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An atom optics approach to studying lattice transport phenomena Bryce Gadway∗ Department of Physics, University of Illinois at Urbana-Champaign, Urbana, IL 61801-3080, USA (Dated: January 25, 2016) We present a simple experimental scheme, based on standard atom optics techniques, to design highly versatile model systems for the study of single particle quantum transport phenomena. The schemeisbasedonadiscretesetoffree-particlemomentumstatesthatarecoupledviamomentum- changing two-photon Bragg transitions, driven by pairs of interfering laser beams. In the effective latticemodelsthatareaccessible,thisschemeallowsforsingle-sitedetection,aswellassite-resolved 6 anddynamicalcontroloverallsystemparameters. Wediscusstwopossibleimplementations,based 1 on state-preserving Bragg transitions and on state-changing Raman transitions, which respectively 0 allow for the study of nearly arbitrary single particle Abelian U(1) and non-Abelian U(2) lattice 2 models. n PACSnumbers: 03.75.Be;61.43.-j;71.23.-k;73.43.Nq a J 1 I. I. INTRODUCTION techniquesforlocalmanipulationviaglobalfieldaddress- 2 ing[27,28]. Inthecontextofstudyingtransportphenom- ] Over the past few decades, atomic, molecular, and ena, however, we consider an inhomogeneous landscape s optical (AMO) systems have played an increasingly im- of site energies, with unique energy differences between a neighboring sites, which defines unique tunneling reso- g portant role in shaping our understanding of complex nances for each site-to-site link. Combined with global - quantum phenomena. Precise knowledge of the micro- t fieldaddressingthatcandrivetransitionsbetweenneigh- n scopic properties of AMO systems, combined with un- boringsites,andinparticularbysimultaneousdrivingof a precedented levels of control and novel diagnostic tools, u have stimulated the development of several platforms - many such transitions in an amplitude, frequency, and q phase-controlled manner, this would allow for local con- basedoncoldatoms[1],trappedions[2],andphotons[3] . trol over the parameters of a discrete lattice model rele- t -forthequantumsimulationofmyriadphysicalphenom- a vant to myriad coherent transport phenomena. ena, especially those related to condensed matter [4, 5]. m For the study of single electron transport phenomena, Atom optics offers a natural candidate system featur- - d photonic [6–11] and cold atom [12–20] simulators have ing a quadratic energy landscape and field-driven transi- n made great progress in the experimental exploration of tions between states. Here, we propose to create a dis- o disorderedandtopologicalsystems,whileofferinglargely crete “lattice of sites” represented by free-particle mo- c complementary capabilities and challenges. Photonic mentumstatesofatomicmatterwaves,havingquadratic [ simulators generally permit control of system parame- energy-momentum dispersion, which can be effectively 1 ters and the detection of probability distributions at the nearest-neighborcoupledviaresonanttwo-photonBragg v microscopic, site-resolved level. However, the use of real transitions [29, 30]. The free particle dispersion allows 8 materials as the medium for light transport makes these for spectrally-resolved control over all parameters of the 1 systemssusceptibletoinherentdisorderinsampleprepa- system at the single-link level, including all site-to-site 8 ration [21] and to absorption in the material [22], and “tunneling” amplitudes and phases, achieved by writing 5 0 makessimulationsinhigherspatialdimensionsandtime- multiple radiofrequency sidebands onto a pair of inter- . dependent control of system parameters non-trivial. For fering laser beams. We describe how this can enable the 1 cold atoms, pristine and dynamically variable potential simulation of near-arbitrary single particle models, in- 0 landscapescanbeconstructedbasedontheirinteraction cluding two-dimensional Abelian U(1) lattice models de- 6 1 withlaserlight. However,amicroscopiccontroloversys- scribing integer Hall systems [31]. Additionally, we show : tem parameters is difficult to realize in atomic systems. how another well established atoms optics tool - stimu- v Moreover, finite temperatures and the absence of hard- lated Raman transitions that change both the internal i X wall system boundaries have limited the observation of state and momentum of atoms [32, 33] - can be used to r topological phenomena. study non-Abelian U(2) gauge fields, which to date have a Here, we propose an atom optics-based [23–26] ap- been difficult to realize in photonic and cold atom set- proach to the study of coherent transport phenomena, tings. whichincorporatesmanyofthedesiredfeaturesofatomic Theproposedschemebuildsonalargebodyofworkin- andphotonicexperimentalplatforms. Theschemewede- volvingthestudyoftransportphenomenausingtheevo- scribeismotivatedinspiritbymagneticresonance-based lution of momentum-space distributions of cold atomic gases [12, 34–37], including recent precision studies of the three-dimensional Anderson insulator-metal transition [38–40]. While the majority of such studies have in- ∗ [email protected] volved time-dependent driving by lattice potentials not 2 fulfilling a resonant Bragg condition, notably in the Fig. 1(b). The (nearly) common frequency detuning ∆ realization [12, 38–41] of quantum kicked rotor mod- of all the laser fields from atomic resonance is assumed els [42, 43], here our proposed method operates deep to be much larger than all other relevant