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Topological lattice using multi-frequency radiation
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New J. Phys. 20 (2018) 055001

https://doi.org/10.1088/1367-2630/aab7a3

PAPER

Topological lattice using multi-frequency radiation
OPEN ACCESS

Tomas Andrijauskas1, I B Spielman2,3 and Gediminas Juzeliūnas1
RECEIVED

20 June 2017
REVISED

26 February 2018
ACCEPTED FOR PUBLICATION

19 March 2018

1
2
3

Institute of Theoretical Physics and Astronomy, Vilnius University, Saulėtekio 3, LT-10222 Vilnius, Lithuania
Joint Quantum Institute, University of Maryland, College Park, MD 20742-4111, United States of America
National Institute of Standards and Technology, Gaithersburg, MD 20899, United States of America

E-mail: gediminas.juzeliunas@tfai.vu.lt
Keywords: artiﬁcial gauge ﬁelds, Floquet, ﬂux lattice

PUBLISHED

3 May 2018

Abstract
We describe a novel technique for creating an artiﬁcial magnetic ﬁeld for ultracold atoms using aperiodically pulsed pair of counter propagating Raman lasers that drive transitions between a pair ofinternal atomic spin states: a multi-frequency coupling term. In conjunction with a magnetic ﬁeld
Any further distribution ofgradient,
this dynamically generates a rectangular lattice with a non-staggered magnetic ﬂux. For athis work must maintainattribution to thewide range of parameters, the resulting Bloch bands have non-trivial topology, reminiscent of Landauauthor(s) and the title of
the work, journal citation levels, as quantiﬁed by their Chern numbers.
Original content from thiswork may be used under
the terms of the Creative
Commons Attribution 3.0
licence.

and DOI.

1. Introduction
Ultracold atoms ﬁnd wide applications in realizing condensed matter phenomena [1–4]. Since ultracold atom
systems are ensembles of electrically neutral atoms, various methods have been used to simulate Lorentz-typeforces, with an eye for realizing physics such as the quantum Hall effect (QHE). Lorentz forces are present inspatially rotating systems [5–11] and appear in light-induced geometric potentials [12, 13]. The magnetic ﬂuxes
achieved with these methods are not sufﬁciently large for realizing the integer or fractional QHE. In opticallattices, larger magnetic ﬂuxes can be created by shaking the lattice potential [14–17], combining static opticallattices along with laser-assisted spin or pseudo spin coupling [12, 13, 18–24]; current realizations of thesetechniques are beset with micro motion and interaction induced heating effects. Here we propose a new methodthat simultaneously creates large artiﬁcial magnetic ﬁelds and a lattice that may overcome these limitations.
Our technique relies on a pulsed atom-light coupling between internal atomic states along with a statedependent gradient potential that together create a two-dimensional periodic potential with an intrinsic
artiﬁcial magnetic ﬁeld. With no pre-existing lattice potential, there are no a priori resonant conditions thatwould otherwise constrain the modulation frequency to avoid transitions between original Bloch bands [25].
For a wide range of parameters, the ground and excited bands of our lattice are topological, with nonzero Chern
number. Moreover, like Landau levels the lowest several bands can all have unit Chern number.
The manuscript is organized as follows. Firstly, we describe a representative experimental implementation of
our technique directly suitable for alkali atoms. Secondly, because the pulsed atom-light coupling is timeperiodic, we use Floquet methods to solve this problem. Speciﬁcally, we employ a stroboscopic technique toobtain an effective Hamiltonian. Thirdly, using the resulting band structure we obtain a phase diagram which
includes a region of Landau level-like bands each with unit Chern number.

