Can we use atom interferometers in searching for gravitational waves?

F.Vetrano - Aspen Winter Conference, FEB 2004

Can we use atom interferometers in searching

for gravitational waves?

• C.J. Bordé, University of Paris N.

• G. Tino, University of Firenze

• F. Vetrano, University of Urbino


Intuitive concept: δΦ ~ 1/λ lower λ as much as you can !!

1210 2

10 10 mc

MW

OW

OW

MW

But only if any other thing is the same. Is it so? Is it easy?

• The Physics (e.g. : mp > 0, vp < c; mL = 0, vL = c different dispersion equations E = E(p))

• The Technology: very different state of evolution for OW and MW “optical components”


What kind of interference are we speaking about ? - 1

A coherent description of quantum particles exists, both for Dirac and K.G. scalar particles, from which the main contribution from gravity fields to the phase can be deduced. Neglecting coupling between spin and curvature we have:

• scalar (newtonian)

• vectorial (e.g. Sagnac, Gale…..) i = 1,2,3

• tensorial (G.W.) i,j = 1,2,3

[C.J.Bordé et al, in “Gyros,Clocks,Interferometers…”, C.Lammerzahl, C.W.F.Everitt, F.W.Hehl (Eds), Springer, Berlin (2001) C.J.Bordé, Gen.Rel.Grav., 36, (2004) in press]

php E(p)

dt

2

c

1

0

t

t

2

dth 2

E 1

0

t

t

oo

dt ph c i

t

t

io

1

0

dt php E2

c jij

t

t

i2 1

0


What kind of interference are we speaking about ? - 2

Ramsey Interference

• Two level atoms (ground, excited)• Two successive electromagnetic interactions• No external momentum exchange (for the moment)

Only internal degrees of freedom are involved in the processes

[w.p.] g g, e

e.m.1 e.m.2

Overlap (g, e): e g

0 t1 t2 t

No possibility to know which are g and which are e along the (time) path: in the finale state there isinterference between the two states: this is the Ramsey Interference.

decay

It involves only internal D.O.F. : we look at the number of decays/s

|e> |g> , which shows interference fringes in the “time” space.


M.W. Evolution - 1

Standard quantum approach (in “rotating” system): first order perturbation theory for a dipole e.m./w.p. interaction. If for simplicity we assume zero detuning δ, the population ofthe e state after the interaction time τ with the e.m. field (by a laser of frequency ωL) is

) cos - (1 2

1 )(c eg

2

e

where Ω eg is the Rabi frequency ( ), being d the dipole momentum

of the atom and E the electric oscillating field. Putting ωeg = ωe – ωg , the detuning is definedby δ = ωL – ωeg. Assumiming δ = 0, the equation (♦) describes the so called resonant Rabi oscillations .

gEde - eg

(♦)

Assume: t = 0 |cg(0)|² = 1; τ such that Ω eg τ = π/2 (π/2 pulse). From (♦) we have :

2

1 )(c

2

e 2

1 )(c

2

g


M.W. Evolution - 2

i

1

2

1

1

0

π/2 pulse

This is a perfect Beam Splitterfor internal states

g

g

ee

0

1/2

1

τ


M.W. Evolution - 3Use two π/2 pulses in order to have 2Ωeg τ = π (π pulse): we have 1 )2(c

2

e

0

1

1

0

π pulse

This is a perfect Mirrorfor internal states

1

0 2 τ

g

e

e

g

If δ ≠ 0, with two π/2 pulses (time interval T between them) same considerations as before with Tcos 1 2

1 )T2(c

2

e

Note:


M.W. Evolution - 4

Take into account external D.O.F. considering the momentum exchange :

) t - rk ( cosE E Ed- V

p g p,g

p e p,e

Lo

gg

ee

• kinetic term in the Hamiltonian (and in the phase too)• recoil term in the detuning δ:

) M 2

k

M

kp (

2

egL

Even if the interference is on internal d.o.f., the recoil opens the path: if we look for interference, we must close it spatially. Anyway, we are always looking at the fringes in the number of decays/s.


