Experimental reconstruction of the Berry curvature in a Floquet Bloch band

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 can be captured by the distribution of Berry curvature, which describes the geometry of the eigenstates across the Brillouin zone. Employing fermionic ultracold atoms in a hexagonal optical lattice, we generate topological bands using resonant driving and show a full momentumresolved 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 that have recently attracted a lot of interest, 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] and the integral 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]. While in a variety of lattice systems ranging from solid state systems to photonic waveguides and even coupled mechanical pendula, edge states have been directly observed [3][4][5][6][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 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 due to external magnetic fields or intrinsic spin-orbit coupling of the material, they can in cold atom systems be engineered by periodic driving analogous to illuminated graphene [10]. Interestingly, the resulting Floquet system can have totally new topological properties [11]. The driving can, for example, be realized by lattice shaking [12][13][14][15][16] or Raman coupling [17][18][19] with high precision control in a large parameter space. In particular, the driving can break time-reversal symmetry [13,14,16] and thus allows for engineering non-trivial topology [16,18]. In quantum gas experiments, topolog- ical properties have been probed via the Hall drift of accelerated wave packets [16,18], via an interferometer in momentum space [20,21]

Experimental measurement of the Berry curvature from anomalous transport

“…We also demonstrate the first full tomography of the Berry curvature of a geometrical energy band from measurements of the anomalous displacement of a coherent classical wavepacket. This approach complements existing methods in ultracold gases to measure the Berry curvature via band projection 26 or using Wilson lines 27 , with the significant advantage that it does not require a full reconstruction of the band eigenstates as an intermediate step. Consequently, our method can be easily applied to complex systems with many bands and degrees of freedom, where a full determination of the eigenstates would be experimentally challenging.…”

Observation of dynamical vortices after quenches in a system with topology

“…1a). The final Hamiltonian after the quench is realized as a Floquet Hamiltonian and is engineered by means of resonant circular lattice shaking 7,11,18,19 and re-establishes tunnelling. The initial Hamiltonian features flat bands with vanishing Berry curvature, while the final Hamiltonian realizes a Haldanelike model with tunable Berry curvature 2,11 .…”

Floquet Symmetry-Protected Topological Phases in Cold-Atom Systems

“…For cold atoms, where Floquet control has so far been applied only to single-particle band strucarXiv:1610.07611v3 [cond-mat.quant-gas] 22 Sep 2017 tures [29][30][31][37][38][39], recent advances in optically controlling interactions [40][41][42][43][44][45][46][47] offer new opportunities for accessing strongly correlated phases [48][49][50][51]. Notably, coherent spin-spin interactions with a range of several microns [42,43,46,47] can be introduced via Rydberg dressing [42-44, 46, 47, 52-55].…”