Jean Dalibard Laboratoire Kastler Brossel ENS Paris & Collège de France

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56.3

Figure 1. The Hofstadter butterfly. The eigenenergies ! are shown as blackdots for different ", and J is the hopping energy as defined in the text. The mostdominant feature is the splitting of the energy band into r subbands for " = 1/r .For further details of this plot see [18].

into a finite number of exactly r bands whereas if " is irrational the energy spectrum breaks upinto infinitely many bands and thus the fractal structure shown in figure 1 emerges (for a detaileddiscussion on the properties of the Hofstadter butterfly see [18]).

Fractal energy band structures are believed to play an important role for a number ofeffects like the quantum Hall effect induced by magnetic fields in strongly correlated electronsystems [14, 19]. Therefore it is desirable to study the Hofstadter butterfly experimentally underwell defined conditions. However, it turns out that the area of the elementary cells in metalswhere fractal energy bands could possibly be seen is so small that huge magnetic fields would berequired to obtain values of " which are on the order of unity [18]. Also in more sophisticatedsuperlattice set-ups with larger area A it is experimentally very difficult to obtain direct clearexperimental evidence of the Hofstadter butterfly [20].

We will consider a 2D system of ultracold atoms trapped in one layer in the xy plane ofa three-dimensional optical lattice. The atoms are in the lowest motional band which can beachieved e.g. by loading the optical lattice from a Bose–Einstein condensate [1, 21]. Hoppingalong the z direction is turned off completely by the lattice potential. Different columns of thelattice trap atoms in internal states |g! (|e!) (denoted in figure 2(a) by open (closed) circles) [2, 5].In addition the optical lattice is either accelerated along the x-axis or an inhomogeneous staticelectric field is applied and two Raman lasers driving transitions between the states |g! and |e!induce hopping along the x-axis while hopping along the y-axis is controlled by the depth of theoptical lattice along this direction. This set-up corresponds to applying a magnetic field with aparameter " = q#/4$ where q is the wavenumber of the Raman lasers along the y direction and# the wavelength of the lasers creating the optical lattice. We also note that the atomic set-up weare going to describe here can be used for a large number of other purposes. It is straightforwardto add terms to the system that correspond to an external electric field. Also, the atoms willinteract with each other via collisional contact interactions [5] and off-site interactions can beengineered by dipolar Rydberg interactions [6]. Therefore the set-up presented here can be used

New Journal of Physics 5 (2003) 56.1–56.11 (http://www.njp.org/)

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lower energy dressed state are shown in Fig. 2. The fluxdensity is maximum on the sites of a honeycomb lattice,and vanishes on the triangular lattice dual to this. The tight-binding limit involves tunneling between the sites of thisdual triangular lattice, Fig. 2(b).

The above optical potentials (8) and (9) can be readilygeneralized to many other cases with nonzero mean flux.(There are also many cases with zero mean, but nonzerolocal flux density.) The central requirements to generate anoptical flux lattice are threefold. First, the coupling laser(Mx;y) must generate optical vortices. A 2D lattice ofoptical vortices can be formed from a minimum of threetravelling waves [20]. The resulting optical field is peri-odic, with an equal number of (single-winding) vortices Nv

and antivorticesNav in a unit cell [21]. Second, the species-dependent potential (Mz) must be nonzero at the cores ofthese vortices, such that there is no degeneracy of thedressed states at these points. A small nonzero Mz causesthe cores of the vortices to have the topology of ‘‘merons’’[18], in which ~n!r" sweeps over half of the Bloch sphere.For a given meron, the sign ofMz at its core times the signof its vorticity determines whether it contributes #1=2 or$1=2 a flux quantum. The total number of flux quantathrough the unit cell isN! % N#

v $ N#av, whereN

#v=av is the

number of vortices or antivortices at which Mz is positive.Thus, the third requirement for a nonzero mean flux is thatMz varies in space such that N#

v ! N#av.

