Abstract
The precision of matterwave sensors benefits from interrogating largeparticlenumber atomic ensembles at high cycle rates. Quantumdegenerate 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 matterwaves using a crossed optical dipole trap and shaping them by means of an alloptical matterwave 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 matterwave sensors. Our method is implemented using timeaveraged optical potentials and hence easily applicable in optical dipole trapping setups.
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Introduction
Ever since their first realization, atom interferometers^{1,2,3,4} have become indispensable tools in fundamental physics^{5,6,7,8,9,10,11,12,13,14,15,16,17} and inertial sensing^{18,19,20,21,22,23,24,25,26,27,28,29,30}. The sensitivity of such matterwave sensors scales with the enclosed spacetime area which depends on the momentum transferred by the beam splitters as well as the time the atoms spend in the interferometer.
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 timeofflights. Nevertheless, colder ensembles with lower expansion rates typically need longer preparation times. Therefore, matterwave sensors require sources with a high flux of large cold atomic ensembles to obtain fast repetition rates.
BoseEinstein 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 matterwave interferometry by enabling (i) ultralow expansion rates through collective mode dynamics with a recent demonstration of a 3D expansion energy of \({k}_{B}\cdot 3{8}_{7}^{+6}\) pK^{44}, and (ii) ultimately the generation of mode entanglement through spinsqueezing dynamics to significantly surpass the standardquantum limit^{45,46,47,48}.
Today’s fastest BEC sources rely on atomchip technology, where nearsurface 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 timeaveraged 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 timeaveraged potentials for efficient BEC generation and demonstrate an alloptical matterwave lens capable of further reducing the ensemble’s residual kinetic energy. Contrary to pulsed schemes of matterwave lensing^{44,60,61,62,63,64,65}, we keep the atoms trapped over the entire duration of the matterwave lens^{37}, which eases implementation in groundbased sensors. Moreover, we show that with this technique one can shortcut the evaporation sequence prior to the matterwave 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 matterwave 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 magnetooptical trap (details in the “Methods” section). The timeaveraged potentials are generated by simultaneuos centerposition 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 centerposition 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 \(\{{x}^{\prime};\,{y}^{\prime};\,z\}\) 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 centerposition 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 centerposition 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.
Alloptical matterwave lens
Our matterwave 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 matterwaves. 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 matterwave 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 Deltakick 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 centerposition 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 matterwave (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 matterwave 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 wellcontrolled time. The efficiency of the matterwave 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 \({({\omega }_{0}/{\omega }_{l})}^{2}\) where a large aspect ratio enables a better collimated ensemble. The atoms are loaded into the timeaverage potential with an optimized centerposition 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 twodimensional acoustooptical 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 inphase oscillations of the atomic ensemble’s size^{67}. When applying our matterwave lens in a dualspecies 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 MachZehnderlike 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 atomlight 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 wavenumber 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 molassescooled ensemble (dashdotted 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^{36,38,39,40,42,43} reducing σ_{a}(τ) or suppressing systematic errors^{20,31,32,33,34,74} 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 matterwave 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 magnetooptical trap (2D/3DMOT) situated in our main chamber. After 2 s we turn off the 2DMOT and compress the atomic ensemble by ramping up the magnetic field gradient as well as the detuning of the cooling laser in the 3DMOT. 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 centerposition modulation of the trapping beams is achieved by modulating the frequency driving the acoustooptical modulator (AOM) (Polytec, ATM1002FA53.24). A voltagecontrolled oscillator (MiniCircuits, ZOS150+) generates the signal for this, which is driven by a programmable arbitrarywaveform generator (Rigol, DG1022Z). We chose the waveform to generate a largevolume parabolic potential based on the derivation shown by Roy et al.^{52}. The amplitude of the displacement of the centerposition of the dipole trap beam, h_{0}, is controlled by regulating the amplitude of the AOM’s frequency modulation. This yields a maximum beam displacement of h_{0} = 200 μm (300 μm) at the position of the atoms for the initial (recycled) beam.
Data acquisition and analysis
We apply our matterwave 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 stepwise 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 matterwave 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 x and ydirection 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 collisionless 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 timedependent 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 swave 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 matterwave can be described following Castin & Dum^{85}. Here, the evolution of the BEC’s ThomasFermi radius, R_{i}(t) = b_{i}(t)R_{i}(0), is described by the timedependent evolution of the scaling parameter:
and R_{i}(0) is the initial ThomasFermi radius of the BEC along the ith direction. It is worth to notice that recent studies^{86,87} extend the analysis of GuéryOdelin^{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 (\({\sigma }_{{r}_{i}}\)) and velocity distribution (\({\sigma }_{{v}_{i}}\)) is determined during the entire sequence of our matterwave lensing sequence by
and
The scaling parameter b_{i} can be applied either on the radius of a gaussian distributed thermal ensemble or the ThomasFermi radius of a BEC.
Estimation of instability in matterwave 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 atomlight interactions during the MachZehnder 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 matterwave 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.
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Acknowledgements
This work is funded 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 Nos. DLR 50WM1641 (PRIMUSIII), DLR 50WM2041 (PRIMUSIV), DLR 50WM2245A (CALII), DLR 50WM2060 (CARIOQA), and DLR 50RK1957 (QGYRO). We acknowledge financial support from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)ProjectID 274200144SFB 1227 DQmat within the projects A05, B07, and B09, and ProjectID 434617780SFB 1464 TerraQ within the projects A02 and A03 and Germany’s Excellence Strategy—EXC2123 QuantumFrontiers—ProjectID 390837967 and from “Niedersächsisches Vorab” through the “Quantum and NanoMetrology (QUANOMET)” initiative within the Project QT3. A.H. and D.S. acknowledge support by the Federal Ministry of Education and Research (BMBF) through the funding program Photonics Research Germany under contract number 13N14875.
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W.E., E.M.R., and D.S. designed the experimental setup and the dipole trapping laser system. H.A., A.H., A.R., and D.S. contributed to the design, operation, and maintenance of the laser system and the overall setup. R.C., E.C. and N.G. set the theoretical framework of this work. H.A., R.C., C.S., and D.S. drafted the initial manuscript. H.A., and R.C. performed the analysis of the data presented in this manuscript. H.A., and R.C. under lead of N.G. and C.S. performed the instability study. C.V., M.W., C.L., S.H. together with the other authors discussed and evaluated the results and contributed to, reviewed, and approved of the manuscript.
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Albers, H., Corgier, R., Herbst, A. et al. Alloptical matterwave lens using timeaveraged potentials. Commun Phys 5, 60 (2022). https://doi.org/10.1038/s42005022008252
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DOI: https://doi.org/10.1038/s42005022008252
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