Demonstration of the temporal matter-wave Talbot effect for trapped matter waves

We demonstrate the temporal Talbot effect for trapped matter waves using ultracold atoms in an optical lattice. We investigate the phase evolution of an array of essentially non-interacting matter waves and observe matter-wave collapse and revival in the form of a Talbot interference pattern. By using long expansion times, we image momentum space with sub-recoil resolution, allowing us to observe fractional Talbot fringes up to 10th order.


Introduction
Interference of matter waves is one of the basic ingredients of modern quantum physics. It has proven to be a very rich phenomenon and has found many applications in fundamental physics as well as in metrology [1] since the first electron diffraction experiments by Davisson and Germer [2]. Matter-wave optics has now developed into a thriving subfield of quantum physics. Many key experiments from classical optics have found their counterpart with matter waves, for example the realization of Young's double slit experiment with electrons [3], the implementation of a Mach-Zehnder-type interferometer with neutrons [4], or, more recently, the observation of Poisson's spot with molecules [5]. The creation of Bose-Einstein condensates (BEC) in 1995 [6,7] opened the door to many more exciting experiments with matter waves, to a large extent in the same way as the laser did in the case of classical light waves.
One remarkable phenomenon in classical optics is the Talbot effect, the self-imaging of a periodic structure in near field diffraction. The effect was first observed by Talbot in 1836 [8] and was later explained in the context of wave optics by Rayleigh in 1881 [9]. When light with a wavelength λ illuminates a material grating with period d, the intensity pattern of the light passing through the grating reproduces the structure of the grating at distances behind the grating equal to odd multiples of the so-called Talbot length L Talbot = d 2 /λ. At even multiples of the Talbot length the intensity pattern again reproduces the structure of the grating, but shifted laterally in space by half of the grating period. In between these recurrences, at rational fractions n/m of L Talbot (with n, m coprime), patterns with smaller period d/m are formed. This effect is known as the fractional Talbot effect. A necessary requirement for the appearance of the Talbot effect and its fractional variation is the validity of the paraxial approximation [10]. Crucial to the Talbot effect is the fact that the accumulated phase differences of the propagating waves behind the grating show a quadratic dependence on lateral distance or grating slit index.
The first observations of the atomic matter-wave Talbot effect [11,12] were based on setups comprising an atomic beam and two material gratings, where the second grating acted as a mask used for detection purposes. The demonstration of the fractional Talbot effect with atomic matter waves used the fact that the interference fringes could be recorded directly by using a spatially resolving detector [13]. The Talbot effect can also be demonstrated with spatially incoherent wave sources by using an additional first grating to create spatial coherence according to Lau [14]. In this way, an interferometer is formed that is made of two or even three gratings. Such Talbot-Lau interferometers [15] are now an important tool in atomic and molecular interferometry [1,16,17]. In the context of macroscopic matter waves, i.e. atomic BECs, the Talbot effect has been observed in the time domain by using pulsed phase gratings formed by standing laser waves [18]. During expansion after release from the trap the BEC was exposed to two short grating pulses separated by a variable time delay and the momentum distribution was measured. At a specific delay, this distribution was observed to rephase to the initial one. In essence, the quadratic dispersion relation of freely propagating, non-interacting matter waves resulted in a quadratic phase evolution for the diffracted momentum states and hence to a temporal version of the Talbot effect. Intriguingly, the Talbot effect is also present for interacting matter waves, as we could show in our previous work [19]. The momentum distribution of a trapped array of decoupled two-dimensional BECs proved to exhibit a regular, time-varying interference pattern. In this case, the quadratic phase evolution was driven by the local mean-field interaction that had a quadratic spatial dependence reflecting the parabolic shape of the initial density distribution.