terms, includ- within the resonant Bragg diffraction regime. ing Doppler shifts of magnitude |p|k/M and the reso- The paper is organized as follows. In Sec. II, we in- nantRabicouplingfrequencies|Ω+|and|Ω−|. Thislarge j troduce the basic experimental scheme based on state- single-photondetuningfromresonancemakesdirectpop- preserving Bragg transitions that allows for the simu- ulation of the atomic excited state |e(cid:105) negligible. In the lation of Abelian U(1) models in discrete lattice sys- following we assume an effective ground-state Hamilto- tems. In Sec. III, we discuss in more depth some rele- nian Hˆ =Hˆ +Hˆ based on adiabatic elimination of eff 0 int vant aspects of the proposed scheme, including how it theexcitedstate|e(cid:105). ThiseffectiveHamiltoniandescribes is extended to higher-dimensional systems, some of its thefree-particlekineticenergiesHˆ andlight-atominter- 0 uniquecapabilities,andsomepracticalexperimentallim- actions Hˆ that drive two-photon processes changing int itations. In Sec. IV, we introduce a second experimental the atomic momenta by ±(cid:126)k = ±2(cid:126)kxˆ while leaving eff scheme based on internal state-changing Raman tran- theinternalstateunchanged,characterizedbyvirtualab- sitions, which allows for the simulation of non-Abelian sorption of a photon from one laser field and stimulated U(2)models. Finally,conclusionsarepresentedinSec.V. emissionintotheother. Assumingthattheatomicsource is a condensate of atoms with small momentum spread 2σ (cid:28) (cid:126)k, we now define a discrete basis of relevant p II. II. ABELIAN U(1) LATTICE MODELS plane-wave momentum states |n(cid:105) (with n an integer), having momenta p =2n(cid:126)kxˆ. n We begin by considering a generic system of two-level atoms,havingasingleinternalground(excited)state|g(cid:105) (|e(cid:105)) with energy (cid:126)ω and having a mass M. These (a) g(e) two-levelatomsandtheirinteractionwithadrivingelec- tric (laser) field E, neglecting spontaneous emission, are  ( )  ( ) E+coskx−ω+t+φ+ ∑E−coskx−ω−t+φ− described in the dipole approximation by the single par- n n n ticle Hamiltonian n pˆ2 E/ħ e Hˆ = 2M +(cid:126)ωe|e(cid:105)(cid:104)e|+(cid:126)ωg|g(cid:105)(cid:104)g|−d·E , (1) (b) ∆ where p is the free particle momentum of the atoms and d=−|e|r is the atomic dipole operator, with r a vector pointingfromtheatomicnucleustotheelectronposition. We assume that, as shown in Fig. 1(a), the electric field + ω E of the driving lasers is composed of two distinct con- ω- ω- tributions – a right-traveling field E+(x,t) with a single -1 ω- 1 frequency component and a left-traveling field E−(x,t) 0 withanumberofdiscretefrequencycomponents. Explic- ω+ ω+ ~ g ω =12E / ħ itly, we take these two fields to be 1 R ~ ω =4E / ħ E+(x,t)=E+cos(k+·x−ω+t+φ+) and (2) 0 R p/2ħk -1 0 1 2 (cid:88) E−(x,t)= E−cos(k−·x−ω−t+φ−) . (3) j j j j FIG. 1. (Color online) Experimental scheme for studying j lattice-driven momentum-space dynamics. (a) Atomic mat- Weassumewithoutlossofgeneralitythatthefieldsprop- ter waves are driven by a pair of counter-propagating laser agatealongthex-axis,andmoreoverthattheyarenearly fields,oneofwhichiscomposedofseveraldifferentfrequency monochromatic such that k+ = kxˆ and k− (cid:39) −kxˆ ∀ j, components, with controllable phase, frequency, and ampli- j tude. (b) Energy-momentum dispersion. All laser fields are with k = 2π/λ the wavevector of the laser light hav- far-detunedbyanamount∆fromatomicresonancebetween ing wavelength λ. Similarly, all laser frequencies are theground|g(cid:105)andexcited|e(cid:105)states. Stimulatedtwo-photon detuned from atomic resonance (ω ≡ ω − ω ) by a eg e g Bragg transitions are driven by the pairs of interfering laser nearly equal amount ∆≡ωeg−ω+ (cid:39)ωeg−ωj− ∀ j. For fields, coherently coupling plane-wave momentum states sep- each frequency component of the driving electric field, arated by two photon momenta (2(cid:126)k). The quadratic free we define the respective resonant Rabi couplings to be particle dispersion defines a unique two-photon Bragg reso- Ω+ =−(cid:104)e|d·E+|g(cid:105)/(cid:126) and Ω−j =−(cid:104)e|d·E−j |g(cid:105)/(cid:126). nance condition (cid:126)ωñ = (2n+1)4ER for each link between Experimentally relevant terms related to the energy- neighboring states. Each frequency component of the multi- momentum dispersion of the atoms are depicted in frequency field addresses a unique state-to-state link. 3 Thisdiscretesetofallowedmomentumstateswillform jth two-photon Bragg resonance. This now brings us to the “lattice of sites” that can be coupled in a controlled the physical picture of building up individual links be- way via two-photon transitions. These states have ki- tween a “lattice” of discrete momentum states, through netic energies E = (cid:104)n|Hˆ |n(cid:105) = n2(4E ), where the the engineering of many interfering laser frequency com- n 0 R single-photonrecoilenergyisgivenbyE =(cid:126)2k2/(2M). ponents. In the limit of “weak-driving”, which for this R In the assumed form of the driving electric field E, off- one-dimensional example we define as (cid:126)Ω˜ (cid:28) 8E ∀ j, j R diagonaltermsthatincreasethemomentumby2(cid:126)kxˆcan thebandwidthoftwo-photontransitionsissufficientlyre- in principle come about by absorption of a photon from duced such that at most one frequency component has a the right-traveling field, followed by stimulated emission substantialcontributiontoeachoff-diagonalelement. We into any of the different frequency fields that constitute then ignore all but the