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Optical Flux Lattices for Ultracold Atomic Gases N. R. Cooper Cavendish Laboratory, University of Cambridge, J. J. Thomson Ave., Cambridge CB3 0HE, U.K. (Dated: January 12, 2011) Weshowthatsimplelaserconfigurationscangiveriseto“opticalfluxlattices”,inwhichoptically dressedatomsexperienceaperiodiceffectivemagneticfluxwithhighmeandensity. Thesepotentials leadtonarrowenergybandswithnon-zeroChernnumber. Opticalfluxlatticeswillgreatlyfacilitate 1 theachievement of thequantumHall regime for ultracold atomic gases. 1 0 PACSnumbers: 67.85.Hj,67.85.-d,37.10.Vz 2 an One of the most important techniques in the ultra- dau level. Since the lengthscale n¯φ−1/2 ∼ λ is similar to J cold atom toolbox is the optical lattice[1]: a periodic the typical interatomic spacing, optical flux lattices will 2 scalar potential formed from standing waves of light. allowthestudyofquantumHallphysicsathighdensities 1 Optical lattices are central to the use of atomic gases where interaction energy scales are large. as quantum simulators, and allow the exploration of We consider an atom moving in optical fields within ] s strong-correlationphenomenarelatedto condensedmat- the rotating wave approximation, with Hamiltonian a ter systems[2, 3]. Their usefulness derives from the fact g pˆ2 - that the scalar potential has a lengthscale, set by the Hˆ = Iˆ+VMˆ(r) (1) t optical wavelength λ, that is similar to the typical inter- 2m n a atomic spacing. where V is the energy scale of the optical potential, of u Largely separate have been the experimental develop- dimensionlessformMˆ(r). Wefocusontwo-levelsystems q mentsofeffectivevectorpotentials,representingthecou- and write . t plingofachargedparticletoamagneticfield. Auniform a M M iM m magnetic field can be simulated for neutral atoms using Mˆ =M~(r) ~σˆ = z x− y (2) rotation[1,4,5]. Gaugefieldsleadingtoaneffectivemag- · (cid:18)Mx+iMy Mz (cid:19) − - d netic field can also be generated by optical dressing[6]. with~σˆ thevectorofPaulimatrices. Additionalscalarpo- n These techniques have allowed experimental studies of o quantizedvorticesincondensedBoseand(paired)Fermi tentialscanbeaddedbyconventionalopticallattices;for c simplicity, we neglect these here. The off-diagonalterms gases[1, 4, 7–9]. An interesting regime of strong corre- [ (M ) arise from the Raman coupling that effects the lation, with connections to the fractional quantum Hall x,y 1 effect, is expected when the effective magnetic flux den- interspecies conversion. The diagonal term (Mz) repre- v sity n is sufficiently high that the lengthscale n 1/2 is sents a species-dependent potential. One possible imple- 2 φ φ− mentationofthetwo-levelsystemiswiththegroundstate comparable to the interatomic spacing[5, 10]. However, 8 and long-lived excited state of an alkaline earth atom or 3 themagneticfluxdensitiesachievableusingexistingtech- ytterbium[14], in which case M can be generated by a 2 niques are relatively small for large systems[6, 9, 11], so z . this strongly correlated regime occurs at very low densi- laser at an “anti-magic” wavelength, λam. As we shall 1 describe below, optical flux lattices can be formed by a 0 ties when the gas is weakly interacting and highly sus- standing wave at λ and three travelling waves of the 1 ceptible to uncontrolled perturbations[5, 11]. am 1 Ramanlaser[16]. Weshallrestrictattentiontoquasitwo- In this paper we describe simple laser configurations : dimensional (2D) systems with r =(x,y). The resulting v that use optical dressing to generate what we refer to fluxlatticesarereadilyadaptedto3D,withnetfluxalong i as “optical flux lattices”. While a conventional optical X one direction. lattice imprints a periodic scalar potential, an optical r The emergence of an effective gauge potential is best a fluxlattice alsoimprintsa periodic magnetic flux density