The simplest “closed” atom interferometer

Source

g

e

g

g

e

Counter (decays/s)

π/2 π π/2

2T2(ckp,e

2T2(ckp,e

)2

- cos - (1 2

1 )2T2(c

2

kp,e

ΔΦ takes into account all that happens to the atoms (GW too)

• the absorption (emission) of momenta modifies both internal and external states simultaneously

• this Mach-Zender atom interferometer is too symmmetric: no hope for effects from GWs on the interference fringes on the number of decays/s


The Ramsey-Bordé Interferometer

Atoms source

1

2

a,0

b,1

a,0

a,0

a,0a,0

b,1

b,-1

b,-1b,-1

b,1

b,1

a,0

a,2

a,2

a = ground s.

b = excited s.

L 1 L 2 L 3 L 4

Atoms source

Laser

Cat eyes


Towards the optics of M.W. - 1

• Atom Interferometers exist (and do work)

• Coherent approaches to quantum particle in curved space (weak field) has been developed

• Find general tools for phase calculation in A.I. in presence of G.W.

• Design a “good” interferometer: optimal configuration, source, detection…..

• Find all possible noise sources and lower them to the best you can

We discuss only the first point with a simple look at the second one


Towards the optics of M.W. - 2

• In a limited interval of velocity the dispersion function E = E(p) can be “simplified” through a power series expansion a Schroedinger type equation is obtained

• If we look at the equation of a (laser) beam in a cavity with some (possibly) non-linear susceptivity, indicating by U the shape of the mode and z the propagation axis, we have:

Ur 2k

k U

2k

1

z

Ui n

n n

2o2

t

which is a Schroedinger-like equation, provided the exchanges t z; M/ћ k

• Gaussian modes are a good basis for light; but gaussian wave packets are a basis too for particle “beams”

[C.J.Bordé in “Fundamental systems in quantum optics”, LXIII Les Houches Session, J.Dalibard,J.M.Raimond,J.Zinn-Justin (Eds),Elsevier, Amsterdam (1992)

C.J.Bordé, Gen.Rel.Grav, 36 (2004) in press]


Towards the optics of M.W. - 3Look at the gaussian (lowest) modes for light and for a particle:

Q

1

X

Y ;

R2

wk i

w

w wk

X

Y

2

i

r 2

wk

X

Yi- exp

X

w i U

2oo

2

2o2

oo

22ooo

2

2

2

2

1

2

qM2

ti1q4

qexp

qM2

ti1q2

1 )t,q(

2

1

4

1

Note : M/ћ k; t z; Δq wo/2 ;


Towards the optics of M.W. - 4• similar equations

• similar basisCan we use the tools of gaussian (light) optics in gaussian (m.w.) optics?And especially the ABCD approach?

ABCD matrices approach for light

Gaussian mode(P1) medium [ABCD matrices] gaussian mode(P2)

D CQ

B AQ Q

1

12

ABCD law for gaussian

(light) optics

Yes, we can; but:

• remember the correspondances• most important: how can we write the ABCD matrices for M.W.s propagation?


The path to reach ABCD matrices - 1w.p.(P1) ? w.p.(P2)

[ABCD]

Suppose the Hamiltonian quadratic at most:

qq 2

M -

M2

pp qp H

��

�

• write the classical action Scl(1,2)• write the quantum propagator for w.p.(1) w.p.(2) (e.g. applying the principle of correspondence to Scl)• Apply the quantum propagator to a basis of gaussian w.p.s from (1) to (2)• Write formally the ABCD law and require it be identically satisfied


The path to reach ABCD matrices - 2We obtain:

where:

and:

τ being a time ordering operator.

)q,p, w.p.(t)t,t(iS

exp )q,p,t.(p.w 11121cl

222

1

1

2121

2121

2

2

1-

1

1

2121

21212

2

Y

X

DC

BA

Y

X

0 -

M

M

pq

DC

BA

M

pq

2

1

t

t2121

2121 dt exp DC

BA


The Beam Splitter influenceStandard 1st order perturbation approach for weak dipole interaction

ttt theorem

The B.S. introduces a multiplicative amplitude and phase factor simply related to the laser beam quantities (indicated by a star):

qk t exp Mbs

[C.Antoine, C.J.Bordé, Phys.Lett.A, 306 , 277-284 (2003) and references therein]


