Topological transitions to chaos in quasi-periodically driven quantum systems

A quantum system with time-translational symmetry has a quasi-periodic response. Driving with two incommensurate tones is the simplest protocol without this symmetry. In this talk I will analyze the response of a finite quantum system to this quasi-periodic driving.

For even the simple case of a spin-1/2 there is a rich phase diagram. For weak driving the response remains quasi-periodic. However for strong driving there is a topological-dynamical transition to a new chaotic response. The chaotic phase is characterised by continuous spectra, non-zero Chern number, and chiral modes which deterministically pump energy between the drives.