understoodwhenthekineticenergyissmallcomparedto with non-zero mean, and large magnitude n¯ 1/λ2. φ ∼ the energyspacingofthe (local)dressedstates,obtained We emphasize that the magnetic flux density is a con- from the eigenvalues of VMˆ. The atom then moves tinuous function of position, so these potentials are dis- through space adiabatically, staying in a given dressed tinct from proposed ways to imprint gauge potentials state. The adiabatic limit is always valid for sufficiently onto deep optical lattices which apply only in the tight- large V provided the spectrum of Mˆ is non-degenerate. bindinglimit[12–14];furthermore,opticalfluxlatticesre- Assuming this to be true (as shall be verified below for quire only a small number of lasers, so are much easier the cases of interest), we consider the adiabatic motion to implement than these tight-binding proposals. We in a normalized dressed state showthatopticalflux lattices leadto narrowbands with non-zero Chern number[15]. In particular the lowest en- φ (r) Φ(r) = 1 . (3) ergy band is topologically equivalent to the lowest Lan- | i (cid:18)φ2(r)(cid:19) 2 Projecting (1) onto adiabatic motion on the state (3) leads to an effective Hamiltonian with both a scalar and a vector potential, the latter given by[6] a qA=i¯h Φ∇Φ (4) h | i for effective charge q. The number density of magnetic flux quanta perpendicular to the xy-plane is therefore qB q n = ∇ A. (5) φ ≡ h h × (a) a (b) For optical fields of wavelength λ, it is natural to as- sume that the vector potential (4) is smoothly varying FIG. 1: Properties of the lower energy dressed state for the with qA <h/λ. Then,themaximummeanfluxdensity squareopticalfluxlattice(8),asafunction ofposition inthe in a r|egio|n∼of space of sides L ,L λ may be found unitcell. (a)LocalBlochvector,asrepresentedbythevector by applying Stokes’ theorem:x ny ≫d2r n¯ L L = (nx,ny)andcontoursofnz (lightshadingfornz <1anddark (q/h) A dr <(Lx+Ly)/λ, leaRdinφgton¯φ≡<1φ/Lxλwyith flionrenszx>=1).a(/b4)aTnhdeyloc=al flau/x4daenndsitvyanniφshiessmaatxfiomuarlpoonintthse. L=mHin(L·x,L∼y). All existing proposals for∼optically in- The lattice±sites and neare±st-neighbour hopping in the tight- duced gauge fields in the continuum[6], and the scheme bindinglimit are indicated bydark circles and dashed lines. implemented in Ref.9, are of this form with the scale L set by the width of the cloud. Since, typically, L λ, this leads to relatively small flux density, n¯ 1/L≫λ. where κ 2π/a. This has square symmetry with Althoughapparentlyverygeneral,thesecoφn∼siderations a1 = (a,0≡),a2 = (0,a). Achieving this high symme- neglect the fact that smoothly varying optical fields can try in experiment may involve tilting the lasers out of induce singularities in qA. These singularities depend the xy-plane to tune the periods of the Raman and the on the gauge used for (3), and cause no singularities in species-dependent fields[18]. The eigenvalues are non- gauge-invariant properties. Such issues arise whenever degenerate at all positions, so the dressed states admit a U(1) gauge field has non-zero flux through a closed anadiabaticlimit. Fig.1(a) showsthat, forthe loweren- manifold,notablyleadingtoDiracstringsforamagnetic ergy dressedstate, the (nx,ny) components of the Bloch monopole. For the optical flux lattices we propose here, vector exhibit two vortices and two anti-vortices in the there is a net flux through a unit cell which (due to the unit cell. These vortices lead to gauge-dependent singu- spatialperiodicity)hasthetopologyofatorus. Weavoid larities in the vector potential. However, the Bloch vec- technicaldifficulties ofthe gauge-dependentsingularities tor varies smoothly, with nz = 1 at the vortex cores, ± by defining the local Bloch vector in such a way that all four of these regions wrap the sphere in the same sense and contribute a flux density ~n(r)= Φ(r)~σˆ Φ(r) (6) of the same sign. The flux density is shown in Fig.1(c). h | | i It is everywhere non-negative and has total flux N = 2 φ for which~n ~n=1. The flux density is then in the unit cell. (This may be seen by noting that the · two vortices and two antivortices cause ~n to wrap the 1 nφ = ǫijkǫµνni∂µnj∂νnk. (7) Bloch sphere twice.) The flux density is not constant, −8π and vanishes at four points in the unit cell. These four Thisis(minus)the“topologicaldensity”ofthemapfrom points coincidewith the locations atwhichthe adiabatic position space, r = (x,y), to the surface of the Bloch energy is minimum, Fig.1(b), so form the lattice sites in sphere, ~n[17]. The number of flux quanta through a re- the tight-binding limit. gion A is Anφd2r = Ω/4π where Ω is the solid angle Triangular lattice: An optical flux lattice with trian- that regionRA maps to on the Bloch sphere. Thus, each gular symmetry is generated by time the Bloch vector wraps the surface of the sphere corresponds to one magnetic flux quantum. Optical flux Mˆ =cos(r κ )σˆ +cos(r κ )σˆ +cos(r κ )σˆ (9) tri 1 x 2 y 3 z · · · latticesarespatiallyperiodicconfigurationsforwhichthe Bloch vector wraps the sphere an integer number, Nφ, where κ = (1,0)κ,κ = 1,√3 κ and κ = κ times ineachunitcell. The latticevectorsa1 anda2 are 1 2 (cid:16)2 2 (cid:17) 3 1 − both of order the optical wavelengthλ, so the mean flux κ2, with κ 4π/(√3a), giving lattice vectors a1 = ≡ density is of order n¯ N /λ2 which is large. We focus (√3/2, 1/2)a and a2 = (0,1)a. Again, the eigenval- on two cases of highφsy∼mmφetry which achieve this goal. ues are−non-degenerate at all positions. The properties Square Lattice: Consider the optical coupling ofthelowerenergydressedstateareshowninFig.2. The fluxdensityismaximumonthesitesofahoneycomblat- Mˆ =cos(κx)σˆ +cos(κy)σˆ +sin(κx)sin(κy)σˆ (8) tice, and vanishes on the triangular lattice dual to this. sq x y z 3 mogeneousinspace,insomeevenchangingsign. Indeed, onecanshowthat,forsmoothlyvaryingopticalfields,the flux density must have at least N +N zeroes in the v av unitcell[21]Theabovecases(8,9)havenon-negativeflux density with the minimum number of zeroes. For three-, a ormore-,levelsystems,anopticalflux lattice canhavea fluxdensitythatnowherevanishes. Wehaveexamplesof opticalpotentialsthatleadtosuchcases. However,these require more involved laser configurations, so we do not pursue this here. Having determined the properties of the optical flux lattices in the adiabatic limit, we now turn to describe their bandstructures, obtained from the eigenvalues of a (a) (b) (1). ThelaserpotentialsMˆ (8)andMˆ (9)areclearly sq tri invariantundertranslationsbytherespectivelatticevec- tors a . In fact, they enjoy higher translational sym- 1,2 FIG.2: (a)Blochvectorand(b)fluxdensityn forthelower metry,beinginvariantundertheunitarytransformations φ energy dressed state of the triangular optical flux lattice (8). The local minima of the adiabatic energy are at the points Tˆ1 σˆye21a1·∇ Tˆ2 σˆxe21a2·∇ (10) ≡ ≡ wheren =0,formingatriangularlatticeinthetight-binding φ limit (dark circles and dashed lines). which effect translations by 12a1,2 and rotations in spin- space. These operators do not commute, but satisfy Tˆ Tˆ = Tˆ Tˆ . (11) The tight-binding limit involves tunneling between the 2 1 − 1 2 sites of this dual triangular lattice, Fig.1(b). This indicates that they represent magnetic translations The above optical potentials (8,9) can be readily gen- aroundaregionofspace(enclosedby 1a and 1a )that 2 1 2 2 eralized to many other cases with non-zero mean flux. contains 1/2 a flux quantum. As is conventional in sys- (Therearealsomanycaseswithzeromean,butnon-zero tems with magnetic translation symmetry[15], we define local flux density.) The centralrequirements to generate a magnetic unit cell that encloses an integer number of an optical flux lattice