In the present work, we report on the demonstration of the temporal Talbot effect using trapped, non-interacting matter waves. Here, the Talbot effect is not driven by interactions but by the (weak) external harmonic dipole-trap confinement, leading to a characteristic quadratic phase evolution. For our measurements we use as before an array of pancake-shaped, two-dimensional BECs in a one-dimensional optical lattice [19]. The optical lattice takes on the role of the grating. Cancelling the effect of interactions in the vicinity of a Feshbach resonance and decoupling the individual BECs by means of a gravitational tilt initiates long-lived Bloch oscillations (BO) in momentum space [20]. These are quickly superimposed by a Talbot-type interference pattern in the presence of the external confinement. The pattern can be directly connected to the (fractional) Talbot effect. In particular, after specific hold times that are multiples of the Talbot time, the time-analogue to the Talbot length, a rephasing of the momentum distribution can be observed.

Preparation of the initial sample
We first produce an essentially pure BEC of Cs atoms (no detectable non-condensed fraction) by largely following the procedure detailed in Ref. [21,22]. The atoms are in the lowest hyperfine sublevel F = 3, m F = 3 trapped in a crossed optical dipole trap and initially levitated against gravity by a magnetic gradient field. As usual, F is the atomic angular momentum quantum number, and m F its projection on the magnetic field axis. For the present experiments, the atom number is set to typically 6×10 4 atoms. The trap frequencies in the crossed dipole trap are chosen to be ω x = 2π × 21.7(3) Hz, ω y = 2π × 26.7(3) Hz, and ω z = 2π × 26.9(3) Hz. The confinement along the vertical axis (z) and the two horizontal axes (x, y) is controlled by two horizontally propagating dipole trap beams with beam waists of 46 µm and 144 µm and one vertically propagating dipole trap beam with a beam waist of 123 µm. The atomic scattering length a s and therefore the strength of interactions in the BEC can be tuned via a magnetic offset field B in a range between a s = 0 a 0 and a s = 1000 a 0 by setting B to values between approximately 17 and 46 G using a magnetically induced Feshbach resonance [23] as illustrated in figure 1(a). Here, a 0 is Bohr's radius. For the initial preparation of the sample, we set a s to positive values, typically between 100 a 0 and 210 a 0 . Later, a s is set to zero as discussed below. We gently load the condensed atomic sample into a vertical standing wave as illustrated in figure 1(b) by exponentially ramping up the power in the standing wave over the course of about 1000 ms. The standing wave is generated by a retro-reflected laser beam at a wavelength of λ = 1064.48(5) nm with a 1/e-waist of about 350 µm. We are able to achieve well depths of up to 40 E R , where E R = 2 k 2 /(2m) =h 2 /(2mλ 2 ) = k B × 64 nK is the atomic photon recoil energy. Here k = 2π/λ, m denotes the mass of the Cs atom, h is Planck's constant, and k B is Boltzmann's constant. The lattice light as well as the light for the dipole trap beams is derived from a single-frequency, narrow-band, highly-stable Nd:YAG laser that seeds a home-built fibre amplifier [24]. The maximum output power is up to 20 W without spectral degradation. The powers in all light beams are controlled by acousto-optical intensity modulators and intensity stabilization servos.

Phase evolution and the Talbot effect
Our system, the BEC loaded into a 1D optical lattice with spacing d = λ/2, can be modelled by a discrete nonlinear equation (DNLE) in one dimension [25], as discussed in our earlier work [19]. In brief, this equation can be obtained by expanding the condensate wave function from the Gross-Pitaevskii equation, Ψ, in a basis of wave functions Here, z is the coordinate along the (vertical) lattice direction, r ⊥ is the transverse coordinate, and c j (t) are time-dependent complex amplitudes. The atoms are restricted to move in the lowest Bloch band and we can write Ψ j (r ⊥ , z) = w 0 (z) are the lowest-band Wannier functions localized at the j-th site and Φ ⊥ (ρ j , r ⊥ ) is a radial wave function depending on the peak density ρ j at each site [25]. By inserting this form into the Gross-Pitaevskii equation and integrating out the radial direction, the DNLE is obtained, Here, J/h is the tunnelling rate between neighbouring lattice sites, V j = Fd j + V trap j describes the combination of a linear potential with force F and an external, possibly time-varying trapping potential V trap j , and E int j (c j ) is the nonlinear term due to interactions.
