Protocol for autonomous rearrangement of cold atoms into low-entropy configurations
PProtocol for autonomous rearrangement of cold atoms into low-entropy configurations
Matthew A. Norcia ∗ Institut f¨ur Quantenoptik und Quanteninformation,¨Osterreichische Akademie der Wissenschaften, Innsbruck, Austria (Dated: January 19, 2021)The preparation of low-entropy starting conditions is a key requirement for many experimentsinvolving neutral atoms. Here, we propose a method to autonomously assemble arbitrary spatialconfigurations of atoms within arrays of optical tweezers or lattice sites, enabled by a combinationof tunneling and ground-state laser cooling. In contrast to previous methods, our protocol doesnot rely on either imaging or evaporative cooling. This circumvents limitations associated withimaging fidelity and loss, especially in systems with small spatial scales, while providing a substantialimprovement in speed relative to evaporative approaches. These features may make it well-suitedfor preparing arbitrary initial conditions for Bose-Hubbard or Rydberg interacting systems.
I. INTRODUCTION
Microscopic control of neutral atoms has lead to a re-cent explosion of interest for applications in quantumsimulation, quantum information processing, and quan-tum enhanced metrology [1–4]. Achieving configurationsof large numbers of atoms with well-defined positions andmotional states is a key capability at the forefront of mod-ern experiments. Currently, two main approaches areused to accomplish this: In the first, evaporative coolingis used to remove entropy from the atomic system, afterwhich point the atoms may be adiabatically loaded intoa desired potential landscape such as an optical lattice[1]. Entropy redistribution techniques can then be usedto further reduce the entropy of a sub-region [5–7]. In thesecond approach, atoms are stochastically loaded into thetightly confining potential of optical tweezers or latticesites, with light-assisted collisions leading to either zeroor one atom in each site [8]. The atoms are then rear-ranged into the desired configuration based on informa-tion gained from imaging [9–13]. Of particular relevancefor this proposal are recent experiments where atoms arealso laser-cooled to their motional ground states [14].While highly successful, these two existing approacheseach have limitations. Evaporation is typically slow,leading to long experimental cycle times. This can beproblematic for quantum simulation experiments, wherelarge numbers of experimental trials are required, and inthe context of metrology, where long cycle times degradethe stability of atomic clocks or other sensors [15]. Ap-proaches based on measurement and rearrangement canprovide a substantial advantage in terms of speed, as lasercooling replaces evaporation as the means of entropy re-moval. However, the resulting level of entropy can belimited by imaging fidelity and loss. Further, these ap-proaches require an imaging system capable of localizingand manipulating atoms at the single-site level, which hasso far prevented the application of rearrangement proto-cols to systems with sub-micron-scale lattices, desirable ∗ E-mail: [email protected] for tunneling.In this work, we propose an alternate approach thatdoes not require either evaporation or imaging. Like theimaging-based approach, our method starts with an arrayof stochastically loaded atoms to be rearranged into adesired configuration. However, rather than relying onimaging to determine the initial site occupations, atomsare rearranged in an autonomous manner. For clarity, wewill leave further motivation and connections to relatedconcepts such as Thouless pumping, algorithmic coolingand autonomous stabilization for later (Section V) andproceed immediately with a description of our proposedmechanism.
II. PROTOCOL OVERVIEW
The goal of our protocol is to transfer atoms into spe-cific “target” sites of an optical potential from stochasti-cally loaded “reservoir” sites by repeated application ofirreversible “shift” operations, after which point atomsremaining in reservoir sites are removed (Fig. 1a). Theshift operation can be understood in a simple subsystemconsisting of a single target site and a single reservoirsite. If initially the target site is empty and the reser-voir site is occupied by an atom, the atom is transferredfrom the reservoir site to the target site. For all otherconfigurations, the state remains unchanged.Our proposed implementation for the shift operation isshown in Fig. 1b. We assume that either zero or one atominitially occupies the motional ground state (denoted by n = 0) of each of two optical potential wells, defininga reservoir and target site. Excited motional states areassumed to be empty. The n = 0 state of the reservoirwell is then brought onto resonance with the first excitedmotional state in the target well ( n = 1), allowing theatom to tunnel between the two wells at a rate that welabel J . The potential could either be held in this reso-nant configuration for a time π/J , or the relative depthof the wells could be adiabatically ramped through reso-nance to gain insensitivity to offset errors at the expenseof reduced speed. In either case, an atom occupying thereservoir site will tunnel into the target site. a r X i v : . [ c ond - m a t . qu a n t - g a s ] J a n P = 0 P = 1/2 P = 2/3 P = 1 a. b.c.a.b. a.c. b.d.a.c. b. even shu ffl eodd shu ffl e
2D even shu ffl e2D odd shu ffl einitial statefinal statefinal stateinitial state J U b. i.ii. iii. shiftcoolmix reservoir remove reservoirstochastic loading repeat a. FIG. 1. a. Conceptual overview of method. An array of lat-tice sites, consisting of target (red circles) and reservoir (nocircles) sites is initially loaded in a stochastic manner. Af-ter subsequent cycles of directional tunneling into target sitesand between reservoir sites, interleaved with dissipative cool-ing steps, target states within the array are occupied. Atomsremaining in the reservoir sites are then removed, leaving atailored low-entropy configuration. b. Basic building blockof implementation. i. An optical potential consisting of twoneighboring wells is imbalanced to create a tunneling reso-nance between the ground state of the reservoir site (rightwell) and the first motionally excited state of the target site(left well). An atom occupying the reservoir site will thentunnel into an empty target site. ii.
If an atom already oc-cupies the target site, contact interactions shift the tunnelingresonance, preventing double occupation of the target site. iii
After tunneling, ground-state laser cooling is applied to returnatoms to the ground state of their respective sites, providingdissipation and preventing future tunneling away from thetarget site.