We have explored the properties of optical potentialsgenerated by simple laser patterns. An optical flux latticecan be generated using just five travelling waves: threetravelling waves of the coupling laser (Mx;y) to effect thevortex lattice, and a standing wave of the species-dependent potential (Mz). [One such example is to removeone of the four travelling waves from the optical couplingin (9).] In all cases the local flux density is inhomogeneousin space, in some even changing sign. Indeed, one can

show that, for smoothly varying optical fields, the fluxdensity must have at least Nv # Nav zeros in the unit cell[22] The above cases (8) and (9) have non-negative fluxdensity with the minimum number of zeros. For three-, ormore-, level systems, an optical flux lattice can have a fluxdensity that nowhere vanishes. We have examples of opti-cal potentials that lead to such cases. However, theserequire more involved laser configurations, so we do notpursue this here.Having determined the properties of the optical flux

lattices in the adiabatic limit, we now turn to describe theirband structures, obtained from the eigenvalues of (1). Thelaser potentials Msq (8) and Mtri (9) are clearly invariantunder translations by the respective lattice vectors a1;2. Infact, they enjoy higher translational symmetry, being in-variant under the unitary transformations

T 1 & "ye!1=2"a1'r T2 & "xe

!1=2"a2'r (10)

which effect translations by 12a1;2 and rotations in spin

space. These operators do not commute, but satisfy

T 2T1 % $T1T2: (11)

This indicates that they represent magnetic translationsaround a region of space (enclosed by 1

2a1 and 12a2) that

contains 1=2 a flux quantum. As is conventional in systemswith magnetic translation symmetry [15], we define amagnetic unit cell that encloses an integer number offlux: we choose a1, a2=2. Writing the eigenvalues of theassociated (commuting) translation operators T2

1 and T2 aseik'a1 and eik'a2=2 defines the Bloch wave vector k and theassociated Brillouin zone. The additional symmetry T1 andthe condition (11) cause the energy spectrum Ek for allbands to be invariant under k ' a2=2 ! k ' a2=2( # withk ' a1 ! k ' a1.For the square optical flux lattice (8), a solution of the

band structure shows that the lowest energy band does notoverlap any higher band for V * 0:1@2$2=m. The Chernnumber [15] of this band is 1, the sign being reversed underan odd number of sign changes to the terms in (8). Thus,the lowest energy band is topologically equivalent to thelowest Landau level of a charged particle in a uniformmagnetic field. It is instructive to consider the band struc-ture for V ) @2$2=m, when the variation in the adiabaticenergy is dominant, and the low energy bands are welldescribed by a tight-binding model [2]. The minima of theadiabatic potential form a square lattice, Fig. 1(b). Nearest-neighbor hopping on this square lattice leads to a model inwhich each plaquette encloses 1=2 a flux quantum. (Notethat the tight-binding lattice has four plaquettes per a* aunit cell, for which N! % 2.) The magnetic unit cell con-tains two lattice sites, so there are two tight-binding bands.The bands touch at two Dirac points [23], so one can speakonly of the Chern number of the two bands together. Thistotal Chern number is zero, consistent with the fact that thisnearest-neighbor tight-binding model is time-reversal

a

a(a) (b)

FIG. 2 (color online). (a) Bloch vector and (b) flux density n!for the lower energy dressed state of the triangular optical fluxlattice (8). The local minima of the adiabatic energy are at thepoints where n! % 0, forming a triangular lattice in the tight-binding limit (dark circles and dashed lines).

PRL 106, 175301 (2011) P HY S I CA L R EV I EW LE T T E R Sweek ending

29 APRIL 2011

175301-3

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N. R. Cooper and J. Dalibard

-0.5 -0.25 0 0.25 0.5k

3

3.5

4

4.5

5

E/E

R

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Fig. 2: (Colour on-line) Left-hand panel: band structure forF = 1/2 for V = 1.8ER along a path k= k(K2! 2K1)!k3/2through the Brillouin zone. For != "= 0 (dotted blue line),the decoupled m=±1/2 states each have two Dirac points,indicated by circles. For weak coupling != "= 0.1 (dashedgreen line) the Dirac points split in a manner that breaks time-reversal symmetry, giving the lower two bands a net Chernnumber of 1. For intermediate coupling != 0.4, "= 0.3 (solidblack line) the lowest energy band has Chern number 1. Right-hand panel: the density of states for V = 1.8ER, != 0.4 and"= 0.3.