2. Pulsed lattice
Figure 1 depicts a representative experimental realization of the proposed method. A system of ultracold atoms
is subjected to a magnetic ﬁeld with a strength B (X ) = B0 + B¢X . This induces a position-dependent splitting
gFμBB between the spin up and down states; gF is the Land g-factor and μB is the Bohr magneton. Additionally,
the atoms are illuminated by a pair of Raman lasers counter propagating along ey, i.e., perpendicular to thedetuning gradient. The ﬁrst beam [up-going in ﬁgure 1(a)] is at frequency w+ = w0, while the second [downngoing in ﬁgure 1(a)] contains frequency components wn = w0 + ( - 1) (dw + nw ) ; the difference frequency© 2018 The Author(s). Published by IOP Publishing Ltd on behalf of Deutsche Physikalische Gesellschaft

New J. Phys. 20 (2018) 055001

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Figure 1. Floquet ﬂux lattice. (a) Experimental schematic depicting a cold cloud of atoms in a gradient magnetic ﬁeld, illuminated by apair of counter-propagating laser beams tuned near two-photon Raman resonance. The down-going beam includes sidebands both to
the red and blue of the carrier (ω0) in resonance at different spatial positions along ex. (b) Level diagram showing even and odd
sidebands linking the ∣ ñ and ∣ ñ states with differing detuning from resonance at X=0. (c) Spatially dependent coupling. Bottom:
different frequency components are in two-photon resonance in different X positions. Top: the recoil kick associated with the Raman
transition is along e y and thus alternates spatially depending on whether the Raman transition is driven from the red or blue
sideband of the down-going laser beam.

between these beams contains frequency combs centered at dw with comb teeth spaced by 2ω, as shown inﬁgure 1(b). In our proposal, the Raman lasers are tuned to be in nominal two-photon resonance with the
Zeeman splitting from the large offset ﬁeld B0 such that gFμBB0=ÿδω0, making the frequency difference+
wn = 0 - w resonant at X = 0, where B = B0. Intuitively, each additional frequency component w n adds aresonance condition at the regularly spaced points Xn = nw gF m B B¢, however, transitions using even-n
sidebands give a recoil kick opposite from those using odd-n sidebands (see ﬁgure 1(c)). Each of these couplinglocations locally realizes synthetic magnetic ﬁeld experiment performed at the National Institute of Standards
and Technology [26], arrayed in a manner to give a rectiﬁed artiﬁcial magnetic ﬁeld with a nonzero average that
we will show is a novel ﬂux lattice.
In practice only a ﬁnite number of lattice teeth are needed, owing to the ﬁnite spatial extent of a trapped
atomic gas. In rough numbers the spatial extent of a quantum degenerate gas is 20 μm, and if we select a verylarge gradient corresponding to a lattice spacing of 0.5 μm, this gives just 40 comb teeth. Generating thefrequency comb is straightforward. In the laboratory, acoustic-optical modulators (AOMs) frequency-shift laser
beams by an amount deﬁned by a laboratory radiofrequency (rf) source. Therefore creating a comb is a matter ofﬁrst creating a frequency comb—simple with rf—and then feeding that signal into the AOM. This sort offrequency synthesis is routine in the ultracold atom labs.
We formally describe our system by ﬁrst making the rotating wave approximation (RWA) with respect to thelarge offset frequency ω0. This situation is modeled in terms of a spin-1/2 atom of mass M and wave-vector
K = -i with a Hamiltonian
H (t ) = H0 + V (t ).

(1)

The ﬁrst term is
H0 = 2K 2
D(X )
+
s3,
2M
2

(2)

where Δ(X)=Δ′X describes the detuning gradient along ex, and s3 = ∣  ⟩⟨  ∣ - ∣  ⟩⟨  ∣ is a Pauli spin
operator. In the RWA only near-resonant terms are retained, giving the Raman coupling described by
V (t ) = V0 å [e i(K0 Y - 2nwt ) + e i(-K0 Y - (2n + 1) wt ) ]∣  ⟩⟨  ∣ + H.c.

(3)n

The ﬁrst term describes coupling from the sidebands with even frequencies 2nω, whereas the second term
describes coupling from the sidebands with odd frequencies (2n + 1) w . The recoil kick is aligned along e ywith opposite sign for the even and odd frequency components. In writing equation (3) we assumed that the
coupling amplitude V0 and the associated recoil wavenumber K0 are the same for all frequency components. The
2

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T Andrijauskas et al

coupling Hamiltonian V(t) and therefore the full Hamiltonian H(t) are time-periodic with period 2p w , and we
accordingly apply Floquet techniques.

