REVIEW

Quantum simulations with ultracoldatoms in optical latticesChristian Gross1 and Immanuel Bloch12

Quantum simulation a subdiscipline of quantum computation can provide valuable insightinto difficult quantum problems in physics or chemistry Ultracold atoms in optical latticesrepresent an ideal platform for simulations of quantum many-body problemsWithin thissetting quantum gasmicroscopes enable single atom observation andmanipulation in largesamples Ultracold atomndashbased quantum simulators have already been used to probe quantummagnetism to realize and detect topological quantummatter and to study quantum systemswith controlled long-range interactions Experiments on many-body systems out of equilibriumhave also provided results in regimes unavailable to the most advanced supercomputersWereview recent experimental progress in this field and comment on future directions

Ultracold atoms offer remarkable opportu-nities for investigating quantum many-body problems that are relevant to fieldsas diverse as condensed matter physicsstatistical physics quantum chemistry and

high-energy physics (1) The high degree of con-trollability and new observation tools whichenable the detection of each individual atommake ultracold atoms ideal as analog quantumsimulators (2ndash5) The aim of such simulators isto solve difficult quantum many-body problemsespecially those so complex that they cannot besolved even on todayrsquos most powerful classicalsupercomputers One such problem is identifyingground states in the dopedFermi-Hubbardmodelwhich is believed to play an important role in theexplanation of high-temperature superconductivity(6ndash8) Others include out-of-equilibrium dynam-ics of quantum matter (9) where even the mostadvanced numerical methods cannot for exam-ple track the time dynamics of modestly sizedtwo-dimensional strongly interacting systemsOver the past 15 years ultracold atoms have

thus increasingly become a tool to investigatesuch complex strongly interacting quantum mat-ter Here we discuss this progress focusing onsome of the latest experimental developments inthe fieldWe concentrate on lattice-basedquantummany-body systems where the study of stronglycorrelated many-body physics started with theobservation of the superfluidndashMott insulatortransition for bosons (10) We note that a largeamount of highly important work has also beencarried out in the continuum most notably therealization of the Bose-Einstein condensationndashBardeen-Cooper-Schrieffer crossover (11)The optical lattices used in the experiments

are typically formed by interfering several laserbeams in order to realize a fully controllableperiodic light structuremeant tomimic the crystallattice of a solid Atoms can be trapped in suchperiodic light fields because of their polarizability

induced by the alternating electric field of thelaser light (12) Such optical lattices offer a tremen-dous amount of versatility Not only can the ge-ometry dimensionality disorder and depth ofthe lattice be controlled lattices can also be en-gineered that carry effective magnetic fields (13)hundreds of times stronger than the strongestmagnetic fields created in solid-state laboratoriesThis has enabled the investigation of topologicalquantummatter using the toolbox offered by ultra-cold atomsFundamentally new observation tools based

on quantum gas microscopes allow observationand control of quantum matter at the level of in-dividual atoms For example being able to takesingle ldquosnapshotsrdquo of a quantum many-body sys-tem provides access to intricate correlations be-tween the constituent particles that are difficult

to observe in other settings Attributes such asmagnetic correlation hiddenmagnetism and topo-logical order can thus be probed in remarkablydirect waysFinally ultracold atoms can be highly isolated

from the environment allowing the explorationof fundamental phenomena in statistical physicsHow does a closed isolated quantum system es-tablish a temperature and thermalize during itsquantum evolution (14ndash16) What type of closedquantum system fails to thermalize even after along quantumevolution Experimentswith ultra-cold atoms and ions are central to exploring thesefundamentally important researchdirections (17 18)that challenge the foundations of classical statis-tical physics and thermodynamics

Quantum gas microscopes

The realization of quantum gas microscopes forbosonic rubidium atoms (19 20) was an impor-tant step forward for quantum simulations basedon optical lattice systems This technology en-abled the single sitendashresolved detection of indi-vidual atoms ormore precisely of the local parityof the onsite atom number in a two-dimensionaloptical lattice thus it realized the ideal tool forthe precision readout of an ultracold atomndashbasedquantum simulator (Fig 1) The detection is basedon in-trap laser cooling techniques pioneeredearlier in large spacing optical lattices (21 22)Sufficientlymany fluorescence photons scatteredduring the laser cooling are collected with a high-resolution imaging objective to allow the distinc-tion of atoms on neighboring lattice sites Thelaser cooling is crucial as it prevents the atomsfrom hopping to nearby sitesWith illuminationtimes of a fraction of a second the achievablesignal-to-noise ratio is significantly higher than

