==== Front Sci Rep Sci Rep Scientific Reports 2045-2322 Nature Publishing Group UK London 78859 10.1038/s41598-020-78859-1 Article Universal atom interferometer simulation of elastic scattering processes Fitzek Florian 12 Siemß Jan-Niclas 12 Seckmeyer Stefan 1 Ahlers Holger 1 Rasel Ernst M. 1 Hammerer Klemens 2 Gaaloul Naceur gaaloul@iqo.uni-hannover.de 1 1 grid.9122.80000 0001 2163 2777Institut für Quantenoptik, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany 2 grid.9122.80000 0001 2163 2777Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstraße 2, 30167 Hannover, Germany 17 12 2020 17 12 2020 2020 10 2212013 7 2020 30 11 2020 © The Author(s) 2020Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.In this article, we introduce a universal simulation framework covering all regimes of matter-wave light-pulse elastic scattering. Applied to atom interferometry as a study case, this simulator solves the atom-light diffraction problem in the elastic case, i.e., when the internal state of the atoms remains unchanged. Taking this perspective, the light-pulse beam splitting is interpreted as a space and time-dependent external potential. In a shift from the usual approach based on a system of momentum-space ordinary differential equations, our position-space treatment is flexible and scales favourably for realistic cases where the light fields have an arbitrary complex spatial behaviour rather than being mere plane waves. Moreover, the solver architecture we developed is effortlessly extended to the problem class of trapped and interacting geometries, which has no simple formulation in the usual framework of momentum-space ordinary differential equations. We check the validity of our model by revisiting several case studies relevant to the precision atom interferometry community. We retrieve analytical solutions when they exist and extend the analysis to more complex parameter ranges in a cross-regime fashion. The flexibility of the approach, the insight it gives, its numerical scalability and accuracy make it an exquisite tool to design, understand and quantitatively analyse metrology-oriented matter-wave interferometry experiments. Subject terms Quantum physicsMatter waves and particle beamsQuantum metrologyAtomic and molecular physicsUltracold gaseshttp://dx.doi.org/10.13039/501100002347Bundesministerium für Bildung und ForschungVDI 13N14838http://dx.doi.org/10.13039/501100001659Deutsche ForschungsgemeinschaftCRC 1227 (projects A05 and B07)DLR/ Bundesministerium für Wirtschaft und Energie50WM2060 (CARIOQA)Förderung von Wissenschaft und Technik in Forschung und Lehre for the initial funding of research in the new DLR institutes (DLR-SI and DLR-QT)Niedersächsisches Vorab, Förderung von Wissenschaft und Technik in Forschung und Lehre Quantum- and Nano Metrology (QUANOMET)QT3EXC-2123 QuantumFrontiers390837967DLR/Bundesministerium für Wirtschaft und Energie50WM1861 (CAL)Gaaloul Naceur Projekt DEALOpen Access funding enabled and organized by Projekt DEAL. issue-copyright-statement© The Author(s) 2020 ==== Body Introduction The commonly used approach for treating light-pulse beam-splitter and mirror dynamics in matter-wave systems consists in solving a system of ordinary differential equations (ODE) with explicit couplings between the relevant momentum states. This formulation starts by identifying the relevant diffraction processes and extracting their corresponding coupling terms in the ODE1,2. In the elastic scattering case, each pair of light plane waves can drive a set of two-photon transitions from one momentum class j to the next neighboring orders j±2. The presence of multiple couplings allows for higher order transitions and the system is simplified by choosing a cutoff omitting small transition strengths. This ODE approach works well for simple cases leading to analytical solutions in the deep Bragg and Raman-Nath regimes1,2. Using a perturbative treatement, it was generalised to the intermediate, so-called quasi-Bragg regime3. A numerical solution in this regime has been extended in the case of a finite momentum width4. In a different approach, Siemß et al.5 developed an analytic theory for Bragg atom interferometry based on the adiabatic theorem for quasi-Bragg pulses. Realistically distorted light beams or mean-field interactions, however, sharply increase the number of plane wave states and their couplings required for an accurate description. The formulation of the ODE becomes increasingly large and inflexible, with a set of coupling terms for each relevant pair of light plane waves. Here, we take an alternative approach and solve the system in its partial differential equation (PDE) formulation following the Schrödinger equation. This time-dependent perspective6 has several advantages in terms of ease of formulation and implementation, flexibility and numerical efficiency for a broad range of cases. Indeed, this treatment is valid for different types of beam splitters (Bloch, Raman-Nath, deep Bragg and any regime in between) and pulse arrangements. Combining successive light-pulse beam-splitters naturally promoted our solver to a cross-regime or universal atom interferometry simulator that could cope with a wide range of non-ideal effects such as light spatial distortions or atomic interactions, yet being free of commonly-made approximations incompatible with a metrological use. The position-space representation seems underutilised in the treatment of atom interferometry problems in favor of the momentum-space description although several early attempts of using it were reported for specific cases7–11. In this paper, we show the unique insights this approach can deliver and, contrary to widespread belief, its great numerical precision and scalability. In addition we illustrate our study with relevant examples from the precision atom interferometry field. Theoretical model Light-pulse beam splitting as an external potential We start with a semi-classical model of Bragg diffraction, where a two-level atom is interacting with a classical light field1,2. This light field consists of a pair of two counter-propagating laser beams realised by a retro-reflection mirror setup for example. Assuming that the detuning of the laser light Δ is much larger than the natural line width of the atom, one may perform the adiabatic elimination of the excited state. This yields an effective Schrödinger equation for the lower-energy atomic state ψ(x,t) with an external potential proportional to the intensity of the electric field 1 iħ∂tψ(x,t)=-ħ22m∂2∂x2+2ħΩcos2(kx)ψ(x,t) with the two-photon Rabi frequency Ω and wave vector k=2π/λ in a simplified 1D geometry along the x-direction. For the present study, we consider a 87Rb atom that is addressed at the D2 transition with λ=780 nm resulting in a recoil frequency and velocity12 of ωr=ħk2/2m=2π·3.8 kHz and vr=ħk/m=5.9 mm/s, respectively. In the context of realistic precision atom interferometric setups, it is necessary to include Rabi frequencies Ω(x,t) and wave vectors k(x, t) which are space and time-dependent. This allows one to account for important experimental ingredients such as the Doppler detuning or the beam shapes including wavefront curvatures13–15 and Gouy phases16–19. Moreover, this