We first load the BEC into the vertical lattice and then allow the gravitational force to tilt the lattice potential. We thus enter the limit Fd J, in which tunnelling between sites is inhibited and the on-site occupation numbers |c j | 2 are constant, determined by the initial density distribution. The time evolution of the system is then given by the time-dependent phases of all c j , and the 1D wave functionΨ(q, t) in quasi-momentum space q acquires a particularly simple form [26]: Here, we have assumed that our external potential is harmonic, given by V trap j = β tr (j − δ) 2 , where β tr = mω 2 z d 2 /2 characterizes the strength of the potential with trapping frequency ω z along z for a particle with mass m. The parameter δ in the interval [−1/2, 1/2] describes a possible offset of the potential centre with respect to the nearest lattice well minimum along the z-direction. For the interaction term α int , the spatial dependence is also parabolic, reflecting the fact that we initially load a (parabolically shaped) BEC in the Thomas-Fermi regime. In our experiments, the offset δ is not well controlled. It is nearly constant on the timescale of a single experimental run (duration of up to 20 s), but its value changes over the course of minutes as the positions of the horizontally propagating laser beams generating the trapping potential and the position of the retro-reflecting mirror generating the vertical standing wave drift due to changes of the ambient conditions.
The phase evolution in equation (2) has a simple interpretation. The term in the exponent linear in j results in Bloch oscillations [27,28,20] with a Bloch period T Bloch = 2π /(Fd). In figure 1(c) a full cycle of one BO, corresponding to a Bloch phase from 0 to 2π, is shown. When restricting ourselves to times that are integer multiples of T Bloch this term can be omitted. The nonlinear exponents proportional to j 2 lead to a dephasing between lattice sites, resulting in a time-varying interference pattern for the quasimomentum distribution [19]. In our experiments we have full control over these nonlinear terms, not only over β tr via the external trapping potential, but also over the interaction term characterized by α int via the scattering length a s . Our previous work [19] has focused on the role of interactions, whereas in this work we focus on the (nonlinear) term caused by the external potential. For this we tune a s in such a way that the term with α int is minimized. Now the phase evolution depends only on the term with β tr . The offset δ slightly modifies the Bloch period, resulting in a global shift of the interference pattern in quasimomentum space when imaged at integer multiples of the original T Bloch . However, as it is irrelevant for the Talbot effect, we set δ to zero here. By including the simplifications and introducing the Talbot with c j (0, q) = c j (0) exp(−iqjd). Now the Talbot effect is evident. For times that are even multiples of T Talbot the original wave function is recovered, whereas for odd multiples the original wave function appears with a shift of k in quasimomentum space. This realisation of the Talbot effect is nearly ideal, since no paraxial approximation is needed and since there is no limitation in time due to decreasing wave packet overlap [18]. For fractions n/m of T Talbot , m copies of the original wave function with a spacing 2 k/m appear, corresponding to the fractional Talbot effect. The evolution of the quasimomentum distribution as a function of time can be visualized in terms of socalled matter-wave quantum carpets [29]. Such a quantum carpet, calculated by solving equation (1) numerically with the parameters typical to our experiment, is shown in figure 2. Note that in this case the more simple calculation based on equation (3) leads to the same result. However, equation (1) gives us more flexibility in relaxing the requirements of harmonic confinement or negligible tunnelling. We plot the distribution as a line density plot with white areas indicating high densities. Only times that are integer multiples of T Bloch are shown. After a fast spreading of the quasimomentum distribution a regular pattern appears at times for which one expects fractional Talbot interferences. The number of peaks in the momentum distributions directly represents the fraction t/T Talbot . At T Talbot a refocusing to the initial distribution occurs, shifted by k in quasimomentum space. The evolution is then repeated until at 2T Talbot the original wave function is recovered.