Gross and Bloch Science 357 995ndash1001 (2017) 8 September 2017 1 of 7

1Max-Planck-Institut fuumlr Quantenoptik 85748 GarchingGermany 2Germany Fakultaumlt fuumlr Physik Ludwig-Maximilians-Universitaumlt 80799 Munich GermanyCorresponding author Email christiangrossmpqmpgde (CG)immanuelblochmpqmpgde (IB)

Optical lattice laser beams

Mirror 1064 nmwindow 780 nm

High-resolution objective

X

YZ

16 microm

Fig 1 Quantum gas microscopes [Adapted from (20)]

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with high-resolution absorptive imaging tech-niques (23) Electron beammicroscopy offers evenhigher resolution but at lower detection efficiency(24) Soon after the single atomndashresolved detec-tion precise manipulation techniques were addedto the microscope toolbox allowing for local con-trol of individual atoms Highly resolved inten-sity patterns were projected onto the atomicplane with the help of strongly focused beamsor spatial light modulators (25ndash27) First exper-iments based on these techniques revealed thelocal entropy distribution of ultracold latticebosons in the Mott-insulating phase and identi-fied coherent particle-hole pairs as the remainingquantum fluctuations in the ground state close tothe superfluidndashMott insulator transition (202829)Six years after the first demonstrationof bosonic

quantum gasmicroscopes the technique was alsosuccessfully implemented for fermionic atoms(30ndash34) Owing to the electronic properties of thealkali atom fermions more elaborate laser cool-ingmethods such as Raman-sideband cooling inthree-dimensional optical lattices needed to bedeveloped to enable the site-resolved detection ofindividual fermions Simultaneous detection oftwo hyperfine spin states has also been demon-strated (35 36) which provides unique access tocorrelations between the spin and charge (density)sectors (37)Such single atomndashresolved fluorescence imag-

ing is not limited to alkali atoms It has also been

demonstrated for ytterbium an atom with twovalence electrons (38 39)

Strongly interacting bosons on a lattice

The Bose-Hubbard model the first strongly cor-related lattice model to have been realized withultracold atoms (10) is still very prominent inexperiments with ultracold atoms It is realizedwhen loading interacting bosonic atoms into op-tical lattices Temperatures below the band gapare routinely obtained such that its physics canbe studied in the lowest band limit extensions tohigher bands are also possible (40) The simple butintriguingmodel contains three basic parametersThe nearest-neighbor tunneling strength t de-scribes themotion of atoms in the lattice Themu-tual interaction of atoms on the same lattice siteis parametrized by U and the (often harmonic)spatial confinement due to the laser beams re-sults in a site indashdependent energy shift Vi Byreducing tU the ratio of the tunneling to theinteractions the system can be driven from thesuperfluid into the strongly correlated Mott in-sulating regimeA prototypical example of a cold atom quan-

tum simulation experiment is the observation ofthe Higgs amplitude mode in a two-dimensionalBose-Hubbard system (41ndash44) This excitationmode corresponds to the oscillations of the mag-nitude of the superfluid order parameter It ispredicted to exist in the vicinity of the superfluidndash

Mott insulator transition where an emergingrelativistic field theory describes the system Intraditional condensedmatter systems this modehad not been observed in two dimensions be-cause a clean excitation method was missing Ina single-layer two-dimensional ultracold systemhowever the modulation of the amplitude of theoptical lattice potential naturally couples only tothemagnitude of the superfluid order parameterwhich made it possible to directly confirm theexistence of the Higgs mode in two dimensionsMotivated by the cold atom experiments recentwork on low-dimensional materials has now alsofound evidence for the existence of theHiggsmodein these systems (45ndash47) TheHiggsmode has alsobeen observed in a quite different syntheticmany-body systemof a Bose-Einstein condensate loadedinto two crossed optical cavities (48 49) This set-ting realizes a many-body system supporting asupersolid phase of matter for which the effectivefield theory description also predicts an ampli-tude modeMagnetic phenomena are ubiquitous in solid-