In order to prevent double occupation of the target site,the contact interactions U associated with atoms in the n = 0 and n = 1 states of the same site must exceed thetunneling rate J . This ensures that tunneling from thereservoir to the target site is not resonant if both sitesare initially occupied, so the atom in the reservoir wellremains in place. To make the process non-reversible – toprevent the atom from tunneling out of the target site ona later cycle – the shift operation concludes by subjectingall atoms to ground-state cooling.In general, a target site can be coupled to several reser-voir sites. In this case, one of the atoms (if present)in any of the reservoir sites may tunnel into the targetsite. Unlike the two-site case, this process is inherentlyprobabilistic, depending on the relative degeneracy of theconfigurations with occupied and unoccupied target sites(see appendix C for details). However, by sufficient rep- etition of the shift protocol, the target sites can still beloaded with high probability. a.c. b. FIG. 2. Applications to different geometries. a. Target sitescan be defined within a two-dimensional lattice by applyinglocal optical potentials to these sites, for example with opticaltweezers. Shift operations transfer atoms into target sitesfrom neighboring reservoir sites. b. The same concept canbe applied to the filling of layers within a three-dimensionallattice. Here, shift operations are applied between layers, andtunneling within layers enables reservoir atoms to find emptytarget sites. c. A continuous region can be filled by repeatedlyshifting atoms towards a potential barrier, represented by thegray region on the left of the array. In all protocols, atomsare removed from the reservoir sites after filling of the targetsites or layers is complete (see appendix D)
III. SPECIFIC IMPLEMENTATIONS
Variations on this simple building block can be usedin different geometrical configurations, for example to fillarbitrary sites within a two-dimensional plane, or to filltarget planes within a three-dimensional lattice. It canalso be used in more complex rearrangement algorithms,for example to fill contiguous regions within one- or two-dimensional lattices. In this section we briefly describethree such variations before turning to a more detailedanalysis of their performance.The conceptually simplest application of our rear-rangement protocol can be used to fill arbitrary targetsites of a two-dimensional lattice, as illustrated in Fig. 2a.In this approach, a background lattice defines a two-dimensional plane of sites among which the atoms cantunnel, and an auxiliary potential (generated using op-tical tweezers or superlattices), is applied to the targetsites. All other sites are considered reservoir sites. The P = 0 P = 1/2 P = 2/3 P = 1 a. b.c.a.b. a.c. b.d.a.c. b. even shu ffl eodd shu ffl e
2D even shu ffl e2D odd shu ffl einitial statefinal statefinal stateinitial state J U b. i.ii. iii. shiftcoolmix reservoir remove reservoirstochastic loading repeat a. FIG. 3. Protocol for filling uniform regions without addi-tional dimensions. By alternately shifting even and odd pairsof atoms, atoms are shuffled to one side, where they bunchup against a barrier (barrier not pictured). The shift opera-tions are implemented in parallel by superimposing an opti-cal lattice (red lines) with an auxiliary potential (blue lines)that both creates an imbalance between neighboring wellsand pushes down the barrier in the combined potential (blacklines), enabling rapid tunneling. The phase of the additionalpotential is shifted relative to the lattice to shift even (upper)or odd (lower) pairs. Right: The shuffle operation can eas-ily be extended to two dimensions simply by applying shuffleoperations to many rows in parallel. Shuffling of each row isindependent of others. shift operation is implemented by increasing the depth ofthe auxiliary potential, adiabatically ramping the energyof the n = 1 state of the target sites across the n = 0states of the reservoir sites. Importantly, the reservoirsites are assumed to be degenerate in energy, so atomscan also tunnel between them. This serves the impor-tant function of mixing the reservoir atoms so that ifa given target site initially lacked neighboring atoms, itmay attain them in future cycles. During the ground-state cooling phase, the depth of the background latticeis increased, localizing each atom on its lattice site, theauxiliary potential decreased, and all atoms cooled to n = 0.The same concept can be used to fill target planes in-stead of target sites, as shown in Fig. 2b. In this case,the background lattice is three-dimensional, and the aux-iliary potential shifts the energy of entire planes. Theplane shifted to the lowest energy becomes the targetplane, and its neighbors the reservoir. The protocol thenproceeds as before, with atoms repeatedly shifted intothe target plane. Again, we assume that the reservoirsites in a single plane are degenerate, so tunneling withinthe planes serves to randomize the location of atoms, en-abling high-probability loading of the target plane.The two preceding variations rely on a high connec-tivity of target sites to reservoir sites, and of reservoir sites to each other. This can be limiting if one wishesto prepare a state that is uniformly filled without usingadditional lattice dimensions. To get around this limi-tation, a procedure of repeated directional shifts can beused to shuffle atoms into a target region of a lattice, asshown in Fig. 2c. This protocol fundamentally operatesin a single dimension, but can be used to generate two- orthree-dimensional regions with simultaneous applicationto all rows in the array.With this in mind, we consider a single row within athree-dimensional optical lattice, which is tightly confin-ing in the directions orthogonal to its length. To create auniformly filled section of the lattice, a potential barrieris applied to one end of a desired region (here,the leftend), and shift operations are applied simultaneously toall non-overlapping pairs of lattice sites with an even in-dex of the left well (even pairs), alternating with shiftoperations applied to all of the pairs with an odd indexof the left well (odd pairs), as shown in Fig. 3. We label apair of such shift operations a shuffle. Repeated shufflescause the atoms to move to the left, until they encountereither the barrier or other atoms, at which point furthertunneling is suppressed either by the barrier or interac-tion potential. After this point, atoms beyond a certaindistance from the barrier can be removed from the sam-ple, leaving a uniformly filled region of the lattice (ap-pendix D).Because the shuffling operations can be applied to allrows of a 2D array simultaneously and independently, weexpect the characteristic time and error rates when fillingan array of M rows and N columns to be the same asthose for a single one-dimensional array of length N . Thisfeature represents a key strength of this technique – whilefilling a single one-dimensional array takes a relativelylarge number of steps, because the atoms are only moveda single site at a time, the shift operations can be appliedin parallel to all atoms at the same time, leading to thefavorable scaling with respect to total number of atomsin the array, especially when M exceeds N . IV. PROTOCOL PERFORMANCE
