Realization of an anomalous Floquet topological system with ultracold atoms
Karen Wintersperger, Christoph Braun, F. Nur Ünal, André Eckardt, Marco Di Liberto, Nathan Goldman, Immanuel Bloch, Monika Aidelsburger
RRealization of an anomalous Floquet topological system with ultracold atoms
Karen Wintersperger , , Christoph Braun , , , F. Nur ¨Unal , , Andr´e Eckardt , ,Marco Di Liberto , Nathan Goldman , Immanuel Bloch , , , Monika Aidelsburger , Fakult¨at f¨ur Physik, Ludwig-Maximilians-Universit¨at M¨unchen, Schellingstraße 4, 80799 M¨unchen, Germany Munich Center for Quantum Science and Technology (MCQST), Schellingstraße 4, 80799 M¨unchen, Germany Max-Planck-Institut f¨ur Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany Max-Planck-Institut f¨ur Physik komplexer Systeme, N¨othnitzer Straße 38, 01187 Dresden, Germany T.C.M. Group, Cavendish Laboratory, 19 JJ Thomson Avenue, Cambridge, CB3 0HE, UK Institut f¨ur Theoretische Physik, Technische Universit¨at Berlin, Hardenbergstraße 36, 10623 Berlin, Germany Center for Nonlinear Phenomena and Complex Systems,Universit´e Libre de Bruxelles, CP 231, Campus Plaine, B-1050 Brussels Belgium
Coherent control via periodic modulation, alsoknown as Floquet engineering, has emerged asa powerful experimental method for the realiza-tion of novel quantum systems with exotic prop-erties. In particular, it has been employed tostudy topological phenomena in a variety of dif-ferent platforms. In driven systems, the topo-logical properties of the quasienergy bands canoften be determined by standard topological in-variants, such as Chern numbers, which are com-monly used in static systems. However, due tothe periodic nature of the quasienergy spectrum,this topological description is incomplete and newinvariants are required to fully capture the topo-logical properties of these driven settings. Mostprominently, there exist two-dimensional anoma-lous Floquet systems that exhibit robust chiraledge modes, despite all Chern numbers are equalto zero. Here, we realize such a system withbosonic atoms in a periodically-driven honeycomblattice and infer the complete set of topological in-variants from energy gap measurements and localHall deflections.
Floquet engineering [1–3] has found widespread appli-cations for the realization of out-of-equilibrium many-body systems with novel properties in systems of ul-tracold atoms [4, 5], photonics [6, 7], superconductingqubits [8] and graphene [9]. It plays a key role in manysuccessful realizations of artificial gauge fields and topo-logical lattice models [10, 11], including the paradigmaticHarper-Hofstadter [12, 13] and Haldane model [6, 14]and more recently the generation of non-trivial Chernbands in a 2D optical Raman lattice [15]. The topolog-ical properties of non-interacting two-dimensional (2D)lattice models without additional symmetries are wellunderstood by a set of Chern numbers C µ – a 2D in-variant defined as the integral of the Berry curvatureΩ µ ( q ) in quasimomentum space for the µ th energy band: C µ = π (cid:82) BZ Ω µ ( q ) d q [16, 17], where BZ denotes theBrillouin zone. In cold-atom systems a number of ex-perimental techniques has been developed to determinethe geometric properties of Floquet quasienergy bandsin analogy to their static counterparts [14, 18–22]. Theproperties of chiral edge modes on the other hand have xy a T ω π xy R e l a t i v e i n t e n s i t y M o d a m p m ω ω - 0 HaldaneAnom. phase Q u a s i e n e r g y ε F Figure 1. Schematics of the periodically-modulated lattice,its Floquet quasienergy spectrum and the topological phasediagram. a . Exemplary Floquet spectrum (reduced zonescheme) with topological invariants: Chern numbers C ± andWinding numbers W j , which determine the number and chi-rality of edge modes (red lines) in gap g j , j ∈ { , π } . b .Periodic modulation of the laser intensities I i ( t ), i = { , , } and the resulting real-space potential over one period of thedriving T = 2 π/ω , with ω the modulation frequency. Redlines indicate larger tunnel couplings. Right panel: Illustra-tion of the three interfering laser beams with frequency ω L and the modulated lattice with constant a = 284 nm. Latticeacceleration is realized with additional time-dependent detun-ings ∆ ω ( t ) (Methods). c . Topological phase diagram. Phaseboundaries are obtained from a Floquet band structure cal-culation (Supplementary Information). The gray shaded areacontains additional phases not discussed in this work. been mostly studied with photonic platforms [6, 7].For static 2D systems, such as Chern insulators, thereis a direct correspondence between the Chern numberof the bulk band and the net number of topologically a r X i v : . [ c ond - m a t . qu a n t - g a s ] O c t protected 1D edge modes at the boundaries of the sam-ple, known as bulk-edge correspondence [23, 24]. Remark-ably, this correspondence survives for certain classes ofperiodically-driven systems in the high-frequency limit,where the modulation frequency is the largest energyscale in the system. In general, however, the bulk-edgecorrespondence is modified and knowledge about theChern numbers is not sufficient to determine the num-ber and chirality of chiral edge modes in these drivensettings [25–27]. Instead this information can be ob-tained from a new bulk topological invariant, the windingnumber W (Fig. 1a), characterizing the topology of thequasienergy gaps [26], which depends on the full timeevolution during one period of the drive (i.e. the micro-motion).In periodically driven systems, the quasienergy ε F isonly defined up to integer multiples of the driving en-ergy quantum (cid:126) ω , with angular modulation frequency ω and reduced Planck’s constant (cid:126) . Hence, the edge-statedispersion can leave the spectrum from the top and re-enter from below. An exemplary spectrum is illustratedin Fig. 1a, which shows a Floquet quasienergy spectrumin the reduced zone scheme with − (cid:126) ω/ < ε F < (cid:126) ω/ (cid:126) ω ]. Here, the presenceof the edge mode is the result of a non-trivial winding ofthe quasienergy spectrum itself. This implies that thereis an anomalous Floquet topological phase [28], wheretopological edge modes are present, although the Chernnumber of the energy bands is zero ( C ± = 0); here C − and C + denote the Chern number of the lower and upperquasienergy band in a two-band model.Anomalous Floquet systems are genuine time-dependent settings without any static counterpart. Inparticular, these anomalous edge modes exhibit a re-markable robustness that can even exceed those of con-ventional quantum Hall systems [29]. Anomalous edgemodes have been observed in photonic [30–33] andphononic experiments [34]. However, a complete exper-imental characterization of the topological properties ofFloquet systems is still lacking. Here we report on ex-perimental results obtained with ultracold bosonic atomsin a periodically-modulated honeycomb lattice, where weuse a combination of energy gap [35, 36] and local Halldrift measurements [14, 18] in order to reveal the full setof bulk topological invariants.In honeycomb lattices anomalous Floquet phases canbe generated via step-wise periodic modulation of thetunnel couplings [25], using step-wise linear phase shak-ing [37] or circular phase shaking near resonant with asublattice energy offset [38]. Here, we employ a continu-ous analog of the step-wise modulation protocol proposedin Ref. [25] using amplitude modulation. This is realizedby sinusoidal modulation of the laser intensities (Fig. 1b): I i ( t ) = I (1 − m + m cos( ωt + φ i )), where m is the relativemodulation amplitude and φ i = π × ( i −
1) denotes themodulation phase for the three laser beams, i = { , , } .This time-dependent lattice model exhibits a rich topo- logical phase diagram as a function of the modulationparameters (Fig. 1c). We study the geometric propertiesof the quasienergy spectrum in the three most robustphases: (cid:172) The topological Haldane phase with C ± = ∓ (cid:173) an anomalous phase with trivial Chern bands C ± = 0,but chiral edge modes and (cid:174) a Haldane-like topologi-cal phase with C ± = ±
1, where the chiral edge modesare located between Floquet zones. Generally, the Chernnumber of a certain energy band is determined by thedifference between the net number of edge modes leav-ing the band at the top and entering from below, i.e., C ± = ∓ ( W − W π ), where W j characterizes the net num-ber of edge modes in the energy gap g j , j ∈ { , π } . Thisset of winding numbers uniquely defines the Chern num-bers of the bulk bands, however, the opposite only holdsfor static systems. In this work we deduce the value of thebulk winding numbers by tracking the topological chargesassociated with each energy-gap-closing point [38] thatoccurs at the topological phase transition, providing a fullclassification of the Floquet topological phase diagram ofour model (Fig. 1c). In addition, we calculated the Flo-quet quasienergy spectrum for a semi-infinite system us-ing an approximate tight-binding model, which directlyreveals the edge modes in the respective quasienergy gaps(Supplementary Information).The experimental setup consists of a Bose-Einstein-condensate (BEC) of K atoms loaded into an opticalhoneycomb lattice, that is created by interfering three s -polarized laser beams with λ L = 736 . ◦ (Fig. 1b). Additional harmonic confine-ment is provided by a crossed dipole trap and a third,vertical dipole beam, all at 1064 nm, with total trappingfrequency ω r = 2 π × . xy -plane and ω z ≈ π ×
200 Hz. Using a Feshbach resonance at 403 . a s = 6 . a . The initial state for allmeasurements described in the following is a condensatein the lowest energy eigenstate at zero quasi-momentum(Γ-point) prepared in a honeycomb lattice with in-planedepth V = 6 . E r , where E r = (cid:126) k L m K = h × .
