How to Directly Measure Floquet Topological Invariants in Optical Lattices

The classification of topological Floquet systems with time-periodic Hamiltonians transcends that of static systems. For example, spinless fermions in periodically driven two-dimensional lattices are not completely characterized by the Chern numbers of the quasienergy bands, but rather by a set of winding numbers associated with the gaps. We propose a feasible scheme for measuring these winding numbers in a periodically driven optical lattice efficiently and directly. It is based on the construction of a one-parameter family of drives, continuously connecting the Floquet system of interest to a trivial reference system. The winding numbers are then determined by the identification and the tomography of the band-touching singularities occurring on the way. As a by-product, we also propose a method for probing spectral properties of time evolution operators via a time analog of crystallography.

Figure 1

Driven honeycomb lattice. (a) Continuous drive via circular shaking. (b) Stepwise drive. (c)–(e) Floquet spectra for continuously driven model on a finite strip with armchair termination, for the quasimomentum $k_{\parallel }$ along periodic direction along the strip for $\nu =1$. The parameters are (c) $\omega =3J$, $\delta =-2J$, $\kappa =2$; (d) $\omega =4J$, $\delta =0.2J$, $\kappa =1$; (e) $\omega =2.5J$, $\delta =-2J$, $\kappa =1.5$.

Figure 2

(a) General scheme of constructing the family of drives. Topologically distinct phases are depicted with different colors in the parameter space given by the drive period ${\tau }^{\lambda }$ and other parameters (symbolized by the horizontal axis). Two ways of reaching the target Hamiltonian with period $T$ [marked by a star in (a)] are (b) increasing the driving period ${\tau }^{\lambda }$ and (c) chopping and repeating the drive at ${\tau }^{\lambda }$.

Figure 3

Quasienergy bands of the circularly shaken honeycomb lattice as the drive frequency is lowered from $\omega =10J$ to target frequency $\omega =3J$ for $\kappa =1.5$, $\mathrm{\Delta }=J$. (b), (c) $\mathit{k}$-space plots at topological transitions marked by vertical lines in (a), with topological charges $q_0=q_{\pi }=1$.

Figure 4

Quasienergy bands of partial drives constructed for the stepwise drive with $\omega =3J$, and $\mathrm{\Delta }=0$, with a topological charge $q_0=1$.

Figure 5

(a) The measured frequency ${\gamma }_{<}$ at $\mathit{k}=0$ and the quasienergy gap ${\gamma }_0$ for the singularity shown in Fig. 3. The cusp indicates that the association of the ${\gamma }_0$ and the ${\gamma }_{\pi }$ gap as the smaller gap changes. (b) Experimental protocol starting from a high frequency with a small period ${\tau }^0\ll |J|,|\mathrm{\Delta }|$, the drive strength ${\kappa }^{\lambda }$ is ramped up to its final value and symmetry-breaking perturbation are smoothly turned off in the target Hamiltonian.