Aberrations in (3+1)D Bragg diffraction using pulsed Gaussian laser beams
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] J a n Aberrations in (3+1)D Bragg diffraction using pulsed Gaussian laser beams
A. Neumann, ∗ M. Gebbe, and R. Walser Technical University of Darmstadt, Institute of Applied Physics, Germany ZARM, Universität Bremen, Germany (Dated: January 20, 2021)We analyze the transfer function of a three-dimensional atomic Bragg beamsplitter formed by twocounterpropagating pulsed Gaussian laser beams. Even for ultracold atomic ensembles, the transferefficiency depends significantly on the residual velocity of the particles as well as on losses intohigher diffraction orders. Additional aberrations are caused by the spatial intensity variation andwavefront curvature of the Gaussian beam envelope, studied with (3+1)D numerical simulations.The temporal pulse shape also affects the transfer efficiency significantly. Thus, we consider thepractically important rectangular-, Gaussian-, Blackman- and hyperbolic secant pulses. For thelatter, we can describe the time-dependent response analytically with the Demkov-Kunike method.The experimentally observed stretching of the π -pulse time is explained from a renormalization ofthe simple Pendellösung frequency. Finally, we compare the analytical predictions for the velocity-dependent transfer function with effective (1+1)D numerical simulations for pulsed Gaussian beams,as well as experimental data and find very good agreement, considering a mixture of Bose-Einsteincondensate and thermal cloud. Keywords: Bragg diffraction, atomic beamsplitter, atom optics, atom interferometer, Bose-Einstein con-densation
I. INTRODUCTION
Atoms represent the ultimate “abrasion free” quan-tum sensors for electro-magnetic fields and gravitationalforces. By a feat of nature, they occur with bosonic orfermionic attributes, but are produced otherwise identi-cally without “manufacturing tolerance”. A beamsplitterbased on Bragg diffraction [1–4] prepares superpositionsof matter wavepackets by transferring photon momentumfrom a laser to an atomic wave. Controlling the diffractedpopulations, one can realize a beamsplitter and a mirror.These devices are the central component of a matter-wave interferometer [1–5]. Due to the well-defined prop-erties of the atomic test masses and their precise controlby laser light, matter-wave interferometry can be usedfor high-precision measurements of rotation and accel-eration. Applications range from tests of fundamentalphysics, like the equivalence principle [6–13] or quantumelectrodynamics [14–16], to inertial sensing [17–21]. Likeall imaging systems, atom optics suffer from imperfec-tions and an accurate characterization is required in orderto rectify them. This is relevant for high-precision exper-iments, for instance gravimetry [17, 22, 23] and extendedfree-fall experiments in large fountains, micro-gravity andspace [24–30]. Such challenging experiments require re-alistic modeling and aberration studies, ideally hintingtowards rectification.For ultra-sensitive atom interferometry a large and pre-cise momentum transfer is essential [31–34]. Bragg scat-tering of atoms from a moving standing light wave [35–38], potentially in a retroreflective geometry [39, 40], pro-vides an efficient transfer of photon momentum withoutchanging the atomic internal state. In contrast, Raman ∗ [email protected] scattering [2, 41] couples different atomic internal states,enabling velocity filtering [42, 43]. While Raman pulseshave lower demands on the atomic momentum distribu-tion [40, 44], Bragg pulses can be used for higher-orderdiffraction, also in combination with Bloch oscillations[16, 32, 34, 45–50].The quasi-Bragg regime of atomic diffraction withsmooth temporal pulse shapes is optimal [16, 31–34, 48,51–54]. It provides a high diffraction efficiency withmoderate velocity selectivity for relevant pulse duration.However, losses into higher diffraction orders and thevelocity dispersion must be considered because atomicclouds do have a finite momentum width.The limit of the deep-Bragg regime with long interac-tion times and shallow optical potentials gives a perfecton-resonance diffraction efficiency but remains very nar-row in momentum [2]. However, it is suitable to generatevelocity filters [26, 32]. In the opposite Raman-Nath limitshort laser pulses provide a vanishing velocity dispersionbut the diffraction efficiency is very low [2]. Despite theirrestrictions, both limits are popular as simple analyticalsolutions can be given for rectangular pulse shapes andplane-wave laser beams.For smooth temporal envelopes there exist modelsbased on adiabatic elimination of the off-resonant coupleddiffraction orders, solving the effective two-level dynam-ics [51] and considering the velocity dispersion [39, 55].The Bloch-band picture is suitable in the quasi-Braggregime for sufficiently slow (adiabatic) pulses [56]. Ananalytic theory for smooth pulses based on the adiabatictheorem for single quasi-Bragg pulses is given in [57].Here, Doppler shifts are considered in terms of perturba-tion theory to take finite atomic momentum widths intoaccount.Besides temporal envelopes, spatial envelopes alsoaffect the beamsplitter efficiency [16, 22, 26, 58], espe-cially for large momentum transfer interferometers. Inparticular, spatial variations due to three-dimensionalGauss-Laguerre beams lead to aberrations.In this article, we will revisit atomic beamsplitters in amoving frame in Sec. II. We compare two common meth-ods to solve the Schrödinger equation with plane-wavelaser beams in Sec. III. This is the Bloch-wave ansatzand an ad-hoc ansatz, which leads to a more convenientextended zone scheme. In Sec. IV, we study aberrationsdue to velocity selectivity, higher diffraction orders, spa-tial variations of the beam intensities, wavefront curva-tures and the influence of four non-adiabatic temporalpulse envelopes in terms of the complex transfer func-tion and the fidelity. We introduce an explicitly solvableDemkov-Kunike type model, which applies to hyperbolicsech pulses. With the full (3+1)D simulations the ef-fects of spatially Gauss-Laguerre laser beams are stud-ied. Finally, we gauge simulations and explicit models toexperimental data in Sec. V.
II. MATTER-WAVE BRAGG BEAMSPLITTERA. Conservation laws
The basic mechanism of an atomic beamsplitter isthe stimulated absorption and emission of two pho-tons from bichromatic, counterpropagating laser beams[1, 59]. This process is depicted in Fig. 1a and satisfiesenergy and momentum conservation p i M + ~ ω = p f M + ~ ω , p i + ~ k = p f − ~ k . (1)Here, p i,f are the initial and final momenta of the particlewith mass M , ± ~ k , are photon momenta and ω , arethe laser frequencies. We choose to work with positivewavenumbers k , > and emphasize the propagationdirections with explicit signs, but retain the directional-ity of p i,f . Frequency and wavenumber are coupled bythe vacuum dispersion relation ω = ck , with the speed oflight c . One chooses counterpropagating beams to maxi-mize the momentum transfer p f − p i = 2 ~ k L , introducingthe average wavenumber and frequency k L ≡ k + k , ω L ≡ ck L . (2)Wave mechanics considers superpositions of momen-tum states | g, p i i and | g, p f i in the internal atomic groundstate g . For atoms initially at rest p i = 0 , energy andmomentum conservation (1) requires laser frequencies ω = ω + ω r ≈ ω + ~ (2 k ) M . (3)Due to the two-photon recoil, we need to introduce ω r ≡ ~ (2 k L ) M = 4 ω r , (4)as the two-photon frequency ω r in terms of the singlephoton frequency ω r . The approximation (3) holds fornon-relativistic energies, just as the kinetic energy in (1). B. Off-resonant response
Releasing ultracold atomic ensembles from traps pro-vides localized wavepackets with a finite momentum dis-persion. Therefore, one needs to study the response ofthe Bragg beamsplitter with finite initial- and final mo-menta ¯ p i = κ ~ k L , ¯ p f = (2 + κ ) ~ k L , introducing a dimen-sionless momentum κ . This opens a frequency gap δ ≡ ¯ p f M ~ + ω − ¯ p i M ~ − ω = ω r κ, (5)shown in Fig. 1 (a).Alternatively, one can also probe the momentum re-sponse by a detuning of the laser frequencies ˜ ω , fromthe resonant values ω , in (3). Conveniently, this detun-ing is measured by δω ≡ ω − ω + ˜ ω − ˜ ω . (6)Dash-dotted arrows mark the deviant frequencies inFig. 1 (a). For a particle, which is initially at rest ˜ p i = 0 and acquires a momentum ˜ p f = ~ (˜ k + ˜ k ) after the mo-mentum transfer, one obtains a frequency gap δ = ˜ p f M ~ + ˜ ω − ˜ ω ≈ δω. (7)The approximation holds for | ˜ ω , − ω , | ≪ ω L , whichis satisfied very well in the present context. ComparingEqs. (5) and (7), one finds a linear relation δω = ω r κ, (8)between laser-frequency mismatch δω and the dimension-less initial particle momentum κ . Therefore, both real-izations are suitable to probe the momentum response ofBragg diffraction and their results are related by Eq. (8).Experimentally, it is advantageous to modify the laser-frequencies (cf. Sec. V) and to prepare atomic wavepack-ets initially at rest in the lab-frame S . Theoretically,it is beneficial to emphasize the symmetries of the sys-tem. Therefore, we will adopt a moving inertial frame S ′ , wherein the Doppler-shifted laser-frequencies coin-cide and the momentum coupled states p ′ i = − ~ k L , p ′ f = + ~ k L are distributed symmetrically (cf. Sec. II C,App. A). This is depicted in Fig. 1b. C. Counterpropagating, bichromatic fields
