Abstract
We present programmable twodimensional arrays of microscopic atomic ensembles consisting of more than 400 sites with nearly uniform filling and small atom number fluctuations. Our approach involves direct projection of light patterns from a digital micromirror device with high spatial resolution onto an optical pancake trap acting as a reservoir. This makes it possible to load large arrays of tweezers in a single step with high occupation numbers and low power requirements per tweezer. Each atomic ensemble is confined to ~1 μm^{3} with a controllable occupation from 20 to 200 atoms and with (sub)Poissonian atom number fluctuations. Thus, they are ideally suited for quantum simulation and for realizing large arrays of collectively encoded Rydbergatom qubits for quantum information processing.
Introduction
Neutral atoms in optical tweezer arrays have emerged as one of the most versatile platforms for quantum manybody physics, quantum simulation, and quantum computation^{1,2,3,4,5,6,7,8,9,10}. This is largely due to their long coherence times combined with flexible configurations and controllable longrange interactions, in particular using highly excited Rydberg states^{11,12,13,14,15,16,17}. To date, substantial experimental efforts have been devoted to create fully occupied atomic arrays with ≃1 atom in each tweezer by exploiting lightassisted inelastic collisions^{18,19,20} and rearrangement to fill empty sites^{4,5,21,22,23,24}. In combination with highfidelity Rydberg blockade gates^{16,25,26,27,28,29,30,31}, these systems have been recently used to demonstrate coherent quantum dynamics of up to 51 qubits^{32} and entangled states of up to 20 qubits in onedimensional (1D) chains^{33}. So far however, achievable array sizes are limited to ≲100 fully occupied sites (including for 2D and 3D systems), in part due to high power requirements and increasing complexity associated with the rearrangement process for larger arrays^{21,23,24}.
In this paper, we demonstrate an alternative approach to prepare large and uniformly filled arrays of hundreds of tweezers with large occupation numbers in a single step. This is exemplified by the 400 site triangular array shown in Fig. 1a, as well as more exotic geometries, such as connected rings (Fig. 1b) and quasiordered geometries (Fig. 1c), which exhibit structures on different length scales making them difficult to produce using other methods. Our approach involves transferring ultracold atoms from a quasi2D optical reservoir trap into an array of optical tweezers produced by a digital micromirror device (DMD). To realize large arrays we optimize the loading process and the homogeneity across the lattice by adapting the DMD light patterns to control the trap depth of each tweezer. Each atomic ensemble is localized well within the typical Rydberg blockade radius, and the typical intertrap separations of several micrometers are compatible with Rydbergblockade gates. Furthermore, we show that the fluctuations of the number of atoms in each tweezer is comparable to, or below the shotnoise limit for uncorrelated atoms. These properties make the system well suited for quantum simulation of quantum spin models^{13,14,17,34,35,36,37,38,39} and dynamics^{40,41,42,43,44,45,46,47} in novel geometries, as well as for realizing quantum registers with collectively enhanced atom–light interactions for quantum information processing^{11,36,48,49,50,51}.
Results
Our experimental cycle starts with a threedimensional magnetooptical trap (MOT) loaded from a beam of ^{39}K atoms produced by a twodimensional MOT. This is overlapped with a far offresonant pancakeshaped reservoir trap created by a 1064 nm single mode laser with a power of 16 W tightly focused by a cylindrical lens. The beam waists are ω_{z} = 7.6 μm, ω_{x} = 540 μm, and ω_{y} = 190 μm and the estimated trap depth is 330 μK. To maximize the number of atoms in the reservoir we apply an 8 ms graymolasses cooling stage on the D_{1} transition^{52}, yielding 3.3 × 10^{5} atoms in the \(\left4{s}_{1/2},F=1\right\rangle\) state at an initial temperature of 45 μK.
Next, we transfer the atoms to the tweezers from the reservoir trap. To generate the tweezers we illuminate a DMD with a collimated 780 nm Gaussian beam of light with a 4.3 mm waist and a peak intensity of 1.44 W cm^{−2}. We directly image the DMD plane onto the atoms (similar optical setup to refs. ^{53,54}) with a calibrated demagnification factor of 53, using a 4f optical setup involving a 1500 mm focal length lens and a 32 mm focal length lens (Fig. 1d). The latter is a molded aspheric lens with a numerical aperture of 0.6, located inside the vacuum chamber. With this setup, each (13 μm)^{2} pixel of the DMD corresponds to (245 nm)^{2} in the atom plane.