spectrum at the locations of the Dirac points. For != 0the lower two bands separate from the upper two bandsin such a way that both pairs of bands are topologicallytrivial, that is each pair has net Chern number of zero.When both ! and " are non-zero, the optical dressing leadsto a net flux through the unit cell, indicating time-reversalsymmetry breaking. Indeed, we find that for !, " != 0 thebands can acquire non-zero Chern numbers. Specifically, inthe perturbative limit (!, "" 1) the Dirac points split insuch a way that the lower two bands have a net Chernnumber of 1 provided "2/! is su!ciently small. When"2/! exceeds a certain value (# 0.19 for V = 1.8ER) thereis a transition to the topologically trivial case describedabove. Beyond the perturbative limit, as the couplings !and " increase, the splitting between the lower two bandsincreases and the lowest energy band evolves into a narrowband with Chern number 1. An example is shown by thesolid lines and density of states in fig. 2, for which thelowest band has a width"E # 0.1ER. The optical dressing(2) a#ords a great deal of freedom to tune parameters toreduce the ratio of the bandwidth of the lowest band to thegap to the next band. For example, for V = 2ER, "= #/4,!= 1.3, the lowest band has a width of only "E # 0.01ERand is separated from the next band by about 0.4ER (seefig. 3).It is important to emphasize that the formation of this

narrow low-energy band is not simply due to compressioninto a tight-binding band2. Rather it is closely related

2For a tight-binding band in the limit of vanishing tunnel couplingwhen the Wannier states become exponentially localized, the Chernnumber of the band must be zero [20], or a set of bands with netChern number zero must become degenerate.

3 3.5 4 4.5 5 5.5E/E

R

DoS

(ar

b.)

x 1/10

Fig. 3: Density of states for F = 1/2 for V = 2ER, "= #/4,!= 1.3. The lowest band has Chern number 1, a width of about0.01ER, and is well separated in energy from the next band.The density of states for the lowest band has been rescaled by1/10.

to the formation of Landau levels in a uniform magneticfield. A continuum Landau level is highly degenerate, withdegeneracy equal to the number of flux quanta piercing theplane. Thus, for the flux density here, of one flux quantumper unit cell, a Landau level would have one state per unitcell: that is one band within the Brillouin zone. The lowestband of figs. 2 and 3, with its narrow width and Chernnumber of 1, is the optical flux lattice equivalent of thelowest Landau level.The above scheme can be generalized to atoms of the

alkali-metal family, whose ground state nS1/2 is split intotwo hyperfine levels I ± 1/2, where I is the nuclear spin.The laser excitation is tuned in this case around theresonance lines D1 (coupling to nP1/2 with detuning "1)and D2 (coupling to nP3/2 with detuning "2). Let usfocus here on the lowest hyperfine level F = I $ 1/2. Forthe configuration of fig. 1 the atom-laser coupling can bewritten

V =!$2tot"1 + F ·B, (5)

where "!1 = (1/3)"!11 +(2/3)"!12 , F is the angular

momentum operator in the ground state manifold in unitsof !, and

Bx+ iBy = %E$0, Bz = % (|$!|2$ |$+|2), (6)

with % = ("!12 $"!11 )!/[3(F +1)]. Under the unitary

transformation U % exp($ik3 · rFz) the Hamiltoniantakes a similar form to (4), now with &z/2 replaced byFz, and again with a coupling V " in which $0 is replacedby $"0 = e

!ik3·r$0 giving the unit cell of the honeycomblattice as before. Adiabatic motion of the atom stillleads to a dressed state with angular momentum alongthe vector n that wraps the Bloch sphere once in theunit cell. However, now the Berry phase acquired islarger by a factor of 2F [14]. Therefore, the unit cellcontains N! = 2F flux quanta. This increase of N! leadsto an important new feature: a continuum Landau levelnow corresponds to N! = 2F states per unit cell. Thus,the analogue of the lowest Landau level is a set of 2Flow-energy bands with a net Chern number of 1. Spatialvariations in the scalar potential and flux density willcause these bands to split and to acquire non-zero widths.We shall illustrate the physics for F > 1/2 by describing

