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Comparison of atom and photon interferometers using 5D optics Comparison of atom and photon interferometers using 5D optics

Christian J. Bordé

Académie des Sciences

FLORENCE 2009

π/2 Pulses Atoms

a b

space time cτ

Total phase=Action integral+End splitting+Beam splitters

Laser beams

Laser beams

Total phase=Action integral+End splitting+Beam splitters

Atoms

BORDÉ-CHU INTERFEROMETER

π/2 π/2 π

a a b b a a b b a a b b a a b b a a b b a a b b

BORDÉ-RAMSEY INTERFEROMETERS

Multiple wave interferometer

Altitude

g

a

Time

b a b a b b b b a a a b b a a a b b a

2 2 2

1 c t ∂ ≡ − Δ ∂

• 2

2 2 2 2

M c p p M c

μ μ

ϕ ϕ = → + =

• g μν

μ ν

≡ ∇ ∇

( ) ( ) ( ) ( ) ( )

τ ϕ τ τ ϕ ϕ τ τ τ ϕ , , , exp ,

2 2

x Mc c x i Mc x Mc i c x − = ∂ ∂ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − =

• (

) ( ) ( )

Mc x Mc i Mc d c x , exp 2 ) ( ,

2

ϕ τ τ π τ ϕ ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − = ∫

• 2

2 2 2 2 2 2 2 2 2 2 2

1 1 1 M c c t c t c ϕ ϕ ϕ ϕ ϕ ϕ τ ∂ − Δ + = ∂ ∂ ∂ − Δ − = ∂ ∂

• 2

2 2 2 2 2

1 1 ˆ c t c ϕ ϕ ϕ ϕ τ ∂ ∂ ≡ − Δ − = ∂ ∂

• FROM 3 TO

4 SPATIAL DIMENSIONS

E(p) p//

a b

Ma c2 Mb c2

p Mc E

x

S=cτ

t

2 2 2 2

= − = dx dt c ds

2 2 2 2 2

= − − = ds dx dt c dσ

ˆ ˆ ˆ ˆ

ˆ ˆ eikonal equation in 5D ( , 0,1,2,3,4): g μν

μ ν

μ ν φ φ = ∂ ∂ =

( )

(4) 00 00 j j

g E dl t t dx h cg c g φ ⎛ ⎞ = − − − + ⎜ ⎟ ⎜ ⎟ ⎝ ⎠

∫ ∫

00 2 2 ) 4 (

g g g g f d c dx dx f dl

j i ij ij j i ij

− = + − = τ

( )

00 4

g c E h = λ

2 2 2

1 ˆ c ϕ ϕ ϕ τ ∂ ≡ − = ∂

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

1 2 3 4 1 2 3 4

E(p) p

* 2

E M c =

p BASICS OF ATOM /PHOTON OPTICS Parabolic approximation

• f

slowly varying phase and amplitude

2 E

* 2 4 4 * * * 4

1 1 2 2 2 ; ; ( / for photons)

j j j j c

M c i p p p p p h p t M M p i p i p M c c

μν μ ν τ

ϕ ϕ ϕ ϕ ω ∂ ⎡ ⎤ = − + + ⎣ ⎦ ∂ = ∂ = ∂ =

• 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 γ α β δ

⇒ ⇒ ⇒ ⇒ ⇒

= − − = − = −

• *

* *

. ( ). . ( ). /2 . ( ). /2 . .

ext

H p t q p t p M M q t q M g q f p α β γ

⇒ ⇒ ⇒

= + − − +

ABCDξφ 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 = − − ⎡ ⎤ ⎣ ⎦

• *

* *

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

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 ξ φ = + + = + + ( ) ( ) ( ) ( ) ( ) ( ) 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 cτ = − − ⎡ ⎤ ⎣ ⎦ = =

Ehrenfest theorem + Hamilton equations

* * *

. ( ). . ( ). /2 . ( ). /2 . .

ext

H p t q p t p M M q t q M g q f p α β γ

⇒ ⇒ ⇒

= + − − +

• (

) ( ) ( ) ( )

, , ( ') ( ') exp ' , , ( ') ( ')

t t

A t t B t t t t dt C t t D t t t t α β γ α ⎛ ⎞ ⎡ ⎤ ⎛ ⎞ = ⎜ ⎟ ⎢ ⎥ ⎜ ⎟ ⎝ ⎠ ⎣ ⎦ ⎝ ⎠

∫

T

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

(5) (5) 1

( . )

N j j j j

k q δϕ δ δϕ

=

= +

∑

• [

]

