Abstract
Sixty years ago, Karplus and Luttinger pointed out that quantum particles moving on a lattice could acquire an anomalous transverse velocity in response to a force, providing an explanation for the unusual Hall effect in ferromagnetic metals^{1}. A striking manifestation of this transverse transport was then revealed in the quantum Hall effect^{2} where the plateaux depicted by the Hall conductivity were attributed to a topological invariant characterizing the Bloch bands: the Chern number^{3}. Until now, topological transport associated with nonzero Chern numbers has only been observed in electronic systems^{2,4,5}. Here we use the transverse deflection of an atomic cloud in response to an optical gradient to measure the Chern number of artificially generated Hofstadter bands^{6}. These topological bands are very flat and thus constitute good candidates for the realization of fractional Chern insulators^{7}. Combining these deflection measurements with the determination of the band populations, we obtain an experimental value for the Chern number of the lowest band ν_{exp} = 0.99(5). This first Chernnumber measurement in a nonelectronic system is facilitated by an alloptical artificial gauge field scheme, generating uniform flux in optical superlattices.
Main
One of the most challenging goals in the context of artificial gauge fields is to implement experimental probes revealing the nontrivial topology of energy bands. This would open the path towards the realization of novel topological states of matter in a wide class of physical settings, ranging from cold atoms to polariton gases^{8,9,10,11}. Considering cold atoms in optical lattices, it has been shown theoretically that transport measurements could allow a determination of the Chern number characterizing topological Bloch bands^{8,12,13}. Although earlier experiments used local cyclotron orbits to detect the artificial gauge field structure at the singleplaquette level^{14,15}, observing the entire cloud dynamics or determining the Chern number of the underlying bands has remained out of reach. Here we load ultracold bosonic atoms into Hofstadter bands, and we report on the direct detection of the cloud’s spatial motion as a response to an applied force. Using a novel bandmapping technique, we track the populations of the Hofstadter bands over time. We observe that the flatness of the bands leads to homogeneous populations within individual bands, through interactions and finite temperatures. For short times, heating and nonadiabatic interaction effects are limited, and the detected transverse Hall drift is in agreement with exact diagonalization studies^{12,13}. Combined with independent bandpopulation measurements, we extract the Chern number of the lowest Hofstadter band from our experimental data. Our work represents the first determination of a topological invariant characterizing twodimensional Bloch bands using ultracold atoms, and complements other studies based on atom interferometric^{16,17} and collectivemode^{18} measurements.
Our opticallattice setup realizes the celebrated Harper–Hofstadter Hamiltonian^{6}, which describes the motion of particles on a square lattice penetrated by a uniform magnetic flux Φ per unit cell (see also the original works by Harper^{19} and Azbel^{20}). An atom that hops around a closed loop of the optical lattice picks up a geometric phase, which mimics the Aharonov–Bohm phase of a charged particle in a magnetic field. The artificial flux is thus produced by engineering complex tunnelling matrix elements with spacedependent (Peierls) phases^{6}, through the laserassistedtunnelling method introduced by Jaksch and Zoller^{21} and subsequent works^{9,22,23,24,25}. The present scheme controls tunnelling along the x direction and uses two main ingredients: a staggered potential with energy offset Δ between neighbouring sites, inhibiting bare tunnelling along x, together with a resonant modulation of frequency ω = Δ/ℏ, restoring the hopping in a controlled way. Using a simple timedependent optical potential, which simultaneously triggers the hopping on all the links, naturally produces a staggered flux distribution^{14}. To rectify the flux, we implement a new alloptical driving scheme that uses two pairs of laser beams to control the laserinduced tunnelling on successive links independently, hence producing a uniform flux Φ = π/2 per unit cell (Fig. 1a). The lowest band of the corresponding energy spectrum has a Chern number ν_{1} = +1 and is thus topologically equivalent to the lowest Landau level. Moreover, this band exhibits a large flatness ratio of E_{gap}/E_{bw} ≃ 7, where E_{gap} is the energy gap to the second band and E_{bw} the bandwidth. In contrast to previous experiments generating uniform flux in optical lattices^{15,26}, the present scheme does not rely on magnetic field gradients, and therefore offers a higher degree of experimental control.