state physics and underlie many intriguingphenomena some of which are still not fully un-derstood theoretically Frustratedmagnets wherefrustration leads to a large degeneracy of thesystemrsquos ground state are one example Classicalfrustrated magnets can be realized in an ultra-cold atom system using shaken triangular lattices(50 51) where the local phase of the bosons rep-resents a classical spin The shaking parametersgovern the phase winding as they induce con-trolled complex tunneling elements between thedifferent sites of a plaquette Time-of-flight mea-surements provide access to the ensemble aver-age of this classical spin and thus the systemstate can be directly measuredThe simulation of quantummagnetism at first

requires the identification of two local quantumstates (the elementarymagnets) representing thetwo spin states of the electrons on which tra-ditional magnetism is based There are variouspossibilities to encode these two states in theultracold lattice system and different magneticmodel Hamiltonians can thereby be realized Forexample the Ising model where only one com-ponent of the spin vector interacts has been re-alized in a tilted lattice by encoding the two spinstates in the occupation difference between twoneighboring sites (52) This method allowed for asingle spinndashresolved observation of the transitionfrom a paramagnet with all spins aligned in thedirection of an external field to the interaction-dominated Ising antiferromagnetwith anti-alignednearest neighbors Nonequilibrium dynamics ofsuch Ising chains have been studied by a con-trolled quench of the Hamiltonian parameters tothe vicinity of this quantumphase transition (53)long-range tunneling resonances over several lat-tice sites have also been observed (54)More complicatedquantumspinmodels emerge

when several directions of the spin vector con-tribute to the interaction between the elementarymagnets The special case of equal interactions inalldirections is realized in the symmetricHeisenbergmodel This model emerges for two-component

Gross and Bloch Science 357 995ndash1001 (2017) 8 September 2017 2 of 7

Fig 2 Probing many-body localization in two dimensions (Left) Almost arbitrary disorderpotentials of light are projected onto an ultracold bosonic atom cloud The quantum evolutionof an initial nonequilibrium state can be tracked using a quantum gas microscope (Right) The initialstate is a domain wall of a bosonic Mott insulator (ldquohalf circlerdquo in microscopy images) Even forlong evolution times of ~250 tunneling times t the system fails to thermalize indicated by theremnant domain wall still visible in the experiment In contrast a thermalized state would not carryany information about the initial state of the system (68)

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bosons with equal interactions among all part-ners when the tunneling t in the lattice is sup-pressed in the Mott insulating phase The twocomponents can be defined by two Zeeman statesin thehyperfinemanifolds of the electronic groundstate of the atoms At unity filling in the latticemdashthat is one atomper lattice sitemdashthe only remain-ing degree of freedom is the local orientation ofthis effective spin The interactions between thespins turn out to be ferromagnetic and restricted tonearest neighbors only resulting in theHeisenbergmodel H frac14 J

PiSi Sithorn1Theinteractionstrength

is given by the so-called superexchange couplingJ = 4t2U (55ndash57) Such a setting is realized with87Rb atoms in an optical lattice the correspondingcoherent spin dynamics were first observed locally(58 59) and later over several sites in one dimen-sion (26)Such two-component bosonic 87Rb atoms in

the lattice can be used to study spin transportmicroscopically Quantum gas microscopes en-able the direct tracking of the spin motion acontrolled initial state canbe prepared for exam-ple by using local spin manipulation to reversethe direction of one (60) or two spins (61) Thequantummotion of the spins through the chainresults in entanglement generationbetweendiffer-ent sites enabling direct detection of the prop-agation of the resulting entanglement wave (60)In the case of two initially flippednearest-neighborspins bound magnon states are excited Thesemagnons consist of the flipped pair bound by theferromagnetic Heisenberg interaction They canregarded as the most elementary form of a mag-netic soliton and were found to propagate asstablemagnetic quasiparticles ballistically throughthe spin chain (61) At higher excitation energiesthe quasiparticle picture is no longer useful andthe exact theoretical solution is no longer easilyaccessible Spin transport can however still beexperimentally studied by starting from spin-spiral states where the transversemagnetizationvector linearly winds along the equator of theBloch sphere as a function of position along thechain Interferometricmeasurements of the trans-versemagnetization relaxationdynamics indicateddiffusive spin transport at these high energies (62)Magnetization relaxation caused by superexchangeand tunneling dynamics has also been studied inbosonic systems in two-dimensional lattices (63)Magnetic many-body models still pose many

open challenges that might be addressed by ultra-cold atom quantum simulators in the future Onthe fundamental side this includes models withfrustrated ground states or topologically non-trivial models that emerge for elementary spinslarger than frac12 On the more technological sidethe control of quantum spin arrays is a centralresource in quantum computation