most near-resonant contribution theleft-travelinglaserfield. Forsuchaprocessdrivenby for each off-diagonal coupling, in the spirit of a rotating therespectivefrequencycomponentlabeledbytheindex wave approximation. This greatly simplifies the effective j, we define a corresponding two-photon Rabi coupling interaction-picture Hamiltonian, leading to weakly time- dependent off-diagonal couplings of the form Ω∗−Ω+ Ω˜jeiφ˜j = j2∆ ei(φ+−φ−j) , (4) (cid:104)n+1|HêIff|n(cid:105)/(cid:126)≈Ωñeiφñeiξnt . (7) For any two coupled modes, this weak time-dependence where Ω˜ is assumed to be real and positive, and the j can be further absorbed into diagonal “site”-energies ε n phase shift associated with this process is determined by (relatedbyξ =ε −ε )byarotatingframetransfor- the phases φ+ and φ− of the two laser fields, which can n n+1 n j mation,permittingafullytime-independentHamiltonian be easily controlled using acoust-optic or electro-optic description with a controlled “potential landscape”. We modulators, for example. will assume the less general case, however, where all fre- We now define the effective ground-state Hamiltonian quency components of the applied fields exactly fulfill a of this system in the interaction picture HÎ , where the eff two-photon Bragg resonance condition, i.e. ξn = 0 ∀ n. time-dependenceduetoHˆ0 ismovedontothesystemop- We then arrive at the desired description of a single par- erators. In the ground-state plane-wave basis, the diago- ticle tight-binding Hamiltonian naltermsarenowallzero(uptoanignoreddiagonalAC Stark shift that is common to all states). The nearest- HêIff ≈(cid:88)tn(eiϕncˆ†n+1cˆn+h.c.) . (8) neighbor off-diagonal elements, described in terms of the n two-photon Rabi couplings for all allowed transitions, Here, arbitrary control over all tunneling amplitudes take the time-dependent form t ≡ (cid:126)Ω˜ and tunneling phases ϕ ≡ φ˜ of the system n n n n areenabledinalink-dependentwaythroughcontrolofa (cid:104)n+1|HêIff|n(cid:105)/(cid:126)=(cid:88)Ω˜jeiφ˜je−iδj(n)t , (5) single global addressing field E−(x,t). This can be sim- j ply accomplished, for example, by passing a single laser beam through a pair of acousto-optic modulators driven where δ(n) describes the two-photon detuning of the jth by tailored radiofrequency signals [44]. Moreover, the j frequency component from the |n(cid:105) to |n+1(cid:105) transition, tailored radiofrequency signal can be smoothly varied in given as δ(n) = (ω+ − ω−) − ω˜ . Here, the term ω˜ time, such that the parameters of the model system can j j n n be made time-dependent. describes the Doppler frequency shift of the transition The scheme as described, with local control over tun- |n(cid:105) → |n+1(cid:105). Given that the free-particle dispersion is nelingamplitudesandphases,permitsthestudyofnear- quadratic, its linear first derivative relates to a linearly arbitrary one-dimensional systems. Of natural interest varying Doppler frequency shift wouldbethestudyofsuperlatticesystemsknowntohave p ·k (cid:126)|k |2 non-trivial topological properties [8, 45–48], in particu- ωñ = nM eff + 2Meff =(2n+1)4ER/(cid:126) , (6) lar when combined with either additional modulation of the tunneling parameters [49–51] or in the presence of which serves to define the two-photon Bragg resonance disorder [52, 53]. condition for the |n(cid:105) to |n+1(cid:105) transition. We can make use of this unique state-to-state frequency shift to achieve the stated goal of controlling the III. III. FURTHER ASPECTS OF THE SCHEME off-diagonal elements in a link-specific manner. We explicitly assume that the two-photon detuning between A. A. Extension to higher dimensions each frequency component j of the left-traveling field and the right-traveling field approximately satisfies a While the ability to simulate arbitrary Hamiltonians unique Bragg resonance condition. Formally, for every describing lattice transport in one dimension would al- frequency component of the field labeled by index j, we low for a number of interesting studies, particularly rel- set ω+−ωj− ≡ω˜j −ξj, with j an integer and ξj a small evant to disordered and symmetry-protected topologi- ((cid:126)ξ (cid:28) 8E ∀ j) and controllable detuning from the cal states, the tunneling phases ϕ are of little physical j R n 4   (a) -k ⋅x (c) in integer quantum Hall systems [60]. 2   (0,0) (1,0) (2,0) Here, we describe the simple extension to realizing k ⋅x=kxˆ two dimensional models that preserve full spectral con- 1 (0,1) (1,1) (2,1) trol over all tunneling links (with straightforward exten-   θ -k ⋅x (0,2) (1,2) (m,n)=(2,2) sionstohigher-dimensionalsystemsaswell). Weconsider 1 ħω~x the case of driving by two independent pairs of counter- k2⋅x=k[cos(θ)xˆ+sin(θ)yˆ] 0 10 20 30 EmR,n propagating laser fields as shown in Fig. 2, where we ne- glect any effect of cross interferences (by choice of polar- (b) (d) ization or an appropriate frequency offset). Elementary (0,2) (1,2) (2,2) (3,2) (0,0) (1,0) (2,0) (3,0) changes to the atomic momentum by ∆p = 2(cid:126)k and 1 1 p (0,1) (1,1) (2,1) (3,1) ∆p =2(cid:126)k result from allowed two-photon Bragg pro- y (0,1) (1,1) (2,1) (m,n)=(3,1) 2 2 (0,0) (1,0) (2,0) (3,0) ∆p =2k cesses as in the earlier-described scheme. Assuming that  2 2 ħω~y we start with population nominally at zero momentum, ∆p =2k m,n 1 1 px 0 10 20 30 ER this defines a set of possible momentum states |m,n(cid:105), having momenta p = 2(cid:126)(mk +nk ). We next as- m,n 1 2 FIG.2. (Coloronline)Atwo-dimensionallatticesystemwith