The superposition of two counterpropagating laserbeams E = E + E , is defined by the constituent fields E i = Re[ E (+) i ] with the positive frequency components E (+) i ( t, r ) = ǫ i e − iφ i ( t,x ) E i ( t, r ) . (9)Here, ǫ i denote the polarization vectors, E i ( t, r ) the slowly varying complex Gaussian envelopes and φ ( t, x ) = ω t − k x , φ ( t, x ) = ω t + k x are the rapidlyoscillating carrier phases for fields propagating along the FIG. 1. Bragg diffraction: energy diagram versus atomic wavenumber k = p/ ~ in units of k L (2) in the lab frame S (a)and an inertial frame S ′ (b) moving with velocity v g (12). Ground- and excited state eigen-frequencies of a free particle are ω g ( k ) , ω e ( k ) , the two-photon and one-photon recoil frequencies ω r and ω r , respectively. In frame S , we show that a deliberatedetuning δω (6) of the laser frequencies ω , ω leads to the same fr6equency gap δ (7) (dashed-dotted arrows), as caused bya finite initial momentum p i = κ ~ k L (5) (dotted arrows). In frame S ′ , the counterpropagating lasers have equal frequencies ω ′ , = ω L (2) and link p ′ i = − ~ k L with p ′ f = ~ k L . The velocity selectivity of Bragg scattering leads to an incomplete transferin the momentum ensembles (red, shadowed). Odd momenta ± k L , ± k L , . . . are populated by higher order diffraction. x-direction [60] (cf. App. A and B). From the superposi-tion of two scalar counterpropagating bichromatic fields E = e − iφ ( t,x ) E + e − iφ ( t,x ) E , (10)one obtains a steady motion of the intensity pattern |E| = |E | + |E | + 2 Re h E ∗ E e i ( k + k )( x − v g t ) i , (11)where nodes move with the group velocity v g = ω − ω ω + ω c, | v g | = ω r ω L c ≪ c. (12)If the lab frame S has the coordinates x , then the mov-ing interference pattern defines another inertial frame S ′ ,where the grating is at rest and the coordinates x ′ = x − v g t, (13)are related to the lab frame coordinates x by a passiveGalilean transformation. D. Interaction energy
The atom is represented by a ground | g i and an excitedstate | e i . These levels are separated by the transition fre-quency ω = ω e − ω g and coupled by the electric dipolematrix element d eg = h e | ˆ d | g i . To neglect spontaneousemissions, the lasers are far-detuned from the atomic res-onance frequencies | ω − ω i | ≫ Γ , where Γ is the naturallinewidth of the transition. In the lab frame S the Hamil-ton operator of an atom with mass M reads ˆ H ( t ) = ˆ p M + ~ ω g ˆ σ g + ~ ω e ˆ σ e + V ( t, ˆ r ) , (14) V ( t, r ) = ~ σ † X i =1 Ω i ( t, r ) e − iφ i ( t,x ) + h.c. , using the spin operators ˆ σ i = e,g = | i ih i | and ˆ σ = | g ih e | .Here, we evaluate the electric dipole interaction energyin the rotating-wave approximation and denote the Rabifrequencies as Ω i ( t, r ) = − ε i · d ge E i ( t, r ) / ~ .If we transform this Hamilton operator to the frame S ′ , comoving with the nodes of the interference pattern(13), and use a corotating internal frame (A9), it reads ˆ H ′′ ( t ) = ˆ p M − ~ ∆ˆ σ e + ~ σ † h ˜Ω ( t, ˆ r ) e ik L ˆ x + ˜Ω ( t, ˆ r ) e − ik L ˆ x i + h.c. (15)In this specific frame the atom responds only to a car-rier wavenumber k L . We measure the laser detuning ∆ ≡ ω L − ω with respect to the common Doppler-shiftedfrequency ω L . The Rabi frequencies ˜Ω i ( t, r ) are given bythe pulsed Gauss-Laguerre beams of Eq. (B10).Dissipative processes are not an issue for large detun-ings, why we can resort to the solution of the Schrödingerequation for t > t i and | ψ i ≡ | ψ ′′ i| ψ ( t ) i = G ( t, t i ) | ψ ( t i ) i , (16)with the propagator G ( t, t i ) (D1).For the numerical solution of this two-component,(3+1)D problem, we use Fourier methods with sym-plectic integrators [61] and operator disentangling [62].Analytical solutions are examined for rectangular pulses(Sec. IV C) and the hyperbolic secant pulse (Sec. IV D). E. Ideal Bragg beamsplitter and mirror
The interaction of a two-state system with laser pulsescan be understood qualitatively by the “pulse area” [63] θ ( t ) = Z t −∞ d t ′ Ω( t ′ ) , (17)which is rather a phase by dimension. In the context ofideal Bragg scattering, the two states are the momentumstates {|− k L i x , | k L i x } . One can visualize the evolutionduring the action of the Bragg pulse as a motion on theBloch sphere [64]. A symmetrical 50:50 Bragg beamsplit-ter corresponds to a θ = π/ rotation from the south poleto the equator at some longitude. This gives equal prob-ability to the outputs channels |± k L i . A θ = π rotationfrom the south pole to the north pole reverses the mo-menta |− k L i → | k L i and thus acts like a mirror. In thefollowing discussion, we will focus on the mirror configu-ration as it is most susceptible to aberrations, due to thelonger interaction time.The polar decomposition of the transition amplitude h k ′ | G ( t, t i ) | k i = √ η k ′ k e iφ k ′ k (18)between initial | k i and final | k ′ i momentum states char-acterizes the diffraction efficiency ≤ η k ′ k ≤ . Foratomic wavepackets, we use the phase sensitive fidelity F = |h ψ ideal | ψ ( t f ) i| , | ψ ideal i = e i k L ˆ x | ψ i i , (19)characterizing the overlap of the final state | ψ ( t f ) i ofEq. (16) and the ideal final state | ψ ideal i . For an initialplane wave, the fidelity is F = η k ′ k with k ′ = k + 2 k L . F. Sources of aberrations
The velocity dispersion of Bragg diffraction [55] issignificant and leads to incomplete population trans-fer atomic wavepackets (cf. Fig. 1, Sec. IV C 1). An-other cause for population loss is off-resonant coupling tohigher diffraction orders (cf. Sec. IV C 2). This signalsthe crossover from the deep-Bragg towards the Raman-Nath regime, referred to as quasi-Bragg regime [51].In general, smooth time-dependent laser pulses (cf.Sec. IV A) lead to equally smooth beamsplitter responses(cf. Sec. IV D and IV F). In contrast, smooth spatial en-velopes lead to aberrations (cf. Sec. IV G). Every Gauss-Laguerre beam exhibits spatial inhomogeneity and wave-front curvature. This is relevant for atomic clouds thatare comparable in size to the laser beam waist, or forclouds displaced from the symmetry axis. Static lasermisalignment further degrades the diffraction efficiency.There are sundry other dynamical sources of aberra-tions, such as mechanical vibrations of optical elementsor stochastic laser noise [65]. The fundamental processof spontaneous emission leads to decoherence and aber-rations, too. Fortunately, this can be suppressed by adetuning | ∆ | ≫ Γ much larger than the linewidth Γ , aswell as limiting the interaction time. III. PLANE-WAVE APPROXIMATION
The basic mechanism of Bragg beamsplitters arisesfrom the momentum transfer of plane waves with a real,constant Rabi frequency ˜Ω ( t, r ) = ˜Ω ( t, r ) = Ω withinthe duration of a rectangular pulse. This model is the reference to gauge more realistic calculations. Conse-quently, the two components { ψ e ( t, r ) , ψ g ( t, r ) } of theSchrödinger field evolve according to i∂ t ψ e = (cid:18) − ~ M ∇ − ∆ (cid:19) ψ e + Ω cos( k L x ) ψ g , (20a) i∂ t ψ g = − ~ M ∇ ψ g + Ω ∗ cos( k L x ) ψ e . (20b)using the Hamilton operator (15). Assuming the excitedstate is initially empty, the atom’s kinetic energy is smalland the lasers are far-detuned | ∆ | ≫ Γ , Ω , ω r , we canadiabatically eliminate the excited state [51, 66] ψ e ≈ Ω ∆ cos( k L x ) ψ g . (21)Then, the ground state Schrödinger equation reads i∂ t ψ g = (cid:18) − ~ M ∇ + V ( x ) (cid:19) ψ g , (22)with the dipole potential V ( x ) = cos ( k L x ) | Ω | / ∆ [67].Stationary solutions of the one-dimensional problem areMathieu functions [68]. Our goal is to formulate asuitable ansatz for the (3+1) dimensional non-separableequation with time-dependent pulses. A. Bloch-wave ansatz
The Bloch picture is suitable for describing the ve-locity selective atomic diffraction by a standing laserwave [1, 69, 70]. The characteristic translation invari-ance of the Hamilton operator (22) by a displacement of a x = λ L / defines a natural length scale. Its reciprocalis the lattice vector q x = 2 π/a x = 2 k L . It is conve-nient to embed the total three-dimensional wavefunctionin an orthorohmbic volume with lengths ( N x a x , a y , a z ) ,with N x ∈ N and to impose periodic boundary conditions ψ g ( x + N x a x , y + a y , z + a z ) = ψ g ( x, y, z ) . Bragg scatter-ing involves at least two photons, one from each of thecounterpropagating lasers. Therefore, the two-photon re-coil frequency ω r (4) emerges as the frequency scale.In terms of the dimensionless length ξ = q x x and time τ = ω r t , the Schrödinger field ψ g ( t, r ) = l Ny m − X r = − j Ny k ⌈ Nz ⌉ − X s = − ⌊ Nz ⌋ e i ( r q y y + s q z z − ¯ ω r,s τ ) h ( r,s ) ( τ, ξ ) , (23)factorizes into one-dimensional fields h ( r,s ) ( τ, ξ ) and two-dimensional plane waves with the transversal lattice vec-tors q y,z = 2 π/a y,z . The integers N y,z ∈ N define themaximal momentum resolution q max i = q i ⌊ N i / ⌋ . Pleasenote the use of the Gauss brackets rounding towards thenearest integer at the “floor” ⌊ ⌋ or the “ceiling” ⌈ ⌉ . Witha detuning dependent shift of the frequency, introducingthe two-photon Rabi frequency Ω , ¯ ω r,s = ~ r q y + s q z M ω r + Ω , Ω = Ω r ω r = | Ω | ω r ∆ , (24)the Schrödinger equation for each amplitude simplifies to i∂ τ h ( τ, ξ ) = (cid:0) − ∂ ξ + Ω cos ξ (cid:1) h ( τ, ξ ) . (25)By construction, the potential is π -periodic and theeigenfunctions h ( τ, ξ ) = e − iτω ( b ) ( q ) h ( b ) ( ξ, q ) are given byBloch-waves h ( b ) ( ξ, q ) [71–74] with the lattice periodicfunction g ( b ) ( ξ, q ) for momentum q and band index bh ( b ) ( ξ, q ) = e iqξ g ( b ) ( ξ, q ) , (26) g ( b ) ( ξ + 