VARENNA 2007
Introduction to 5D-Optics for Space-Time Sensors
Introduction to 5D-Optics for Space-Time Sensors
Christian J. BordéChristian J. Bordé
A synthesis between optical interferometry
and matter-wave interferometry
ATOMS ARE QUANTA OF A MATTER-WAVE FIELD
JUST LIKE PHOTONS ARE QUANTA OF THE MAXWELL FIELD
QM FOR SPACE / ONERA 2005
2
2 2
1
c t
2 22 2
20
M cp p M c
g
MOMENTUM
E(p)
p
atomslope=v
photonslope=c
rest mass
ENERGY
Mc2
h
h / h dB/
h dB
K
2 22 2
20
M cp p M c
2 4 2 2( )E p M c p c
CHEMIN OPTIQUE & PRINCIPE DE FERMAT
0
(3)00
jj
j j
dB
S p dx E dt p dx
gdl EE dt h dx
c g
2 2
0, 0,1,2,3
g p p M c
E p c
00
002
g
gggfdxdxfdl ji
ijijji
ij
222
00
2
3
cMcg
E
hdB
2 2 2Hamilton-Jacobi /
équation d'iconale si 0
g M c
M
E(p)
p//
h
h /
Recoil energy 22 2/ Mh
ab
ATOMES
b
a
a
b
b
a*
b*
a
b
b*
a
b*
a*
a*
ab
temps
espace
Optical clocks
Laser beams
Atom
beam
Stimulated Raman transitions
Raman pulses act as mirrors and beam splitters for matter waves
k2, 2k1, 1
|i >
|b
|a
~ 1 GHz
Alkali atoms (Rb, Cs)
|a and |b Hyperfine states
Tra
nsiti
on P
roba
bilit
y
Rabi
Effective two level system Quantum superposition => Rabi oscillations
pulseAtomic mirror
/2 pulse Atomic
beam Splitter
k2
k1
|a, p
ħkeff=ħ(k1-k2)
|b, p+ħkeff
Laser beams
Total phase=Action integral+End splitting+Beam splitters
Atoms
2 2
20
M c
1/ 2 1/ 2g g g g
KLEIN-GORDON EQUATION(Curved space-time)
with 1
1 with 2
g h h
hg h h
(espace) 3,2,1,(temps) 0,
33
2322
131211
03020100
1
1
1
1
h
hhsym
hhh
hhhh
g
dxdxgds 2Elementary interval
Metric tensor
Analogy with: ),( AVA
Post-Newtonian parameters (PPN):ijij c
Uh
c
U
c
Uh
2
2
2200
21
..22
00 2 2- gravitation field: 2 . / . . /
- rotation field: . /
- gravitational wave:
h g q c q q c
h q c
h
25 July 2003 BIPM metrology summer school 2003
2 4 2 2( )E p M c p c
ATOM WAVES
- Non-relativistic approximation:
2 2( ) / 2E p Mc p M
- Slowly-varying amplitude and phase approximation:
2 4 2 2 2 4 2 2 2 2 2
2 2 2 2 22 4 2 2
02 2 20
2 4 2 22 2 *0
0 0
( ) ( )
( ) ( )11 ...
2
1... / 2 ...