Thermalization and many-body localization

The use of statistical physics and thermodynamicsto describe classical and quantummany-body sys-tems is ubiquitous It is therefore of fundamentalinterest to understand why and when the prin-ciples of statistical physics hold and whetherthere are genuine robust scenarios where this

approach completely fails For the first questionthe notions of thermalization and ergodicity inquantummany-body systems are relevant Ther-malization occurs if local observables can be char-acterized by thermal expectation values after longtime evolution Thermalizing systems are some-times also called ergodic because during theirevolution they explore all configurations inHilbertspace that are allowed by global conservation laws(64) To tackle the question of thermalization theso-called eigenstate thermalization hypothesis hasbeen proposed and tested both numerically andexperimentally with increasing success (14ndash16 65)For most generic Hamiltonians the hypothesis in-dicates that for an arbitrarily chosen high-energymany-body eigenstate of an isolated quantumsystem any subsystem (after taking a trace overthe rest of the system) will look thermal that isits density matrix will be given by a thermal den-sity matrix To some extent this fact explains thetremendous success of quantumstatistical physicsand thermodynamics even for fully isolated quan-tum systems whose initial state can be far awayfrom any thermal equilibrium state but will even-tually evolve into such a thermal state

The second question concerning when such sce-narios fail dates back to early work by Anderson(66) This question remained dormant and openfor quite some time but in 2006 a theoreticalbreakthrough was achieved that asserted thateven for interacting quantummatter in disorderedpotentials a new state ofmatter should exist (67 )for which the conductivity strictly vanishes evenat finite temperatures This so-called many-bodylocalized phase defies description by traditionalstatistical physics as even for arbitrarily long evolu-tion times no thermal equilibriumstate is reachedThe associated phase transition from a thermaliz-ing state to an insulating nonthermal state is adynamical phase transition whose properties stillremain in large part unclear (17 18)Ultracold atoms (as well as ultracold ions) offer

an ideal scenario to investigate these fundamentalquestions (68ndash72) Their minimal coupling to anythermal baths and the outside worldmakes themalmost ideal isolated quantum systems in contrastto normal low-temperature condensedmatter ex-periments in dilution refrigerators where the sam-ple is a priori in thermal contact and in thermalequilibriumwith the connected thermal reservoir

Gross and Bloch Science 357 995ndash1001 (2017) 8 September 2017 3 of 7

Fig 3 Spin correlations in Fermi-Hubbard systems [Bottom images adapted with permissionfrom (84)]

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Experiments probing such many-body localizedphases of matter have so far typically proceededby setting up a nonequilibrium initial state suchas an artificial charge density wave where onlyeven lattice sites are occupied (69) a spatial do-main wall boundary (68) (Fig 2) or an atomcloud that received a momentum kick (70) Thesubsequent long-term evolution over asmany asseveral hundred tunneling times showed thatindeed no thermal steady state is reached andthat the system retains a memory of the initialstate even after a long-term quantum evolution(68 69) Experiments in one and two dimensionsalso suggest that the transition between the ther-malizing and nonthermalizing phase takes placerather abruptly around a specific critical disorderstrength The experiments not only support theexistence of such many-body localized phases inone and very likely also in twodimensions but alsoindicated several other interesting effects For ex-ample domainwall experiments in two-dimensionalsystems found evidence for a diverging lengthscale around the transition and exhibited behaviorstrikingly different from that of an Anderson local-ized transition for noninteracting particles (68)Many fundamental questions remain open

Quantum evolution to even longer times of thou-sands or tens of thousands of tunneling timeswould be quite useful in exploring the unique fea-tures of this novel phase transition and wouldhelp to distinguish the dynamics from those ofclassical glasses which do thermalize albeit onextremely long time scales Complemented by the-oretical research there is much hope that thisapproach will unravel many of the secrets of themany-body localization transition specificallyto obtain a better understanding of the charac-teristics of this dynamical phase transition thatdefies classification by the usual schemes of sta-tistical physics In addition the role of small ther-malizing inclusions inamany-body localized system

needs to be better understood as these can cru-cially affect the stability of this intriguing phasein higher dimensions