are threefold. Firstly, the Raman flux: we choose a , a /2. Writing the eigenvalues of the 1 2 laser (Mx,y) must generate optical vortices. A 2D lat- associated(commuting) translationoperatorsTˆ12 and Tˆ2 ticeofopticalvorticescanbeformedfromaminimumof aseika1 andeika2/2 definestheBlochwavevectorkand · · three travelling waves[19]. The resulting optical field is the associated Brillouin zone. The additional symmetry periodic, with an equal number of (single-winding) vor- Tˆ1 and the condition (11) cause the energy spectrum Ek ticesNv andantivorticesNav inaunitcell[20]. Secondly, forallbandstobeinvariantunderk a /2 k a /2 π. 2 2 the species-dependent potential (M ) must be non-zero · → · ± z For the square optical flux lattice (8), a solution of at the cores of these vortices, such that there is no de- the bandstructure shows that the lowest energy band generacy of the dressed states at these points. A small does not overlap any higher band for V > 0.1h¯2κ2/m. non-zero Mz causes the cores of the vortices to have The Chern number[15] of this band is 1, t∼he sign being the topology of “merons”[17], in which~n(r) sweeps over reversed under an odd number of sign changes to the half of the Bloch sphere. For a given meron, the sign terms in (8). Thus, the lowest energy band is topologi- of M at its core times the sign of its vorticity deter- z cally equivalent to the lowest Landau level of a charged mines whether it contributes +1/2 or 1/2 a flux quan- particle in a uniform magnetic field. It is instructive to − tum. The total number of flux quanta through the unit consider the bandstructure for V ¯h2κ2/m, when the cell is N = N+ N+, where N+ is the number of ≫ φ v − av v/av variation in the adiabatic energy is dominant, and the vortices/antivortices at which Mz is positive. Thus, the low energy bands are well described by a tight-binding third requirement for a nonzero mean flux is that Mz model[2]. The minima of the adiabatic potential form varies in space such that Nv+ 6=Na+v. asquarelattice, Fig.1(b). Nearest-neighbourhoppingon We have explored the properties of optical potentials thissquarelatticeleadstoamodelinwhicheachplaque- generated by simple laser patterns. An optical flux lat- tte encloses 1/2 a flux quantum. The magnetic unit cell tice can be generated using just five travelling waves: contains two lattice sites, so there are two tight-binding three travelling waves of the Raman laser (M ) to ef- bands. The bands touch at two Dirac points[22], so one x,y fectthevortexlattice,andastandingwaveofthespecies- canspeakonlyoftheChernnumberofthetwobandsto- dependent potential (M ). [One such example is to re- gether. This total Chernnumber is zero,consistent with z move one of the four travelling waves from the Raman the fact that this nearest-neighbour tight-binding model couplingin(9).] Inallcasesthelocalfluxdensityisinho- is time-reversal symmetric[23]. In the physical model, 4 E is the ν = 1/2 bosonic Laughlin state on the triangu- tb lar lattice (9), for which the lowest energy chiral band is 3 narrow and well-separated from higher bands. There is 2 1/2afluxquantumperlatticesite,sotheLaughlinstate appears at 1/4 filling. It will be interesting also to ex- 1 plore strong correlation phenomena in 3D settings, with k a -2 Π -Π -1 Π 2 Π y anopticalfluxlattice providingnetflux inonedirection. -2 I am grateful to Jean Dalibard for many helpful com- -3 ments. This work was supported by EPSRC Grant EP/F032773/1. FIG. 3: Lowest energy bands for the triangular flux lattice (9) in the nearest-neighbour tight-binding limit (for uniform spacing of 2π/√3 kxa 2π/√3). The energy Etb is − ≤ ≤ relativetotheatomiclimit, inunitsofthenearest-neighbour [1] I. Bloch, J. Dalibard, and W.Zwerger, Rev.Mod. Phys. hopping. The bandshaveChern numbers 1. 