Quantum Theory of Atom-Wave Beam Splitters and Application ...christian.j.borde.free.fr/GRG4.pdf · to two-photon transitions and derive the corresponding recoil corrections. The

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General Relativity and Gravitation (GERG) PP1066-gerg-477704 December 12, 2003 16:2 Style file version May 27, 2002

General Relativity and Gravitation, Vol. 36, No. 3, March 2004 ( C© 2004)

Quantum Theory of Atom-Wave Beam Splittersand Application to Multidimensional AtomicGravito-Inertial Sensors

Christian J. Bordé1,2

Received September 18, 2003

We review the theory of atom-wave beam splitters using atomic transitions induced byelectromagnetic interactions. Both the spatial and temporal dependences of the e.m.3

fields are introduced in order to compare the differences in momentum transfer whichoccur for pulses either in the time or in the space domains. The phases imprinted on thematter-wave by the splitters are calculated in the limit of weak e.m. and gravitationalfields and simple rules are derived for practical atom interferometers. The frameworkis applicable to the Lamb-Dicke regime. Finally, a generalization of present 1D beamsplitters to 2D or 3D is considered and leads to a new concept of multidimensional atominterferometers to probe inertial and gravitational fields especially well-suited for spaceexperiments.

KEY WORDS: Gravito-inertial sensor; atom-wave beam splitter.

1. INTRODUCTION

A very convenient beam splitter for atom waves, easily and accurately controlled,is realized through the interaction of atoms with resonant laser beams and moregenerally resonant e.m. waves [1]. This interaction leads to the absorption ofboth the energy and the momentum of an effective photon in a one-photon ormultiphoton process such as a Raman process [2–6, 27]. It was demonstrated

1 Laboratoire de Physique des Lasers, UMR 7538 CNRS, Université Paris-Nord, 99 avenue J.-B.Clément, 93430 Villetaneuse, France; e-mail: [email protected]

2 Equipe de Relativité Gravitation et Astrophysique, LERMA, UMR 8112 CNRS-Observatoire de Paris,Université Pierre et Marie Curie, 4 place Jussieu, 75005 Paris, France.

3 e.m. = electromagnetic

475

0001-7701/04/0300-0475/0 C© 2004 Plenum Publishing Corporation

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476 Bordé

recently [7–10] that the main contribution to the phase shift in atom interferometerscomes from the phase imprinted on the matter-wave by the beam splitters (seeAppendix C). A good understanding of this phase is thus essential to give a properdescription of atom interferometry. Many papers have been devoted already to thetheory of beam splitters covering various aspects [11–17]. The present paper hasessentially a tutorial ambition but tries also to answer some specific questions andto suggest some new directions for the future. For example, it was recognized thate.m. pulses in the time domain (separated in time) and pulses in the space domain(spatially separated) have a different action on an extended atom wave and leadto different expressions for the phase shift. To understand these differences, it isnecessary to give a quantum description of the splitting process without assumingany classical point of intersection where the interaction takes place. To keep easilytractable expressions and focus on the previous point, we have limited ourselvesto a first-order theory leaving the strong-field case for a future publication [18]. Inthis limit, we derive the ttt theorem, which gives simple expressions for the phaseshift introduced by the beam splitter.

In Appendix A, a Schrödinger-type equation valid for both massive and non-massive particles is briefly rederived from the Klein-Gordon equation in curvedspace-time. Appendix B is a short reminder on the ABCD matrices used to writethe propagators of atom waves and, in Appendix C, we recall the general formulafor the phase difference in atom interferometers. In each of these last two appen-dices, we give the example of the action of a gravitational wave as an illustration.The calculation of the first-order scattered amplitude, in a one-photon process, isdetailed in Appendix D. Finally, in Appendix E, we show how to extend this resultto two-photon transitions and derive the corresponding recoil corrections.

The simple model of 1D atom beam splitters provided by this weak-fieldapproach is a first basis to understand the principles of 2 or 3D atom beam splittersin which atom waves are diffracted from an initial atomic cloud in orthogonaldirections of space. With such splitters one could build a coherent superposition ofatomic clouds, images of the initial cloud and forming a macroscopic 3D figure inspace, such as a trihedron, a cube, an octahedron or an extended grating, expandingor at rest. This macroscopic quantum superposition would be an ideal inertialreference system that could be used to probe simultaneously several componentsof the gravitational field through an interference with itself at a later time. Suchpossibilities are clearly offered today by ultra-cold atomic clouds, Bose-Einsteincondensates or atom lasers for future space experiments.

2. SCHRÖDINGER EQUATION AND INTERACTION HAMILTONIAN

We start with the Schrödinger equation in gravitational and inertial fields (seeAppendix A and references [7, 8]):

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Quantum Theory of Atom-Wave Beam Splitters 477

i h̄∂|�(t)〉

∂t=

[H0 + 1

2M�pop ·

⇒β (t) · �pop − ��(t) · (�Lop + �Sop)

−M �g(t) · �rop − M2

�rop ·⇒γ (t) · �rop + V (�rop, t)

]|�(t)〉, (1)

where H0 is an internal Hamiltonian of the atom with eigenvalues Ea, Eb.., whereV (�rop, t) is the electric or magnetic dipole interaction Hamiltonian with the elec-tromagnetic field in the beam splitters and where the other terms contribute to ageneral external motion Hamiltonian4 in the presence of various gravito-inertialfields including a rotation term (with angular velocity ��(t)), a gravity field �g(t)and its gradients

⇒γ (t) and possibly other contributions coming from the metric ten-

sor in⇒β (t) (representing for example the effect of gravitational waves in a given

gauge. . .). We have used the usual Dirac bra and ket notation in which �rop, �pop,�Lop and �Sop are respectively the position, linear momentum, angular momentumand spin operators.

A series of unitary transformations:

|�̃(t)〉 = U−10 (t, t1)|�(t)〉, (2)where t1 is an arbitrary time (which will disappear from the final result) and where(T is a time-ordering operator)

U0(t, t1) = UE (t, t1)e−i H0(t−t1)/h̄T exp[

i

h̄

∫ tt1

��(t ′) · �Sopdt ′]

(3)

UE (t, t1) = UR(t, t1)U1(t, t1) . . . U6(t, t1) (4)(see Appendix 2 of [8]), eliminates one term after the other and brings theSchrödinger equation to the simple form:

i h̄∂|�̃(t)〉

∂t= Ṽ (�rop, �pop, t)|�̃(t)〉, (5)

with

Ṽ (�rop, �pop, t) = V̂ ( �Rop(t, t1), t) (6)(V̂ = ei H0(t−t1)/h̄ V e−i H0(t−t1)/h̄ in the absence of spin-rotation interaction) and

�Rop(t, t1) = U−1E (t, t1)�ropUE (t, t1) (7)= A(t, t1) · �rop + B(t, t1) · �pop/M + �ξ (t, t1) (8)

4 This means relative to the motion of the center of mass. If this motion is relativistic, M should bereplaced by M∗ as discussed in Appendix A.

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478 Bordé

which reduces to �rop + �pop(t − t1)/M + �ξ (t, t1) in the absence of rotation and fieldgradient. In the general case, the ABC D matrices and the vector �ξ are given inAppendix B.