2 2 2
c c
c cc
c
E p M c p c M c p c p p c
p p p p cM c p c E
M c p E
E M c p cMc p M
E E
1 2 3 4
1.5
2
2.5
3
3.5
4E(p)
Mc2 p
* 20E M c
2 42 0
02 2
E M cMc
E
cp
BASICS OF ATOM /PHOTON OPTICSParabolic approximation of slowly varying phase and amplitude
Massive particles
1 2 3 4
1
2
3
4E(p)
p
/ 2
k
Photons
2* *
* 22 2 *
02*
1 1
2 2
; ;2 2
jj
j j
i Mc p p p h pt M M
M MMc c p i p M c
M
phase shift*
1
2p h p dt
M
Schroedinger-like equation for the atom (photon) field:
BASICS OF ATOM /PHOTON OPTICS
25 July 2003 BIPM metrology summer school 2003
2 4 2 2( )E p M c p c
0 0
3( ) ( ) /
3/ 2( , ) ,2 2
i p r r E t tdE d pa r t e E E p a p E
E p
ATOM WAVES
2 2 *0 0
* 2 *2 2 *0 0 00 0
2 * 20
3( ) ( / 2 )( ) /
3/ 2
3' / ( ) ' / 2 )( ) /( ) ( / 2 )( ) /
3/ 2
/ 2 ( ) /
( , )2
''
2
cc c
c
i p r r Mc p M t t
c
i p r r p M t t p M t ti p r r Mc p M t t
i p M Mc t t
d pa r t e a p p
d pe e a p
e
*
0 0( / ( )) / *0 0/ ( )c ci p r r p M t t
ce F r r p M t t
*
0 0 00( / ( )) /( , ) / *
0 0
2 2 20 0
/ ( )
with ( , ) 1 v / ( )
ccli p r r p M t tiS t t
c
cl c
e e F r r p M t t
S t t Mc c t t
Minimum uncertainty wave packet:
2 v1( , ) exp exp
2c
c
M z zM Yz t z z i
iXX
0( ) 2 /iX t z i t t M z
0 0 0( ) ( ) v( )c cz t z t t t t center of the wave packet
complex width of the wave packet in physical space
velocity of the wave packet
width of the wave packet in momentum space( ) /Y t M z
0v( ) v( )t t
* 2Im YX
M
conservation ofphase space volume
z=
ABCD PROPAGATION LAW
0
0
( )( )
v( )v( )cc z tz t A B
tt C D
0
0
( )( )
( )( )
X tX t A B
Y tY t C D
01
0 1
A B t t
C D
00 @
_ ( , )
exp ( , ) / _ ( ( ), v( ), ( ), ( ))cl t c
wave packet z t
iS t t wave packet z z t t X t Y t
Framework valid for Hamiltonians of degree 2 in position and momentum
0 0 0( , ) v( ) ( ) v( ) ( ) / 2cl c cS t t M t z t t z t
is the classical action
where
ABCD LAW OF ATOM/PHOTON OPTICS
( , )
exp / exp ( ) ( ) / ( ), ( ), ( )cl c c c
wavepacket q t
iS ip t q q t F q q t X t Y t
*
0 0 0
* *
0 0 0
( ) ( ) ( ) / ( , )
( ) / ( ) ( ) / ( , )
c c c
c c c
q t Aq t Bp t M t t
p t M Cq t Dp t M t t
0 0
0 0
( ) ( ) ( )
( ) ( ) ( )
X t AX t BY t
Y t CX t DY t
* * *. ( ). . ( ). / 2 . ( ). / 2 . .extH p t q p t p M M q t q M g q f p
Ehrenfest theorem+
Hamilton equations
* * *. ( ). . ( ). / 2 . ( ). / 2 . .extH p t q p t p M M q t q M g q f p
0
0 0
0 0
, , ( ') ( ')exp '
, , ( ') ( ')
t
t
A t t B t t t tdt
C t t D t t t t
T
Hamilton’s equations for the external motion
*
( ) ( ) ( )( ) ( )
( ) ( ) ( )1
ext
ext
dH
t t f tdpdt t
dH t t g tdt
M dq
* * *. ( ). . ( ). / 2 . ( ). / 2 . .extH p t q p t p M M q t q M g q f p
*/
q
p M
0 0 0
00 0 0
, , ,( )
, , ,
A t t B t t t tt t
C t t D t t t t
0
0 00
0 0
, , ( ') ( '), exp '
, , ( ') ( ')
t
t
A t t B t t t tt t dt
C t t D t t t t
M T
0
0
0
,, ' ( ') '
,
t
t
t tt t t dt
t t
M
kβ1 kβ2
kα1kα2
β1
α1
β2
α2
Mα1
Mβ1
Mα2
Mβ2
t1 t2
βN
kβN
MβN
βD
αDαN
tN tD
MαN
kαN