The experimental setup consists of an ultracold gas of ^{87}Rb atoms that is loaded into a twodimensional lattice created by two orthogonal standing waves with wavelength λ_{s} = 767 nm. The atoms are confined in the perpendicular direction by a weak harmonic potential using an optical dipole trap. A further standing wave with twice the wavelength λ_{L} = 2λ_{s} is superimposed along x to create the staggered potential (Fig. 1a), with an energy offset Δ much larger than the bare tunnelling J_{x}. The modulation restoring resonant tunnelling is created by two further pairs of fardetuned laser beams, each pair generating a moving potential of the form V_{i}(x, y, t) = κ cos(k_{L}x + φ_{i})cos(−k_{L}y ± ωt), where κ is the driving amplitude, k_{L} = 2π/λ_{L}, and ω = Δ/ℏ. The relative phases φ_{i} are adjusted so as to control successive links independently (Fig. 1a). In the highfrequency limit ℏω ≫ J_{x}, J_{y}, this system can be described by an effective timeindependent Hamiltonian^{27,28,29,30,31}, whose dominant contributions reproduce the Harper–Hofstadter Hamiltonian (Methods)
where the Landau gauge was chosen to describe the system^{6}. Here annihilates (creates) a particle on site (m, n), where the position in the lattice is defined as , with m, n integers and the unit vectors. In the limit Δ ≫ κ, the effective coupling strengths are given by and J_{y}^{eff} ≃ J_{y}; the experimental parameters were chosen such that J_{x}^{eff} ≈ J_{y}^{eff} ≡ J. Higherorder corrections to the effective Hamiltonian lead to a local renormalization of the hopping along y, which for our experimental parameters κ/(ℏω) can lead to modifications of the tunnelling up to 0.3J_{y} (Methods and Supplementary Information). In the presence of the effective flux Φ = π/2, the magnetic unit cell is four times larger than the standard unit cell (Fig. 1a). Consequently the first magnetic Brillouin zone (FBZ) is reduced, and the energy bands split into four subbands^{3,6}. Because the two middle bands touch at the Dirac points (Fig. 2b), the energy spectrum is partitioned into three isolated bands, labelled as E_{μ}, with Chern numbers ν_{μ}. We stress that the central ‘superband’ contains twice the number of states as compared to the other two bands.
To load the atoms into the lowest Hofstadter band, we developed an experimental sequence using an auxiliary superlattice potential (Methods), which introduces a staggered detuning δ along both directions: along x, the offset between neighbouring sites is increased away from the resonance condition to Δ + δ, whereas it is simply given by δ along y (Fig. 2a). Importantly, the unit cell of the square lattice with staggered potentials along both directions is equivalent to the magnetic unit cell of the Harper–Hofstadter model; thus, the number of energy bands is preserved during the loading sequence. For δ > 2J the topology of the bands is trivial, all Chern numbers are zero, and the atoms populate the lowest band (lowenergy sites of the superlattice). When crossing the topological phase transition at δ = 2J, the spectral gaps close at a single point in the FBZ and the system enters the topologically nontrivial regime, where the lowest band E_{1} has a Chern number ν_{1} = +1. Finally, for δ = 0 we realize the Harper–Hofstadter model with flux Φ = π/2 (Fig. 2b).