The solution of the Schrödinger equation is

|�(t)〉 = U0(t, t1)T exp[

1

i h̄

∫ tto

dt ′V̂ ( �Rop(t ′, t1), t ′)]

U0(t1, t0)|�(t0)〉. (9)

In the position representation

Kα(�r , �r1, t, t1) = 〈�r , α|U0(t, t1)|�r1, α〉 (10)is the propagator of state α in the absence of laser field and

α(�r , t) = 〈�r , α|�(t)〉= 〈�r , α|U0(t, t1)|�(t1)〉= ei Sα (t,t1)/h̄ei �pα (t) · (�r−�rc(t))/h̄F(�r − �rc(t), X (t), Y (t)), (11)

where the action Sα(t, t1), the momentum �pα , the wave-packet center posi-tion �rc(t) and the widths matrices X (t), Y (t) are given by the ABCDξ the-orem for 3D Hermite-Gauss envelopes F [8]. The time-ordered exponentialT exp

[1i h̄

∫ tto

dt ′V̂ ( �Rop(t ′, t1), t ′)]

is a transition operator between internal states

α, that we shall evaluate now.For one-photon transitions in a two-level system, the matrix element of the

Hamiltonian of interaction with the e.m. waves is5

Vba(�r , t) = −∑±

h̄�±baei(ωt∓kz+ϕ±) F(t − tA)U±0 (�r − �rA) + c.c. (12)

where �ba is a Rabi frequency, where

U±(�r ) = w20

4π

∫d3k exp

[−

(k2x + k2y

)w20

4

]

ei(kx x+ky y+kz z)δ

(kz ± k ∓

k2x + k2y2k

)(13)

= w20

4π

∫dkx dky exp

[−

(k2x + k2y

)w20

4

(1 ∓ i 2

kw20z

)]

ei(kx x+ky y)e∓ikz (14)

5 For simplicity, we have not introduced the dispersion k(ω) within the field envelope F .

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= L±(z) exp [ − L±(z)(x2 + y2)/w20]e∓ikz (15)= U±0 (�r )e∓ikz (16)

reflects the Gaussian beam geometry (see e.g. [19] for the expression of the complexLorentzian L±(z)), and where

F(t − tA) =∫

dω′√2π

F̃(ω′ − ω)ei(ω′−ω)(t−tA) (17)

is a temporal envelope. Thus the Fourier representation of the interaction Hamil-tonian matrix element is

Vba(�r , t) =∑±

∫d3k ′dω′

(2π )2V ±ba( �k ′, ω′)ei �k

′ · �r+iω′t

= −∑±

h̄�±ba

√2πw20

2ei(ωt+ϕ

±)e∓ikz A

∫d3k ′dω′

(2π )2exp

[−

(k ′2x + k ′2y

)w20

4

]ei

�k ′ · (�r− �rA)

δ

(k ′z ± k ′ ∓

k ′2x + k ′2y2k ′

)F̃(ω′ − ω)ei(ω′−ω)(t−tA) + c.c. (18)

with a positive and negative temporal frequency component

V ±+ba ( �k ′, ω′) = −h̄�±ba√

2πw202

e−i( �k′ · �rA+ω′tA)ei(ωtA∓kz A+ϕ

+)

F̃(ω′ − ω)Ũ 0( �k ′⊥)δ(

k ′z ± k ∓k

′2⊥

2k

)(19)

V ±−ba ( �k ′, ω′) = − h̄�±ba√

2πw202

e−i(�k′ · �rA+ω′tA)e−i(ωtA∓kz A+ϕ

+)

F̃(ω′ + ω)Ũ 0( �k ′⊥)δ(

k ′z ∓ k ±k

′2⊥

2k

)(20)

(here F̃ and Ũ 0 are supposed to be real and even, but this assumption is easilyremoved).

With the rotating-wave approximation (RWA)

V̂ ( �Rop(t, t1), t) =(

0 V−ba(t, t1)V+ab(t, t1) 0

)(21)

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480 Bordé

and

|�(t)〉 = U0(t, t1)T exp[

1

i h̄

∫ tt0

dt ′(

0 V−ba(t ′, t1)V+ab(t ′, t1) 0

)]U0(t1, t0)|�(t0)〉,

(22)where we abbreviated

V−ba(t, t1) =∑±

V ±−ba ( �Rop(t, t1), t)eiωba (t−t1) (23)

V+ab(t, t1) =∑±

V ±+ab ( �Rop(t, t1), t)e−iωba (t−t1). (24)

The time-ordered exponential has been calculated in a number of cases in refer-ences [6, 11, 12] but, here, we shall rather outline the weak-field approach, whichis more transparent for a tutorial.

3. FIRST-ORDER PERTURBATION THEORY AND ttt THEOREM

In the weak-field limit, the first-order excited state amplitude is simply relatedto the lower-state unperturbed amplitude by:

〈b|�(1)(t)〉 = 1i h̄

〈b|U0(t, t1)|b〉∫ tto

dt ′V +−ba ( �Rop(t ′, t1), t ′)eiωba (t′−t1)

〈a|U0(t1, t0)|a〉〈a|� (0)(t0)

〉(25)

This amplitude is calculated in the position representation in Appendix D.In the temporal beam splitter case, the excited state amplitude at (�r , t) is found

to be

b(1)(�r , t) = Mbaei Sb(t,tA)/h̄ei �pb(t) · (�r−�rc(t))/h̄e−i(ωtA−kzc(tA)+ϕ+)

ei Sa (tA,t0)/h̄F(�r − �rc(t), X (t), Y (t)) (26)with the momentum change

�pb(tA) = �pa(tA) + h̄kẑ, (27)and where Mab is a constant factor defined in Appendix D, Sα the classical ac-tion and where �rc(t), X (t), Y (t) are, respectively, the central position and widthparameters of the atomic wave packet given by the ABC Dξ law [7, 8, 19].

In the spatial beam splitter case we get

b(1)(�r , t) = Mbaei Sb(t,t ′A)/h̄ei �pb(t) · (�r−�rc(t))/h̄e−i(ωt ′A−kzc(t ′A)+ϕ+)

ei Sa (t′A,t0)/h̄F(�r − �rc(t), X (t), Y (t)), (28)

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where t ′A is such that xc(t′A) = xA. This is the same formula as in the temporal case

but with the momentum change

�pb(t ′A) = �pa(t ′A) + h̄k0x x̂ + h̄kẑ (29)i.e. with an additional momentum δ�k = k0x x̂ = (� − kv0z − δ) x̂/v0x in the lon-gitudinal direction defined by the unit vector x̂ and proportional to the detuning6

This proves the t t t theorem (where t t t stands for t0tAt), which is thebasis for the calculation of exact phase shifts in atom interferometry [9, 10](see Appendix C): When the dispersive properties of a laser beam splitter areneglected (i.e. the wave packet shape is preserved) its effect may be summarized,besides an obvious momentum change, by the introduction of both a phase and anamplitude factor for the atom wave

Mbae−i(ω∗t∗−k̃∗q∗+ϕ∗) (30)

where t∗ and q∗ depend on tA and qA, the central time and central position of theelectromagnetic pulse used as an atom beam splitter7: for a temporal beam splitter

t∗ ≡ tAq∗ ≡ qcl (tA)k∗ ≡ kω∗ ≡ ωϕ∗ ≡ ϕ (laser phase), (31)

and for a spatial beam splitter

q∗ ≡ qAt∗such that qcl(t∗) ≡ qA

k∗ ≡ k + δkω∗ ≡ ωϕ∗ ≡ ϕ + δϕ, (32)

6 As mentioned in the footnotes of Appendix D, it is preferable to transfer the term kvz as a shift in thez coordinate of the wave packet. See reference [8]. In this case δ�k = (� − δ) x̂/v0x .