3. Theoretical analysis
The outline of this section is as follows. (1) We begin the analysis of the Hamiltonian given by equation (1) bymoving to dimensionless units; (2) subsequently derive an approximate effective Hamiltonian from the singleperiod time evolution operator; (3) provide an intuitive description in terms of adiabatic potentials; and (4)ﬁnally solve the band structure, evaluate its topology and discuss possibilities of the experimental
implementation.
3.1. Dimensionless units
For the remainder of the manuscript we will use dimensionless units. All energies will be expressed in units of
w , derived from the Floquet frequency ω; time will be expressed in units of inverse driving frequency ω−1,
denoted by τ=ωt; spatial coordinates will be expressed in units of inverse recoil momentum K−1
0 , denoted bylowercase letters (x, y)=K0(X, Y). In these units, the Hamiltonian (1) takes the form
H (t w )1= E r k 2 + W (t ) · s ,w222
where E r =  K0 (2Mw ) is the dimensionless recoil energy associated with the recoil wavenumber K0;
k = K K0 is the dimensionless wavenumber. The dimensionless couplingh (t ) =W (x , y , t ) = (2 Re u ( y , t ) , 2 Im u ( y , t ) , bx )

(4)

(5)

includes a combination of position-dependent detuning and Raman coupling. Here b = D¢ (wk 0) describes
the linearly varying detuning in dimensionless units; the function u ( y , t ) = v0 ån{exp [i ( y - 2nt )]+ exp [i (-y - (2n + 1) t )]} is a dimensionless version of the sum in equation (3) with v0 = V0 (w ).
In the time domain the coupling given by equation (5) is
1
1
W (t ) · s = bxs3 + å vl ( y ) d (t - pl ) ,
(6)
2
2
l
with
vl ( y ) = pv0 [e iy + ( - 1)l e-iy]∣ ñá∣ + H.c.

(7)

In this way we have separated the spatial and temporal dependencies in the coupling (6).
3.2. Effective Hamiltonian
We continue our analysis by deriving an approximate Hamiltonian that describes the complete time evolution
over a single period from τ=0−ò to t = 2p -  with   0 . This evolution includes a kick v0 at the
beginning of the period τ+ = 0 and a second kick v1 in the middle of the period t- = p; between the kicks theevolution includes the kinetic and gradient energies. In the full time period, the complete evolution operator is a
product of four terms:(1)(0)
U (2p , 0) º lim U (2p -  , 0 -  ) = U0 U kick
U0 U kick
.

(8)

⎧
⎡
⎤⎫
1
U0 = exp ⎨ - ip ⎢Er k 2 + s3 bx ⎥ ⎬
⎣
⎦⎭
⎩
2

(9)

 0

Here

is the evolution operator over a half period, generated by kinetic energy and gradient. The operator(l )U kick= exp [ - ivl ( y )](10)

describes a kick at t = lp .
We obtain an effective Hamiltonian by assuming that the Floquet frequency ω greatly exceeds the recoil
frequency, 1  E r , allowing us to ignore the commutators between the kinetic energy and functions ofcoordinates in equation (8). We then rearrange terms in the full time evolution operator (8) and obtain (seeappendix)Ueff = exp [{- i2p ([Er k 2 + veff )]]},

3

(11)

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where veff is an effective coupling deﬁned by(1) -ips3 bx 2 (0)exp ( - i2pveff ) = e-ips3 bx 2U kicke
U kick.

(12)(l )The function vl(y) entering the kick operators Ukick
is spatially periodic along the y direction with a period
2π. This period can be halved to π by virtue of a gauge transformation U = exp (-iys3 2). Subsequently, when
exploring energy bands and their topological properties, this prevents problems arising from using a twice larger
elementary cell. Following this transformation the evolution operator becomesUeff = exp {- i2p [Er (k + s3 e y 2)2 + veff ]},

(13)

(l )
where vl(y) featured in the kick operators Ukick
has now the spatial periodicity π along the y direction, i.e. it
should be replace tovl ( y ) = pv0 [e i2y + ( - 1)l ]∣  ⟩⟨  ∣ + H.c.