The effectiveness of our protocol can be quantified indifferent situations by the number of cycles required toproduce a desired filling (which would dictate experimen-tal cycle times), and by the error rate in the final con-figuration. We quantify the former in terms of n , thetypical number of cycles to reach 99% filling, as such afilling would enable assembly of perfect arrays of sizesthat can be difficult to simulate classically. We definethe error rate ξ as the minimum achievable fraction ofempty target sites.Both n and ξ may be limited by both fundamental as-pects of the protocol and by experimental imperfections.Fundamental limits include the probabilistic distributionof atoms in reservoir sites, and the probabilistic nature oftunneling into a target site that has more than one neigh- P = 0 P = 1/2 P = 2/3 P = 1 a. b.c.a.b. a.c. b.d.a.c. b. even shu ffl eodd shu ffl e
2D even shu ffl e2D odd shu ffl einitial statefinal statefinal stateinitial state J U b. i.ii. iii. shiftcoolmix reservoir remove reservoirstochastic loading repeat a. FIG. 4. Effects of errors on target site (left column) and layerfilling (right column) protocols. Red (black) points represent50% initial filling, with F = 1 , (0 . F = 1. a. Filling fraction 1 − ξ versus number of cycles. Vertical lines represent the numberof steps n required to reach 99% filling for each condition.Loss not included. Inset: effect of tunneling fidelity on n .For moderate infidelities, the effect is minor. Gray line isintuitive approximation discussed in text. b. Minimum holefraction ξ fraction versus per-cycle loss, for same conditionsas (a). boring reservoir site. Experimental imperfections can bebroken into two categories: fidelity errors and loss er-rors. Fidelity errors represent imperfect tunneling (forexample due to imperfections in the optical potential orimperfect adiabatic transfers), leading to a probabilityof desired tunneling events reduced by a factor F belowtheir idealized value. Loss errors, quantified by the per-cycle atom loss probability (cid:15) , represent atom loss fromthe system due to effects such as collisions with back-ground gas due to imperfect vacuum, double-occupancyof a lattice site, or imperfect ground state cooling. De-tailed estimates for experimentally achievable fidelitiesand loss rates can be found in appendix B, indicatingthat fidelities F ≥ . (cid:15) ≤ .
01 should be attain-able.For all variations of our protocol, we find through sim-ulation that for small loss rates and infidelity, the mini-mum occupancy error ξ (cid:39) α(cid:15)/F . α then provides a figureof merit that can be used to assess the performance ofa given protocol, target state, and initial filling fraction.Intuitively, α/F is related to the number of cycles re-quired to repair a defect caused by loss. Because (cid:15) mustbe kept small for any useful application of our protocol,the effect of loss is minor in determining the timescalerequired to reach a quasi-equilibrium condition, and wecompute n (which is of course not a good metric forconditions where ξ ≥ .
01) in the absence of loss.We estimate the performance of our protocols using aclassical simulation of the whole system with with pa- rameters informed by master-equation-based simulationsperformed on much smaller subsystems, details of whichare provided in appendix C. We break each filling cycleinto three steps: a filling step in which atoms may tunnelinto an unoccupied target site from neighboring reservoirsites, a mixing step where the atoms are allowed to re-arrange between reservoir sites (where applicable), and aloss step. In the experiment, filling and mixing could oc-cur simultaneously, and be interleaved with ground statecooling (which is not represented in the simulation aswe explicitly track only site occupation, but include theeffects of imperfect cooling as effective loss).To benchmark the performance of our first protocolvariation, we consider a specific target state – a grid withquarter filling of the base lattice, as shown in Fig. 2a.When the filling fraction of the target state is much lowerthan the typical filling of stochastically loaded lattices,there is a high probability of having enough atoms tofill the target sites for systems of at least moderate size.The filling process for such a target state is shown inFig. 4a. In the absence of imperfections, we find that n = 6 for 50% initial filling. Very roughly, this indicatesthat on each cycle, the chance that a target site remainsunoccupied is reduced by a factor or about 2 (though thisfactor changes slightly over the filling process as atomsare moved to the target sites, reducing the density of thereservoir). The effect of imperfect tunneling fidelity isto reduce this factor, leading to slower filling. For half-filling, we find that an intuitively derived formula: n = Log − P (0 . P = 0 . F represents the chance offilling an unfilled target site on any cycle, provides goodagreement with the results of our simulation.The filling of layers is limited by conceptually similarfactors to the filling of target sites. However, the numberof neighbors connected to each target site, and the con-nectivity of the reservoir sites are not the same for the twogeometries, leading to quantitative differences. Further,because the performance of these protocols benefits froma high density of reservoir atoms, the layer filling config-uration presents an opportunity: atoms can be broughtinto the reservoir planes that neighbor the target planefrom farther away planes. This leads to a higher den-sity of atoms in reservoir planes that neighbor the targetplane, allowing for more rapid filling of vacancies in thetarget plane. This both reduces the number of cycles n required to fill the target plane, and allows errors associ-ated with loss to be repaired more quickly, reducing theerror rate ξ .As a specific example, we consider a implementationbased on five layers, numbered one through five, that areinitially stochastically loaded with half filling, with planenumber three the target plane. First, shift operations areapplied simultaneously from plane one to two, and fromfive to four. This increases the density of atoms in planestwo and four. After this, repeated shift operations areapplied into plane three from both of its neighbors, untilthe desired filling fraction is reached.The filling performance of this protocol is shown infigure 4b, along with the same prediction for the effectsof imperfect fidelity as for the target sites. In this case,we find n = 7 for perfect fidelity and 50% initial fill-ing, and that for lower fidelities our intuitive predictionslightly underestimates the number of cycles required.This is because fewer atoms are transferred from planesone and five into the central three in the first cycle, re-ducing the density of available reservoir atoms. We notethat this quantitative similarity with the target statesprotocol is specific to the example configurations that wechose, and does not represent a general property of thetwo approaches.For both benchmark cases described above (for targetsites and target layers with 50% initial filling), we findfrom simulation that α (cid:39) .