generalisation allows one to effortlessly include the superposition of more than two laser fields interacting with the atoms as in the promising case of double Bragg diffraction20–22, and to model complex atom-light interaction processes where spurious light reflections or other experimental imperfections are present23. Atom interferometer geometries The light-pulse representation presented in the previous section is the elementary component necessary to generate arbitrary geometries of matter-wave interferometers operating in the elastic diffraction limit. Indeed, since the atom-light interaction in this regime conserves the internal state of the atomic system, a scalar Schrödinger equation is sufficient to describe the physics of the problem in contrast to the model adopted in Ref.10. For example, a Mach–Zehnder-like interferometer geometry can be generated by a succession of π2-π-π2 Bragg pulses (beam-splitter, mirror, beam-splitter pulses) of order n separated by a free drift time of T between each pair of pulses. In the case of Gaussian temporal pulses, this leads to a time-dependent Rabi frequency 2 Ω(t)=Ωbse-t22τbs2+Ωme-(t-T)22τm2+Ωbse-(t-2T)22τbs2, where Ωbs, τbs and Ωm, τm are the peak Rabi frequencies and their respective durations associated to the beam-splitter and mirror pulses, respectively. We numerically solve the corresponding time-dependent Schödinger equation using the split-operator method24 to propagate the atomic wave packets along the two arms. The populations in the two output ports |+⟩=|0ħk⟩ and |-⟩=|2nħk⟩ are evaluated after the last recombination pulse waiting for a time of flight τToF long enough that the atomic wave packets spatially separate. They are obtained by the integration 3 P±unnormalised=∫±dx|ψ(x,τToF)|2, where the integration domains extend over a space interval with non-vanishing probability density of the states |±⟩. These probabilities are further normalised to account for the loss of atoms to other parasitic momentum classes 4 P±=P±unnormalisedP+unnormalised+P-unnormalised. Using Feynman’s path integral approach, the resulting phase shift between the two arms can be decomposed as25,26 5 Δϕ=Δϕpropagation+Δϕlaser+Δϕseparation. The propagation phase is calculated by evaluating the classical action along the trajectories of the wave packet’s centers. The laser phase corresponds to the accumulated phase imprinted by the light pulses at the atom-light interaction position and time. Finally, the separation phase is different from zero if the final wave packets are not overlapping at the time of the final beam splitter, t=2T. To extract the relative phase Δϕ between the two conjugate ports and the contrast C, one can scan a laser phase ϕ0∈[0,2π] at the last beam splitter and evaluate the populations1 varying as 6 P±=121±Ccos(Δϕ+nϕ0). The resulting fringe pattern is then fitted with Δϕ and C≤1 as fit parameters. This method, analogous to experimental procedures, allows one to determine the relative phase modulo 2π. Results Raman-Nath beam splitter The Raman-Nath regime, characterised by a spatially symmetric beam splitting, is the limit of elastic diffraction for very short interaction times of τ≪12Ωωr. The dynamics of the system can, in this case, be analytically captured following Refs.1,2 7 |gn(t)|2=Jn2(Ωt), where gn(t) describes the amplitude of the momentum state |2nħk⟩ and Jn the Bessel functions of the first kind. Such experiments are at the heart of investigations as the one reported in Ref.27 where a Raman-Nath beam splitter was used to initialise a three-path contrast interferometer offering the possibility of measuring the recoil frequency ωr. To demonstrate the validity of our position-space approach, we contrast our results to the analytical ones obtained adopting the parameters of Ref.27. Figure 1 shows the outcome of a symmetric Raman-Nath beam splitter targeting the preparation of three momentum states: 50% into |0ħk⟩ and 25% in each of the |±2ħk⟩ momentum classes. As a feature of our solver, we directly observe the losses to higher momentum states (p=±4ħk and p=±6ħk) due to the finite pulse fidelity. An excellent agreement is found with the analytical predictions (green filled circles) of the populations of the momentum states.Figure 1 Probability density after a Raman-Nath pulse with Ω=50 ωr, τ=1 μs and a rectangular temporal profile as implemented in27. This shall create a beam splitter of roughly 50% in |0ħk⟩ and 25% in each of the |±2ħk⟩ momentum states with an added time of flight of τToF=20 ms to clearly separate the wavepackets in position space. The left and right panels show the position- and momentum-space probability density. The initial momentum width of the Gaussian wavepacket is chosen to be σp=0.01 ħk. Numerical results of this work (continuous blue lines) agree well with the analytical solution of the Raman-Nath regime (green dots, momentum space) given by the Bessel functions of the first kind. Bragg-diffraction Mach–Zehnder interferometers To simulate a Mach–Zehnder atom interferometer based on Bragg diffraction, we consider a pair of two counter-propagating laser beams with a relative frequency detuning Δω=ω1-ω2=2nkvr and a phase jump ϕ0∈[0,2π]. This gives rise to the following running optical lattice 8 VBragg(x,t)=2ħΩ(t)cos2(k(x-nvrt)+ϕ02). For sufficiently long atom-light interaction times, i.e. in the quasi- and deep-Bragg regimes2,3,28,29, the driven Bragg order n with momentum transfer Δp=2nħk is determined by the relative frequency detuning Δω of the two laser beams. The relative velocity between the initially prepared atom and the optical lattice is v=nvr. In the rest frame of the optical lattice, the atom has a momentum p=-nħk. The difference of kinetic energy between the initial (p=-nħk) and target state (p=+nħk) is vanishing and therefore this transition is energetically allowed and leads to a Δp=nħk-(-nħk)=2nħk momentum transfer. We now realise beam splitters and mirrors by finding the right combination of peak Rabi frequency and interaction time (Ω,τ), either by numerical population optimisation or analytically, when we work in the deep Bragg regime. Recent advances by Siemß et al.5 generalise this to the quasi-Bragg regime in an analytical description of Bragg pulses based on the adiabatic theorem. For the pulses used in this paper, the two approaches give the same result for the optimised Rabi frequencies and pulse durations.Figure 2 (a) Rabi frequency Ω(t) over time of a 2ħk-Bragg Mach–Zehnder interferometer according to Eq. (2). (b) Corresponding space-time diagram of the probability density |ψ(x,t)|2. The initial momentum width is chosen to be σp=0.1 ħk and the splitter and mirror Gaussian pulses have peak Rabi frequencies of Ω=1.0573 ωr with pulse lengths of τbs=25 μs and τm=50 μs, respectively. The separation time between the pulses is T=10 ms with a final time of flight after the exit beam splitter of τToF=20 ms. Due to the velocity selectivity of the Bragg pulses, several trajectories can be observed after each pulse. Inset: Additional insight into the dynamics of the mirror pulse of the upper arm. The peak amplitude of the Gaussian pulse is reached at t=11 ms. The interference fringes of the density plot indicate the overlap between