sume, without loss of generality, that k1 = k1xˆ and spectrally resolved link resonances. (a) Two pairs of inter- k =k [cosθxˆ+sinθyˆ] as depicted in Fig. 2. If k (cid:54)=k , 2 2 1 2 fering laser fields (set 1 shown in red, set 2 shown in blue), this can allow for effectively higher-dimensional systems intersecting in a plane at an angle θ, are shone onto a col- (of finite extent) to be realized even for θ =0 [37]. Here lection of atomic matter waves. Cross interferences between we consider instead the case of driving by lattices with the two pairs of beams can be avoided by choice of laser po- near-identicalwavevectorsalongtwodifferentdirections, larizations or by introducing a frequency offset between the i.e. k (cid:39) k = k and θ (cid:54)= 0,π. The resulting kinetic twopairs. (b)Thediscrete“lattice”ofmomentumstatesthat 1 2 energies of the |m,n(cid:105) states will be given by canbepopulatedbystimulatedtwo-photonBraggtransitions, starting from zero momentum. (c) The spectral positions of E =4E [m2+n2+2mncosθ] . (9) the nearest-neighbor Bragg resonances ω˜x driven by the m,n R m,n laser pair 1, relating to the finite-sized set of states labeled So long as the lattice directions are not orthogonal (θ (cid:54)= (m,n)withmomentap =2(cid:126)(mk +nk )asshownin(b). m,n 1 2 π/2,3π/2), there will exist unique Bragg resonance con- (d) Same as in (c), but for the Bragg transition resonances ω˜y addressed by the second pair of lasers. ditionsforeachlinkofafinite-sizedtwo-dimensionalsys- m,n tem. Similar to the unique Bragg transition frequencies ω˜ n betweenadjacentstates|n(cid:105)and|n+1(cid:105)inonedimension, consequence when applied only to one-dimensional sys- described in Eq. 6, in two dimensions we have unique tems with nearest-neighbor couplings. In higher dimen- Bragg transition frequencies that depend on the initial sions,anaturalapplicationoftheabilitytoengineerlink- state|m,n(cid:105)andinwhichdirectionthemomentumisim- specific phases would be to mimic the Aharonov-Bohm parted. For a momentum change of ∆p , this gives the 1 phase φAB acquired by charged particles (with charge condition q) moving along a path P in an electromagnetic vector potential A(cid:126), φAB = (q/(cid:126))(cid:82)P A(cid:126) ·d(cid:126)x. This would allow ω˜mx,n =[2m+1+2ncosθ]4ER/(cid:126) . (10) the study of topologically non-trivial (2+1)-dimensional A similar condition (ω˜y = [2n+1+2mcosθ]4E /(cid:126)) Abelian U(1) models, such as those describing the inte- m,n R exists for a momentum change ∆p , and because there ger quantum Hall effect exhibited by electrons confined 2 is no cross interference between the pairs of laser fields, in two dimensions under the influence of strong trans- unique spectral control of tunneling terms along all links verse magnetic fields [31, 54]. The local manipulation of can still be preserved even if there exist overlapping res- phases could also allow the study of random magnetic onancesω˜x =ω˜y alongthetwodifferentdirections. flux models [55, 56], which are believed to exhibit metal- m,n m(cid:48),n(cid:48) Following the procedure as in Sec. II, through the appli- licbehaviorandprovideaninterestingcounterexampleto cation of spectral sidebands to one laser from each pair, Anderson’s theorem [57, 58] in two dimensions. Higher- being controlled in amplitude and phase and offset in dimensional studies allow access to novel lattice geome- frequencyfromthecounter-propagatingpartnertofulfill tries as well, where link-specific control over tunneling particular resonance conditions, one may realize a two- amplitudescanbeusedtotransformasimplesquarelat- dimensional Abelian U(1) lattice model of the form tice into a brick-wall honeycomb lattice [59] by setting certainlinkstozerotunneling. Ingeneralthiscontrolal- lows one to impose hard-wall boundary conditions, and HêIff ≈(cid:88)[txm,n(eiϕxm,ncˆ†m+1,ncˆm,n+h.c.) a two-dimensional scheme with tailored links would al- m,n (11) low one to create one-dimensional systems with periodic + tym,n(eiϕym,ncˆ†m,n+1cˆm,n+h.c.)] . boundary conditions. Recently, researchers have used suchalocalmanipulationinphotonicsimulatorstoprobe As a concrete example, we analyze in Fig. 3 the effec- novelquestionsaboutthebulk-boundarycorrespondence tive dynamics that can be driven in a two-dimensional 5 Time = 0 ħ/t 1.875 ħ/t 3.75 ħ/t 7.5 ħ/t 3 exact (a)5 t t = 0.01 ER n0 /y - E j 20 ψˆm 2 tt == 00..0048 EERR (b)5 rg ψ 1 e n en-2 (e) (g) 0 0 4 (c)5 0.1 m=0,n=0 (f) (h) n 2 0,0 +i0,1 3 0 ψin 2 ψ ˆn 2 (d)5 ψj ψ exact n 1 t = 0.01 ER t = 0.04 ER 0 0 t = 0.08 ER 0 0 m 5 0 m 5 0 m 5 0 m 5 10 20 30 0 2 4 6 8 0 1P = m,nψ 2 eigenstate index - j Time (units of ħ/t) m,n FIG. 3. (Color online) Simulated momentum-space dynamics of a small (6 site × 6 site) two-dimensional (2D) lattice system. (a-d) Probability distributions of the different momentum modes |m,n(cid:105), following dynamics initiated from a state |ψ (cid:105), at in times of 0, 1.875, 3.75, and 7.5 in units (cid:126)/t, with t the tunneling energy. (a) Shown for a regular lattice with homogeneous tunneling energies t and no tunneling phases, starting from |ψ (cid:105) = |0,0(cid:105). The dynamics shown relate to evolution governed in by Eq. (11) from the text. (b) Same, but for an enclosed synthetic magnetic flux of 2π/3 per lattice plaquette, set through a non-trivial tunneling phase along one direction, ϕy =2mπ/3. In this case, the particles avoid entering the bulk or interior m,n of the system, and instead propagate along the system boundaries. Because