All-optical matter-wave lens using time-averaged potentials

The precision of matter-wave sensors benefits from interrogating large-particle-number atomic ensembles at high cycle rates. Quantum-degenerate gases with their low effective temperatures allow for constraining systematic errors towards highest accuracy, but their production by evaporative cooling is costly with regard to both atom number and cycle rate. In this work, we report on the creation of cold matter-waves using a crossed optical dipole trap and shaping them by means of an all-optical matter-wave lens. We demonstrate the trade off between lowering the residual kinetic energy and increasing the atom number by reducing the duration of evaporative cooling and estimate the corresponding performance gain in matter-wave sensors. Our method is implemented using time-averaged optical potentials and hence easily applicable in optical dipole trapping setups. Matter-wave sensors benefit from high flux cold atomic sources. Here, a time-averaged optical dipole trap is reported that generates Bose-Einstein condensates by fast evaporative cooling and further reduces the expansion by means of an all-optical matter-wave lens.

The expansion of the atomic clouds, used in interferometers, needs to be minimized and well controlled to reach long pulse separation times, control systematic shifts, and create ensembles dense enough to detect them after long time-of-flights. Nevertheless, colder ensembles with lower expansion rates typically need longer preparation times. Therefore, matter-wave sensors require sources with a high flux of large cold atomic ensembles to obtain fast repetition rates.
Bose-Einstein condensates (BECs) are well suited to perform interferometric measurements. They are investigated to control systematic effects related to residual motion at a level lower than a few parts in 10 9 of Earth's gravitational acceleration 20,[31][32][33][34] . In addition, due to their narrower velocity distribution 35 , BECs offer higher beam splitting efficiencies and thus enhanced contrast 23,36,37 , especially for large momentum transfer 36,[38][39][40][41][42][43] . Finally, the inherent atomic collisions present in BECs can enhance matter-wave interferometry by enabling (i) ultra-low expansion rates through collective mode dynamics with a recent demonstration of a 3D expansion energy of k B Á 38 þ6 À7 pK 44 , and (ii) ultimately the generation of mode entanglement through spin-squeezing dynamics to significantly surpass the standardquantum limit [45][46][47][48] .
Today's fastest BEC sources rely on atom-chip technology, where near-surface magnetic traps allow for rapid evaporation using radio frequency or microwave transitions. This approach benefits from constant high trapping frequencies during the evaporative cooling process, thus leading to repetition rates on the order of 1 Hz with BECs comprising 10 5 atoms 49 .
Anyway, since magnetic traps are not suitable in certain situations optical dipole traps become the tool of choice 50 . Examples are trapping of atomic species with low magnetic susceptibility 51,52 , or molecules 53,54 and composite particles 55,56 . In optical dipole traps external magnetic field allow tuning parameters, e.g., when using Feshbach resonances 57 .
Here, the intrinsic link between trap depth and trap frequencies in dipole traps 58 inhibits runaway evaporation. Cold ensembles can be only produced in shallow traps, leading to drastically increased preparation time t P . This long standing problem has been recently overcome through the use of time-averaged potentials, where trap depth and trap frequencies can be controlled independently, thus allowing for more efficient and faster evaporation while maintaining high atom numbers 52,59 .
In this work, we use dynamic time-averaged potentials for efficient BEC generation and demonstrate an all-optical matterwave lens capable of further reducing the ensemble's residual kinetic energy. Contrary to pulsed schemes of matter-wave lensing 44,60-65 , we keep the atoms trapped over the entire duration of the matter-wave lens 37 , which eases implementation in ground-based sensors. Moreover, we show that with this technique one can short-cut the evaporation sequence prior to the matter-wave lens, which increases the atomic flux by enhancing atom number and reducing cycle time while simultaneously reducing the effective temperature. Our method can largely improve the matter-wave sensor's stability in various application scenarios.

Results
Evaporative cooling. We operate a crossed optical dipole trap at a wavelength of 1960 nm loaded from a 87 Rb magneto-optical trap (details in the "Methods" section). The time-averaged potentials are generated by simultaneuos center-position modulation of the crossed laser beams in the horizontal plane. Controlling the amplitude of this modulation and the intensity of the trapping beams enables the dynamic control and decoupling of the trapping frequencies and depth. We chose the waveform of the center-position modulation to generate a parabolic potential 52 .
Up to 2 × 10 7 rubidium atoms are loaded into the trap with trapping frequencies ω/2π ≈ {140; 200; 780} Hz in fx 0 ; y 0 ; zg direction (definition of coordinate systems in the "Methods" section) with a trap depth of 170 μK. For this we operate the trap at the maximum achievable laser intensity of 12 W and the center-position modulation at an amplitude of h 0 = 140 μm.