43 kHzis the recoil energy, k L = 2 π/λ L and m K is the mass ofan atom.In a first set of measurements we locate the phaseboundaries shown in Fig. 1c by probing the quasienergygaps. This uniquely determines the topological phasetransition points, since the topology of the bands canonly change via a gap-closing in the spectrum. We re-solve the energy gap locally as a function of quasimomen-tum using St¨uckelberg interferometry [35, 36] (Methods):The quasimomentum of the condensate is changed non-adiabatically using lattice acceleration, which results ina coherent superposition of population in the first andsecond band. Holding at a specific final quasimomen-tum q and subsequently driving back with the sameforce, produces oscillations of the relative band popu-lation with a frequency ∆ E ( q ) / (cid:126) , which can be mea-sured using bandmapping [39]. Note that in principlethere are two different energy gaps g and g π that could q x q y E ω ∆ E ω ∆ E ω ∆ E ω πω π ω π M o d a m p m Modulation amplitude m Modulation amplitude m q x q y Γ K M ω ε F Figure 2. Energy gaps at Γ, K and M and energy bands in the different topological phases shown in the extended zone scheme. a . Minimal energy gap ∆ E = min( g , g π ) at Γ as a function of modulation frequency and amplitude. The white arrows markthe parameter scan used for the measurements in b and c . b . Measured energy gaps (points) at Γ with F a/h = 4086 Hz andtheoretical values (solid line) obtained from a band structure calculation including the six lowest energy bands of the modulatedlattice (Supplementary Information). Error bars denote fitting errors; every data point represents a St¨uckelberg interferometrymeasurement with 23 points each being averaged over 3-4 single experimental realizations. The blue shaded areas indicate thedifferent topological phases, deduced from the gap-closing points. Upper panels: Lowest two energy bands in the extended zonescheme along the high-symmetry path in the first BZ calculated at ω/ (2 π ) = 30 kHz, m = 0 . ω/ (2 π ) = 10 kHz, m = 0 . ω/ (2 π ) = 6 . m = 0 . c . Measured and calculated energy gaps at K and M with errorbars similar to b . be probed using this method. However, since we probethe system at a fixed quasimomentum q at stroboscopictimes, we always measure the minimal quasienergy gapmin( g ( q ) , g π ( q )) (Methods), which can be chosen to liein the interval [0 , (cid:126) ω/
2] as discussed in [38].Figure 2 shows experimental data for various modula-tion parameters along the path illustrated by the whitearrows in Fig. 2a, which covers all three topologicalphases (Fig. 1c). The experimental results of the energygaps at the high-symmetry points are shown in Fig. 2bfor the Γ-point and in Fig. 2c for the M - and K -points.Probing the high-symmetry points is sufficient in two di-mensions to detect gap-closing points in a honeycomblattice with our modulation scheme [40]. The obtaineddata is in excellent agreement with an ab-initio Floquet-bandstructure calculation, which includes the first sixbands. While the two lowest quasienergy bands deter-mine the nature of the topological phase diagram, theirshape is modified due to hybridization between the s - and p -bands during one modulation period, which needs to betaken into account for quantitative comparisons (Supple-mentary Information). This parameter scan allows us to identify the phase transitions from gap-closing points atΓ. The energy gaps at the M - and K -points remain finitefor all modulation parameters (Fig. 2c).Since we always determine the minimal gapmin( g ( q ) , g π ( q )), we can use this data to unam-biguously identify whether the gap closes within orbetween Floquet zones [38]. In the high-frequencylimit, where (cid:126) ω is much larger than any other energyscale, one always measures the gap around zero energy g , because g π (cid:29) g for all quasimomenta. Followingthe parameter scan shown in Fig. 2b, the energy gap g (Γ) increases for smaller modulation frequencies until g π (Γ) = g (Γ) = (cid:126) ω/ g π (Γ).We then continue to probe g π (Γ), until a second cuspappears, indicating that g (Γ) is now the smaller gap.From this we conclude that the first phase transitionbetween phases (cid:172) and (cid:173) occurs via a band touchingat the Γ-point between Floquet zones, while the secondone between (cid:173) and (cid:174) appears via a gap-closing atΓ around zero energy (upper panels in Fig. 2b). For ω →
0, additional phase transitions occur, which are notdiscussed in this work.The change of topological invariants across the phasetransition is determined by the signed topological charge Q js associated with the band touching singularity, whichoccurs at ( q s , λ s ), in the abstract 3D parameter spacespanned by the quasimomentum q and the modulationparameter λ , which parametrizes the path through thephase diagram (white arrows in Fig. 2a). For a genericcase of linear band touching points as in our model, thetopological charge is Q js = ± W jλ s + ε = W jλ s − ε + Q js (1)across the phase transition, where ε is a small param-eter. As a result, we can determine the winding num-bers of any topological phase by tracking the numberof gap-closing points and characterizing the associatedtopological charge along a smooth path parametrized by λ . Note that there can be several degenerate singulari-ties Q js at the same parameters ( q s , λ s ), which results in | ∆ W jλ s | = | W jλ s + ε − W jλ s − ε | >
1. Such a situation couldbe always identified experimentally by applying a smallperturbation, which lifts the degeneracy and results inisolated phase transitions [27, 42].The high-frequency limit ω → ∞ can always bemapped onto a static Hamiltonian via the rotating-waveapproximation, in which case the winding number W π between Floquet zones is necessarily trivial, W π ω →∞ −−−−→ C ± = ∓ W , W π ) = (1 , W j along the path shown inFig. 2a by extracting the topological charges Q js and us-ing Eq. (1). As derived in the Supplementary Informa-tion, the value of the topological charge is determinedby the sign of the local Berry curvature Ω ± ( q s ), con-centrated at the band touching singularity. The concen-trated Berry curvature is associated with a π -Berry fluxin quasimomentum space near the gap-closing singular-ity and the sign of the topological charge is given by thechange of sign of this π -flux across the phase transition(upper panels in Fig. 3d). In the experiment we investi-gate the local Berry curvature distribution via Hall driftmeasurements [14, 18] close to the singularity. Due to thefinite width of the momentum distribution of the BEC,we obtain a signal that is averaged and weighted accord-ing to the momentum-space profile (Supplementary In-formation). Nonetheless, the change in sign of the Berryflux across the phase transition can be unambiguouslydefined from a change in sign of the measured deflections s µ ⊥ across the phase transition Q s = sgn (cid:0) ∆ s −⊥ ( q s ) (cid:1) , Q πs = − sgn (cid:0) ∆ s −⊥ ( q s ) (cid:1) , (2)∆ s −⊥ ( q s ) = sgn[ s −⊥ ( q s , λ s + ε )] − sgn[ s −⊥ ( q s , λ s − ε )] , where the transverse deflection s µ ⊥ is defined with respectto the energy band the atoms are prepared in.The sequence starts by applying a force to the atomsby linear acceleration of the lattice (Methods), to adia-batically move the wavepacket in quasimomentum space.The Berry curvature acts as an effective magnetic field inquasimomentum-space, adding a transverse componentto the atom’s velocity [14, 18, 46, 47]. This anomalous ve-locity is directly proportional to the Berry curvature andgives rise to a deflection perpendicular to the direction ofthe force (Eq. (S.3) in the Supplementary Information).We typically move the atoms along high-symmetry paths:Γ − M − Γ (Γ-direction) and Γ − K − K (cid:48) ( K -direction)between q and q f (Fig. 3a), probing regions with non-vanishing Berry curvature at the Γ- and K -points.Since the force is generated by lattice acceleration,there is an additional longitudinal velocity component,which gives rise to large displacements in the directionof the force F . This leads to a restoring force fromthe harmonic trap, which reduces the final longitudinalquasimomentum to q eff < q f , whereas the effect alongthe transverse direction can be neglected (Supplemen-tary Information). We record the final position of thecloud after applying the force by taking insitu absorptionimages and determine the center-of-mass (CoM) positionby fitting a two-dimensional (2D) Gaussian function. Weperform the same measurement for opposite chirality ofthe modulation to evaluate the differential deflection s µ ⊥ (Fig. 3b), which is more robust to systematic deviationsin the CoM position of the cloud.To characterize the Haldane regime and quantitativelyvalidate our experimental approach, we probed the Berrycurvature almost in the entire first BZ by measuringdeflections along all K - and Γ-directions in the range q = 0 . √ k L → q eff ≈ . √ k L (Fig. 3c). The Berrycurvature around Γ as well as around both Dirac points istraversed once by all atoms. We find a good agreementbetween our experiment and ab initio numerical simu-lations taking into account the finite Gaussian width σ of the momentum distribution (Supplementary Informa-tion), which was independently calibrated for each dataset and lies in the range σ ∈ [0 . , . k L . In particu-lar, we observe positive deflections throughout the wholeBZ, which coincide well with theoretical calculations thatemploy a band with C − = 1. In contrast, we find distinctnegative deflections along all Γ-directions in the anoma-lous regime, which is consistent with a quasienergy bandwith trivial Chern number C − = 0.In Fig. 3d we show a complete scan of the transversedeflections at the Γ-point, where the gap closings occur,across the three different regimes. We traverse the Berrycurvature around the Γ-point approximately once withthe full cloud. The start and end point of the path in q x q y q q f q f Modulationdirection xy q x q y q x q y s ω ∆ E ω π ω π Modulation amplitude m M o d a m p m s a − s a − s q x q y Γ Γ KK’ MMK’
Γ Γ
M M K Γ Figure 3. Schematics and experimental results for the local Hall deflections s −⊥ to probe the local Berry curvature distributionΩ. a . Schematics of the paths traversed in quasimomentum space for measuring the transverse deflections along the K -and Γ-direction ( q → q f ) and the first BZ (grey shaded area). b . Definition of the differential transverse deflection s µ ⊥ :The CoM positions for left- and right circular modulation as well as the starting position are measured as the mean valuesover 30-40 individual experimental realizations for each point. The bisecting line of the opening angle formed by the twopaths defines an axis and the differential deflection is the distance of each final position to this axis (being the same forboth modulation directions per definition). c . Measured transverse deflections s −⊥ along all K - and Γ-directions (inset) with F a/h = 204 Hz for: ω/ π = 16 kHz, m = 0 .
