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