2 π, q ) = g ( b ) ( ξ, q ) . (27)From the periodic boundary conditions for the wavefunc-tion h ( b ) ( ξ + 2 πN x , q ) = h ( b ) ( ξ, q ) , one obtains a quanti-zation of the wavenumber q n = n/N x with n ∈ Z . Theinterval − / ≤ q n < / defines the first Brillouin zonein the reduced zone scheme, whose extent equals the crys-tal momentum Q = 1 .Bloch wavefunctions are also periodic in momentumspace h ( b ) ( ξ, q + Q ) = h ( b ) ( ξ, q ) , provided we define g ( b ) ( ξ, q ) = N − X m = −N e imξ g ( b ) ( m + q ) , (28)by a Fourier series for a maximal diffraction order N ∈ N with boundary condition g ( b ) ( q + N ) = g ( b ) ( q − N ) = 0 .From a superposition of these Bloch waves, one obtainsthe ansatz h ( τ, ξ ) = ⌈ Nx ⌉ − X n = − ⌊ Nx ⌋ N − X m = −N e i ( m + q n ) ξ g ( τ, m + q n ) , (29)for the time-dependent solution of Eq. (25), compatiblewith the Bloch theorem and suitable for numerical com-putation. This ansatz transforms the partial differentialequation into the parametric difference equation i∂ τ g m ( τ, q ) = ( m + q ) g m + Ω2 ( g m +1 + g m − ) . (30)The q -dependence of the m th -order scattering amplitude g m ( τ, q ) ≡ g ( τ, m + q ) leads to the velocity dispersionof Bragg diffraction. Assuming Dirichlet boundary con-ditions, one can use a (2 N − -dimensional representa-tion g e = ( g − ( N − , . . . , g N − ) , to study the initial valueproblem i ˙ g e = H e ( q ) g e , H e = D e + L + L † . (31)For the indices − N ≤ m ≤ N − , the Hamilton ma-trix H e is formed by a diagonal matrix D e and a lowertriangular matrix LD em,n = ( m + q ) δ m,n , L m,n = Ω2 δ m,n +1 . (32)In order to study the discrete Bloch energy bands ω ( b ) ( q ) , one has to solve the eigenvalue problem g e ( τ, q ) = e − iτω ( q ) g e ( q ) , ω ( q ) g e = H e ( q ) g e . (33) − . − . . . . q − . . . . . ω ( b ) | ψ ( k ) | (2 m − k L m k L (2 m +1) k L FIG. 2. Energy bands ω (0 , , ( q ) of a periodic lattice in theextended zone scheme versus quasi-momentum q , with empty lattice Ω = 0 (dotted) and finite depth
Ω = 1 (solid), where Ω r = Ω ω r = 4 ω r . Initial wavepackets with odd momenta (2 m + 1) k L are located at the edges q = ± / of the st Bril-louin zone, while even momenta mk L are at the center q = 0 . In Fig. 2, we present the lowest few energy bands ω ( b ) ( q ) versus the lattice momentum q in an extended momen-tum zone scheme. For reference, we depict the quadraticdispersion relation of the empty lattice Ω = 0 and the dis-persion relation for
Ω = 1 ( Ω r = Ω ω r = 4 ω r ), a moder-ately deep lattice. Narrow momentum wavepackets ψ ( k ) with σ k ≪ k L are ideal for beamsplitters. If they arelocated at the band edges k = q q x = ( ± / m )2 k L , thetwo-photon process covers at least three Brillouin zones.For wavepackets at the center k = q q x = 2 mk L , only twoBrillouin zones are coupled by a Bragg pulse. B. Ad-hoc ansatz
There are alternatives formulations [51, 55] to theBloch-wave ansatz, if we define a Fourier series on theperiodic lattice h ( x + N x a x ) = h ( x ) as h ( x ) = ∞ X l = −∞ e i πlNxax x g l , πlN x a x = 2 lN x k L . (34)By decomposing the index l = N x m + r into a quotient m = ⌊ l/N x ⌋ and a remainder ≤ r < N x , one obtains h ( x ) = ⌈ Nx ⌉ − X n = − ⌊ Nx ⌋ N − X m = −N g m +1 ( κ n ) e ik n m +1 x , (35)with n = r − ⌊ N x / ⌋ . In this series, we use a momentum k nµ = ( µ + κ n ) k L and a quasimomentum κ n − ≤ κ n = 2 nN x − (cid:6) N x (cid:7) − (cid:4) N x (cid:5) N x < , (36)in an extended Brillouin zone. As the Schrödinger equa-tion (25) has even parity, parity is a conserved quantity.An ansatz with sin and cos functions would lead to adecoupling of (35) with respect to parity manifolds.The decomposition of the index l = N x m + n is notunique, if we admit signed integral remainders within thelimits − ⌊ N x / ⌋ ≤ n < ⌈ N x / ⌉ . This implies a quotient m = ⌊ ( l + ⌊ N x / ⌋ ) /N x ⌋ . Now, the Fourier series reads h ( x ) = ⌈ Nx ⌉ − X n = − ⌊ Nx ⌋ N − X m = −N g m ( κ n ) e ik n m x , (37)with the quasimomentum κ n − ≤ κ n = 2 nN x ≤ − N x . (38)The definitions of the quasimomenta in Eqs. (36) and(38), agree exactly for even number N x = 2 u of lat-tice sites or coincide asymptotically for N x → ∞ . Theeven/odd ambiguity of number of lattice sites can notbe of physical significance as the periodic boundary con-dition are mere mathematical convenience. Therefore,assuming an even number of lattice sites is no limitation.Using time-dependent amplitudes g µ ( τ, κ n ) in the se-ries (35) and (37), transforms the Schrödinger equation(25) into a single difference equation ∀ µ ∈ Z i∂ τ g µ ( τ, κ ) = ( µ + κ ) g µ + Ω2 ( g µ +2 + g µ − ) . (39)Due to the two-photon transfer, there is no coupling be-tween even and odd solution manifolds. Consequently, itis advantageous to use Eq. (35) for wavepackets locatedaround odd multiples of k L or Eq. (37) for even multi-ples of k L (cf. Fig. 2). As in the comoving frame S ′ (13)mainly |− k L i is coupled with | + k L i , we focus on the oddsolution manifold with µ = 2 m + 1 . Therefore, Eq. (39)can be cast into a tridiagonal system of linear differentialequations i ˙ g o = H o g o , H o = D o + L + L † , (40)for g o = ( g − N +1 , g − N +3 , . . . g N − ) with L from (32)and a diagonal matrix D oµ,ν = ( µ + κ ) δ µ,ν ≡ D µ,ν + ̟δ µ,ν . (41)In the following, it will be prudent to adopt a rotatingframe g o ( τ ) = e − i̟τ g ( τ ) with a frequency offset denotedby ̟ = ( − κ ) / i ˙ g = H g , H = D + L + L † , (42) D µ,ν = ω µ δ µ,ν , ω µ = ( µ + κ ) − ̟. (43)This grounds the frequency ω − = 0 . IV. ABERRATION ANALYSIS
Using the ad-hoc ansatz for Bragg scattering, we willsuccessively consider more realistic processes to assesstheir contribution to aberrations. We begin with theplane-wave approximation and consider four temporalBragg-pulse shapes f i ( τ ) . We will analyze their influenceon the velocity dispersion as well as losses into higherdiffraction orders. Finally, we will add the spatial en-velopes of the Gaussian-Laguerre beams and consider thecumulative effect. − ∆ τ/ τ/ τ f τ S τ B τ G τ R RGSB
FIG. 3. Temporal envelopes f ( τ ) for rectangular-, Gaussian-,hyperbolic secant- and Blackman pulses for equal nominaltime T = T j , j ∈ { G, R, S, B } and total pulse length ∆ τ =8 τ G . The vertical lines indicates the pulse widths τ j . A. Bragg-pulse shapes
We examine temporal Gaussian- (G), rectangular- (R),hyperbolic secant- (S) and Blackman- (B) Rabi pulses Ω( τ ) = Ω f j ( τ ) , j ∈ { G, R, S, B } . (44)The shape functions f j , depicted in Fig. 3, are all nor-malized to unity at maximum and characterized by awindow width τ j . Different Rabi pulses (44) can be com-pared physically, if they cover the same pulse area (17) θ ≡ θ ( τ = ∞ ) = Ω T, (45a) T ≡ T ( −∞ , ∞ ) , T ( τ i , τ f ) = Z τ f τ i d τ f j ( τ ) , (45b)for equal nominal time T = T G = T R = T B = T S . Rectangular pulses are popular in theory as they are con-stant during the interaction time and lead to simple an-alytical approximations. They read f R ( | τ | ≤ τ R ) = 1 , T R = 2 τ R (46)and f R ( | τ | > τ R ) = 0 , elsewhere. Gaussian pulses are the standard shapes in pulsed laserexperiments f G ( τ ) = e − τ τ G , T G = √ π τ G , (47)with Gaussian width τ G . Blackman pulses are characterized by a window function f B ( τ ) = w B (cid:18) ττ B (cid:19) , T B = 21 π τ B , (48) w B ( | φ | ≤ π ) = [21 + 25 cos( φ ) + 4 cos(2 φ )] (49)and w B ( | φ | > π ) = 0 elsewhere. Hyperbolic secant pulses are defined with f S ( τ ) = sech (cid:18) ττ S (cid:19) , T S = πτ S . (50)They are amenable for analytical solutions [75, 76]. B. Definition of π - and π -pulses The symmetrical 50:50 beamsplitter pulse and the0:100 mirror pulse are the two most relevant applicationsof atomic Bragg diffraction (cf. Sec. II E). Irrespectiveof the shape, a symmetrical beamsplitter pulse is definedby a pulse area of θ = π/ , while a complete specular re-flection in momentum space is achieved for θ = π . Thisdefines the nominal times T π = π | Ω | , T π/ = T π . (51)In particular, the four pulse shapes yield mirror widths τ Gπ = √ π √ | Ω | , τ Rπ = π | Ω | , τ Bπ = 2521 | Ω | , τ Sπ = 1 | Ω | . (52)Due to the linearity, the symmetric beamsplitter widthis just a half of the mirror time i. e., τ π/ = τ π / . C. Diffraction efficiency of a rectangular pulse
1. Velocity selective Pendellösung
In the deep-Bragg regime N = 1 , off-resonant diffrac-tion orders are negligible. Thus, for first order diffraction N = 1 the state vector in the beamsplitter manifold k ± ≡ ( ± κ ) k L , (53)simplifies to the amplitude tuple g ∓ ( τ ) = ( g − , g +1 ) with g ∓ ( τ i ) = (1 , . The well known Pendellösung [77, 78] g − ( τ ) = e − iϕ (cid:18) cos ϑ − κi Ω κ sin ϑ (cid:19) ,g +1 ( τ ) = e − iϕ Ω i Ω κ sin ϑ, (54)depends on ϕ = κ ( τ − τ i ) / , ϑ = Ω κ ( τ − τ i ) / and thegeneralized two-photon Rabi frequency Ω κ = √ κ + Ω .It follows from (42) for the rectangular pulse shape (46) i ˙ g ∓ ( τ ) = H ∓ g ∓ , H ∓ = (cid:18) Ω2Ω2 κ (cid:19) . (55)With this solution the mirror pulse width (52) can begeneralized for arbitrary κ = 0 . Maximal efficiency η + − ( τ Rπ ) = | g +1 ( τ π ) | is achieved for ϑ = π/ , whichdetermines the mirror pulse width τ Rπ ( κ ) = π κ . (56)On resonance ( κ = 0 ), we recover Eq. (52). Finally, thediffraction efficiency reads η + − ( τ Rπ ) = Ω Ω κ sin ϑ π , ϑ π = π κ Ω . (57)The relative phase of the transfer function (18), betweenthe final k − and k + components is ∆ φ ≡ φ −− − φ + − = arctan (cid:18) κ Ω κ tan ϑ (cid:19) − π . (58)For ϑ = ϑ π , one obtains the phase shift after a mirrorpulse ∆ φ ( τ Rπ ) .