the properties for bosonic atoms with F = 1. This is a very

66004-p4

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N. R. Cooper and J. Dalibard

interactions and to the formation of strongly correlatedFQH states. We show that, even for fermions interactingwith contact interactions, there remain significant inter-particle interactions within this low-energy band. Thus,our scheme will allow experiments on cold atomic gases toexplore strong correlation phenomena related to the frac-tional quantum Hall e!ect for both fermions and bosons.In the first part of this paper we consider an atomic

species with a ground level g of angular momentum Jg =1/2. Examples of atoms in this category that have alreadybeen laser-cooled are 171Yb or 199Hg (level 6 1S0) [9,10].The atoms are irradiated by laser waves of frequency !Lthat connect g to an excited state e also with angularmomentum Je = 1/2. For ytterbium and mercury atoms,we can choose e to be the first excited level 6 3P0 enteringin the so-called “optical clock” transition. The very longlifetime of e (!10 s for Yb [11] and !1 s for Hg [12]) guar-antees that heating due to random spontaneous emissionsof photons is negligible on the time scale of an experi-ment. Another possible choice could be 6Li atoms, but weestimated in this case a photon scattering rate that is toolarge to maintain the gas at the required low temperature.We assume that the atomic motion is restricted to thexy-plane and described by the Hamiltonian

H =p2

2M1 + V (r), (1)

where M is the atomic mass and p its momentum. Thematrix V acts in the Hilbert space describing the internalatomic dynamics. For an o!-resonant excitation, we canassume that the population of e is negligible at all times, sothat V is a 2" 2 matrix acting on the g± manifold [13]. Itscoe"cients depend on the local value of the laser electricfield, which we characterize by the Rabi frequencies "m,m= 0,±1, where m! is the angular momentum along zgained by the atom when it absorbs a photon.In order to increase our control on the spatial variations

of V , we suppose that a magnetic field parallel to the z-axislifts the degeneracy between the states g±. The resultingsplitting # is supposed to be much larger than the "m’s.Hence for a monochromatic laser excitation at frequency!L, the o!-diagonal matrix elements V+! and V!+ arenegligible compared to the diagonal ones. However wealso assume that another laser field at frequency !L+ #,propagating along the z-axis with $! polarization (i.e.m=#1 with the notation above), is shone on the atoms.The association of this field with the % component (m= 0)of the light at !L provides the desired resonant Ramancoupling between |g±$ (fig. 1(a)). Using standard angularmomentum algebra we find in the {|g+$, |g!$} basis:

V =!"2tot3#1 +

!

3#

!

|"!|2# |"+|2 E"0E""0 |"+|2# |"!|2

"

. (2)

Here "2tot =#

m |"m|2, #= !L#!A, where !A isthe atomic resonance frequency, and we assume|#|% |#|, |"m|. The quantity E characterizes the field

g!

g+

Je = 1/2

!L + "

#!

pol.

!L

!L!L

$

$

$(a () b)

Fig. 1: (Colour on-line) (a) A ground level with angularmomentum Jg = 1/2 is coupled to an excited level also withangular momentum Je = 1/2 by laser beams at frequency !Land !L+ ". The Zeeman splitting between the two groundstates g± is ". (b) Three linearly polarized beams at frequency!L with equal intensity and with wave vectors at an angleof 2#/3 propagate in the xy-plane. The beams are linearlypolarized at an angle $ to the z-axis. The fourth, circularlypolarized beam at frequency !L+ " propagates along thez-axis.

of the additional laser at !L+ #. This beam is assumedto be a plane wave propagating along z, so that E is auniform, adjustable coupling. The ac Stark shift due tothis additional laser is incorporated in the definition of #.We consider the laser configuration represented in

fig. 1(b). The laser field at frequency !L is formed bythe superposition of three plane travelling waves ofequal intensity with wave vectors ki in the xy-plane.We focus on a situation of triangular symmetry, inwhich the three beams make an angle of 2%/3 witheach other, k1 =#k/2 (

&3, 1, 0), k2 = k/2 (

&3,#1, 0) and

k3 = k(0, 1, 0). Each beam is linearly polarized at an angle& to the z-axis, which leads to

