Floquet analysis of a quantum system with modulated periodic driving

TL;DR

This paper develops a formalism for analyzing quantum systems with periodic driving whose strength varies slowly, introducing a time-dependent Floquet Hamiltonian and micromotion operators, and demonstrating applications to spin systems and cold atom setups.

Contribution

It generalizes Floquet theory to include slowly varying driving parameters, deriving a systematic high-frequency expansion for the effective Hamiltonian and micromotion operators.

Findings

Effective Hamiltonian captures long-term dynamics with slow parameter changes.

Micromotion operators exhibit both rapid oscillations and slow temporal dependence.

Application to spin in oscillating magnetic field reveals non-Abelian geometric phases.

Abstract

We consider a quantum system periodically driven with a strength which varies slowly on the scale of the driving period. The analysis is based on a general formulation of the Floquet theory relying on the extended Hilbert space. It is shown that the dynamics of the system can be described in terms of a slowly varying effective Floquet Hamiltonian that captures the long-term evolution, as well as rapidly oscillating micromotion operators. We obtain a systematic high-frequency expansion of all these operators. Generalizing the previous studies, the expanded effective Hamiltonian is now time-dependent and contains extra terms appearing due to changes in the periodic driving. The same applies to the micromotion operators which exhibit a slow temporal dependence in addition to the rapid oscillations. As an illustration, we consider a quantum-mechanical spin in an oscillating magnetic field with a slowly changing direction. The effective evolution of the spin is then associated with non-Abelian geometric phases reflecting the geometry of the extended Floquet space. The developed formalism is general and also applies to other periodically driven systems, such as shaken optical lattices with a time-dependent shaking strength, a situation relevant to the cold atom experiments.