2. Losses into higher diffraction orders
The transfer function h k ′ | G ( t, t i ) | k i (18) exhibits res-onances at k ′ = k + 2 N k L . On the one hand, resonanceswith N = 1 lead to a population loss from the N = 1 beamsplitter manifold { k ± } and reduce the diffractionefficiency. On the other hand, they diminish the cou-pling strength within the beamsplitter manifold. Conse-quently, this increases the optimal π -pulse time ˜ τ π > τ π of a Bragg mirror compared to the prediction of the Pen-dellösung (52). Gochnauer et al. [56] have demonstratedthis effect experimentally for Gaussian pulses, provingthat the effective coupling strength is given by the en-ergy bandgap in the quasimomentum space. a. Renormalized π -pulse time The influence ofhigher order resonances on the beamsplitter manifold canbe calculated perturbatively in terms of the generalizedtwo-photon Rabi frequency Ω κ . For Ω κ → all momen-tum states are doubly degenerate with respect to theirenergies. We employ Kato’s perturbation theory [79],as it can describe the generalized degenerate eigenvalueproblem (C2). Remarkably, Kato’s 1 st order perturba-tion theory coincides with the Pendellösung (cf. App. C)From a third order perturbation calculation O (Ω κ ) , wefind the renormalized Rabi frequency ˜Ω = p κ (1 + 2 I ) + Ω (1 − I ) I≪ −→ Ω κ (59)within the beamsplitter manifold using the abbreviation I = Ω / . For weak dressing I ≪ , it reduces to thegeneralized Rabi frequency of the Pendellösung. From(52), one can evaluate the π -pulse time stretching factor ζ κπ = ˜ τ Rπ τ Rπ = Ω κ ˜Ω , ζ π ≡ ζ κ =0 π = 11 − I ≈ I . (60)Figure 4 depicts a contour plot of the fidelity F ( I , ζ ) (19)for a Bragg-mirror pulse versus the bare two-photon in-tensity I and the inverse pulse stretching factor ζ − = τ jπ /τ j . This representation uncovers a linear relation.The numerically calculated fidelity (19) considers fouroff-resonant diffraction orders ( N = 5 ). As initial condi-tion, we consider 1D Gaussian wavepackets (B1) centeredat k = − k L with momentum width σ k , localized in thecenter of the laser beams x = 0 . Here, in the plane-wave approximation, the results are independent of theexpansion size. This size σ x = (2 σ k ) − follows from theHeisenberg uncertainty.Clearly, the π -pulse stretching factor ζ π (60) traversesthe optimal fidelity regions for all pulse shapes and mo-mentum widths, as a universal rule, motivating the effec-tive π -pulse widths ˜ τ jπ = ζ π τ jπ , j ∈ { G, R, B, S } , (61)with τ jπ from Eq. (52). b. Renormalized π -pulse efficiency In Fig. 5 the ve-locity dispersion of the response of an atomic mirror isvisualized for typical parameters used in experiments (cf.Tab. II) and a two-photon Rabi frequency Ω r = Ω ω r = . . . . . ζ − (a) . . . . . . . GaussianΩ r /ω r (b) . . . . . . . BlackmanΩ r /ω r (c) . . . . . . . SechΩ r /ω r (d) . . . RectangularΩ r /ω r . . . . . . . . . . F .
00 0 .
05 0 . I . . . . . ζ − (e) . . . . .
00 0 .
05 0 . I (f) . . . . .
00 0 .
05 0 . I (g) . . . .
00 0 .
05 0 . I (h) . . . . . . . . . . . . . F FIG. 4. Fidelity F versus two-photon intensity I = Ω / , respectively two-photon Rabi frequency Ω r = Ω ω r , and inverse π -pulse stretching factor ζ − = τ jπ /τ j , j ∈ { G, B, S, R } , for Gaussian (a, e), Blackman (b, f), sech (c, g) and rectangular pulses(d, h). The initial state is a 1D Gaussian wavepacket (B1), initially centered at ( x, k x ) = (0 , − k L ) with momentum width σ k = 0 . k L (top), σ k = 0 . k L (bottom). The optimal stretching factor ζ π (60) (solid line) traverses the regions of maximalfidelity. For the numerical (1+1)D integration (16) with pulse widths ζτ jπ , and total pulse length ∆ τ j = 8 ζτ G , typical laserand atom parameters, used in experiments (Tab. II), are applied. ω r . The Pendellösung (54), valid in the deep-Braggregime ( N = 1 ), applying the pulse width τ Rπ (Ω) (52),is compared to the eigenvalue solution (42) with pulsewidth ˜ τ Rπ (Ω) (61). Therefore, the diffraction efficiency η k ′ k reveals the velocity selectivity of the Bragg condi-tion and the population loss into higher diffraction or-ders, here in the quasi-Bragg regime ( N = 5 ). The phasedifference ∆ φ (58) shows a π jump at resonance. The per-turbative Kato solution (C14) describes the beamsplitterresponse very well, only at the band edges κ → ± , thereare small deviations. For weak coupling Ω , the diffrac-tion efficiency after a mirror pulse of width ˜ τ πR (Ω) (61),exhibits a sinc behavior [cf. Fig. 6 (a)]. It is the typi-cal Fourier-response to a rectangular pulse. Increasingthe Rabi frequency Ω , the response is power broadened,in conjunction with a reduced efficiency. Simultaneously,the Kato solution becomes less accurate for | κ | > , whilethe resonant efficiency η ≡ η + − ( κ = 0) = η k L , − k L can beapproximated further. This is also depicted in Fig. 6 (b),together with the efficiency’s full width half maximum ∆ η of the Bragg mirror. For an ideal mirror, η = 1 and ∆ η → ∞ are desirable, but impossible.In addition, we study the optimal interaction time inFig. 6 (b). The approximation τ Rπ (52) for the deep-Bragg regime and ˜ τ Rπ (61) for the quasi-Bragg regime,considering higher diffraction orders, are compared tothe optimal interaction time, defined by the maximumnumerical transfer efficiency at resonance κ = 0 . Withincreasing Ω , in a regime where the losses into higherdiffraction orders are important, the approximation with τ Rπ is less accurate, while ˜ τ Rπ can be used further. Pleasenote that for the maximized transfer efficiency the veloc-ity acceptance ∆ η is reduced, while for ˜ τ Rπ it remainslarger, for increasing Ω . From the Kato solution (C14)a simple analytic equation for the diffraction efficiencyon resonance, for the effective π -pulse time ˜ τ Rπ can bederived (cf. App. C) to η K (˜ τ Rπ ) = (1 − I ) (cid:20) | Ω |I sin (cid:18) π | Ω | I − I (cid:19)(cid:21) , (62)also depicted in Fig. 6 (b). This expression predicts lossesinto higher diffraction orders within the convergence ra-dius Ω r = Ω ω r < ω r ( I < . ), very well. The ap-proximation remains positive for Ω r < √ ω r ( I = 0 . . D. Diffraction efficiency of a sech pulse
1. Velocity selective Demkov-Kunike Pendellösung
For hyperbolic secant pulses Ω( τ ) = Ω f S ( τ ) (50), onecan solve Eq. (55) also in a closed form [75, 76]. A decou-pling of the first order differential equation system with g +1 = 2 i Ω( τ ) − ˙ g − , leads to Hill’s second order differen-tial equation [68] g − − (cid:18) ˙Ω( τ )Ω( τ ) − iκ (cid:19) ˙ g − + Ω( τ ) g − . (63)With the nonlinear map z ( τ ) = [1 + tanh ( τ /τ S )] / , thedifferential equation for γ ( z ) ≡ g − ( τ ) emerges as z (1 − z ) γ ′′ + [ c − z (1 + a + b )] γ ′ − abγ = 0 , (64) . . . η + − (a) 1 . . . ∆ φ / π − . − . . . . κ . . η k ′ k (b) N=2N=-1Kato
FIG. 5. (a) Diffraction efficiency η + − ( (cid:3)(cid:3)(cid:3) ) in the beamsplittermanifold N = 1 , together with the relative phase shift ∆ φ (58)( ◦ ) and (b) losses into higher diffraction orders N = 1 versusdetuning κ , after a rectangular mirror pulse. For the numer-ical solution (solid), considering four off-resonant diffractionorders (with k = ( − κ ) k L and k ′ = k + 2 Nk L ) the ap-plied pulse width is ˜ τ Rπ (Ω) (61) and for the Pendellösung(57), (58) (dotted), considering only the resonant diffractionorder, τ Rπ (Ω) (52) for Ω r = Ω ω r = 3 ω r . In (a), the Pendel-lösung overestimates the efficiency and phase shift, while theKato corrections (C14) (dashed) match the numerical results(solid) much better. There are only deviations at the bandedges, especially for N = 2 (b). with a = Ω τ S / , b = − a and c = (1+ iκτ S ) / . This is thehypergeometric differential equation with solutions f = F ( a, b ; c ; z ) , f = z − c F (1+ a − c, b − c ; 2 − c ; z ) andWronski determinant w = (1 − z ) c − z − c . Straightforwardanalysis (cf. App. D) leads to the Demkov-Kunike (DK)solution with unitary propagator G ∓ ( τ, τ i ) g ∓ ( τ ) = G ∓ ( τ, τ i ) g ∓ ( τ i ) , G ∓ ( τ i , τ i ) = . (65)For the initial datum g ∓ ( τ i ) = (1 , , one obtains g − ( τ ) = [ f ( τ ) f ′ ( τ i ) − f ( τ ) f ′ ( τ i )] /w ( τ i ) . (66)For a pulse beginning in the remote past τ i ≪ − τ S , thissimplifies to g − ( τ ) = F ( a, − a ; c, z ) , (67) g +1 ( τ ) = aic p z (1 − z ) F (1 − a, a ; 1 + c, z ) . (68)Now, the diffraction efficiency of a beamsplitter reads η DK + − ( κ, τ ) = | g +1 ( τ ) | = 1 − | g − ( τ ) | . (69)Furthermore, for very long pulse durations τ S ≪ τ f , | τ i | ,the diffraction efficiency simplifies to η DK + − ( κ, Ω , T ) = sech (cid:18) κT (cid:19) sin (cid:18) Ω T (cid:19) , (70)with the nominal time T (45). In order to achieve fulldiffraction efficiency η DK = η DK + − ( κ = 0) = 1 , one should − . − . . . . κ . . . η + − (a) Ω r = 1 ω r Ω r = 3 ω r Ω r = 5 ω r Ω r /ω r . . . η ∆ η (b) FIG. 6. (a) Diffraction efficiency η + − after a mirror pulseof width ˜ τ Rπ (Ω) (61) versus detuning κ for different two-photon Rabi frequencies Ω r = Ω ω r , numerical results (solid)and Kato (C14) solution (dashed). (b) Resonant transfer ef-ficiency η ( × ) and efficiency width ∆ η ( ⊲ ) versus Ω r . Thenumerically optimal interaction time, for maximal efficiency(dash-dotted) is compared to the approximations for the π -pulse width τ Rπ (52) (dotted) and ˜ τ Rπ (61) (solid). The an-alytical Kato approximation η K (˜ τ Rπ ) (62) (dashed) providesmeaningful predictions. choose the π -pulse width as τ Sπ = | Ω | − , in agreementwith the pulse area (52). Waiting indefinitely long ishardly ever an option [80]. Therefore, the finite timeapproximation η DK ( τ ) ≈ z = (1 + tanh Ω τ ) (71)reveals the exponential convergence past several π -pulsetimes τ ≫ τ S . It requires Ω r = Ω ω r < ω r .