Recently, several methods were proposed to probe the topological nature of energy bands with cold atoms, exploiting Bloch oscillations and other transport measurements^{12,13,28}. In the presence of a constant force , atoms on a lattice undergo Bloch oscillations along the direction of the force; this longitudinal motion is captured by the band velocity v_{μ}^{band} = ∂_{k}E_{μ}/ℏ. Moreover, when the energy bands have nonzero Berry curvature, the cloud also experiences a net perpendicular (Hall) drift (Fig. 1b); this transverse motion is described by a further contribution to the velocity, denoted v_{μ}^{x} (ref. 32). For a particle in a state u_{μ}(k)〉 of the μth band, this ‘anomalous’ contribution to the velocity reads
where Ω_{μ}(k) is the Berry curvature of the band^{32}. The effects associated with the anomalous velocity v_{μ}^{x} can be isolated by uniformly populating the bands, which averages out any contribution from the band velocity, ∫∂E_{μ}/∂k_{x, y}d^{2}k = 0. This could be directly realized with fermionic atoms by setting the Fermi energy within a spectral gap^{13}, in analogy with the integer quantum Hall effect. Here we consider an incoherent distribution of bosonic atoms, where the population within each band is homogeneous in kspace, an assumption which has been validated independently (Supplementary Information). In the absence of interband transitions, the contribution of the μth band to the centreofmass (COM) motion perpendicular to the force can be written in terms of the Chern number of the μth band ν_{μ} = ∫ _{FBZ}Ω_{μ}d^{2}k/(2π),
where the factor 4a^{2} is the area of the magnetic unit cell (Fig. 1a) and τ_{B} = h/(Fa) is the characteristic timescale for Bloch oscillations. In our experiments, we applied an optical dipole force along y (Methods) and measured the COMevolution of the atomic cloud in situ for opposite directions of the flux Φ, which were then subtracted to obtain the differential shift x(t, Φ) − x(t, − Φ) = 2x(t). For short evolution times, an almost linear differential displacement is observed (Fig. 3a), whereas for longer times it saturates owing to band repopulation (Fig. 3b, c). We note that the deflection is symmetric with respect to the direction of the applied force (black and grey data points in Fig. 3b), as expected from theory. Furthermore, we measured the COMmotion for Φ = 0 (light blue data points in Fig. 3b) and for a staggeredflux distribution (dark blue data points in Fig. 3b); neither measurement shows any significant displacement, which is consistent with a Chern number of zero (Methods).
The bandmapping sequence, which is basically the reversed loading sequence as illustrated in Fig. 2b, allows us to measure the band populations of the different Hofstadter bands during the dynamics (Fig. 3c). For large detuning δ, tunnelling is inhibited along both directions and the populations of the Hofstadter bands map onto those of the twodimensional superlattice, where standard detection techniques can be used to evaluate the band populations η_{μ} (Methods and Supplementary Information)^{33}. The contribution of all atoms in different bands to the mean COMdisplacement can be written as
where the filling factor γ_{0} is given by the initial band populations η_{μ}^{0} (Fig. 3c). This result is obtained by invoking the particle–hole symmetry inherent to the Harper–Hofstadter model (that is, ν_{1} = ν_{3}), together with the fact that the sum of Chern numbers necessarily vanishes, ∑_{μ}ν_{μ} = 0; we have also taken into account that the middle band contains twice the number of states as compared to the other two bands; its contribution to x(t) is thus proportional to ν_{2}(η_{2}/2) = − ν_{1}η_{2} (Supplementary Information). As a result, the COMdisplacement in equation (2) is determined only by the initial band populations η_{μ}^{0} and the Chern number of the lowest band ν_{1}, under the assumption that the band populations are constant over time, which is reasonable for short times. Comparing the shorttime trajectories of the atomic cloud x(t) in Fig. 3a with the equation of motion equation (2), together with the measured initial filling factor γ_{0}, provides a first reasonable experimental value for the Chern number of the lowest band as ν_{exp} = 0.9(2). In particular, for short times, we find good agreement between the theoretical predictions based on the effective Hamiltonian and the experimental data.
We now present a more precise Chernnumber measurement based on a longtime analysis, which takes into account the repopulation of atoms between the three Hofstadter bands (Fig. 3c). One possible reason for this repopulation is the presence of Landau–Zener transitions, which are neglected in equation (2) but captured well by the numerical simulations (green shaded area in Fig. 3a). However, we observe similar repopulation timescales in the absence of the force, most probably due to heating of the atoms caused by the periodic driving. To capture the bandrepopulation effects, we benefit from the measured filling factor γ(t) = η_{1}(t) − η_{2}(t) + η_{3}(t) and model the dynamics according to the modified equations of motion
By fitting this equation to the experimental data x(t), with the Chern number being the only fit parameter, we obtain an experimental value for the Chern number of the lowest band
Here we averaged over four independent Chernnumber measurements, two for each direction of the gradient to avoid systematic errors (see inset of Fig. 4a). The stated uncertainty is the standard deviation obtained from these measurements. This shows that including our timeresolved bandmapping data into our modelling of the transverse Hall drift leads to a very good understanding of the full time dynamics, and allows us to extract the value of the Chern number with high accuracy. The applied force was chosen to be strong enough to accurately detect the displacement, but weak enough to limit nonlinear effects and Landau–Zenerinduced interband transitions.
