A flow equation approach to periodically driven quantum systems

TL;DR

This paper introduces a flow equation method to derive accurate, effective time-independent Hamiltonians for periodically driven quantum systems, outperforming traditional high-frequency expansions especially at lower frequencies and strong drives.

Contribution

The authors develop a flow equation approach that extends the validity of effective Hamiltonian derivations into lower frequency regimes, surpassing the Magnus expansion and similar methods.

Findings

Improved accuracy over Magnus expansion in low-frequency regimes

Effective Hamiltonians for both interacting and non-interacting systems

Better global approximation especially under strong external driving

Abstract

We present a theoretical method to generate a highly accurate {\em time-independent} Hamiltonian governing the finite-time behavior of a time-periodic system. The method exploits infinitesimal unitary transformation steps, from which renormalization group-like flow equations are derived to produce the effective Hamiltonian. Our tractable method has a range of validity reaching into frequency regimes that are usually inaccessible via high frequency $\omega$ expansions in the parameter $h/\omega$, where $h$ is the upper limit for the strength of local interactions. We demonstrate our approach on both interacting and non-interacting many-body Hamiltonians where it offers an improvement over the more well-known Magnus expansion and other high frequency expansions. For the interacting models, we compare our approximate results to those found via exact diagonalization. While the approximation generally performs better globally than other high frequency approximations, the improvement is especially pronounced in the regime of lower frequencies and strong external driving. This regime is of special interest because of its proximity to the resonant regime where the effect of a periodic drive is the most dramatic. Our results open a new route towards identifying novel non-equilibrium regimes and behaviors in driven quantum many-particle systems.