Experimental realisation
For the present experiments we choose a lattice depth of 8 E R . For lattice loading the interaction strength is set to a s = 100 a 0 and the external trap frequencies are changed adiabatically to populate about 40 lattice sites. After loading, we change ω z to the final value. This change is done sufficiently quickly (within 3 ms) to avoid a change in the initial distribution due to tunnelling, but sufficiently slowly to avoid motional excitations along the z-direction. Then, within 0.1 ms, we switch off the levitating magnetic field gradient to decouple the individual lattice sites and set the scattering length to the value near a s = 0 a 0 that gives minimal dephasing [20]. Note that the point of minimal dephasing does not correspond exactly to 0 a 0 as residual magnetic dipoledipole interactions have to be taken into account [30]. The shift is calculated to be about −0.7 a 0 . After a variable hold time t hold , which typically corresponds to hundreds of Bloch cycles with T Bloch = 0.575 ms, we switch the levitation field back on in 0.1 ms and ramp down the optical lattice and the dipole trap responsible for trapping in the vertical direction in 0.3 ms. The ramp is adiabatic with respect to the trap frequency of the individual lattice sites, ensuring that the atoms stay in the lowest Bloch band and thus mapping quasimomentum onto real momentum [31]. Before taking an absorption picture we let the sample expand for 80 ms while it remains levitated and thus map momentum to real space. The dipole trap responsible for horizontal trapping is not turned off immediately, but instead it is ramped down slowly over the course of 50 ms to reduce spreading of the sample in the horizontal direction. At the same time, a s is kept at the value that gives minimal interactions to avoid broadening of the sample in the vertical direction. From the absorption pictures we calculate the momentum width ∆p as two times the second moment of the momentum distribution along the vertical direction. Note that the presence of the horizontal trap during expansion leads to additional broadening in vertical direction. This broadening plus some residual incoherent background limits the observable values of ∆p. Nevertheless, with our ability to image the quasimomentum with high resolution [19] we are able to compare not only the momentum width but also the substructure in the momentum distribution to theory. Figure 3 shows the measured momentum distribution of the atom cloud at specific hold times t hold that are fractions of the calculated Talbot time T Talbot . For this measurement we choose a vertical trap frequency of ω z = 2π × 22.0(2) Hz, which gives T Talbot = 555(10) ms. Figure 3(a) shows the absorption images as density plots (white areas indicate regions with high density), while figure 3(b) plots the horizontally integrated densities from the corresponding images of figure 3(a). Initially, the momentum distribution is singly peaked, as expected for a non-dephased BEC. After a rapid coherent dephasing (corresponding to a rapid broadening of the momentum distribution, not shown here) regularly structured patterns appear. The number of peaks within the first Brillouin zone [− k, + k] corresponds exactly to the fraction t hold /T Talbot , as expected from the theoretical considerations. A small fraction of the atoms is detected outside the first Brillouin zone, likely caused by imperfections in the mapping of quasimomentum onto real momentum. Finally, at the Talbot time, the momentum distribution rephases again to the initial distribution. In general, apart from an overall shift of each individual distribution in quasimomentum space due to variations in δ as discussed below, we find very good qualitative agreement with the results of the calculation shown in figure 2. Figure 4 illustrates the effect of δ on the observed patterns in quasimomentum space. For two different hold times t hold = T Talbot and t hold = T Talbot /2, absorption images for several individual experimental realisations and the corresponding horizontally integrated densities are shown. The expected single-and double-peaked momentum patterns are reproduced from one experimental realisation to the next, but they experience a varying shift in quasimomentum space. As a consequence of the periodic structure of quasimomentum space, a peak that is located near one edge of the Brillouin zone also reappears at the opposite edge. The maximum possible shift of the pattern in quasimomentum space due to δ increases with hold time and is calculated to be ± k × t hold /T Talbot . This is why the patterns shown in figure 3, e. g. at t hold = T Talbot or at t hold = T Talbot /2, agree with the calculated patterns only modulo the shift in quasimomentum space. Note that, alternatively, we could have chosen to present in figure 3 selected patterns from a sufficiently large sample of measurements, e.g. the one from experimental run 4 for t hold = T Talbot or the one from experimental run 1 for t hold = T Talbot /2 shown in figure 4.