GENERAL FORMULA FOR THE PHASE SHIFT OF AN ATOM INTERFEROMETER
1 11
1
, , /N
j j j jj
N
j j j j j j j j jj
D D D D
S t t S t t
k q k q t
p q q p q q
1q
DC
BA1p
The quantity:
is conserved by the ABCD transformations
21111
2222
)'(2/)')('(
2/)')('('
cMMqqpp
qqppSS
THE LAGRANGE INVARIANT IN ATOM OPTICS
SpaceorTime
“Optical System”
1'q1'p
M
pq
M
pq
M
pq
M
pq 2
22
21
11
1 ''
''
'
'
2p
2q
2'q
2'p
SM ,
',' SM
Then the action difference cancels the mid-point phase shift
The four end-points theoremCh. Antoine and Ch.J. Bordé, Exact phase shifts for atom
interferometry, Phys. Lett. A306, 277-284 (2003)
T= t2- t1
β 1
β 2
α 1α 2
Mβ
Mα
t1 t2
2 12 2 1 1
2 12 2 1 1
2
2 2
2 2
S p pq q q q
M M M
S p pq q q q
M M M
c
22 2 2 2 1 1 1 1
1 1
2 2S S p p q q p p q q M M c
kβ1 kβ2
kα1kα2
β1
α1
β2
α2
Mα1
Mβ1
Mα2
Mβ2
βN
kβN
MβN
βD
αDαN
MαN
kαN
GENERAL FORMULA FOR THE PHASE SHIFT OF AN ATOM INTERFEROMETER
1
1(0)
11 1
1 1 1 1
/ 2
1/
2 2
N
j j j j j j j jj
N N
j j j j j j jj j
D DD D
k q k q k k q q
t t
q qp p q p p q q
(0) 2
1; /
j
j k k jkM M c
Atom Interferometers as Gravito-Inertial
Sensors: Analogy between gravitation and electromagnetism
1 0000 hg g
T T
Metric tensor
Newtonian potential
Gravitoelectric field
gUhc
2/002
e.m.22
00 ~/.2/2 VcxgcUh
Atom Interferometers as Gravito-Inertial Sensors: I - Gravitoelectric field case
g
Gravitational phase shift:
k
T T
with light: Einstein red shiftwith neutrons: COW experiment (1975)with atoms: Kasevich and Chu (1991)
2/1
002hMcdt
Phaseshift
Circulation of potential
Mass independent (time)2
Ratio of gravitoelectric flux to quantum of flux
2.k g T
2/.
/ 00
2
hxdtdM
c
ABCD matrices for matter-wave optics
)(cosh)(sinh
)(sinh1
)(cosh
00
00
tttt
tttt
DC
BA
We add a quadratic potential term (gravity gradient):
2 / 2U Mgz M z
2 2/ / 2 / 2M z g Mg
Atomic Gravimeter
ecap
Seta
nidrooc
z
Time coordinate t
T T'z0 v0
v0' z1 v1
z1' v1'
z2' v2'
z2 v2
arm I
arm II
1S
2S3S
4S
2 21 3 2 4 2 2
'v v '
2
z zS S S S S M k M
2 2' sinh ' 2sinh 'k
z z T T TM
2 1 1 0 2 2( ' ) ( ' ) / 2k z z z z k z z
1 0 0( )( / ) ( )vz A T z g B T
1 0 0v ( )( / ) ( )v /C T z g D T k M
/v)()/)(( 001 gTBgzTAz
2 1 1 0 2 2
0
0
( ' ) ( ' ) / 2
sinh ' 2sinh v2
1 cosh ' 2cosh
k z z z z k z z
k kT T T
M
gT T T z
Exact phase shift for the atom gravimeter
which can be written to first-order in with T=T’
2 2 20 0
7v
12 2
kkgT k T gT T z
M
Reference: Ch. J. B., Theoretical tools for atom optics and interferometry, C.R. Acad. Sci. Paris, 2, Série IV, p. 509-530, 2001
31
Atom Interferometric Gravimeter
• Performances:– Resolution: 3x10-9 g after 1 minute
– Absolute accuracy: g/g<3x10-9
• From A. Peters, K.Y. Chung and S. Chu
32
Gradiometer with cold atomic clouds
Yale university
Sensitivity: 3.10-8 s-2/Hz
30 E/Hz
Potential on earth:
1E/Hz
Atoms
Atoms
MirrorR
aman lasers
~1m
Stanford/Yale Gravity Gradiometer: Measurement of G
Pb mass translated vertically along gradient measurement axis.