25 (regime (cid:172) ) and ω/ π = 10 kHz, m = 0 .
24 (regime (cid:173) ). The solid lines aretheoretical calculations (Supplementary Information) and the errorbars denote the standard error of the mean (SEM). Lowerpanels: Calculated Berry curvature for the same modulation parameters in the first BZ. d . Main panel: Measured transversedeflections along the Γ-direction along the path in Fig. 2a with F a/h = 170 Hz, for q → q eff ≈ . √ k L (regime (cid:172) and (cid:173) ) andfor Γ → q eff ≈ . √ k L (regime (cid:174) ). Errorbars indicate the SEM. Upper panels: Calculated Berry curvature distributions inthe first BZ. Inset: Change of modulation amplitude and frequency during the ramp-up in the three different regimes (denotedby the numbers) while changing the quasimomentum from Γ to the point denoted at the end of the ramp-up path (Methods),the parameters are ω/ (2 π ) = { , , } kHz, m = { . , . , . } . Lower panel: Winding numbers deduced from the measuredtransverse deflections. Right panel: Schematics of the energy bands in the different topological phases with the correspondingChern numbers and edge modes (red lines). reciprocal space as well as the ramp-up of the modula-tion depend on the parameter regime (inset of Fig. 3d andMethods). At larger modulation frequencies, we measurea slight positive deflection, as expected in the Haldaneregime, which then grows when approaching the gap clos-ing point. At the phase transition it suddenly changes tonegative values, indicating a change of the winding num-ber by sgn (cid:0) ∆ s −⊥ (Γ) (cid:1) = −
1. Combined with the previousenergy gap measurements, which indicated a gap closingpoint in the π -gap, we conclude that Q πs = +1, accord-ing to Eq. (2), and that the winding number W π changesfrom 0 → (cid:172) - (cid:173) . This signals the transition to the anomalousphase, where C ± = 0 and ( W , W π ) = (1 , (cid:173) - (cid:174) , the de-flection s −⊥ starts to decrease due to the spreading ofthe negative Berry curvature in quasimomentum space. Shortly before the gap-closing point at g , the deflec-tion turns positive and jumps to negative values after thetransition. According to Eq. (2) the topological chargeat this transition is Q s = −
1, signaling a change of W from 1 →
0. At this transition we enter the third topo-logical phase with C ± = ± W , W π ) = (0 , E vanishes. We attribute these to non-adiabaticexcitations to higher bands, which become important if F a (cid:38) ∆ E .To confirm our observations, we further probed the en-ergy gap closings as well as the local Berry curvature for
123 123 ω∆ E ω π ω πω π ω π Modulation amplitude m M o d u l a t i o n a m p li t u d e m M o d u l a t i o n a m p li t u d e m s a + s a − Γ M Γ M Γ K Γ K Γ Figure 4. Transverse deflections in both bands in the threedifferent regimes. a . Measured transverse deflections s −⊥ atthe Γ- (upper panel) and K -point (lower panel); the strengthof the force was adapted to the different bandgaps, also chang-ing the effective final quasimomentum (Methods). The SEMis on average ¯Σ Γ = 0 . a and ¯Σ K = 0 . a . Thedark gray data points correspond to the phase transitionpoints obtained from linear fits to bandgap measurements atΓ where the errorbars represent the stepsize of the modula-tion frequency used in the St¨uckelberg interferometry (Sup-plementary Information). The solid lines show the theoreti-cal phase boundaries. b . Calculated transverse deflections,including effects due to the harmonic trap and the calibratedmomentum-space widths. Solid lines are the same as in a . c . Left panel: Transverse deflections s + ⊥ in the second bandfor q → q eff ≈ . √ k L and for Γ → q eff ≈ . k L inregime (cid:173) along Γ; F a/h = 170 Hz. The modulation pa-rameters are varied along the path depicted in Fig. 2a, theshadings and numbers indicate different topological phases.Errorbars indicate the SEM. Right panel: Variation of themodulation frequency and amplitude and final quasimomen-tum during the ramp-up and loading into the second band inthe different regimes, identified by the numbers on the right; ω/ (2 π ) = { , , } kHz, m = { . , . , . } (top to bottom). a larger range of modulation parameters (Fig. 4a). Overthe full parameter regime shown here and in Fig. 3d,the measured deflections are in quantitative agreementwith numerical simulations (Fig. 4b) taking into accountthe finite momentum-space width and the harmonic trap,without free parameters. We further show the experi-mentally determined phase boundaries, which were ex-tracted from energy gap measurements at the Γ-point(Supplementary Information).The periodic nature of the Floquet bands further al-lows for direct loading of the atoms into the second band,when the modulation parameters lie in the anomalous regime during the complete ramp-up of the modulation(right panel of Fig. 4c and Methods). This can be under-stood as follows: In the limit of m → (cid:126) ω . Increasing the modulation amplitude opensgaps at the crossing points and hybridizes the two low-est bands. A condensate located at the Γ-point initiallyand during the ramp-up of the modulation amplitude, isnow adiabatically transferred in the upper Floquet en-ergy band, if the modulation frequency ω is kept within[3 . , .
6] kHz. To also probe the other two regimes wedivided the ramp-up into two parts, first loading the sec-ond band in the anomalous regime as described above,but with a smaller or larger amplitude, and then ramp-ing to the final amplitude while moving by q to avoidthe gap closing points. Using these different ramp-upschemes we were able to probe the Berry curvature ofthe second band in the different regimes around the Γ-and K -points, plotted in the left panel of Fig. 4c. Weindeed observe an inversion of the Berry curvature, asexpected from theory.In this work we have presented the first experimen-tal realization of anomalous Floquet systems with coldatoms and its complete characterization using bulk wind-ing numbers W , which were deduced by combining en-ergy gap measurements with local Hall deflections. Theexperimental data is in excellent agreement with numer-ical band structure calculations involving the first sixbands of the modulated lattice over a wide range ofparameters explicitly probing three distinct topologicalregimes. Moreover, the large degree of control of our ex-perimental setup and observables facilitated an indepen-dent probe of the geometric properties of the first excitedband. The successful realization of different topologicalsystems with rather long lifetimes in the non-interactinglimit, opens the door to a variety of interesting phenom-ena, for instance, the properties of edge modes at theboundaries [48–50] or the interplay of disorder and topol-ogy in driven systems [51, 52], to name only a few. Themany-body regime provides a particularly rich experi-mental and theoretical playground [28]. Intriguingly, ithas been shown that an anomalous Floquet insulator mayexhibit a remarkable robustness against disorder [29],which holds great promise for realizing quantized trans-port at high temperatures. This is due to the non-zerowinding number of the Floquet spectrum, which cannotbe annihilated even if all bulk states are localized, in con-trast to conventional quantum Hall insulators. Moreover,in the anomalous phase one of the quasienergy bands ex-hibits a moatlike dispersion, i.e. a ring-shaped minimumof near-degenerate states, which can give rise to exoticmany-body phenomena [53–55]. Acknowledgements
We thank Jean Bellissard,Erez Berg, Jean Dalibard, Eugene Demler and Ne-tanel Lindner for inspiring discussions. The researchin Munich and Dresden was funded by the DeutscheForschungsgemeinschaft (DFG, German Research Foun-dation) via Research Unit FOR 2414 under project num-ber 277974659. The work in Munich was further sup-ported under Germany’s Excellence Strategy – EXC-2111– 39081486. F.N. ¨U. further acknowledges support fromEPSRC Grant No. EP/P009565/1. The work in Belgiumwas supported by the ERC Starting Grant TopoCold,and the Fonds De La Recherche Scientifique (FRS-FNRS,Belgium).