The algebra of Pauli matrices allows us to write the effective coupling veff (r ) featured in the evolution
equations (12) and (13) as:1
veff (r ) = Weff (r ) · s ,
2

(14)

(15)

where Weff = (Weff,1, Weff,2 , Weff,3) is a position-dependent effective Zeeman ﬁeld which takes the analytic form
exp ( - i2pveff ) = q0 - iq1 s1 - iq2 s2 - iq3 s3.

(16)Here q0, q1, q2 and q3 are real functions of the coordinates (x, y), allowing to express the effective Zeeman ﬁeld asqarccos q0,(17)
Weff = p-1∣∣q∣∣where q is a shorthand of a three dimensional vector (q1, q2, q3). In general the equation (16) gives multiple
solutions that correspond for different Floquet bands. Our choice (17) picks only to the two bands that lie in theenergy window from -1 2 to 1/2 covering a single Floquet period.
Comparing (12) and (16) and multiplying four matrix exponents give explicit expressions
q0 = cos f1 cos f2 cos (pbx ) ,(18)

q1 = sin f1 cos f2 cos ( y + pbx ) - cos f1 sin f2 sin ( y ) ,(19)

q2 = sin f1 cos f2 sin ( y + pbx ) + cos f1 sin f2 cos ( y ) ,(20)

q3 = cos f1 cos f2 sin (pbx ) - sin f1 sin f2(21)

f1 ( y ) = 2pv0 cos ( y ) ,(22)

f2 ( y ) = 2pv0 sin ( y ).

(23)

with

These explicit expressions show that the resulting effective Zeeman ﬁeld (17) and the associated effective
coupling (15) are periodic along both ex and e y , with spatial periods a x = 2/b and a y = p respectively.
Therefore, although the original Hamiltonian containing the spin-dependent potential slope µxs3 is notperiodic along the x direction, the effective Floquet Hamiltonian is. The spatial dependence of the Zeeman ﬁeld
components Weff,1, Weff,2 and Weff,3 is presented in the ﬁgure 2 for β=0.6 giving an approximately square unit
cell. In ﬁgure 2 we select v0=0.25 where the absolute value of the Zeeman ﬁeld Weff is almost uniform, as isapparent from the nearly ﬂat adiabatic bands shown in ﬁgure 3 below.
3.3. Adiabatic evolution and magnetic ﬂux
Before moving further to an explicit numerical analysis of the band structure, we develop an intuitive
understanding by performing an adiabatic analysis of motion governed by effective Hamiltonianh eff (r ) = Er (k + s3 e y 2)2 +1
Weff · s2(24)

featured in the evolution operator Ueff, equation (13). The coupling ﬁeld Weff (r ) is parametrized by the sphericalangles q (r ) and f (r ) deﬁned by
Weff,3,Weff
Weff,2tan f =.
Weff,1
cos q =

4

(25)
(26)

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Figure 2. Coupling components (a) Weff,1(r ), (b) Weff,2 (r ) and (c) Weff,3 (r ) for v0=0.25 and β=0.6 calculated using equations (17)–(23). The corresponding eigenvalues of the coupling v(r ) = Weff 2 are presented by the thick red solid lines in the ﬁgure 3.

Figure 3. Adiabatic Floquet potentials given by equation (29) for β = 0.6 ms. (a) Thin black dotted lines denote the spin-dependent
gradient slopes without including the Raman coupling (v0 = 0); (b) thin blue solid lines denote effective adiabatic potentials for weak
Raman coupling (v0=0.05) (c) red solid lines denote nearly ﬂat adiabatic potentials that are achieved for stronger Raman coupling(v0=0.25). All the curves are projected into x plane for various y values. A weak y dependence of the adiabatic potentials is seen to
appear in the strong coupling case (c) making the superimposed red lines thicker.

