Piecewise Adiabatic Following: General Analysis and Exactly Solvable Models

TL;DR

This paper explores the adiabatic following dynamics in non-Hermitian, periodically driven quantum systems, revealing conditions for stable cyclic states to follow or switch between eigenstates, supported by exact solutions and universal analysis.

Contribution

It provides a systematic analysis of piecewise adiabatic following in non-Hermitian systems, including the discovery of a universal mechanism based on the sign change of a critical exponent.

Findings

Stable cyclic states can follow instantaneous eigenstates or exhibit sudden switches.

The sign change of a critical exponent determines the adiabatic behavior.

The work extends previous studies with exact solutions and broad analysis.

Abstract

The dynamics of a periodically driven system whose time evolution is governed by the Schr\"{o}dinger equation with non-Hermitian Hamiltonians can be perfectly stable. This finding was only obtained very recently and will be enhanced by many exact solutions discovered in this work. The main concern of this study is to investigate the adiabatic following dynamics in such non-Hermitian systems stabilized by periodic driving. We focus on the peculiar behaviour of stable cyclic (Floquet) states in the slow-driving limit. It is found that the stable cyclic states can either behave as intuitively expected by following instantaneous eigenstates, or exhibit piecewise adiabatic following by sudden-switching between instantaneous eigenstates. We aim to cover broad categories of non-Hermitian systems under a variety of different driving scenarios. We systematically analyse the sudden-switch behaviour by a universal route. That is, the sign change of the critical exponent in our asymptotic analysis of the solutions is always found to be the underlying mechanism to determine if the adiabatic following dynamics is trivial or piecewise. This work thus considerably extends our early study on the same topic [Gong and Wang, Phys. Rev. A 97, 052126 (2018)] and shall motivate more interests in non-Hermitian systems.