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Comparison of atom and photon Comparison of atom and photon interferometers using 5D optics interferometers using 5D optics Christian J. Bordé Académie des Sciences FLORENCE 2009

Laser beams b a Atoms space c τ π /2 Pulses time Total phase=Action integral+End splitting+Beam splitters

Laser beams Atoms π π /2 π /2 BORDÉ-CHU INTERFEROMETER Total phase=Action integral+End splitting+Beam splitters

BORDÉ-RAMSEY INTERFEROMETERS a a a b b b a a a a a a b b b b b b b b b a a a

Multiple wave interferometer Altitude a g b b a a a b b b b a a a a b b b a a b Time

∂ 2 1 g μν ≡ ∇ ∇ � ≡ − Δ � μ ν ∂ 2 2 c t 2 2 M c μ = → ϕ + ϕ = 2 2 0 p p M c μ 2 �

FROM 3 TO 4 SPATIAL DIMENSIONS ⎡ ⎤ 2 ( ) Mc ( ) ( ) ϕ τ = τ − τ ϕ , exp , ⎢ ⎥ x c i x Mc 0 � ⎣ ⎦ ( ) ∂ ϕ τ , x c ( ) 2 = − ϕ τ � , i Mc x ∂ τ ∂ ϕ 2 2 2 1 M c − Δ ϕ + ϕ = 0 ∂ 2 2 2 � c t ∂ ϕ ∂ ϕ 2 2 1 1 ∂ ϕ ∂ ϕ 2 2 1 1 − Δ ϕ − = 0 ϕ ≡ − Δ ϕ − = ˆ � 0 ∂ ∂ τ 2 2 2 2 ∂ ∂ τ 2 2 2 2 c t c c t c ⎡ ⎤ 2 = ∫ ( ) ( ) d Mc Mc ( ) ( ) ϕ τ τ − τ ϕ , exp , ⎢ ⎥ x c i x Mc 0 π � � 2 ⎣ ⎦

b E(p) a c 2 M b c 2 M a p //

Mc p E

d σ = − − = 2 2 2 2 2 t 0 c dt dx ds S =c τ = − = 2 2 2 2 0 ds c dt dx x

OPTICAL PATH & FERMAT’S PRINCIPLE IN (4+1)D ∂ ϕ 2 1 ϕ ≡ ϕ − = ˆ � � 0 ∂ τ 2 2 c μ ν = ˆ ˆ eikonal equation in 5D ( , 0,1,2,3,4): g μν ∂ φ ∂ φ = ˆ ˆ 0 μ ν ˆ ˆ ⎛ ⎞ (4) g E dl ( ) ∫ ∫ 0 φ = − − − + ⎜ ⎟ j j t t dx ⎜ ⎟ 0 h cg c g ⎝ ⎠ 00 00 h ( 4 ) = − + 2 τ 2 ( ) i j dl f dx dx c d 4 λ = ij c g 00 g g E 0 0 = − i j f g ij ij g 00

BASICS OF ATOM /PHOTON OPTICS Parabolic approximation of slowly varying phase and amplitude E(p) 4 = * 2 E M c 3 0 2 E 0 2 1 p 1 2 p 3 4 0

BASICS OF ATOM /PHOTON OPTICS Schroedinger-like equation for the atom /photon field: ∂ ϕ * 2 1 1 M c ⎡ ⎤ = ϕ − + ϕ + μν ϕ 4 � j i p p p p p h p ⎣ ⎦ μ ν 4 ∂ * * j 2 2 2 t M M = ∂ = ∂ = ω * � � � ; ; ( / for photons) p i p i p M c c τ 4 0 j j c � � � � ⇒ = − − γ 00 2 2 - gravitation field: 2 . / . . / h g q c q q c � � ⇒ = − α - rotation field: . / h q c ⇒ ⇒ ⇒ = β − δ - gravitational wave: h � � � � � � � � � � ⇒ ⇒ ⇒ = α + β − γ − + * * * . ( ). . ( ). /2 . ( ). /2 . . H p t q p t p M M q t q M g q f p ext

ABCD ξφ LAW OF ATOM/PHOTON OPTICS = ( , ) wavepacket q t = ( , ) wavepacket q t ( ) ( ) ⎡ − ⎤ − exp ( ) ( ) / � ( ), ( ), ( ) ( ) ip t ( q q t ) F q ( q t X t Y t ) ⎣ ⎦ ⎡ − ⎤ − exp / � exp ( ) ( ) / � ( ), ( ), ( ) c c c iS ip t q q t F q q t X t Y t ⎣ ⎦ cl c c c = = x y z c τ ( , , , ); ( , , , ) p p p p Mc q x y z = + + ξ ( ) ( ) ( ) / ( , ) * q t Aq t Bp t M t t 0 0 0 c c c = + + φ ( ) / ( ) ( ) / ( , ) * * p t M Cq t Dp t M t t 0 0 0 c c c = + ( ) ( ) ( ) X t AX t BY t 0 0 = + ( ) ( ) ( ) Y t CX t DY t 0 0

Ehrenfest theorem + Hamilton equations � � � � � � � � � � ⇒ ⇒ ⇒ = α + β − γ − + * * * . ( ). . ( ). /2 . ( ). /2 . . H p t q p t p M M q t q M g q f p ext ( ) ( ) ⎛ ⎞ ⎡ α β ⎤ , , ⎛ ⎞ ( ') ( ') A t t B t t t t ∫ t 0 0 = exp ' ⎜ ⎟ T ⎢ ⎥ ⎜ ⎟ dt ( ) ( ) γ α , , ( ') ( ') ⎝ ⎠ ⎝ C t t D t t ⎠ ⎣ t t ⎦ t 0 0 0

