Scaling atom array assembly with grey molasses

We show that with a purely blue-detuned cooling mechanism we can densely load single neutral atoms into large arrays of shallow optical tweezers. With this ability, more efficient assembly of larger ordered arrays will be possible - hence expanding the number of particles available for bottom-up quantum simulation and computation with atoms. Using Lambda-enhanced grey molasses on the D1 line of 87Rb, we achieve loading into a single 0.63 mK trap with 89% probability, and we further extend this loading to 100 atoms at 80% probability. The loading behavior agrees with a model of consecutive light-assisted collisions in repulsive molecular states. With simple rearrangement that only moves rows and columns of a 2D array, we demonstrate one example of the power of enhanced loading in large arrays.

In quantum simulation and computing, the assembly of large arrays of individually-controllable particles is a frontier challenge. Ultracold gases of neutral atoms have long simulated quantum physics on a macroscopic scale, and quantum gas microscopes are now a window to microscopic dynamics [1,2]. However, the desire for control of individual atoms, in particular for quantum computing, motivates pursuing a bottom-up engineering approach [3][4][5][6]. Implementations of a Maxwell's demon based upon single atom imaging and rearranging have presented new opportunities in studies of multi-particle quantum dynamics [7][8][9][10][11][12][13][14][15], but compared to trapped ions, neutral atoms are still difficult to trap individually. In our work, we combine dense loading of large optical tweezer arrays using Λ-enhanced grey molasses (ΛGM) with atom imaging and rearrangement to form ordered atom arrays [16,17]. With ΛGM we load single atoms with 89(1)% efficiency in traps shallower than required for standard sub-poissonian loading [18] and nearly an order of magnitude shallower than required for previous enhanced loading [19]. Our key insight was to use blue-detuned cooling to photoassociate atoms exclusively to repulsive molecular states. Our technique will scale up neutral-atom array assembly by expanding rearrangement algorithms and by enabling considerably larger ordered arrays.
To isolate single atoms in optical tweezers or lattices, it is standard practice to drive light-assisted collisions in the collisional blockade regime using red-detuned light [18,20]. Here, atoms are photoassociated to attractive molecular states in which they accelerate towards each other and gain kinetic energy that predominantly expels both from the trap [ Fig. 1(a)]. If the collisions occur quickly enough to dominate the dynamics, as is the case in microtraps, a single atom is left about half the time. In the pioneering work of Ref. [19], after adding a blue-detuned laser to drive atoms into repulsive molecular states, the energy gained in the collision was tuned to induce single atom loss [19,[21][22][23]. Loading efficiency was enhanced to 90%, but at the cost of requiring large trap depths (U/k B ∼ 3 mK compared to 1 mK for red-  A common theme in enhanced loading to-date is that the red-detuned lasers required for cooling still drive lossy collisions in conflict with desired blue-detuned collisions. Here we resolve this conflict by cooling and photoassociating with the same blue-detuned laser [ Fig. 1(a)]. After the blue-detuned cooling modifies the atom number dis-tribution, we apply red-detuned light, which both assures only single atoms remain and images the atoms. A single optical tweezer with a trap depth of U/k B = 0.63 mK is loaded with 89(1)% efficiency, and a 10 × 10 array is loaded with 80.49(6)% efficiency [ Fig. 1(b)]. Dense loading could also be used in optical lattices or in microtraps in 3D [14,15]. We also demonstrate a proofof-principle rearrangement technique that relies on the enhanced loading to create a 6 × 6 defect-free array using a simplified sequence of parallel moves of entire rows and columns [ Fig. 1(c)] [9].
We first present results from loading a single optical tweezer using ΛGM, and compare to standard loading using red-detuned polarization gradient cooling (RPGC) [ Fig. 2(b,c)]. We capture 87 Rb atoms in a magneto-optical trap (MOT), cool them into a spatiallyoverlapped optical tweezer with depth U with either ΛGM or RPGC, and then image the atoms with RPGC [ Fig. 2(a)]. See Fig. 1(a) and the Appendix for laser configurations. The procedure is repeated to determine average loading efficiencies (see Appendix) [34]. Fig. 2(b,c) shows the loading probability P as a function of both detuning and trap depth for RPGC and ΛGM. With ΛGM we observe 89(1)% loading efficiency at (∆ ΛGM , U/k B ) = (45 MHz, 0.55 mK), and we can still load with ∼ 80% efficiency at trap depths of U/k B ≈ 0.27 mK. These findings are remarkable as with the same optical power we can load tweezer arrays that are more densely filled and two to three times larger compared to RPGC loading. The maximum RPGC loading of 64(1)% for (∆ RPGC , U/k B ) ≈ (−14 MHz, 1.1 mK) is among the highest reported for RPGC [9,10,22,23]. In the simplest picture of RPGC, one expects 50% loading, but, in agreement with other studies [23], additional processes result in ∼ 35% of the collisions causing only one atom to leave the trap.
