周期驱动谐振子的一种非厄米哈密顿和经典-量子对应

The non-Hermitian Hamiltonian for periodically driven harmonic oscillator and classical-quantum correspondence

本文构造一非幺正厄米变换算符和广义规范变换,定义了非厄米哈密顿算符双正交基矢的度规算符,证明了周期驱动谐振子的一种非厄米哈密顿是赝厄米哈密顿算符;从双含时薛定谔方程出发将其厄米化,从而得到其解析量子波函数;基于经典正则变换,得到非厄米哈密顿的经典对应;将系统演化一个周期,得到量子LR相和经典角变化率,以及它们之间的对应关系.本文通过对周期驱动谐振子非厄米哈密顿的分析,得出结论:哈密顿算符的厄米性是其有实数本征值的充分,但非必要条件,非厄米哈密顿算符都可以通过构造一非幺正的厄米变换算符和广义规范变换得到其厄米对应,此结论为研究非厄米系统提供了一种新思路.

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 timedependent.Analytic quantum wave functions of the corresponding Hermitian Hamiltonian are obtained from the dual Schrodinger 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.