7 Here and in Appendices B and C, instead of the usual vector notation �q, we use the simplified notation

q , which is the matrix of the components of the vector in a given coordinate system q = xy

z

and

the notation k̃ which stands for the transposed matrix (kx , ky , kz). So that, the scalar product �k · �q iswritten k̃q . The same notation is used for tensors.

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482 Bordé

where qcl is the central position of the incoming atomic wave packet, where δk isthe additional momentum transferred to the excited atoms out of resonance, andwhere δϕ is the phase δϕ ≡ −δk̃qA.

Let us emphasize that, in this calculation, we have never assumed that thesplitter was infinitely thin or that the atom trajectory was classical.

4. CONCLUSIONS AND PERSPECTIVES: MULTIDIMENSIONALATOM INTERFEROMETERS

We have derived simple phase factors for the beam splitters that display ex-plicitly the difference between the spatial and temporal cases. These phase factorshave to be combined with the phase factors coming from the action integral andfrom the end-points splitting as discussed in Appendix C for any given interfer-ometer geometry. This procedure has been applied in previous publications to thecases of gravimeters [7], gyros and atomic clocks [8, 10].

We have kept the calculations as simple as possible by assuming weak e.m.interactions and free-motion in the beam splitters:

A(t ′, t1) = 1, B(t ′, t1) = t ′ − t1, �ξ (t ′, t1) = 0. (33)

It is clear that in realistic calculations these two assumptions have to be abandonedat the expense of more cumbersome expressions. Strong fields lead to the Borrmanneffect and new corrections to the phase shifts induced by other fields have to beintroduced.

In some atomic clocks, the atoms (or ions) are confined to a small regionin space by an external e.m. trapping potential. This leads to a suppression ofthe first-order Doppler shift and of the recoil shift known as the Lamb-Dicke orMössbauer effect. In our approach it is easy to recover such effects by the inclusionof the relevant A and B matrices in Eq. (83). If ωT is the trap frequency

A(t ′, t1) = cos[ωT (t ′ − t1)], B(t ′, t1) = 1ωT

sin[ωT (t′ − t1)] (34)

then the factor

ei�k ′ · A(t ′,t1) · �r1 ei �k

′ · B(t ′,t1) · �p/M eih̄ �k′ AB̃ �k ′/2M (35)

can be expanded in Bessel functions Jn and it is clear that the term associated withJ0 will be free of first-order and recoil shifts.

If, on the contrary, atoms are falling in a constant gravitational field �g, then

�ξ (t ′, t1) = 12

�g(t ′ − t1)2 (36)

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and ∫ +∞−∞

dt ′ei �k′ · �g(t ′−t1)2/2+i[ωba+ω′+ �k ′ · �p/M+h̄ �k ′2/2M](t ′−t1)

=√

2π√−i �k ′ · �g

exp

− 2i�k ′ · �g

(ωba + ω′ +

�k ′ · �pM

+ h̄�k ′2

2M

)2 (37)

replaces the δ(ωba + ω′ + �k ′ · �p/M + h̄ �k ′2/2M) function in Eq. (85) and is easilycombined with Gaussians in k

′x or ω

′ to give the lineshape.The previous calculations also assume that the beam splitters consist only of

one laser beam in a specific privileged direction ẑ. We may extend this conceptto a 2 or 3D atom-wave splitter comprising two or three laser beams in differ-ent directions (orthogonal or not). From the results of this paper, we may inferthat the set of two or three beam splitters will generate clouds propagating in or-thogonal directions, which have a well-defined phase relationship imposed by theorthogonal laser beams (that may come from a single laser source). The diffractedatom wave will then consist of a coherent superposition of excited state ampli-tudes e.g.: bx (�r , t), by(�r , t), bz(�r , t) which differ by their additional momentumh̄k x̂, h̄k ŷ, h̄kẑ. After some time the two or three excited state wave-packets canbe deflected and later recombined thus forming a multi-arms multi-dimensionalinterferometer. For example, if the atom wave packet travels with some initial ve-locity in the x̂ direction two orthogonal laser beams in the ŷ and ẑ directions willgenerate a set of four beams (α, py, pz) = (a, −h̄k/2, −h̄k/2), (b, −h̄k/2, h̄k/2),(b, h̄k/2, −h̄k/2), (a, h̄k/2, h̄k/2). Two more identical beam splitters will gen-erate a diamond-shaped interferometer. If, on the other hand, one starts with anatomic cloud at rest, three orthogonal travelling laser waves will generate a set ofthree diffracted clouds in the excited state

(α, px , py, pz) = (b, h̄k, 0, 0), (b, 0, h̄k, 0), (b, 0, 0, h̄k), thus forming an ex-panding inertial trihedron with the initial wave packet (a, 0, 0, 0). After some timethe three excited wave packets can be stopped by a second interaction while the(a, 0, 0, 0) wave packet is again split into three moving pieces that will later in-terfere with the three previous ones. In this way a 3D version of the usual atomgravimeter can be generated. If the initial cloud is cold enough (sub-recoil) or byaccumulating many recoils [20, 21], the three interfering clouds can be resolvedin space and give three independent fringe patterns. Alternatively, phases, polar-izations, frequencies and time delays of each one of the laser beams can be usedto discriminate between the various interferometers formed by the each pair ofatomic paths. One can also use counterpropagating laser beams to bring back thethree diffracted clouds to the origin and generate a 3-D Bordé-Ramsey opticalclock. By varying the orientations many spurious phases [22] can be cancelled.

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484 Bordé

APPENDIX A

A Relativistic Schrödinger-Type Equation for Atom Waves

Atoms in a given internal energy state can be treated as quanta of a matter-wave field with a rest mass M corresponding to this internal energy and a spincorresponding to the total angular momentum in that state. To take this spin intoaccount one can use, for example, a Dirac [24, 25], Proca or higher-spin waveequation. Here, for simplicity, we shall ignore this spin and start simply with theKlein-Gordon equation for the covariant wave amplitude of a scalar field:

[� + M

2c2

h̄2

]ϕ = 0, (38)

where the d’Alembertian is related to the curved space-time metric gµν by theusual expression

�ϕ = gµν∇µ∇νϕ = (−g)−1/2 ∂µ[(−g)1/2 gµν∂νϕ

]. (39)

We assume that space-time admits a coordinate system (xµ) in which themetric tensor takes the form

gµν = ηµν + hµν, |hµν | � 1. (40)

In what follows, the hµν’s will be considered as first-order quantities and all cal-culations will be valid at this order, e.g.

