Generalized gauge transformation with $PT$-symmetric non-unitary operator and classical correspondence of non-Hermitian Hamiltonian for a periodically driven system

TL;DR

This paper demonstrates how a $PT$-symmetric non-Hermitian Hamiltonian for a periodically driven system can be derived from a Hermitian kernel Hamiltonian via a generalized gauge transformation, establishing a quantum-classical correspondence.

Contribution

It introduces a generalized gauge transformation framework for $PT$-symmetric non-Hermitian Hamiltonians and explores their classical counterparts and quantum-classical relations.

Findings

Analytical wave functions and Berry phases obtained for the system.

Classical Hamiltonian expressed as a complex function of canonical variables.

Quantum-classical correspondence between Berry phase and Hannay's angle established.

Abstract

We in this paper demonstrate that the $PT$-symmetric non-Hermitian Hamiltonian for a periodically driven system can be generated from a kernel Hamiltonian by a generalized gauge transformation. The kernel Hamiltonian is Hermitian and static, while the time-dependent transformation operator has to be $PT$ symmetric and non-unitary in general. Biorthogonal sets of eigenstates appear necessarily as a consequence of non-Hermitian Hamiltonian. We obtain analytically the wave functions and associated non-adiabatic Berry phase $\gamma_{n}$ for the $n$th eigenstate. The classical version of the non-Hermitian Hamiltonian becomes a complex function of canonical variables and time. The corresponding kernel Hamiltonian is derived with $PT$ symmetric canonical-variable transfer in the classical gauge transformation. Moreover, with the change of position-momentum to angle-action variables it is revealed that the non-adiabatic Hannay's angle $\Delta \theta_{H}$ and Berry phase satisfy precisely the quantum-classical correspondence,$\gamma_{n}=$ $(n+1/2)\Delta \theta_{H}$.