Abstract
We analyze 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.
Highlights
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The aim of the present paper is to provide a systematic approach to the analysis of effective resonant transitions in quantum systems obeying the SU (2), SU (1, 1), and
We construct the Lie-type all-order perturbation theory allowing to determine the order of every possible resonance that may emerge in the effective Hamiltonians. We consider both single and coupled quantum systems and determine the efficient resonant transitions emerging as a combination of the time modulation and intrinsic non-linearities, especially relevant in interacting systems
Summary
The most famous examples of time-dependent systems with an infinite number of effective resonances are the Rabi model in classical field [1,2,3] and the parametric quantum oscillator [24] Even in these simplest systems, where the Hamiltonian is a linear form on the SU (2) and SU (1, 1) Lie algebras, it turns out that the general expressions for the effective interaction constants and the frequency shifts in the vicinity of each resonance, are not easy to obtain. We consider both single and coupled quantum systems and determine the efficient resonant transitions emerging as a combination of the time modulation and intrinsic non-linearities, especially relevant in interacting systems.
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