aa r X i v : . [ qu a n t - ph ] D ec Two-dimensional imaging of gauge fields in optical lattices
Jaeyoon Cho and M. S. Kim
QOLS, Blackett Laboratory, Imperial College London, London SW7 2BW, UK (Dated: June 7, 2018)We propose a scheme to generate an arbitrary Abelian vector potential for atoms trapped in a two-dimensional optical lattice. By making the optical lattice potential dependent on the atomic state,we transform the problem into that of a two-dimensional imaging. It is shown that an arbitrarily finepattern of the gauge field in the lattice can be realized without need of diffraction-limited imaging.
Recently, many-body systems of trapped atoms haveoffered a new avenue towards understanding strongly cor-related matters [1]. In this context, trapped atoms ingauge fields have attracted much attention [2–5]. In atwo-dimensional (2D) confinement, such systems wouldexhibit intriguing phenomena, such as the fractionalquantum Hall effect and the anyonic statistics, whichhave opened up a new era of modern condensed matterphysics [6, 7].The aim of this paper is to propose a concrete frame-work to realize the following Hamiltonian in an opticallattice system H = − J X j,k (cid:0) c † j +1 ,k c j,k e iθ j,k + c † j,k +1 c j,k + H.c. (cid:1) , (1)which describes a tight-binding particle in the lowestband of a 2D square lattice in a spatially slow-varyingmagnetic field B ( r ) = ▽ × A ( r ) [8]. Here, the positionsof lattice sites are r j,k = r ( j ˆ x + k ˆ y ) with r being thelattice spacing, c j,k denotes a particle annihilation oper-ator at the site, and θ j,k = (2 π/ Φ ) R r j +1 ,k r j,k A · d l withΦ = h/e being the magnetic flux quantum. We havechosen a gauge such that the vector potential is writtenas A ( r ) = A ( x, y )ˆ x . While the lattice structure givesrise to intriguing physics in its own right, such as Hofs-tadter butterfly [9], this model Hamiltonian also restoresthe fractional quantum Hall physics in the continuumlimit when a strong on-site interaction is considered inaddition [5]. Such an optical lattice system would thusprovide an ideal testbed to study the physics in the pres-ence of gauge fields, along with established techniques forits microscopic control and measurement [10, 11].Before proceeding further, we highlight the underly-ing idea and important features of the present work.Differently from earlier schemes in this context [3, 4],our scheme is devised to make it feasible to realize anarbitrary A ( r ), which enables the generation of quasi-particles/quasiholes by adiabatically changing A ( r ) [6].This may open up an intriguing possibility of directlyand microscopically inspecting the fractional charge andstatistics of the quasiparticles, as well as other modes ofmeasurement discussed in earlier literature [3]. Further-more, we achieve this goal along with overcoming theearlier experimental problems, such as the stability issueor complicated experimental setup [3, 4]. Unlike most of the earlier schemes based on engineering Raman-inducedhopping, we simply make use of the ordinary tunneling inan optical lattice. We employ the state-dependent latticepotential that naturally comes about by adjusting the de-tuning and polarization of the trapping laser [12]. On topof this simple setup, we show that the spatial phase pro-file of a static driving field, which induces on-site Ramantransition of atoms with different phases, is mapped intothe Abelian gauge field A ( r ). Our scheme should thusbe well within state-of-the-art technology and compati-ble with currently prevalent setups based on alkali-metalatoms. Moreover, as we use the atomic dark states de-coupled from the Raman field, the recoil heating is min-imized. While the diffraction limit seems problematic inimaging an arbitrary phase pattern site-by-site on thelattice, we show that a 2D array of only moderately wellfocused beams, each centered on a lattice site, in factsuffices for any A ( r ), aside from the available technologyof subdiffraction imaging [13].We first present our (abstract) Hubbard model,putting off the detailed explanation of its realization un-til later. Let us consider a situation where two hyperfinelevels | a i and | b i of an atom experience different opti-cal lattice potentials, while