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Physical Review A

Driven harmonic oscillators in the adiabatic Magnus approximation

Serge Klarsfeld, José Ángel Oteo

Published Mar 1, 1993• 6 citations
Publisher

Abstract

The time evolution of driven harmonic oscillators is determined by applying the Magnus expansion in the basis set of instantaneous eigenstates of the total Hamiltonian. It is shown that the first-order approximation already provides transition probabilities close to the exact values even in the intermediate regime.
DOI
10.1103/physreva.47.1620

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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.
The temporal evolution of the state vector relative to a harmonic oscillator with time-dependent frequency is examined. The Schrödinger equation is solved by choosing the instantaneous eigenstates of the Hamiltonian as the basis, thus getting an infinite set of coupled linear differential equations. This formulation is particularly suitable for studying the cases in which the Hamiltonian undergoes a very slow or a sudden variation from an initial constant value into a final one. A rigorous proof of adiabatic invariance to all orders in the slowness parameter is given for the transition probabilities. An application to the evolution of an initially coherent state is made.
We study a time-dependent harmonic oscillator based on the dynamics of instantaneous phonons, which have obvious physical meaning and direct experimental relevance. We find simple analytic solutions for an important class of evolution and identify two parameter-changing-rate regimes with qualitatively different oscillator behaviors. We show that rapid adiabatic processes are possible if the frequency and the mass of the oscillator change in opposite directions. The state vector in the Schrödinger picture is handily achieved by use of the eigenstates of the instantaneous phonon operators that are analytically known for arbitrary frequency and mass values.
The convergence of the Magnus expansion in the Schr\"odinger representation is investigated with the aid of two exactly solvable models. A baffling observation regarding the dynamics of a spin-(1/2 system driven by a superposition of a constant and a rotating magnetic field is elucidated. Perturbation theory is applied to the exponential solution to the Schr\"odinger equation for a time-dependent harmonic oscillator. The first terms of the perturbation expansion for the singularity of the exponent are exactly calculated. The scope and limitations of perturbation theory in obtaining the range of validity of the Magnus expansion are discussed.