25 (Fig.4c, d, red points).This number roughly reflects the inverse of the probabil-ity that a defect is filled on a given filling cycle. It isinfluenced both by the availability of atoms in neighbor-ing reservoir sites, and the inherently probabilistic na-ture of filling when a target site is coupled to multiplereservoir sites (see appendix C). As mentioned above,these numbers can be dramatically improved by increas-ing the density of atoms in the reservoir states (Fig.4c,d, blue points), either by shifting atoms into those sites,or through protocols that can lead to filling fractions sig-nificantly above 50% [16–18].The filling process of a one-dimensional sub-array un-der our shuffling protocol is shown in Fig.5. In the ab-sence of errors (Fig.5a, b), and starting from an ini-tial randomly half-filled array long enough to containat least N atoms with high probability, we make an in-formed guess that the typical number of shuffles requiredto fill a sub-array of length N is well approximated by n = N + C ( N ), where C ( N ) is a logarithmic correctionfactor. This guess is motivated by thinking of holes beingshuffled out of the sub-array — each hole can move upto one sites per cycle (either even or odd shuffle), whichcontributes the linear term. However, a hole cannot be-gin to move until there is an adjacent atom to fill it, soif we start with several adjacent holes, the one that mustmove the farthest does not begin to move until its neigh-bors have been filled. Because the longest typical numberof adjacent holes within the sub-array should scale withlog ( N ) [19], we expect this to set the scale of C ( N ),and find empirically that C ( N ) = log ( N ) indeed pro-vides reasonable agreement with simulation (Fig. 5b).In the shuffle protocol, the per-site error rate increaseswith the length of the region being filled. For an targetregion of N by M lattice sites, where shuffling is per-formed in the direction whose size is N , we find fromsimulation that the achievable error rate is given approx-imately by ξ (cid:39) N (cid:15)/ F , corresponding to α = N/
2. In-tuitively, this can be understood because a hole in thetarget region must be moved N/ N/ F shuffling cycles (Fig.5c, d). Note againthat while the performance scales unfavorably with N ,the number of columns in the target region, it is inde-pendent of the number of rows M . As a benchmark case, given per-shift losses of (cid:15) = 0 .
01, one could expect toachieve perfect filling of a five-by-five array in approxi-mately half of attempts. In experiments where perfectarrays are not needed (for example, for those targetingentanglement-enhanced metrology [20]), the system sizecould be dramatically increased. P = 0 P = 1/2 P = 2/3 P = 1 a. b.c.a.b. a.c. b.d.a.c. b. even shu ffl eodd shu ffl e
2D even shu ffl e2D odd shu ffl einitial statefinal statefinal stateinitial state J U b. i.ii. iii. shiftcoolmix reservoir remove reservoirstochastic loading repeat a. FIG. 5. Shuffling performance in idealized case and with im-perfect fidelity and loss. a. Example of lattice site occupationversus number of shuffling cycles, with perfect tunneling andno loss. Each cycle consists of either an even or an odd shift.After an initial traffic jam, the atoms shuffle at a rate of onesite per step until they are blocked by other atoms. b. Num-ber of cycles required to reach 99% filling ( n ) for differenttarget region lengths N . Dashed black line is the predictionfrom the text, and dashed grey line is N . c. Example oflattice site occupation versus number of shuffling cycles inthe presence of finite shift fidelity ( F = 0 .
9) and loss percycle( (cid:15) = 0 . d. Prevalence of holes in the final arrayversus loss (cid:15) and N . V. DISCUSSION AND OUTLOOK
So far, we have described several ways in which low-entropy arrays of atoms may be autonomously assembledfrom stochastically populated initial conditions, poten-tially combining advantages of evaporative techniques,and those based on measurement and rearrangement.Because entropy is removed through the cooling pro-cess, rather than imaging, our techniques are suitablefor systems where imaging may have limited fidelity orbe accompanied by loss, for example in atomic specieswith complex level structures or in systems with smalllattice spacings. This may make these techniques es-pecially suitable for systems where preparing atoms inthe motional ground state of a short wavelength opticallattice is already required, such as those seeking to ex-plore Hubbard physics. Further, because the definitionof the target sites is can be chosen at will (up to certainconstraints), this approach could be useful for preparinginitial conditions for Hubbard-regime experiments thatrequire or benefit from arbitrary, non-uniform startingconditions, such as sampling problems [21] or quantumwalk experiments [22], or for simulators of spin-models[2, 3], where interactions can be mediated by Rydberginteractions with range larger than the lattice spacing.There are some similarities between our method andpreviously proposed and demonstrated methods for algo-rithmic cooling, in that both rely on contact interactionsbetween atoms in a lattice to allow or prevent coherenttransitions that ultimately enable the reduction of fill-ing errors [23–26]. However, algorithmic cooling relieson overfilling the lattice and then removing the excessatoms, so typically requires an evaporatively preparedsample to begin with. In contrast, because our methodworks by irreversibly moving atoms towards the targetsites, it is specifically suitable for the relatively sparse ini-tial fillings typically associated with direct laser coolinginto micron-scale optical potentials. The directional tun-neling employed here is similar to the topological chargepumping employed in a Thouless pump, which has beendemonstrated to be highly robust to experimental im-perfections [27–30]. However, the addition of ground-state cooling makes the pumping irreversible, enablingthe compression of particles in the sites of an optical lat-tice and the removal of spatial entropy.Our scheme also bears resemblance to autonomousquantum stabilizers for bosonic systems, for examplethose used to stabilize highly correlated photonic statesin arrays of superconducting qubits [31, 32]. With suit-able modifications, the basic ideas behind our approachcould also be used to stabilize analogous atomic systemsagainst the effects of atom loss, and to provide a well-controlled means of environmental coupling. This willbe a direction of future exploration, and may benefitfrom the use of even narrower transitions, such as theclock transitions in alkaline earth atoms. In a very re-cent proposal [33], a related mechanism is explored toautonomously assemble and stabilize atoms within opti-cal lattices, with dissipation provided by interaction withan atomic bath, rather than laser cooling as is used here.If extended to molecules in tweezers or optical lattices[34–36], variations on the method proposed here could behighly advantageous to increase the filling fractions andreduce disorder, as nondestructive imaging of moleculesis challenging.
