m-c. angoninand ph. tourrenc- cold atom interferometer in a satellite : orders of magnitude of the...

8/3/2019 M-C. Angoninand Ph. Tourrenc- Cold atom interferometer in a satellite : orders of magnitude of the tidal effects

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Cold atom interferometer

in a satellite :orders of magnitude of the

tidal effects

M-C. Angonin and Ph. Tourrenc

ERGA / LERMA

Universit Pierre et Marie CurieObservatoire de Paris

and

CNRS


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Goal

atomic wave interferometer in a free falling satellite

+ telescope pointing toward far away star

acceleration and rotation

general parametrized description


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Principle

* Determination of the local metric to the third order

comoving with an idealized device

* Define someperturbations* Compute thephase shift

* Order of magnitude of each term

* Select the terms that cannot be modelized

norcancelled


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Do we have a possibility to observe

Lense-Thirring effect ?

If Yes : How shall we proceed ?

If No : Why not ?

And what can be done presently ?

The case of HYPER


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Atomic Sagnac Unit

2 Ramsey-Bordinterferometers

Shift between the two centres : OSOs

HYPER

Spin around u 1 : angle t ,

angular velocity ddt


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Local metric

00 1 2U

0k2

r3J r 1

2U w

ds2 g00 dt2 2g0kdx

kdt gikdxidx k

Geometrical units : G c 1

U)(h ikikikik 21+=+=

Lense-Thirring

21 J2Mr Rr UU P2 cos S M,P, etc,

J1

J*

1

82


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Parameters

MrOrbital velocity : 2.5105( HYPER)O1

rl=Size of the orbit :

: X d lSize of the experimental set -up 60 cm

vAtom velocity :

Orders of magnitude

Eccentricity of the orbit < l1/2 O1


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)()0( , kuu

= Tetrad comoving with the satellite


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The metric comoving with the set-up (Li & Ni 1979)

locityangular velocal:onacceleratilocal:

r

r

a

ab 0

g 0 0 1 2 a x a x

2

x

2

U, i k xi x k 1

3U, i k j x

i x kx j

g 0 k xk

23 R 0 i k j x i x j

gm n m n1

3R i m j n x

i x j .

+ d2 l3 O6

+ d2 l5/2 O5

+ d2 l2 O4

measured by comovingaccelerometers and gyroscopes

0 T d S TSpin

Aberration + deflexion


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B 2 t

t

t dt with

The phase shift on one atomic beam(WKB approximation L & T 1976)

t h 00 2h0kvatk h kjvat

k vatj

and matomc2

Lense-Thirring effect c Jr3

10 14 rad s 1( HYPER)

ccuracy on Lense Thirring phase shift within the present framework :

%)( 182 vMJ

d

HYPER


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can be calculated at the required accuracywith the assumption (??)

O6accuracy of the present expansion

xxrr // 410r

d2 l3mod

2J

r3

3 J n

r3 n1 M

4 r2 n w x vg A Lense-Thirring

2 u 1 x vat B spin of the satellite

2

M

r2 u 1 n x v at 2 u 1 a x v at C aberration

M

r3x2 n

23 x D tidal effect

2 a OS x E residual acceleration

v at atomic velocity 20 cm s 1 O2 c

1 w : coupling to a preferred frame


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First conclusion :

Tidal effects on HYPER

Perfectly symmetric interferometers superimposed

with precision < 2 nm


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SunJupiterEarth

10-14 rad.s-1 10-12 rad.s-1 10-13 rad.s-1 Lense-Thirring

18 % 1.5 % 90 %Accuracy

2 10-12 10-12 6 10-13Tidal effects

U/U < l d O2

No !


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We measure and11

2

212


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Spinning satellite

: spin angular position: orbital angular position

1 2vatc 2

2 cTD2

r

K1 K2

Lense-Thirring ; 2

Geometry of the set-up 2 ; 2; 2+; 22 ;2+2

Change of the local field 2; 2+; 22 ;2+2

during the flight time of the atoms

Preferred frame


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Conclusion

Spinning the satellite

A polar, circular orbit with (?)

The high sensitivity to geometrical defect.

de Sitter/Thomas effect,54

1 10,10

2/ =

Bibliography

Angonin et al., GRG, 2004

Tourrenc et al., GRG, 2004


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= Tetrad comoving with the satellite)()0( , kuu

M R J J J 2 r

4,4 mm 6400 km 145 cm 2 10 3 10 6 7000 km

Geometry of the set up2 T 2


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2 + 2

2 2

2

cteM

r 2cos 2

3Mr 1 sin2

4 sin 22

r cos 23

r

3Mr

1 sin 2

8sin 2 2

2

r cos 2 23

r

Mr 4

1 sin 2 sin 2 2

Mr 4

1 sin 2 sin 2 2

3Mr

1 sin 2

8sin 2 2

2

r cos 2 23

r

: spin angular position

: orbital angular position

Geometry of the set-up

Change of the local fields during

the flight time of the atoms2 cTD2

r 1017

1 2v atc 2

2 cTD2

rK1 K2

r 6 10

10v atTD cTD

r2 r

3/2

1010

v t 2 cTD2

2 cT 2


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2

2 +

2vat

r 2 1024

1 2v atc 2

2 cTDr

K1 K22 cTD

r10 17Lense-Thirring

Preferred frame

r 6 1010

Geometry

vat

r 1014

Aberration

Change of the localfields during the flight

time of the atom

2vatJ

r2

cos J32

M

rcos sin sin

1

r

6vatJ

r2 cos 2 J 2

M

r 1 sin cos cos 2

32

M

rcos 1 sin

2sin 2

1

r

vatM

r 1w

3 2 2 sin sin

2

M

r1 sin cos cos 2

3M

2 rcos 1 sin

2sin 2

1

r

10 10

m-c. angoninand ph. tourrenc- cold atom interferometer in a satellite : orders of magnitude of the...

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