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arXiv:1807.06755 · Annals of Physics

A Magnus approximation approach to harmonic systems with time-dependent frequencies

C.D. Fosco, F.C. Lombardo, F.D. Mazzitelli

Published Oct 25, 2018• 0 citations
arXivHTMLPDF
DOI
10.1016/j.aop.2018.10.016

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We derive a Magnus expansion for a frequency chirped quantum two-level system. We obtain a time-independent effective Hamiltonian which generates a stroboscopic time evolution. At lowest order the according dynamics is identical to results from using a rotating wave approximation. We determine, furthermore, also the next higher-order corrections within our expansion scheme in correspondence to the Bloch-Siegert shifts for harmonically driven systems. Importantly, our scheme can be extended to more complicated systems, i.e., even many-body systems.
We derive and discuss a formula, due to Magnus, for the exponential representation of the operator solution to Schrödinger's equation when the Hamiltonian is time dependent. The formula gives a unitary time-displacement operator in every order of approximation. We study the usefulness of the first- and second-order approximations for the kind of problem posed by the semiclassical theory of inelastic collisions, basing our discussion on two exactly soluble two-state problems. The algebraic structure of the Magnus formula is in itself useful; to illustrate this, we solve exactly the problems of the linearly forced harmonic oscillator and the harmonic oscillator with time-dependent force constant.
The Floquet-Magnus expansion is a widely used tool to derive effective descriptions of time-periodic quantum systems by approximating their dynamics with a time-independent Hamiltonian. However, its standard formulation is, strictly speaking, restricted to bounded Hamiltonians. In this work, we extend its definition and analysis to a broad class of time-periodic unbounded Hamiltonians. Our approach is based on an a priori distinct nonperturbative framework for the construction of effective Hamiltonians, which we show to reproduce the Floquet-Magnus expansion. A particular strength of our framework is that it allows us to prove that the resulting effective dynamics approximates the original time evolution propagators to arbitrary order in the high-frequency limit without requiring convergence of the Floquet-Magnus expansion, a condition that is already highly restrictive even in the bounded setting. We illustrate the scope of the method on representative models: the quantum Rabi Hamiltonian in the interaction picture, and the periodically driven quantum harmonic oscillator.