## Cold atoms do geometry

Electrons in solids populate energy bands, which can be simulated in cold atom systems using optical lattices. The geometry of the corresponding wave functions determines the topological properties of the system, but getting a direct look is tricky. Fläschner *et al.* and Li *et al.* measured the detailed structure of the band wave functions in hexagonal optical lattices, one resembling a boron-nitride and the other a graphene lattice. These techniques will make it possible to explore more complex situations that include the effects of interactions.

## Abstract

Topological properties lie at the heart of many fascinating phenomena in solid-state systems such as quantum Hall systems or Chern insulators. The topology of the bands can be captured by the distribution of Berry curvature, which describes the geometry of the eigenstates across the Brillouin zone. Using fermionic ultracold atoms in a hexagonal optical lattice, we engineered the Berry curvature of the Bloch bands using resonant driving and show a full momentum-resolved measurement of the ensuing Berry curvature. Our results pave the way to explore intriguing phases of matter with interactions in topological band structures.

Topology is a fundamental concept for our understanding of many fascinating systems, such as topological superconductors or topological insulators, which conduct only at their edges (*1*). The topology of the bulk band is quantified by the Berry curvature (*2*), the integral of which over the full Brillouin zone is a topological invariant called the Chern number. According to the bulk boundary correspondence principle, the Chern number determines the number of chiral conducting edge states (*1*). Although edge states have been directly observed in a variety of lattice systems—ranging from solid-state systems to photonic waveguides, and even coupled mechanical pendula (*3*–*7*)—the underlying Berry curvature as the central measure of topology is not easily accessible. In recent years, ultracold atoms in optical lattices have emerged as a platform with which to study topological band structures (*8*, *9*), and these systems have seen considerable experimental and theoretical progress. Whereas in condensed-matter systems, topological properties arise thanks to external magnetic fields or intrinsic spin-orbit coupling of the material, in cold atom systems they can be engineered by periodic driving analogous to illuminated graphene (*10*, *11*). The resulting Floquet system can have topological properties very different from those of the original system (*12*). The driving can, for example, be realized through lattice shaking (*13*–*17*) or Raman coupling (*18*–*20*) with high-precision control in a large parameter space. In particular, the driving can break time-reversal symmetry (*14*, *15*, *17*) and thus allows for engineering nontrivial topology (*17*, *19*). In quantum gas experiments, topological properties have been probed via the Hall drift of accelerated wave packets (*17*, *19*), via an interferometer in momentum space (*21*, *22*), and via edge states (*23*, *24*), but so far, the full underlying Berry curvature was not measured quantitatively.

We measured the Berry curvature with full momentum resolution based on a method proposed in (*25*, *26*). We performed a full tomography of the Bloch states across the entire Brillouin zone by observing the dynamics at each momentum point after a projection onto flat bands. The topological bands were engineered through resonant dressing of the two lowest bands of an artificial boron nitride lattice and feature a rich distribution of Berry curvature. Other relevant quantities such as the Berry phase or the Chern number can easily be obtained from the Berry curvature, which is thus the central concept for the description of topology.

Our system consists of ultracold fermionic atoms in a hexagonal optical lattice (*27*) formed by three interfering laser beams. With an appropriate polarization (*28*), a variable energy offset *h*Δ_{AB} between the A and B sites (Fig. 1A), which breaks inversion symmetry, can be engineered. With the emerging band gap *hv*_{AB}, the Dirac points at K and K′ become massive, and for a large offset, the bands are flat (Fig. 1B) (*28*). This is a key ingredient for our tomography, because the flat band acts as the reference frame in which we reconstruct the eigen states. Then as a central experimental method, we could accelerate the lattice on circular trajectories in real space by modulating the phases of the three lattice beams, thus realizing circular shaking (*13*–*17*). When the shaking frequency is near resonant with respect to a band transition, the two bands couple and form two new dressed Floquet bands. In Fig. 1C, we show the dressed Floquet bands for different accessible driving amplitudes. Apart from the dramatic change in the dispersion relation, the topological properties of the bands are changed. This manifests itself in the creation of a new Dirac point at the Γ-point and the annihilation of a Dirac point at the K point (Fig. 1D). A threefold symmetry also becomes visible in the dispersion relation (Fig. 1E).