The dependence of our Chernnumber measurement with respect to the force was studied in more detail, as shown in Fig. 4a. For gradient strengths smaller than the bandgap, Fa < E_{gap} ≈ 1.5J, the experimental value for the Chern number saturates to ν_{exp} ≈ 1, indicating that it can be determined reliably for small forces. For very strong forces, Fa > E_{gap}, our model breaks down and the experimental value ν_{exp} decreases to zero.
Finally, we characterized the topological phase transition, which is expected for a staggered detuning of δ = 2J (Fig. 2). For this analysis, we set the gradient amplitude to the value Fa = 38.4(8) Hz × h = 0.51(1)J, which is well below the bandgap for δ = 0. In agreement with theory, we observe that the experimental value for the Chern number decreases to zero across the phase transition (Fig. 4b). We note that Landau–Zener transitions to higher bands become more important when approaching the transition (gapclosing point); however, this should not affect the measurement because the band repopulation is taken into account, according to equation (3). The smoothened transition is most probably due to the experimental uncertainties in the resonance condition (green solid line in Fig. 4b). Furthermore, secondorder corrections to the effective Hamiltonian shift the transition point to a mean value of δ ≈ 1.8J for our experimental parameters κ/(ℏω) = 0.58(2) (solid blue line and shaded region in Fig. 4b, see Methods and Supplementary Information).
In conclusion, we have successfully implemented a method to measure the Chern number in a coldatom setup, which can be generalized to a wide range of nonelectronic systems, including ion traps^{9}, photonic crystals^{10} and polaritons^{11}. Although our measurement accommodates dynamical transitions to higher bands, which we attribute to the lattice modulation used to engineer the topological band structure, our results highlight the necessity to further deepen the understanding of heating processes in periodically driven quantum systems. Minimizing heating effects and clarifying the role of interactions in these modulated systems will be crucial in view of reaching topological strongly correlated states in Chern bands, such as fractional Chern insulators^{7}.
Recently we have become aware of related measurements showing signatures of the Berry curvature in periodically modulated honeycomb optical lattices^{34}.
Methods
Loading sequence.
The experimental sequence started by loading a Bose–Einstein condensate of ^{87}Rb atoms within 150 ms into a twodimensional optical superlattice. Along each of the axes two standing waves were superimposed with λ_{s} = 767 nm and λ_{L} = 2λ_{s}. The relative phase between them was chosen such that a lattice potential with staggered energy offsets Δ + δ_{x} along x and δ_{y} along y, with δ_{x} ≈ δ_{y} ≡ δ and δ < Δ, was created. The lattice depths were V_{y} = 10(1) Er_{s}, V_{yL} = 1.75(5) Er_{L}, V_{x} = 6.0(2) Er_{s} and V_{xL} = 5.25(16) Er_{L}, with Er_{α} = ℏ^{2}k_{α}^{2}/(2m), α = {s, L}. At this point of the sequence all atoms were loaded into the lowenergy sites. The two pairs of beams used for the modulation were then switched on in 30 ms, with a frequency difference ω_{r} = −ω_{b} = ±Δ/ℏ; at this stage, no resonant tunnelling between neighbouring sites was induced, owing to the offset detuning δ. After that, we ramped down the long lattices within 30 ms to V_{yL} = 0 Er_{L} and V_{xL} = 3.25(10) Er_{L}, which corresponds to δ = 0. For these values, resonant laserassisted tunnelling along x and bare tunnelling along y occurred, simultaneously creating a homogeneous flux Φ = ±π/2 (aligned along ) depending on the sign of the modulation frequency. We checked that all lattice sites were equally populated after the loading sequence.