Fermions in optical lattices

One of themost active problems in optical latticendashbased quantumsimulation is the realization of thefermionic Hubbard model (73) This is motivatedby thestrongconnection tohigh-temperature super-conductors (6ndash8) where theHubbardmodel is rou-tinely used tomodel the effective electronic degreesof freedomBinary spinmixtures of ultracold latticefermions provide a defect-free and fully controlledimplementation of the Hubbard model in whichboth the interactions and the doping can be con-tinuously tuned The former is controlled by the

external magnetic field via Feshbach resonances(74) whereas the latter is determined by the atomnumber setting the chemical potential in thetrapped system Ultracold atoms and quantumgas microscopes provide unique observation andmanipulation opportunities such as the charac-terization of the quantum state by local or non-local correlation functions that allowdirect probesof hidden order parameters in a many-body set-ting (Fig 3)As the interaction Ut is increased finite-

temperature half-filled repulsively interactingFermi-Hubbard systems undergo a crossover fromthe metallic to the Mott insulating state in this

context half-filling means that the system is inthe undoped regime of one-atom-per-site latticefilling and has equal numbers of both spin com-ponents TheMott-insulating phasewas observedin 2008 (75 76) using global observables such asthe compressibility or the doublon fraction Withtheuse of high-resolution absorption imaging tech-niques the equation of state in the density sectorof the two-dimensionalHubbardmodelwasmea-sured (77) and extended density correlationsweredetected (78) The rich phase diagramof the Fermi-Hubbard model at low temperatures results fromthe interplay of the density (charge) and the mag-netic degrees of freedom Although the magneticenergy scale is given by the superexchange coupl-ing J even at higher temperaturesTgt J magneticshort-range correlationsmdashespecially next-neighborcorrelationsmdashare present Such nearest-neighborcorrelationshavebeendetected first in lowerdimen-sions (79 80) and later also in three-dimensionalsystems (81)The realization of fermionic quantum gas mi-

croscopes was undoubtedly an important stepforward toward the quantum simulation of theFermi-Hubbard model in computationally in-accessible regimes It resulted in the in situ observa-tion of band andMott-insulating states (33 82 83)and medium-range antiferromagnetic spin cor-relations (36 84ndash87) with the effective temper-atures of the respective systems being at or onlyslightly below the superexchange energy scaleVery recent developments demonstrate the po-tential of these systems In twodimensions tailoredoptical traps have been used to realize entropyredistribution in the inhomogeneous system asproposed in (88 89) Here a large low-densityregion surrounding the higher-density core servesas an entropy reservoir Because the systems canbe manipulated while keeping the total entropyconstant the redistribution of entropy to the largereservoir leads to a decrease of the entropy in the

Gross and Bloch Science 357 995ndash1001 (2017) 8 September 2017 4 of 7

Fig 4 Topological matter in cold atom systems From left to rightAharonov-Bohm effect of a charged particle in a magnetic field Whentraveling around a closed loop enclosing magnetic flux F the electron wavefunction acquires a phase shift proportional to F Neutral atoms do not carrycharge but in engineered lattices with complex tunneling matrix elementsbetween lattice sites an atom hopping around a closed trajectory can also bemade to pick up a phase shift thereby emulating the effect of a magneticfield [Adapted with permission from (155)] Berry curvature can be directlymeasured via an Aharonov-Bohmndashstyle interferometer in momentum space

The wave packet of an atom is split and moved around a region includingBerry curvature (eg Dirac cone with singular Berry flux located at thetip of the cone) such that the phase shift after interfering the wavepackets will be proportional to the Berry flux enclosed in the interferometer[Adapted from (104)] Alternatively by rapidly switching the Hamiltonianbetween two band structures with coupled (weak lattice) and decoupled(strong lattice) sites enables direct mapping of the Berry curvaturedistribution in the Bloch bands via an alternative interferometric technique[Adapted with permission from (105)]

ldquoIt isof fundamentalinterest to understand whyand when the principles ofstatistical physics holdrdquo

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central part This technique has been recently dem-onstrated and effective core temperatures wellbelow the superexchange scale have been realizedThe resulting long-range correlations extendedover the full extent of the low-entropy region (90)Simultaneously progress on one-dimensional sys-tems has also been made and the direct de-tection of spin-charge multipoint correlationswhich is important for the characterization of theFermi-Hubbard system in the doped regime hasbeen reported (37) Measuring such multipointcorrelators opens the path to the direct detectionof many hidden and topological order parame-ters in one-dimensional systems these can becharacterized by string correlation functions