VI. ACKNOWLEDGEMENTS
M.A.N. thanks Adam Kaufman and Hannes Pichler forinsightful discussions and feedback on the manuscript.This project has received funding from the EuropeanUnion’s Horizon 2020 research and innovation pro-gramme under the Marie Sk(cid:32)lodowska-Curie grant agree-ment No 801110 and the Austrian Federal Ministry ofEducation, Science and Research (BMBWF). It reflectsonly the author’s view, the EU Agency is not responsi-ble for any use that may be made of the information itcontains [1] C. Gross and I. Bloch, “Quantum simulations with ul-tracold atoms in optical lattices,” Science , 995–1001(2017).[2] A. Browaeys and T. Lahaye, “Many-body physics withindividually controlled rydberg atoms,” Nature Physics ,1–11 (2020).[3] D. Weiss and M. Saffman, “Quantum computing withneutral atoms,” Physics Today (2017).[4] L. I. R. Gil, R. Mukherjee, E. M. Bridge, M. P. A. Jones,and T. Pohl, “Spin squeezing in a Rydberg lattice clock,”Phys. Rev. Lett. , 103601 (2014).[5] C. Chiu, G. Ji, A. Mazurenko, D. Greif, and M. Greiner,“Quantum state engineering of a Hubbard system withultracold fermions,” Phys. Rev. Lett. , 243201 (2018).[6] A. Kantian, S. Langer, and A. Daley, “Dynamical disen-tangling and cooling of atoms in bilayer optical lattices,”Phys. Rev. Lett. , 060401 (2018).[7] B. Yang, H. Sun, C.-J. Huang, H.-Y. Wang, Y. Deng,H.-N. Dai, Z.-S. Yuan, and J.-W. Pan, “Cooling andentangling ultracold atoms in optical lattices,” Science(2020).[8] N. Schlosser, G. Reymond, I. Protsenko, and P. Grang-ier, “Sub-poissonian loading of single atoms in a micro-scopic dipole trap,” Nature , 1024–1027 (2001).[9] D. S. Weiss, J. Vala, A. V. Thapliyal, S. Myrgren,U. Vazirani, and K. B. Whaley, “Another way to ap-proach zero entropy for a finite system of atoms,” Phys.Rev. A , 040302(R) (2004).[10] Y. Miroshnychenko, W. Alt, I. Dotsenko, L. F¨orster,M. Khudaverdyan, D. Meschede, D. Schrader, andA. Rauschenbeutel, “An atom-sorting machine,” Nature , 151–151 (2006).[11] H. Kim, W. Lee, H. Lee, H Jo, Y. Song, and J. Ahn,“In situ single-atom array synthesis using dynamic holo-graphic optical tweezers,” Nat. Comm. , 1–8 (2016).[12] D. Barredo, S. de L´es´eleuc, V. Lienhard, T. Lahaye, andA. Browaeys, “An atom-by-atom assembler of defect-freearbitrary two-dimensional atomic arrays,” Science ,1021–1023 (2016).[13] M. Endres, H. Bernien, A. Keesling, H. Levine, E. R.Anschuetz, A. Krajenbrink, C. Senko, V. Vuleti´c,M. Greiner, and M.D. Lukin, “Atom-by-atom assemblyof defect-free one-dimensional cold atom arrays,” Science , 1024–1027 (2016).[14] A. Kumar, T.-Y. Wu, F. Giraldo, and D.S. Weiss, “Sort-ing ultracold atoms in a three-dimensional optical latticein a realization of Maxwell’s demon,” Nature , 83–87(2018).[15] G. Dick, “Local oscillator induced instabilities in trappedion frequency standards,” Proc. of Precise Time andTime Interval , 133–147 (1987).[16] T. Grunzweig, A. Hilliard, M. McGovern, and M. F. An-dersen, “Near-deterministic preparation of a single atomin an optical microtrap,” Nat. Phys. , 951 (2010).[17] B.J. Lester, N. Luick, A.M. Kaufman, C.M. Reynolds,and C.A. Regal, “Rapid production of uniformly filledarrays of neutral atoms,” Phys. Rev. Lett. , 073003(2015).[18] M.O. Brown, T. Thiele, C. Kiehl, T.-W. Hsu, and C.A.Regal, “Gray-molasses optical-tweezer loading: Control-ling collisions for scaling atom-array assembly,” Phys. Rev. X , 011057 (2019).[19] M.F. Schilling, “The longest run of heads,” The CollegeMathematics Journal , 196–207 (1990).[20] J. Van Damme, X. Zheng, M. Saffman, M.G. Vavilov,and S. Kolkowitz, “Impacts of random filling on spinsqueezing via rydberg dressing in optical clocks,” arXivpreprint arXiv:2010.04776 (2020).[21] G. Muraleedharan, A. Miyake, and I.H. Deutsch, “Quan-tum computational supremacy in the sampling of bosonicrandom walkers on a one-dimensional lattice,” New Jour-nal of Physics , 055003 (2019).[22] J. Kempe, “Quantum random walks: an introductoryoverview,” Contemporary Physics , 307–327 (2003).[23] P. Rabl, A.J. Daley, P.O. Fedichev, J.I. Cirac, andP. Zoller, “Defect-suppressed atomic crystals in an op-tical lattice,” Phys. Rev. Lett. , 110403 (2003).[24] M. Popp, J.-J. Garcia-Ripoll, K.G. Vollbrecht, and J.I.Cirac, “Ground-state cooling of atoms in optical lat-tices,” Phys. Rev. A , 013622 (2006).[25] Waseem S Bakr, Philipp M Preiss, M Eric Tai, RuichaoMa, Jonathan Simon, and Markus Greiner, “Orbitalexcitation blockade and algorithmic cooling in quantumgases,” Nature , 500–503 (2011).[26] Malte C Tichy, Klaus Mølmer, and Jacob F Sherson,“Shaking the entropy out of a lattice: Atomic filtering byvibrational excitations,” Phys. Rev. A , 033618 (2012).[27] O. Romero-Isart and J.J. Garcia-Ripoll, “Quantumratchets for quantum communication with optical super-lattices,” Phys. Rev. A , 052304 (2007).[28] M. Lohse, C. Schweizer, O. Zilberberg, M. Aidelsburger,and I. Bloch, “A thouless quantum pump with ultracoldbosonic atoms in an optical superlattice,” Nat. Phys. ,350–354 (2016).[29] S. Nakajima, T. Tomita, S. Taie, T. Ichinose, H. Ozawa,L. Wang, M. Troyer, and Y. Takahashi, “Topologicalthouless pumping of ultracold fermions,” Nat. Phys. ,296–300 (2016).[30] J. Koepsell, S. Hirthe, D. Bourgund, P. Sompet, J. Vi-jayan, G. Salomon, C. Gross, and I. Bloch, “Robust bi-layer charge-pumping for spin-and density-resolved quan-tum gas microscopy,” arXiv preprint arXiv:2002.07577(2020).[31] R. Ma, C. Owens, A. Houck, D.I. Schuster, and J. Simon,“Autonomous stabilizer for incompressible photon fluidsand solids,” Phys. Rev. A , 043811 (2017).[32] R. Ma, B. Saxberg, C. Owens, N. Leung, Y. Lu, J. Si-mon, and D.I. Schuster, “A dissipatively stabilized mottinsulator of photons,” Nature , 51–57 (2019).