Floquet resonant states and validity of the Floquet-Magnus expansion in the periodically driven Friedrichs models

TL;DR

This paper investigates Floquet states in driven Friedrichs models, revealing differences between discrete and continuous cases and examining the validity of the Floquet-Magnus expansion across frequency regimes.

Contribution

It provides a detailed analysis of Floquet bound and resonant states in Friedrichs models, highlighting the conditions under which the Floquet-Magnus expansion is valid or leads to metastable states.

Findings

Floquet bound state exists in high-frequency regime for discrete models

In continuous models, the Floquet-Magnus expansion predicts metastable states with diverging lifetime

Low-frequency regime features Floquet resonant states with exponentially small decay rates

Abstract

The Floquet eigenvalue problem is analyzed for periodically driven Friedrichs models on discrete and continuous space. In the high-frequency regime, there exists a Floquet bound state consistent with the Floquet-Magnus expansion in the discrete Friedrichs model, while it is not the case in the continuous model. In the latter case, however, the bound state predicted by the Floquet-Magnus expansion appears as a metastable state whose lifetime diverges in the limit of large frequencies. We obtain the lifetime by evaluating the imaginary part of the quasi-energy of the Floquet resonant state. In the low-frequency regime, there is no Floquet bound state and instead the Floquet resonant state with exponentially small imaginary part of the quasi-energy appears, which is understood as the quantum tunneling in the energy space.