A physically rich picture can be gained from studying the detuning dependence of ΛGM loading shifted resonance, is a key energy scale for the physics of the enhanced loading. At shifts smaller than 2U/h, the collision does not give a pair of zero-temperature atoms sitting at the bottom of the trap enough energy for either to escape, while at larger detunings both atoms will be expelled. A finite temperature, and hence an initial center of mass motion, will blur the transition, and indeed is necessary for inducing the desired single-atom loss. Although our data are roughly consistent with this picture, we look more closely by plotting the data of Fig. 2(c) against a dimensionless detuning h(∆ ΛGM −δ trap )/U . We do this for all data traces U/k B ≥ 0.65 mK [ Fig. 2(d)], and observe a number of interesting features, one of which is that the maximum loading peaks below the 2U/h shift (blue line).
To elucidate detailed trends, we have carried out a Monte-Carlo calculation of the collision dynamics. Most generally, we expect loading to be affected by both collisions and the ΛGM cooling performance, and both may be influenced by the non-trivial light shifts and polarization gradients in the tweezer traps. Modeling the interplay of these effects is beyond the scope of this letter, but we can understand the collisional process quantitatively if we assume the continuous ΛGM cooling can load at least a few atoms per trap, and re-thermalizes any atoms remaining after a collision. The simulation starts by preparing a Poisson-distributed number of atoms N atom with a mean numberN atom = 5 and temperature T , whereN atom was chosen > 2.5 to avoid loading zero atoms initially. To simulate the finite experiment cooling time, we calculate a finite number of 5000 time steps each having two atoms collide once if they are closer than 100 nm. A collision might eject none, one, or both atoms out of the trap depending on the final energy of each atom, which is determined by their pre-collision energy and the collisional energy gain E = h [∆ ΛGM − δ atom ]. This process continually reduces N atom in each time step. At the end, the RPGC imaging is simulated by assuming that it entails a fast collisional process at the start of the image where red-detuned collisions reduce atom numbers in a manner consistent with our red loading -namely we reduce any remaining N atom > 1 by 2 with a chance of 65% and by 1 with a chance of 35% until N atom ≤ 1. Fig. 2(d) shows the result of the Monte-Carlo simulation: During ΛGM-loading, the initial atom number (red dashed line) is reduced (cyan line). During RPGC imaging theN atom is further reduced [red line in Fig. 2(d)]. Fig. 2(e) shows how ΛGM and RPGC modify the Poisson distribution. We observe three physical regimes: For E 2U , little atom loss occurs during the ΛGM, but the imaging step reduces the number of atoms to 0 or 1, yielding a RPGC-like 65% loading. In contrast, for E 2U , two-body losses dominate resulting in ∼ 50% loading efficiency. At the transition E ≈ 2U , both single atom and two-body losses occur with roughly equal probability, and the reason why the maximal loading probability is found at E < 2U is because of the finite temperature and the fact that lower detunings favor one-body loss over two-body loss. Our model indicates no fundamental limitations to the loading efficiency and that by optimizing the trap size, atom temperature, and related parameters, it may be possible to reach higher loading fractions. Note that the simulation allows us to fit a mean atom temperature of T = 120(10) µK to our measured data. This value is consistent with the free-space ΛGM temperature we measure of T ≈ 50 µK, which is higher than typical values, likely due to non-ideal beam geometries (see Appendix).
We have also performed a loading study for an array of 10 × 10 optical tweezers, of which we display the measurement at U/k B = 0.55 mK [purple in Fig. 2(f)]. Compared to the single-trap data at similar U (black), the data are shifted to smaller detunings, and we observe a maximum loading of 80.49(6)%. These effects could be due to a variety of consequences of the larger array: variations in trap shape and depth or overall degradations of the optical spot sizes (see Appendix).
We now present an example of how dense loading opens up new opportunities in rearrangement algorithms for array assembly. After densely loading the array, we first obtain the location of each atom using a single image [left panel Fig. 3(a)]. Even with dense loading, the probability of loading a specific set of 6 × 6 traps is exponentially small [dashed lines in Fig. 3(b)]. However, there are many potential sets of (sometimes disjoint) 6 × 6 traps embedded in the 10 × 10 array. We then search for such a configuration of completely loaded 6 × 6 traps. If successful, we turn off the extra traps to remove the excess atoms, and then contract and shift the identified disjoint array in a single move (right panel) [9]. Currently, successful rearrangement to a square n = 36 array only works in 0.1% of cases due to unexpected loss observed when turning off rows and columns, in which an atom is effectively lost with a 17%-chance. But, as illustrated in Fig. 3(b), observing this array with this parallel technique would have been impossible without enhanced loading. In going from P = 60% to P = 80%, the percentage of experiment runs in which one could possibly extract a defect-free 6 × 6 array goes from 0.02% to 37%. Notably, this entire procedure is completed using only a pair of acousto-optic modulators to control the optical tweezers.