[33] V. Sharma and E. Mueller, “Driven-dissipative controlof cold atoms in tilted optical lattices,” arXiv preprintarXiv:2101.00547 (2021).[34] Amodsen Chotia, Brian Neyenhuis, Steven A Moses,Bo Yan, Jacob P Covey, Michael Foss-Feig, Ana MariaRey, Deborah S Jin, and Jun Ye, “Long-lived dipolarmolecules and feshbach molecules in a 3d optical lattice,”Physical review letters , 080405 (2012).[35] Lo¨ıc Anderegg, Lawrence W Cheuk, Yicheng Bao, SeanBurchesky, Wolfgang Ketterle, Kang-Kuen Ni, andJohn M Doyle, “An optical tweezer array of ultracoldmolecules,” Science , 1156–1158 (2019). [36] William B Cairncross, Jessie T Zhang, Lewis RB Picard,Yichao Yu, Kenneth Wang, and Kang-Kuen Ni, “Assem-bly of a rovibrational ground state molecule in an opticaltweezer,” arXiv preprint arXiv:2101.03168 (2021).[37] Y. N. Martinez de Escobar, P. G. Mickelson, P. Pellegrini,S. B. Nagel, A. Traverso, M. Yan, R. Cˆot´e, and T. C. Kil-lian, “Two-photon photoassociative spectroscopy of ul-tracold Sr,” Phys. Rev. A , 062708 (2008).[38] M.A. Norcia, A.W. Young, and A.M. Kaufman, “Mi-croscopic control and detection of ultracold strontium inoptical-tweezer arrays,” Phys. Rev. X , 041054 (2018).[39] A. Cooper, J.P. Covey, I.S. Madjarov, S.G. Porsev, M.S.Safronova, and M. Endres, “Alkaline-earth atoms in op-tical tweezers,” Phys. Rev. X , 041055 (2018).[40] J. Sebby-Strabley, M. Anderlini, P.S. Jessen, and J.V.Porto, “Lattice of double wells for manipulating pairs ofcold atoms,” Phys. Rev. A , 033605 (2006).[41] I. Bloch, J. Dalibard, and W. Zwerger, “Many-bodyphysics with ultracold gases,” Rev. Mod. Phys. , 885–964 (2008).[42] D.J. Wineland, W.M. Itano, J.C. Bergquist, and R.G.Hulet, “Laser-cooling limits and single-ion spectroscopy,”Phys. Rev. A , 2220 (1987). Appendix A: Experimental Implementation
The success of this method relies on strong contact in-teractions between atoms to prevent multiple-occupancyerrors, and a level structure compatible with high-fidelityground-state laser cooling. One atomic species of in-terest that meets these criteria would be Sr. Sris a boson (for fermionic species, the contact interac-tion vanishes for atoms of the same spin state in dif-ferent motional states of the same well, so our protocolwould not be applicable without somehow using multiplespin states), and has an unusually high s-wave scatteringlength of roughly 800 times the Bohr radius [37], lead-ing to large interaction shifts U . Strontium also hasa narrow-linewidth optical transition with a linewidthof 7.5 kHz, suitable for ground-state sideband cooling[38, 39]. Other bosonic species could be used as well, pro-vided that strong enough interactions can be achieved,for example with magnetic Feshbach resonances, and thatground-state cooling is possible, for example using Ra-man sideband cooling.In practice, the optical potentials could be imple-mented in several ways, with appropriate modificationsdepending on the variant of the protocol. In any case, weassume the atoms to be confined in a three-dimensionaloptical lattice potential, whose depth can be tuned inde-pendently in all three directions. This defines the “base”potential. The ability to tune the lattice depths indepen-dently enables one to increase the confinement along di-rections where tunneling is not required, which increasesthe interaction strength for two particles occupying thesame site. For the different protocols, we assume that theappropriate number of planes within the base potentialare initially loaded, or that atoms remaining in undesiredplanes can later be removed. On top of the base potentialis superimposed an auxiliary potential, whose form de-pends on the variant of the protocol to be implemented.For loading target sites within a two-dimensionalplane, the auxiliary potential could simply consist oftightly focused optical tweezers, incident from a directionorthogonal to the plane and projected through a high-numerical aperture optical system. These serve both tocreate the required offset between target sites and neigh-boring reservoir sites, and by choosing the size of thetweezer spot, can also push down the potential barriersto enable faster tunneling.For loading target planes, the role of the auxiliary po-tential is simply to shift the energy of the target plane rel-ative to its neighbors, and potentially of those neighborsrelative to their outward neighbors if initial shift opera-tions are applied to increase the reservoir density. Practi-cally, this could be implemented by superimposing a long-wavelength standing-wave auxiliary lattice formed fromtwo beam intersecting a small angle onto the base lattice.The planes of the base lattice that experience the lowestenergy potential from the auxiliary lattice define the tar-get planes. Favorable initial conditions could be achievedby loading a single plane of a variable-wavelength “accor- dion” auxiliary lattice at long wavelength, then transfer-ring these atoms into several planes of the base latticeprior to ground-state cooling and parity projection onindividual sites. The ratio of the wavelengths of the aux-iliary and base lattices then defines how many planes ofatoms the target plane has available to draw atoms from.Finally, for the shuffle protocol, the auxiliary potentialconsists of a non-sinusoidal potential with twice the pe-riod of the base potential along the tunneling direction,and variable relative phase to the base potential. Theauxiliary potential could be created either by projectinga pattern onto the lattice using a high numerical aper-ture optical system, or by combining two polarizations oflight in a bowtie-configuration lattice [40]. In either case,the wells of the auxiliary potential can be aligned to thebase potential with a phase such that they create bothan offset between selected neighboring wells, and to pushdown the potential barrier between the wells to facilitatetunneling [40]. In addition to these two periodic poten-tials, we assume an additional potential to be presentthat creates a wall on the left side of the array (for ex-ample by pushing a specific potential well off resonance),and a linear potential gradient applied along lattice toavoid additional unwanted resonances. Appendix B: Estimates of experimental parameters