√−g = 1 + h2

with h = hµµ = ηµνhµν. (41)

Then the Klein-Gordon equation becomes

[∂µ∂µ + M

2c2

h̄2

]ϕ + 1

2(∂µh)∂

µϕ − ∂µhµν∂νϕ = 0. (42)

We shall furthermore assume that the covariant amplitude has the form

ϕ = ϕ0 exp[−i E0t

h̄

], (43)

where ϕ0 varies slowly with time. Then

∂2ϕ

∂t2� −2i E0

h̄

∂ϕ

∂t+ E

20

h̄2ϕ (44)

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and one obtains a Schrödinger-like equation (after renormalization to take intoaccount the change in scalar product)8:

i h̄∂ϕ

∂t=

(E02

+ M2c4

2E0

)ϕ − h̄

2c2

2E0∇2ϕ − h̄

2c2

2E0∂µh

µν∂νϕ (47)

or in the momentum representation

i h̄∂ϕ

∂t=

(E02

+ M2c4

2E0

)ϕ − c

2

2E0p j p jϕ + c

2

2E0pµh

µν pνϕ (48)

This means that the usual hyperbolic dispersion curve is locally approx-imated by the parabola tangent to the hyperbola for the energy E0. This ap-proximation scheme applies to massive as well as to massless particles (e.g.for quasi-monochromatic light M = 0 and E0 = h̄ω [19]). However, in thislimit, only the group velocity of a wave packet is correct, wheras the longitu-dinal wave-packet spreading requires higher-order terms (p4) in the expansion of√

1 + (p2 − p20)c2/E20 . This slowly varying phase and amplitude approximationcan even be used when the weak-field approximation is not valid. To first-order,the Linet-Tourrenc phase shift [26] is immediately recovered. If we introduce themass M∗ defined by:

E0 = M∗c2 (49)the field equation can be written as an ordinary Schrödinger equation in flat space-time

i h̄∂ϕ

∂t= M

∗c2

2

(1 + M

2

M∗2

)ϕ − 1

2M∗p j p jϕ + 1

2M∗pµh

µν pνϕ. (50)

The non-relativistic limit is obtained for M∗ → M . This equation can also bewritten as

i h̄∂ϕ

∂t− 1

2

(E0h

00 + c2

(pi h

i0 + hi0 pi))

ϕ =(

E02

+ M2c4

2E

)ϕ

− 12E0

(cp j − 1

2

(E0h

j0 + cpi hi j)) (

cp j − 12

(E0h j0 + chi j pi

))ϕ (51)

to display the analogs of the scalar and vector e.m. potentials as in [25].

8 The rule∂t → −i E0/h̄ (45)

p0 = E/c → E0/c (46)is used in the terms associated with hµν .

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486 Bordé

Eq.(42) is invariant under the infinitesimal coordinate transformation (gaugeinvariance)

xµ → xµ + ξµ (52)i.e. under the simultaneous changes of ϕ → [1 − ξµ∂µ]ϕ and hµν → hµν − ξµ,ν −ξν,µ. The corresponding finite gauge transformation

T exp[

i

h̄

∫pµ Xµνdx

ν

]ϕ, (53)

where T is an ordering operator and where the quantities Xµν are gauge functions,suggests the general transformation

U = exp[

i

h̄�(t)

]T exp

[i

h̄E0

∫pµ Xµν(x)p

νdt

](54)

in order to remove the gravito-inertial interaction terms in Schrödinger equation.This is, indeed, what is performed in references [8, 19].

APPENDIX B

Background on the ABCD Matrices9

In most cases of interest for atom interferometry, the external motion Hamil-tonian (i.e. relative to the center-of-mass motion) can be expressed as a quadraticpolynomial of momentum and position operators

Hext = 12

�pop · ⇒α (t) · �qop + 12M∗

�pop ·⇒β (t) · �pop − 1

2�qop ·

⇒δ (t) · �pop

− M∗

2�qop ·

⇒γ (t) · �qop + �f (t) · �pop − M∗�g(t) · �qop. (55)

The evolution of wave packets under the influence of this Hamiltonian has beenstudied in detail and is given by the ABC D law. But, we know from Ehrenfesttheorem, that the motion of a wave packet is also obtained in this case from classicalequations. The equations satisfied by the ABC D matrices can be derived eitherfrom the Hamilton-Jacobi equation (see [7]) or from Hamilton’s equations. Forthe previous Hamiltonian, Hamilton’s equations can be written as an equation forthe two-component vector

χ =(

q

p/M∗

)(56)

as

dχ

dt=

( d Hextdp

− 1M∗ d Hextdq

)= �(t)χ + �(t), (57)

9 Based on [7, 8, 19].

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where

�(t) =(

α(t) β(t)

γ (t) δ(t)

)(58)

is a time-dependent 6 × 6 matrix, with (Hermiticity of the Hamiltonian)δ(t) = −α̃(t) (59)

(for a pure rotation we have α(t) = δ(t) = i �J · ��), and where

�(t) =(

f (t)

g(t)

). (60)

The integral of Hamilton’s equation can thus be written as

χ (t) =(

A(t, t0) B(t, t0)

C(t, t0) D(t, t0)

)χ (t0) +

(ξ (t, t0)

φ(t, t0)

), (61)

where

M(t, t0) =(

A (t, t0) B (t, t0)

C (t, t0) D (t, t0)

)= T exp

[∫ tt0

(α(t ′) β(t ′)γ (t ′) δ(t ′)

)dt ′

], (62)

with T as time-ordering operator, and where(ξ (t, t0)

φ(t, t0)

)=

∫ tt0

M(t, t ′)�(t ′)dt ′. (63)

One can easily show that

φ = β−1(ξ̇ − αξ − f ). (64)As an illustration, one can calculate the ABC D matrix in the case of gravi-

tational waves:

� in Einstein coordinates:⇒β (t) = ⇒h cos(ωgwt + ϕ),

⇒γ (t) = 0, (65)

where⇒h= {hi j } and where ωgw is a gravitational wave frequency.

� in Fermi coordinates:⇒β (t) =⇒1, ⇒γ (t) =

(ω2gw/2

) ⇒h cos(ωgwt + ϕ), (66)

where the z dependence of the wave is contained in ϕ.

Then, from the formulas given above, to first-order in h:

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488 Bordé

� in Einstein coordinates:

A = 1, B = t + hωgw

[sin(ωgwt + ϕ) − sin ϕ] (67)� in Fermi coordinates:

A = 1 − h2

[cos(ωgwt + ϕ) − cos ϕ] − hωgwt2

sin ϕ, (68)

B = t + hωgw

[sin(ωgwt + ϕ) − sin ϕ] − ht2

[cos(ωgwt + ϕ) + cos ϕ].