The DMD can be programmed with arbitrary binary patterns to control the illumination in the atom plane. To generate the tweezer arrays shown in Fig. 1a–c we create different patterns of spots where each spot is formed by a small diskshaped cluster of typically A = 20–100 pixels. To detect the atoms we use the saturated absorption imaging technique^{55,56}. The probe laser is resonant to the 4s_{1/2} → 4p_{3/2} transition of ^{39}K at 767 nm and with an intensity of \(I\approx 2.1{I}_{{\rm{sat}}}^{{\rm{eff}}}\). The atoms are exposed for 10 μs and the absorption shadow is imaged onto a chargecoupleddevice camera using the same optics as for the DMD light patterns (Fig. 1d). The resulting optical depth image is well described by a sum of twodimensional Gaussian distributions. By fitting and integrating each distribution we determine the number of atoms in each tweezer. By analyzing a background region of the images we infer a singleshot detection sensitivity of 3.9(5) atoms.
We found that optimal loading of the tweezers is achieved by evaporatively cooling the atoms in the reservoir trap while superimposing the DMD light pattern (Fig. 2a). At the end of the evaporation ramp the reservoir trap can be switched off leaving the atoms confined by the tweezers alone. The overall cycle time including MOT loading, evaporative cooling, transfer to the tweezers and imaging is <4 s. Figure 2b, c show the characterization of the loading process for a single tweezer with A = 100 pixels, corresponding to an optical power of 90 μW. There is little difference if the tweezer is switched on suddenly or ramped slowly, however we found it is beneficial to turn on the tweezers at least 200 ms before the end of the evaporation ramp (Fig. 2b), indicating a significant enhancement of the loading through elastic collisions with the reservoir atoms^{57}. Figure 2c shows that the mean occupation number \(\bar{N}={\langle N\rangle }_{i}\) (with \({\left\langle \cdot \right\rangle }_{i}\) denoting an average over different experimental realizations) strongly depends on the final temperature of the reservoir, with the maximal \(\bar{N}=120(5)\) found for \({T}_{{\rm{res}}}\approx 2\) μK. The temperature of the atoms measured using the timeofflight method for a single tweezer with A = 100 pixels, corresponding to a relatively deep trap, is 17(1) μK. Figure 2d shows the lifetime of the atoms in the tweezer. An exponential fit describing pure onebody decay (dashed curve) is clearly ruled out by the data while a model which includes both threebody and onebody decay^{58} provides an excellent fit (solid curve). From this model we extract both the threebody and onebody decay constants \({k}_{3}^{1}=110\ {\rm{ms}}\), \({k}_{1}^{1}=3100\ {\rm{ms}}\), which are both orders of magnitude longer than typical timescales in Rydberg atom experiments.
We now show that it is straightforward to scale up to a large number of sites while maintaining a uniformly high occupation of each tweezer. For arrays with more than approximately M = (10 × 10) sites we found it is beneficial to adapt the DMD pattern to compensate the Gaussian illumination profile. We adapt the number of pixels in each cluster according to its distance from the center of the illumination region following an inverted Gaussian dependence \({A}_{m}={A}_{\min }\exp (g{r}_{m}^{2})\) where m indexes a tweezer at position r_{m} in the DMD plane. The parameters \({A}_{\min }=20\) pixels and g = 2.3 × 10^{−5} were manually adapted to obtain approximately equal optical depths for each tweezer, where \({A}_{\min }=20\) pixels corresponds to an optical power per tweezer of 18 μW. The maximum array size of M = (22 × 22) was fixed such that \(\max ({A}_{m})=66\) pixels, due to larger spots increasing the apparent size of the atomic ensembles near the edge of the array. Once the optimal compensation profile is found it can be applied to different geometries without further adjustment.
