The non-Hermitian Hamiltonian for periodically driven harmonic oscillator and classical-quantum correspondence
Abstract
<p indent="0mm">By constructing a non-unitary but Hermitian transformation operator, we prove that the non-Hermitian Hamiltonian for a periodically driven harmonic oscillator is a pseudo-Hermitian Hamiltonian, characterized by real eigenvalues. For the time-dependent non-Hermitian Hamiltonian, both the metric and transformation operators are shown to be time-dependent. Analytic quantum wave functions of the corresponding Hermitian Hamiltonian are obtained from the dual Schrödinger equations respectively for the “bra” and “ket” states. Moreover, the classical correspondence of the non-Hermitian Hamiltonian is revealed through classical canonical transformations. The relation between quantum LR phase and classical angle is found in one period of the driving field. By analyzing the non-Hermitian Hamiltonian of periodic driven harmonic oscillator, it is concluded while a Hermitian Hamiltonian is sufficient for having real eigenvalues, it is not a necessary condition. The Hermitian counterpart of the non-Hermitian Hamiltonian can be obtained by a generalized gauge transformation that utilizes a non-unitary, but Hermitian transformation operator. The results of this work pave the way for new explorations into the non-Hermitian Hamiltonian system.</p>