This gives the effective coupling [12]⎡11
q e-if sin q ⎤ ,Weff · s = Weff ⎢ icos⎥f
⎣ e sin q - cos q ⎦
2
2

(27)

characterized by the position-dependent eigenstates⎛ cos (q / 2) ⎞∣ + ⟩ = ⎜ if⎟,
⎝ e sin (q / 2)⎠⎛- e-if sin (q / 2)⎞∣-⟩ = ⎜⎟.
⎝ cos (q / 2) ⎠(28)

The corresponding eigenvalues
1v(r ) =  Weff ,2(29)

are shown in ﬁgure 3 for various value of the Raman coupling v0. As one can see in ﬁgure 3, for v0=0.25 theresulting bands v(r ) (adiabatic potentials) are ﬂat and have a considerable gap »w 2, a regime suitable for a
description in terms of an adiabatic motion in selected bands [27].
As in [28], we consider the adiabatic motion of the atom in one of these ﬂat adiabatic bands with the
projection Schrödinger equation that includes a geometric vector potential
1A (r ) =  (cos q - 1) f.
2(30)This provides a synthetic magnetic ﬂux density B(r ) =  ´ A(r ). The geometric vector potential A(r )may contain Aharonov–Bohm type singularities, that give rise to a synthetic magnetic ﬂux over an elementarycell5New J. Phys. 20 (2018) 055001

T Andrijauskas et al

Figure 4. Geometric ﬂux density B(r ) =  ´ A(r ) computed for v0 = 0.25 and β = 0.6 using equation (30). The overall spatial
structure of this ﬂux density does not depend on the gradient β; rather it scales with the corresponding lattice constant a x = 2/b .

a = - å∮singul dr · A(r ).

(31)

The singularities appear at points where q = p , where the angle f and its gradient f are undeﬁned andcos q = -1. The term cos q - 1 in (30) is nonzero and does not remove the undeﬁned phase f . Our unit cell
contains two such singularities located at r = (a x , 3a y ) 4 and r = (3a x , a y ) 4, containing the same ﬂux, so thatthey do not compensate each other, giving the synthetic magnetic ﬂux 2p in each unit cell. Note that usually
the optical lattices are sufﬁciently deep, and the 2p ﬂux per elementary is topologically trivial. In that case thetight binding model can be applied, with the tunneling taking place only between the nearest-neighboring sitesof the square plaquette. The 2p ﬂux over the square plaquette can then be eliminated by a gaugetransformation. Yet if the lattice is shallow enough, the tight binding model is not applicable and the above
arguments do not work. In the present situation, the most interesting topological lattice appears for a ﬂat
adiabatic trapping potential shown by a solid red curve in ﬁgure 3. In such a situation there are no well deﬁned
lattice sites, and the 2p ﬂux per elementary cell results in topologically non-trivial bands explored in the next
subsection.
For a weak coupling (such as v = 0.05) the geometric ﬂux density B (r ) º B(r ) is concentrated around theintersection points of the gradient slopes shown in in ﬁgure 3 and has a very weak y dependence. With increasing
the coupling v, the ﬂux extends beyond the intersection areas and acquires a y dependence. Figure 4 shows thegeometric ﬂux density B (r ) º B+(r ) for the strong coupling (v0 = 0.25) corresponding to the most ﬂat
adiabatic bands. In this regime the ﬂux develops stripes in the x direction and has a strong y dependence. For thewhole range of coupling strengths 0  v0  1 2 the total synthetic magnetic ﬂux per unit cell is 2π and isindependent of the Floquet frequency ω and the gradient β.
Now let us discuss the effect of an extra spin-independent trapping potential. The present scheme requires alarge spin-dependent energy gradient which would have a huge inﬂuence on the relative trapping for the two
spin states without the Raman coupling or for a weak Raman coupling. In that case one would expect that thestable positions for any trapped sample of the two spin states would live at entirely distinct locations, possiblywith no overlap. Yet we are interested mostly in a sufﬁciently strong Raman coupling where the two spin states
get mixed, and the atomic motion takes place in almost ﬂat adiabatic potentials shown in red in ﬁgure 3.
Therefore the atoms are no longer affected by the steep spin-dependent potential slopes, and the spinindependent trapping potential would not cause separation of different spin states. Instead, the extra spinindependent parabolic trapping potential would simply make the ﬂat adiabatic trapping potentials parabolic. Of
course, one needs to be all the time in the regime where the Raman coupling is strong compared to thecharacteristic energy of the spin-dependent potential slope. That is why we propose to introduce the spindependent potential gradient only at the ﬁnal stage of the adiabatic protocol discussed in section 3.5.
3.4. Band structure and Chern numbers
We analyze the topological properties of this Floquet ﬂux lattice by explicitly numerically computing the band
structure and associated Chern number using the effective Hamiltonian (24) without making the adiabatic
approximation introduced in section 3.3. Again the gradient of the original magnetic ﬁeld is such that weapproximately get a square lattice, β=0.6. Furthermore, we choose the Floquet frequency to be ten times larger
6New J. Phys. 20 (2018) 055001