Typical data:
~1x10-8 g change in acceleration due to gravitational forces for different Pb positions
Present sensitivity/accuracy:
G = 3 x 10-3 G
Measurement consistent with accepted value
Experimental Set-Up
/2
/2
2D-MOT
atom interferometer
Raman 2
Detectionof |a et |b
3D-MOT107 Rb-atoms in 50 ms
Tatoms~2 µK
Raman 1
Mirror
2D MOT
Passive isolation Plateform
Sismometer
Magnetic shields
Schéma de principe du gravimètre
PMO : Piège Magnéto-Optique
Faisceau pousseur
/2
/2
Faisceaux laser Raman
Détection
Impulsions Raman stimuléestemporelles 2T=100 ms
=interféromètre
Jet d’atomes de 87Rbrefroidis dans un PMO-2D
108 atomes piégés en 100 msrefroidis à quelques K
dans un PMO-3D
109 à 1010 atomes.s-1
Sélection de l’état |5S1/2, F=1, m F=0 >
2.. Tgkeff
Faisceaux laser Raman
Enceinte à vide
Chambre à vide du PMO 2D
Chambre à vide
Vanne d’isolation de
la réserve
Réserve de rubidium
PMO 3D
Detection
• 2 paires de faisceaux Raman dans l'enceinte
• Plans équiphases solidaires de la position du miroir
Mesure des déplacements des atomes par rapport au miroir
Montage expérimental
Deux faisceaux superposés et retroréfléchis :
Raman 1 Raman 2
miroir
σ+
σ-
σ-
σ+
PMO 3DTps capture : 50 ms
Tatomes~2 µK
blindage magnétique
DétectionNat détectés
2.105
Interferometer fringes
Parameters
2T=100 ms = 6 µsv ~ vr
Ndet = 106 Tc = 250 ms
Contrast ~ 45 %-180 0 180 360 540 720 900 1080 1260 1440 1620 1800 1980
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Phase (degrees)
Tra
nsit
ion
prob
abil
ity
Sources of noise- laser phase noise (Phase lock : 3.5 mrad /shot) - mirror vibrations
SNR = 25σΦ = 1/SNR = 40 mrad/shot
g/g = 10-7 /shot
• Free fall → Doppler shift of the resonance condition of the Raman transition
= keff.g.T2 - aT2
• Dark fringe : independent of T
0
eff
ag
k=
• Ramping of the frequency difference to stay on resonance :
Principle of g measurement
π/2 π π /2t0 a
-25.1455 -25.1450 -25.1445 -25.1440 -25.1435 -25.14300.2
0.3
0.4
0.5
0.6
0.7
0.8 80 ms 90 ms 100 ms
Tra
nsit
ion
Prob
abil
ity
Raman frequency chirp rate (MHz/s)
C=45%
Long term stability
Bias fluctuations : ± 15.10-9 g
Fluctuations of the systematic effects
0 100000 200000 300000-2.0x10-7
-1.0x10-7
0.0
1.0x10-7
g/g
Time (s)
22-26 december 2006 Earth tides : ± 1.10-7 gModel accurate to a few 10-9 g
Continuous measurements
Earthquake!
2007, January 13 - 04:23 UTCKuril Islands Magnitude 8.1
Period 17 s
07:00 07:05-10
0
10
Acc
ele
ratio
n (m
/s2 )
UTC
04:00 08:00 12:00
-20
-10
0
10
20
Acc
ele
ratio
n (m
/s2 )
UTC
Period 17s
Intégration sans isolation passive
- Fonctionnement hors du régime linéaire
- Nouvel algorithme d’asservissement : trois mesures consécutives de (P,vib
s) permettent de déduire l’erreur de phase
- Robustesse vis-à-vis des modifications du bruit de vibration
1000 1100 1200 1300 1400 1500-60
-40
-20
0
20
40
60
Acc
eler
atio
n (
m/s
2 )
Time (s)
0 500 1000 1500 2000 2500 3000 3500-60
-40
-20
0
20
40
60
Acc
eler
atio
n (
m/s
2 )
Time (s)
Séisme du 20 Mars 2008, Chine, Magnitude 7.7
Atom Interferometers as Gravito-Inertial Sensors: Analogy between gravitation and electromagnetism
Metric tensor
Gravitomagnetic field
Pure inertial rotation
e.m.0 ~ Ahh i
cxh /
chc 22
with light: Sagnac (1913)with neutrons: Werner et al.(1979)with atoms: Riehle et al. (1991)
Atom Interferometers as Gravito-Inertial Sensors: II - Gravitomagnetic field case
dtphc
.