( )

(5) (5)

(0) (5) (5) (5) (5) (5) (5)

( , , , ), ; ( , , , ), ; / 2

j

x y z j j j j

k k k k q x y z c ct c c k k k q q q

β α β α

ω ω τ δ ⎡ ⎤ = = ⎢ ⎥ ⎣ ⎦ = − = +

Atomic Gravimeter

e c a p S e t a n i d r

• c

z Time coordinate t T T' z0 v0 v0' z1 v1 z1' v1' z2' v2' z2 v2 arm I arm II

1

τ

2

τ

3

τ

4

τ

( ) ( )

2 2 2 1 3 2 4 2 2

' v v ' 2 z z Mc M k M τ τ τ τ − ⎛ ⎞ + − − + + + = ⎜ ⎟ ⎝ ⎠

• 2

1 1 2 2

( ' ) ( ' ) / 2 k z z z z k z z δϕ = − − − + + −

1

( )( / ) ( )v z A T z g B T γ = − +

1

v ( )( / ) ( )v / C T z g D T k M γ = − + +

γ γ / v ) ( ) / )( (

1

g T B g z T A z + + − =

=

∫

j jdx

p

( )

( ) ( )

( )

( ) ( )

2 1 1 2 2

( ' ) ( ' ) / 2 sinh ' 2sinh v 2 1 cosh ' 2cosh k z z z z k z z k k T T T M g T T T z δϕ γ γ γ γ γ γ γ = − − − + + − ⎧ ⎛ ⎞ ⎡ ⎤ = + − + ⎨ ⎜ ⎟ ⎣ ⎦⎝ ⎠ ⎩ ⎫ ⎛ ⎞ ⎡ ⎤ + + + − − ⎬ ⎜ ⎟ ⎣ ⎦⎝ ⎠⎭

• Exact phase shift for the

atom gravimeter

which can be written to first-order in γ, with T=T’:

2 2 2

7 v 12 2 k kgT k T gT T z M δϕ γ ⎡ ⎤ ⎛ ⎞ = + − + − ⎜ ⎟ ⎢ ⎥ ⎝ ⎠ ⎣ ⎦

• Reference: Ch. J. B., Theoretical

tools for atom

• ptics

and interferometry, C.R. Acad. Sci. Paris, 2, Série IV, p. 509-530, 2001

ARBITRARY 3D TIME-DEPENDENT GRAVITO-INERTIAL FIELDS

* *

Hamiltonian: . ( ). . ( ). / 2 . ( ). / 2 Hamilton's equns: exp H p t q p t p M M q t q A B dt C D α β γ α β γ α

⇒ ⇒ ⇒

= + − ⎛ ⎞ ⎛ ⎞ = ⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠

∫

• T

Example: Phase shift induced by a gravitational wave

( )

{ } ( )

( )

2

Einstein coord.: 1 cos , 0, with Fermi coord.: 1, / 2 cos

ij

h t h h h t β ξ φ γ β γ ξ ξ φ

⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒ ⇒

= + + = = = = +

( ) ( ) ( ) ( )

1 Einstein coord.: sin sin 1 cos cos sin 2 2 Fermi coord.: sin sin cos cos 2 A h B t t h h t A t h ht B t t t ξ φ φ ξ ξ ξ φ φ φ ξ φ φ ξ φ φ ξ = ⎧ ⎪ ⎨ = + + − ⎡ ⎤ ⎣ ⎦ ⎪ ⎩ ⎧ = − + − − ⎡ ⎤ ⎣ ⎦ ⎪ ⎪ ⎨ ⎪ = + + − − + + ⎡ ⎤ ⎡ ⎤ ⎣ ⎦ ⎣ ⎦ ⎪ ⎩

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

( ) ( ) ( ) ( )

1 2

/ 2 cos 2 2cos cos cos 2 cos 2 khq T T khV T T T ξ φ ξ φ φ ξ φ ξ φ ϕ ϕ ϕ − + − + + ⎡ ⎤ ⎣ ⎦ − + − + + − + ⎡ ⎤ ⎣ ⎦

( ) ( )

2 2

sin sinc / 2 khV T T T δϕ ξ ξ φ ξ = − +

/ 2 k V p M ⎛ ⎞ = + ⎜ ⎟ ⎝ ⎠

Bordé-Ramsey interferometers Laser beams Atom beam

( ) ( )

2 * 2 *

cos sinc M c k T h T T M c δϕ ξ φ ξ ⎛ ⎞ = − + ⎜ ⎟ ⎝ ⎠