the atoms in momentum class p=2ħk and the atoms lost due to velocity selectivity remaining at p=0ħk. In Fig. 2, we simulate a Mach–Zehnder geometry and illustrate the diffraction outcome by showing a space-time diagram of the density distribution |ψ(x,t)|2. For the parameters chosen here, a clear feature of the dynamics is the appearance of additional atomic channels after the mirror pulse, which can be attributed to the velocity selectivity arising from a pulse with a finite duration characterised by τ. The finite velocity acceptance can, indeed, be estimated over the Fourier width σf of the applied pulse as 9 F(Ωe-t22τ2)=2πτ2Ω2e-2(πfτ)2, with σf=1/(2πτ) and f being the frequency variable. This yields the velocity acceptance30 10 σvpulse=18ωrτvr=0.11vr. With an initial velocity width of the atomic probability distribution of σvatom=0.1 vr, it is clear that velocity components with |v|=σvpulse will have a much smaller excitation probability than the components at the centre of the cloud, which leads to the characteristic double well densities of the parasitic trajectories. With momenta pupper=0ħk and plower=2ħk, both parasitic trajectories still fulfill the resonance condition with the final Bragg beam splitter, which leads to the emergence of ten trajectories after the exit beam splitter. For a measurement in position space, it is now important that a sufficiently long time of flight τToF is applied that the ports of the Mach–Zehnder interferometer do not overlap with the parasitic ports and bias the relative phase measurement. For large densities, the parasitic trajectories at the Mach–Zehnder ports should not overlap since this may already lead to density interaction phase shifts Hint∝|ψ(x,t)|2. To circumvent these problems it is important to choose σvpulse≫σvatoms. An example of state-of-the-art experiments23 with delta-kick collimated BEC sources31–36 uses σvatoms=0.03 vr≪0.14 vr=σvpulse, for strongly suppressed parasitic trajectories due to velocity selectivity. Implementing high-order Bragg diffraction is a natural avenue to increase the momentum separation of an atom interferometer, and therefore its sensitivity. In Fig. 3, we run our solver to observe the population distribution across the different ports of a Mach–Zehnder configuration with Bragg orders up to n=3. This is done in a straightforward way by scanning the laser phase ϕ0. We fit the data points corresponding to the population in the fast port |2ħk⟩ for the different Bragg orders according to Eq. (6) and observe a clear sinusoidal signal of the simulated fringes, as expected. The resulting contrasts and phase shifts are directly found by our theory model and numerical solver which include the ideal phase shifts commonly found25,26, and go beyond to comprise several non-ideal effects as (i) finite momentum widths, (ii) finite pulse timings and (iii) multi-port Bragg diffraction37,38 and the resulting diffraction phase. The natural occurrence of these effects and the possibility to quantify them are a native feature of our simulator.Figure 3 Scan of Mach–Zehnder interferometer phase for different Bragg transition orders of 2ħk (red dots), 4ħk (green dots) and 6ħk (blue dots). The phase shift is applied as a laser phase jump ϕ0∈[0,2π] at the last Bragg pulse. The lengths of the Gaussian splitting and mirror pulses are τbs=25 μs and τm=50 μs, respectively. The initial momentum width of the atomic sample is σp=0.01 ħk. The corresponding Rabi frequencies for the higher order Bragg transitions were found by optimising for an ideal 50 : 50 population splitting of the π2 pulse. This leads to Ω4ħk=3.7 ωr and Ω6ħk=8.4 ωr. The Rabi frequency for the 2ħk transition is Ω2ħk=1.0573 ωr. The solid lines are the respective fringe scan fits from which the phase shifts and contrasts are directly extracted. Symmetric double Bragg geometry Scalable and symmetric atom interferometers based on double Bragg diffraction were theoretically studied22 and experimentally demonstrated21. This dual-lattice geometry has particular advantages, including an increased sensitivity due to the doubled scale factor compared to single-Bragg diffraction, as well as an intrinsic suppression of noise and certain systematic uncertainties due to the symmetric configuration21. Combining this technique with subsequent Bloch oscillations applied to the two interferometer arms led to reaching momentum separations of thousands of photon recoils as was recently shown in Ref.23. In double Bragg diffraction schemes, two counter-propagating optical lattices are implemented in such a way that the recoil is simultaneously transferred in opposite directions, leading to a beam splitter momentum separation of Δp=4nħk21,22. To extend our simulator to this important class of interferometers, we merely have to add a term to the external potential 11 VdoubleBragg(x,t)=2ħΩ(t)(cos2(k(x-nvrt))+cos2(k(x+nvrt))). The procedures of realising a desired 4nħk momentum transfer, as well as mirror or splitter pulses, are identical to the case of single-Bragg diffraction. A simple scan of the Rabi frequency and pulse timings was enough to obtain a full double Bragg interferometer as shown in Fig. 4. The different resulting paths are illustrated in this space-time diagram of the density distribution |ψ(x,t)|2. Similarly to the single-Bragg Mach–Zehnder interferometer, we observe additional parasitic interferometers due to the finite velocity filter of the Bragg pulses after the mirror pulse of the interferometer. Due to a finite fidelity of the initial beam splitter, some atoms remain in the |0ħk⟩ port and recombine at the last beam splitter with the trajectories of the interferometer. In a metrological study, these effects are highly important to quantify. Our simulator gives access to all the quantitative details of such a realisation in a straightforward fashion.Figure 4 (a) Rabi frequency Ω(t) over time of a symmetric double Bragg interferometer according to Eq. (2). (b) The corresponding probability density |ψ(x,t)|2 is plotted for an initial momentum width of σp=0.1 ħk. The timings of the Gaussian splitter and mirror pulses are set to τbs=25 μs and τm=50 μs, respectively. The corresponding Rabi frequencies are found by optimising the desired population transfer. The first π2 pulse corresponds to a 2ħk transfer in two directions, realised by two counter-propagating optical lattices which results in a 4ħk separation between the two interferometer arms. The mirror pulse is a 4ħk Bragg transition with a Rabi frequency of Ω=1.9 ωr such that both arms make a transition from |±2ħk⟩→|∓2ħk⟩. The last recombination pulse now realises a 50 : 50 split of the upper trajectory to |-2ħk⟩ and |0ħk⟩ and the lower trajectory to |+2ħk⟩ and |0ħk⟩. This leads to a final population of 25% in the |±2ħk⟩ ports and 50% in the |0ħk⟩ port. The separation time between the pulses is T=10 ms with a final time of flight after the exit beam splitter of τToF=20 ms. Due to velocity selectivity of the Bragg pulses and a non-ideal fidelity of the initial beam splitter pulse, several parasitic interferometers can be observed. Gravity gradient cancellation for a combined Bragg and Bloch geometry Precision atom interferometry-based inertial sensors are sensitive to