Author contributions
K.W., C.B., M.D.L., N.G.and M.A. designed the modulation scheme and bench-marked the deflection measurement. F.N. ¨U., A.E. andM.A. devised the protocol to extract the set of topologi-cal invariants. K.W. and C.B. performed the experiment,analyzed the data and performed the numerical simula-tions with M.A. All authors contributed to the writingof the manuscript and the discussion of the results.
Data availability
The data that support the plotswithin this paper and other findings of this study areavailable from the corresponding author upon reasonablerequest.
Code availability
The code that supports the plotswithin this paper are available from the correspondingauthor upon reasonable request.
Competing interests
The authors declare no com-peting interests.
METHODS
St¨uckelberg interferometry:
The energy gaps betweenthe two lowest bands are measured using St¨uckelberg in-terferometry [35, 36]. We start by loading the atomsinto the lowest band of the static lattice at Γ and thenchange their quasimomentum non-adiabatically to thepoint where the bandgap should be probed, which leadsto coherent population of the second band. The changein quasimomentum was carried out via linear frequencysweeps of two laser beams, enabling large forces in arbi-trary directions in the 2D quasimomentum space. Themodulation amplitude was ramped up linearly at the de-sired modulation frequency within five modulation cy-cles T and the acceleration started meanwhile, such thatthe final quasimomentum was reached at the end of theramp-up. Then, the atoms were held at the final quasi-momentum for integer multiples of the driving cycle andsubsequently accelerated back to Γ in the first BZ us-ing the same force as before while ramping down themodulation inversely to the ramp-up. During the holdtime, the atoms acquire a dynamical phase dependingon the energy band and quasimomentum they are oc-cupying. Driving back non-adiabatically recombines thepopulations in the two bands leading to oscillations ofthe band populations in time with a frequency given bythe bandgap at the probed quasimomentum.Considering the first order Floquet copies of the s -bands, there are two energy gaps that can be probed,the gap at zero energy, g ( q ), and between Floquet zones, g π ( q ), with g ( q ) + g π ( q ) = (cid:126) ω at each point in quasi-momentum space. Since the hold times are always in- teger multiples of the modulation cycle, the maximumgap frequency that can be measured with St¨uckelberginterferometry is ω max = ω/
2, which would correspondto sampling the cosine-wave with two points per oscil-lation. This means that at every q we always measurethe smaller gap out of g ( q ) and g π ( q ), enabling us todifferentiate between them as described in the main text.At certain modulation parameters in the anomalousregime excitations to the second band can occur duringthe ramp-up due to energy gap closings. This leads onlyto an offset phase in the St¨uckelberg oscillations and doesnot change their frequency. The relative population inthe lowest band n is measured by taking absorption im-ages after performing bandmapping and a time-of-flight(TOF) of 3 . p -bands, being suppressed withthe corresponding Floquet order. This would lead to anoscillation with multiple frequencies which we take intoaccount by fitting a sum of two cosine functions. Wealso include a possible damping of the oscillation due todynamical instabilities and atom loss: n ( t ) = e − ( t − t ) γ A cos( ω ( t − t ))+ A cos( ω ( t − t )) + n , (3)where A , A , ω , ω , t , t and γ are free fit parametersand the main oscillation frequency is defined as havingthe larger relative amplitude. For most parameters, thecontribution from the second frequency is negligible, itmainly plays a role in the anomalous regime close to thephase transition. In Fig. S4 an example of populationoscillations at Γ is shown together with its Fast Fouriertransform (FFT), which clearly shows the closing andopening of the energy gaps. Deflection measurements:
For the deflection mea-surements the force is also applied using lattice accel-eration but now it is small compared to the energy gaps,in order to adiabatically move inside a single band. Here,the ramp-up of the modulation depends on the band andtopological regime that is probed. When measuring inthe first band at modulation frequencies ω/ (2 π ) ≥ q = 0 . √ k L (the M -pointwhen driving along the Γ-direction). At this point the fi-nal modulation parameters are reached and we continueto accelerate to the final quasimomentum q f for the lo-cal deflection measurement. The ramp-up time is chosenas the time needed to change the quasimomentum of thecloud from Γ to q , with the given force rounded up tofull cycles of the modulation. We have verified indepen-dently that the magnitude of the force leads to negligi-ble excitations (Fig. S2b). By moving away from the Γ-point we avoid the gap closing point, which would resultin excitations to the second band. We verified that thepoints in quasimomentum space where the bands havecrossed and hybridized (the ring-shaped minimum in theanomalous phase) are always located away from the outeredge of the moving cloud defined by its Gaussian widthin reciprocal space, meaning that during the ramp-upthe cloud does not traverse regions with non-zero Berrycurvature. Hence, we effectively probe the Berry curva-ture along paths in quasimomentum space starting at adistance of q from the Γ-point, which was also used inthe calculations. For smaller modulation frequencies thephase transition takes place at earlier times during theramp-up and the lowest frequency for which we used thisscheme was set to 8 kHz to ensure that the energy gapsthe atoms see during the ramp-up are always larger than∆ E/h ≈
500 Hz. From the good agreement between themeasured deflections and the theoretical calculations wecan confirm that for most modulation parameters therecan only be minor excitations to higher bands and theBerry curvature probed during the ramp-up is negligible.We also verified the latter experimentally by measuringthe transverse deflection when accelerating by q alongthe Γ- and K -directions (see Fig. S2a).For measurements in the anomalous regime with ω/ (2 π ) < f = 13 . m → q , and thefrequency was changed exponentially (see also inset ofFig. 3d): ω ( t ) / (2 π ) = f f − f e m f · p − (cid:16) e mftr t · p − (cid:17) + f , (4)where the final frequency and amplitude are denoted by f f and m f , respectively, t r is the ramp-up time and p = 20 for the ramp-up in the anomalous regime. Thefunctional form was motivated by the shape of the phasetransition lines, which approximately follow f ( m ) ∝ e m .When using this chirped ramp-up, the time was chosen asbeing close to the traversing time to q but ending at fullcycles of the modulation. We verified numerically thatfor all modulation parameters probed here, the minimalgap and Berry curvature during the ramp-up fulfill therequirements stated above.In the third regime we start at f = 2 . p is fitted tomaximize the (absolute) energy gap during the ramp-upand the atoms are held at Γ. When probing the Berrycurvature around Γ, we moved by q eff ≈ . √ k L , so along Γ − M − Γ, which is equivalent to M − Γ − M forthe modulation parameters used here, as verified numer-ically. This is not true along the K -directions, where wedetermined the final deflection from q to q f by measur-ing along the whole path and subtracting the measureddeflection when only moving up to q . Here, the ramp-up time was chosen to be similar to the traversing timesused in the other regimes ( ≈ .
96 ms) and ending with afull modulation cycle again.To probe the second band, we always started in theanomalous regime, where it is connected to the first bandof the static lattice for m → f = 4 . f = 4 . m = 0 . f f at Γ with p being fitted to maximize the gap,and then decreased the amplitude linearly to m f , whilemoving to q , avoiding the gap closing points. In thiscase, the atoms do traverse Berry curvature during thesecond part of the ramp-up, so we determined again thefinal deflection by subtracting the deflection measured upto q , now along all directions. In the third regime weused a similar procedure, now ramping up to m = 0 . m f .For the frequency scans presented in Fig. 4a, theforce was adapted to the energy gaps: For m = 0 . ω/ (2 π ) ≥ F a/h = 272 Hz probing q → q eff ≈ . √ k L ; for m = 0 . F a/h = 341 Hz and q → q eff ≈ . √ k L . Inregime (cid:174) we probed Γ → q eff ≈ . √ k L along theΓ-direction using F a/h = 204 Hz. In all other cases,
F a/h = 204 Hz and q → q eff ≈ . √ k L .For all modulation parameters (including the param-eter scans in Fig. 3d and Fig. 4c) except for m = 0 . ω/ (2 π ) ≥ (cid:173) or the second band in regime (cid:174) along Γ, the fi-nal quasimomentum was set to q f = 1 . √ k L by the lat-tice acceleration, but the effective values q eff are reduceddue to the restoring force of the harmonic trap (Supple-mentary Information), which increases for larger longi-tudinal displacements and hence smaller forces. In thecase of m = 0 . ω/ (2 π ) ≥ q f = 1 . √ k L was pro-grammed yielding q eff ≈ . √ k L similar to the othermeasurements. The effective distance traversed duringthe ramp-up remains q ≈ . √ k L , since the real spacedisplacement is still small here. For the theoretical calcu-lations, the full equations of motion were solved numer-ically, including the harmonic trap and the band disper-sion as well as the measured momentum space widths inall cases (Supplementary Information). [1] N. 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Here, we present calibration measurements and addi-tional data (S1), the detailed theoretical model (S2), thecalculation of the edge states in a tight-binding model(S3) and the connection between the topological chargeand the Berry curvature (S4).
S1. CALIBRATIONS AND ADDITIONALMEASUREMENTSA. Influence of the harmonic trap
We apply a force on the cloud by accelerating the lat-tice, which leads to a longitudinal velocity of the atomsin the lab frame. Detuning the frequency of one laserbeam by ∆ f = ∆ ω/ (2 π ) changes the quasimomentum ofthe atoms by ∆ q = 2 λ L m K ∆ f (cid:126) . (S.1)Changing the laser frequency linearly for a time ∆ t givesrise to the force: F = (cid:126) ∆ q ∆ t = m K a L . (S.2)The force is varied by changing the time ∆ t and keepingthe final detuning fixed. For the bandgap measurementsthe applied forces are large and ∆ t is small, leading onlyto minor displacements in real space, so in this case theeffect of the harmonic trap can be neglected. The trans-verse deflections were probed with smaller forces to en-sure that we adiabatically move within a single band,yielding real-space displacements up to ≈ µ m. In thepresence of the harmonic trap the semiclassical equationsof motion read:˙ x = 1 (cid:126) ∂ε∂q x ( q ) − (cid:126) (cid:18) F y − ∂V trap ∂y (cid:19) Ω( q ) + F x m K t ˙ y = 1 (cid:126) ∂ε∂q y ( q ) + 1 (cid:126) (cid:18) F x − ∂V trap ∂x (cid:19) Ω( q ) + F y m K t ˙ q x = 1 (cid:126) (cid:18) F x − ∂V trap ∂x (cid:19) ˙ q y = 1 (cid:126) (cid:18) F y − ∂V trap ∂y (cid:19) , (S.3)where the trapping potential is given by V trap =0 . m K ω r ( x + y ) with the mean trapping frequency inthe xy -plane being ω r = 2 π × . q f setby the lattice acceleration according to Eq. (S.1). At q x = q = 0 . √ k L , the changes are minor, even for thesmallest force of F a/h = 170 Hz that was used for the pa-rameter scan in Fig. 3 in the main text. But at the finalvalue of q f = 1 . √ k L the quasimomentum is reducedto q eff ≈ . √ k L . F a h Figure S1. Calculated longitudinal quasimomentum center ofmass (CoM) and calibration of the momentum space width. a . Calculated quasimomentum CoM q eff along the directionof the force as a function of the programmed quasimomentum q x in the presence of the harmonic trap and a static latticewith V = 6 E r (see Sec. S2). The force is applied along theΓ-direction and varied between F a/h = 170 Hz and
F a/h =545 Hz, as indicated by the colorbar, and σ = 0 . k L . Theblack line is the solution without the harmonic trap. Thedashed lines mark the quasimomentum q = 0 . √ k L up towhich we accelerate during the parameter ramp-up in manycases, and the final quasimomentum q f = 1 . √ k L . b . Mea-surement of the momentum space width by observing the pop-ulation transfer when driving adiabatically across the borderof the first BZ. Each point is an average over five individualexperimental realizations, errorbars indicate the standard er-ror. The solid line is an errorfunction fitted to the data todetermine the Gaussian width σ . Along the transverse direction the real space displace-ments are small leading only to minor changes in thequasimomentum due to the trap. To calculate the trans-verse deflections we numerically solved the set of equa-tions in (S.3) including the band dispersion and the har-monic trap. The resulting transverse quasimomentumcomponents were q ⊥ ≤ . k L for all modulation pa-rameters used in this work, meaning that the transverseband derivative is negligible, since the paths in recipro-cal space are still well directed along the high-symmetrylines of the lattice. Hence, it is justified that that thetransverse deflection measured in the experiments is in-deed proportional to the Berry curvature.The trapping frequency was measured in the presenceof a static lattice with V = 6 E r by observing the breath-ing mode of the BEC insitu after a quench of the in-plane harmonic confinement. We fitted a 2D Gaussian tothe absorption images, with the principle axes directedalong the propagation directions of the trapping beams2in the xy -plane, to extract the oscillation of the real-space width. In our system, the trapping frequency alongthe vertical direction is about 8-times larger than the in-plane frequency. According to [S1], the in-plane trappingfrequency can thus be extracted from the frequency f b ofthe breathing mode as: f = (cid:114) f b . The corresponding trapping frequencies along thedipole axes were f X = 27 . f Y = 26 . f = 27 . B. Momentum space width
Due to finite temperatures, harmonic confinement andon-site interactions, the BEC is broadened in recipro-cal space, which we describe by a symmetric Gaussianmomentum distribution with width σ . The width wasdetermined experimentally by performing a knife-edgemeasurement in reciprocal space: The quasimomentumis changed adiabatically by one reciprocal lattice vectoralong the Γ-direction using a force of F a/h = 204 Hz andperforming bandmapping [39] at certain quasimomentaalong the path. Since the velocity component imposed bythe moving lattice is directed opposite to the Bloch oscil-lation, the atoms appear, when bandmapping, at the Γ-point within the first BZ. When some of the atoms reachthe edge of the BZ they appear at the Γ-point in the nextBZ, so we count the relative population in the first BZ(similar to the bandgap measurements) depending on thequasimomentum. The amount of atoms in the first BZ isgiven by the integral over the Gaussian distribution andhence described by an errorfunction. An exemplary mea-surement is depicted in Fig. S1b along with the resultingfit. The width of the error function was obtained fromthe fit, all other parameters were fixed. For every mea-surement of the transverse deflections we determined thewidth in reciprocal space immediately before or after themeasurement and used this to calculate the correspond-ing theory values (see Sec. S2 B).
C. Deflections during ramp-up and test of the usedforces
To probe the Berry curvature in the Haldane andanomalous regime the modulation amplitude and partlythe modulation frequency were ramped up while drivingto q . Using the band structure calculations we verifiedthat during the ramp-up the points where the two lowestbands potentially had touched and hybridized, which isthe location of the additional negative Berry curvaturein the anomalous phase, is always located away from the Fa h ω π s a − s a − K Γ K Γ Figure S2. Transverse deflections s −⊥ in the Haldane andanomalous regime. a . Deflections vs. modulation frequencyup to q along the Γ- and K -directions with F a/h = 204 Hzfor m = 0 .
25. Errorbars indicate the SEM. b . Deflectionsin the anomalous regime ( m = 0 . , ω/ (2 π ) = 10 kHz) for q → q eff ≈ . √ k L depending on the applied force. Thesolid lines denote the corresponding theoretical values includ-ing the momentum space width. Errorbars indicate the SEM. edge of the moving cloud in reciprocal space (see Meth-ods). Since we are accelerating along high-symmetrylines in reciprocal space, the band derivatives along thetransverse direction average to zero. Hence, there shouldbe no deflection during the ramp-up and the measuredtransverse deflection can be assumed to correspond to apath in reciprocal space starting at q . This is confirmedby the data in Fig. S2a, showing the measured transversedeflections along the Γ- and K -direction up to a distanceof q for m = 0 .
25 and different modulation frequencies.We also verified that the forces we used to measurethe transverse deflections were sufficiently small to avoida reduction of the deflections due to excitations to thesecond band. We measured the deflections depending onthe applied force when driving by q eff ≈ . √ k L alongthe Γ- and K -directions for modulation parameters inthe anomalous regime (see Fig. S2b). The final quasimo-menta for the lattice acceleration were chosen such thatthe effective length of the traversed path in reciprocalspace was similar for all forces. For F a/h >
300 Hz, thedeflections along both directions are smaller than pre-dicted by the theoretical calculations due to excitationsto the second band. The modulation parameters chosenhere lie close to the phase transition with energy gaps∆ E ( K ) /h = 1500(30) Hz and ∆ E (Γ) /h = 1110(70) Hz.The measured deflections saturate for smaller forceswhich happens earlier along the K -direction, also indi-cating the larger energy gap compared to Γ. In total, thechosen forces of F a/h = 170 Hz and
F a/h = 204 Hz usedfor these modulation parameters are sufficiently small,which is also confirmed by the overall good agreementbetween the measured deflections and the theoretical cal-culations, where we assume population in a single band.
D. Bandgap measurements for frequency scans
To explore the phase diagram shown in Fig. 1c weprobed the bandgaps and Berry curvature for a broad3 m = 0.2 m = 0.27 m = 0.22 m = 0.3 m = 0.25 m = 0.3 ω π ω πω π K Γ ∆ E / h ∆ E / h Figure S3. Energy gaps ∆ E at Γ and K depending on the modulation frequency for modulation amplitudes m = { . , . , . , . , . } measured with St¨uckelberg interferometry using F a/h = 1360 Hz. The energy gap at K (graycircles) remains open over the full parameter range. The gap at Γ shows multiple closings, indicating the transitions from theHaldane to the anomalous and third phase with decreasing frequency. With increasing amplitude an avoided crossing appearsat modulation frequencies around ω/ (2 π ) = 15 kHz. The solid green and grey lines are the corresponding theoretical minimalgaps calculated using six bands, the solid blue lines in the last two panels are the theoretical gaps from a two-band model(see Sec. S2). The red dashed lines are fits ∝ | ω | to the bandgaps at Γ to determine the phase transitions (see text). Theerrorbars indicate fitting errors from the oscillation fits, every oscillation consists of 23 points each averaged over 3-4 individualexperimental realizations. range of modulation parameters in different topologi-cal regimes. The measured transverse deflections alongthe Γ- and K -directions are shown in the main text inFig. 4a accompanied by the experimentally determinedphase transitions. The corresponding gap measurementsat Γ and K are displayed in Fig. S3 together with thetheoretical values from our model including the six low-est energy bands. For m = 0 . E/h = n · | ω − ω | / (2 π ) tothe slope on the left and right of the gap closings, with n = 1 and n = 2 for the first and second phase transi-tion. The second phase transition could only be obtainedfor m ≥ .
25. The errors for the phase transitions are σ tot = (cid:113) σ + σ with the fit errors σ fit and the sys-tematic errors σ sys . The latter are given by the step size∆ ω/ (2 π ) = 300 Hz used in the energy gap measurementswhich is dominating the fit errors σ fit ∈ [20 ,
70] Hz.We also measured the energy gaps at K to validate ourtheoretical calculations and pick the forces for the deflec-tion measurements appropriately. For large modulationfrequencies and amplitudes the influence of the p -bandsbecomes significant which can be seen in the jumps of theenergy gap around ω/ (2 π ) = 15 kHz: Due to the coupling between the different bands, gaps open at avoided cross-ings, which increase with modulation amplitude. Thesemanifest in discontinuities in the effective Floquet bandsand the corresponding energy gaps. The experimentaldata is well reproduced by a six-band model, signalingthat coupling to even higher bands with µ > m = 0 . p -bands. At K , thetheoretical curves also coincide at small frequencies but inthe Haldane regime the deviations are larger, especiallythe jumps at the avoided crossings are not captured bya two-band model. The general mechanism of the phasetransitions is captured by a simple two-band model but toquantitatively describe the experiments performed here,a six-band model is necessary. E. St¨uckelberg interferometry
All energy gaps presented in this work were mea-sured using St¨uckelberg interferometry as described in4the Methods. To quantify the amount of atoms in thefirst and second band, we take absorption images afterperforming bandmapping at Γ giving distinct peaks cor-responding to the different bands, as shown in the insetsof Fig. S4a. The atoms in the lowest band appear in thecenter, whereas atoms in the second to sixth band aredistributed over the outer peaks. The forces were cho-sen sufficiently large to ensure population of the secondband but not too large to avoid excitations to the p -bandswhich can also be assumed to be small due to the goodagreement of the measured energy gaps with the calcu-lated minimal gaps. We sum up the pixels inside eachof the seven regions of interest (ROIs) drawn as yellowcircles with radius R in the insets of Fig. S4a. To ac-count for inhomogeneities in the background due to thefinite size of the imaging beam, we also count the pixelsin a larger ROI with radius √ R (grey circles). The pixelcounts for each peak are then obtained as 2Σ R − Σ √ R and the relative population in the lowest band is givenby the counts in the central peak divided by the totalcounts.An example of the population oscillations at Γ for dif-ferent modulation frequencies and m = 0 .
25 is presentedin Fig. S4a, already showing the decrease of the oscilla-tion frequency towards the phase transitions. To obtainthe points in Fig. 2 and Fig. S3 we fit a sum of cosinesto each population curve as described in the Methodssection of the main text. However, the change in theoscillation frequency can also be seen directly by per-forming a Fast Fourier transform (FFT) of the popula-tion oscillation (Fig. S4b) where the gap closings at thetwo phase transitions are clearly visible as well as ad-ditional small frequency components appearing around ω/ (2 π ) = 10 kHz probably arising from weak coupling toFloquet copies of the p -bands. F. Lifetimes
We measured the lifetime of the BEC at Γ in all threetopological regimes probed in this work. As described inthe main text, in the anomalous phase the first band ofthe static lattice is adiabatically connected to the secondband of the modulated lattice which has an energy mini-mum at Γ. Hence we probed the lifetime for the anoma-lous regime in the second band by ramping the modu-lation frequency and amplitude simultaneously in a non-linear fashion (see Methods) to directly access the anoma-lous regime. The third regime was probed in the firstband, using a similar ramp-up but starting at a smallermodulation frequency. After ramping up the modulationwe held the atoms at the Γ-point in the modulated lat-tice for different times t = nT with n ∈ N , then rampeddown the modulation and performed bandmapping after10 ms TOF. In the Haldane regime (for amplitude andphase modulation) the ramp time was fixed to 5 T , in theanomalous and third regime we used 12 T and 7 T -8 T re-spectively, corresponding to ≈ ω π Figure S4. Raw data used to obtain the energy gaps. a .Relative population n in the lowest band measured at Γ de-pending on the hold time for different modulation frequenciesand m = 0 .
25. The reduction of the energy gap and thus oscil-lation frequency is clearly visible illustrating the gap openingand closing. Each population is an average over 3-4 individualexperimental realizations. The two insets show raw images for ω/ (2 π ) = 15 . ω/ (2 π ) = 15 . b . Fast Fourier transform (FFT)of the signal in a as a function of the St¨uckelberg oscillationfrequency. the lowest band was then counted as Σ = 2Σ R − Σ √ R for the central peak using the main and background ROIsdescribed above for the St¨uckelberg oscillations. Thepopulation exhibited an exponential decay as a functionof the hold time for most modulation parameters, so wefitted the function Σ ( t ) = A e − t/τ + y to it and ex-tracted the parameters A, y and the lifetime τ , whereasall of them were constrained to be real and positive. Thefitted offset was negligible in most cases, since we mea-sured up to times t at which almost no atoms were left.In the Haldane regime (Fig. S5a) we compared the life-time for different modulation amplitudes at ω/ (2 π ) =10 kHz and for m = 0 . ω/ (2 π ) = 20 kHz. The life-times increase linearly for smaller modulation amplitudesand larger frequencies moving away from the first phasetransition. The value for m = 0 . ω/ (2 π ) = 20 kHzis comparable to the lifetime in the static lattice. Inthe anomalous regime (Fig. S5b) the lifetimes are muchsmaller and depend mainly on the modulation frequency.For ω/ (2 π ) = 7 kHz the system is deep in the anomalousregime and exhibits similar lifetimes for all amplitudes,whereas the lifetime is reduced significantly for ω/ (2 π ) =10 kHz and slightly decreases with the modulation ampli-tude. In the third regime (Fig. S5c) the lifetimes increaseagain and strongly depend on the modulation amplitudeand frequency. The parameters were chosen such thatthey have equal distance to the phase transition and thelifetimes are reduced by almost two orders of magnitudefor larger amplitudes and higher frequencies. Based onthe observed dependence of the lifetimes on the modu-lation parameters, we assume that these effects mainlyoriginate from excitations to higher quasienergy bands,5which are favored for larger modulation amplitudes andfrequencies, increasing the coupling between the Floquetzones. In the Haldane regime, the lifetime increases for f = 20 kHz compared tor f = 10 kHz, which can beunderstood as follows: At f = 20 kHz, g π (Γ) > g (Γ),reducing the coupling to the first Floquet copy of thesecond band, whereas the gap to the corresponding p -bands is still large. Increasing the modulation frequencyfurther, reduces the lifetime again, since excitations tothe p -bands are favored: For f = 30 kHz and m = 0 . f = 10 kHz (notshown in the plot).The last panel of Fig. S5 shows the lifetimes in theHaldane regime as a function of the scattering lengthwhich we can tune using a Feshbach resonance (see maintext). All measurements so far were performed at a s =6 . a . Increasing the on-site interaction considerablyreduces the lifetimes in the static lattice, but even morein the modulated case, where the minimal lifetime is τ ≈
15 ms for a s = 80 . a . This suggests that there arealso two-particle processes involved increasing the rateof excitations to higher Floquet bands.Overall, the smallest lifetimes measured are on the or-der of 2 ms in the second band and the anomalous regimebeing comparable to the maximal duration of ≈ ω M / (2 π ) = 8 kHz with an ampli-tude b M / (2 π ) = 6 . E/h = 160 Hz at the K -points which is similar to thecorresponding gap for m = 0 . ω/ (2 π ) = 20 kHz.The phase shaking leads to a reduced lifetime comparedto the amplitude modulation, which is nevertheless largewith respect to the experimental times used here. S2. NUMERICAL CALCULATIONSA. Effective Hamiltonian and energy bands
In the experiments, we directly probe the properties ofthe bulk from which we can deduce the topological wind-ing numbers associated with the band gaps. These topo-logical invariants determine the existence of chiral edgemodes in the system (not measured in the experiment).To obtain the bulk energy bands and corresponding Berrycurvatures in the modulated lattice we numerically cal-culated the effective Hamiltonian H eff , which is definedvia the time-evolution operator U ( T ) over one full period τ ( m s ) τ ( m s ) τ ( m s ) τ ( m s ) Modulation amplitude m Modulation amplitude m Modulation amplitude m Scattering length a Figure S5. Measured lifetimes τ at Γ vs. modulation param-eters. a . Haldane regime in the first band for a s = 6 . a . b . Anomalous regime in the second band for a s = 6 . a . c . Third regime in the first band for a s = 6 . a whereasthe modulation amplitude and frequency were chosen suchthat the gap at Γ is similar. d . Haldane regime in the firstband for ω/ (2 π ) = 20 kHz vs. scattering length. The reddata point is measured for phase modulation of the latticewith a frequency of 8 kHz and an amplitude of 6 . K calcu-lated to be ∆ E/h = 160 Hz similar to the measured valuefor ω/ (2 π ) = 20 kHz and m = 0 .
