Exact-WKB Analysis of Two-level Floquet Systems

TL;DR

This work develops an exact-WKB framework for two-level Floquet systems to compute the quasi-energy $\epsilon$ and Floquet effective Hamiltonian $H_{\rm eff}$ from the one-period monodromy $M$, expressing these quantities through Voros cycle integrals that encode monodromy data. By transforming the time-dependent Schrödinger problem into a second-order equation and employing Borel-resummed WKB wave functions with Airy-type and degenerate Stokes-curve connections, the authors construct the median-resummed monodromy $M_{\rm med}$ and extract $\epsilon$ from $\det(M_{\rm med}-e^{i\epsilon T/\hbar})=0$, while $U(T)$ yields $H_{\rm eff}$ via $H_{\rm eff} = \frac{i\hbar}{T}\log U(T)$. The approach reveals perturbative (adiabatic) contributions plus nonperturbative corrections of the form $e^{-\beta/\hbar\omega}$, which lift quasi-energy degeneracies and produce resonant transitions; these corrections are validated against numerical solutions and shown to capture resonance frequencies and oscillation dynamics in the low-frequency regime. The paper also systematically extends the method to general Floquet systems with multiple turning-point pairs and provides explicit formulas for the nonperturbative corrections through cycle data, paving the way for applications to multi-level driven quantum systems. The results demonstrate that exact-WKB offers a principled, nonperturbative toolkit for analyzing slow Floquet dynamics with quantitative accuracy and clear physical interpretation of resonance phenomena.

Abstract

We explore the application of the exact Wentzel-Kramers-Brillouin (WKB) analysis to two-level Floquet systems and establish a systematic procedure to calculate the quasi-energy and Floquet effective Hamiltonian. We show that, in the exact-WKB analysis, the quasi-energy and Floquet effective Hamiltonian can be expressed in terms of cycle integrals (Voros symbol), which characterize monodromy matrices for Schrödinger-type differential equations governing two-level Floquet systems. We analytically evaluate the cycle integrals using the low-frequency expansion and derive both perturbative and non-perturbative corrections to the quasi-energy and Floquet effective Hamiltonian. To verify the accuracy of our results, we compare them with numerical calculations and analyze resonant oscillations, which reveal non-perturbative features that cannot be captured by the perturbative expansion.

Figures (24)

• Figure 1: Stokes graph for $Q(t) = at$. The wavy line represents the branch cut of $W_0(t)$. The signs ($\pm$) attached to each Stokes curve indicate whether $W_0(t)/\hbar$ is increasing ($+$) or decreasing ($-$) along that line towards infinity $|t| \rightarrow \infty$.
• Figure 2: Integration contours for Airy functions. The integral converges as long as the contours extend to the infinity in the unshaded region.
• Figure 3: Stokes graph for the Landau-Zener problem.
• Figure 4: Stokes graph for $\arg \hbar > 0$
• Figure 5: Stokes graph for $\arg \hbar < 0$