Here, we provide estimates for the relevant parame-ters achievable in a realistic experimental system. As abenchmark case, we consider Sr atoms confined withina base lattice formed by retro-reflection of 813 nm light,as this is the “magic” wavelength that causes zero differ-ential shift to the strontium clock transition. While thisfeature is not required for our protocol, it makes it a de-sirable wavelength for other applications of a strontiumsystem.
Tunneling rate:
The tunneling rate J between theground state of one well and the first excited state ofits neighbor is a critical quantity for the success of thisprotocol. Analytical expressions exist for the tunnelingrates within a given band of a uniform lattice, but to ourknowledge not for our nonuniform inter-band situation.We thus calculate the tunnelling rate numerically for ourdesired potential landscape by simply integrateing theSchr¨odinger equation. We find the ratio of J to J tobe approximately four for depths of the base lattice ofat least 10 E R , where E R = (cid:126) k / m . Here, k is thewavenumber of the light used to form the lattice, and m is the mass of the atoms. An analytical expression existsfor J [41]: J (cid:39) √ π E R ( VE R ) / exp[ − VE R ) / ] (S1)Where V is the depth of the lattice. For our benchmarksystem with a lattice depth of V = 13 E R (correspondingto a harmonic oscillator frequency of ω = 2 π ×
23 kHz),we calculate a tunneling rate J (cid:39) π ×
175 Hz. Thiscan easily be made slower by increasing the depth of thelattice. In principle, it could be made faster as well ina shallower lattice, but after applying the required shiftto neighboring wells, the potential becomes substantiallydistorted at the energy of the occupied states, and ourexpressions are no longer valid.
Interaction shifts:
The interaction shifts that preventdouble occupation of sites in our protocols result fromthe contact interaction between atoms in the first mo-tional excited state and the ground state. These shifts(under the assumption of unperturbed eigenstates) areapproximated by: U = 12 (cid:114) π kaE R ( V x E R ) / ( V y E R ) / ( V z E R ) / (S2)where V i represents the depth of the lattice in the i thdirection. The factor of 1 / U is a factor of 2larger).For our benchmark system (a = 800 a ) in cases wherethe tunneling occurs along a single direction (shuffle andlayer protocols), depths of V x , V y , V z = 13, 30, 30 E R give an interaction shift U = 2 π ×
10 kHz. When si-multaneous tunneling in two dimensions is desired, thesame interaction shift can be achieved using V x , V y , V z =13, 13, 70 E R . For systems with more typical scatteringlengths of a (cid:39)
100 a , the lower interaction shifts couldbe at least partially compensated by increasing the depthof the lattices in directions orthogonal to tunneling. Tunneling fidelities:
In principle, our protocols couldbe implemented purely with resonant tunneling – thedepths of neighboring wells could be quickly brought intothe desired resonance condition, then held there for atime π/J . This approach would have the advantage ofspeed, but suffers from a high sensitivity to the relativedepths of the neighboring wells, and requires preciselytimed operations. For total depths of the optical poten-tial near 100 E R (accounting for the tight confinementrequired in non-tunneling directions in order to ensurelarge interaction shifts), this would require neighboringwells to be balanced at the part-per-thousand level intotal lattice depth. Because most of the intensity is as-sociated with the base-potential, which can be createdusing retro-reflected lasers, this may not be unreason-able. Roughly 10% of the total potential in this scenariowould be contributed by the auxiliary potential, so thiswould have to be controlled at the 1% level. The pro-tocols will be substantially more robust however if anadiabatic ramp through resonance is used, so we assumethis condition in all subsequent analysis.We numerically calculate the probability of an atomtunneling either into an empty well, or into an occupiedwell (represented by a shift of U = 2 π ×
10 kHz, as esti-mated above). For a linear sweep over a range of 10 J ,centered about resonance, and over a duration 2 π × /J (roughly 30 ms for a tunneling rate J (cid:39) π ×
175 Hz), we expect a transfer fraction of approximately 95% intothe empty well, and below 2 × − for the occupied well.Such ramps only require control of the total potential atthe 1% level, and of the auxiliary potential at the 10%level.To compare with prior work, tunneling fidelities atthe 99% level or higher have been demonstrated inthe ground band of systems with optical superlattices[7, 28, 30]. Given the additional requirements of speedand tunneling between bands for this proposal, we thinkthat fidelities above 90% would be readily achievable inpractice, and would not present a major performance lim-itation. P = 0 P = 1/2 P = 2/3 P = 1 a. b.c. FIG. S1. Tunneling properties within nearest-neighbor sub-system. a., b.