!= "3$

i=1

eiki·r[cos & z+sin & (z" ki)], (3)

where " is the Rabi frequency of a single beam. In thefollowing we denote V = !"2/(3#) the energy associatedwith the atom-light interaction and '=E/" the relativeamplitude of the !L+ # field with respect to the !L field.The recoil energy ER = !2k2/2M sets the characteristicenergy scale of the problem.The coupling V is written in eq. (2) as the sum of

the scalar part !"2tot/(3#) 1 and a zero-trace componentthat can be cast in the form W = " ·B/2, where the$i are the Pauli matrices (i= x, y, z). For E '= 0 andsin 2& '= 0, the coupling B is everywhere non-zero. Sup-pose that the atom is prepared in the local eigenstate|((r)$ of W , with a maximal angular momentum projec-tion along n=#B/|B|. Suppose also that it moves su"-ciently slowly to follow adiabatically this eigenstate,which is valid when V%ER. This leads to the Berry’s-phase–related gauge potential i!((|!($, representing anon-zero e!ective magnetic flux density [14]. For mostoptical lattice configurations, the periodic variation of

66004-p2

✓ = ⇡/4

N. R. Cooper and J. Dalibard

interactions and to the formation of strongly correlatedFQH states. We show that, even for fermions interactingwith contact interactions, there remain significant inter-particle interactions within this low-energy band. Thus,our scheme will allow experiments on cold atomic gases toexplore strong correlation phenomena related to the frac-tional quantum Hall e!ect for both fermions and bosons.In the first part of this paper we consider an atomic

species with a ground level g of angular momentum Jg =1/2. Examples of atoms in this category that have alreadybeen laser-cooled are 171Yb or 199Hg (level 6 1S0) [9,10].The atoms are irradiated by laser waves of frequency !Lthat connect g to an excited state e also with angularmomentum Je = 1/2. For ytterbium and mercury atoms,we can choose e to be the first excited level 6 3P0 enteringin the so-called “optical clock” transition. The very longlifetime of e (!10 s for Yb [11] and !1 s for Hg [12]) guar-antees that heating due to random spontaneous emissionsof photons is negligible on the time scale of an experi-ment. Another possible choice could be 6Li atoms, but weestimated in this case a photon scattering rate that is toolarge to maintain the gas at the required low temperature.We assume that the atomic motion is restricted to thexy-plane and described by the Hamiltonian

H =p2

2M1 + V (r), (1)

where M is the atomic mass and p its momentum. Thematrix V acts in the Hilbert space describing the internalatomic dynamics. For an o!-resonant excitation, we canassume that the population of e is negligible at all times, sothat V is a 2" 2 matrix acting on the g± manifold [13]. Itscoe"cients depend on the local value of the laser electricfield, which we characterize by the Rabi frequencies "m,m= 0,±1, where m! is the angular momentum along zgained by the atom when it absorbs a photon.In order to increase our control on the spatial variations

of V , we suppose that a magnetic field parallel to the z-axislifts the degeneracy between the states g±. The resultingsplitting # is supposed to be much larger than the "m’s.Hence for a monochromatic laser excitation at frequency!L, the o!-diagonal matrix elements V+! and V!+ arenegligible compared to the diagonal ones. However wealso assume that another laser field at frequency !L+ #,propagating along the z-axis with $! polarization (i.e.m=#1 with the notation above), is shone on the atoms.The association of this field with the % component (m= 0)of the light at !L provides the desired resonant Ramancoupling between |g±$ (fig. 1(a)). Using standard angularmomentum algebra we find in the {|g+$, |g!$} basis:

V =!"2tot3#1 +

!

3#

!

|"!|2# |"+|2 E"0E""0 |"+|2# |"!|2

"

. (2)

Here "2tot =#

m |"m|2, #= !L#!A, where !A isthe atomic resonance frequency, and we assume|#|% |#|, |"m|. The quantity E characterizes the field

g!

g+

Je = 1/2

!L + "

#!

pol.

