Exact Floquet solutions of quantum driven systems

TL;DR

This paper introduces a new method to obtain exact Floquet solutions for periodically driven quantum systems, providing analytical wave functions and Berry phases for specific models, with potential applications in mathematics.

Contribution

A novel approach combining Floquet theorem and unitary transformation to find exact solutions of time-dependent Schrödinger equations for driven systems.

Findings

Exact Floquet solutions for linear potential, harmonic oscillator, and coupled harmonic oscillator models.

Analytic expressions for quasienergy and Berry phase in the harmonic oscillator model.

Method applicable to solving partial differential equations in mathematics.

Abstract

How to accurately solve time-dependent Schr\"odinger equation is an interesting and important problem. Here, we propose a novel method to obtain the exact Floquet solutions of the Schr\"odinger equation for periodically driven systems by using Floquet theorem and a time-dependent unitary transformation. Using the method, we give out the exact Floquet solutions of wave function for three interesting physical models -- linear potential model, harmonic oscillator model, and the coupled harmonic oscillator model in the presence of a periodic driving. In addition to the quasienergy, we also give out the analytic expression of Berry phase for the harmonic oscillator model. Moreover, the idea presented in this paper can be used in mathematics to solve partial differential equations.