higher order terms of the gravitational potential, including gravity gradients. In particular, for atom interferometric tests of Einstein’s equivalence principle (EP), gravity gradients pose a challenge by coupling to the initial conditions, i.e. position and velocity of the two test isotopes39. A finite initial differential position or velocity of the two species can, if unaccounted for, mimic a violation of the EP. By considering a gravitational potential of the form 12 V(x)=-mgx-12mΓx2, where Γ=Γxx is the gravity gradient in the direction normal to the Earth’s surface, the relative phase of a freely falling interferometer can be calculated as40 13 Δϕ=keff|g-aBragg|T2+keffΓ(x0+v0T)T2, with keff=2nk. In Ref.40, it was shown that introducing a variation of the effective wave vector Δkeff=ΓkeffT2/2 at the π pulse can cancel the additional phase shift due to the gravity gradient. This was experimentally demonstrated in Refs.41,42. The same principle applies to the gradiometer configuration of left panel in Fig. 5 where the effect of a gravity gradient is compensated by the application of a wave vector correction. This is reminiscent of another experimental cancellation of the gravity gradient phase shifts41. In our example, we first consider a set of two Mach–Zehnder interferometers vertically separated by h=2 m, realised with 4ħk Bragg transitions where the atoms start with the same initial velocities v0. Choosing a Doppler detuning according to aBragg=g, the gradiometric phase reads 14 Φ=4kΓhT2. By scanning the momentum of the applied π pulse, one can compensate the gradiometric phase. This is observed in our simulations at the analytically predicted value of Δkeff=ΓkeffT2/2 (red dashed curve crossing the zero horizontal line). It is particularly interesting to use our simulator to find this correction phase in the context of more challenging situations, such as a combined scalable Bragg and Bloch Mach–Zehnder interferometer or a symmetric Bloch beam splitter43 where analytic solutions are not easily found. Bloch oscillations can be used to quickly impart a momentum of p=2nBlochħk on the atoms44,45. This adiabatic process can be realised by loading the atoms into a co-moving optical lattice, then accelerating the optical lattice by applying a frequency chirp and finally by unloading the atom from the optical lattice. In our model, this corresponds to the following external potential 15 VBloch(x,t)=2ħΩ(t)cos2(k(x-x(t))) 16 x(0)=0 17 x˙(0)=2nvr 18 x¨(t)=0099%, which is N≤6·104 in this case. In Fig. 6b, the absolute value of the difference between the numerical and the analytical solutions of Δϕ is plotted. For an imbalance of the order of 10%, we observe an agreement at the mrad level. We could point to different possible sources for the relative phase difference. First, the assumptions of the Thomas–Fermi approximation at the heart of the analytical method are not necessarily satisfied here with N≤6·104. Moreover, the analytical treatment neglects all time-dependent effects occurring during the light-atom interactions at the mirror and beam-splitter pulses. These effects, combined with a non-vanishing mean-field would lead to additional phase shifts and shape deformations of the wave functions that are absent from a simple Thomas–Fermi assumption.Figure 6 (a) Mean-field-driven phase shifts as a function of the particle imbalance δN. The analytic solution is given by Eq. (24) (blue line). For the numerical solution (orange dots), we modeled the imbalance by considering a first π2 beam-splitter with a finite fidelity. The Gaussian splitter and mirror pulses have peak Rabi frequencies of Ω=1.0573 ωr with pulse lengths of τbs=25 μs and τm=50 μs, respectively. The transverse trapping frequency is realised with an angular frequency of 2π×50 Hz and the initial trap frequency in which the BEC is condensed is set to 2π×1 Hz with a number of atoms of N=6·104 and a scattering length of as=a0 with a0 being the Bohr radius. (b) Absolute value of the phase difference between the analytic solution and the numerical simulation. Discrepencies with respect to the analytical model stem from the assumptions of the Thomas–Fermi approximation being not satisfied here (see main text). Scalability and numerics Numerical accuracy and precision To gain a better understanding of the numerical accuracy of the simulations, we plot in Fig. 7 the dependency of the phase shift |Δϕ| on the momentum width of the atomic sample σp for a 2ħk Bragg Mach–Zehnder interferometer. We study two realisations which differ in the peak Rabi frequency with corresponding pulse lengths to perform beam splitter and mirror pulses. For both cases we observe a similar characteristic qualitative behaviour of |Δϕ| scaling with σp. Going to smaller initial momentum widths systematically decreases the phase shift until it reaches a plateau of 1×10-7 rad for Ω=1.06ωr and 2.5×10-14 for Ω=0.53ωr. This qualitative behaviour can be explained by considering the effect of parasitic trajectories. In Fig. 2 it is clearly visible that after the time of flight of τToF=2T, there is no clear separation between the parasitic trajectories and the main ports of the Mach–Zehnder interferometer, which leads to interference between them. We choose the integration borders by setting up a symmetric interval around the peak value of each of the ports (see Eq. (3)), ensuring a minimal influence of the parasitic atoms on the interferometric ports. Nevertheless, the interference between the interferometric ports and the parasitic trajectories modifies the measured particle number and therefore also the inferred relative phase. This effect decreases with smaller initial momentum width since less atoms populate the parasitic trajectories overlapping with the main ports, which explains the decrease of relative phase |Δϕ| between σp=0.1ħk to σp=0.05ħk (Ω=1.06 ωr) and σp=0.03ħk (Ω=0.53 ωr). Another important contribution to the relative phase |Δϕ| which is not captured by Feynman’s path integral approach25,26 is the diffraction phase, which is fundamentally linked to the excitation of non-resonant momentum states37,38. Using smaller Rabi frequencies leads to a reduced population of non-resonant momentum states (after a beam splitter pulse we find P(-2ħk,Ω=1.06ωr)+P(4ħk,Ω=1.06ωr)=1.3×10-7 and P(-2ħk,Ω=0.53ωr)+P(4ħk,Ω=0.53ωr)=1.9×10-18) and therefore to a reduced diffraction phase which explains that operating a Mach–Zehnder interferometer at Ω=0.53 ωr leads to a much smaller residual phase shift than at Ω=1.06 ωr. These results indicate that our simulator reaches at least a relative phase accuracy at the level of 2.5×10-14 rad. It is worth mentioning, that the numerical parameters chosen to reach this performance are very accessible on modestly powerful desktop computers. The computation took τCPUtime=12.7 s on an Intel Xeon X5670 processor using four cores (2.93 GHz, 12 MB last level cache). Modeling precision atom interferometry problems with this method is therefore a practical, flexible and highly accurate approach. Using improved resolutions in position and time or higher order operator splitting schemes55 leads to even better numerical precision and accuracy.Figure 7 Phase shift of a 2ħk Mach–Zehnder interferometer