Floquet Schrieffer-Wolff transform based on Sylvester equations

TL;DR

The paper introduces a Floquet Schrieffer-Wolff transform (FSWT) that systematically derives effective Hamiltonians for periodically driven many-body systems, surpassing traditional high-frequency expansion methods by solving operator Sylvester equations.

Contribution

It develops a novel perturbative method (FSWT) based on Sylvester equations to analyze non-resonant driven quantum systems, extending Floquet theory beyond high-frequency approximations.

Findings

FSWT reduces to Floquet-Magnus at high frequencies.

Demonstrated on driven Fermi-Hubbard model.

Potential applications in quantum gate design and quantum simulations.

Abstract

We present a Floquet Schrieffer Wolff transform (FSWT) to obtain effective Floquet Hamiltonians and micro-motion operators of periodically driven many-body systems for any non-resonant driving frequency. The FSWT perturbatively eliminates the oscillatory components in the driven Hamiltonian by solving operator-valued Sylvester equations with systematic approximations. It goes beyond various high-frequency expansion methods commonly used in Floquet theory, as we demonstrate with the example of the driven Fermi-Hubbard model. In the limit of high driving frequencies, the FSWT Hamiltonian reduces to the widely used Floquet-Magnus result. We anticipate this method will be useful for designing Rydberg multi-qubit gates, controlling correlated hopping in quantum simulations in optical lattices, and describing multi-orbital and long-range interacting systems driven in-gap.