Artificial gauge fields and topology

The investigation and realization of novel topo-logical phases of matter is of great current in-terest in condensed matter physics Topologicalquantum phases of matter typically arise in asolid under the action of spin-orbit coupling orstrong externalmagnetic fields (91ndash93) They defythe standard Landau classification of quantumphases in terms of a local order parameter ratherthey are typically characterized through a non-local topological invariant such as for examplethe Chern number Realizing orbital magnetismthat lies at the heart of many topological quan-tum phases with neutral cold atoms seems apriori impossible as the atoms do not carry anycharge However the effects of orbital magnet-ism can be mimicked by emulating the quantummechanical effect of a charged particle in a mag-netic field [ie the Aharonov-Bohm effect (94)](see Fig 4) A charged particle completing a closed-loop trajectory picks up a phase shift which isproportional to the appliedmagnetic field strengthOptical lattice structures in which a neutral par-ticle picks up a phase shift when traveling arounda closed loopwould emulate the effect of a chargedparticle in a magnetic field and hence topologicalBloch bands could arise (95 96)Such phase shifts have been realized in two

distinct ways either by laser-induced tunnelingor by lattice shaking In the former case tunnel-ing between lattice sites is first suppressed by forexample applying a strong potential gradientThe tunneling can then be restored by introduc-ing a pair of laser beams whose difference en-ergy is tuned to bridge the energy differencebetween neighboring lattice sites induced by thestrong potential gradient However the new tun-neling matrix element can be complex-valuedandmay acquire a phase term depending on theeffective wave vector of the two laser beams Anatom completing a closed loop can therefore pickup a net phase shift mimicking the Aharonov-Bohm phases picked up by electrons on closedtrajectories in amagnetic field Interestingly andin contrast to condensedmatter experiments coldatom experiments thereby choose a certain gaugeof the effective magnetic field An alternative ap-proach is based on lattice shaking where the en-tire optical lattice is moved around in space on acircular trajectory thereby realizing effective com-plex tunneling matrix elements in the resulting

time-independent FloquetHamiltonian (509798)The former approach has been used to realize theHofstadter model (99 100) for different fluxvalues in optical lattices whereas the latter wasapplied to realize staggered magnetic fluxes intriangular lattices (50) or the celebrated Haldanemodel (101) in a hexagonal lattice setup (98)The geometry of the resulting topological Bloch

bands is characterized by the so-called Berry cur-vature defined over the magnetic Brillouin zoneof the lattice W frac14 iethhqx ujqy ui hqyujqxuiTHORN(102 103) where ju(qx qy)i denotes the Blochwave functions in a single band of the lattice ThisBerry curvature can be directly measured in amomentum-resolved way using an interferomet-ric approach similar to an Aharonov-Bohm in-

terferometer in which an atomic wave packet issplit and coherently moved around a region inmomentum space that has a finite Berry curvaturedistribution (104) (Fig 4C) After recombiningthe two atomic wave packets the wave functionwill have acquired a phase shift proportional tothe enclosed Berry flux of the interferometer Al-ternatively bringing the atomic wave packet intoa superposition of different energy bands hasenabled direct mapping of the Berry curvaturedistribution in the Brillouin zone (105) (Fig 4D)The topological invariantmdashor Chern numbermdash

of the band is simply given by the integral of theBerry curvature over the entire Brillouin zoneC frac14 1

2p intWd2q It canbe accessed through the trans-verse bulk displacement of the atom cloud underthe action of a force Using semiclassical trans-port theory that includes the Berry curvature ofthe band (103) one finds that the transverse dis-placement of the cloud after one Bloch period tB =hFya of the applied force Fy is given by Dx =CAuc a where a denotes the lattice constant andAuc is the area of the unit cell In contrast tointeger quantum Hall measurements in con-densed matter where the topological invariantcan be determined through edge state transportandmaking use of the bulk-edge correspondence(102) here the Chern number can be measureddirectly through the bulk displacement of the atomcloud yielding a value of C = 099 plusmn 005 for thelowest energy band of theHofstadtermodel (106)A dynamical version of the integer quantum