as a function of the initial momentum width of an atomic sample. We evaluate the phase shift for pulse lengths of τbs=25 μs and τm=50 μs (blue dots) and for τbs=50 μs and τm=100 μs (red dots), using peak Rabi frequencies of Ω25μs=1.06 ωr and Ω50μs=0.53 ωr. The dashed lines are a guide to the eye. We find a systematic decreasing behaviour of the relative phase offset |Δϕ| starting from an initial momentum width of σp=0.1ħk (far right) to σp=0.05ħk (red dots) and σp=0.03ħk (blue dots). Reaching those critical initial momentum widths both curves show fixed relative phase offsets |Δϕ|, which in the case of the interferometer with smaller Rabi frequency of Ω=0.53ωr (red dots) reaches a value of 2.5×10-14 rad (see text). The numerical simulations were performed with 65,536 grid points, an interaction time step of dtint=1 μs and a free evolution time step of dtfree=10 μs, leading to a computational time of τCPUtime=12.69 s on four cores of an Intel Xeon X5670 processor with 2.93 GHz frequency and 12 MB of cache. Numerical convergence To analyse the numerical convergence as well as the connected numerical precision and accuracy of the split-operator method applied to the previously presented systems, we simulate three different interferometer settings for different space and time grids. We consider a 2ħk Mach–Zehnder interferometer, a 2ħk Mach–Zehnder interferometer in a waveguide and as a last example a (2+2)ħk Bragg+Bloch Mach–Zehnder interferometer in order to quantify the necessary resolution and grid sizes that can be derived from these results. In Figs. 8 and 9 we extract the relative phase over successively decreasing spatial and temporal steps and we compare them to simulations with sufficiently fine spatial and temporal resolutions by plotting the absolute value of the difference of the relative phases, i.e. |Δϕ-Δϕ(dt=4ns)| and |Δϕ-Δϕ(dx=0.01λ)|. For the presented cases we compare to dt=4 ns and dx=0.01λ. The choice of the steps fulfilling the necessary resolution is motivated in the following, relating them to the physical quantities of the problem (optical lattice and atomic wavepacket).Figure 8 Numerical convergence analysis of three different interferometer realisations given by a 2ħk Mach–Zehnder interferometer (blue and orange dots corresponding to the 0ħk and 2ħk ports), a 2ħk Mach–Zehnder interferometer in a waveguide (green dots) and a (2+2)ħk Bragg+Bloch Mach–Zehnder interferometer (red dots). We analyse the numerical convergence behaviour when changing the position step dx expressed in units of λ=780 nm of the numerical simulation using the third-order split-operator method with temporal steps of dtint=1μs and dtfree=10μs. The Gaussian splitter and mirror pulses have peak Rabi frequencies of Ω=1.0573 ωr with pulse lengths of τbs=25 μs and τm=50 μs, respectively. The Bloch sequence is implemented with an adiabatic loading time of τload=0.5 ms, a frequency chirp time of τchirp=0.5 ms during which the momentum transfer occurs and an adiabatic unloading time of τunload=0.5 ms. The Rabi frequency of the Bloch lattice is Ω=4 ωr. In the case of the 2ħk Mach–Zehnder and (2+2)ħk Bragg+Bloch Mach–Zehnder interferometers, we use an initial momentum width of the atomic sample of σp=0.01 ħk. In the case of the 2ħk Mach–Zehnder interferometer in a waveguide the transverse trapping frequency is realised with an angular frequency of 2π×50 Hz and the initial trap frequency in which the BEC is condensed is set to 2π×1 Hz with a number of atoms of N=6·104 and a scattering length of as=a0 with a0 being the Bohr radius. Figure 9 Numerical convergence analysis of three different interferometer realisations given by a 2ħk Mach–Zehnder interferometer (blue and orange dots corresponding to the 0ħk and 2ħk ports), a 2ħk Mach–Zehnder interferometer in a waveguide (green dots) and a (2+2)ħk Bragg+Bloch Mach–Zehnder interferometer (red dots). We analyse the numerical convergence behaviour when changing the temporal step dt of the numerical simulation using the third-order split-operator method with a spatial step of dx=4×10-2λ. The same parameters as Fig. 8 are used. The fast Fourier transform (FFT) efficiently switches between momentum and position representations to apply kinetic and potential propagators. The corresponding position and momentum grids are defined by the number of grid points Ngrid and the total size of the position grid Δx as 27 dx=ΔxNgrid-1,dp=2πħΔxandΔp=2πħdx, where dp and dx are the steps in momentum and position, respectively, and Δp the total size of the momentum grid. To resolve a finite momentum width of the atomic cloud we are restricted to (λ=780 nm) 28 dp≪ħk⇔Δx≫λ, which sets a bound to the size of the position grid. Finally, Δx has to be chosen according to the maximal separation of the atomic clouds Δxsep. With this we find 29 Δx≳Δxsep≫λ. To include all momentum orders necessary to simulate the considered atom interferometric sequences, we are naturally bound by 30 Δp=2πħdx⇔Δpħk=λdx. Hence, we find that 31 Δpħk=λdx≫1⇔dx≪λ, which is the natural condition imposed by the necessity of resolving the atomic dynamics in the optical lattice nodes and anti-nodes of the Bragg and Bloch beams. The absence of data points of the (2+2)ħk Bragg+Bloch Mach–Zehnder interferometer in Fig. 8 shows the limits given by Eq. (31). Choosing position steps at dx=0.07λ leads to a maximal computed momentum of ±7.1 ħk, which results in the impossibility to find probabilities at 8ħk. For this specific interferometer, however, it is critical to resolve those momenta, since they are residually populated during the atom-light interaction processes. Imposing that the position step is roughly one order of magnitude smaller than the wavelength (dx≲0.06 λ) results in a reasonable momentum truncation and resolution of the light potential and therefore in the convergence of the numerical routine. Additionally, we can observe that reaching spatial resolutions of dx=0.01λ, one finds, for all three studied cases, satisfying numerical accuracy and precision which in the worst case is approximately 1×10-9 rad for the (2+2)ħk Bragg+Bloch Mach–Zehnder interferometer. The typical time scales we need to consider are set on the one hand by the velocities of the optical lattice beams and the atomic cloud, and on the other hand by the duration of the atom-light interaction τ. The beams, as well as the atomic cloud, move with velocities which are proportional to the recoil velocity vr. Given that we want to drive Bragg processes of the order of n, we find the following bound on the time step dt 32 dt≪λnvr≈100μsn. The typical duration of a pulse in the quasi-Bragg regime is typically adapted to the momentum width due to the spectral properties of the finite pulse. Here, we assume a lower bound of τ=10μs, which leads to dt<τ. It is worth noting that this time step is only necessary during the atom-light interaction. One can simulate the free evolution between the pulses with a much larger time step (without external and interaction potentials a single step suffices) or using scaling techniques34,56–59. Figure 9 shows that depending on the specific form of the simulated light potential or the consideration of two-particle interactions, we observe a characteristic convergence