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Figure 5. Left: band structure calculated using the effective Hamiltonian (24) for v0=0.25, β=0.6 and Er=0.1. Right: the band gap
D12 between the ﬁrst and second bands for Er=0.1 and various values of v0 and β. The letters A, B and C indicate the regions where
the Chern numbers of the ﬁrst three bands correspond to the right, middle and left parts of the ﬁgure 6.

Figure 6. Dependence of Chern number for the bands calculated using the effective Hamiltonian (24) on the coupling strength v0 for
b = 0.6 and Er=0.1. Here we present the Chern numbers c1, c2 and c3 of the three lowest bands.

than the recoil energy, Er=0.1. Note that one can alter the length of the plaquette along the x direction (andthus the ﬂux density) by changing β representing the potential gradient along the x axis.
First, let us consider the case where v0=0.25 corresponding to the most ﬂat adiabatic potential. In thissituation the Chern numbers of the ﬁrst ﬁve bands appear to be equal to the unity, as one can see in the left part ofﬁgure 5. Thus the Hall current should monotonically increase when ﬁlling these bands. This resembles the QHE
involving the Landau levels. Second, we check what happens when we leave the regime v0=0.25 where theadiabatic potential is ﬂat, and consider lower and higher values of the coupling strength v0. Near v0=0.175 we
ﬁnd a topological phase transition where the lowest two energy bands touch and their Chern numbers change toc1=0 and c2=2, while the Chern numbers of the higher bands remain unchanged, illustrated in ﬁgure 6. In avicinity of v0=0.3 there is another phase transition, where the second and third bands touch, leading to a new
distribution of Chern numbers: c1=1, c2=−1, c3=3, c4=1. Interestingly the Chern numbers of thesecond and the third bands jump by two units during such a transition.
Finally, we explore the robustness of the topological bands. The right part of ﬁgure 5 shows the dependenceof the band gap Δ12 between the ﬁrst and second bands on the coupling strength v0 and the potential gradient β.
7New J. Phys. 20 (2018) 055001

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One can see that the band gap is maximum for v0=0.25 when the adiabatic potential is the most ﬂat. The gap
increases by increasing the gradient β, simultaneously extending the range of the v0 values where the band gap isnonzero. Therefore to observe the topological bands, one needs to take a proper value of the Raman couplingv0≈0.25 and a sufﬁciently large gradient β, such as β=0.6.
We now make some numerical estimates to conﬁrm that this scheme is reasonable. We consider an
ensemble of 87Rb atoms, with ∣ ⟩ = ∣ f = 2, mF = 2⟩ and ∣ ⟩ = ∣ f = 1, mF = 1⟩ ; the relative magnetic
moment of these hyperﬁne states is »2.1 MHz G-1, where 1 G=10−4 T. For a reasonable magnetic ﬁeld
gradient of 300 G/cm, this leads to the D¢  » 2p ´ 600 MHz cm-1 = 2p ´ 60 kHz mm-1 detuning
gradient. For 87Rb with λ = 790 nm laser ﬁelds the recoil frequency is wr 2p = 3.5 kHz . Along with the driving
frequency ω=10ωr, this provides the dimensionless energy gradient b = D¢ (wk 0) » 1.3, allowing easy
access to the topological bands displayed in ﬁgure 5.
3.5. Loading into dressed states
Adiabatic loading into this lattice can be achieved by extending the techniques already applied to loading into
Raman dressed states [29]. The loading technique begins with a Bose–Einstein condensate (BEC) in the lower
energy ∣  ⟩ state in a uniform magnetic ﬁeld B0. Subsequently one slowly ramps on a single off resonance RF
coupling ﬁeld and the adiabatically ramp the RF ﬁeld to resonance (at frequency δω). This RF dressed state can betransformed into a resonant Raman dressed by ramping on the Raman lasers (with only the ω0+δω frequency
on the k− laser beam) while ramping off the RF ﬁeld. The loading procedure then continues by slowly ramping
on the remaining frequency components on the k− beam, and ﬁnally by ramping on the magnetic ﬁeld gradient(essentially according in the lattice sites from inﬁnity). This procedure leaves the BEC in the q = 0 crystal
momentum state in a single Floquet band.