GENERAL FORMULA FOR THE PHASE SHIFT OF AN ATOM/PHOTON INTERFEROMETER N ∑ � δϕ = δ + δϕ (5) (5) ( . ) k q j j j = 1 j ⎡ ⎤ ω ω (0) [ ] � = = τ (5) (5) � ( , , , ), ; ( , , , ), ⎢ ⎥ k k k k q x y z c ct x y z ⎣ ⎦ c c ( ) � � � (5) (5) δ = − = + (5) (5) (5) (5) ; / 2 k k k q q q β α β α j j j j j

Atomic Gravimeter z 2 ' v 2 ' τ arm II 4 z z 1 ' v 1 ' z 2 v 2 e t − a ⎛ ⎞ ' z z ( ) ( ) n τ + τ − τ − τ + + + = 2 v � v ' 2 2 0 ⎜ ⎟ Mc M k M i d 1 3 2 4 2 2 ⎝ 2 ⎠ r o o τ c ∫ τ = 2 0 j dx e p 3 c a j p S = − γ + + γ = ( )( − / ) γ + ( ) v / ( )( / ) ( )v z A T z g B T g v 0 ' z 1 v 1 z A T z g B T 1 0 0 1 0 0 τ = − γ + + � v ( )( / ) ( )v / C T z g D T k M arm I 1 1 0 0 T T' v 0 z 0 Time coordinate t δϕ = − − − + + − ( ' ) ( ' ) / 2 k z z z z k z z 2 1 1 0 2 2

Exact phase shift for the atom gravimeter δϕ = − − − + + − ( ' ) ( ' ) / 2 k z z z z k z z 2 1 1 0 2 2 ⎧ ( ) ( ) ⎛ � ⎞ k k ( ) ⎡ ⎤ = γ + − γ + sinh ' 2sinh v ⎨ ⎜ ⎟ T T T ⎣ ⎦⎝ 0 γ 2 ⎠ ⎩ M ⎫ ⎛ ⎞ ( ) ( ) g ( ) ⎡ ⎤ + γ + γ + − γ − 1 cosh ' 2cosh ⎬ ⎜ ⎟ T T T z ⎣ ⎦⎝ 0 γ ⎠⎭ in γ, with T=T’ : which can be written to first-order ⎡ ⎤ ⎛ ⎞ 7 � k δϕ = + γ − + − 2 2 2 v ⎜ ⎟ kgT k T ⎢ gT T z ⎥ 0 0 12 ⎝ 2 ⎠ ⎣ ⎦ M Reference: Ch. J. B., Theoretical tools for atom optics and interferometry, C.R. Acad. Sci. Paris, 2, Série IV, p. 509-530, 2001

ARBITRARY 3D TIME-DEPENDENT GRAVITO-INERTIAL FIELDS � � � � � � ⇒ ⇒ ⇒ = α + β − γ * * Hamiltonian: . ( ). . ( ). / 2 . ( ). / 2 H p t q p t p M M q t q α β ⎛ ⎞ ⎛ ⎞ A B ∫ = Hamilton's equns: exp ⎜ ⎟ T ⎜ ⎟ dt γ α ⎝ ⎠ ⎝ ⎠ C D Example: Phase shift induced by a gravitational wave ⇒ ⇒ ⇒ ⇒ ⇒ { } ( ) β = + ξ + φ γ = = Einstein coord.: 1 cos , 0, with ij h t h h ⇒ ⇒ ⇒ ⇒ ( ) ( ) β = γ = ξ ξ + φ 2 Fermi coord.: 1, / 2 cos h t = ⎧ 1 A ⎪ Einstein coord.: ⎨ h ( ) = + ⎡ ξ + φ − φ ⎤ sin sin B t t ⎣ ⎦ ⎪ ξ ⎩ ξ ⎧ h h t ( ) = − ⎡ ξ + φ − φ ⎤ − φ 1 cos cos sin A ⎣ t ⎦ ⎪ ⎪ 2 2 Fermi coord.: ⎨ h ht ( ) ( ) ⎪ = + ⎡ ξ + φ − φ ⎤ − ⎡ ξ + φ + φ ⎤ sin sin cos cos B t t t ⎣ ⎦ ⎣ ⎦ ⎪ ξ 2 ⎩

Atomic phase shift induced by a gravitational wave ( ) ( ) δϕ = − ξ ξ + φ ξ 2 2 sin sinc / 2 khV T T T 0 ( ) ( ) − ⎡ ξ + φ − ξ + φ + φ ⎤ / 2 cos 2 2cos cos khq ⎣ T T ⎦ 0 ( ) ( ) − ⎡ ξ + φ − ξ + φ ⎤ + ϕ − ϕ + ϕ cos 2 cos 2 khV T T T ⎣ ⎦ 0 0 1 2 ⎛ ⎞ � k = + / ⎜ ⎟ V p M 0 0 ⎝ 2 ⎠ 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

Bordé-Ramsey interferometers Laser beams Atom beam 2 * 2 ⎛ ⎞ � M c k ( ) ( ) δϕ = − ξ + φ ξ cos sinc ⎜ ⎟ T h T T * � ⎝ ⎠ M c