Effective and efficient resonant transitions in periodically modulated quantum systems

TL;DR

This paper develops a method to classify and analyze resonant transitions in periodically modulated quantum systems with $SU(2)$ and $SU(1,1)$ symmetries, offering explicit procedures for both weak and strong modulations.

Contribution

It introduces a Lie transformation-based approach to identify effective resonant transitions and provides an iterative method to determine interaction constants for various resonances.

Findings

Classified all effective resonant transitions in modulated quantum systems.

Proposed explicit iterative procedure for determining interaction constants.

Identified resonant transitions in coupled systems due to modulation and nonlinearities.

Abstract

We analyse periodically modulated quantum systems with $SU(2)$ and $SU(1,1)$ symmetries. Transforming the Hamiltonian into the Floquet representation we apply the Lie transformation method, which allows us to classify all effective resonant transitions emerging in time-dependent systems. In the case of a single periodically perturbed system, we propose an explicit iterative procedure for the determination of the effective interaction constants corresponding to every resonance both for weak and strong modulation. For coupled quantum systems we determine the efficient resonant transitions appearing as a result of time modulation and intrinsic non-linearities.