2. Losses into higher diffraction orders
To consider losses into the higher diffraction orders, weuse time-dependent perturbation theory in Eq. (42) i ˙ g = H ( τ ) g , H ( τ ) = H ( τ ) + H ( τ ) . (72)The free evolution H ( τ ) consist of a direct sum H ( τ ) = H ∓ ( τ ) N M µ = −N +1 µ =0 , ω µ − (73)of the DK-generator H ∓ ( τ ) (55) in the beamsplitter man-ifold and the unperturbed energies ω µ (43) in the highermomentum states. The perturbation H ( τ ) is simply thecomplement of the complete Hamilton operator.The free retarded propagator is defined for τ ≥ τ i as G ( τ, τ i ) = G ∓ ( τ, τ i ) N M µ = −N +1 µ =0 , e − iω µ − ( τ − τ i ) (74)and vanishes elsewhere (cf. App. D). It involves the DK-Pendellösung G ∓ (65) and the free time evolution of off-resonant momentum states. The complete solution0 . . . η + − (a) Ω r =1 ω r Ω r =3 ω r Ω r =7 ω r − . − . . . . κ . . . ∆ φ / π (b) FIG. 7. Velocity dispersion of (a) the diffraction efficiency η + − and (b) the phase shift ∆ φ for sech-pulses with pulsewidth τ S = ˜ τ Sπ (61) and different Rabi frequencies Ω r =Ω ω r . The DK-Pendellösung (67) (dotted) is suitable for Ω r < ω r while the extended model (76) (dashed) matchesthe numerical results (16) (solid) very well also for larger Ω r . g ( τ ) = G ( τ, τ i ) g ( τ i ) (75)follows from the solution G ( τ, τ i ) of the integral equa-tion (D3). A second order approximation couples to the ± k L , ± k L momentum states and shifts the frequenciesof the beamsplitter manifold G ( τ, τ i ) = G − i Z ∞−∞ d t G ( τ, t ) H ( t ) G ( t, τ i ) (76) − Z ∞−∞ d t d t ′ G ( τ, t ) H ( t ) G ( t, t ′ ) H ( t ′ ) G ( t ′ , τ i ) . This is required to observe the stretching of the π -pulsetime. An explicit analytical approximation can be ob-tained. It is numerically efficient and useful for the in-terpretation, but remained unwieldy for display [81].In Fig. 7, we compare the simple and the extended DK-model after a π pulse, with the corresponding numerical(1+1)D simulations (16). The diffraction efficiency is de-picted in Fig. 7 (a) and the phase shift ∆ φ between thecoupled states in Fig. 7 (b). The simple DK-Pendellösung(67) is valid for Ω r = Ω ω r < ω r . For Ω r > ω r , lossesinto higher diffraction orders are significant, but the ex-tended solution (76) still matches the numerical solution. a. Adiabaticity The crossover from the deep- to thequasi-Bragg regime at Ω ≈ ω r for atomic mirrors using ˜ τ jπ (61) is related to the adiabaticity criterium [82] max τ ∈ [ τ i ,τ i +∆ τ ] (cid:12)(cid:12)(cid:12)(cid:12) ddτ (cid:18) g on ( τ ) ∗ ˙ g om ( τ ) ω n ( τ ) − ω m ( τ ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ∆ τ ≪ , (77) ∀ m = n , with the eigenvalues ω m ( τ ) and eigenvectors g om ( τ ) of H o (40). Equation (77) results in Ω r = Ω ω r ≪ ω r for ˜ τ Sπ at κ = 0 . This is confirmed by the resultsof Gochnauer et al. [56] and visible in Figs. 9 and 10.Therefore, while the DK-Pendellösung (67) is valid in the − . − . . . . κ . . . η + − Ω r =1 ω r Ω r =3 ω r Ω r =7 ω r FIG. 8. Velocity dependent diffraction efficiency η + − ( κ ) for aGaussian pulse ( j = G , solid: numerical, dotted: deep-Bragglimit (78)) and the sech pulse [ j = S , dashed: analytical (76)].A mirror pulse of width ˜ τ jπ (61) with total pulse duration ∆ τ = 8˜ τ Gπ is applied for three Rabi frequencies Ω r = Ω ω r . adiabatic regime, the extended model (76) can be evenused for non-adiabatic pulses. E. Diffraction efficiency of a Gaussian pulse in thedeep-Bragg limit
Due to the similarity of the Gaussian- to the sech-pulses [cf. Eqs. (47) and (50)], one can estimate thevelocity selective diffraction efficiency for infinitely longGaussian pulses in the deep-Bragg regime. The differentpulses have equal nominal times (45). Therefore, ap-proximating sech ( a ) from Eq. (70), with a similar ex-ponential form, providing the same integration area as R ∞−∞ d a sech ( a ) = R ∞−∞ d a exp (cid:0) − πa / (cid:1) = 2 , leads to η G + − ( κ, Ω , T ) = exp (cid:18) − π (cid:16) κT (cid:17) (cid:19) sin (cid:18) T Ω2 (cid:19) . (78)The results are discussed in the next section. F. Diffraction efficiency for all pulses in (1+1)D
In beamsplitter experiments, Gaussian laser pulsesare ubiquitous. There is a good reason for it, as theyare self-Fourier-transform functions. This is evident inthe numerical simulations of first order diffraction effi-ciency in Fig. 8, which is free of the side lobes of rect-angular pulses, seen in Fig. 6 (a). The diffraction effi-ciency becomes power-broadened for increasing Rabi fre-quency. Beyond Ω r > ω r , scattering into higher diffrac-tion order depletes the population in the beamsplittermanifold. However, in the deep-Bragg regime, the ap-proximation (78) matches the numerical solutions verywell. Sech-pulses [extended DK-model (76)] behave sim-ilarly, as shown in Fig. 8 and 9. The explicit solutionfor the sech-pulse [extended DK-model (76)] deviatesslightly from Gaussian- and Blackman-pulses, but pro-vides very detailed forecasts. Indeed, all smooth pulseshapes ( j = G, B, S ) with pulse widths ˜ τ jπ are very sim-ilar and exhibit almost identical phase shifts and effi-ciencies as depicted in Fig. 9. Here, for finite total in-teraction times ∆ τ , the π -pulse conditions are not metexactly Ω T j ( − ∆ τ / , ∆ τ / ≈ π (45). One could adjustthe pulse width ˜ τ jπ for each pulse shape j to obtain a π − . − . . . . κ . . . η + − (a) . . . ∆ φ / π . . . . η (b) Ω r /ω r . . . ∆ η (c) FIG. 9. Comparison of the Bragg diffraction for a mirror pulsewidth ˜ τ iπ , for rectangular- (dash-dotted × ), Gaussian- (solid (cid:3)(cid:3)(cid:3) ), Blackman- (dotted ◦ ) and sech-pulses (dashed, numer-ical: ▽▽▽ , DK (67) ⊳ , DK (76) ⊲ ). (a) Velocity dispersion ofthe numerical diffraction efficiency η + − (without plotmarkers)and phase shift ∆ φ (with plotmarkers) for Ω r = Ω ω r = 3 ω r .(b) On-resonance diffraction efficiency η and (c) width of thediffraction efficiency ∆ η versus Ω r . pulse individually Ω T j ( − ∆ τ / , ∆ τ /
2) = π , but this leadsto unequal nominal times T j = T (45) and results in sig-nificant phase differences. Thus, we consider the same π -pulse time ∆ τ = 8˜ τ Gπ for all pulses and the widths τ j = ˜ τ jπ connected via T j = T , the resulting differencesin the pulse areas Ω T j ( − ∆ τ / , ∆ τ / (45) are negligible.The phase sensitive fidelity (19) for different pulseshapes and momentum widths σ k of an initial Gaus-sian wavepacket in 1D (B1) are compared in Fig. 10.For the smooth envelopes, an increasing σ k reduces therange of admissible Rabi frequencies Ω r = Ω ω r , whichshifts the optimum to higher values. Evidently, theDK-Pendellösung (67) matches numerical simulations for Ω r < ω r , while the extended DK-model (76) remainsfurther valid. The explicit Kato solution (C14) matchesthe results for rectangular pulses very well, demonstrat-ing its applicability for wavepackets with finite momen-tum width. G. Diffraction efficiency for spatial Gauss-Laguerremodes with pulse shapes in (3+1)D
1. Gauss-Laguerre modes
The experimental beamsplitter beams are pulsed,bichromatic, counterpropagating Gauss-Laguerre modes[83]. In the specific frame S ′ , comoving with the nodes ofthe interference pattern, there is only a single wavenum- . . . . F (a) (b)0 1 2 3 4 5 6 7 8 Ω r /ω r . . . . F (c) 0 1 2 3 4 5 6 7 8 Ω r /ω r (d) FIG. 10. Fidelity F (Ω r , σ k ) after a mirror pulse ofwidth ˜ τ iπ (Ω) (61) versus the two-photon Rabi frequency Ω r = Ω ω r for different initial atomic momentum widths σ k = { . , . , . , . } k L , { × , ▽▽▽ , (cid:3)(cid:3)(cid:3) , ◦ }; for (a) Gaussian,(b) Blackman, (c) sech and (d) rectangular pulses. The totalinteraction time is ∆ τ = 8 ˜ τ Gπ (a-c) and ∆ τ = 2˜ τ Rπ (d), cf.Eq. (61). The 1D initial Gaussian wavepacket (B1) is centeredat ( x, k x ) = (0 , − k L ) . The DK-Pendellösung (67) [dotted, (c)]matches the results of the numerical integration (16) (solid)very well for Ω r < ω r , considering population loss to higherdiffraction orders (76) (dashed) also for larger Ω r . The Katosolution (C14) (dashed) is depicted in (d), matching the nu-merical results.FIG. 11. Two counterpropagating, bichromatic Gauss-Laguerre beams form a travelling, standing wave (12) with anintensity pattern in cylindrical coordinates ( x, ̺ ) . The grayarrows are the local wavevectors, w ( x ) is the local waist and R ( x ) the local radius of curvature. The distance between thetwo beam waists is ℓ . The atomic cloud, generally localizedat r is indicated as red ellipse. ber k L (cf. (15) and Apps. A, B). The slowly varyingamplitude of the electric field leads to Rabi frequencies Ω j ( t, r ) = Ω j ( t, ̺ ) e i Φ( ̺ ) , (79) Ω j ( t, ̺ ) = Ω j ( t ) w w j e − ̺ w j , Φ( ̺ ) = k L ̺ R j − ξ j (80)with beam parameters w , = w ( ℓ/ , R , = ± R ( ℓ/ , ξ , = ± ξ ( ℓ/ and the distance ℓ between both lasersbeam waists, as depicted in Fig. 11.2 . . . . . . F ( r = ( , , )) (a) σ x /w = 1 / Ω r /ω r . . . . . . F ( r = ( , w , )) (e) (b) σ x /w = 1 / Ω r /ω r (f) (c) σ x /w = 1 / Ω r /ω r (g) (d) σ x /w = 1 / Ω r /ω r (h) FIG. 12. Fidelity F (Ω r , σ k , σ x ) after a mirror pulse versus two-photon Rabi frequency Ω r = Ω ω r for different atomic initialmomentum widths σ k = { . , . , . } × k L , {solid blue, dashed red, dashed-dotted green} of a 3D ballistically expandedGaussian wavepacket (B3) for Gauss-Laguerre beams ( ◦ ) in comparison to plane waves ( (cid:3)(cid:3)(cid:3) ), using the (3+1)D numericalintegration (16). Gaussian temporal pulses of width ˜ τ Gπ (Ω) (61) and total duration ∆ τ = 8 ˜ τ Gπ (61) are applied. Each columnrepresents a different ratio σ x /w between spatial width of the initial state σ x and the beam waist w . In the bottom row theatomic initial state is displaced in the radial direction of the Gaussian beams to ̺ = y = w / .
2. Local plane-wave approximation
To isolate the momentum kick of the beamsplitter fromthe momentum imparted by the dipole force, we considera local plane-wave approximation of the Gauss-Laguerrebeam at the initial position r = (0 , ̺ ) , ̺ = ( y , z ) ofthe atomic cloud Ω j ( t, r ) ≈ Ω j ( t, r ) = Ω j ( t, ̺ ) e i Φ( ̺ ) . (81)Thus, the atomic cloud feels only a reduced Rabi fre-quency but experiences no spatial inhomogeneity. There-fore, simulations with plane waves must be independentof the ratio σ x /w for σ x > λ .
