On the validity of many-mode Floquet theory with commensurate frequencies

TL;DR

This paper investigates the validity of many-mode Floquet theory for systems with commensurate frequencies, clarifying its correct application and spectral interpretation for quantum control engineering.

Contribution

It demonstrates the conditions under which many-mode Floquet theory accurately describes time evolution and clarifies the physical meaning of the Floquet Hamiltonian spectrum.

Findings

Floquet theory correctly predicts unitary evolution for commensurate frequencies.

Eigenvalues and eigenvectors require careful interpretation.

Spectrum analysis reveals physically relevant and irrelevant parts.

Abstract

Many-mode Floquet theory [T.-S. Ho, S.-I. Chu, and J. V. Tietz, Chem. Phys. Lett., v. 96, 464 (1983)] is a technique for solving the time-dependent Schr\"odinger equation in the special case of multiple periodic fields, but its limitations are not well understood. We show that for a Hamiltonian consisting of two time-periodic couplings of commensurate frequencies (integer multiples of a common frequency), many-mode Floquet theory provides a correct expression for unitary time evolution. However, caution must be taken in the interpretation of the eigenvalues and eigenvectors of the corresponding many-mode Floquet Hamiltonian, as only part of its spectrum is directly relevant to time evolution. We give a physical interpretation for the remainder of the spectrum of the Hamiltonian. These results are relevant to the engineering of quantum systems using multiple controllable periodic fields.