Loading sequence for lattice setup with trivial topology.
The loading sequence described above is also used to load the atoms into the staggered flux lattice. The only difference is that the modulation is switched on with a frequency difference ω_{r} = ω_{b} = ±Δ/ℏ, which results in a flux Φ = ±(−1)^{m}π/2.
The sequence for the lattice without flux was performed in a similar manner. It started by loading the atoms into a twodimensional superlattice with lattice depths V_{x} = V_{y} = 10(1) Er_{s}, V_{xL} = V_{yL} = 1.75(5) Er_{L} within 150 ms. Subsequently, the long lattices were decreased to zero within 30 ms. In this way a simple square lattice configuration without flux and a tunnel coupling of J/h = 75(3) Hz along both directions was realized.
Optical gradient.
The optical gradient used to induce Bloch oscillations along y was realized using a further laser beam with wavelength λ_{z} = 844 nm. It was focused at the atom position to a waist of about 125 μm and aligned such that the atomic cloud is located at the maximum slope of the Gaussian beam profile along y. Along x the beam was centred on the atom position. The strength of the gradient was determined through independent measurements of Bloch oscillations in a onedimensional lattice with V_{y} = 10(1) Er_{s}.
Bandmapping sequence.
To measure the populations in different Hofstadter bands, we reversed our loading sequence described above and ramped up the long lattices to V_{yL} = 1.75(5) Er_{L} and V_{xL} = 5.25(16) Er_{L}, respectively, within 30 ms. At this point of the sequence, tunnelling is offresonant along both directions, and the Hofstadter bands map onto the bands of the usual 2D superlattice. We then suddenly switched off the modulation and used standard detection techniques to infer the momentum distribution and band populations (Supplementary Information).
The effective Hamiltonian.
The setup is described by a timedependent Hamiltonian , where the static part includes the nearestneighbour hopping terms, the staggered lattice potential (with energy offset Δ + δ along x, and δ along y) and the external confinement. The main offset Δ ≫ J_{x} is chosen so as to inhibit the tunnelling along the x direction. The timeperiodic modulation with frequency ω is created by two pairs of laser beams, and it is resonant with the offset ω = Δ/ℏ; it simultaneously restores the tunnelling along x and generates spacedependent Peierls phases, effectively creating a uniform flux Φ = 2π(1/4) per plaquette. The modulation also modifies the tunnelling matrix elements along the y direction. In the highfrequency limit, ℏω ≫ J_{x}, J_{y}, the system is described well by the effective Harper–Hofstadter Hamiltonian in equation (1). Adding weak higherorder corrections to the latter, and considering a nonzero staggered detuning δ ≪ Δ, leads to a Hamiltonian, whose momentum representation is given by the 4 × 4 matrix
where κ is the modulation strength, J/h ≈ 75 Hz is the effective tunnelling amplitude, ϕ_{0} is the phase of the modulation relative to the underlying lattice (Supplementary Information), and the lattice spacing is a = 1. In the experiment, κ/(ℏω) ≈ 0.58, so that the inhomogeneous corrections to the hopping along the y direction (that is, h_{1} and h_{2}) are not negligible.
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Acknowledgements
We acknowledge fruitful discussions with J. Dalibard and also with A. Dauphin, P. Gaspard, F. Gerbier, F. Grusdt, I. Carusotto, T. Ozawa and H. Price. This work was supported by NIM, the EU (UQUAM, SIQS) and EPSRC Grant No. EP/K030094/1. M.Aidelsburger was further supported by the Deutsche Telekom Stiftung, M.L. by ExQM and N.G. by the Université Libre de Bruxelles and the FRSFNRS (Belgium).
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M.Aidelsburger, M.L. and C.S. performed the experiment. All authors contributed to the design of the experiment, the theoretical and data analysis, and to the writing of the paper. I.B. and N.G. supervised the project.
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Aidelsburger, M., Lohse, M., Schweizer, C. et al. Measuring the Chern number of Hofstadter bands with ultracold bosonic atoms. Nature Phys 11, 162–166 (2015). https://doi.org/10.1038/nphys3171
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DOI: https://doi.org/10.1038/nphys3171
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