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Physical Review A

Higher-order adiabatic approximations for classical evolution

Zhaoyan Wu

Published Dec 1, 1989• 7 citations
Publisher

Abstract

The higher-order adiabatic approximation method of solving the Schr\"odinger equation for slowly transported quantum systems, which contains Berry's improved adiabatic theorem as its zeroth-order step, is generalized to solve evolution equations for a wide class of slowly transported classical systems. It is shown how the adiabatic condition is broken and transitions between different modes take place. Selection rules for such transitions are shown to be a topological property of the evolution path in parameter space.

DOI
10.1103/physreva.40.6852

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The higher-order adiabatic approximation method of solving the Schr"odinger equation for slowly transported quantum systems, which contains Berry's improved adiabatic theorem as its zeroth-order step, is generalized to solve evolution equations for a wide class of slowly transported classical systems. It is shown how the adiabatic condition is broken and transitions between different modes take place. Selection rules for such transitions are shown to be a topological property of the evolution path in parameter space.
We introduce a perturbative approach to solving the time-dependent Schr\"odinger equation, named adiabatic perturbation theory (APT), whose zeroth-order term is the quantum adiabatic approximation. The small parameter in the power series expansion of the time-dependent wave function is the inverse of the time it takes to drive the system's Hamiltonian from the initial to its final form. We review other standard perturbative and nonperturbative ways of going beyond the adiabatic approximation, extending and finding exact relations among them, and also compare the efficiency of those methods against the APT. Most importantly, we determine APT corrections to the Berry phase by use of the Aharonov-Anandan geometric phase. We then solve several time-dependent problems, allowing us to illustrate that the APT is the only perturbative method that gives the right corrections to the adiabatic approximation. Finally, we propose an experiment to measure the APT corrections to the Berry phase and show, for a particular spin-1∕21∕21∕2 problem, that to first order in APT the geometric phase should be two and a half times the (adiabatic) Berry phase.
In this paper high-order adiabatic approximate method is proposed to study the evolution of degenerate quantum systums with slowly-changing parameters and varying symmetry. We not only prove the quantum adiabatic theorem for degenerate cases and discuss the corresponding non-abelian induced gauge structure, but also point out the non-adiabatic corrections resulting from higher-order approximations. With nuclear quadrupole resonance as an explicit example, we use this method to analyse observable effects of non-abelian Berry's phases in adiabatic and non-adiabatic experimental process.
In this paper the evolution of a quantum system driven by a non-Hermitian Hamiltonian depending on slowly-changing parameters is studied by building an universal high-order adiabatic approximation (HOAA) method with Berry's phase, which is valid for either the Hermitian or the non-Hermitian cases. This method can be regarded as a non-trivial generalization of the HOAA method for closed quantum system presented by this author before. In a general situation, the probabilities of adiabatic decay and non-adiabatic transitions are explicitly obtained for the evolution of the non-Hermitian quantum system. It is also shown that the non-Hermitian analog of the Berry's phase factor for the non-Hermitian case just enjoys the holonomy structure of the dual linear bundle over the parameter manifold. The non-Hermitian evolution of the generalized force harmonic oscillator is discussed as an illustrative example.