A Thouless quantum pump with ultracold bosonic …p^ qA^ 2 = p^2 x 2m a + 1 2m a (^p y qB^x) 2 (S.7) with the vector potential A^ = B^xe y given in the Landau gauge. As the above Hamiltonian

SUPPLEMENTARY INFORMATIONDOI: 10.1038/NPHYS3584

NATURE PHYSICS | www.nature.com/naturephysics 1

M. Lohse1,2, C. Schweizer1,2, O. Zilberberg3, M. Aidelsburger1,2, I. Bloch1,2

1 Fakultät für Physik, Ludwig-Maximilians-Universität,

Schellingstrasse 4, 80799 München, Germany2 Max-Planck-Institut für Quantenoptik,

3Hans-Kopfermann-Strasse 1, 85748 Garching, Germany

Institut für Theoretische Physik, ETH Zürich, 8093 Zürich, Switzerland

MAPPING TO THE HOFSTADTER MODEL IN

THE DEEP TIGHT-BINDING REGIME

In the tight-binding approximation, the dynamics ofultracold atoms in an optical superlattice potential witharbitrary lattice constants dl and ds = αdl, α < 1 can bedescribed by the following Hamiltonian:

H(ϕ) =−∑(( )

a†m+1am + h.c.)

+

m∑m

J0 + δJm(ϕ)

∆m(ϕ)a†mam

(S.1)

Here, a†m (am) is the creation (annihilation) operator act-ing on the m-th lattice site and J denotes the hoppingamplitude between neighboring sites in the short lattice.δJm is the modulation of the hopping amplitude and∆m the on-site energy of the m-th site, both of whichare induced by the long lattice and depend on ϕ. ForVs � Vl

2/(4Er,s), Eq. S.1 reduces to a generalized 1DHarper model [S1, S2] with

δJm(ϕ) =δJ

2cos(2παm− ϕ

)∆m(ϕ) = −∆

2cos(2πα(m− 1/2)− ϕ

) (S.2)

Following the approach of dimensional extension [S3],the periodic parameter ϕ can be regarded as an extradimension, i.e. a realization for a given ϕ corresponds toa 1D subset of a 2D system. The pumping then samplesthe full 2D parameter space which is the reason why thedisplacement can be related to a 2D topological invari-ant. The 2D Hamiltonian onto which the 1D superlat-tice maps in this limit can be found by thinking of the1D Hamiltonian as a single Fourier component of the 2Done and performing an inverse Fourier transform. Forthis, we relabel the creation and annihilation operatorsas a†m,ϕ and am,ϕ and de�ne a 2D Hamiltonian

H2D =1

2π

∫ 2π

0

H(ϕ) dϕ (S.3)

(m,n) (m+1,n)

(m+1,n+1)(m,n+1) Jx

Jx

Jy e+i2πα(m-1/2) Jy e

+i2πα(m+1/2)

Jd e+i2παm

Jd e-i2παm

Figure S1. Square lattice in the presence of a uniform mag-netic �eld with nearest-neighbor hopping Jx = J0, Jy = ∆/4and next-nearest-neighbor hopping Jd = δJ/4 using the gaugeof Eq. S.4.

Expand g a†m,ϕ and am,ϕ into the Fourier components,

a†m,ϕ =

in∑n e

iϕna†m,n and am,ϕ =

ir∑n e−iϕnam,n, gives

H2D = −∑m,n

J0a†m+1,nam,n

−∑m,n

∆

4e+i2πα(m−1/2)a†m,n+1am,n

−∑m,n

δJ

4e+i2παma†m+1,n+1am,n

−∑m,n

δJ

4e−i2παma†m+1,nam,n+1

+ h.c.

(S.4)

which is precisely the 2D Harper-Hofstadter-Hatsugaimodel describing non-interacting particles on a 2D squarelattice in the presence of a uniform magnetic �ux 2πα persquare plaquette with nearest and next-nearest neighbortunneling as illustrated in Fig. S1 [S4]. Under this map-ping, the superlattice phase ϕ corresponds to the trans-verse quasi-momentum ky and in this sense the pumpingprotocol is equivalent to performing Bloch oscillations in

A Thouless quantum pump with ultracold bosonic atoms in an optical superlattice

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http://dx.doi.org/10.1038/nphys3584

2

the 2D lattice by applying a gradient. For our particularcase of ds = dl/2, one obtains a �ux of π per plaquette.

