Extended degenerate perturbation theory for the Floquet-Hilbert space

TL;DR

This paper introduces an extended degenerate perturbation theory (EDPT) for periodically driven systems, improving quasienergy spectrum accuracy in resonant conditions by including all Floquet zone levels, balancing computational efficiency and precision.

Contribution

The paper develops EDPT, an extension of DPT that incorporates all Floquet zone levels, offering a more accurate and computationally feasible method for analyzing resonant driven quantum systems.

Findings

EDPT yields more accurate quasienergy spectra than conventional DPT.

The computational complexity of EDPT is intermediate between DPT and exact methods.

Application to a driven Bose-Hubbard model demonstrates the effectiveness of EDPT.

Abstract

We consider construction of effective Hamiltonians for periodically driven interacting systems in the case of resonant driving. The standard high-frequency expansion is not expected to converge due to the resonant creation of collective excitations, and one option is to resort to the application of degenerate perturbation theory (DPT) in the Floquet-Hilbert space. We propose an extension of DPT whereby the degenerate subspace includes not only the degenerate levels of interest but rather all levels in a Floquet zone. The resulting approach, which we call extended DPT (EDPT), is shown to resemble a high-frequency expansion, provided the quasienergy matrix is constructed such that each $m$th diagonal block contains energies reduced to the $m$th Floquet zone. The proposed theory is applied to a driven Bose-Hubbard model and is shown to yield more accurate quasienergy spectra than the conventional DPT. The computational complexity of EDPT is intermediate between DPT and the numerically exact approach, thus providing a practical compromise between accuracy and efficiency.