3. Simulations
Beamsplitters perform best, if the atomic cloud (of size σ x ∼ µ m − mm ) is well localized within the beam waist w . For w ∼ mm and optical wavelengths λ ∼ µ m theRayleigh lengths x R are several meters, thus x R ≫ w > σ x > λ. (82)Therefore, one can expect that the transversal dipoleforces will be stronger than the forces along the prop-agation direction x . Small clouds centered at the sym-metry point r = (0 , , will feel the least degradationof the beamsplitter fidelity [cf. Fig. 12 (a) and (b)] dueto dipole forces. This will be confirmed by displacing theinitial cloud transversely to r = (0 , w / , , leading tolarger aberrations [cf. Fig. 12 (e)-(h)]. In these simulations of a Bragg mirror, depicted inFig. 12, we use the effective π -pulse width ˜ τ Gπ (Ω( r )) (61) in the local plane-wave approximation (81) for dif-ferent Rabi frequencies Ω r = Ω ω r and a longitudinallaser displacement ℓ = 0 . x R , like in the experiment(cf. Tab. II). As initial states of the atomic cloud, weconsider ballistically expanded 3D Gaussian wavepack-ets (B3) with different widths in real space σ x and inreciprocal space σ k .For atoms located at the center of the Gaussian laserbeams, the spatial inhomogeneity (B10) leads to sig-nificant aberrations only for large atomic clouds [cf.Fig. 12 (c), (d)]. By contrast, even small displacedclouds [cf. Fig. 12 (e), (f)] show a significant reductionof the fidelity in realistic Gaussian beams compared toideal plane waves. The latter uses a reduced Rabi fre-quency according to the local plane-wave approximation(81). For large clouds this reduction is detrimental [cf.Fig. 12 (g), (h)]. Please note that we use parameters,where the simulation results for the fidelity only dependon the ratio σ x /w < .Besides the phase sensitive fidelity, the aberrationsdue to Gaussian beams are already apparent in thediffraction efficiency. In Fig. 13 the momentum density ˜ n ( k x , k y ) is shown for the (3+1)D simulation with (a)Gauss-Laguerre laser beams and (b) the idealized lo-cal plane-wave approximation, after a mirror pulse with Ω r = 3 ω r . In the momentum space, the splitting isvisible directly after the π -pulse. We study a ballis-tically expanded Gaussian wavepacket (B3) as initialstate with σ k = 0 .
05 k L and σ x = 1 / w , located at r = (0 , w / , . The logarithmic scale highlights the3 − . . . k y / k L (a) − k x /k L − . . . k y / k L (b) − − − − FIG. 13. Column integrated atom density in momentumspace ˜ n ( k x , k y ) = R d k z | ˜ ψ ( k x , k y , k z ) | after a π pulse for (a)Gaussian laser beams and (b) plane waves. The initial stateis a temporally evolved Gaussian wavepacket (B3) located at r = (0 , w / , with momentum width σ k = 0 . k L andexpansion size σ x = w /
25 = 2 . µ m . Gaussian pulses with Ω r = Ω ω r = 3 ω r , τ G = ˜ τ Gπ (Ω) (61), ∆ τ = 8 ˜ τ G π and beamwaist w = 62 . µ m are applied. The final momentum ex-pectation value in y -direction h ˆ p y i = 0 . ~ k L is highlightedwith grey lines. imperfections of the Bragg diffraction, using Gaussianlaser beams. Even for the tiny momentum width, thediffraction efficiency is reduced to . in comparisonto . for idealized plane waves. In addition, thedipole force leads to a rogue, transversal momentum com-ponent h ˆ p y i = 0 . ~ k L . As opposed to the diffractionefficiency and the fidelity, this momentum component de-pends not only on the relation σ x /w but on the beamwaist, here w = 62 . µ m . Further studies of the me-chanical light effects of the dipole force are subjects ofour present research.Locating the initial state at the center r = (0 , , re-duces the aberrations due to Gauss-Laguerre laser beams.The diffraction efficiency of . reaches almost theefficiency of idealized plane waves with . and thetransverse momentum component vanishes. V. PROVING THE DEMKOV-KUNIKE MODELEXPERIMENTALLY
Experimentally, we employ an atom chip apparatus toBose-condense Rb [34, 84] with a condensate fraction of N c = (10 ± × and a quantum depletion (thermalcloud) of N t = (7 ± × . After release from the trap(lab frame S ), with trap frequencies listed in Tab. II, theyexpand ballistically and fall vertically towards Nadir.The Bragg-laser beams are aligned horizontally. It is suf-ficient to consider inertial motion during the short Braggpulses (< ms ). After
10 ms time-of-flight (TOF), at thebeginning of the diffraction pulses, the temperature ofthe thermal cloud is obtained from a bimodal fit [85] as T = (20 ±
3) nK . So far, the cloud σ x = 20 µ m is muchsmaller than the beam waist w = 1386 µ m and permitsthe plane-wave approximation.Experimentally, the first order diffraction efficiency inthe deep-Bragg limit η = N + N − + N + (83) is obtained from the number of atoms N + diffracted intothe first order k ′ = k + and the undiffracted atoms N − remaining in the initial state k ′ = k − . The diffractionefficiency is either a function of the detuning δω (6) of thelaser from the two-photon resonance with atoms initiallyat rest h ˆ p x ( τ i ) i = 0 , or it is the response for resonantlasers and an initial wavepacket centered at h ˆ p x ( τ i ) i = ( − κ ) ~ k L , ¯ κ = δωω r , (84)using Eq. (8) (cf. Sec. II B).Theoretically, we compute the diffraction efficiency(83) in the laser plane-wave approximation from the num-ber of diffracted atoms N ± (¯ κ ) = Z − d κ η ±− ( κ ) n ( κ, ¯ κ ) , (85)following from a reaction equation derived in App. E,which completely encloses the wavepacket with the ef-fectively one-dimensional momentum density n ( κ, ¯ κ ) andthe average initial momentum ¯ κ . Please note that forideal plane matter-waves with wavenumber ¯ κ the diffrac-tion efficiency (83) reduces to η = η + − (¯ κ ) . In the deep-Bragg regime, theoretically N + + N − = N A = N c + N t and the diffraction efficiency simplifies to η = N + (¯ κ ) N A = p c n c + (¯ κ ) + p t n t + (¯ κ ) , (86)splitting into a condensate and a thermal cloud fractionwith p c = N c /N A , p t = 1 − p c . Approximating the nor-malized initial momentum distributions n c ( κ, ¯ κ ) , n t ( κ, ¯ κ ) by Gaussian functions (E10) of widths σ ck = 0 . k L and σ tk = (0 . ± . k L (cf. App. E 1) and using theGaussian approximation (78) for the diffraction efficiency η ±− ( κ ) , one obtains the analytical model η = sin (cid:18) Ω T (cid:19) X a = { c,t } p a ˜ σ ak ( ˜ T ) e − (¯ κ ˜ T )22˜ σak ( ˜ T )2 , (87)with ˜ σ ak ( ˜ T ) = q T ˜ σ ak ) , ˜ T = T p π/ , σ ak = ˜ σ ak k L .In Fig. 14, the diffraction efficiency (83) is depicted fortwo different laser powers P • = 20 mW , P × = 30 mW ofa Gaussian pulse of width τ G (47) and total interactiontime ∆ τ = 8 τ G . In the experiment, the atoms are dis-placed axially to z = (1165 ± µ m = (0 . ± . w ,while x = y = 0 µ m . This reduces the effective Rabifrequency at the location of the atoms (81).Fits using the model (87) describe the experimentaldata already very well and provide starting parameters[ p c , Ω( r ) ] for the effective (1+1)D numerical simulationswith Gaussian pulses, fully matching the experimentaldata. The experimental, numerical and fit parametersare listed in Tab. I.In Fig. 14 (a), the velocity dispersion of the diffrac-tion efficiency uncovers an initial motion k Sx = ¯ κ S k L = τ G /ω r ( µs ) . . . η (b) − . − . . . . ¯ κ . . . η (a) FIG. 14. Experimental diffraction efficiency η (83) for differ-ent laser powers P • = 20 mW and P × = 30 mW of Gaussianpulses of width τ G with numerical simulations (solid, blue)and fits (87) (dashed, red) based on the DK-model. (a) Ve-locity selectivity for Ω T G = 0 . π pulses (45) versus detuning ¯ κ of the initial central momentum h ˆ p x ( τ i ) i = ( − κ S +¯ κ ) ~ k L ,were ¯ κ S k L = 0 . k L is a small initial velocity of the atomsin the lab frame S and ¯ κ = δω/ω r (8). (b) Rabi oscillationsof the diffraction efficiency versus pulse width τ G , with totalinteraction time ∆ τ = 8 τ G and highlighted pulse widths of(a). Other parameters cf. Tab. I, II. . k L of the atomic cloud in the lab frame S . Consider-ing this in h ˆ p x ( τ i ) i = ( − κ S + ¯ κ ) ~ k L with ¯ κ = δω/ω r leads to a very good match of the fit model (87), thesimulations and the experimental data.In Fig. 14 (b), the diffraction efficiency displaysdamped Rabi oscillations versus the pulse width τ G . Thisis a typical inhomogeneous line-broadening caused bythe momentum widths σ ck , σ thk , the two-photon detuning δω = ¯ κω r = 0 and a residual velocity ¯ κ S = 0 . It is alsorevealed by the Gaussian approximation (87). The fit re-sults for the two-photon Rabi frequency are also optimalfor the numerical simulations matching the experimentwithin the error level.It is worth mentioning that the velocity dispersion ofthe efficiency [Fig. 14 (a)] is less sensitive to the conden-sate ratio p c than the Rabi oscillations [Fig. 14 (b)]. TheGaussian approximation (87) underestimates the secondmaxima, but the fit of p c matches the experimental valuewithin its uncertainty. The numerical simulations predicta condensate ratio at the lower bound of the experimen-tal ratio, still within the uncertainty. The reduction ofcondensate fraction p c in simulations and Gaussian ap-proximation is equivalent to increasing the momentumwidth of the condensate or thermal cloud.Thus, the Gaussian approximation (87) of the DK-model gives an unbiased prediction of the experimen-tal data. It assumes weak two-photon Rabi frequencies Ω r ( r ) < ω r , justifying the Pendellösung (70) and smallatomic clouds σ x ≪ w to approximate Gaussian beamsby plane-waves. Table I. Parameters of Fig. 14 for the experiment (e), the nu-merical simulation (n) and the approximation (87) (a). P • =(20 ±
2) mW P × =(30 ±
3) mW e p c . ± .
08 0 . ± . e Ω (6 . ± . ω r (9 . ± . ω r e Ω( r ) (1 . ± . ω r (2 . ± . ω r e ¯ κ S k L (0 . ± . k L (0 . ± . k L (a) e τ G /ω r . µ s 98 . µ s e Ω( r ) T G (45) . π . π a p c . ± .
06 0 . ± . a Ω( r ) (1 . ± . ω r (2 . ± . ω r n p c .
59 0 . n Ω( r ) 1 . ω r . ω r (b) e δω/ π − − . a p c . ± .
03 0 . ± . a Ω( r ) (1 . ± . ω r (2 . ± . ω r n p c .