behaviour which we directly connect to the propagation error of the split-operator routine55,60. For the 2ħk Mach–Zehnder interferometer one can observe the already found level of convergence around 1×10-13 rad (see Fig. 7). Interestingly, the 2ħk port reaches a level of 1×10-13 rad at a time step of dt≈0.1 μs, whereas the 0ħk port already convergences to that level at a time step of dt=1 μs. Note, that the diffraction phase and therefore the relative phase in the slow port vanishes but reaches a finite value of approximately 1×10-7 rad in the fast port37,38 (see Fig. 7). The next analysed case is the 2ħk Mach–Zehnder interferometer in a waveguide whereby introducing the non-linear interaction term (see Eq. (20)) one can observe a more demanding convergence behaviour, leading to an initial precision of 1 μrad at dt=1 μs, which converges to 1×10-10 rad at a time step of dt≈10-8 s. Introducing additional Bloch oscillations shifts the convergence curve again by two orders of magnitude at a minimal time step of dt=1μs and reaches a level of approximately 1 μrad at dt=10-8 s. Note that the precision and accuracy of the split-operator algorithm strongly depends on the potential that is simulated60 and that in the case of a Bloch oscillation the optical potential linearly changes its velocity, where in the case of a Bragg transition the optical lattice moves with a constant velocity during the atom-light interaction process. Additionally, the atom-light interaction time of a Bloch oscillation is typically one order of magnitude larger compared with a Bragg transition which explains the need for finer temporal grids in order to achieve reasonable precision and accuracy. Time complexity analysis In this section, we compare the time complexity behaviour of the commonly-used method of treating the beam splitter and mirror dynamics given by the ODE approach with the PDE formulation presented in this paper, based on a position-space approach to the Schrödinger equation. To assess the time complexity of the ODE treatment, we re-derive it from the Schrödinger equation 33 iħ∂tψ(x,t)=-ħ22m∂2∂x2+2ħΩcos2(kx)ψ(x,t). We decompose the wave function in a momentum state basis as done in Refs.2–4 34 ψ(x,t)=∑j,δgj+δ(t)ei(j+δ)kx, where j denotes the momentum orders considered and δ the discrete representation of momenta in the interval [kj-k/2,kj+k/2] which captures the finite momentum width of the atoms around each momentum class kj. Making the two exponential terms appear in cos2(kx), one obtains 35 iħg˙j+δ(t)=ħ((j+δ)2ωr+Ω)gj+δ(t)+ħΩ2(gj+δ+2(t)+gj+δ-2(t)), which is a set of Neq coupled ordinary differential equations. This number Neq of equations to solve is equal to NjNδ, set by the truncation condition restricting the solution space to Nj momentum classes, each discretised in Nδ sub-components. Using standard solvers for such systems as Runge-Kutta, multistep or the Bulirsch–Stoer methods61, we generally need to evaluate the right hand side of the system of equations over several iterations. With Neq differential equations, where each one has only two coupling terms, one finds a time complexity of O(Neq). In Fig. 10 we present a visualisation of different possible momentum couplings starting from 0ħk to other momentum components. One starts with only three momentum states and two coupling elements in Fig. 10a corresponding to a vanishing momentum width (δ=0 in Eq. (35)). If the momentum width is introduced (Fig. 10b), the number of coupling elements increases since every δ sub-momentum class of j is connected to the same sub-momentum class of j-2 and j+2 as suggested by Eq. (35). In order to reduce visual complexity we are only showing couplings that start from the 0ħk wavepacket, while dropping coupling elements starting from ±2ħk. We also fixed the number of momentum states per integer momentum class kj to three, which in a realistic example is at least an order of magnitude larger.Figure 10 Visualisation of different momentum couplings from the 0ħk momentum wavepacket corresponding to different levels of complexity. (a) Zero momentum width and two coupling elements from 0ħk to ±2ħk. (b) Finite momentum widths with coupling elements for each momentum component in the 0ħk wavepacket to the corresponding momentum component in the ±2ħk wavepackets with a momentum difference for each transition of Δp=2ħk. The different colours indicate the separate momentum subspaces in which transitions can occur. (c) Finite momentum widths with multiple possible coupling elements from the 0ħk wavepacket to the ±2ħk wavepacket with a broadening of the possible momentum difference Δp. (d) Finite momentum widths with higher order coupling elements from the 0ħk wavepacket to momentum components of the ±4ħk,±6ħk,⋯ wavepackets. In a next step, the coupling terms are calculated for more general potentials with time and space-dependent Rabi frequencies Ω(x,t) and wave vectors k(x, t). For this purpose, the momentum-space representation of the Schrödinger equation is more appropriate and can be written for the Fourier transform of the atomic wave function g(p, t) 36 iħg˙(p,t)=p22mg(p,t)+V(p,t)∗g(p,t), where 37 V(p,t)∗g(p,t):=∫dxe-ipħx2πħV(x,t)ψ(x,t). Expressing the wave function in momentum space gives 38 V(p)∗g(p,t)=12πħ∫dp′∫dxeix(p′-p)ħV(x,t)⏟=:F(p,p′,t)∈Cg(p′,t). Discretising p→(j+δ)ħk and p′→(l+γ)ħk, one finds 39 V(p)∗g(p,t)≈12πħ∑l,γF((j+δ)ħk,(l+γ)ħk,t)gl+γ(t), where l and γ span the same indices ensembles as j and δ. The new equations to solve read 40 iħg˙j+δ(t)≈((j+δ)ħk)22mgj+δ(t)+12πħ∑l,γF((j+δ)ħk,(l+γ)ħk,t)gl+γ(t), which yields the necessary momentum couplings for an arbitrary potential V(x, t). In the worst case, the sum in Eq. (40) runs over Neq nonzero entries (NlNγ=NjNδ=Neq) which leads to a time complexity of O(Neq2). This, however, is an extreme example that contrasts with commonly operated precision interferometric experiments since it would correspond to white light with speckle noise. Realistic scenarios rather involve time-dependent potentials with a smaller number of momentum couplings, i.e. Neq≫#couplingterms≳2 as would be the case in Fig. 10c. To evaluate the momentum couplings, it is necessary to calculate the integral F(p,p′,t) at each time step using the FFT, which leads to a final time complexity class for solving the ODE of O(NeqlogNeq). The next important generalisation aims to include the effect of the two-body collisions analysed in the mean-field approximation, i.e. Hint=g1D|ψ(x,t)|2. In this case, the equation describing the dynamics of the system and the couplings can be written as 41 iħg˙j+δ(t)=ħ((j+δ)2ωr+Ω)gj+δ(t)+ħΩ2(gj+δ+2(t)+gj+δ-2(t)) 42 +g1D∑l,γ,o,νgl+γ∗(t)g2o-l+2ν-γ(t)gj+δ(t), where ν and o are running indices over the same values as l and γ. One ends up with Neq differential equations where each has more than Neq2 coupling terms, and finds a time complexity class of O(Neq3). This shows the growth in numerical operations of the ODE treatment as reflected by the number of couplings in Fig. 10d. We analyse now the time complexity class for the PDE approach, using the split-operator method24. Based on the application of the FFT, it is known that the complexity class of this method is scaling as O(NgridlogNgrid), where Ngrid is the number