Phase shift for a sequence of pairs of homologous paths (an interferometer geometry) - 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

αD

t

q


Phase shift for a sequence of pairs of homologous paths (an interferometer geometry) - 2

q q 2M

k p - q q

2M

p

M

S

q q 2M

k p - q q

2M

p

M

S

2

M M

t - 2

q q k k q q

2

pp -

iii

iiii

i

1i

i

i

iii

ii1i1i

i

1i

i

iN

1 i

ii

N

1 iiiiii

iiii11

11

From ABCD law, ttt theorem, and properties of classical action Scl(P1,P2)for N beam-splitters :

[C.Antoine, C.Bordé, J.Opt. B: Quantum Semiclass.Opt., 5, 199-207 (2003)]


Build the simplest A.I. for G.W. - 1

Atom source

Counter

BS1 BS2 BS3 BS4

T1 T’ T2

a,0 a,0 a,0 a,0

b,k b, -k

a,0

Put T1 = T2 , T’ 0 the simplest unsymmetrical A.I.



The machine

• Choose FNC (It’s better!!!)

• Calculate ABCD matrices in presence of GW at the 1st order in the strain amplitude h (e.g.: a wave with polarization “+” impinging perpendicularly on the plane of the interferometer)

• Apply ΔΦ expression (slide 19) to the interferometer

• Use ABCD law to substitute all qj coordinates

• Fully simplify

• Print ΔΦ

• End

Note : the job should be worked in the frequency space: use Fourier transform, please!


Build the simplest A.I. for G.W. -3

The phase shift: f T h p m

k - 22

1

where: 2/m = 1/Ma + 1/Mb ; kћ = transversal momentum (by BS);

p1 = initial (longitudinal) momentum;

f(ω) = complex frequency response of the interferometer

2

2

2T2T

sin Tcos -

T

Tsin - Tsin2 i

T

Tcos - Tcos2

2T2T

sin Tsin

2 f



The sensitivity (1):

Suppose the A.I. “shot noise” limited, with η a kind of efficiency of the decay process we use:

At the level of S.N.R. = 1 we have: N

noise shot

2

2T

2T2T

sin

T

Tsin 2 - 1

2T2T

sin

2 f

f

1

T Lp N

2 h

Note: |f(ω)| ~ T (ω→0); |f(ω)| ~ (2/ω) |sin (ωT/2)/ (ωT/2)| → 0 (ω→∞)


Build the simplest A.I. for G.W. - 5The sensitivity (2): h (1/Hz)

0.1 1 10 100 1000 Hz

20 10

22 10

24 10

26 10

m/s 10 vm; 10 Ls; 10 T

(H)/s; atoms 10 N m/s; 10v

T3 3-

186L

s 10 T

m 10 L m/s; 10 v2 -

5 7 L

s/)Cs( atoms10 N

m; 50 L ; m/s 5 v; m/s 10 v 18

TL


Final Remarks

• comprehensive approach to the problem (A.I. + G.W.)

• “automatic” tool for solving atom interferometers

• realistic values of physical parameters (someone on the borderline…)

• interesting value of the sensitivity even for a “minimal” atom interferometer

• don’t forget BEC

• noise budget ?

• from the idea to the experiment

Looking at a future (not too far from now, hopefully) and at a very hard work, bearing in mind the title of this talk, in my opinion we can say…………….


Conclusion

….be optimistic: we can!!!


Some general references:

• P.Berman (Ed), Atom Interferometry, Ac.Press, N.Y. (1997)• S.Chu, in “Coherent atomic matter waves”, 72° Les Houches Session, R.Kaiser, C.Westbrook, F.David (Eds), Springer Verlag, N.Y. (2001)• C.J. Bordé, CR Acad.Sc.Paris, t2-SIV, 509-530 (2001)• C.J.Bordé, Metrologia, 39, 435-463 (2002)

Please Note:

Not all of the Authors have had the possibility of revising the final form of this talk; theyshare obviously the scientific content, but mistakes, misunderstandings, misprints are totally under my own responsibility alone (F.V.).

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Can we use atom interferometers in searching for gravitational waves?