the two lattices coincide. Inparticular, let us assume the atoms in | a i ( | b i ) hop onlyin the ˆ x (ˆ y ) direction. The Hamiltonian for this state-dependent hopping can be written as H l = − J X j,k (cid:0) a † j +1 ,k a j,k + b † j,k +1 b j,k + H.c. (cid:1) , (2)where a j,k ( b j,k ) denotes the annihilation operator foratoms in | a i ( | b i ) and for simplicity the hopping rate J isassumed to be the same for each direction. On top of this,Raman fields for transition | a i ↔ | b i are applied withposition-dependent phases φ j,k . Note that these phasesare determined unambiguously by the difference betweenthe phases of two Raman fields. The Hamiltonian for theRaman transitions is written as H r = ω X j,k (cid:0) a † j,k b j,k e − iφ j,k + H.c. (cid:1) , (3)where ω is the (real-valued) Raman transition rate as-sumed to be the same for every lattice site. For con-venience, let us perform a local gauge transformation (a) (b)FIG. 1. The smallest loop in a square lattice (a) and a bilayerlattice we consider (b). In (b), the upper and lower layers trapatoms in | a i and | b i , respectively. Both loops enclose the samenumber of magnetic flux quanta (mod 1) ( θ j,k − θ j,k +1 ) / π . a j,k → a j,k e − iφ j,k . The single-particle Hamiltonian H s = H l + H r then reads H s = − J X j,k (cid:0) a † j +1 ,k a j,k e iθ j,k + b † j,k +1 b j,k + H.c. (cid:1) + ω X j,k (cid:0) a † j,k b j,k + H.c. (cid:1) , (4)where θ j,k = φ j +1 ,k − φ j,k . Note that θ j,k can be chosenarbitrarily and independently for each pair of sites. Weare going to choose θ j,k as defined in Eq. (1) (conversely,the laser phases determine the vector potential A ( r ) ofthe simulated system). Note that the smallest loop of thissystem as shown in FIG. 1(b), which is not 2D, enclosesthe same amount of magnetic fluxes as that in FIG. 1(a).A further progress is made by performing a trans-formation such that c j,k ≡ √ ( a j,k − b j,k ) and d j,k ≡ √ ( a j,k + b j,k ). Eq. (4) then reads H s = H + H , where H = − J X j,k (cid:0) c † j +1 ,k c j,k e iθ j,k + c † j,k +1 c j,k + H.c. (cid:1) − J X j,k (cid:0) d † j +1 ,k d j,k e iθ j,k + d † j,k +1 d j,k + H.c. (cid:1) + ω X j,k (cid:0) d † j,k d j,k − c † j,k c j,k (cid:1) , (5) H = − J X j,k (cid:0) c † j +1 ,k d j,k e iθ j,k + d † j +1 ,k c j,k e iθ j,k + H.c. (cid:1) + J X j,k (cid:0) c † j,k +1 d j,k + d † j,k +1 c j,k + H.c. (cid:1) , (6)where J ≡ J/ H alone isdivided into two bands centered at ± ω , each correspondsexactly to that of our desired Hamiltonian (1), whereas H only contains terms that give rise to transitions be-tween the two bands. These transitions are energeticallycostly for 2 ω ≫ J , in which case they are suppressedand hence H can be treated as a perturbation. This be-comes clearer when we consider a uniform magnetic field θ j,k = 2 παk , where α is the number of magnetic fluxquanta passing through the lattice cell. In FIG. 2, we plotthe energy spectrum of Hamiltonian (4). The horizontalaxis represents the energy and the vertical axis α , where rational numbers α = p/q with p , q coprime integers and q <
50 are taken. As the width of each energy band is4 J , two energy bands are separated when ω > J . Asis expected, for a large ω (e.g., see FIG. 2(d)), the origi-nal shape of the Hofstadter butterfly is restored in eachband, albeit slightly deformed due to the perturbation.This deformation gets smaller for larger ω .For a system of N atoms, one can choose the lowestenergy band centered at − N ω , wherein all the particlesare approximately in c j,k modes. It is noteworthy that c j,k particles are in dark states when the correspondingtwo Raman fields have the same Rabi frequency. As theseparticles are decoupled from the Raman fields, they are,in principle, not influenced by the spontaneous emission.In order to see that this system can produce the fractionalquantum Hall physics, we again employ a uniform mag-netic field θ j,k = 2 παk and consider the following on-siteinteraction Hamiltonian on top of the above single-atomHamiltonian: H i = U X j,k (cid:2) ( a † j,k ) a j,k +( b † j,k ) b j,k + a † j,k a j,k b † j,k b j,k (cid:3) , (7)where for simplicity all the interaction rates are assumedto be the same as U . As an example, we have numericallydiagonalized the Hamiltonian for two atoms in a 8 × U = ω = 10 J and α = 1 /