state preparation is based on mode projection, with no explicit energy dependence, a combination of clockwise and counter-clockwise propagating edge states are populated. √ (c) As in (b), but starting from the state |ψ (cid:105) = (|0,0(cid:105)+i|0,1(cid:105))/ 2. The populated state propagates with essentially only in one chirality. (d) Exactly as in (c), but including all tunneling contributions due to the entire sideband spectrum [i.e. with dynamics governed by the 2D equivalent of Eq. (5) and not Eq. (11), with t/E = 0.01]. (e) Energy spectrum relating to R the systems of (b) and (c), with 2π/3 flux enclosed per lattice plaquette. The system is split into 3 bulk energy bands, and featuresadditionaldispersiveedgestates(shadedinblueasaguidetotheeye). Insetsshowthemodaldistributionofdifferent energy eigenstates. (f) Probability distribution of eigenstates populated by projection from |ψ (cid:105) = (|0,0(cid:105) (solid blue) and √ in |ψ (cid:105)=(|0,0(cid:105)+i|0,1(cid:105))/ 2 (dashed red). (g,h) Center-of-mass position dynamics, in terms of mode numbers m and n along in thetwodirections,forenclosedfluxandinitialstateasin(c)and(d). Theblacklinesshowtheexactdynamicsasin(c). From darker to lighter colors, the red [blue] lines in (g) [(h)] show dynamics for t/E = 0.01, 0.04, 0.08. The smallest energy gap R between spectral resonances ω˜x(y) in the system is 0.97E . m,n R system with non-trivial tunneling phases, relating to an ulated by beginning with population in a superposition effective Aharonov–Bohm phase acquired by particles of multiple momentum states. Given the similarities to evolving in the system of momentum states. We show photonicsystems,withrespecttoprojectivestateinitial- that far in the weak-driving limit, the effective dynam- ization and out-of-equilibrium dynamics, we expect that ics that emerge from Eq. (5) exactly coincide with those many of the techniques developed for studying topologi- of Eq. (11). In the case of a non-zero synthetic mag- cal properties of photonic simulators should prove useful netic flux, these dynamics show insulating behavior in in the envisioned atom optics setting [61]. the bulk of the system and transport along the edge of thesystem. ThedynamicsillustratedinFig.3alsohigh- Oneissuetonoteinaccessinghigher-dimensionalmod- light an important aspect of the simplest studies that elsisthatthefrequencyspacingbetweenthelink-specific can be performed using the proposed scheme - those in- Bragg resonances, found in one dimension to have the volving population initiated in one or a few momentum value 8E /(cid:126), is reduced as the number of links in each states, with laser-driving turned on suddenly. Similar to R directionisincreased. Thisingeneralrequireslowertun- the case of many photonic simulators [7], spatial projec- neling rates (two-photon Rabi rates) to remain in the tion onto the system’s eigenmodes dictates the ensuing weak-driving limit where individual resonances are spec- dynamics, and there is no explicit energy selection or trally resolved. Practically, a more realistic approach to preparationinthesystem’sgroundstate. Foranon-zero studyinghigherdimensionalsystemswhilepreservingar- synthetic magnetic flux, population initiated in the bulk bitrarycontrolofallparametersmaybefoundinsystems ofthesystemwillremainstationary,whilepopulationon extendedinonedirectionandwithonlytwoorafewsites the systems edge will undergo transport. Furthermore, alongasecond[17–19,62]orsecondandthird[63]direc- Fig. 3 shows how particular edge modes can be pop- tion. 6 B. B. Unique features and the zero-point motion associated with the ground state of their confining potential [73]. The momentum The suggested atom optics-based approach, which al- spread of trapped Bose–Einstein condensates is typically lows for precise and time-dependent control of a sin- muchsmallerthantherecoilmomentum,2σp (cid:28)(cid:126)k,such gle particle lattice model at a link-specific level, affords that the picture of a discrete lattice of states is justified. many unique experimental capabilities relevant to quan- However, even a small but finite momentum spread will tum simulation. Furthermore, the fact that the effective introducerestrictionsontheexperimentaltimescalesover “tunneling” transitions between sites are explicitly field- which coherent momentum-space dynamics can be ob- drivenanddonotresultfromquantumtunnelingthrough served. Coherent dynamics in momentum-space requires a barrier allows in principle for several unique features. that momentum states with direct off-diagonal coupling As discussed in the previous section, this allows for the occupy indistinguishable spatial modes. In other words, simulation of higher-dimensional systems of finite extent thelaser-drivendynamicswilloccuronlyinthenear-field in three or fewer physical dimensions. It additionally regime [36, 74], before the populated momentum states allows for direct and independent control of tunneling havetimetospatiallyseparateintodistinctwavepackets. terms beyond nearest-neighbor. For example, one can While this imposes a strong limit on the timescales access next-nearest-neighbor hopping terms by driving over which coherent transport phenomena can be ex- second-order Bragg processes with resonances given by pected to occur, a significant number of coherent tun- 2ω˜(NNN) = (4n+4)4E , which are spectrally distinct neling events can still be achieved. Moreover, this ef- n R from the first-order resonances [29]. Controlled access to fect will be less relevant