(69)

APPENDIX C

Phase-Shift Formula for Atom Interferometers

The total phase difference between both arms of an interferometer is the sumof three terms: the difference in the action integral along each path, the differencein the phases imprinted on the atom waves by the beam splitters and a contributioncoming from the splitting of the wave packets at the exit of the interferometer [7].If α and β are the two branches of the interferometer

δφ(q) = 1h̄

N∑j=1

[Sβ(t j+1, t j ) − Sα(t j+1, t j )]

+N∑

j=1(k̃β j qβ j − k̃α j qα j ) − (ωβ j − ωα j )t j + (ϕβ j − ϕα j )

+ 1h̄

[ p̃β,D(q − qβ,D) − p̃α,D(q − qα,D)] (70)

where Sα j = Sα(t j+1, t j ) and Sβ j = Sβ(t j+1, t j ).In the case of quadratic Hamiltonians, the four end-points theorem derived

in [9] states that along homologous segments of the two branches (where τ j is aproper time)

Sα jMα j

+ p̃α, j+12Mα j

(qβ, j+1 − qα, j+1) − p̃α j + h̄k̃α j2Mα j

(qβ j − qα j )

= Sβ jMβ j

+ p̃β, j+12Mβ j

(qα, j+1 − qβ, j+1) − p̃β j + h̄k̃β j2Mβ j

(qα j − qβ j )

= −c2τ j (71)

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from which we get

Sβ j − Sα j = 12

( p̃β, j+1 + p̃α, j+1)(qβ, j+1 − qα, j+1)

− 12

( p̃β, j + p̃α, j )(qβ, j − qα, j ) − h̄2

(k̃β, j + k̃α, j )(qβ, j − qα, j )

− (Mβ j − Mα j )c2τ j (72)and

δφ(q) =N∑

j=1(k̃β j qβ j − k̃α j qα j ) − 1

2(k̃β j + k̃α j )(qβ j − qα j )

+N∑

j=1

[ωβα j (t j+1 − t j ) − ω(0)βα jτ j

]+

N∑j=1

(ϕβ j − ϕα j )

+(

p̃β D − p̃αD)

h̄

(q − qβ D + qαD

2

)− p̃α1 + p̃β1

2h̄(qβ1 − qα1) (73)

with ωβα j =∑ j

k=1 ωβk − ωαk and ω(0)βα j = (Mβ j − Mα j )c2/h✥.Usually qβ1 = qα1 and we may use the mid-point theorem [8] which states

that the phase difference for the fringe signal integrated over space is given by thephase difference before integration at the mid-point (qβ,D + qα,D)/2, so that thelast line of the previous equation drops out. In the case of identical masses, wesee that the contributions of the action and of the end points splitting (except forsmall recoil corrections proportional to k2) have cancelled each other and we areleft with the contributions from the beam splitters only.

For a symmetric Bordé interferometer (Mach-Zehnder diamond geometry)kβi + kαi = 0, ∀i ∈ [2, N − 1] , and with the approximation of equal massesMβi = Mαi = M the following simple result is obtained

δφ =N∑

j=1

[(k̃β j qβ j − k̃α j qα j ) + (k̃βN + k̃αN )qαN − qβN

2

− (ωβ j − ωα j )t j + (ϕβ j − ϕα j )]

=N∑

j=1

[(k̃β j − k̃α j )qα j + qβ j

2− (ωβ j − ωα j )t j + (ϕβ j − ϕα j )

](74)

which is manifestly gauge-invariant. The coordinates qα j and qβ j are finally cal-culated with the ABC D matrices. As an example, in the case of three beam

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490 Bordé

splitters only

δφ = [k̃1 − 2k̃2 A(t2, t1) + k̃3 A(t3, t1)]q1+ [k̃3 B(t3, t1) − 2k̃2 B(t2, t1)]

(p1M

+ h̄k12M

)+ ϕ1 − 2ϕ2 + ϕ3 (75)

which, for equal time intervals T , frequencies and wave vectors k, gives

δφ = k̃ [1 − 2A (T ) + A (2T )] q1+ k̃ [B (2T ) − 2B (T )]

(p1M

+ h̄k2M

)+ ϕ1 − 2ϕ2 + ϕ3. (76)

As an illustration, one can calculate this phase shift in the case of gravitationalwaves

δφ = − k̃hq12

[cos(2ωgwT + ϕ) − 2 cos(ωgwT + ϕ) + cos ϕ]

+ k̃hωgw

V1[sin(2ωgwT + ϕ) − 2 sin(ωgwT + ϕ) + sin ϕ]

− k̃hV1T [cos(2ωgwT + ϕ) − cos(ωgwT + ϕ)] + ϕ1 − 2ϕ2 + ϕ3

= k̃γ q1T 2 sin2(ωgwT/2)

(ωgwT/2)2− k̃hV1ωgwT 2 sin(ωgwT + ϕ) sin

2(ωgwT/2)

(ωgwT/2)2

− k̃hV1T [cos(2ωgwT + ϕ) − cos(ωgwT + ϕ)] + ϕ1 − 2ϕ2 + ϕ3, (77)where

V1 = 1M

(p1 + h̄k

2

)and γ = ω

2gw

2h cos(ωgwT + ϕ). (78)

The first term is the phase shift already derived in [28]. It corresponds to the actionof the gravitational wave on the light beam connecting the two atomic clouds in agradiometer set-up. The formula satisfies the equivalence principle. It reduces tothat derived for the atom gravimeter in [7] in the static limit and is very similar tothe formula derived for the Sagnac effect in [8].

APPENDIX D

First-Order Excited State Amplitude for One-Photon Transitions

In the position representation, the first-order excited state amplitude

〈b|�(1)(t)〉 = 1

i h̄〈b|U0(t, t1)|b〉

∫ tto

dt ′V +−ba ( �Rop(t ′, t1), t ′)eiωba (t′−t1)

〈a|U0(t1, t0)|a〉〈a|� (0)(t0)

〉(79)

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gives the following amplitude for the scattered wave packet

b(1)(�r , t) = 〈�r , b|� (1)(t)〉=

∫d3r1〈�r , b|U0(t, t1)|�r1, b〉 1

i h̄

∫ +∞−∞

dt ′eiωba (t′−t1)

∫d3 p〈�r1|V +−ba ( �Rop(t ′, t1), t ′)|�p〉

〈�p, a|U0(t1, t0)|� (0)(t0)〉, (80)where we have let t and t0 go to infinity10 (bounded interaction in space or time).Let us introduce, as an intermediate step

b(1)eff (�r1, t1) =1

i h̄

∫ +∞−∞


∫d3 p

〈�r1∣∣V +−ba ( �Rop(t ′, t1), t ′)∣∣�p〉〈�p, a|U0(t1, t0)|� (0)(t0)〉= 1

i h̄

∫ +∞−∞


∫d3 p

∫d3k ′dω′

(2π )2V +−ba ( �k ′, ω′)〈�r1∣∣ei �k ′ · (A(t ′,t1) · �rop+B(t ′,t1) · �pop/M+�ξ (t ′,t1))+iω′t ′ ∣∣�p〉〈�p, a|U0(t1, t0)|� (0)(t0)〉. (81)

This effective scattered amplitude term will be later propagated in the absence ofV from (�r1, t1) to (�r , t)

b(1)(�r , t) = ∫ d3r1〈�r , b|U0(t, t1)|�r1, b〉b(1)eff (�r1, t1). (82)

We check that this final amplitude is indeed independent of t1 in the case of freepropagation (we shall drop the subscript “eff” in what follows)

b(1)(�r1, t1) = 1i h̄

∫ +∞−∞


∫d3 p

∫d3k ′dω′

(2π )2V +−ba ( �k ′, ω′)

ei�k ′ · (A(t ′,t1) · �r1+�ξ (t ′,t1))+iω′t ′ei �k

′ · B(t ′,t1) · �p/M eih̄ �k′ AB̃ �k ′/2M

〈�r1|�p〉〈�p, a|U0(t1, t0)|� (0)(t0)〉

= 1i h̄

∫d3k ′dω′

(2π )2V +−ba ( �k ′, ω′)e−iω

′t1∫

d3 p∫ +∞−∞

dt ′e[i(ωba+ω′)(t ′−t1)+i �k ′ · B(t ′,t1) · �p/M+i �k ′ · �ξ (t ′,t1)+i h̄ �k ′ AB̃ �k ′/2M]

ei�k ′ · A(t ′,t1) · �r1〈�r1|�p〉

〈�p, a|U0(t1, t0)|� (0)(t0)〉. (83)10 Calculations could also be pursued with a time integral from −∞ to t as in references [7, 8, 13], see

the calculation in the two-photon case in Appendix E.