We perform evaporative cooling by reducing the trap depth exponentially in time while keeping the trapping frequencies at a high level by reducing the amplitude of the center-position modulation. This method allows us to generate BECs with up to 4 × 10 5 atoms within 5 s of evaporative cooling. By shortening the time constant of the exponential reduction we generate BECs with 5 × 10 4 (2 × 10 5 ) particles within 2 s (3 s) of evaporative cooling. At the end of the evaporation sequence the trap has frequencies of ω/2π ≈ {105; 140; 160} Hz and a depth of about 200 nK. The expansion velocity of the condensate released from the final evaporation trap is 2 mm s −1 , which corresponds to an effective temperature of 40 nK.
All-optical matter-wave lens. Our matter-wave lens can be applied in any temperature regime explorable in our optical trap. We investigate the creation of collimated atomic ensembles for different initial temperatures of the matter-waves. To this aim, the evaporation sequence is stopped prematurely at different times to generate input atomic ensembles at rest with initial trap frequency ω 0 and initial temperature T 0 . We then initiate the matter-wave lens by a rapid decompression 66 of the trap frequency in the horizontal directions from ω 0 to ω l . Here we denote by ω l the lensing potential in analogy with the Delta-kick collimation technique. The reduction of the trapping frequencies from the initial ω 0 to ω l depends on experimental feasibility, such as the maximum achievable amplitude of the center-position modulation and the modulation amplitude right before the rapid decompression. With ongoing evaporative cooling this amplitude is reduced and thus the trap can be relaxed much further for more continued sequences. However, we need to maintain the confinement in the vertical direction by adjusting the dipole trap's intensity to suppress heating or loss of atoms.
Subsequent oscillations in the trap result in a manipulation in phase space (Fig. 1a, b) for focusing, diffusion, and, importantly collimation of the matter-wave (Fig. 1c). Figure 1c depicts the expansion of a thermal ensemble in 1D for three different holding times (t hold ) to highlight the importance of a well chosen timing for the lens. Figure 2 shows exemplary expansion velocities (colored circles) depending on the holding time t hold . The colored curves in this graph display the simulated behavior following the scaling ansatz (details in the "Methods" section) with an error estimation displayed by shaded areas. Only for the final measurement (also shown in the inset in Fig. 2) we create a BEC with a condensed fraction of 92.5% of the total atom number and apply the matter-wave lens to it.
With the presented method we observe oscillations of the expansion rate, which are in good agreement with the simulations for different ensemble temperatures. For all investigated temperatures an optimal holding time exists for which the final expansion rate is minimized (Fig. 3a). The ratios of σ v,l /σ v,0 and ω l /ω 0 for each measurement is shown in Fig. 3b.
The change in atom number from the initial to the lensing trap ( Fig. 3a) lies within the error bars and arises mainly due to pointing instabilities of the crossed optical dipole trap beams. The lowest expansion rate is achieved with 553(49) μm s −1 with a related effective temperature of 3.2(0.6) nK and an atom number of 4.24(0.02) × 10 5 . With this method we achieve a more than one order of magnitude lower effective temperature while maintaining a comparable atom number compared to evaporative cooling.

Discussion
In this paper, we demonstrate a technique to reduce the expansion velocity of an atomic ensemble by rapid decompression and subsequent release from an dipole trap at a well-controlled time. The efficiency of the matter-wave lens for higher temperatures is mainly limited experimentally by the limited ratio between the initial and the lensing trap frequency ω l /ω 0 ( Fig. 3b) which is constrained by the maximum possible spatial modulation amplitude of the trapping beams. In general, according to the Liouville theorem, the expansion speed reduction of the matterwave is proportional to ðω 0 =ω l Þ 2 where a large aspect ratio enables a better collimated ensemble. The atoms are loaded into the time-average potential with an optimized center-position modulation amplitude of 140 μm, while the maximum is 200 μm. During the evaporation sequence this amplitude is decreased. Consequently, the relaxation of the trap is less efficient at the beginning of the evaporative sequence or directly after the loading of the trap.
Another constraint is that the trap's confinement in the unpainted vertical direction is required to remain constant. If the vertical trap frequency is increased we observe heating effects and suffer from atom loss when it is decreased. To compensate for the trap depth reduction during the switch from the initial to the lensing trap we increase the dipole trap laser's intensity accordingly.
