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 nonperturbative 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 nonperturbative features that cannot be captured by the perturbative expansion.

Japan

Toshiaki Fujimori

Syo Kamata

Tatsuhiro Misumi

Naohisa Sueishi

Hidetoshi Taya

Exact Wentzel–Kramers–Brillouin (WKB) Analysis of Two-Level Floquet Systems

Quantum deformed potentials arise naturally in quantum mechanical systems of one bosonic coordinate coupled to <math><msub><mi>N</mi><mi>f</mi></msub></math> Grassmann valued fermionic coordinates, or to a topological Wess-Zumino term. These systems decompose into sectors with a classical potential plus a quantum deformation. Using exact Wentzel-Kramers-Brillouin, we derive exact quantization condition and its median resummation. The solution of median resummed form gives physical Borel-Ecalle resummed results, as we show explicitly in quantum deformed double- and triple-well potentials. Despite the fact that instantons have finite actions, for generic quantum deformation, they do not contribute to the energy spectrum at leading order in semiclassics. For certain quantized quantum deformations, where the alignment of levels to all order in perturbation theory occurs, instantons contribute to the spectrum. If deformation parameter is not properly quantized, their effect disappears, but higher-order effects in semiclassics survive. In this sense, we classify saddle contributions as fading and robust. Finally, for quantum deformed triple-well potential, we demonstrate the perturbative/nonperturbative relation, by computing period integrals and Mellin transform.

Poland

United States

Mithat Ünsal

Exact WKB analysis for SUSY and quantum deformed potentials: Quantum mechanics with Grassmann fields and Wess-Zumino terms

We investigate the exact-WKB analysis for quantum mechanics in a periodic potential, with N minima on S 1. We describe the Stokes graphs of a general potential problem as a network of Airy-type or degenerate Weber-type building blocks, and provide a dictionary between the two. The two formulations are equivalent, but with their own pros and cons. Exact-WKB produces the quantization condition consistent with the known conjectures and mixed anomaly. The quantization condition for the case of N-minima on the circle factorizes over the Hilbert sub-spaces labeled by discrete theta angle (or Bloch momenta), and is consistent with ’t Hooft anomaly for even N and global inconsistency for odd N. By using Delabaere-Dillinger-Pham formula, we prove that the resurgent structure is closed in these Hilbert subspaces, built on discrete theta vacua, and by a transformation, this implies that fixed topological sectors (columns of resurgence triangle) are also closed under resurgence.

Exact-WKB, complete resurgent structure, and mixed anomaly in quantum mechanics on S 1

There are two well-known approaches to studying nonperturbative aspects of quantum mechanical systems: saddle point analysis of the partition functions in Euclidean path integral formulation and the exact-WKB analysis based on the wave functions in the Schrödinger equation. In this work, based on the quantization conditions obtained from the exact-WKB method, we determine the relations between the two formalism and in particular show how the two Stokes phenomena are connected to each other: the Stokes phenomenon leading to the ambiguous contribution of different sectors of the path integral formulation corresponds to the change of the “topology” of the Stoke curves in the exact-WKB analysis. We also clarify the equivalence of different quantization conditions including Bohr-Sommerfeld, path integral and Gutzwiller’s ones. In particular, by reorganizing the exact quantization condition, we improve Gutzwiller’s analysis in a crucial way by bion contributions (incorporating complex periodic paths) and turn it into an exact result. Furthermore, we argue the novel meaning of quasi-moduli integral and provide a relation between the Maslov index and the intersection number of Lefschetz thimbles.

China

On exact-WKB analysis, resurgent structure, and quantization conditions

Abstract This paper considers the  problem, which is the divergent behavior of the ground-state energy of the asymmetric potential in quantum mechanics, which is calculated with semi-classical expansion and the resurgence technique. Using the resolvent method, it is shown that including not only one complex bion but a multi-complex bion and multi-bounce contributions solves this problem. This result indicates the importance of summing all possible saddle-point contributions and also the relationship between the exact Wentzel–Kramers–Brillouin approximation and path-integral formalism.