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492 Bordé

Let us assume free-motion in the beam-splitter

A(t ′, t1) = 1, B(t ′, t1) = t ′ − t1, �ξ (t ′, t1) = 0. (84)

Then

b(1)(�r1, t1) = 1i h̄

∫ +∞−∞


∫d3 p

〈�r1∣∣V +−ba ( �Rop(t ′, t1), t ′)∣∣�p〉〈�p, a|U0(t1, t0)|� (0)(t0)〉= 1

2π i h̄

∫d3k ′dω′V +−ba ( �k ′, ω′)ei �k

′ · �r1+iω′t1∫

d3 p

δ(ωba + ω′ + �k ′ · �p/M + h̄ �k ′2/2M

)〈�r1|�p〉

〈�p, a|U0(t1, t0)|� (0)(t0)〉. (85)With the expression Eq. (20) for V +−ba ( �k ′, ω′)

b(1)(�r1, t1) = i�+bae−i(ωtA+ϕ+) w

20

2

∫d3 p

∫dω′√

2πF̃(ω′ + ω)eiω′(t1−tA)

∫d2k ′⊥Ũ 0( �k ′⊥)ei

�k ′⊥ · (�r1− �rA)eikz1−ik′2⊥ (z1−z A)/2k

δ(ωba + ω′ +

(k − k ′2⊥

/2k

)pz/M + �k ′⊥ · �p/M + δ

)〈�r1|�p〉

〈�p, a|U0(t1, t0)|� (0)(t0)〉, (86)where the recoil term h̄ �k ′2/2M is approximated by δ = h̄k2/2M . Next we performthe ω′ integration

b(1)(�r1, t1) = i�+bae−i(ωtA+ϕ+) w

20

2√

2π

∫d3 p

∫d2k ′⊥

F̃(ω − ωba −

(k − k ′2⊥

/2k

)pz/M − �k ′⊥ · �p/M − δ

)e−i[ωba+(k−k

′2⊥/2k)pz/M+ �k ′⊥ · �p/M+δ](t1−tA)

Ũ+0 ( �k ′⊥)ei�k ′⊥ · (�r1− �rA)eikz1−ik

′2⊥ (z1−z A)/2k

〈�r1|�p〉〈�p, a|U0(t1, t0)|� (0)(t0)〉. (87)

This result may be simplified with the choice t1 = tA. If we neglect also the dis-persive character coming from the momentum dependence in the envelope F̃ , then

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the following simple result is obtained

b(1)(�r1, t1) = i�+bae−i(ωtA+ϕ+)eikz1

w20

2√

2π

∫d2k ′⊥

F̃(ω − ωba −

(k − k ′2⊥

/2k

)v0z − �k ′⊥ · �v0 − δ

)Ũ+0 ( �k ′⊥)e−ik

′2⊥ (z1−z A)/2kei

�k ′⊥ · (�r1− �rA)〈�r1, a|� (0)(t1)〉. (88)However, we shall postpone these two choices and first show how the k ′⊥ integrationcan be performed. To simplify, we keep only the k ′x term (assuming py = 0) andneglect the quadratic correction to k in F̃

b(1)(�r1, t1) = i�+baw0√

2e−i(ωtA+ϕ

+)eikz1 G+∗ (y1 − yA, z1 − z A)∫d3 pe−i[ωba+kpz/M+δ](t1−tA)

∫dk ′x

F̃(ω − ωba − kpz/M − k ′x px/M − δ)G̃0(k

′x )e

−ik ′2x (z1−z A−pz/M(t1−tA))/2keik′x (x1−xA−px /M(t1−tA))

〈�r1|�p〉〈�p, a|U0(t1, t0)|� (0)(t0)〉, (89)

where we have introduced factorized x and y dependences

G+ (x − xA, z − z A) =√

L+(z − z A) exp[−L+(z − z A)(x − xA)2/w20]

= w02√

π

∫dkx G̃0(kx )e

ik2x (z−z A)/2keikx (x−xA)

= w02√

π

∫dkx G̃

+(kx )eikx (x−xA), (90)

whhich is consistent with

U±0 (�r ) = G± (x, z) G± (y, z) (91)Ũ 0( �k ′⊥) = G̃0(kx )G̃0(ky) real. (92)

In order to evaluate (89) we will use the convolution theorem

w0√2

∫dk ′x F̃(ω − ωba − kvz − k ′xvx − δ)

G̃0(k′x )e

−i k′2

x2k (z1−z A−vz (t1−tA))eik

′x (x1−xA−vx (t1−tA)) =∫ +∞

−∞dθei(ω−ωba−kvz−δ)θ F(θ )

G+∗ (x1 − xA − vx (t1 − tA) − vxθ, z1 − z A − vz (t1 − tA)) . (93)

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494 Bordé

To proceed with a concrete example, we assume that the temporal envelope is arectangular pulse (this is a frequent choice in actual experiments; another realisticchoice is a pulse with a Gaussian shape)

F (t − tA) = ϒ(

t − tA + τ2

)− ϒ

(t − tA − τ

2

), (94)

where ϒ is Heaviside step function and

F̃(ω′ + ω) =√

2

π

sin[(ω′ + ω)τ/2]ω′ + ω , (95)

then ∫ +∞−∞

dθei(ω−ωba−kvz−δ)θ F(θ )

G+∗ ((x1 − xA) − vx (t1 − tA) − vxθ, z1 − z A − vz (t1 − tA))

=∫ +τ/2

−τ/2dθei(�−kvz−δ)θ

√L+∗ exp

[−L+∗ ((x1 − xA) − vx (t1 − tA) − vxθ )2 /w20]=

√πw0

2vxeikx (x1−xA−vx (t1−tA)) exp

[− (kxw0)

2

4L+∗

][erf(L+) − erf(L−)]

(96)

with

kx = (� − kvz − δ) /vx (97)and the abbreviation

L± =√

L+∗x1 − xA − vx (t1 − tA) ± 12vxτ

w0+ i (� − kvz − δ) w0

2√

L+∗vx. (98)

Spatial Beam Splitter. For τ −→ +∞ the θ integral yields√

πw0

vxeikx (x1−xA−vx (t1−tA)) exp

[− (kxw0)

2

4L+∗

](99)

and we obtain for the continuous spatial beam splitter

b(1)(�r1, t1) = i�+bae−i(ωtA+ϕ+)eikz1 G+∗ (y1 − yA, z1 − z A)∫

d3 pe−i[ωba+kpz/M+δ](t1−tA)