The motional excited state into which theatoms tunnel is two-fold degenerate, with a node either alongthe vertical (a.) or horizontal (b.) axis. Atoms tunneling intothe target site populate one of these two states, depending onwhether they tunneled horizontally or vertically. During thetunneling process, an atom is thus confined to its original rowor column, but can tunnel through the target site to the reser-voir site directly opposite (not shown) c. The probability ofa target site being filled depends on the number and configu-ration of filled neighboring reservoir sites. Here we show theunique filling possibilities (omitting those that are equivalentup to reflection or rotation of the system), along with theassociated probability for filling the target site, assuming aperfect adiabatic ramp of the target site depth.
Atom loss:
Atom loss is the core limitation for ourprotocol. Atom loss can occur either directly, thoughcollisions with background gas, or as a result of our rear-rangement protocol, through light-assisted collisions withother atoms. The latter occurs if two atoms occupy thesame site, which could be caused either by imperfecttunneling suppression from of the interaction shift, orthrough errors in ground state cooling.The effect of background gas collisions simply dependson the duration of the shift cycle relative to the vacuumlifetime in the experiment. For our predicted tunnelingrates of several hundred Hz, and associated ramp timesof a few 10’s of ms, we expect each cycle of the rear-rangement to take well below 100 ms, as ground statecooling and changes in trap configuration for the coolingshould be possible on a timescale of several milliseconds.As atomic lifetimes of 100 seconds or more are now some-what common, we expect this loss to be at the part perthousand level per cycle, and could in principle be betterif resonant tunneling is used instead of adiabatic ramps.The impact of imperfect ground-state cooling can haveseveral impacts on our protocol. If an atom in a targetsite begins a shift cycle in the n=1 state, it may tun-nel into a neighboring empty reservoir state, leading toan effect similar to loss of the atom. More concerningwould be the possibility of an atom tunneling into a welloccupied by a neighbor, enabled by an accidental res-onance between higher motional states (whose spacingbecomes rapidly nonlinear with increasing N due to therelatively shallow potentials required for appreciable tun-neling rates) and causing a double-occupancy. The ten-dency for this to happen would likely depend sensitivelyon the exact optical potential used, but we consider thesomewhat pessimistic case that imperfect ground statecooling leads to the loss of the non-cooled atom. Thefundamental limit on the performance of ground statecooling is given by ¯ n = 5( γ/ω ) /
16, where ¯ n representsthe average number of motional quanta along a given di-rection, γ is the decay linewidth used for cooling, and ω isthe trap frequency [42]. This expression assumes coolingfrom three directions in an isotropic trap. For realistictrap frequencies of ω = 2 π ×
100 kHz during cooling, thisimplies that motional excitation should be limited to thelevel of a few parts per thousand.As described above, we find from simulation that thechances of an atom tunneling onto an occupied latticesite in a single rearrangement cycle can be below 2 × .Because this would lead to the correlated loss of bothatoms, the effect of this loss differs by protocol. In thecase of the shuffling protocol, both lost atoms are fromthe target state, so the effect on the final filling fractionis larger than single-atom loss. However, because the lossis correlated, if we are only interested in the probabilityof preparing a perfect array, the effect is similar to lossof a single atom. In the target sites and target planesprotocols, only one of the atoms is from the target site,so the loss of the second atom is far less consequential.In principle, it may be possible to avoid the loss associ-ated with double-occupation of lattice sites, as loss occursonly as a result of subsequent light-assisted collisions.However, this would likely lead to a permanent double-occupation of the site in the final configuration, so forsimplicity, we assume here that light-assisted collisionsoccur after each shuffling step (either through deliberateapplication of photoassociation light or as a byproduct ofground state cooling) and causes the loss of both atoms.While the loss rates from double occupation may beacceptable for these parameters, they could be improvedin several ways. First, because of the quadratic scalingof the double occupation probability with tunneling rate,a relatively minor reduction in tunneling rate (deeperlattice) could dramatically reduce these errors, though atthe expense of longer cycle times and greater sensitivityto lattice inhomogeneity. Further, it may be possible tomodify the light-assisted collision step to preferentially eject only a single atom, for example by using light blue-detuned light [16–18].Adding these effects together, and acknowledging thatthey are rough estimates that will vary depending on thedetails of experimental implementation, these predictionsindicate that a loss per cycle of less than 1% should bepossible, so we use this value as a benchmark for esti-mating the likely performance of our protocols. Appendix C: Simulating rearrangement protocols: