Geometrical Amplitude factors in the the adiabatic evolution

TL;DR

This paper introduces the concept of geometric amplitude factors in adiabatic quantum evolution, extending the understanding of phase and amplitude contributions, especially in systems with imaginary eigenvalues, exemplified by a generalized harmonic oscillator.

Contribution

It presents a novel derivation of geometric amplitude factors in adiabatic evolution, including systems with imaginary eigenvalues, and demonstrates their effect using a generalized harmonic oscillator.

Findings

Additional purely geometric amplitude factors arise in cyclic adiabatic evolution.

The concept is extended to systems with imaginary eigenvalues.

Application to a generalized harmonic oscillator illustrates the theoretical findings.

Abstract

In a quantum system initially in the n-th eigenstate, an adiabatic evolution of the Hamiltonian ensures that the system remains in the corresponding instantaneous eigenstate while acquiring a phase factor. This phase has two components: one resulting from standard time evolution and another associated with the dependence of the eigenstate on the varying Hamiltonian, known as the Berry phase. In this work, we explore the concept of geometric amplitudes in the context of a Hermitian Hamiltonian with imaginary eigenvalues. We introduce the notion of geometric amplitude and provide a novel derivation of this concept. Our study reveals that a system undergoing cyclic evolution under adiabatic conditions acquires an additional amplitude factor of purely geometric origin. To illustrate this idea, we apply it to a concrete case: a generalized harmonic oscillator.