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√πw0

vxexp [ikx (x1 − xA − vx (t1 − tA))] exp

[− (kxw0)

2

4L+∗

]

〈�r1|�p〉〈�p, a|U0(t1, t0)|� (0)(t0)〉. (100)

We check that tA disappears

b(1)(�r1, t1) = i�+bae−i(ωt1+ϕ+)eikz1 G+∗ (y1 − yA, z1 − z A)∫

d3 p

√πw0

vxeikx (x1−xA) exp

[− (kxw0)

2

4L+∗

]

〈�r1|�p〉〈�p, a|U0(t1, t0)|� (0)(t0)〉 (101)

and if we neglect the dispersion of the splitter:

b(1)(�r1, t1) = i√

π

(�+baw0

v0x

)e−i(ωt1−kz1+ϕ

+)G+∗ (y1 − yA, z1 − z A)

eik0x (x1−xA) exp[− (k0xw0)

2

4L+∗

]a(0)(�r1, t1) (102)

where

a(0)(�r , t) = 〈�r , a|� (0)(t)〉 (103)is the unperturbed (that is, for the absence of the e.m. field) ground-state wavepacket amplitude, and where

k0x = � − kv0z − δv0x

(104)

is the momentum communicated to the atom out of resonance. Here v0x and v0zare the velocity components of the wave packet center11.

Temporal Beam Splitter. If vx and vz −→ 0 (or w0 −→ +∞), then the θintegral gives

sin ((� − kvz − δ)τ/2)(� − kvz − δ)/2 G

+∗ (x1 − xA, z1 − z A) (105)

the momentum induced out of resonance disappears and the following re-sult is obtained for the temporal beam splitter (rectangular pulse in the timedomain)

11 A better approximation is to neglect the dispersion of the first-order Doppler shift only in the envelopeand to write a(0)(x1, y1, z1 − h̄kMv0x (x1 − xA), t1).

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496 Bordé

b(1)(�r1, t1) = i�+bae−iϕ+eikz1U+∗0 ( �r1 − �rA)∫

d3 pe−i(ωba+kvz+δ)t1 e−i(�−kvz−δ)tA

sin ((� − kvz − δ)τ/2)(� − kvz − δ)/2 〈�r1|�p〉

〈�p, a|U0(t1, t0)|� (0)(t0)〉 (106)and if we neglect the dispersion of the splitter12

b(1)(�r1, t1) = i(�+baτ

)e−i(ωt1−kz1+ϕ

+)U+∗0 ( �r1 − �rA)ei(�−kv0z−δ)(t1−tA)sin ((� − kv0z − δ)τ/2)

(� − kv0z − δ)τ/2 a(0)(�r1, t1). (107)

In both cases the incident wave packet given by the ABC Dξ theorem[7, 8, 19]

a(0)(�r1, t1) =〈�r1, a|� (0)(t1)〉

= 〈�r1, a|U0(t1, t0)|� (0)(t0)〉= ei Sa (t1,t0)/h̄ei �pa (t1) · (�r1−�rc(t1))/h̄F(�r1 − �rc(t1), X (t1), Y (t1)) (108)

is multiplied by space-dependent Gaussians that we shall assume either centeredabout the same position as the wave packet or broad enough to be ignored. Whenmultiplied by these, the wave-packet envelope will keep its Gaussian or Hermite-Gauss character. In all cases we shall write the multiplication factor introduced bythe splitter as:

Mbae−i(ωt1−kz1+ϕ+)ei(�−kv0z−δ)(t1−tA) (109)

with

Mba = i(�+baτ

)U+∗0

sin ((� − kv0z − δ)τ/2)(� − kv0z − δ)τ/2 (110)

or

Mbae−i(ωt1−kz1+ϕ+)eik0x (x1−xA) (111)

with

Mba = i√

π

(�+baw0

v0x

)G+∗ exp

[− (k0xw0)

2

4L+∗

](112)

The same phase factors also appear in the strong field theory of beam splitters([1, 11]).

12 A better approximation is to neglect the dispersion of the first-order Doppler shift only in the envelopeand to write a(0)(x1, y1, z1 − h̄kM (t1 − tA), t1).

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In order to apply the ABCDξ theorem, space-dependent phase factors likeeik0x (x1−xA) or eikz1 will be rewritten as:

eik0x (x1−xc(t1))eik0x (xc(t1)−xA) (113)

or

eik(z1−zc(t1))eikzc(t1) (114)

In the temporal beam splitter case, the excited state amplitude at (�r , t) willthus be:

b(1)(�r , t) = ∫ d3r1〈�r , b|U0(t, t1)|�r1, b〉b(1)(�r1, t1)

=∫

d3r1Kα(�r , �r1, t, t1)b(1)(�r1, t1)

= Mbae−i(ωt1+ϕ+)ei Sb(t,t1)/h̄ei �pb(t) · (�r−�rc(t))/h̄

ei(�−kv0z−δ)(t1−tA)eikzc(t1)

ei Sa (t1,t0)/h̄F(�r − �rc(t), X (t), Y (t)) (115)or with the choice t1 = tA

b(1)(�r , t) = Mbaei Sb(t,tA)/h̄ei �pb(t) · (�r−�rc(t))/h̄e−i(ωtA−kzc(tA)+ϕ+)

ei Sa (tA,t0)/h̄F(�r − �rc(t), X (t), Y (t)) (116)with

�pb(tA) = �pa(tA) + h̄kẑ. (117)In the spatial beam splitter case:

b(1)(�r , t) =∫

d3r1〈�r , b|U0(t, t1)|�r1, b〉b(1)(�r1, t1)

=∫

d3r1Kα(�r , �r1, t, t1)b(1)(�r1, t1)= Mbae−i(ωt1+ϕ+)ei Sb(t,t1)/h̄ei �pb(t) · (�r−�rc(t))/h̄eik0x (xc(t1)−xA)

eikzc(t1)ei Sa (t1,t0)/h̄F(�r − �rc(t), X (t), Y (t)) (118)or with the choice of t1 = t ′A such that xc(t ′A) = xA

b(1)(�r , t) = Mbaei Sb(t,t ′A)/h̄ei �pb(t) · (�r−�rc(t))/h̄e−i(ωt ′A−kzc(t ′A)+ϕ+)ei Sa (t

′A,t0)/h̄F(�r − �rc(t), X (t), Y (t)) (119)

which is the same formula as in the previous case but now with

�pb(t ′A) = �pa(t ′A) + h̄k0x x̂ + h̄kẑ (120)

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498 Bordé

i.e. with an additional momentum in the longitudinal direction and proportional tothe detuning.