16, hence the filling factor is ν = 1 /
2. Dueto the torus geometry, this system has twofold degen-erate ground states. We have obtained the two lowestenergy eigenstates ρ µ ( µ = 1 ,
2) by tracing out the inter-nal state. The purities of these states are Tr( ρ ) = 0 . ρ ) = 0 . c j,k modes isTr( P c † j,k c j,k ρ µ ) = 2 . P L = (cid:12)(cid:12) Ψ L (cid:11) (cid:10) Ψ L (cid:12)(cid:12) + (cid:12)(cid:12) Ψ L (cid:11) (cid:10) Ψ L (cid:12)(cid:12) and calculatethe overlap of the above numerical ground states to thissubspace, which is found out to be Tr( P L ρ µ P L ) = 0 . α , the ground state is not expectedto be altered significantly. Our numerical calculation in-dicates that for the second-nearest-neighbor hopping rateset to be J/
10, the overlap of the ground state is still ashigh as 0.998.The remaining question is how to realize Hamiltoni-ans (2) and (3). To be concrete, we introduce below aparticular setup based on alkali-metal atoms, generaliz-ing the conventionally used one-dimensional (1D) setups[12]. However, the idea would be applicable to differentatomic species or laser configurations. -1-3 1 3010.5 0-2-4 2 4 -3-5 -1 1 3 5 -12 -10 -8 (a) ω = J (b) ω = 2 J (c) ω = 3 J (d) ω = 10 J Energy (J) Energy (J) Energy (J) Energy (J) α (e) FIG. 2. The energy spectrum of the system for different parameter choices. In (d), only one of the two bands, with the energyranging from − J to − J , is plotted, which resembles the original Hofstadter butterfly shown in (e).(a) (b)FIG. 3. (a) The fine structure of alkali-metal atoms, wherethe transitions by ˆ σ − polarization, equivalent to those by ˆ σ + with m J → − m J , are omitted for brevity, and (b) a laserconfiguration for the lattice potential in ˆ x direction. Hamiltonian (2) requires the lattice potential to de-pend on the atomic state. This can be achieved byexploiting the polarization dependence of atomic dipoletransitions. Let us first consider the hopping in ˆ x di-rection. The basic idea is as follows [12]. Consider thefine structure of alkali-metal atoms and suppose that thefrequency of the trapping laser is chosen between thoseof D (S / ↔ P / ) and D (S / ↔ P / ) transitions,as shown in FIG. 3(a). If the trapping laser is in ˆ σ + polarization, atoms in | + i would experience a negativeac Stark shift V + as the laser is red-detuned from theD transition. On the other hand, atoms in |−i wouldexperience both negative and positive ac Stark shiftsdue to the D and D transitions, respectively. Whenthese two Stark shifts are summed, the resulting en-ergy shift V − can range from negative to positive val-ues, depending on the laser frequency, while V + is stillnegative in any case. The ratio V + /V − can thus beadjusted to a great extent. The energy shift for eachhyperfine level is now obtained as a linear combinationof V ± , determined by the Clebsch-Gordan coefficients.As an example, let us take | a i ≡ | F = 1 , m F = +1 i and | b i ≡ | F = 2 , m F = +1 i among the ground hyper-fine levels of Rb. The ratio between the correspond-ing lattice potentials is then given by V b ( x ) /V a ( x ) =(3 V + + V − ) / ( V + + 3 V − ). For example, suppose the fre-quency of the laser is chosen such that V + = − V − ,leading to V b ( x ) /V a ( x ) = 5 (note that this can be donewhen the frequency is red-detuned compared to that cor- responding to V − = 0). In 1D cases, the hopping rateis proportional to E r ( V /E r ) / exp( − p V /E r ), where E r = ~ k / m is the recoil energy and V is the poten-tial depth [10]. If we take moderately V a ( x ) = − E r at the minima, the above choice leads to a hopping ratefor | b i being 0.013 times that for | a i in ˆ x direction, wellapproximating our model.In an analogous fashion, one could use ˆ σ − polariza-tion in ˆ y direction. In a 2D geometry, however, thiscannot be done without introducing ˆ π polarization whenthe beam in ˆ x direction is chosen to be ˆ σ + polarized.When the ˆ π polarization is involved, as well as the acStark shift, a Raman transition can also take place, ascan be seen in FIG. 3(a). Although the Raman transi-tion could be suppressed by applying a magnetic field,it would be advantageous to exploit