varenna 2007 introduction to 5d-optics for space-time sensors introduction to 5d-optics for...

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

QM FOR SPACE / ONERA 2005

2

2 2

1

c t

2 22 2

20

M cp p M c

g

Laser beams

Total phase=Action integral+End splitting+Beam splitters

Atoms

MOMENTUM

E(p)

p

atomslope=v

photonslope=c

rest mass

ENERGY

Mc2

h

h / h dB/

h dB

KQM FOR SPACE

2 22 2

20

M cp p M c

2 4 2 2( )E p M c p c

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 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

25 July 2003 BIPM metrology summer school 2003Minimum 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


Hamilton’s equations for the external motion

*

( ) ( ) ( )( ) ( )

( ) ( ) ( )1

ext

ext

dH

t t f tdpdt t

dH t t g tdt

M dq


*/

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

25 July 2003 BIPM metrology summer school 2003ABCD 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

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

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

ARBITRARY 3D TIME-DEPENDENT GRAVITO-INERTIAL FIELDS

COSPAR 2004

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.: 1 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

CLASSICAL ACTION AND PROPER TIME

2S Ldt Mc d Mc ds

,,

,exp,

2

0

2

xMccx

i

McxMc

icx

McxMc

iMcd

x ,exp2

)(, 0

2

2 2 2

2 2 2

2 2

2 2 2 2

10

1 10

M c

c t

c t c

McMccMcdc )(

2 2

2 2 2 2

1 1ˆ 0

c t c

jop

j

i px

44

44op op op

x c x

p M c p

0 jjdxpInvariant de Lagrange

2 2 22

2 2 2 2

10 0

x

x

ikx

kc t y z

E(p)

p//

ab

0i H

Mac2

Mbc2

2 2

2 2 2 2

1 1

c t c

p

Mc

E

x

s

t

02222 dxdtcds

022222 dsdxdtcd

OPTICAL PATH & FERMAT’S PRINCIPLE IN (4+1)D

Landau and Lifshitz, vol. 2, §88

40 4

00 0 0

0000

jj

j j

S p dx p c dt p dx p dx

gdlp c dt p p dx

gg

2 2 444 4 0g p p M c g p p g p p

00

004

42

g

gggdxdxdxdxdl ji

ijijji

ij

22

00

20

300

0

4

cMgp

hg

p

hdB

1 2 3 4

1

2

3

4

E(p)

p

* 20E M c

0p

BASICS OF ATOM /PHOTON OPTICSParabolic approximation

of slowly varying phase and amplitude

0

2

E

HAMILTONIAN & LAGRANGIAN

iiL p x p x H

2 2 * 2 0

0 * 00 * 00

1

2 2 2

iij

i j i

M c M c gH p c p p cp

M g M g

00

000*

g

gggcpM

jiijij

2 2

20

M c

2

2 2

1

c t

avec =(1,-1,-1,-1) et 1g h h

h

2 , 0,1,2,3ds g dx dx

2 2

2 2 2 2

1 1

c t c

KLEIN-GORDON EQUATIONin presence of weak gravito-inertial fields

* 24

4* *

*4 0

1 1

2 2 2

; ; ( / for photons)

jj

j j c

M ci p p p p p h p

t M M

p i p i p M c c

Schroedinger-like equation for the atom /photon field:

BASICS OF ATOM /PHOTON OPTICS

00 2 2- gravitation field: 2 . / . . /

- rotation field: . /

- gravitational wave:

h g q c q q c

h q c

h


ABCD LAW OF ATOM 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

( , )

exp ( ) ( ) / ( ), ( ), ( )

( , , , ); ( , , , )

c c c

x y z

wavepacket q t

ip t q q t F q q t X t Y t

p p p p Mc q x y z c

Ehrenfest theorem+

Hamilton equations


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

The four end-points theorem

T= t2- t1

β1

β2

α1α2

Mβ

Mα

t1 t2

2 2 2 2

2 2 1 1 22 2 1 1

( )2 2

2 2 2

S S M c M c M M c M M c

q q q qp p p p M M c

2 2 1 1 22 2 1 1

( , , )

02 2 2

x y z

x

p p p p q y

z

q q q qp p p p M M c

0 jjdxpLagrange Invariant

kβ1 kβ2

kα1kα2

β1

α1

β2

α2

Mα1

Mβ1

Mα2

Mβ2

βN

kβN

MβN

βD

αDαN

MαN

kαN

1

(0)

/ 2

2

N

j j j j j j j jj

j j

j j j j j j

k q k q k k q q

t

(0) 2 /j M M c

GENERAL FORMULA FOR THE PHASE SHIFT OF AN ATOM/PHOTON INTERFEROMETER

GENERAL FORMULA FOR THE PHASE SHIFT OF AN ATOM/PHOTON INTERFEROMETER

(5) (5)

1

( . )N

j j jj

k q

(5) (5)

(0)(5) (5)

(5) (5) (5) (5)

( , , , ), ; ( , , , ),

; / 2j

x y z

j j j j

k k k k q x y z c ctc c

k k k q q q

a

a

b

bApplication to fountain clocks

q1

1 1 2 2 2

(0)1 2

/ 2b a

ba b a

k q k q q

T

2 1 1

2 1 1 1

1a

b ab

q q v T

q q M v k TM

a

a

b

b

k

2

xz

2

1

v/)v(

gT

kk ba

Metrologia 39, 435-463 (2002)

ATOMES

b

a

a

b

b

a*

b*

a

b

b*

a

b*

a*

a*

ab

temps

espace

23 Novembre 2004

Collège de France

Optical clocks

Laser beams

Atom

beam

23 Novembre 2004

Collège de France

Christian J. Bordé, M. Weitz and T.W. Hänsch, Laser Spectroscopy XI (1993) p.76

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

http://christian.j.borde.free.frhttp://christian.j.borde.free.fr

23 Novembre 2004

Collège de France

References:

Ch.J. Bordé, Atomic clocks and inertial sensors, Metrologia 39 (2002) 435-463

Ch.J. Bordé, Theoretical tools for atom optics and interferometry, C.R. Acad. Sci. Paris, t.2, Série IV (2001) 509-530

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 (2003) 199-207 Ch.J. Bordé, Quantum theory of atom-wave beam splitters and application to multidimensional atomic gravito-inertial sensors, General Relativity and Gravitation, 36 (2004) 475-502

Atom Interferometry, ed P. Berman, Academic Press (1997)

Ch.J. Bordé, Atomic Interferometry and Laser Spectroscopy,in Laser Spectoscopy X, World Scientific (1991) 239-245

varenna 2007 introduction to 5d-optics for space-time sensors introduction to 5d-optics for...

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