Hidden time-nonlocal Floquet symmetries

TL;DR

The paper addresses why exact crossings occur in the Floquet spectrum of a detuned driven two-level system by uncovering a hidden time-nonlocal parity, valid at integer detunings $\epsilon = n\Omega$. It constructs a symmetry operator $J(t)=Q(t)P$ with a Liouville-type equation for $Q(t)$ and uses anti-unitary symmetries to derive a finite Fourier-content solution via a break condition, yielding a parity-based decomposition of the Floquet spectrum. The authors verify the mechanism with numerical Floquet spectra, showing exact crossings between modes of opposite parity, and they present an independent numerical scheme to compute the symmetry operator from Floquet modes, demonstrating the method’s generality. The findings offer a principled explanation for degeneracies in driven quantum systems and provide practical tools to identify hidden time-nonlocal symmetries in broader Floquet models, with potential implications for controllability and robust quantum dynamics.

Abstract

We investigate the Floquet spectrum of a detuned, driven two-level system and show that it exhibits exact quasienergy crossings when the detuning is an integer multiple of the energy quantum of the driving field. This behavior can be explained by a hidden time-nonlocal parity, which allows the Floquet modes to be classified as even or odd. Then a generic feature is the emergence of exact crossings between quasienergies of different parity. A constructive proof of the existence of the symmetry is based on a scalar recurrence relation. Moreover, we present a general scheme for its numerical computation, which can be applied to models beyond the two-level system. Analytical results are illustrated with numerical data.

Figures (3)

• Figure 1: Minimal quasienergy splitting as a function of the detuning and the driving amplitude for tunnel coupling $\beta=1.3\,\Omega$. The color scale is chosen such that regions with particularly small values are highlighted. The vertical dashed lines mark detunings that are integer multiples of the driving frequency, $\epsilon = n\Omega$.
• Figure 2: Floquet spectrum of the driven two-level system as a function of the driving amplitude $\alpha$ for tunneling matrix element $\beta=2.7\,\Omega$ and detuning $\epsilon=\Omega$ (i.e., $n=1$). The gray box marks the first Brillouin zone. The parity $j=\pm 1$, derived below, is indicated by the color---red for $+1$ and blue for $-1$. Equivalent states in neighboring Brillouin zones have opposite parity. The zooms indicate that only quasienergies with the same parity form avoided crossings, while those with different parity cross exactly.
• Figure 3: Floquet spectra as a function of the driving amplitude for various integer detunings $\epsilon = n\Omega$. The tunnel matrix element $\beta = 2.7\Omega$ and the color code of the parity are as in Fig. \ref{['fig:spectrum']}. As there, some crossings of levels with equal parity appear exact, but in fact are narrowly avoided.