the large fine split-ting to obtain a stronger potential, as we discuss below.Let us take the convention that ˆ σ ± = √ (ˆ x ± ˆ y ) andˆ π = ˆ z . In FIG. 3(b), we depict the laser configurationin ˆ x direction. The lattice potential in ˆ y direction canbe created analogously. In ˆ x direction, two tilted stand-ing waves with wavevectors ± ˆ k = ± k (cos η ˆ x − sin η ˆ z )and ± ˆ k = ± k (cos η ˆ x + sin η ˆ z ) and polarizations ˆ ǫ ∝√ η ˆ σ + + cos η ˆ z and ˆ ǫ ∝ √ η ˆ σ + − cos η ˆ z , re-spectively, are applied (note ˆ k · ˆ ǫ = ˆ k · ˆ ǫ = 0).Choosing the phases appropriately, the electric fieldcan be written as E ( r ) = ˆ σ + E + ( r ) + ˆ πE π ( r ) with E + ( r ) ∝ √ η cos( kx cos η ) cos( kz sin η ) and E π ( r ) ∝ cos η sin( kx cos η ) sin( kz sin η ). Note that E + ( r ) and E π ( r ) are π/ σ + ( ∝ −| E + ( r ) | ) correspond tothe maxima of the potential for ˆ π ( ∝ −| E π ( r ) | ), and viceversa. If we choose a 2D plane formed, e.g., at z = 0 witha strong confining potential in ˆ z direction, the influenceof the ˆ π polarization to the lattice potential is negligible.Furthermore, denoting by W ± ( r ) the (real and symmet-ric) Wannier functions for state |±i , the Raman transi-tion rate ∝ (cid:12)(cid:12)R d r E + ( r ) E π ( r ) W + ( r − r j,k ) W − ( r − r j,k ) (cid:12)(cid:12) always vanishes for every site r j,k because the electricfield part is an odd function around r j,k . Note that thelattice spacing is 1 / cos η times larger than usual λ/ V , this decreasesthe hopping rate, hence the energy scale of the system.Whereas this can be mitigated by decreasing η , smaller η results in a shallower potential. This trade-off relationwould be important in choosing η in experiments.Finally, we discuss the realization of Hamiltonian (3).For the hyperfine levels | a i and | b i chosen above, theRaman transition in Hamiltonian (3) can be achievedby applying two ˆ σ + polarized Raman fields with differ-ent frequencies. As the 2D lattice is on the x - y plane,this can be naturally done by applying the light in ˆ z direction. Suppose the spatial profiles of the Ramanfields are given by Ω { a,b } ( x, y ) e ikz . The Raman tran-sition rate at site r j,k is then given by ω j,k e iφ j,k ∝ R Ω ∗ a ( x, y )Ω b ( x, y ) W a ( x − jr , y − kr ) W b ( x − jr , y − kr ) dxdy , where W a ( x, y ) and W b ( x, y ) are the Wan-nier functions for | a i and | b i , respectively, and the z dependency is assumed to be decoupled. SupposeΩ ∗ a ( x, y )Ω b ( x, y ) ≃ | Ω | e iφ ( x,y ) . We require φ ( x, y ) tobe slowly varying compared to W { a,b } ( x, y ) so that ω j,k to be the same for every site. φ j,k is then approxi-mately given by φ ( jr , kr ). Note that any nonlinearterm in φ ( x, y ) can lead to non-vanishing magnetic flux( θ j,k − θ j,k +1 ) = 0. For example, computer generatedholography would be relevant for adjusting φ ( x, y ) arbi-trarily as far as its resolution limit permits [15].A truly arbitrary magnetic field would be possiblewhen θ j,k (or φ j,k ) can be controlled site by site. Atfirst sight, this seems to require diffraction-limited imag-ing. For example, this is the case for a uniform magneticfield φ j,k = 2 παjk (mod 2 π ) in the sense that for lat-tice sites with j = 1 / α , the phase φ j,k = ( − k π shouldbe flipped at every adjacent site. As will be shown be-low, however, this seemingly necessary requirement ofdiffraction-limited imaging is in fact not essential in ourscheme, although it should also be noted that subdiffrac-tion imaging technologies are already available [13]. Forconvenience, let us represent the sites ( j, k ) with a sin-gle index ξ or λ . We apply at each site ξ a moder-ately well focused Raman field for | b i with spatial pro-file ω ′ ξ e iφ ′ ξ A ( r − r ξ ), where A ( r ) is a common normalizedmode function, e.g., a Gaussian function, while the otherRaman field for | a i is applied globally and uniformly. TheRaman transition rate at site λ is then given by the sum-mation of all the contributions ωe iφ λ ∝ P ξ T λξ ω ′ ξ e iφ ′ ξ ,where T λξ = R A ( r − r ξ ) W a ( r − r λ ) W b ( r − r λ ) dxdy . Forwell focused beams, T λξ would be non-vanishing only forsmall | r λ − r ξ | . 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