of grid points in the position or momentum representations. Since the discretisation of the problem for the ODE and PDE (Schrödinger equation) approaches is roughly the same (Neq≈Ngrid), a direct comparison between the two treatments is possible. The time complexity analysis is summarised in Table 1. It shows that the standard ODE approach is only better suited in the case of ideal light plane waves. In every realistic case where the light field is allowed to be spatially inhomogeneous, the amount of couplings increases and it is preferable to employ the PDE approach with a scaling of O(NgridlogNgrid), independently of any further complexity to be modelled.Table 1 Comparison of the different time complexity classes of the commonly-used ODE treatment with the position-space approach developed in this work (PDE-based). Including more and more realistic features of the atom-light system leads to an ODE time complexity unfavourably scaling. The PDE formulation, however, routinely scales with O(NgridlogNgrid). Feature Numerical operations ODEs Numerical operations PDE Infinitely sharp momentum widths (δ→1) O(Nj) – Finite momentum width and ideal light potential O(Ngrid) O(NgridlogNgrid) Inhomogeneous light potential O(NgridlogNgrid)→O(Ngrid2) O(NgridlogNgrid) Mean-field interaction O(Ngrid3) O(NgridlogNgrid) Conclusion In this paper, we have shown that the position-space representation of light-pulse beam splitters is quite powerful for tackling realistic beam profiles in interaction with cold atom ensembles. It was successfully applied across several relevant regimes, geometries and applications. We showed its particular fitness in treating metrologically-relevant investigations based on atomic sensors. Its high numerical precision and scalability makes it a flexible tool of choice to design or interpret atom interferometric measurements without having to change the theoretical framework for every beam geometry, dimensionality, pulse length or atomic ensemble property. We anticipate the possibility of accurately implementing this approach to analyse important systematic effects in the field of precision light-pulse matter-wave interferometry such as the ones related to wavefront aberrations, large momentum transfer and inhomogeneity and fluctuations of the Rabi pulses. Finally, we would like to highlight the possibility to generalise this method to Raman or 1-photon transitions if we account for the internal state degree of freedom change during the diffraction. Publisher's note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Acknowledgements We thank Sven Abend, Sina Loriani, Christian Schubert for insightful discussions and Eric Charron for carefully reading the manuscript. N.G. wishes to thank Alexander D. Cronin for fruitful indications about previous publications related to our current work. We also thank Matthew Glaysher and Heather Glaysher for proofreading the manuscript. This work was funded by the Deutsche Forschungsgemeinschaft (German Research Foundation) under Germany’s Excellence Strategy (EXC-2123 QuantumFrontiers Grants No. 390837967) and through CRC 1227 (DQ-mat) within Projects No. A05 and No. B07, the Verein Deutscher Ingenieure (VDI) with funds provided by the German Federal Ministry of Education and Research (BMBF) under Grant No. VDI 13N14838 (TAIOL). We furthermore acknowledge financial support from “Niedersächsisches Vorab” through “Förderung von Wissenschaft und Technik in Forschung und Lehre” for the initial funding of research in the new DLR-SI Institute and the “Quantum- and Nano Metrology (QUANOMET)” initiative within the project QT3. Further support was possible by the German Space Agency (DLR) with funds provided by the Federal Ministry of Economic Affairs and Energy (BMWi) due to an enactment of the German Bundestag under grant No. 50WM1861 (CAL) and 50WM2060 (CARIOQA). Author contributions F.F. implemented the numerical model, performed all numerical simulations, and prepared the figures. J.-N.S. and H.A. helped with the interpretation of the results. H.A. and N.G. designed the research goals and directions. E.M.R. and K.H. contributed to scientific discussions. F.F. and N.G. wrote the manuscript. S.S. critically reviewed the manuscript. All authors reviewed the results and the paper and approved the final version of the manuscript. Funding Open Access funding enabled and organized by Projekt DEAL. Competing interests The authors declare no competing interests. ==== Refs References 1. Berman PR Atom Interferometry 1997 London Academic Press 2. Meystre P Atom Optics 2001 New York Springer Science & Business Media 3. Müller H Chiow S-W Chu S Atom-wave diffraction between the Raman-Nath and the Bragg regime: Effective Rabi frequency, losses, and phase shifts Phys. Rev. A 2008 77 023609 10.1103/PhysRevA.77.023609 4. Szigeti SS Debs JE Hope JJ Robins NP Close JD Why momentum width matters for atom interferometry with Bragg pulses N. J. Phys. 2012 14 023009 10.1088/1367-2630/14/2/023009 5. Siemß J-N Analytic theory for Bragg atom interferometry based on the adiabatic theorem Phys. Rev. A 2020 102 033709 10.1103/PhysRevA.102.033709 6. Tannor DJ Introduction to Quantum Mechanics 2018 Mill Valley University Science Books 7. Simula TP Muradyan A Mølmer K Atomic diffraction in counterpropagating Gaussian pulses of laser light Phys. Rev. A 2007 76 063619 10.1103/PhysRevA.76.063619 8. Stickney JA Kafle RP Anderson DZ Zozulya AA Theoretical analysis of a single- and double-reflection atom interferometer in a weakly confining magnetic trap Phys. Rev. A 2008 77 043604 10.1103/PhysRevA.77.043604 9. Liu C-N Krishna GG Umetsu M Watanabe S Numerical investigation of contrast degradation of Bose–Einstein-condensate interferometers Phys. Rev. A 2009 79 013606 10.1103/PhysRevA.79.013606 10. Stuckenberg, F., Marojević, Z. & Rosskamp, J. H. Atus2. https://github.com/GPNUM/atus2/tree/master/doc. Accessed 12 Dec 2019 11. Blakie PB Ballagh RJ Mean-field treatment of Bragg scattering from a Bose–Einstein condensate J. Phys. B Atom. Mol. Opt. Phys. 2000 33 3961 3982 10.1088/0953-4075/33/19/311 12. Steck, D. A. Rubidium 87 D Line Data. http://steck.us/alkalidata (Revision 2.2.1, 21 November 2019). 13. Louchet-Chauvet A The influence of transverse motion within an atomic gravimeter N. J. Phys. 2011 13 065025 10.1088/1367-2630/13/6/065025 14. Schkolnik V Leykauf B Hauth M Freier C Peters A The effect of wavefront aberrations in atom interferometry Appl. Phys. B 2015 120 311 316 10.1007/s00340-015-6138-5 15. Zhou M-K Luo Q Chen L-L Duan X-C Hu Z-K Observing the effect of wave-front aberrations in an atom interferometer by modulating the diameter of Raman beams Phys. Rev. A 2016 93 043610 10.1103/PhysRevA.93.043610 16. Bade S Djadaojee L Andia M Cladé P Guellati-Khelifa S Observation of extra photon recoil in a distorted optical field Phys. Rev. Lett. 2018 121 073603 10.1103/PhysRevLett.121.073603 30169104 17. Wicht A Hensley JM Sarajlic E Chu S A preliminary measurement of the fine structure constant based on atom interferometry Phys. Scr. 2002 T102 82 10.1238/physica.topical.102a00082 18. Wicht A Sarajlic E Hensley JM Chu S Phase shifts in precision atom interferometry due to the localization of atoms and optical fields Phys. Rev. A 2005 72 023602 10.1103/PhysRevA.72.023602 19. Cladé P Precise measurement of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h / {m}_{\rm Rb}$$\end{document} h / m Rb using Bloch oscillations in a vertical optical lattice: Determination of the fine-structure constant Phys. Rev. A 2006 74 052109 10.1103/PhysRevA.74.052109 20. Küber, J., Schmaltz, F. & Birkl, G. Experimental realization of double Bragg diffraction: robust beamsplitters, mirrors, and interferometers for Bose–Einstein condensates (2016). arXiv:1603.08826. 