Figure 3 shows the mean occupation number \(\bar{N}\) for square arrays with different numbers of sites. The solid orange symbols show the array averaged mean occupation number \({\langle \bar{N}\rangle }_{m}\), where we average over all 20 experimental repetitions and each lattice site m. The experiments show that \({\left\langle \bar{N}\right\rangle }_{m}\simeq 40\) is approximately constant for tweezer arrays with different numbers of sites 4 ≤ M ≤ 484. As an example, for M = 400 the standard deviation is \(0.17{\left\langle \bar{N}\right\rangle }_{m}\) (excluding shottoshot fluctuations), compared to \(0.55{\langle \bar{N}\rangle }_{m}\) without compensation. The uniformity could be further improved by adapting A_{m} for each tweezer individually, but it is already better than the expected intrinsic shottoshot fluctuations of the atom number in each site due to atom shotnoise (with standard deviation \(\sqrt{\bar{N}}\)) for \(\bar{N}\lesssim 36\).
The apparent size of each atomic ensemble, found by analyzing the averaged absorption images, ranges from 0.64 μm near the center of the field to 0.93 μm at the edges (e^{−1/2} radii of each absorption spot). This can be understood as a convolution of several effects, the most important are recoil blurring, the finite resolution of the imaging system including off axis blurring and the finite size of each DMD spot. By projecting the DMD light pattern onto a camera in an equivalent test setup we independently determine the beam waist of each tweezer to be 0.9 μm. Assuming each tweezer is described by a Gaussian beamlet and approximating the atomic cloud by a thermal gas with a temperature ~V_{0}/5 (with trap depth V_{0}), we infer a cloud size of σ_{r,z} = {0.2, 1.0} μm. This is reasonably close to an independent estimate \(\root{3}\of{{\sigma _r^2{\sigma _z}}} \approx 0.6\) μm based on the experimentally measured threebody loss rate and a theoretical calculation of the zero field threebody loss coefficient for ^{39}K^{59}.
The small spatial extent of each ensemble is encouraging for experiments which aim to prepare a single Rydberg excitation at each site, as it is significantly smaller than the nearestneighbor distance and the typical Rydberg blockade radius of R_{bl} ~ 3–6 μm (corresponding ns_{1/2} Rydberg states with principal quantum number n between 40 and 60). Approximating the density distribution as quasi1D we estimate the weightedaverage fraction of blockaded atoms, assuming random initial positions, as \({f}_{\mathrm {{bl}}}={\rm{erf}}({R}_{\mathrm {{bl}}}/2{\sigma }_{z})\), where \({\rm{erf}}(x)\) is the Gausserror function. For R_{bl}/σ_{z} ≥ 3, f_{bl} ≥ 0.97 which suggests that the blockade condition within a single tweezer should be well satisfied. To serve as effective twolevel systems (comprised of the collective ground state and the state with a single Rydberg excitation shared amongst all atoms in the ensemble), it is additionally important that the fluctuations of the atom number from shottoshot are relatively small otherwise the \(\sqrt{{N}_{i}}\) collective coupling^{49,60,61} will reduce the single atom excitation fidelity when averaging over different realizations. Previous theoretical estimates have assumed Poisson distributed atom shotnoise^{36}, which we generalize to the case of a stretched Poissonian distribution and imperfect blockade. However, we neglect other possible imperfections such as spectral broadening due to laser linewidth or interactions between ground state and Rydberg atoms^{36}. For a simple estimate, we assume that within the blockade volume, the probability to create a single excitation undergoes collective Rabi oscillations \({p}_{i}=1{\cos }^{2}(\sqrt{{N}_{i}}\Omega t/2)\) where Ω is the single atom Rabi frequency^{36}. In contrast, the nonblockaded fraction of atoms 1 − f_{bl} undergoes Rabi oscillations at the frequency Ω. By expanding around the time for a collective π pulse: \(t=\pi /(\sqrt{\bar{N}}\Omega )\) and averaging over a stretched Poissonian statistical distribution for atom number fluctuations we find that the infidelity for producing exactly one excitation is \(\epsilon \approx ({\pi }^{2}/4)[{\rm{var}}(N)/(4{\bar{N}}^{2})+(1{f}_{\mathrm {{bl}}})]\). For f_{bl} → 1 the infidelity is proportional to the relative variance \({\rm{var}}(N)/{\bar{N}}^{2}\).