An additional modulation in the vertical direction, e.g., by means of a two-dimensional acousto-optical deflector, as well as an intersection angle of 90°would enable the generation of isotropic traps. In such a configuration, the determination of the optimal holding time will benefit from the in-phase oscillations of the atomic ensemble's size 67 . When applying our matter-wave lens in a dual-species experiment, isotropy of the trap will also improve the miscibility of the two ensembles 68 .   To illustrate the relevance for atom interferometers, we discuss the impact of our source in different regimes (details in the "Methods" section) operated at the standard quantum limit for an acceleration measurement. In a Mach-Zehnder-like atom interferometer 1,18 , the instability reads after an averaging time τ, neglecting the impact of finite pulse durations on the scale factor [69][70][71] . Eq. (1) scales with the interferometer contrast C, the atom number per cycle N, the effective wave number nℏk eff indicating a momentum transfer during the atom-light interaction corresponding to 2n photons, and the separation time between the interferometer light pulses T I . The cycle time of the experiment t cycle = t P + 2T I + t D includes the time for preparing the ensemble t P , the interferometer 2T I , and the detection t D . In Eq. (1), the contrast depends on the beam splitting efficiency. This, in turn, is affected by the velocity acceptance and intensity profile of the beam splitting light, both implying inhomogeneous Rabi frequencies, and consequently a reduced mean excitation efficiency 35,72,73 . Due to expansion of the atomic ensemble and inhomogeneous excitation, a constrained beam diameter implicitly leads to a dependency of the contrast C on the pulse separation time T I , which we chose as a boundary for our discussion. We keep the effective wave-number fixed and evaluate σ a (1 s) for different source parameters when varying T I . Figure 4 shows the result for collimated (solid lines) and uncollimated (dotted lines) ensembles in our model (details in the "Methods" section) and compares them to the instability under use of a molasses-cooled ensemble (dash-dotted line). Up to T I = 100 ms and σ a (1 s) = 10 −8 m s −2 , the molasses outperforms evaporatively cooled atoms or BECs due the duration of the evaporation adding to the cycle time and associated losses. In this time regime, the latter can still be beneficial for implementing large momentum transfer beam splitters 36,38-40,42,43 reducing σ a (τ) or suppressing systematic errors 20,31-34,74 which is not represented in our model and beyond the scope of this paper. According to the curves, exploiting higher T I for increased performance requires evaporatively cooled atoms or BECs. This shows the relevance for experiments on large baselines 23,37,[74][75][76][77] or in microgravity 78,79 . We highlight the extrapolation for the Very Long Baseline Atom Interferometer (VLBAI) 76,80 , targeting a pulse separation time of T I = 1.2 s 81 . Here, the model describing our source gives the perspective of reaching picokelvin expansion temperatures of matter-wave lensed large atomic ensembles.

Methods
Experimental realization. The experimental apparatus is designed to operate simultaneous atom interferometers using rubidium and potassium and is described in detail in references 9,10,82 .
For the experiments presented in this article only rubidium atoms were loaded from a two dimensional to a three dimensional magneto-optical trap (2D/3D-MOT) situated in our main chamber. After 2 s we turn off the 2D-MOT and compress the atomic ensemble by ramping up the magnetic field gradient as well as the detuning of the cooling laser in the 3D-MOT. Subsequent to compression, the atoms are loaded into the crossed dipole trap by switching off the magnetic fields and increasing the detuning of the cooling laser to about − 30Γ, with Γ being the natural linewidth of the D 2 transition. Figure 5 depicts the setup of our crossed optical dipole trap. The center-position modulation of the trapping beams is achieved by modulating the frequency driving the acousto-optical modulator (AOM) (Polytec, ATM-1002FA53.24). A voltagecontrolled oscillator (Mini-Circuits, ZOS-150+) generates the signal for this, which is driven by a programmable arbitrary-waveform generator (Rigol, DG1022Z). We chose the waveform to generate a large-volume parabolic potential based on the derivation shown by Roy et al. 52 Fig. 2. The different source parameters are described in more detail in the "Methods" section.