MAPPING OF THE SLIDING LATTICE TO

LANDAU LEVELS

In the opposite limit of the sliding lattice with Vs → 0,a similar analogy can be made to the Landau levels ofa free particle in an external magnetic �eld in 2D. TheHamiltonian of a single particle in the long lattice reads

Hl =p2x

2ma+ Vl sin2 (παx/ds − ϕ/2) (S.5)

For a deep lattice and when only considering the lowestbands, the lattice potential can be expanded in orders ofx− xm around each lattice site xm (i.e. the minimum ofthe potential). In second order this gives:

Hl =p2x

2ma+ Vl

π2α2

d2s

∑m

(x− xm)2 (S.6)

Neglecting the coupling between neighboring sites, thissplits into a series of decoupled Hamiltonians for eachlattice site, each of which is a 1D harmonic oscillator.On the other hand, the motion of a particle with charge

q in an external magnetic �eld B = Bez along the z-direction is described by the Hamiltonian

HLL =1

2ma

(p− qA

)2

=p2

x

2ma+

1

2ma(py − qBx)

2(S.7)

with the vector potential A = Bxey given in the Landaugauge. As the above Hamiltonian commutes with theoperator for the transverse momentum py, one can de�ne

a common set of eigenstates of both HLL and py. For agiven state with momentum ~ky, Eq. S.7 can also berewritten as a 1D harmonic oscillator

HLL =p2

x

2ma+

1

2maω

2c

(x− ~ky

maωc

)2

(S.8)

where ωc = qB/ma is the cyclotron frequency. Compar-ing this with Eq. S.6, one can see that the state in then-th band localized at the lattice site xm corresponds tothe state in the n-th Landau level with a transverse mo-mentum ~ky = maωcxm. As in the tight-binding regime,the number of magnetic �ux quanta per unit cell of the(non-existent) short lattice is given by the ratio of thelattice constants α = ds/dl in this mapping.

Vy,z

Vl

Vs

50ms

50ms

50ms

50ms

50ms

50ms

50ms

150ms

10ms

50ms

initial state 1st pump cycle 2nd pump cycle

t (ms)3602601600

1

40

20

0

0

ϕ (2π)

Vi (

Er,i

)

0.31

0.75

0.50

Figure S2. Schematic of the experimental sequence with thelattice and phase ramps for the initial state preparation as wellas for two pumping cycles. The light and dark lines indicatethe two di�erent lasers used sequentially to overcome tuningrange limitations. The phase ϕ is shown modulo 2π.

TOPOLOGICAL TRANSITION

For ds = dl/2, the transition to the Wannier tunnelingregime, where the pump is characterized by the sameChern number distribution as the 2D HHH model, occurswhen the second and third band cross at the staggeredcon�guration ϕ = π/2 during the pumping cycle. In thetight binding limit, the crossing point can be determinedby comparing the energy of the �rst excited state on thelower site with the one of the ground state on the highersite. For a su�ciently deep short lattice, they can beapproximated by harmonic oscillator states with En =(n + 1/2)~ω. The on-site trapping frequency ω is givenby

ω =

√2π2Vs

mad2s

=2

~√VsEr,s (S.9)

At ϕ = π/2, the tilt is largest with ∆ = Vl and the bandstouch if ∆ = ~ω. Hence, the transition occurs at

Vl = 2√VsEr,s (S.10)

A similar derivation can be made for ds = dl/(2j), j ∈N which gives the same transition point. For all otherratios, Eq. S.10 gives a lower bound of Vl(Vs) and thetransition will in general take place at slightly larger Vl.

EXPERIMENTAL SEQUENCE

Fig. S2 illustrates the detailed experimental sequencefor the initial state preparation of the atoms in the low-est band of the superlattice potential and two exem-plary pump cycles. As discussed in the Methods of the


3

main text, the sequence started by loading an n = 1Mott insulator in 150ms in a 3D optical lattice createdby the long lattice and two orthogonal standing waveswith depth Vi = 30(1)Er,i, i ∈ {l, y, z}. Subsequently,the long lattice sites were split symmetrically by ramp-ing up the overlapped short lattice with λs = λl/2 toVs = 10.0(3)Er,s at ϕ = 0. Simultaneously, the longlattice was lowered to Vl = 20(1)Er,l such that J1 � J2.For the pumping, the phase of the long lattice was

moved by changing the laser frequency slightly. Due tothe limited tuning range of a single laser, a successivehand over between two independent laser systems (in-dicated with light and dark lines in Fig. S2 for the dif-ferent lasers) was implemented in order to extend thepumping range to arbitrary values. Each pump cycleconsists of four segments with s-shaped phase rampsϕ ∈ [0, 0.62π], [0.62π, π], [π, 1.5π] and [1.5π, 2π] of 50msduration to minimize the probability for non-adiabatictransitions to higher bands at the symmetric con�gura-tions ϕ = lπ, l ∈ Z. For the measurements in Fig. 3and Fig. 5 of the main text, the ramp time was scaledwith Vs as the tunneling rates and thus the band gap de-crease with larger Vs. The switching between the lasersat 0.62π and 1.5π was done instantaneously within a 3mshold time between the ramps. It was con�rmed experi-mentally that this switch does not lead to any measur-able excitation to other bands. This scheme was used forall measurements with the exception of the points withϕ < 1.5π in Fig. 2a of the main text, where a single laserwas used with one ramp for ϕ ≤ π and two ramps forϕ > π, with a duration of 100ms for each ramp.

CENTER-OF-MASS POSITION IN THE

PERPENDICULAR DIRECTION

In addition to determining the COM position alongthe pumping direction, we also veri�ed that the pumpingalong x does not lead to any measurable displacement ofthe cloud along the perpendicular y-direction. For themeasurement depicted in Fig. 2b of the main text, theCOM positions along both axes are shown in Fig. S3. Asexpected, we do not observe any signi�cant displacementof the cloud along y over multiple cycles in both pumpingdirections.

MODELS FOR FINITE PUMPING EFFICIENCY

When transitions to other bands occur during thepumping process, the de�ection of the cloud can bechanged due to the di�erent Chern numbers of the bands.For the measurements in the Wannier tunneling limit,we model the band occupations in a simple two-bandmodel assuming that a small fraction of atoms is ex-cited non-adiabatically to the other band during each

ϕ (2π)-4 -2 0 2 4

-4

-2

0

2

4

x (d

l ) /

y (d

l )

J1-J2

∆

J1-J2

∆

Figure S3. Evolution of the COM position along the x- andy-axis when pumping is performed along the x-axis. The greypoints depict the COM displacement in the y-direction for thedata points of Fig.2b in the main text, which are illustratedin red. The positions along y were evaluated using the sameprocedure as for the ones along x. Each point is averaged overten data sets and the error bars depict the standard deviation.

cycle. In the two-band model, these transitions will pre-dominantly take place when ramping over the symmet-ric double well con�guration where the gap between thetwo lowest bands is smallest and atoms tunnel from onesite to the next. This situation occurs twice within onepump cycle and we de�ne the transfer e�ciency as thefraction of atoms staying in the initially populated bandduring one ramp over a symmetric double well, whichcorresponds to half a pump cycle and thus one step inthe COM position.The band populations after the m-th step can be ex-

pressed as

n(m) = εm · n(0) (S.11)

with

n(m) =

(n

(m)1

n(m)2

), ε =

(1− ε1 ε2ε1 1− ε2

)Here, n1 (n2) denotes the fraction of atoms in the �rst(second) band, ε1 is the fraction of atoms excited fromthe �rst to the second band during one step and ε2 theone transferred from the second to the �rst. While thecontribution from non-adiabatic transitions induced bythe pumping is expected to be the same for ε1 and ε2,the �nite lifetime of atoms in the second band generallyleads to ε2 > ε1.When both ε1 and ε2 are very small, one can expand

εm in powers of ε1 and ε2 and obtains to �rst order

εm ≈(

1−mε1 mε2mε1 1−mε2

)(S.12)