APPENDIX E

Case of Two-Photon Transitions

In this appendix, we extend the results of the first-order amplitude calcula-tions obtained in the previous appendix for the one-photon case to two-photontransitions. We shall not consider the temporal dependence of the e.m. field, whichleads to formulas similar to the one-photon case and, for simplicity, we assume alsoequal frequencies for both fields. The formulas are easily generalized to Ramantransitions and fields. The example treated here corresponds to Doppler-free two-photon Ramsey fringes with counterpropagating fields in a cascade configuration(with an application to hydrogen in mind). The matrix element of the interactionHamiltonian is given by:

Vba(�r , t) = −h̄�effei(2ωt+ϕ++ϕ−)U+(�r − �r1)U−(�r − �r1) + c.c. + (+ ↔ −)(121)

where �eff is an effective Rabi frequency and

W (�r ) = U+(�r )U−(�r )

= L+(z)L−(z) exp[− L

+(z) + L−(z)w20

(x2 + y2)]

= w20

w2(z)exp

[−2(x

2 + y2)w2(z)

]

= w20

8π

∫dkx dkye

i(kx x+ky y) exp

[−

(k2x + k2y

)w2(z)

8

]

= w30

4 (2π )3/2

∫d3k

k

k⊥ei(kx x+ky y+kz z)

exp

[−k

2⊥w

20

8

]exp

[−k2z

k2w202k2⊥

]. (122)

In the case of copropagating fields U− is replaced by U+ and there is an addi-tional e−2ikz factor. For Raman transitions U− would be replaced by U−∗ with anadditional e−i(k1+k2)z factor.

Note that now k2eff = k2x + k2y + k2z �= k2

Vba(�k, t) = − h̄�effei(2ωt+ϕ++ϕ−) w30

4

k

k⊥exp

[−k

2⊥w

20

8

]exp

[−k2z

k2w202k2⊥

]+ c.c. + (+ ↔ −) (123)

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and we write the atomic energy factor as (γb is the upper state decay rate)

ei[Eb(�p+h̄�keff)−Ea (�p)−i h̄γb/2](t′−t)/h̄ = ei[ωba+�keff · �v+h̄k2eff/2M−iγb/2](t ′−t). (124)

With the rotating-wave approximation

b(1)(�r , t) = 1i h̄

∫ t−∞

dt ′∫

d3 p

(2π h̄)3/2

∫d3k

(2π )3/2Vba(�k, t ′)ei�k · (�r− �r1)

ei[Eb(�p +h̄�k)−Ea (�p)−iγb/2](t′−t)/h̄

ei[�p · (�r−�r0)−Ea (�p)(t−t0)]/h̄〈a, �p|� (0)〉 (125)

= i�effe−i(2ωt+ϕ++ϕ−) w30

4 (2π )3/2

∫d3 p

(2π h̄)3/2(126)

∫d3k

k

k⊥ei

�k · (�r− �r1) exp[−k

2⊥w

20

8

]exp

[−k2z

k2w202k2⊥

](127)

∫ t−∞

dt ′e−i[2ω−ωba−�keff · �v−h̄k2eff2M +iγb/2](t ′−t) (128)

ei[�p · (�r−�r0)−Ea (�p)(t−t0)]/h̄〈a, �p|� (0)〉 (129)

If we neglect the longitudinal recoil term h̄k2z /2M , then the kz integral has a simpleexpression

w0

∫dkz

(2π )1/2k

k⊥exp

[−k2z

k2w202k2⊥

]eikz (z−z1−vz (t−t

′))

= exp[−(z − z1 − vz(t − t ′))2 k

2⊥

2k2w20

](130)

and

b(1)(�r , t) = i�effe−i(2ωt+ϕ++ϕ−) w20

8π

∫d3 p

(2π h̄)3/2

∫dkx dkye

i�k⊥ · (�r− �r1)

∫ t−∞

dt ′e−i[2ω−ωba−�k⊥ · �v−h̄k2⊥/2M+iγb/2](t ′−t)

exp

[−k

2⊥w

20

8

]exp

[−(z − z1 − vz(t − t ′))2 k

2⊥

2k2w20

]

ei[�p · (�r−�r0)−Ea (�p)(t−t0)]/h̄〈a, �p|� (0)〉 (131)

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500 Bordé

or

b(1)(�r , t) = i�effe−i(2ωt+ϕ++ϕ−) w20

8π

∫d3 p

(2π h̄)3/2

∫dkx dkye

i�k⊥ · (�r− �r1)

∫ t−∞

dt ′e−i[2ω−ωba−�k⊥ · �v−h̄k2⊥/2M+iγb/2](t ′−t)

exp

[−k

2⊥8

w2(z − z1 − vz(t − t ′))]

ei[�p · (�r−�r0)−Ea (�p)(t−t0)]/h̄〈a, �p|� (0)〉 (132)

with

w2(z) = w20[

1 + 4 z2

k2w40

]. (133)

We could let t → +∞ for a field bounded in space as in Appendic D to introducea δ function expressing energy conservation but, for the illustration we prefer hereto proceed with the exact calculation for finite times. If the recoil shift is smallenough, we may use a first-order expansion

b(1)(�r , t) = i�effe−i(2ωt+ϕ++ϕ−)∫

d3 p

(2π h̄)3/2ei[�p · (�r−�r0)−Ea (�p)(t−t0)]/h̄

〈a, �p|� (0)〉

∫ t−∞

dt ′ei[2ω−ωba+iγb/2](t−t′)[

W (�r − �r1 − �v(t − t ′))

+ i h̄2M

(t − t ′)∇2⊥W (�r − �r1 − �v(t − t ′))], (134)

where∫ t−∞

dt ′ei[2ω−ωba+iγb/2](t−t′)W (�r − �r1 − �v(t − t ′))

=∫ +∞

0dτ

w20

w2(z − z1 − vzτ )

exp

[−2

[(x − x1 − vxτ )2 + (y − y1 − vyτ )2

]w2(z − z1 − vzτ )

]ei[2ω−ωba+iγb/2]τ . (135)

If the longitudinal transit-time broadening is neglected, this integral is easily cal-culated as in the one-photon case. For vy = 0 and γb = 0 and with � = 2ω − ωba ,one finds

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b(1)(�r , t) = i√

π

2√

2�eff

w

vxe−i(2ωt+ϕ

++ϕ−) exp(

−w2�2

8v2x

)

ei�(x−x1)

vx

[1 + erf

(i

2√

2

w

vx� +

√2

(x − x1)w

)][

1 + i h̄Mvxw

[�

w

vx− 1

8

(�

w

vx

)3]]a(0)(�r , t) (136)

where the last two terms in the final bracket give the recoil correction to thelineshape (only terms leading to a shift have been conserved). These terms scalewith the ratio of the de Broglie wave to the laser beam radius. When the wavepacket has left the interaction zone the error function −→ 1. Here again we findan additional momentum, communicated to the atom wave, proportional to thedetuning, which will lead to the formation of Ramsey fringes, which can be seenin the crossed term of the modulus squared b(1)(�r , t)b(1)∗(�r , t) corresponding tofield zone centers x1 and x2.

ACKNOWLEDGMENTS

Most of the material presented in this publication has been prepared duringtwo stays of the author as a guest of the Institute of Quantum Optics of the Univer-sity of Hannover within the Sonderforschungsbereich 407 and has been deliveredas lectures during August 2002 and August 2003 [23]. The author is very gratefulto Prof. Dr. Wolfgang Ermer for his hospitality in his research group. He wishesalso to acknowledge many stimulating discussions with Dr. Claus Lämmerzahl,Dr. Ernst Rasel and Christian Jentsch and a very fruitful collaboration with CharlesAntoine on numerous aspects of atom interferometry.

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Quantum Theory of Atom-Wave Beam Splitters and Application ...christian.j.borde.free.fr/GRG4.pdf · to two-photon transitions and derive the corresponding recoil corrections. The