Topological micromotion of Floquet quantum systems

Abstract

The Floquet Hamiltonian has often been used to describe a time-periodic system. Nevertheless, because the Floquet Hamiltonian depends on a micro-motion parameter, the Floquet Hamiltonian with a fixed micro-motion parameter cannot faithfully represent a driven system, which manifests as the anomalous edge states. Here we show that an accurate description of a Floquet system requires a set of Hamiltonian exhausting all values of the micro-motion parameter, and this micro-motion parameter can be viewed as an extra synthetic dimension of the system. Therefore, we show that a d-dimensional Floquet system can be described by a d + 1-dimensional static Hamiltonian, and the advantage of this representation is that the periodic boundary condition is automatically imposed along the extra-dimension, which enables a straightforward definition of topological invariants. The topological invariant in the d + 1-dimensional system can ensure a d − 1-dimensional edge state of the d-dimensional Floquet system. Here we show two examples where the topological invariant is a three-dimensional Hopf invariant. We highlight that our scheme of classifying Floquet topology on the micro-motion space is different from the previous classification of Floquet topology on the time space.

Floquet space exploration for the dual-dressing of a qubit

“… 20 , 21 , as a synthetic dimension for the characterization of topological properties in Ref. 22 , for the realization of entangled gates in Refs. 23 , 24 .…”