Data acquisition and analysis. We apply our matter-wave lens subsequent to loading the dipole trap and evaporative cooling. The duration of the complete evaporative sequence is 5 s for the measurements presented here we interrupt this sequence after 0 s, 0.2 s, 1 s, 2 s, 3.5 s, 4.3 s and 5 s. Before the step-wise change of the trap frequency (ω 0 → ω l ) we hold the ensemble in the trap given by the respective evaporation step configuration for 50 ms. During the matter-wave lens, the rapid decompression of the trap causes oscillations of the ensemble's radius in the lensing trap. Depending on the release time we observe oscillations by performing absorption imaging with iterating t hold for different times after the release from the trap. For each holding time the expansion velocity is extracted by fitting a ballistic expansion. This expansion can be transformed into an effective temperature using: along each direction. The measurement is performed for different starting temperatures in the thermal regime as well as the BEC. The simulations shown in Fig. 2 use the scaling ansatz as described in the "Scaling Ansatz" section. Here, the trapping frequencies of the lens potential in xand y-direction have been extracted by fitting two damped oscillations to the measured data. The starting expansion velocity was set by choosing a reasonable initial radius of the ensemble ( Table 1). The other parameters arise from the measurements or simulations of the trapping potentials. The shaded areas in fig. 2 depict an error estimation of the expansion velocity oscillations obtained from performing the simulation by randomly choosing input parameters from within the error bars for 1000 simulation runs and calculating the mean value as well as the standard deviation for each t hold .
Scaling Ansatz. In the case of a thermal ensemble in the collision-less regime, the dynamics of a classical gas can be described using the scaling ansatz 83,84 , which we briefly recall here for sake of simplicity. Here, the size of the ensemble scales with the time dependent dimensionless factor b i (t).
where θ i acts as an effective temperature in the directions i ∈ x, y, z.
Here ω 0,i denotes the initial angular trap frequency and ω i (t) denotes the time-dependent angular trap frequency defined such as: ω i (t) = ω l,i for 0 < t < t hold , with ω l,i being the lensing potential, and ω i (t) = 0 after the release (see Fig. 1). This system of coupled differential equations contains the mean field interaction, given by the factor: with where a s is the s-wave scattering length, n 0 the peak density and m the mass of a single particle. Collision effects are also taken into account through with the relaxation time and 84 In the special case of a BEC, the mean field energy is large compared to the thermal ensemble's energy (ξ ≈ 1) and the time scale on which collisions appear goes to zero (τ ≈ 0). In this case the time dependent evolution of the matter-wave can be described following Castin & Dum 85 . Here, the evolution of the BEC's Thomas-Fermi radius, R i (t) = b i (t)R i (0), is described by the time-dependent evolution of the scaling parameter: and R i (0) is the initial Thomas-Fermi radius of the BEC along the i-th direction. It is worth to notice that recent studies 86,87 extend the analysis of Guéry-Odelin 83 and Pedri et al. 84 to the BEC regime described by Castin & Dum 85 . With this set of equations the time evolution of the ensemble's size (σ r i ) and velocity distribution (σ v i ) is determined during the entire sequence of our matterwave lensing sequence by  Source parameters for the instability estimation for molasses cooled, released from the optical dipole trap (ODT) with and without evaporation, and Bose-Einstein condensates (BEC and Advanced). The values for the ensemble radius (σ r ) and expansion velocity (σ v ) are given for the horizontal (h) and vertical (v) direction, which corresponds to the transverse and longitudinal direction of the beam splitter respectively. N is the number of atoms and t P the preparation time to calculate the cycle time t cycle . and σ v i ðtÞ ¼ dσ r i ðtÞ dt : ð12Þ The scaling parameter b i can be applied either on the radius of a gaussian distributed thermal ensemble or the Thomas-Fermi radius of a BEC.
Estimation of instability in matter-wave sensors. The instability of a matterwave sensor operating at the standard quantum limit can be estimated using Eq.
(1). Here we assume Raman beam splitters (n = 1) with a 1/e 2 -radius of 1.2 cm and a pulse duration of t π = 15 μs. The contrast (C) is taken into account as the product of the excitation probabilities of the atom-light interactions during the Mach-Zehnder type interferometer following Loriani et al. 72 . Table 1 shows the source parameters used for the estimation of the instability. We chose three parameter sets from the here presented measurements of two thermal ensembles released from the optical dipole trap with starting temperatures of T 0 = 41μK and 4 μK and the BEC. Besides that we simulated the performance of the interferometer operated with a molasses cooled ensemble combined with a velocity selective Raman pulse of 30 μs 73 , based on typical parameters in our experiment, and an advanced scenario. For this we assume a BEC with 1 × 10 6 atoms after a preparation time t P = 1 s with a starting expansion velocity of 2 mm s −1 , as anticipated for the VLBAI setup 76,80 . We extrapolate the performance of our matter-wave lens for this experiment, resulting in expansion velocities of 0.135 mm s −1 corresponding to an equivalent 3D temperature of 200 pK.

Data availability
The data used in this manuscript are available from the corresponding author upon reasonable request.