-
8/3/2019 M-C. Angoninand Ph. Tourrenc- Cold atom interferometer in a satellite : orders of magnitude of the tidal effects
1/19
1
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
-
8/3/2019 M-C. Angoninand Ph. Tourrenc- Cold atom interferometer in a satellite : orders of magnitude of the tidal effects
2/19
2
Goal
atomic wave interferometer in a free falling satellite
+ telescope pointing toward far away star
acceleration and rotation
general parametrized description
-
8/3/2019 M-C. Angoninand Ph. Tourrenc- Cold atom interferometer in a satellite : orders of magnitude of the tidal effects
3/19
3
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
-
8/3/2019 M-C. Angoninand Ph. Tourrenc- Cold atom interferometer in a satellite : orders of magnitude of the tidal effects
4/19
4
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
-
8/3/2019 M-C. Angoninand Ph. Tourrenc- Cold atom interferometer in a satellite : orders of magnitude of the tidal effects
5/19
5
Atomic Sagnac Unit
2 Ramsey-Bordinterferometers
Shift between the two centres : OSOs
HYPER
Spin around u 1 : angle t ,
angular velocity ddt
-
8/3/2019 M-C. Angoninand Ph. Tourrenc- Cold atom interferometer in a satellite : orders of magnitude of the tidal effects
6/19
6
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
-
8/3/2019 M-C. Angoninand Ph. Tourrenc- Cold atom interferometer in a satellite : orders of magnitude of the tidal effects
7/19
7
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
-
8/3/2019 M-C. Angoninand Ph. Tourrenc- Cold atom interferometer in a satellite : orders of magnitude of the tidal effects
8/19
8
)()0( , kuu
= Tetrad comoving with the satellite
-
8/3/2019 M-C. Angoninand Ph. Tourrenc- Cold atom interferometer in a satellite : orders of magnitude of the tidal effects
9/19
9
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
-
8/3/2019 M-C. Angoninand Ph. Tourrenc- Cold atom interferometer in a satellite : orders of magnitude of the tidal effects
10/19
10
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
-
8/3/2019 M-C. Angoninand Ph. Tourrenc- Cold atom interferometer in a satellite : orders of magnitude of the tidal effects
11/19
11
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
-
8/3/2019 M-C. Angoninand Ph. Tourrenc- Cold atom interferometer in a satellite : orders of magnitude of the tidal effects
12/19
12
First conclusion :
Tidal effects on HYPER
Perfectly symmetric interferometers superimposed
with precision < 2 nm
-
8/3/2019 M-C. Angoninand Ph. Tourrenc- Cold atom interferometer in a satellite : orders of magnitude of the tidal effects
13/19
13
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 !
-
8/3/2019 M-C. Angoninand Ph. Tourrenc- Cold atom interferometer in a satellite : orders of magnitude of the tidal effects
14/19
14
We measure and11
2
212
-
8/3/2019 M-C. Angoninand Ph. Tourrenc- Cold atom interferometer in a satellite : orders of magnitude of the tidal effects
15/19
15
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
-
8/3/2019 M-C. Angoninand Ph. Tourrenc- Cold atom interferometer in a satellite : orders of magnitude of the tidal effects
16/19
16
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
-
8/3/2019 M-C. Angoninand Ph. Tourrenc- Cold atom interferometer in a satellite : orders of magnitude of the tidal effects
17/19
17
= 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
-
8/3/2019 M-C. Angoninand Ph. Tourrenc- Cold atom interferometer in a satellite : orders of magnitude of the tidal effects
18/19
18
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
-
8/3/2019 M-C. Angoninand Ph. Tourrenc- Cold atom interferometer in a satellite : orders of magnitude of the tidal effects
19/19
19
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