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

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 $\mathrm{\Delta }{\theta }_H$ and Berry phase satisfy precisely the quantum-classical correspondence,${\gamma }_n=$ $(n+1\mathrm{/}2)\mathrm{\Delta }{\theta }_H$.