Floquet Engineering Topological Dirac Bands

We experimentally realized a time-periodically modulated 1D lattice for ultracold atoms featuring a pair of linear bands, each with a Floquet winding number. These bands are spin-momentum locked and almost perfectly linear everywhere in the Brillouin zone: a near-ideal realization of the 1D Dirac Hamiltonian. We characterized the Floquet winding number using a form of quantum state tomography, covering the Brillouin zone and following the micromotion through one Floquet period. Last, we altered the modulation timing to lift the topological protection, opening a gap at the Dirac point that grew in proportion to the deviation from the topological configuration.

Figure 1

Concept. (a) Left: switching protocol, with the $j=0$ unit cell marked in gray. In C-I atoms tunnel between neighboring unit cells (bold green links); in C-II, they tunnel within the same unit cell (bold black links). Right: adiabatic lattice potentials colored by $⟨{\widehat{F}}_x⟩$. (b) Floquet band structure colored according to $⟨{\widehat{F}}_x⟩$ sampled stroboscopically. The lowest BZ and quasienergy zone is marked in gray. Left: idealized SSH model. Right: numerical lattice model [11]. (c) Schematic. The BEC was illuminated by counterpropagating Raman lasers and an radio frequency (rf) magnetic field. (d) Static lattice tunneling with data (markers) and simulations (black curves). Upper: magnetization $⟨{\widehat{F}}_x⟩$. Middle: group velocity. Bottom: displacement colored according to $⟨{\widehat{F}}_x(t)⟩$ using the color scale in (a).

Figure 2

Floquet protocols. (a),(b) switching protocol and (c),(d) the single-configuration protocol. (a),(c) Computed intercell (black) and intracell (green) tunneling strengths and displacement (colored according $⟨{\widehat{F}}_x⟩$). Gray and white bands indicate the different configurations. (b),(d) Floquet quasienergies [using the same color scale as in Fig. 1].

Figure 3

Crystal momentum resolved pseudospin micromotion and corresponding Berry curvature. Upper three panels: Numerical model (left) and unfiltered experimental data (right) for pseudospin components for configuration switching protocol (initial states $|\uparrow ⟩$). Lower three panels: Berry curvature based on filtered experimental data (right) and numerical simulation (left) for configuration switching protocol with initial states $|\uparrow ⟩$ and $|\downarrow ⟩$ and single-configuration protocol (initial state $|\uparrow ⟩$).

Figure 4

Breaking of chiral symmetry. (a) Time evolution with Floquet period $T=330\mu \mathrm{s}$, away from the optimal point $T_0=438\mu \mathrm{s}$, colored according to the instantaneous magnetization using the color scale in Fig 1. Configurations (gray rectangles) are plotted along with the data. (b) Computed spectrum for data in (a), and circled in orange in (c). (c) Zitterbewegung frequency as a function of $\delta T/T_0$ showing the gap closing at the symmetry point.