the FR behaviour: move the first non-zero pole towards very low values (at expenses of reduced bandwidth)? Different, more complex configurations? (e.g.: asymmetric interferometers; multiple interferometers)
• S.N. very hard limit: balance it with LMT? Heisenberg limit?
• Terrestrial solution: the true noise budget has to be investigated (thermal noise; seismic wall;….) in the low- and intermediate-frequency range;
• Space solution: removing seismic wall is of great advantage but in any case S.N. limit is hard : balance it with LMT and large dimension (but divergency problem) ? Or very slow atoms (but decay problem)?
In any case, in my opinion required numbers are leaving the realm of forbidden dreams and are entering the world of exciting challenges
optimistic