!L

!L!L

$

$

$(a () b)

Fig. 1: (Colour on-line) (a) A ground level with angularmomentum Jg = 1/2 is coupled to an excited level also withangular momentum Je = 1/2 by laser beams at frequency !Land !L+ ". The Zeeman splitting between the two groundstates g± is ". (b) Three linearly polarized beams at frequency!L with equal intensity and with wave vectors at an angleof 2#/3 propagate in the xy-plane. The beams are linearlypolarized at an angle $ to the z-axis. The fourth, circularlypolarized beam at frequency !L+ " propagates along thez-axis.

of the additional laser at !L+ #. This beam is assumedto be a plane wave propagating along z, so that E is auniform, adjustable coupling. The ac Stark shift due tothis additional laser is incorporated in the definition of #.We consider the laser configuration represented in

fig. 1(b). The laser field at frequency !L is formed bythe superposition of three plane travelling waves ofequal intensity with wave vectors ki in the xy-plane.We focus on a situation of triangular symmetry, inwhich the three beams make an angle of 2%/3 witheach other, k1 =#k/2 (

&3, 1, 0), k2 = k/2 (

&3,#1, 0) and

k3 = k(0, 1, 0). Each beam is linearly polarized at an angle& to the z-axis, which leads to

!= "3$

i=1

eiki·r[cos & z+sin & (z" ki)], (3)

where " is the Rabi frequency of a single beam. In thefollowing we denote V = !"2/(3#) the energy associatedwith the atom-light interaction and '=E/" the relativeamplitude of the !L+ # field with respect to the !L field.The recoil energy ER = !2k2/2M sets the characteristicenergy scale of the problem.The coupling V is written in eq. (2) as the sum of

the scalar part !"2tot/(3#) 1 and a zero-trace componentthat can be cast in the form W = " ·B/2, where the$i are the Pauli matrices (i= x, y, z). For E '= 0 andsin 2& '= 0, the coupling B is everywhere non-zero. Sup-pose that the atom is prepared in the local eigenstate|((r)$ of W , with a maximal angular momentum projec-tion along n=#B/|B|. Suppose also that it moves su"-ciently slowly to follow adiabatically this eigenstate,which is valid when V%ER. This leads to the Berry’s-phase–related gauge potential i!((|!($, representing anon-zero e!ective magnetic flux density [14]. For mostoptical lattice configurations, the periodic variation of

66004-p2

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sin ✓ ei� = cos(~1 · ~r)� i cos(~2 · ~r)cos ✓ = cos [(~1 � ~2) · ~r]

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dependence for hyperfine states [15]. Off-diagonal entriesrepresent interstate transitions which can involve single-photon [17] or two-photon processes [16].

In all cases, the coupling V!r" involves absorption oremission of photons from laser beams, so it naturallyappears as couplings V!0!

k0#k $ h!0; k0jVj!; ki, where j!;kiis the momentum k eigenstate for component !. We focuson periodic lattices, such that the set of momentum trans-fers f!g for which V!0!

! is nonzero forms a regular lattice.The momentum k of any given component ! is then onlyconserved up to the addition of reciprocal lattice vectors,the basis vectors of which we denote Gi, with i % 1 . . . dwhere d is the dimensionality of the lattice. (We focus onlattices in d % 2, but the results can be extended to d % 3,with the Berry curvature and magnetic flux density becom-ing pseudovectors.)

By Bloch’s theorem, the energy eigenstates of Eq. (1)can be assigned crystal momentum q and band index n,and be decomposed as jc nqi % P

!;Gcnq!Gj!; q# G# g!i,

where ! runs over allN internal states,G runs over all sitesof the reciprocal lattice, and the vectors g! account forpossible momentum offsets involved in the interstate tran-sitions. The energy eigenvalues Enq follow from

Enqcnq!G % "q#G#g!c

nq!G &

X

!0;G 0V!0!G&g!#G0#g!0

cnq!0G0 ; (2)

where "k $ @2jkj2=!2M".Deep lattice or adiabatic limit.— In the limit ER=V ! 0

the kinetic energy can be neglected. The effects of theoptical coupling in Eq. (1) can then be fully understoodin terms of the local dressed states, the eigenstatesof V!r". Provided these dressed states are everywherenondegenerate (as in the cases of interest below), theatom moves through space adiabatically, remaining in agiven dressed state. For an atom in the nth dressed state,the changing dressed state wave function un!!r" causes theatom to acquire a Berry phase [18], which mimics theAharonov-Bohm phase of an effective magnetic field,and causes the atom to experience a magnetic flux density[1] n#!r" % z'

2$i

P!run(! ) run!. In an optical flux lattice,

the lowest energy dressed state experiences N# ! 0 mag-netic flux quanta through each unit cell [13].

