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

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 quasienergy 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 quasienergy appears, which is understood as the quantum tunneling in the energy space.

Figure 1

Discrete and continuous Friedrichs models. A particle moves around the lead and dot.

Figure 2

(a) In the discrete model, when $\omega$ is sufficiently large, the sequence of $\{\varepsilon +n\omega \}$ can avoid the continuous spectrum, but when $\omega <4g$, it cannot. (b) In the continuous model, $\varepsilon +n\omega$ for $n>0$ lies on the continuous spectrum no matter how large $\omega$ is.

Figure 3

A schematic picture of $U_n=-n\omega +{\lambda }^2\mathrm{Re}v(\varepsilon +n\omega )$ as a function of $n\omega$. The minimum of the continuous spectrum is denoted by $E_{\mathrm{cs}},E_{\mathrm{cs}}=-2g$ in the discrete model and $E_{\mathrm{cs}}=0$ in the continuous model. The particle does the Bloch oscillation in the range $E_{-}\le n\omega \le E_{+}$, which is depicted as the shaded region.