to the observation of phenom- such terms would allow tunable symmetry breaking (in- ena involving localized states or ballistic, non-dispersive version or particle-hole) of topological insulator systems. propagation in momentum-space. Still, we can provide Additionally, it has been shown [64] that the combina- a lower estimate for the limiting timescale based on the tionofnearest-neighbor(NN)andnext-nearest-neighbor worst case scenario, the Ramsey decoherence time in the (NNN) tunneling in one dimension can be used to re- absence of continuous coupling between two states. For alize systems analogous to the two-dimensional Haldane nearest-neighborstatesdifferinginvelocityby2vR (with model [65], allowing study of the anomalous quantum vR ≡(cid:126)k/M therecoilvelocity), theirspatialoverlapwill Hall effect in an experimentally simple setting. Such a belostroughlyonthetimescaleTcoh =Lc/2vR,whereLc combination of terms may also allow for the study of isthecloud’sspatialcoherencelengthalongthedirection Lifshitz-type behavior [66], e.g. as found in axial next- ofmomentumtransfer. WeassumethatLc isdetermined nearest-neighbor Ising (ANNNI) models [67–69]. at ultracold temperatures and low densities by the finite Another relatively unique aspect of the proposed sys- system size in a trapping potential, and we relate this tem stems from the combination of local and time- to the number of lattice sites Ns (of the interfering laser dependent parameter control. To note, either local con- fields) over which the atomic distribution would extend, trol or time-dependent control of the system parameters with Lc =Nsλ/2. This description allows for the simple would allow, e.g., for the controlled implementation of relation Tcoh =Nsτ0, where τ0 =h/8ER. We recall that quenched disorder or of a time-varying Hamiltonian for in one dimension the tunneling rates are restricted to be quantum annealing to novel ground states [70, 71], re- much less than 8ER/(cid:126) to spectrally resolve individual spectively. In this context, their specific combination link resonances. Assuming tunneling rates t∼8ER/10(cid:126), could allow for the study of annealed disorder, with ran- we can expect Ramsey coherence times corresponding to domly distributed system parameters that are addition- roughly Ns/10 tunneling events. ally modulated in time [72]. In essence, such modulation Thisexpectedlimitationtotheschememotivatessome of the lattice parameters over an appropriate range of practical considerations. When implementing higher- frequenciescanmimicthecouplingofparticlestoather- dimensional “lattices” of momentum states, because the mal phonon bath. Through the modulation of disorder tunneling rates necessary to achieve complete spectral at frequencies corresponding to relevant energy scales of control of all tunneling parameters become severely re- the model system being studied, such annealed disorder stricted, the dynamics will remain coherent for far fewer could allow access to the thermodynamic properties of tunneling events. We thus expect that it will be more an otherwise intrinsically out-of-equilibrium system. realistic to pursue studies of one-dimensional lattice and superlatticesystems, aswellasladder-typesystemswith only a few sites along a second direction [17–19, 62] or C. C. Limitations two additional directions [63]. Additionally, an active increase of the relevant experimental timescales may be There exist several practical limitations to the achievedbyincreasingthespatialcoherencelengthofthe timescalesoverwhichtheproposedschemecanbeusedto atomicsamplepriortothelattice-drivendynamics. This simulatecoherentdynamics. Themajorlimitationcomes can be achieved by an adiabatic decrease of the trap- fromthefactthatultracoldatomsarenotidealizedzero- ping depth and stiffness, leading to an increase of the momentumplanewaves,buthaveaspreadinmomentum atoms’ spatial extent [29, 75]. One can also pursue still duetofinitetemperature,interactionsbetweenparticles, more active methods for increasing of the atomic sam- 7 ple’s size based on analogies to Gaussian beam optics, (having mass M) with an electric field E, governed by namely by using matter-wave lensing techniques [76] for theconstructionofanatomicbeamexpanderorGalilean Hˆ = pˆ2 +(cid:126)ω |e(cid:105)(cid:104)e|+ (cid:88) (cid:126)ω |g (cid:105)(cid:104)g |−d·E. (12) telescope. Such techniques have recently been employed 2M e gα α α α∈{1,2} to create mm-scale atomic clouds of 87Rb with pK-scale temperatures [77], which for lattice light tuned near the We assume from the outset that the electric field is D transition would allow for a few hundred coherent formed by two laser fields counter-propagating along the 2 tunneling events in one dimension. x-axis,aright-travelingfieldE+(x,t)andaleft-traveling fieldE−(x,t), havingnearlyidenticalwavevectormagni- If these studies are performed in atomic free fall or tudes k. As in the previous scheme, the right-traveling free expansion, so as to minimize any influence of trap- fieldismonochromatic(ω+)andfar-detunedfromatomic ping potentials on the ensuing matter-wave dynamics, resonancebyanamount∆≡(ω −ω )−ω+ (cid:29)ω . The another practical limitation is found. Assuming a ge- e g1 12 left-traveling field contains a number of spectral compo- ometry of lattice-driving along a direction perpendicular nents with frequencies ω−,α. The fields are explicitly to gravitational acceleration, to avoid additional compli- n given by cations due to time-varying Doppler shifts, then grav- ity