Hall effect the Thouless charge pump (107) hasalso been realized in the context of ultracold atoms(108 109) Here two lattices with different periodare superimposed in one dimension with a phasef characterizing the relative position of the twolattices This phase together with the momen-tum coordinate in one dimension defines a two-dimensional parameter space for which thequantum geometry of the system can again bedefined using a generalized notion of the Berrycurvature (107) Such Thouless pumps provide arobust pumping of charge stable to imperfec-tions which is rooted in quantum mechanicaltunneling motion of the particles and the to-pology of the band Whenever the band is com-pletely filled one effectively integrates over theentire Berry curvature of the system resulting ina quantized transport of charge (density) in thesystem Such quantized transport was recentlyrealized in (108 109) in addition to geometricpumpingwithnoquantized charge (density) trans-port whenever the band is not entirely filled (110)Ultracold atoms also offer possibilities for realiz-

ing synthetic dimensions (111) in addition to thephysical dimensions present in the systemsHereseveral internal states in an atom can be viewedas an extra dimensionwhen coupled to each otherwith the extension of the extra dimension deter-mined by the number of substates available Inprinciple the use of such synthetic dimensionsallows exploration of topological quantummatterbeyond three-dimensional settings Synthetic di-mensionshave also beenused to realizeHofstadterribbons in which the chiral edge states could bedirectly observed in the experiment (112 113)

Gross and Bloch Science 357 995ndash1001 (2017) 8 September 2017 5 of 7

Fig 5 Dipolar interactions Top Ultracoldheteronuclear alkali molecules in their rotationaland vibrational ground state can feature largeelectric dipole moments on the order of severaldebye For a binary mixture of two different internalstates of the molecules (red and blue) this can beused to realize sizable effective spin interactionsover several lattice sites in optical lattices With anexternally set quantization axis (eg set by amagnetic B field) the dipolar interaction strengthdepends not only on the distance but also on thedirection of the vector connecting two moleculesrelative to the quantization axis Bottom Sche-matic indicating the relative interaction strengthfor nearby sites in a three-dimensional cubicoptical lattice [Adapted with permission fromNature (129) copyright 2013 Macmillan Publish-ers Ltd]G

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Systems with dipolar interactionsIn most cold atom experiments atoms interactvia short-range contact interactions Howevermany-body systems with longer-range interac-tions are predicted to feature intriguing quantumphases and exhibit interestingdynamical behaviorthat can be very different from that of systemswith contact interactions only (114ndash116) State-selective long-range interactions can be used toinduce adirect spin-spin interaction (Fig 5) whichcould also lead to exotic quantummagnetism fea-turing frustration or nontrivial topologies (117 118)Including the atomicmotion as a relevant degreeof freedom extended Hubbard models (119) orspin-ice models described by dynamical gaugetheories (120) might be realizedThree approaches to inducing long-range in-

teractions via ultracold atoms are under studyOne is to use strongly magnetic atomic speciessuch as chromium (121) dysprosium (122) orerbium (123) where the magnetic dipole-dipoleinteraction is sizable In such a system a shift ofthe superfluidndashMott insulator transition has beendirectly observed as a consequence of the dipolarinteraction (124) and spin dynamics driven by thedipolar interactions have been studied (125) Asecond approach is based on ground-state polarmolecules (126ndash128) for which the characteristicdipolar interactionhasbeenobserved in thedephas-ing dynamics of a collective spin ensemble ofmolecules (129) The third possibility is to makeuse of the extremely strong van der Waals in-teractions betweenRydberg states (130) In laser-excited samples this interaction exceeds all other

energy scales onmicrometer distances leading tothe Rydberg blockade effect (131 132) whichresults in the formation of spatially correlatedstructures during the excitation dynamics (133)Crystalline low-energy states of the emerging

spinmodel can be prepared using tailored opticalexcitation in combinationwith the spatial controlover the shape ofMott insulators offered by quan-tum gas microscopes (134) This technique how-ever is so far limited to rather small atomic samplesof a fewhundred atoms closely packed at a spacingof 05 mm in an atomic Mott insulator Conse-quently only a few many-body states actuallyparticipate in the excitation dynamics whereasmost configurations are far detuned owing totheir high intrinsic interaction energy This limi-tationcanbeavoidedbyusingoff-resonantRydbergcoupling The ground state is only ldquodressedrdquo bytheRydberg state resulting in a tunable andmuchweaker dipolar interaction (135ndash140) The firstattempts to observe effects attributable to these in-duced interactions failed in large three-dimensionalsamples because of interaction-induced resonancebroadening (141ndash143) Nonetheless coherent evo-lution of a Rydberg-dressed many-body systemwas recently demonstrated to be observable insmall one- and two-dimensional systems hencethis technique indeed allows for coherent inter-action control (144 145) and is a promising basisfor a future coherent quantum annealing plat-form (146)For spin models coupled via state-selective