A simple quantitative comparison between experiment and calculations can be done by considering the time evolution of the momentum width ∆p. The distribution of this quantity across several experimental realisations is evidently sensitive to the deand rephasing of the matter wave. In fact, we can relax the choice of the Bloch phase and allow its value to be random. For example, for a non-dephased BEC the momentum width ∆p is measured to range from (∆p) min ≈ 0.6 k, corresponding to the singly peaked momentum distribution, to (∆p) max ≈ 1.7 k when the momentum distribution is evenly peaked at both edges of the Brillouin zone (e. g. at half the first Bloch period, see figure 1(c)). For a completely dephased sample corresponding to a uniform distribution over the first Brillouin zone we measure a value of ∆p ≈ 1.25 k.
Accordingly, the range for the momentum width ∆p at a given hold time t hold shows a distinct behaviour as a function of t hold , in particular indicating the revival at T Talbot by maximizing the difference D ∆p = (∆p) max − (∆p) min between the extrema of ∆p. Figure 5(a) shows (∆p) max and (∆p) min as a function of t hold as calculated from equation (1). Initially and at T Talbot the extrema lie far apart (at these times the calculation gives values for (∆p) min that are close to zero in accordance with the fact that the momentum width is determined only by the spread in position space, which is large), whereas at intermediate times the difference is drastically reduced, only increasing slightly at rational fractions of t hold /T Talbot . In figure 5(b) we plot the measured momentum width extrema. These are determined from samples of 10 single measurements at each chosen value for t hold . The initial rapid collapse agrees well with the fact that the sample dephases. Then, near the calculated value for T Talbot , a clear increase in D ∆p can be seen. The difference recovers almost completely to the initial value. We attribute the slight reduction to additional dephasing mechanisms not included in our simple model as discussed below.
The behaviour of D ∆p offers a simple method to test the dependence of the Talbot time T Talbot on the vertical trap frequency ω z . Evidently, D ∆p has a maximum at T Talbot . Figure 6(a) shows the momentum width ∆p in the vicinity of the calculated T Talbot , here for a specific trap frequency of ω z = 2π × 26.9(2) Hz. Again we evaluate 10 experimental realisations for each hold time and select (∆p) max and (∆p) min to calculate D ∆p . We locate the position of its maximum by a simple gaussian fit, as shown in figure 6(b). We then vary ω z and determine T Talbot accordingly. In Figure 6(c) T Talbot is plotted as a function of ω z . The experimental values are in excellent agreement with the calculated values for the Talbot time according to T Talbot = h/(mω 2 z d 2 ). We finally discuss the main limitations for our experiment. We believe that the total number of subsequent revivals that we can observe (we detect up to 4 revivals) is mainly limited by three-body loss and by the anharmonicity of the trapping potential. Threebody loss heats the two-dimensional BECs residing at each lattice site. This leads to a loss of phase coherence and thus decreases the visibility of the revivals. Perhaps more interestingly, the anharmonicity of the trapping potential along the vertical direction causes deviations from the quadratic phase evolution required for the Talbot effect. In order to test this effect we generate the vertical trapping potential with a more tightly focused dipole trap beam, which enhances the effect of anharmonicity. We then observe non-perfect Talbot revivals followed by subrevivals as can be seen in figure 7(a). This is in qualitative agreement with calculations shown in figure 7(b), for which the real gaussian shape of the trapping potential instead of a simple harmonic one has been used. The full calculated time evolution of the momentum distribution is shown in figure 7(c). The distortion of the matter-wave quantum carpet can clearly be seen.

Conclusion
We have demonstrated the temporal Talbot effect with trapped, non-interacting matter waves. High resolution imaging in quasimomentum space allows us to resolve Talbot fringes up to the 10th order. We have tested the dependence of the Talbot time on the strength of the confinement and have found very good agreement with the calculated value. We find that the interference pattern is sensitive to the anharmonicity of the trapping potential. In principle, the detailed structure of the interference pattern and the precise revival times are sensitive probes for force gradients and interactions between atoms. The weak magnetic dipole-dipole interaction, for example, has recently been investigated in the context of matter-wave interferometry [30]. Matter-wave interferometry in the Talbot regime could potentially be used to examine in detail the effect of the long-range nature of such an interaction. Similarly, a spatially dependent force like the Casimir-Polder force [32,33,34] near a surface could be investigated through its influence on the Talbot interference pattern.