Topological classification of quasi-periodically driven quantum systems

TL;DR

This paper explores the topological properties of quasi-periodically driven quantum systems, revealing how their band structures can be classified by Chern numbers and how these topological phases manifest in dynamical behaviors.

Contribution

It introduces a topological classification of quasi-energy bands in multi-frequency driven quantum systems and demonstrates their physical signatures and stability conditions.

Findings

Bands are classified by integer Chern numbers.

Topological bands exhibit energy pumping and chaotic dynamics.

Topological phases can be realized as pre-thermal states or with counter-diabatic methods.

Abstract

Few level quantum systems driven by $n_\mathrm{f}$ incommensurate fundamental frequencies exhibit temporal analogues of non-interacting phenomena in $n_\mathrm{f}$ spatial dimensions, a consequence of the generalisation of Floquet theory in frequency space. We organise the fundamental solutions of the frequency lattice model for $n_\mathrm{f}=2$ into a quasi-energy band structure and show that every band is classified by an integer Chern number. In the trivial class, all bands have zero Chern number and the quasi-periodic dynamics is qualitatively similar to Floquet dynamics. The topological class with non-zero Chern bands has dramatic dynamical signatures, including the pumping of energy from one drive to the other, chaotic sensitivity to initial conditions, and aperiodic time dynamics of expectation values. The topological class is however unstable to generic perturbations due to exact level crossings in the quasi-energy spectrum. Nevertheless, using the case study of a spin in a quasi-periodically varying magnetic field, we show that topological class can be realised at low frequencies as a pre-thermal phase, and at finite frequencies using counter-diabatic tools.