Download - F. Vetrano – GW&AI – Feb 24, 2009 – GGI Firenze 1 Università di Urbino Italy F. Vetrano Università di Urbino & INFN Firenze, Italy Atom interferometers.

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



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



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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)



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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



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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)?



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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



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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



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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



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.)



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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



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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



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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



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• 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



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



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



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




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)




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



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



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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



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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.



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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)



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



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



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



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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)



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)



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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



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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



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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