52 0 . n Ω( r ) 1 . ω r . ω r VI. CONCLUSION
We present (3+1)D simulations and analytical modelsof a pulsed atomic Bragg beamsplitter. Thereby, we char-acterize ubiquitous imperfections, like the velocity dis-persion and the population losses into higher diffractionorders. We study the influence of four common tempo-ral pulses (rectangular-, Gaussian-, Blackman- and hy-perbolic sech pulse). Clearly, the diffraction efficiencyand the fidelity benefit from Fourier-limited, smooth en-velopes. Much insight is gained from the analyticalDemkov-Kunike model for a hyperbolic secant pulse (67).It reveals the explicit dependence on the multitude ofphysical parameters. Due to its similarity with a Gaus-sian pulse, the diffraction efficiency (70) can also be usedfor it (78). For a large parameter regime, the model isverified experimentally and matches the velocity disper-sion. The extended DK-model (76) matches also lossesinto higher diffraction orders.For a rectangular pulse, we have obtained explicithigher order diffraction results from Kato degenerateperturbation theory, which provide insight in the quasi-Bragg regime. Due to a renormalization of the effectiveRabi frequency in the beamsplitter manifold, one findssignificant stretching of the optimal π -pulse time, whichhas been seen experimentally [56]. We find this stretch-ing for all considered pulses in the quasi-Bragg regimeand assume it is universal.Comparing Gauss-Laguerre beams with plane wavesreduces the diffraction efficiency and transfer fidelity, ingeneral. The beam inhomogeneity becomes relevant for σ x > w / . But even for smaller decentered clouds,the fidelity suffers significantly. Currently, we investigatethe aberrations due to laser misalignment and transversalconfinement, which will be reported elsewhere.5 ACKNOWLEDGMENTS
We like to thank Jan Teske for (3+1)D simulation ofthe initial Bose-Einstein-condensate, Sven Abend and themembers of the QUANTUS collaboration for fruitful dis-cussions. This work is supported by the DLR GermanAerospace Center with funds provided by the FederalMinistry for Economic Affairs and Energy (BMWi) underGrant No. 50WM1957.
Appendix A: Comoving rotating frame
In quantum mechanics, a Galilean transformation isrepresented by the displacement operator [86] ˆ G ( t ) = e i ~ ( p ˆ r − r ( t ) ˆ p ) = e − i ~ pr ( t ) e i ~ p ˆ r e − i ~ r ( t ) ˆ p (A1)with a time-dependent coordinate r ( t ) = r + v t and amomentum p = m v . It transforms the correspondingHeisenberg operators as (cid:18) ˆ r ′ ˆ p ′ (cid:19) = ˆ G (cid:18) ˆ r ˆ p (cid:19) ˆ G † = (cid:18) ˆ r − r ( t )ˆ p − p (cid:19) . (A2)In the Schrödinger picture, ˆ G ( t ) transforms the lab framestate | ψ ( t ) i = ˆ G ( t ) | ψ ′ ( t ) i into the state | ψ ′ ( t ) i of the co-moving frame. Evaluating the comoving-frame Hamiltonoperator ˆ H ′ the Schrödinger equation reads i ~ ∂ t | ψ ′ i = ˆ H ′ | ψ ′ i = ˆ G † ( ˆ H − i ~ ∂ t ) ˆ G | ψ ′ i , (A3) ˆ H ′ = ˆ p M + ~ ω g ˆ σ g + ~ ω e ˆ σ e + V ( t, ˆ r + r ( t )) . (A4)In the frame, moving with the group velocity v = v g e x (12) in the x-direction, the Doppler shifted laser phases φ ′ = ω t − k (ˆ x + x + v g t ) = ω L t − k (ˆ x + x ) , (A5) φ ′ = ω t + k (ˆ x + x + v g t ) = ω L t + k (ˆ x + x ) (A6)oscillate synchronously with ω L = ω + ω (cid:0) − β (cid:1) ≈ ω + ω . (A7)The second order correction in β = v g /c can be neglectedsafely in our nonrelativistic scenario.Another local frame transformation | ψ ′ i = ˆ F | ψ ′′ i ,eliminates the rapid temporal oscillations and establishesa single spatial period λ = 2 π/k L of the optical potential ˆ F ( t ) = e − iω g t − iω L t ˆ σ e + i [ k (ˆ x + x ) − χ ]ˆ σ z . (A8)Now, the transformed Schrödinger equation reads i ~ ∂ t | ψ ′′ i = ˆ H ′′ | ψ ′′ i , (A9) ˆ H ′′ = (ˆ p x + ~ k ˆ σ z ) M + ˆ p y + ˆ p z M − ~ ∆ˆ σ e + ~ σ † (cid:16) ˜Ω ( t, ˆ r ) e ik L ˆ x + ˜Ω ( t, ˆ r ) e − ik L ˆ x (cid:17) + h.c. (A10) with a laser detuning ∆ = ω L − ω , a common wavenum-ber k L = ( k + k ) / and a relative wavenumber mis-match k = ( k − k ) / . Global phases of the Rabifrequencies Ω i ( t, r ) = ˜Ω i ( t, r ) e − iχ i do vanish with theproper gauge χ = ( χ + χ ) / and the shifted coordi-nate origin x = ( χ − χ ) / k L .Please note, k = ( ω − ω ) /c ∼ × − µ m − ∼ × − k L is tiny in comparison to other relevant mo-menta. We will consider Bose-Einstein condensates withThomas-Fermi radii in the trap of a few microns (cf.Sec. V and E 1), the momentum width can be approx-imated with the Heisenberg width ∆ k HTF = 3 / r TF =0 . µ m − , considering r TF = 10 µ m , while the Rayleighwidth gives ∆ k RTF = 0 . µ m − [87]. In our simulations,we consider atomic initial states as Gaussian wavepack-ets with momentum widths σ k ∈ [0 . , . , . , . k L ,with k L ≈ µ m − , to compare ∆ k RTF corresponds to σ R k ≈ ∆ k RTF / . k L . After release out of the trap themomentum width of the BEC increases. With tempera-tures T ≤
20 nK this gives rise for momentum widths ofa thermal cloud σ k = √ k B T M/ ~ ≤ . k L . Therefore, k can be neglected safely. Appendix B: Spreading Gaussian waves a. Matter waves
Ballistically spreading Gaussianwavepackets are useful input states to test a beamsplit-ter. Using different expansion times t , one can vary theposition width σ x , while keeping the momentum width σ k constant. A n -dimensional Gaussian unnormalizedwavepacket is defined as ψ ( r ) = e i k ( r − r ) − ( r − r )(2Σ ) − ( r − r ) (B1) = Z d n k (2 π ) n e i kr p | | e − i kr − ( k − k )(2Σ )( k − k ) and centered at ( r , k ) = ( h r i , h− i ∇i ) . The wavepacketis normalized to R d n r | ψ | = p | π Σ | with the covari-ance matrix Σ = h ( r − r ) ⊗ ( r − r ) i . The three di-mensional free Schrödinger equation i∂ t ψ ( t, r ) = − α r ψ, α = ~ M (B2)describes the spreading of a matter-wave using theFourier-transformed field ˜ ψ ( k ) implicitly defined in (B1) ψ ( t, r ) = Z d n k (2 π ) n e − it α k e i kr ˜ ψ ( k ) (B3) = A ( t ) e − i Θ( t ) e i k [ r − r ] − [ r − r ( t )][2Σ( t )] − [ r − r ( t )] . The evolving center position r ( t ) , spreading covariance Σ( t ) , dynamical phase Θ( t ) and scale-factor A ( t ) read Σ( t ) = Σ + it α , r ( t ) = r + tα k , (B4) Θ( t ) = t αk , A ( t ) = s | Σ || Σ( t ) | . (B5)6In the simulations, we assume an isotropic initial statewith Σ ij = δ ij σ x and σ x ( t ) = σ x p t/t H ) , (B6)with the Heisenberg time t H = 2 σ x M/ ~ . b. Gaussian laser beams The scalar mode of a circu-larly symmetric Gauss-Laguerre beam propagating alongthe x-direction follows from the two-dimensional n = 2 paraxial approximation of the Helmholtz equation i∂ x u ( x, ̺ ) = − β ̺ u, β = k − L , ̺ = ( y, z ) . (B7)The spatially evolved mode u ( x, ̺ ) follows analogouslyfrom (B3), (B4), substituting ( t, α ) ↔ ( x, β ) u ( x, ̺ ) = x R iq ( x ) e i kL̺ q ( x ) = U e i Φ (B8) U ( x, ̺ ) = w w ( x ) e − ̺ w ( x )2 , Φ( x, ̺ ) = k L ̺ R ( x ) − ξ ( x ) , where ̺ = p y + z is the normal distance to the sym-metry axis and q ( x ) = x − ix R is the complex beamparameter [83]. It is characterized by the Rayleigh range x R = πw /λ , the beam waist w ( x ) = w (1+( x/x R ) ) / ,the minimum waist w = 2 σ , the radius of wavefrontcurvature R ( x ) = x (1 + ( x R /x ) ) , the Gouy phase ξ =arctan( x/x R ) and the wavelength λ L = 2 π/k L .We consider two counterpropagating Gaussian laserbeams, which are symmetrically displaced with respectto their waists by a distance ℓ . Then, the dipole interac-tion energy in the comoving, rotating frame (A10), reads ˆ V ′′ = ~ σ † (cid:2) Ω ( t, r ) e ik L x + Ω ( t, r ) e − ik L x (cid:3) + h.c. , (B9)with pulse amplitudes Ω j ( t ) and spatial envelopes Ω j ( t, r ) = Ω j ( t ) U ( x j , ̺ ) e i Φ( x j ,̺ ) . (B10)We use shifted coordinates x / = ± ( x + v g t + ℓ/ andbeam parameters w j = w ( x j ) , R j = R ( x j ) and ξ j = ξ ( x j ) , which are slowly varying for x ≪ x R . Beamsplitterpulses are typical short and one can neglect the ballisticdisplacement v g t ∼ µ m ≪ ℓ, x R . For small atomic clouds σ x < w / , one can approximate x ≈ − x ≈ ℓ/ . Appendix C: Degenerate perturbation theory