long-range interactions the motional degree offreedom of the atoms is frozen that is the atoms

are pinned to their lattice sites Hence cooling tothe ground state is not always required Arrays ofoptical microtraps are an attractive alternativeplatform for such experiments because they re-quire a less involved optical setup and featuremuch faster experimental cycle times than opticallatticendashbased experiments (147 148) Similar toquantum gas microscopes these microtraps canbe optically read out with single-atom precisionFurthermore these setups allow for arbitrary ge-ometrical arrangement of the microtraps in twodimensionsmdasha great advantage when studyingmany-body models that for example are pre-dicted to feature frustration (117) The realizationof Rydberg spin models in suchmicrotrap geom-etries has been demonstrated for the XY-model(149) as well as for the transverse Ising model(150) The loading of the microtrap arrays is apriori probabilistic which hinders the determi-nistic production of larger atom arrays This limi-tationhas beenovercome first for few-trap systemsby laser-assisted blue shielding techniques (151 152)and recently in larger arrays by techniques that re-locate the initially randomlyplaced atoms (153 154)intoperfectly orderedanddefect-free arrays (Fig 6)These developmentsmake themicrotrap platforman ideal tool to study controlled spin systems in-teracting via Rydberg-induced interactions

Outlook

Recent years have seen tremendous scientific pro-gress in the field of quantum simulations withultracold atoms Quantum gas microscopes haverevolutionized our way of controlling and prob-ing many-body systems atom-by-atom and withalmost perfect control over the underlying po-tential This has resulted in important advancesin the realization of the low-temperature phasesof the doped Fermi-Hubbard models experi-ments now stand at a point where they are aboutto outperform even themost sophisticated quan-tumMonteCarlo algorithmsonclassical computersLikewise dynamical nonequilibrium simulationscan be performed in regimes far beyond the reachof the most advanced supercomputers and aretargeting fundamental questions in statisticalphysics and quantum information science Webelieve that over the next years we will witnessfurther improvements of cold atomndashbased quan-tum simulators larger system sizes and longercoherent evolution times will enable even morecomplex quantum simulations In addition anenhanced local control over individual atomsand engineering of individual coupling strengthsin a lattice will lead to highly versatile and pro-grammable quantum simulators allowing one toprobe quantum matter in previously unexploredregimes with a unique degree of control It willalso establish cold atoms as a highly promisingplatform for coherent quantum annealing algo-rithms at the interface of quantum simulationand quantum computation with applications inthe field of hard optimization problems At thesame time the scientific reach of quantum simu-lations with cold atoms will very likely open upnew paths in the field of dynamical quantumfield theories relevant toquantumelectrodynamics

Gross and Bloch Science 357 995ndash1001 (2017) 8 September 2017 6 of 7

Fig 6 Deterministic atom array preparation Left Schematic of the optical setup to generatearbitrary two-dimensional trap arrays via the spatial light modulator (SLM) The position of themoving tweezer which sorts the initially randomly loaded atoms into the target array is computer-controlled (control system) using an acousto-optical deflector (2d-AOD) The charge-coupleddevice (CCD) camera allows for nondestructive position monitoring of the atoms Right Example ofone realization of deterministic array loading The initially randomly loaded sample (top) is sortedin about 100 ms to a uniformly filled two-dimensional array (bottom) The detected atomicfluorescence is encoded from dark to light in the color scale each spatially distinct area of highfluorescence signal corresponds to a single atom [Adapted with permission from (153)]

GRAPHIC

ADAPTED

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Erratum 21 September 2017 See Erratum on July 15 2020

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quantum chromodynamics and quantum chem-istry perhaps such simulations will even resultin the discovery of new phases of matter

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long-range interacting Ising chain arXiv170508372(2017)

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(2015)

101126scienceaal3837

Gross and Bloch Science 357 995ndash1001 (2017) 8 September 2017 7 of 7

ULTRACOLD MATTER Erratum 21 September 2017 See Erratum

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Quantum simulations with ultracold atoms in optical latticesChristian Gross and Immanuel Bloch

DOI 101126scienceaal3837 (6355) 995-1001357Science

ARTICLE TOOLS httpsciencesciencemagorgcontent3576355995

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