The full potential of dense loading using grey molasses will likely come when combined with the most advanced atom-by-atom rearrangement algorithms in 2D or 3D [10,14,15]. Because ΛGM loading is efficient in shallow traps, more loadable traps can be created with the same amount of optical power. Additionally, 2D algorithms fill defects in a target array of size n with a sequence of m ∝ (1−P )n 1.4 single moves, which we verified with Monte-Carlo simulations [10]. In scaling up array sizes, the time and number of moves required becomes lengthy, lowering the probability of successful rearrangement (S P ) as errors due to finite move fidelities and background collisions suppress this success rate as e −m [10]. Increasing the loading probability from P = 60% to P = 90% would decrease m by a factor of 4, making larger array sizes more obtainable and exponentially improving the success probability S P .
In conclusion, by gaining control over photoassociation to molecular states we have demonstrated enhanced loading of arrays of shallow optical tweezers. We have studied one particular blue-detuned cooling mechanism -ΛGM on the 87 Rb 5S 1/2 −5P 1/2 transition -but it is also known that grey molasses is effective on the 5S 1/2 − 5P 3/2 transition [33], and future studies could compare the salient molecular physics in each manifold [35]. Further, we expect our work will be the start of explorations of the interplay of collisions and cooling in microtraps for a host of blue-detuned cooling mechanisms both for alkali atoms, as well as a variety of other atomic species and molecules. ually and dynamically adjusted to control the position (intensity) of different tweezer-rows and columns. The relative phases of the tones are set to minimize intermodulation in the RF setup. The array of deflections created by the AOMs is then imaged by a 0.6-NA-objective lens into a glass cell. This creates a trap with a 0.68 µm waist for a single tweezer, and traps with an average waist of 0.75 µm for a 10 × 10 array. The standard deviation of the trap depths was minimized to 8% by optimizing the RF amplitudes. Trap depths are calibrated by measuring light-shifts of in-trap atomic transitions as a function of trap power [36]. The lifetime of atoms in the traps is limited to 5 s by the background pressure.

Laser Cooling and Loading
In all experiments, three beam paths are used to address the atoms. Two (diagonal) paths are along the diagonals of the xy-plane, and a third (acute) path in the xz-plane is at an angle of 55°from the z-axis to avoid the objective [34]. All lasers along these paths are retroreflected and in a σ + σ − polarization configuration.
Our magneto-optical trap (MOT) is spatially overlapped with the trap array and cools atoms for 500 ms to a temperature of ∼ 100 µK, measured by imaging its ballistic expansion. The cooling (repump) laser is reddetuned from the D 2 |F = 2 → |F = 3 (|F = 1 → |F = 2 ) transition, and applied on all three beam paths (on only the diagonal paths). In the case of the 20-mslong RPGC stage we cool the atoms to ∼ 10 µK. For this, we detune the cooling (repump) laser by ∆ RPGC (20 MHz), set the intensities at 1.3 I sat (0.1 I sat ) on the diagonal paths and 4.5 I sat (0 I sat ) on the acute path, and zero the magnetic fields.
In the case of the 200-ms-long Λ-enhanced gray molasses (ΛGM) stage, we apply a cooling laser that is detuned by ∆ ΛGM from the D 1 |F = 2 → |F = 2 tran-sition at 2.5 I sat (0.4 I sat ) on the acute (diagonal) paths. We create the coherent repump beam from the cooling laser on the acute path using an electro-optic modulator. The repump beam is detuned by ∆ ΛGM + 0.14 MHz from the D 1 |F = 1 → |F = 2 transition and at 1.5 I sat . Note that the optimal ΛGM free-space temperature of 50 µK is reached for ∆ ΛGM ≈ 15 MHz and is likely limited by the beam path geometry and repump light configuration.

Imaging, Data, and Statistics
Regardless of the loading configuration, we image the atoms using another RPGC stage with the cooling beam (∆ RPGC = 19 MHz at 3 I sat ) only on the acute path.
We alternate the tweezer-light with the imaging light at 2 MHz to scatter light when atoms are experiencing no light shifts. This configuration is maintained for 20 ms during which we collect scattered photons on an EMCCD camera, superbinned to 4 × 4 pixels to reduce readout noise.
At every atom location individually, to determine a count threshold that indicates the presence of an atom in the trap, we create a histogram of all counts during an experiment and fit it with a sum of two Gaussians. The threshold with maximal fidelity F is found, where F = 1 − (E f p + E f n ), with E f p (E f n ) being the expected rate of false positives (false negatives) from the fits. This converts a sequence of counts to a sequence of Booleans which is averaged to determine the loading probability. All errors reported indicate 1σ equal-tailed Jeffrey's prior confidence intervals [37]. The loading efficiencies reported in the main text for RPGC (64(1)%), ΛGM (89(1)%), and 10 × 10-ΛGM (80.49(6)%) were obtained by analyzing 2000, 1000, and 1000-per-atom repetitions with threshold fidelities 0.987, 0.998, and 0.993 respectively.