To estimate the relative variance of the atom number fluctuations we prepare tweezer arrays with different numbers of sites and trap depths, corresponding to mean occupation numbers from \(\bar{N}=20\) to \(\bar{N}=200\). For each set of experimental conditions we take 20 absorption images from which we compute the mean occupation and the variance of the atom number in each tweezer. Figure 4 shows the relative variance \({\rm{var}}(N)/{\bar{N}}^{2}\) calculated for 75,880 tweezer realizations. We see that for smaller atom numbers the relative variance is consistent with the expected Poissonian atomshot noise for independent particles \({\rm{var}}(N)=\bar{N}\) (shown by the solid black line), while for \(\bar{N}\,\gtrsim \,50\) the fluctuations are subPoissonian, reaching the lower limit expected for threebody loss (\({\rm{var}}(N)=0.6\bar{N}\), shown by the dashed black line)^{58}. For \(\bar{N}\,>\,40\) the expected infidelity due to atom number fluctuations would be below 0.03, showing that this system should be compatible with high fidelity preparation of individual Rydberg excitations and quantum logic gates, and could still be further improved using adiabatic or composite pulse techniques^{48}. Precise coherent manipulations of collective qubits could also be achieved by measuring the number of atoms in each ensemble and adapting the excitation pulses accordingly^{62}.
Discussion
In conclusion, we have demonstrated an approach for realizing hundreds of ultracold atomic ensembles in programmable twodimensional arrays, where each tweezer has approximately uniform filling, small spatial extent and small fluctuations of the atom number between realizations. Compared to stochastic loading of the tweezers via lightassisted collisions from a MOT, this has several advantages. First, it is possible to achieve very high occupation numbers N ≫ 1 with relatively low power requirements per tweezer, since the temperature of the initial reservoir trap can be lower than the typical temperatures in a MOT and elastic, rather than inelastic, collisions lead to a high filling probability. This is beneficial for scaling up to hundreds or even thousands of tweezers as the large volume of the reservoir trap makes it possible to simultaneously fill many tweezers in parallel without the need for additional lasers and complex rearrangement protocols to fill empty sites. Atomic ensembles also offer greater robustness against particle loss, since the loss of one or several particles from each microtrap does not result in defects that would otherwise be difficult to repair. Additionally, atomic ensembles present the possibility to evaporatively cool the atoms in each tweezer to reach high phase space densities or as a more controlled starting point for quasideterministic single atom preparation schemes using controlled inelastic collisions^{19,23} or the Rydberg blockade effect^{36,63,64}. The observation that the number of atoms inside each tweezer exhibits fluctuations below the Poissonian atom shotnoise limit is especially promising for quantum information processing based on small Rydberg blockaded atomic ensembles benefiting from fast collectively enhanced lightmatter couplings^{11,36,50,51,63}.
Methods
The methods used to generate the optical tweezer arrays and to load them are described in the first half of “Results“ section. The methods used to characterize the atomic ensembles are described in the second half of “Results“ section.
Data availability
The experimental data that support the findings of this study are available from the corresponding author on reasonable request.
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Acknowledgements
This work is supported by the ‘Investissements d’Avenir’ program through the Excellence Initiative of the University of Strasbourg (IdEx), the University of Strasbourg Institute for Advanced Study (USIAS) and is part of and supported by the DFG Collaborative Research Center ‘SFB 1225 (ISOQUANT)’. S.S., T.M.W., and M.M. acknowledge the French National Research Agency (ANR) through the Program d’Investissement d’Avenir under contract ANR17EURE0024.
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Y.W. and S.S. performed experiments. T.M.W., G.L., and M.M. made contributions to the experimental setup. Y.W., S.S., and S.W. analyzed the data. Y.W. and S.W. wrote the manuscript with input from all the authors.
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Wang, Y., Shevate, S., Wintermantel, T.M. et al. Preparation of hundreds of microscopic atomic ensembles in optical tweezer arrays. npj Quantum Inf 6, 54 (2020). https://doi.org/10.1038/s4153402002851
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