Quasiperiodic Floquet-Thouless Energy Pump

We study a disordered one-dimensional fermionic system subject to quasiperiodic driving by two modes with incommensurate frequencies. We show that the system supports a topological phase in which energy is transferred between the two driving modes at a quantized rate. The phase is protected by a combination of disorder-induced spatial localization and frequency localization, a mechanism unique to quasiperiodically driven systems. We demonstrate that an analogue of the phase can be realized in a cavity-qubit system driven by two incommensurate modes.

Figure 1

(a) The quasiperiodic Floquet-Thouless energy pump for a fermionic chain driven by incommensurate frequencies ${\omega }_1$ and ${\omega }_2$. When one end of the chain is fully occupied (green), energy is pumped between the two modes at the rate $\nu {\mathcal{P}}_0$ (${\mathcal{P}}_0={\omega }_1{\omega }_2/2\pi$), where $\nu \in \mathbb{Z}$ is the topological winding number of the phase. The quantization persists as long as the system is localized in both real and frequency space. (b) An analogue of the phase can be realized with a bichromatically driven qubit-cavity system. (c) 4-step driving protocol that realizes the phase. A particle tunneling along (against) the arrow acquires a phase $e^{-i{\omega }_{2}t}$ ($e^{i{\omega }_{2}t}$).

Figure 2

Simulation of the QFTEP with 200 unit cells and open boundary conditions, as detailed in the numerics section. (a) Average energy absorption rate over 10000 periods of mode 1 when the left half of the chain is initially occupied. (b) Estimated spatial localization length of the generalized Floquet states, $\xi$. (c) Effective spectral density of the system, $\gamma$. (d) Spectral density ${\rho }_{N_0}(\omega )$ resulting from a wave packet initialized on a single site in the middle of the chain for parameters within the topological plateau (bottom), and at the transition (top).