It is instructive to examine this limit ER=V ! 0 alsofrom the point of view of Eq. (2). Dropping the kineticenergy causes this eigenvalue problem to reduce to that of auniform tight-binding model in reciprocal space, definedby the couplings V!0!

G&g!#G0#g!0between sites at positions

G& g!. The energy spectrum of this reciprocal-spacetight-binding model—its ‘‘band structure’’—consists ofN bands, since there are N sites g! associated with eachreciprocal lattice point G. The eigenstates are extendedBloch waves in reciprocal space, and can be written

cnr!G / ei!G&g!"'run!!r". (For ER=V ! 0 the energy is

independent of q and the states are naturally labeledby the band index n % 1; . . . ; N and a conserved ‘‘momen-tum’’ r.)Crucially, we identify this conserved momentum of the

reciprocal-space tight-binding model with the real-spaceposition r (and thus its ‘‘Brillouin zone’’ with the real-space unit cell). The state in the nth band simply corre-sponds to the nth local dressed state of V!r". The power ofthis viewpoint emerges upon considering the magnetic fluxdensity n#: the Berry curvature associated with the adia-batic motion of any dressed state in real space equals theBerry curvature of the associated band of the reciprocal-space tight-binding model. In particular, the number N# ofmagnetic flux quanta through the real-space unit cell equalsthe Chern number of that band. Hence, the criterion for anoptical flux lattice is that the lowest energy band of thereciprocal-space tight-binding model has a nonzero Chernnumber. Since much is known about the Chern bands ofvarious tight-binding models, this criterion can be used todesign optical flux lattices with specific properties.One highly desirable feature is to generate a magnetic

flux density that is uniform. The resulting energy bandswill then closely reproduce Landau levels: topologicalbands with exact degeneracy, highly susceptible to strong-correlation physics. For all previously proposed opticalflux lattices [11,13,15] the magnetic flux density is verynonuniform, vanishing at a set of points. This nonuniform-ity is a mathematical necessity for N % 2 [13] and forgeneral N when V consists of the generators of SU(2), asin Ref. [15]. The above considerations show how to over-come this limitation: one should form a reciprocal-spacetight-binding model for which the lowest energy band has auniform Berry curvature.An optical flux lattice that achieves this goal is illus-

trated in Fig. 1. The N internal states are arranged on thesites of a triangular lattice, marked by the state label !.The links between lattice sites indicate optical couplings,and the displacement in reciprocal space the momentumtransfer, with the label marking amplitude and phase.Explicitly, Fig. 1 encodes the coupling matrix

!1

!3 !2G =2 !3

!Ve2i /N"

!Ve3i /N"!Vei /N"!Ve4i /N"

G1

!V!V

!V !V

!V*

**

**

*

=N !1

1 2

1 2

3

1

1

3

FIG. 1 (color online). One primitive unit cell of the reciprocal-space tight-binding model corresponding to the triangularlattice optical flux lattice with N internal states [Eq. (3)], show-ing the optical couplings #V ei#. The anticlockwise sum of thephases # around any one of the triangular plaquettes is $=Nmodulo 2$.

PRL 109, 215302 (2012) P HY S I CA L R EV I EW LE T T E R Sweek ending

21 NOVEMBER 2012

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!3 !2 !1

'1'2'3

|Xi

|Y i|Zi

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2g

0

0.01

0.02

0.03

Δµ/E

R

ν=1/2 (Nφ=6x3)

ν=1 (Nφ=6x2)

1/4 2/3 .1/2 3/4 10ν

0

0.02

0.04

Δµ/E

R

g=0.15g=0.30g=0.46g=0.76

~

~~

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