The topological properties are not captured by the mere dispersion relation but by the Berry curvature, which describes the winding of the eigenstates across the Brillouin zone. Therefore, a complete tomography of the eigenstates of a Bloch band is mandatory for a measurement of the Berry curvature. The key idea behind our tomography is to reconstruct the eigenvectors from dynamics after a projection onto flat bands (*26*). Consider the Bloch sphere (Fig. 2A), whose poles are given by and , which are the Bloch states restricted to the A and B sublattice, respectively. The lower band can be written as , and after a projection onto flat bands, the state oscillates around , with the frequency *v _{k}* given by the energy difference of the flat bands. Then, the momentum distribution after time-of-flight is given by
(1)from which both θ

*and φ*

_{k}*can be easily obtained, yielding the desired tomography of the eigenstates for each quasimomentum. The method we use hence allows for a direct reconstruction of the Berry curvature according to (2)with (*

_{k}*26*). Our experimental sequence for this state tomography is sketched in Fig. 2. We start with a cloud of 5 × 10

^{4}single-component fermionic

^{40}K atoms forming a noninteracting band insulator in the undressed lattice. Thanks to a large offset between the A and B sites, leading to a band gap of ν

_{AB}= 11.65(11) kHz, the undressed bands are flat, so that for all quasimomenta . We adiabatically ramped up the shaking amplitude to 223 nm within 5 ms at a shaking frequency of

*v*= 9 kHz and then ramped the frequency to

*v*= 11 kHz within 2 ms. By suddenly switching off the dressing, we projected onto the bare flat bands, so that Eq. 1 can be applied. In Fig. 2B, we show typical time-of-flight images for different hold times in the flat bands. The images feature dynamics with very large contrast. Time evolutions for different quasimomenta are shown in Fig. 2C, revealing the pure sinusoidal oscillations with clearly distinct amplitudes sin(θ

*) and phases φ*

_{k}*, which are obtained by a simple fit to Eq. 1. We observed very large and long-lived oscillations after the projection, yielding relative amplitudes of up to 0.8. Additionally, with more than 2800 pixels in the first Brillouin zone, the resolution in momentum space is very high.*

_{k}As the central result, we reconstructed the Berry curvature of the dressed band structure from these fits, which fully visualize the Bloch states (Fig. 3A). The amplitude map features a pronounced threefold symmetry, illustrating the breaking of equivalence between the K and K′ points. The amplitude has a maximum at the K point and is zero at the K′ and points. Even more striking is the very distinct threefold symmetry of the phase map with nearly discrete values of 0, 2π/3, and 4π/3. Where the amplitudes are zero at the K′ and Γ points, the phase map correspondingly displays vortices. The phase vortices are clear signatures of Dirac points, which constitute topological defects. Furthermore, the data clearly show that we annihilated the Dirac point of the undressed hexagonal lattice at K and created a Dirac point in the dressed system at the Γ point, changing the topology of the band. The resulting Berry curvature is localized at the new Dirac points and also shows this clear threefold symmetry. It has opposite signs at the two Dirac points, which results from the opposite chirality of the phase vortices. By inverting the chirality of the shaking, we instead annihilated the Dirac point at the K′ point and inverted the chirality of the phase windings, which also resulted in an inverted symmetry in the Berry curvature. All quantities agree well with a Floquet theory calculation (Fig. 3B), based on a tight-binding model as described in (*28*). In Fig. 3C, we plot the Berry curvature pixelwise evaluated along a high-symmetry path, illustrating the very good agreement with the theory.

As mentioned above, with the fully momentum-resolved Berry curvature, we can easily obtain the further relevant quantities such as the Berry phase or Chern number. A discussion of the respective Berry phases is available in (*28*). The integral over the closed area of the full first Brillouin zone must be quantized to 2π times the integer Chern number *C*. From our data, we obtain *C* = 0.005(6) and *C* = –0.016(8) for the two different shaking chiralities (*28*), which clearly confirms this quantization within the experimental errors. Our measurements demonstrate that even when the global topology has Chern number zero, the distribution of Berry curvature can be very rich.

Our measurement scheme can be readily extended to characterize bands with Chern numbers different from zero (*17*, *19*). In principle, one could start in a shallow lattice, where reaching nonzero Chern numbers is feasible, and for the tomography project onto flat bands, which can be reached, such as by dynamical control over the offset. Our method for generating the topological bands is spin-independent and does not couple different spin states. It therefore can be extended to high-spin systems (*29*) or to strongly interacting spin mixtures, which are expected to lead to interesting many-body phases (*30*–*32*).

## Supplementary Materials

## References and Notes

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**Acknowledgments:**We acknowledge stimulating discussions with A. Eckardt, M. Lewenstein, L. Mathey, and C. Sträter. This work has been supported by the excellence cluster “The Hamburg Centre for Ultrafast Imaging—Structure, Dynamics and Control of Matter at the Atomic Scale” and the GrK 1355 of the Deutsche Forschungsgemeinschaft.