1. Errorbars indicate fittingerrors in all cases, every decay measurement consists of 23points each averaged over 10 experimental realizations. of the drive: H eff = i (cid:126) T ln( U ( T )) , U ( T ) = T e − i (cid:126) (cid:82) T H ( t ) dt , (S.4)where T denotes time-ordering and ln the matrix loga-rithm. To numerically calculate H eff , the time-dependentHamiltonian is evaluated at 300 discrete timesteps t l within one driving period. We set T = 1 for the inte-gration over one driving period to simplify the numerics.For each set of parameters ( q , m ) we calculated the in-stantaneous Hamiltonian H ( t l , q , m ) at each timestep t l in the basis of plane waves and projected it to its sixlowest eigenstates: H ∗ p ( t l , q , m ) = M † ( t l , q , m ) · H ( t l , q , m ) · M ( t l , q , m ) , where the columns of the matrix M are the eigenstates of H corresponding to the six lowest eigenvalues and · de-notes matrix multiplication. The resulting 6 × H ∗ p are then transferred to a common basis consisting ofthe six lowest eigenstates of H ( t = 0 , q = 0 , m = 0), be-ing the columns of the Matrix M . The basis change isdone as: H p ( t l , q , m ) = B ( t l , q , m ) · H ∗ p ( t l , q , m ) · B − ( t l , q , m ) ,B ( t l , q , m ) = M † · M ( t l , q , m ) . U ( T, q , m, f ) = Π l e − i (cid:126) H p ( t l , q ,m ) ∆ tf , (S.5)with f = ω/ (2 π ), and the effective Hamiltonian (inunits of (cid:126) ω ) is given by H eff ( q , m, f ) = i π ln( U ( T, q , m, f )) . (S.6)Due to the periodic driving, the energies are notbounded any more and the band of H eff that is connectedto the lowest band of the static Hamiltonian not necessar-ily appears as the lowest. In our case, we are interested inthe two lowest bands, which are adiabatically connectedto the two s -bands of the static lattice.To extract the two lowest bands, we scanned the quasi-momentum across the first BZ, calculated the six eigen-states and eigenenergies of each H eff ( q , m, f ) and deter-mined which of the states had the maximal overlap withthe first and second eigenstate from the last q -step. Forthe initial step we considered the overlap with the firsttwo unit vectors, being the eigenstates of the two lowestbands in the static lattice. The state overlap is definedas the fidelity F ij : F ij = | (cid:104) φ ( q i , m, f ) | | φ ( q j , m, f ) (cid:105) | . (S.7)Especially at high modulation frequencies and ampli-tudes, all six bands couple and many avoided crossingsappear. In the vicinity of these points, the eigenstate-overlap decreases and there can be several states havingan overlap of similar magnitude with the first or secondstate of the last step. If the overlap with the previouseigenstate dropped below a certain threshold, we usedthe overlap with the unit vectors instead to avoid falseattributions. The threshold value depends on the modu-lation parameters, i.e., for low modulation frequencies itcould be set to 0 .
5, using the eigenstate-overlap mostlyeverywhere. By checking the bands in the first BZ aswell as on a 1D-high-symmetry line (Γ − M − K − Γ),we determined the optimal limits for the fidelity for eachband and set of modulation parameters. The results fortwo bands shown in Fig. S3 were obtained by the sameprocedure but projecting the instantaneous Hamiltonianat each time step to its two lowest eigenstates.
B. Transverse deflections
From the eigenstates of the two lowest bands we nu-merically calculate the Berry curvature according toRef. [S5] on a rhombic grid spanning the first BZ. For thenumeric integration of Eq. (S.3) we interpolate the Berrycurvature and the band derivatives on a large quadraticgrid spanning several BZs to be able to simulate the full trajectory including the momentum space extent of theBEC. The stepsize of the quadratic grid is dq ≈ . k L which was the maximal value at which the resulting realspace positions did not change when decreasing the step-size further.For most modulation parameters in the experiment, weeffectively probed the deflections starting at a distanceof q in reciprocal space after ramping up the modula-tion parameters. The quasimomentum after the ramp-up drive remains as q || ≈ q and q ⊥ = 0, and hence s ⊥ = 0 (see Sec. S1 C). The longitudinal offset in realspace was calculated by solving Eq. (S.3) with Ω( q ) = 0,starting at Γ and applying the respective force along theΓ- and/or K -direction for a time ∆ t corresponding to q . The integration was performed for about 7300 initialpoints in quasimomentum space lying on a circle withradius 0 . k L around Γ to each of which we assigned aGaussian weight according to the normalized momentumdistribution of the BEC with width σ . The CoM posi-tion and quasimomentum after the ramp-up were thengiven as the weighted average over the corresponding fi-nal values. Note that we used the band derivatives forthe final modulation parameters here, which turned outto give similar results as directly simulating the ramp-upby using the band derivatives for the different modulationparameters taken in between.The deflections were then calculated by integratingEq. (S.3) along the Γ- and K -direction with the initialpoints lying on a circle centered around the starting pointafter the ramp-up, given either by the values above or,when measuring along the Γ-direction in the first bandand third regime or the second band and anomalousregime, by q || = 0 = q ⊥ (see Methods). The time spanwas determined by the corresponding force and the pro-grammed quasimomentum distance of 1 . √ k L − q (or1 . √ k L , respectively). The CoM deflection and quasi-momentum were obtained as the average over the finalvalues, again using the Gaussian weights of the indepen-dently calibrated density distribution. For modulationparameters lying in between the experimental points,the Gaussian width of the closest measured point wasused. The approximate final values q eff for the longitu-dinal quasimomentum mainly depend on the magnitudeof the applied force and the time for which it is applied,whereas the influence of the band derivatives and theforce direction is negligible. S3. CALCULATION OF EDGE STATES IN ATIGHT-BINDING MODEL
As described in the main text, in a periodically-drivensystem the net number and chirality of edge modes perenergy gap is given by the winding number of the re-spective gap, allowing for a full characterization of thesystem using topological invariants of the bulk only. Inaddition to the determination of the winding numberswe also calculated the energy spectrum of the effective7 xy
1A 1B2A2BNA NB a δ δ δ ω ε F ω ε F Figure S6. Schematic of the stripe-geometry and calculated quasienergy dispersions in the three topological phases plotted inthe reduced zone scheme. a . Schematic of the stripe geometry described in the text, terminated by an armchair-edge along y and with periodic boundary conditions along x . The unit cell (gray shaded area) consists of N dimers and has a width of 3 a , thered arrows denote the vectors δ i connecting nearest neighbours. b . Quasienergy dispersions from the two-band tight-bindingmodel with N = 50 for a modulation amplitude of m = 0 .