4. Conclusions
Initial proposals [30–32] and experiments [26] with geometric gauge potentials were limited by the small spatial
regions over which these existed. Here we described a proposal that overcomes these limitations using laser
coupling reminiscent of a frequency comb: temporally pulsed Raman coupling. Typically, techniques relying on
temporal modulation of Hamiltonian parameters to engineer lattice parameters suffer from micro-motion
driven heating. Because our method is applied to atoms initially in free space, with no optical lattice present,
there are no a priori resonant conditions that would otherwise constrains the modulation frequency to avoid
transitions between original Bloch bands [25].
Still, no technique is without its limitations, and this proposal does not resolve the second standing problemof Raman coupling techniques: spontaneous emission process from the Raman lasers. Our new scheme extends
the spatial zone where gauge ﬁelds are present by adding sidebands to Raman lasers, ultimately leading to a
µ N increase in the required laser power (where N is the number of frequency tones), and therefore thespontaneous emission rate. As a practical consequence it is likely that this technique would not be able reach thelow entropies required for many-body topological matter in alkali systems [13], but straightforward
implementations with single-lasers on alkaline-earth clock transitions [33, 34] are expected to be practical.

Acknowledgments
We thank Immanuel Bloch, Egidijus Anisimovas and Julius Ruseckas for helpful discussions. This research was
supported by the Lithuanian Research Council (Grant No. MIP-086/2015). IBS was partially supported by the
ARO’s Atomtronics MURI, by AFOSR’s Quantum Matter MURI, NIST, and the NSF through the PCF at
the JQI.

Appendix. Stroboscopic evolution operator
The stroboscopic evolution operator (8) reads explicitly
⎡

1

⎤

⎡

1

⎤

(1)
(0)
U (2p , 0) = U0 U kick
U0 U kick= e-ip ⎣E r k + 2 s3 bx ⎦ e-iv1 ( y ) e-ip ⎣E r k + 2 s3 bx ⎦ e-iv0 ( y ).
2

2

(32)

In the main we have approximated the evolution operator by equation (11). To estimate the validity of the latter
equation, let us make use of the Baker–Campbell–Hausdorff formula1e X e Y = e Z with Z = X + Y + [X , Y ] + ¼
(33)
2

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and consider this expansion up to the leading term 2 [X , Y ], essentially the second term in the Magnus
expansion.
Neglecting the commutation between E r k 2 and σ3βx, one can write⎡

1⎤

U0 = e-ip ⎣E r k + 2 s3 bx ⎦ » e-ipE r k e-i 2 s3 bx .
2

2

p

(34)

The error in doing so is approximately -i p4 E r bs3 [x , k 2] = p2 E r bk x s3. Since E r b  1, this provides a small
momentum shift along the x direction. Furthermore, we shall neglect the commutation between E r k 2 and vl(y).
The error in doing so is approximately -i p2 E r sx, y [ y , k 2] = pE r k y s3. Since the Floquet frequency ω greatly
exceeds the recoil frequency E r  1 and β<1, this also provides a small momentum shift along the y direction.
With these assumptions, one has(1)(0)
U (2p , 0) = U0 U kick
U0 U kick» e-i2pE r k e-i2pveff ,2where
e-i2pveff = e-ips3 bx 2e-iv1 ( y ) e-ips3 bx 2e-iv0 ( y ).

Finally under the above assumptions one can merge the exponents in U (2p, 0), giving equation (11).

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