80, 885 (2008). ± [2] D. Jaksch et al.,Phys. Rev.Lett. 81, 3108 (1998). [3] M. Greiner et al.,Nature 415, 39 (2002). with mV/¯h2κ2 large but finite, time-reversal symmetry [4] A. L. Fetter, Rev.Mod. Phys. 81, 647 (2009). is broken by next nearest-neighbour hopping across di- [5] N. R.Cooper, Advancesin Physics 57, 539 (2008). [6] J. Dalibard, F. Gerbier, G. Juzeliu¯nas, and P. O¨hberg, agonals of the square lattice. This leads to closed loops arXiv:1008.5378. around plaquettes which contain 1/4 of a flux quantum. [7] K. W. Madison, F. Chevy, W. Wohlleben, and J. Dal- ThisperturbationactstosplitthebandsatthetwoDirac ibard, Phys. Rev.Lett. 84, 806 (2000). points,andthe twobandsacquireChernnumbersof 1. [8] M. W. Zwierlein et al.,Nature435, 1047 (2005). ± The bandstructure of the triangular optical flux lat- [9] Y.-J. Lin et al.,Nature 462, 628 (2009). tice (9) has the same qualitative properties, the lowest [10] N. R. Cooper, N. K. Wilkin, and J. M. F. Gunn, Phys. energy band having a Chern number of 1. In this case, Rev. Lett.87, 120405 (2001). time-reversalsymmetry is brokenevenin the tight bind- [11] V. Schweikhard et al., Phys. Rev. Lett. 92, 040404 (2004). ing limit with nearest-neighbour hopping. The energy [12] D. Jaksch and P. Zoller, New J. Phys. 5, 56 (2003). minima are at the sites of a triangular lattice, Fig.2(b), [13] E. J. Mueller, Phys. Rev.A 70, 041603 (2004). the elementary plaquettes of which enclose 1/4 of a flux [14] F. Gerbier and J. Dalibard, New J. Phys. 12, 033007 quantum. The energy spectrum of the resulting tight- (2010). binding model, shown for a convenient gauge in Fig.3, [15] D. J. Thouless, M. Kohmoto, M. P. Nightingale, and M. has two narrow bands that are well separated in energy den Nijs, Phys. Rev.Lett. 49, 405 (1982). and have Chern numbers of 1. [16] Fortwo-photonRamantransitions,oneneedslasersthat ± allow threedifferent momentum transfers. Optical flux lattices will allow experiments on ultra- [17] S.M.Girvin,inTopological Aspects of Low Dimensional cold gases to explore many very interesting phenomena. Systems, edited by A. Comtet, T. Jolicoeur, S. Ouvry, Since they lead to a lowest energy band with non-zero and F. David (Springer-Verlag, Berlin, 2000). Chernnumber,non-interactingfermionsfilling thisband [18] For Yb these natural periods are λ0 = 578nm and (with one fermion per magnetic unit cell) will exhibit λam/2=560nm respectively[14]. the integer quantum Hall effect. Signatures of the re- [19] J. Masajada and B. Dubik, Opt. Commun. 198, 21 sulting chiraledgestate couldbe observedinthe density (2001). excitations (collective modes), which will rotate with a [20] Defining the three wavevectors k1,2,3, the reciprocal lattice vectorsof this cell are k2 k1 and k3 k1. handednessdeterminedbythesignoftheChernnumber. − − [21] The proof follows by noting that n = 0 at the saddle φ Thesquarelatticeinthenearest-neighbourtightbinding pointsofnz(r).FromMorsetheoryappliedtonz onthe limit also offers the possibility to study fermionic Dirac unit cell, the number of saddle points is the sum of the physics. Within mean-field theory, interacting bosons numbersof maxima and minima; asum at least as large loaded into the chiral band will develop vortex lattices as Nv + Nav. with very high flux density. These typically break trans- [22] L.-K. Lim, C. M. Smith,and A.Hemmerich,Phys. Rev. Lett. 100, 130402 (2008). lational symmetries of the lattice[24]. Owing to the very [23] Each plaquette contains 1/2 a flux quantum, which is high flux density, it should be possible to reacha regime gauge-equivalent to 1/2 a flux quantum. where the 2D boson density is comparable to the mean [24] G. Möller and N. R−. Cooper, Phys. Rev. A 82, 063625 flux density, n¯φ 1/λ2, where strongly correlated frac- (2010). tional quantum ∼Hall states of bosons[5, 10] or related [25] G. Möller and N. R. Cooper, Phys. Rev. Lett. 103, states on lattices[25] can appear. A leading candidate 105303 (2009).