21. Ahlers H Double Bragg interferometry Phys. Rev. Lett. 2016 116 173601 10.1103/PhysRevLett.116.173601 27176520 22. Giese E Roura A Tackmann G Rasel EM Schleich WP Double Bragg diffraction: A tool for atom optics Phys. Rev. A 2013 88 053608 10.1103/PhysRevA.88.053608 23. Gebbe, M. et al. Twin-lattice atom interferometry (2019). arXiv:1907.08416. 24. Feit M Fleck J Steiger A Solution of the Schrödinger equation by a spectral method J. Comput. Phys. 1982 47 412 433 10.1016/0021-9991(82)90091-2 25. Hogan, J., Johnson, D. & Kasevich, M. Light-pulse atom interferometry. Proc. Int. School Phys. Enrico Fermi168. 10.3254/978-1-58603-990-5-411 (2008). 26. Storey P Cohen-Tannoudji C The Feynman path integral approach to atomic interferometry. A tutorial J. Phys. 1994 II 4 1999 2027 10.1051/jp2:1994103 27. Gupta S Dieckmann K Hadzibabic Z Pritchard DE Contrast interferometry using Bose–Einstein condensates to measure \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h/m$$\end{document} h / m and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document} α Phys. Rev. Lett. 2002 89 140401 10.1103/PhysRevLett.89.140401 12366031 28. Keller C Adiabatic following in standing-wave diffraction of atoms Appl. Phys. B 1999 69 303 309 10.1007/s003400050810 29. Giltner DM McGowan RW Lee SA Theoretical and experimental study of the Bragg scattering of atoms from a standing light wave Phys. Rev. A 1995 52 3966 3972 10.1103/PhysRevA.52.3966 9912708 30. Kovachy T Chiow S-W Kasevich MA Adiabatic-rapid-passage multiphoton Bragg atom optics Phys. Rev. A 2012 86 011606 10.1103/PhysRevA.86.011606 31. Chu S Bjorkholm JE Ashkin A Gordon JP Hollberg LW Proposal for optically cooling atoms to temperatures of the order of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$10^{-6}$$\end{document} 10 - 6 K Opt. Lett. 1986 11 73 75 10.1364/OL.11.000073 19730537 32. Ammann H Christensen N Delta kick cooling: a new method for cooling atoms Phys. Rev. Lett. 1997 78 2088 2091 10.1103/PhysRevLett.78.2088 33. Morinaga M Bouchoule I Karam J-C Salomon C Manipulation of motional quantum states of neutral atoms Phys. Rev. Lett. 1999 83 4037 4040 10.1103/PhysRevLett.83.4037 34. Müntinga H Interferometry with Bose–Einstein condensates in microgravity Phys. Rev. Lett. 2013 110 093602 10.1103/PhysRevLett.110.093602 23496709 35. Kovachy T Matter wave lensing to picokelvin temperatures Phys. Rev. Lett. 2015 114 143004 10.1103/PhysRevLett.114.143004 25910118 36. Corgier R Fast manipulation of Bose–Einstein condensates with an atom chip N. J. Phys. 2018 20 055002 10.1088/1367-2630/aabdfc 37. Büchner M Diffraction phases in atom interferometers Phys. Rev. A 2003 68 013607 10.1103/PhysRevA.68.013607 38. Estey B Yu C Müller H Kuan P-C Lan S-Y High-resolution atom interferometers with suppressed diffraction phases Phys. Rev. Lett. 2015 115 083002 10.1103/PhysRevLett.115.083002 26340186 39. Aguilera DN STE-QUEST—test of the universality of free fall using cold atom interferometry Class. Quantum Gravity 2014 31 115010 10.1088/0264-9381/31/11/115010 40. Roura A Circumventing Heisenberg’s uncertainty principle in atom interferometry tests of the equivalence principle Phys. Rev. Lett. 2017 118 160401 10.1103/PhysRevLett.118.160401 28474953 41. D’Amico G Canceling the gravity gradient phase shift in atom interferometry Phys. Rev. Lett. 2017 119 253201 10.1103/PhysRevLett.119.253201 29303327 42. Overstreet C Effective inertial frame in an atom interferometric test of the equivalence principle Phys. Rev. Lett. 2018 120 183604 10.1103/PhysRevLett.120.183604 29775337 43. Pagel, Z. et al. Symmetric Bloch oscillations of matter waves (2019). arXiv:1907.05994. 44. Ben Dahan M Peik E Reichel J Castin Y Salomon C Bloch oscillations of atoms in an optical potential Phys. Rev. Lett. 1996 76 4508 4511 10.1103/PhysRevLett.76.4508 10061309 45. Wilkinson SR Bharucha CF Madison KW Niu Q Raizen MG Observation of atomic Wannier–Stark ladders in an accelerating optical potential Phys. Rev. Lett. 1996 76 4512 4515 10.1103/PhysRevLett.76.4512 10061310 46. Ketterle W Nobel lecture: When atoms behave as waves: Bose–Einstein condensation and the atom laser Rev. Mod. Phys. 2002 74 1131 1151 10.1103/RevModPhys.74.1131 47. Cornell EA Wieman CE Nobel lecture: Bose–Einstein condensation in a dilute gas, the first 70 years and some recent experiments Rev. Mod. Phys. 2002 74 875 893 10.1103/RevModPhys.74.875 48. Debs JE Cold-atom gravimetry with a Bose–Einstein condensate Phys. Rev. A 2011 84 033610 10.1103/PhysRevA.84.033610 49. Sugarbaker A Dickerson SM Hogan JM Johnson DMS Kasevich MA Enhanced atom interferometer readout through the application of phase shear Phys. Rev. Lett. 2013 111 113002 10.1103/PhysRevLett.111.113002 24074082 50. Pethick C Smith H Bose–Einstein Condensation in Dilute Gases 2002 Cambridge Cambridge University Press 51. Chin C Grimm R Julienne P Tiesinga E Feshbach resonances in ultracold gases Rev. Mod. Phys. 2010 82 1225 1286 10.1103/RevModPhys.82.1225 52. Olshanii M Atomic scattering in the presence of an external confinement and a gas of impenetrable bosons Phys. Rev. Lett. 1998 81 938 941 10.1103/PhysRevLett.81.938 53. Salasnich L Parola A Reatto L Effective wave equations for the dynamics of cigar-shaped and disk-shaped Bose condensates Phys. Rev. A 2002 65 043614 10.1103/PhysRevA.65.043614 54. Watanabe S Aizawa S Yamakoshi T Contrast oscillations of the Bose–Einstein-condensation-based atomic interferometer Phys. Rev. A 2012 85 043621 10.1103/PhysRevA.85.043621 55. Javanainen J Ruostekoski J Symbolic calculation in development of algorithms: split-step methods for the Gross–Pitaevskii equation J. Phys. A Math. Gen. 2006 39 L179 L184 10.1088/0305-4470/39/12/l02 56. Castin Y Dum R Bose–Einstein condensates in time dependent traps Phys. Rev. Lett. 1996 77 5315 5319 10.1103/PhysRevLett.77.5315 10062773 57. Kagan Y Surkov EL Shlyapnikov GV Evolution of a Bose gas in anisotropic time-dependent traps Phys. Rev. A 1997 55 R18 R21 10.1103/PhysRevA.55.R18 58. van Zoest T Bose–Einstein condensation in microgravity Science 2010 328 1540 1543 10.1126/science.1189164 20558713 59. Meister M Arimondo E Lin CC Yelin SF Efficient description of Bose–Einstein condensates in time-dependent rotating traps, Chapter 6 Advances In Atomic, Molecular, and Optical Physics, Advances in Atomic, Molecular, and Optical Physics 2017 New York Academic Press 375 438 60. Bandrauk AD Shen H Improved exponential split operator method for solving the time-dependent Schrödinger equation Chem. Phys. Lett. 1991 176 428 432 10.1016/0009-2614(91)90232-X 61. Press WH Teukolsky SA Vetterling WT Flannery BP Numerical Recipes in Fortran 77: The Art of Scientific Computing 1992 Cambridge Cambridge University Press