4

Assuming a perfect preparation of the atoms in one band,i.e. n(0) = (1, 0) or n(0) = (0, 1), the resulting bandoccupations depend only on ε1 and ε2, respectively. Thismodel was used to �t both the measured COM positionas well as the site-resolved band mapping data with asingle free parameter.The expected displacement for both bands can be cal-

culated by evaluating the COM evolution of the corre-sponding Wannier functions for one half of a pump cyclefrom ϕ = 0 to π. For each segment ϕ ∈ ](m− 1)π,mπ],the combined motion is obtained by adding up the con-tributions from the two bands weighted with the respec-

tive population fraction n(m)1 and n

(m)2 . After each half

of a pump cycle (ϕ = mπ), the COM position can beexpressed in terms of the Chern numbers of the bands:

x(m) =dl2ν ·

m∑j=1

sign(m)ε|j|n(0)

(S.13)

with ν = (ν1, ν2).For the site occupations, the even-odd distribution was

determined by integrating the probability density of theWannier functions over the even and odd site of the cor-responding double well. These were then weighted in the

same way with n(m)1 and n

(m)2 to obtain the fractions on

even and odd sites as a function of ε1 and ε2.

EFFECTS OF THE TRAPPING POTENTIAL

In addition to the superlattice potential, a weak har-monic con�nement with a trapping frequency of ωx =2π · 83(2)Hz is present along the pumping direction inthe experiment. Locally, this creates a small, position-dependent gradient δ ·x/ds, leading to an additional con-tribution to the double-well tilt ∆ as well as an energyo�set between neighboring double wells.On the scale of a single double well, the pumping is

extremely robust against variations of the tilt. In the(J1 − J2)-∆ plane, the gradient predominantly causes ashift of the pump path along the ∆-axis. But as long asit encloses the degeneracy point at the origin, an atomin this double well will still be transfered to the neigh-boring site, illustrating the topological protection of thequantized transport against perturbations. The pumpingtherefore only breaks down once the gradient δ becomeslarger than the largest possible tilt ∆max at ϕ = π/2.For our experimental parameters δ � ∆max and there-fore even atoms at the edge of the cloud are pumpedwith a very high �delity. For very large gradients, thetunneling rates are modi�ed slightly, leading to a smallreduction of the band gap in the symmetric double wellcon�guration. The resulting increase of the probabilityfor non-adiabatic transitions, however, is negligible in ourcase.

-8

-4

0

E/h

(kH

z)

0

ϕ (2π)-0.04 0.080.04-0.08

Figure S4. Pumping in the presence of a gradient. Eigenen-ergies of two atoms in two double wells versus superlatticephase for Vs = 10Er,s and Vl = 20Er,l with a linear gradientδ = h · 100Hz. The gradient decreases the gap between theground state (red line) and �rst excited state (orange line)at ϕ = 0 and can thereby facilitate non-adiabatic transitionswhere an atom tunnels to the neighboring double well in thedirection of the gradient. No COM transport occurs in thiscase. Transitions to the second band within one double well(blue lines) are barely a�ected by the gradient

In an extended system, on the other hand, the trap cana�ect the pumping via another process which can be un-derstood by considering two double wells. While in gen-eral the repulsive on-site interaction will prevent atomsfrom hopping onto the neighboring, occupied double well,a potential gradient will reduce this energy gap and, ifsu�ciently large, can even make the tunneling processresonant. When ramping over the symmetric double wellcon�guration, this can lead to additional non-adiabatictransitions as illustrated in Fig. S4 where no transportoccurs, thereby reducing the overall pumping e�ciency.The gap scales as Ueff − 2δ for small gradients with Ueff

being the e�ective on-site interaction energy of two par-ticles in a double well at ϕ = 0. Ueff is determined byan interplay of the interaction energy U on one (shortlattice) site and the band gap, with the latter giving anupper bound. For our experimental parameters, whichare close to the hard-core limit U → ∞, Ueff is onlyslightly smaller than the band gap at ϕ = 0 which isEgap = h · 474Hz for Vs = 10Er,s and Vl = 20Er,l.