1
Phaseshift
Circulation of potential
Ratio of gravitomagnetic flux to quantum of flux
2
1 2 ..curl
/ /
AdS h
c E c E
Sagnac phase shift:
Laser beams
Atoms
1S
4S
2S
3S
3 4 4 4v v ' / 2M r r
4 'r
4r
1r
2r
3r
COSPAR 2004
Laser beams
Atoms
1 1'
0
exp . ( )t
ti J t dt
k k
R t,t'
R t,t'
T
COSPAR 2004
4 44
1,4
'. .
2Sagnac j jj
r rk r k
Reference: Ch. J. B., Atomic clocks and inertial sensors, Metrologia 39 (5), 435-463 (2002)
SAGNAC PHASE IN THE ABCD FORMALISM
4 44
1,4
'. .
2Sagnac j jj
r rk r k
Ec
ASagnac /
.22
To first order in
First atom-wave gyro: Riehle et al. 1991
49
Atomic Beam Gyroscope
Sensitivity: 6.10-10 rad.s-1/Hz (Yale University)
Magnetic shield
Cs oven
Wave packetmanipulation
Atomic beams
Statepreparation
Lasercooling
Detection
Rotation rate (x10-5) rad/s-10 -5 0 5 10 15 20
Nor
mal
ized
sig
nal
-1
0
1
Interference fringes
50
COLD CESIUM ATOM SENSOR
GYROSCOPEInterferometer’s area : ~ 10 mm²expected sensitivity: 10-8 rad.s-1 /Hzfirst signal expected for spring 2001
ACCELEROMETERexpected sensitivity: 10-8 m.s-2 /Hz
One RAMAN beam
3 temporal pulses
~ 3 0
cm
MOT
Detection
Collaboration between severallaboratories in Paris:
LHA/LPTF, LPL, IOTA, LKB
= 223
0 600 1200 1800 2400
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Saut de phase 3 (°)
Pro
babi
lité
de t
rans
ition
2T= 20 ms
Cv= 0,55
Cf = 0,46
F.Leduc, D. Holleville, J.Fils, A. Clairon, N. Dimarcq, A. Landragin, P. Bouyer and Ch.J. Bordé, ICOLS 2003
Gyro-accéléromètre à césium froid du SYRTE
MOT 1 MOT 2
Z
X
Y
Cold atomsGood control of the mean velocitySmall velocity dispersion
Unique laser beam modulated on timeGood stability and knowledge of the scaling factor
probe
Experimental setup
PARAMETERS
Cs atomsTatoms~1 µK
Launch velocity 2.4 m/sAngle 8°
Tc = 0.58 s
Six axes of inertia
Vertical measurements
y
Z
XY /2
/2
az
2T = 80 ms
Sum of the signals: Acceleration
Difference of the signals: Rotation
1000 2000 3000 4000 5000 6000-1.0x10-5
-8.0x10-6
-6.0x10-6
-4.0x10-6
-2.0x10-6
0.0
2.0x10-6
9.809304
9.809306
9.809308
9.809310
9.809312
9.809314
Acc
eler
atio
n (m
.s-2)
Rot
atio
n (r
ad/s
)
Time (s)
Rotation
Acceleration50 mrad
Rejection of the acceleration
Sensitivity
Rot
atio
n no
ise
(rad
/s) 2.4 10-7 rad/s @ 1 second
1.4 10-8 rad/s
Integration Time [sec]
Acc
eler
atio
n (
m.s
-2)
2.7 10-9 g
5.5 10-8 g @ 1 second
Integration Time [sec]
Acceleration limited by vibrations
Best signal to noise to rotation: 200 With seismometer correction : 3.5 10-8 g in 1 s
Rotation is limited by QPN
Competitive with best commercial FOG
Sensitivity characterized by the Allan standard deviation
Test of the linearity of the scale factor
Changing the orientation of the experiment - modulates the projection of the Earth rotation- changes the rotation rate in a controled way
-4.0x10-5 -2.0x10-5 0.0 2.0x10-5 4.0x10-5-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
Phas
e sh
ift (
rad)
Rotation rate (rad/s)
Excellent linearity No quadratic term at the 10-5 level
-90 -60 -30 0 30 60 90-0.004
0.000
0.004
Res
idua
ls (
rad)
Angle (degree)
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
Rot
atio
n ph
ase
shif
t (ra
d)
Fit with a free offset : 29 mrad
North
South
E
x
y
z
East
West
y
Testing the scale factor vs T2
0 200 400 600 800 1000 1200 1400 1600 1800-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8 + 900
- 900
