Converging Perturbative Solutions of the Schroedinger Equation for a Two-Level System with a Hamiltonian Depending Periodically on Time

TL;DR

This paper develops a convergent perturbative approach for solving the Schrödinger equation in two-level quantum systems with periodic time-dependent Hamiltonians, avoiding secular terms and ensuring uniform convergence.

Contribution

It introduces a new method for solving the Schrödinger equation that guarantees convergence and avoids secular terms, improving upon existing expansion techniques.

Findings

Proves absolute and uniform convergence of Fourier series for wave functions.

Establishes convergence of epsilon-expansions for secular frequency and Fourier coefficients.

Provides a robust perturbative solution method for periodically driven two-level systems.

Abstract

We study the Schroedinger equation of a class of two-level systems under the action of a periodic time-dependent external field in the situation where the energy difference 2epsilon between the free energy levels is sufficiently small with respect to the strength of the external interaction. Under suitable conditions we show that this equation has a solution in terms of converging power series expansions in epsilon. In contrast to other expansion methods, like in the Dyson expansion, the method we present is not plagued by the presence of ``secular terms''. Due to this feature we were able to prove absolute and uniform convergence of the Fourier series involved in the computation of the wave functions and to prove absolute convergence of the epsilon-expansions leading to the ``secular frequency'' and to the coefficients of the Fourier expansion of the wave function.