High-Order Adiabatic Approximation for Non-Hermitian Quantum System and Complexization of Berry's Phase

TL;DR

This paper develops a high-order adiabatic approximation method for non-Hermitian quantum systems, extending Berry's phase concept and providing explicit decay and transition probabilities, with applications to a generalized harmonic oscillator.

Contribution

It introduces a universal high-order adiabatic approximation method for non-Hermitian systems, generalizing previous approaches for closed systems and analyzing the complex Berry's phase structure.

Findings

Explicit probabilities for adiabatic decay and non-adiabatic transitions

Generalization of Berry's phase to non-Hermitian systems with holonomy structure

Application to a generalized forced harmonic oscillator

Abstract

In this paper the evolution of a quantum system drived 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 forced harmonic oscillator is discussed as an illustrative examples.