will cause the atoms to fall away from the region of E+(x,t)=E+cos(kx−ω+t+φ+) and (13) light-atom interaction. Restricting the atoms to fall less than d = 1 mm, for example, will restrict the experi- mental0timescales to T = (cid:112)2d /g ∼ 14 ms (where E−(x,t)= (cid:88) E−,αcos(−kx−ω−,αt+φ−,α). (14) grav 0 n n n g = 9.81 m/s2 is the assumed gravitational acceleration n,α∈{1,2} due to free fall), or roughly 270 tunneling events in the case of one-dimensional simulations. These timescales are generally less restrictive than those due to the near- (a) fieldconstraint,andcanbelargelyassuagedthroughlev- itation in a magnetic field gradient without introducing  ( )  ( ) E+coskx−ω+t+φ+ ∑E−,αcoskx−ω−,αt+φ−,α significant external confinement. n n n n,α∈{1,2} Lastly, we remark that spontaneous photon scattering can in principle provide an additional limitation to E/ħ e ∆ -,2 the observation of coherent momentum-space dynamics (b) ω 1 driven by stimulated photon scattering [78]. Practically, ω+ however, the heating rates due to off-resonant absorp- -,2 tion and re-emission events can be mitigated by setting ω 0 tthhee sspinognltea-npehooutosndedceatyunraintge Γ∆otfothbeeelxacrigteedcsotmatpear|ee(cid:105)d. to ω+ -,1 ω+ ω-1,1 ω 0 g 2 ω+ g 1 IV. IV. NON-ABELIAN U(2) LATTICE MODELS ω 12 We now describe a straightforward extension to the p/2ħk schemedescribedinSec.II,whichisbasedonusinginter- 0 1 2 nalstate-changingtwo-photonRamantransitions[32,33] as opposed to state-preserving Bragg transitions. This FIG.4. (Coloronline)LaserdrivingschemeforstudyingU(2) modified scheme requires the use of two low energy in- lattice dynamics. (a) Counter-propagating laser fields drive ternal ground states |g1(cid:105) and |g2(cid:105), such as two |mF =0(cid:105) a sample of atomic matter waves, where the field along one Zeemansublevelsofdifferenthyperfinemanifolds,astyp- direction is composed of multiple spectral components, hav- ically used in Raman atom interferometers [24, 32]. At ing frequencies ω−,α. (b) Energy-momentum dispersion dia- n low magnetic field, these states have an energy differ- gram. Two low-energy internal states |g1(cid:105) and |g2(cid:105) are cou- ence (cid:126)ω ≡ (cid:126)ω −(cid:126)ω determined by their hyperfine pledthroughstimulatedstate-andmomentum-changingtwo- 12 g2 g1 photonRamantransitions. Alllaserfieldsarefar-detunedby splitting,whichweassumegreatlyexceedsthelargestki- an amount ∆ (cid:29) ω from atomic resonance, so that the ex- netic energy scales in the problem. As we show below, 12 cited state |e(cid:105) is only virtually driven. For the left-traveling, this extra internal ground state degree of freedom, when multi-frequency field, two distinct sets of frequency compo- combinedwithalaser-drivingprotocolsimilartothatde- nents(labeledα=1and2),areusedinconjunctionwiththe scribed in Sec. II, will allow for the study of U(2) lattice right-traveling field to drive unique state- and momentum- models with near-arbitrary parameter control. changingtransitionsthatdependontheinitialinternalstate, We consider interaction of these three-level atoms as described in the text and shown in the figure. 8 As before, the index n will relate to transitions between provides arbitrary control over all tunneling amplitudes plane-wave states with momenta 2n(cid:126)k and 2(n+1)(cid:126)k. t+ ≡ (cid:126)Ω˜1 and t− ≡ (cid:126)Ω˜2 and tunneling phases ϕ+ ≡ φ˜1 n n n n n n The index α = 1 relates to processes where atoms un- and ϕ− ≡φ˜2. For every site-to-site transition, there ex- n n dergo a transition from |g1(cid:105) to |g2(cid:105) as their momentum ist two possible pathways involving non-commuting op- increases by 2(cid:126)k (|g1,n(cid:105) ↔ |g2,n+1(cid:105)), while α = 2 re- erations on the internal (pseudo)spin degree of freedom. lates to momentum-increasing processes that transition By coordination of the tunneling amplitudes and phases from |g2(cid:105) to |g1(cid:105) (|g2,n(cid:105) ↔ |g1,n+1(cid:105)), as depicted in relating to each of these pathways, a tunable U(2) lat- Fig. 4. Making the restrictive assumption that every tice model can be constructed. To be explicit, if we as- frequency component is exactly resonant with a unique sume equal tunneling amplitudes for the two pathways momentum-changing Raman transition, the frequencies (t+ = t− ≡ t ), the effective Hamiltonian can be recast n n n of the left-traveling field’s various components are given as by HÎ ≈(cid:88)t (cˆ† Uˆ cˆ +h.c.) , (21) ω−,1 =ω+−ω −(2n+1)4E /(cid:126) and (15) eff n n+1 n n n 12 R n ωn−,2 =ω++ω12−(2n+1)4ER/(cid:126) . (16) wΦher=e Uϕˆ+n +=ϕe−iΦna/n2d[coΘs(Θ=n/2ϕ)+σˆx−−ϕs−in.(ΘTnh/i2s)σaˆlyl]o,wwsiuths n n n n n n The relevant one-photon Rabi frequencies relating to in- to vary the U(1) phase and SU(2) internal state spin- teraction with the different field components are given rotation associated with every individual tunneling link. by Ω+,α =−(cid:104)e|d·E+|g (cid:105)/(cid:126), Ω−,1 =−(cid:104)e|d·E−,1|g (cid:105)/(cid:126), Further inclusion of state-preserving Bragg transitions α n n 2 and Ω−,2 = −(cid:104)e|d·E−,2|g (cid:105)/(cid:126). As in the previous case, associated with each link would allow for an even more n n 1 we assume that we are in the limit where all one-photon generalized form of the Uˆ matrices. n Rabifrequenciesaremuchlessthanthesingle-photonde- Followingtheprocedureoutlinedearlier,thisU(2)lat- tuning∆. Thisrestrictionallowsustoagainconsideran ticemodelcanalsobeperformedinmorethanonespatial adiabatic elimination of the excited state |e(cid:105), with only dimension, allowing for a model of the form stimulated two-photon processes allowed. For processes characterized by absorption of a photon from the right- HÎ ≈(cid:88)[tx (cˆ† Uˆx cˆ +h.c.) eff m,n m+1,n m,n m,n traveling laser field and stimulated emission into the fre- m,n (22) quency component of the left-traveling field with indices +ty (cˆ† Uˆy cˆ +h.c.)] . n and α, the effective two-photon Rabi frequency and m,n m,n+1 m,n m,n phase shift are given by This allows for the study of genuine non-Abelian U(2) Ω∗−,αΩ+ models,wheremotionalongclosedpathscanleadtonon- Ω˜αneiφ˜αn = n2∆ ei(φ+−φ−n,α) . (17) trivial operations on the atoms’ internal degree of freedom. For the smallest counter-clockwise path around a We again make the stronger restriction that the two- foursiteplaquette, thiscanleadtoanoperationdistinct photon Rabi frequencies are all smaller in magnitude from identity I, thanthefrequencyspacingbetweenuniquespectralcom- ponents, (cid:126)Ω˜αn (cid:28) 8ER/(cid:126) ∀ n,α. In this weak-driving Uˆ(cid:9) ≡Uˆ†y Uˆ†x Uˆy Uˆx (cid:54)=eiβI , (23) limit, the off-diagonal elements of the interaction Hamil- m,n m,n m,n+1 m+1,n m,n tonian HÎ have only one dominant contribution eff such that the Wilson loop variable associated with this closedpath, tr(Uˆ(cid:9) ), isnotequalto2, thedimensionof (cid:104)g2,n+1|HêIff|g1,n(cid:105)/(cid:126)≈Ω˜1neiφ˜1n and (18) the internal statme,nspace. Independent control over all tunneling amplitudes and phases allows for the study (cid:104)g1,n+1|HêIff|g2,n(cid:105)/(cid:126)≈Ω˜2neiφ˜2n . (19) oelfemmeondtealrsywlaitthticheomploagqeuneettoeuss, aWsiwlsoelnl alososppsat[7ia9l]lyfovrarayl-l The dynamics of this system, neglecting differential AC ing and disordered configurations. In particular, it has Stark shifts of the two ground states, can again be de- been suggested that the U(2) random flux model may scribed by an effective tight-binding Hamiltonian in the be of direct relevance to the effect of giant magnetoresis- limit of weak-driving, given by tance displayed in manganese oxides [56]. Furthermore, while the described setup is clearly restricted to the sim- HêIff ≈(cid:88)t+n(eiϕ+ncˆ†n+1σˆ+cˆn+h.c.) ulation of matter interacting with classical Abelian and non-Abeliangaugefields, Ref.[72]recentlyraisedthein- n (20) +(cid:88)t−n(eiϕ−ncˆ†n+1σˆ−cˆn+h.c.) , taevreersatgininggporvoesrpeacnt aopfpursoinpgriasutechdsisimtriubluattiioonnso-fasltoantgicwainthd n annealed classical gauge field configurations - to gain in- with σˆ = (σˆ + iσˆ )/2 = |g (cid:105)(cid:104)g | and σˆ = (σˆ − sight into certain properties of lattice gauge theories de- + x y 2 1 − x iσˆ )/2 = |g (cid:105)(cid:104)g |, where σˆ and σˆ are the Pauli matri- scribing the interaction of matter with dynamical gauge y 1 2 x y ces. Controlofthelasersidebandamplitudesandphases fields. 9 V. V. CONCLUSIONS atomic condensates evolving in momentum space would naturallyplayhosttosignificantnonlinearprocesses[80], such as cross-phase modulation, self-phase modulation, Inconclusion,wehavepresentedasimpleexperimental andfour-wavemixing[81]. Moreover,thegeneralscheme schemeforstudyingnearlyarbitrarysingleparticletrans- of developing link-resolved control of tunneling by use of port phenomena based on well established atom optics an inhomogeneous potential and global field addressing techniques. We described two variations of this scheme, may be transportable to strongly-correlated studies. In based on internal state-preserving Bragg transitions and an optical lattice simulator, for example, tunable inho- internal state-changing Raman transitions, which enable mogeneouspotentialsmaybecreatedbyprojectivemeth- the study of Abelian U(1) and non-Abelian U(2) lattice ods [82–84], and global addressing via laser-assisted tun- models, respectively. Some unique features of this plat- neling [85, 86] may be used to reintroduce site-to-site form were discussed, including the possibilities of study- coupling in a link-dependent fashion, allowing local con- ing annealed disorder and variable-range hopping. We trol over tunneling amplitudes and phases. have discussed practical limitations to the timescales of Lastly, as a natural consequence of developing a new coherent evolution that this scheme allows, which relate atom optics-based system for simulating coherent trans- toseveraltenstoseveralhundredsoftunnelingeventsfor port phenomena, the atom optics toolset will be ex- realistic system parameters. We neglected discussion of panded to include unique new capabilities for the ma- further extensions, such as the use of additional internal nipulation of atomic matter waves. ground states for the simulation of U(N) models with N > 2, and we neglected a discussion of the important and intriguing role of nonlinear interactions between the VI. ACKNOWLEDGMENTS atoms themselves. WethankTaylorHughesforhelpfulconversations,and In contrast to many photonic simulators, a system of Brian DeMarco for helpful conversations and comments. [1] I.Bloch, J.Dalibard, andS.Nascimbéne,Nat.Phys.8, [16] G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat, 267 (2012). T. Uehlinger, D. Greif, and T. Esslinger, Nature 515, [2] R. Blatt and C. F. Roos, Nat. Phys. 8, 277 (2012). 237 (2014). [3] A. Aspuru-Guzik and P. Walther, Nat. Phys. 8, 285 [17] M. Atala, M. Aidelsburger, M. Lohse, J. T. Barreiro, (2012). B. Paredes, and I. Bloch, Nat. Phys. 10, 588 (2014). [4] I.Bloch,J.Dalibard, andW.Zwerger,Rev.Mod.Phys. 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