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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
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
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
25 July 2003 BIPM metrology summer school 2003
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
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
25 July 2003 BIPM metrology summer school 2003
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
25 July 2003 BIPM metrology summer school 2003
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
* * *. ( ). . ( ). / 2 . ( ). / 2 . .extH p t q p t p M M q t q M g q f p
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
* * *. ( ). . ( ). / 2 . ( ). / 2 . .extH p t q p t p M M q t q M g q f p
*
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
* * *. ( ). . ( ). / 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
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
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