To rectify the Pendellösung (54) with contributionsfrom higher order diffraction, we employ Kato’s methodfor the stationary eigenvalue problem in the presence ofdegeneracy [79]. All eigenvalues of the diagonal part D = D ( κ = 0) of the Bragg Hamilton operator (42) aredoubly degenerated ≤ α ≤ on resonance. Therefore,we consider the flow of the eigensystem H ( λ ) v i,α ( λ ) = ω i,α ( λ ) v i,α ( λ ) with H = D + λ V , V = D ( κ ) − D + L + L † , (C1) for ≤ λ ≤ in the degenerate subspace E i . If we de-note the orthonormal eigenvectors of D with v (0) i,α andtheir eigenvalues ω (0) i , the eigenvectors of the interact-ing Hamilton operator H i ( λ ) restricted to the subspace E i , are v i,α ( λ ) = P i ( λ ) v (0) i,α . Now, all efforts are putin the perturbative evaluation of the projection opera-tor P i ( λ ) , which evolves from the unperturbed projection P (0) i . This results in the generalized eigenvalue problem H i v (0) i,α = ω i,α K i v (0) i,α , (C2) H i = P (0) i H P i P (0) i , K i = P (0) i P i P (0) i , (C3)with power series expressions for the operators P i ( λ ) = P (0) i + ∞ X n =1 λ n A ( n ) i , (C4) A ( n ) i = − X ( n ) S ( k ) i V S ( k ) i V . . . V S ( k n +1 ) i , (C5) H P i ( λ ) = ω (0) i P i ( λ ) + ∞ X n =1 λ n B ( n ) i , (C6) B ( n ) i = X ( n − S ( k ) i V S ( k ) i V . . . V S ( k n +1 ) i . (C7)Here P ( n ) denotes a sum over all combinations of inte-gers k i ∈ N satisfying k + k + . . . + k n +1 = n and S (0) i = − P (0) i , S ( k> i = ( S i ) k , S i = − P (0) i ω (0) i − D . (C8)It is straight forward to evaluate H i and K i from (C3)for the ground-state manifold i = 1 to order O ( λ n ) . Wefind that a third order truncation of the series H = (cid:18) Ω2Ω2 κ (cid:19) − I (cid:18) (cid:19) − I (cid:18) κ ΩΩ 0 (cid:19) ,K = (1 − I ) (cid:18) (cid:19) − I (cid:18) κ Ω2Ω2 − κ (cid:19) , I = Ω (C9)agrees very good with the numerical results. The rootsof the characteristic equation |H − ( ω − ω (0)1 ) K | = 0 ,determine the corrected eigenfrequencies of the Pendelö-sung. As the frequency shifts ω ( λ ) − ω (0)1 , are already O ( λ ) , it is consistent to use a lower approximation for K , which leads to better results at the specified order.In particular, we have evaluated ˜ H = K − H and Tay-lor expanded it at the specified order ˜ H = (cid:18) −I (2 + κ ) Ω2 (1 − I ) Ω2 (1 − I ) κ − I (2 − κ ) (cid:19) + O (cid:0) λ (cid:1) . (C10)This leads to the succinct expression for the eigenvaluesand -vectors ω , ± = κ − I ± ˜Ω2 , v (0)1 , ± = (cid:18) I − √I− κ (1 + 2 I ) ± ˜Ω (cid:19) , ˜Ω = p κ (1 + 2 I ) + Ω (1 − I ) (C11)7in terms of a corrected Rabi frequency ˜Ω (59).Analogous to that the eigenvalues of the next subspace,coupling µ = ± and representing the most importantloss channel, can be calculated from ˜ H = K − H ˜ H = (cid:18) I ) − κ
00 2(1 + I ) + 2 κ (cid:19) + O (cid:0) λ (cid:1) , (C12)skipping the λ terms, which overestimate the losses into µ = ± . Including higher expansion orders would cor-rect this, but we find that the lower expansion (C12) issufficient. The eigenvalues and -vectors of ˜ H are ω , ± = 2(1 + I ) + κ ± κ , ( v (0)3 , + , v (0)3 , − ) = . (C13)With the eigenvectors v i,j = P i v (0) i,j , defined by the pro-jections (C4), also expanded up to λ for µ = ± and λ for µ = ± , the time-dependent solution of the Schrö-dinger equation with the Hamiltonian (C1) results in g K ( τ ) = ˜ g K ( τ ) | ˜ g K ( τ ) | , (C14) ˜ g K ( τ ) = X i = { , } X j = { + , −} c i,j e − iω i,j ( τ − τ i ) v i,j , (C15)where the integration constants c ( j ) i are defined by theinitial condition ˜ g K ( τ ) = (0 , , , .The population of the µ = 1 state is of special inter-est, because it defines the diffraction efficiency η + − . Onresonance ( κ = 0 ), already ˜ g K ( τ ) is approximately nor-malized. Therefore, it can be approximated η K ( τ ) ≈ | (cid:0) ˜ g K ( τ ) (cid:1) | (C16) = A (cid:16) B cos[4 τ ′ ( I − √I ] + C cos θ + + D cos θ − (cid:17) , with θ ± = 2 τ ′ (1 ± √I + 2 I − I / ) , τ ′ = τ − τ i andcoefficients expanded up to the suited order O (cid:0) I (cid:1) A = 12 − I − I O (cid:0) I (cid:1) , B = − O (cid:0) I (cid:1) ,C = − D = − I / + O ( I / ) . (C17)After the effective π -pulse time ˜ τ Rπ = π/ [2 | Ω | (1 − I )] (61) the diffraction efficiency (C16) results in Eq. (62). Appendix D: Demkov-Kunike model
The retarded Green’s function is defined as G ( τ, τ i ) = T e − i R ττi d t H ( t ) θ ( τ − τ i ) , (D1) [ i∂ τ − H ( τ )] G ( τ, τ i ) = iδ ( τ − τ i ) , (D2)which hold equally for the free evolution G ( τ, τ i ) by sub-stituting H → H . This leads to the Dyson-Schwingerintegral equation G ( τ, τ i ) = G − i Z ∞−∞ d t G ( τ, t ′ ) H ( t ′ ) G ( t ′ , τ i ) , (D3) which is central to time-dependent perturbation theory.The two-dimensional Green’s function G ∓ of the DK-model can be expressed completely for Ω , κ = 0 with thehypergeometric basis functions f , f form Eq. (64) G ∓ ( τ, τ i ) = M ( z ) S ( z ) S − ( z i ) M − ( z i ) , (D4) M = (cid:18) ia p z (1 − z ) (cid:19) , S = (cid:18) f f f ′ f ′ (cid:19) . (D5)In the important case of exact resonance κ = 0 , furthersimplifications are possible and lead to G ∓ ( τ, τ i ) = (cid:18) cos ∆ ϕ − i sin ∆ ϕ − i sin ∆ ϕ cos ∆ ϕ (cid:19) , (D6) ϕ ( z ) = Ω τ S arcsin √ z, ∆ ϕ = ϕ ( z ) − ϕ ( z i ) . (D7)The integrals (D3) can be solved approximately analyti-cally. However, the expressions are bulky, why we forgoshowing them [81]. Appendix E: Diffraction efficiency for partiallycoherent bosonic fields
The bosonic amplitude ˆ a g ( k ) describes the groundstate atoms in momentum space and obeys the commu-tation relation [ˆ a g ( k ) , ˆ a † g ( q )] = δ ( k − q ) . For a Bose-condesed sample, the single-particle density matrix ρ ( k , q ) ≡ h ˆ a † g ( q )ˆ a g ( k ) i = ρ c ( k , q ) + ρ t ( k , q ) , (E1)separates into a condensate ρ c ( k , q ) = α ∗ ( q ) α ( k ) and aquantum depletion ρ t ( k , q ) . The momentum density n ( k ) ≡ ρ ( k , k ) = N A (cid:2) p c n c ( k ) + p t n t ( k ) (cid:3) , (E2)is the observable in a beamsplitter. It is normalized to thetotal number of N A = R ∞−∞ d k n ( k ) = N c + N t atoms,the densities n c , n t are probability normalized, thus defin-ing a condensate fraction p c = N c /N A and a thermalfraction p t = N t /N A . Dynamically, the classical field α ( t ) obeys the Gross-Pitaevskii equation and extensionsthereof for ρ t ( t ) [88–90].During the short beamsplitter pulse ( < ), only sin-gle particle dynamics (16) are relevant ρ ( τ ) = G ( τ, τ i ) ρ ( τ i ) G † ( τ, τ i ) , (E3)for the condensate and the thermal cloud. In the plane-wave approximation, the three-dimensional Fourier prop-agator G ( k , q ) = G k G ⊥ (18) factorizes into the transversepropagator G ⊥ ( τ, k ⊥ , q ⊥ ) = e − i ~ ( k y + k z )2 M τ δ (2) ( k ⊥ − q ⊥ ) , (E4)and the longitudinal Greens function in x-direction G k ( τ, x, ξ ) = X µ,ν,n G µ,ν ( κ n ,τ ) N x a x e i ( k nµ x − k nν ξ ) , (E5)8using definitions (35), (36). The discrete Greens-matrix G µ,ν ( τ, κ n ) satisfies (40) with initial condition G µ,ν (0 , κ n ) = δ µ,ν . In the continuum limit, one un-covers the momentum conservation on a lattice with k x = ( µ + κ ) k L and q x = ( ν + κ ′ ) k L , from the Fouriertransformation G k ( τ, k x , q x ) = δ ( κ − κ ′ ) G µ,ν ( τ, κ ) . (E6)All observables are along the x-direction. Thus, weaverage over the transversal directions and introduce themarginal momentum densities at time τn ( τ, k x ) = Z ∞−∞ d k y d k z n ( τ, k ) . (E7)We assume that the initial ensemble is well localizedaround k x = ( ν + κ ) k L with ν = − , and denote the den-sity by n i ( κ ) = n ( τ i , k x ) . From the propagation equation(E3), one obtains the final density n f ( κ ) = n ( τ f , k x ) ,with k x = ( µ + κ ) k L at diffraction order µn f ( µ, κ ) = |G µ, − ( τ f , κ ) | n i ( κ ) . (E8)Now, we can identify the diffraction efficiency as η + − ( κ ) = |G , − ( τ f , κ ) | and η −− ( κ ) = |G − , − ( τ f , κ ) | .Thus, for atomic clouds with initial momentum h ˆ p x i =( − κ ) ~ k L (84), the number of diffracted atoms read N ± (¯ κ ) = Z − d κ η ±− ( κ ) n i ( κ, ¯ κ ) , (E9)which are the observables in st order diffraction theory.
1. Initial momentum distribution
After release from the trap, the width of the BEC inmomentum space increases due to atomic mean-field in-teraction [91]. The momentum distribution is determined by solving the (3+1)D Gross-Pitaevskii equation for thegiven parameters of Tab. II and
10 ms time-of-flight be-fore the diffraction pulses. The result is confirmed bythe scaling approach [92–95] applied to the numericalGross-Pitaevskii ground state. Finally, the marginal,one-dimensional momentum density distribution of theBEC to begin of the diffraction pulses n ci ≈ ˜ n c (E7), canbe approximated with a Gaussian distribution ˜ n ( κ, ¯ κ ) = 1 √ π ˜ σ k e − ( κ − ¯ κ )22(˜ σk )2 , Z ∞−∞ d κ ˜ n ( κ, ¯ κ ) = 1 , (E10)with the dimensionless momentum width ˜ σ k = σ k /k L and σ ck = 0 . k L , as depicted in Fig. 15.The thermal cloud is also approximately a Gaussiandistribution [85], where the one-dimensional momentumwidth σ tk = √ M k B T / ~ introduces a temperature T . Ex-perimentally, time-of-flight measurements of σ x ( t ) (B6)lead to the momentum width σ tk = (0 . ± . k L of n t (E10) (cf. Fig. 15) and temperature T = (20 ±
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Number of atoms in condensate N c (10 ± × Number of atoms in thermal cloud N t (7 ± × Atomic mass M ω π × .
230 484 468 5(62) THz [97]Lifetime τ (26 . ± . [98]Decay rate Γ 2 π × (6 . ± . D ( S / → P / ) transition dipole matrix element D . × − C m [98]Rabi-frequency Ω E D / ~ √ Scattering length a . a [99]Trap frequencies [ ω x , ω y , ω z ] 2 π × [46 ± , ± , ±
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Thomas-Fermi radii inside trap [ r x , r y , r z ] [4 . , . , . µ m Laser
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Wavenumber k L . µ m − Detuning to atomic resonance ∆ 97 .
875 GHz
Beam waist w .
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