The e�ect of the gradient can be estimated by numer-ically calculating the time evolution of this four-site sys-tem. For our experimental parameters, the probability ofstaying in the ground state while ramping over the sym-metric double well is not a�ected for δ < h · 100Hz andonly drops below the experimentally determined pump-ing e�ciency of 97.9(2)% for δ > h · 150Hz. In our sys-tem, the latter occurs at distances larger than 17 ds fromthe center of the trap, whereas in the experiment the ini-tial radius of the atom cloud in the x-direction is about10 ds (where δ = h · 88Hz). Trap e�ects should there-


5

fore only start to become relevant after a couple of pumpcycles and even then only the outer edge of the cloudis a�ected. We estimate that for our measurements theoverall reduction of the pumping e�ciency due to thetrap is below 1%. Nonetheless this could contribute tothe small di�erence between the e�ciencies determinedfrom the COM positions and the site populations whichwas observed in the experiment.

FINITE TEMPERATURE EFFECTS

The dominant e�ect of a �nite temperature/entropyin the initial Mott-insulating state is the appearance ofholes, i.e. non-occupied double wells. Holes, however, donot contribute to the measured center-of-mass positionand do not a�ect the pumping process as long as theirnumber is small enough such that the system remainsinsulating, which was veri�ed experimentally.

In addition to this, two other low-lying types of ex-citations in the Mott-insulating bulk are the creation ofdoublons and excitations to the �rst excited band. Inthe Wannier tunneling regime, the latter would reducethe pumping e�ciency since the Chern number of theexcited band is opposite to the one of the lowest band.But the energy scale for these types of excitations is givenby the band gap which is very large compared to all otherenergy scales (including the temperature) after the ini-tial loading into the long lattice (Egap = h · 9.5 kHz forVl = 30Er,l). While the subsequent splitting into sym-metric double wells by ramping up the short lattice lowersthe band gap substantially (h·474Hz for Vs = 10Er,s andVl = 20Er,l), the band populations should not be a�ectedassuming an adiabatic evolution. Experimentally, an up-per bound for the initial population in the second bandcan be obtained from the band mapping data in Fig. 2bof the main text. After the �rst step, the fraction onthe higher-lying site, which for large tilts corresponds to

the population in the excited band, is 0.011(6) when av-eraging over both pumping directions. Comparing thiswith the �tted pumping e�ciency from the band map-ping data of 98.7(1)%, one can conclude that the initialnumber of excited atoms is negligible.Doublons, on the other hand, are created when two

atoms end up on the same double well. For localizeddoublons, the pumping scheme still works as long as thetwo atoms stay in the two-particle ground state of thedouble well. This is due to the fact that the maximumtilt between the two sites is much larger than the on-siteinteraction such that the two atoms both localize on thelower-lying site and thus move in the pumping direction.Delocalized doublons as they appear in Mott states at �-nite temperatures can be considered as an additional su-per�uid atom on top of the Mott-insulating background.Even though the de�ection of such an atom in a given k-state is not quantized anymore, the deviation from unityis less than 20% for all values of k for our experimentalparameters. The energy gap for the creation of doublonsis given by the on-site interaction energy which initiallyis h · 1.45 kHz for Vl = 30Er,l. In the superlattice, thecorresponding quantity is the e�ective interaction energyon a double well Ueff , which is almost as large as the bandgap for our experimental parameters U � J1, J2. We es-timate that the fraction of doubly occupied sites in oursystem after the initial loading procedure is at most 1%.The e�ect of doublons on the pumping e�ciency shouldtherefore be negligible in our experiment.

[S1] P. G. Harper, Proc. Phys. Soc. A 68, 879 (1955).[S2] G. Roux, T. Barthel, I. P. McCulloch, et al., Phys. Rev.

A 78, 023628 (2008).[S3] Y. E. Kraus and O. Zilberberg, Phys. Rev. Lett. 109,

116404 (2012).[S4] Y. Hatsugai and M. Kohmoto, Phys. Rev. B 42, 8282

(1990).


http://stacks.iop.org/0370-1298/68/i=10/a=305

http://dx.doi.org/10.1103/PhysRevA.78.023628

http://dx.doi.org/10.1103/PhysRevA.78.023628

http://dx.doi.org/ 10.1103/PhysRevLett.109.116404

http://dx.doi.org/ 10.1103/PhysRevLett.109.116404

http://dx.doi.org/10.1103/PhysRevB.42.8282

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A Thouless quantum pump with ultracold bosonic …p^ qA^ 2 = p^2 x 2m a + 1 2m a (^p y qB^x) 2 (S.7) with the vector potential A^ = B^xe y given in the Landau gauge. As the above Hamiltonian