25 and modulation frequencies ω/ (2 π ) = { , , } kHz correspondingto the Haldane regime, anomalous regime and third regime hosting chiral edge modes in the g -gap, both gaps and the g π -gap,respectively. Hamiltonian on a stripe-geometry displaying the disper-sion of the edge states directly. We employed a two-bandtight-binding model defined on a stripe terminated byan armchair-edge in the y -direction and with periodicboundary conditions along x (Fig. S6a).Due to hybridization between the s - and p -bands forlarge modulation frequencies, the lowest six energy bandshave to be taken into account to quantitatively under-stand the position of the phase transitions for the modelrealized in our experiment (Sect. S2 A). Considering onlythe two lowest bands results in a shift of the transitionpoints (see last two panels of Fig. S3), but the gen-eral nature of the topological phase diagram remains un-changed. In this section we present an approximate de-scription based on a two-band tight-binding model withtime-dependent nearest-neighbor hoppings that allows usto compute the dispersion of the edge modes directly.The modulation of the relative intensities leads to amodulation of the distance between neighbouring latticesites, which can be expressed as time-dependent tunnel-ing matrix amplitudes in the tight-binding limit. Theunit cell of the stripe consists of N dimers along the y -direction and has a width of 3 a in the x -direction.Due to the periodicity along x , the Hamiltonian canbe Fourier-transformed along this direction with quasi-momentum q x ∈ [ − π a , π a ]. Including time-dependentnearest-neighbour tunneling along the directions δ i withamplitudes J i ( t ) , i = { , , } and setting the energy off-set between the A - and B -sites to zero, the Hamiltonianreadsˆ H tb ( q x , t ) = − (cid:88) n,q x J ( t ) (cid:16) e − iq x a ˆ α † q x ( n ) ˆ β q x ( n ) + c.c. (cid:17) + J ( t ) (cid:16) e iq x a ˆ α † q x ( n ) ˆ β q x ( n + 1) + c.c. (cid:17) + J ( t ) (cid:16) e iq x a ˆ α † q x ( n + 1) ˆ β q x ( n ) + c.c. (cid:17) , (S.8) where ˆ α † q x ( n ) and ˆ β † q x ( n ) create a particle with quasimo-mentum q x on the n th A - and B -site within the stripe.To extract the time-dependent tunneling amplitudes,we fitted the energy bands of the two-band tight-bindingmodel for the system without boundaries to the twolowest energy bands of the full Hamiltonian for a fixedmodulation amplitude at every timestep t l within onedriving period. The fit was performed on both en-ergy bands in the entire 2D-BZ, yielding the values of J i ( t l ) , i = { , , } within one modulation cycle. In thefull six-band Hamiltonian of our time-dependent honey-comb lattice model, particle-hole symmetry is broken re-sulting in an asymmetry of the two s -bands. This couldbe accounted for by including next-nearest-neighbourhoppings and coupling to p -orbitals, however, for a con-ceptual understanding of the phase diagram, the simpletwo-band model of Eq. (S.8) is sufficient. In general, thenearest-neighbour hopping amplitude between two sitesis expected to depend exponentially on the height of thepotential barrier between the sites. Hence, we describedthe time-dependence of the hopping amplitudes as J i ( t ) = A e B cos( ωt + φ i ) + C i = { , , } , (S.9)with φ i = π × ( i −
1) and A , B and C are free vari-ables that depend on the modulation amplitude. Thisfunction was fitted to the extracted hopping amplitudes.Using the time dependent hoppings we calculated the ef-fective Hamiltonian by integration of ˆ H tb ( q x , t ) over onedriving period according to Eq. (S.4) for every q x . Theresulting 2 N quasienergies are shown in Fig. S6b as afunction of the quasimomentum q x for N = 50, m = 0 . ω/ (2 π ) = 16 kHzcorresponds to the Haldane regime, where a pair of chi-ral edge modes is visible in the gap at zero quasienergy.For a system with an armchair edge being periodic along x , the Γ- and K -point are both displayed at q x = 0. At ω/ (2 π ) = 8 kHz the system is in the anomalous phase ex-hibiting an additional pair of edge modes in the g π -gap8 q y q x λ q y q x λ Q s Q s ε ε ba Figure S7. Schematic drawing of the topological charge Q s and the integration surfaces discussed below. a . Cube withheight 2 ε around Q s in the three-dimensional parameter space( q , λ ). b . Two two-dimensional surfaces in q -space separatedby 2 ε . between FBZs. In the third regime with ω/ (2 π ) = 4 kHzthe are no edge modes at zero quasienergy in the g -gap,but there exist chiral edge modes in the π -gap, charac-terizing a Haldane-like topological system. S4. WINDING NUMBERS ANDTOPOLOGICAL CHARGE
The change in winding number across a topologicalphase transition is defined via the topological charge Q js of the band touching singularity as defined in Eq. (1)in the main text. We consider a three-dimensional pa-rameter space spanned by the quasimomentum q and λ , which smoothly connects a family of Hamiltonians H ( q , λ ) (white arrows in Fig. 2a). For two-band mod-els this Hamiltonian can be expressed as H ( q , λ ) = (cid:80) α = x,y,z h α ( k ) σ α + h ( k ) (cid:49) , where = ( σ x , σ y , σ z ) is thevector of Pauli matrices and (cid:49) is the identity matrix.In generic cases we can make a Taylor expansion ofthe dispersion relation in the vicinity of the gap clos-ing point [42] that occurs at ξ := ( q s , λ s ). The result-ing Hamiltonian can be expressed in the form of a WeylHamiltonian H s = v βα ξ β σ α , (S.10)where v is a 3 × Q s = sgn(det( v )) . (S.11)The Hamiltonian (S.10) has the form of a general three-parameter Hamiltonian H s = v (cid:48) · σ , whose Berry curva-ture of the upper and lower state is described by mag-netic monopoles Ω ± = ∓ v (cid:48) v (cid:48) and there is a singularityat v (cid:48) s = 0. The Berry flux through a closed surface Σ c containing the singularity v (cid:48) s = 0 is given by φ ± = ∓ (cid:90) Σ c dΣ · Ω ± = ∓ πQ s . (S.12) Note that here we defined the topological charge Q s bythe Berry flux of the energy band below the respectiveenergy gap. In Floquet systems there are two indepen-dent gaps and the topological charge Q s in the gap g is defined via the Berry curvature of the lower band Ω − and accordingly, the topological charge Q πs in the gap g π is defined via the Berry curvature of the upper band Ω + .One possibility to measure the topological charge as-sociated with the singularity is to determine the fluxthrough a sphere containing the band touching pointor equivalently through a cube as depicted in Fig. S7a,which would require Berry flux measurements throughthe six surfaces of the cube in ( q , λ )-space. This idea,however, can be simplified, if we consider the limit of2 ε → ξ is at the origin ξ = 0. In this limit,the Berry flux through the two surfaces just before andjust after the phase transition (Fig. S7b) in q -space isdetermined by φ −± ε = (cid:90) q x − q x d q x (cid:90) q y − q y d q y Ω − λ s ± ε ( q s ) φ −± ε ε → −−−→ ± π (S.13)Infinitesimally away from the band-touching singular-ity, the Berry curvature is perfectly localized in the q x q y -plane giving rise to a flux of ± π . Thus, in order to deter-mine the sign of the topological charge we simply needto detect the sign of the π Berry flux on both sides of thephase transition:12 π ∆ φ − ( q s ) ε → −−−→ Q s , (S.14)with ∆ φ − ( q s ) = (cid:0) φ − + ε − φ −− ε (cid:1) . The Berry flux φ is pro-portional to our measured deflections s ⊥ , however, inthe experiment we perform a weighted average accordingto the momentum distribution of our condensate as dis-cussed in Section S2 B. Nonetheless, if the spread is nottoo large and if we can perform the measurement closeenough at the phase transition point, we can identifythe topological charge of the singularity by determiningthe sign of the local Hall drifts across the phase transi-tion [14, 18]: Q s = sgn (cid:0) ∆ s −⊥ ( q s ) (cid:1) = − sgn (cid:0) ∆ s + ⊥ ( q s ) (cid:1) (S.15)Equivalently, the topological charge of the π -gap is de-termined by Q πs = − sgn (cid:0) ∆ s −⊥ ( q s ) (cid:1) = sgn (cid:0) ∆ s + ⊥ ( q s ) (cid:1) . (S.16)9 [S1] S. Stringari, “Collective Excitations of a Trapped Bose-Condensed Gas,” Phys. Rev. Lett. , 2360–2363 (1996).[S2] M. Greiner, I. Bloch, O. Mandel, T. W. H¨ansch, andT. Esslinger, “Exploring Phase Coherence in a 2D Lat-tice of Bose-Einstein Condensates,” Phys. Rev. Lett. ,160405 (2001).[S3] M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer,D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Sza-meit, “Photonic Floquet topological insulators,” Nature , 196–200 (2013).[S4] G. Jotzu, M. Messer, R. Desbuquois, M. Lebrat,T. Uehlinger, D. Greif, and T. Esslinger, “Experimentalrealization of the topological Haldane model with ultra-cold fermions,” Nature , 237–240 (2014).[S5] T. Fukui, Y. Hatsugai, and H. Suzuki, “Chern Numbersin Discretized Brillouin Zone: Efficient Method of Com- puting (Spin) Hall Conductances,” J. Phys. Soc. Jpn. ,1674–1677 (2005).[S6] J. Bellissard, “Change of the Chern number at bandcrossings,” arXiv cond-mat/9504030 (1995).[S7] S. M. Young, S. Zaheer, J. C. Y. Teo, C. L. Kane, E. J.Mele, and A. M. Rappe, “Dirac Semimetal in Three Di-mensions,” Phys. Rev. Lett. , 140405 (2012).[S8] N. P. Armitage, E. J. Mele, and A. Vishwanath, “Weyland Dirac semimetals in three-dimensional solids,” Rev.Mod. Phys. , 015001 (2018).[S9] M. Aidelsburger, M. Lohse, C. Schweizer, M. Atala, J. T.Barreiro, S. Nascimb`ene, N. R. Cooper, I. Bloch, andN. Goldman, “Measuring the Chern number of Hofstadterbands with ultracold bosonic atoms,” Nature Phys.11