Rot
atio
n si
gnal
(ra
d)
Interaction time T2 (ms2)
Rotation signal vs interaction timeFor two opposite orientations
0 200 400 600 800 1000 1200 1400 1600 18000.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Rot
atio
n ph
ase
shif
t (ra
d)
Interaction time T2 (ms2)
Difference : rot
10 20 30 40 50 60 70 80 90-0.01
0.00
0.01
0.02
0.03
0.04
Bia
s on
the
rota
tion
sig
nal (
rad)
Interaction time 2T (ms)
Sum : bias
Conclusions
GRAVIMETER
• Short term stability 2 10-8 g/Hz1/2 (under noisy environment)
• Systematic shifts many controlled at the 10-9 g level, Coriolis & aberrations remain a challenge
• First comparison showed g ~16 10-9 g difference
GYROSCOPE
• Short term sensitivity 2.4 10-7 rad/s/√Hz (limited by atomic shot noise)
• Long term stability 0.6 to 2 10-8 rad/s (limited by wavefront aberrations and fluctuation of the sources)
• Linearity of the scale factor
3-D COMBINATION OF GRAVITO-INERTIAL FIELDSExact phase shift for Gravitation+Field gradient+Rotation
0 0 0 0 0
0 0 0 0
10 0
10 0 0 0 0 0
2
' ' '
1 2 '
with:
and /2
kQ k T A T Q B T V
k T T A T T Q B T T V
k T T T g
kQ q g V p M
R
R
R R
Ch. Antoine and Ch.J. Bordé, Exact phase shifts for atom interferometryPhys. Lett. A 306 (2003) 277-284and Quantum theory of atomic clocks and gravito-inertial sensors: an updateJourn. of Optics B: Quantum and Semiclassical Optics, 5 (April 2003) 199-207
HYPERHYPER-precision cold atom interferometry
in space
61HYPER
Atomic Sagnac UnitInterferometer length 60 cm
Atom velocity 20 cm/s
Drift time 3 s
109 atoms/shot
Sensitivity 2x10-12 rad/s
Area 54 cm2
LENSE-THIRRING FIELD
5
2
21
).(3
4
11
2
1
r
rrr
c
GILT
hc
xx
txtxxd
c
Gtxh
2
1
'
),'(v),'('
4),( 3
3
at rotation earth
Gravitomagneticfield lines
Gravitomagnetic field generated by a massive rotating body:
Field lines ~ to magnetic dipole:
63HYPER
HYPER Lense-Thirring measurement
Signal vs time
Hyper carries two atomic Sagnac interferometers, each of them is sensitive to rotations around one particular axis. The two units will measure the vector components of the gravitomagnetic rotation along the two axes perpendicular to the telescope pointing to a guide star.
TOrbit
0 . 5 1 1 . 5 2
- 2
-
1
1
2
3
10 rad/s-14
-
ARBITRARY 3D TIME-DEPENDENT GRAVITO-INERTIAL FIELDS
Hamiltonian: . ( ). . ( ). / 2 . ( ). / 2
Hamilton's equns: exp
H p t q p t p M Mq t q
A Bdt
C D
T
Example: Phase shift induced by a gravitational wave
2
Einstein coord.: cos , 0, with
Fermi coord.: 1, / 2 cos
ijh t h h
h t
1
Einstein coord.: sin sin
1 cos cos sin2 2
Fermi coord.: sin sin cos cos
2
A
hB t t
h h tA t
h htB t t t
2 20
0 0
sin sinc / 2
with: / 2
khV T T T
kV p M
Atomic phase shift induced by a gravitational wave
Ch.J. Bordé, Gen. Rel. Grav. 36 (March 2004)Ch.J. Bordé, J. Sharma, Ph. Tourrenc and Th. Damour,Theoretical approaches to laser spectroscopy in the presence of gravitational fields J. Physique Lettres 44 (1983) L983-990
0
0 0 1 2
/ 2 cos 2 2cos cos
cos 2 cos 2
khq T T
khV T T T
RELATIVISTIC PHASE SHIFTS
http://christian.j.borde.free.frgr-qc/0008033
for Dirac particles interacting with weak gravitational fieldsin matter-wave interferometers
21 Linet-Tourrenc phase
2
cdt p h p
E
. / . spin-gravitomagnetic field2
ch h pc E s
2
2. generalized Thomas precession
1 2
c p h pp s
M E
mean spin vectors
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