[formula omitted]-symmetric non-Hermitian Hamiltonian and invariant operator in periodically driven [formula omitted] system

Abstract

We study in this paper the time evolution of PT-symmetric non-Hermitian Hamiltonian consisting of periodically driven SU(1,1) generators. A non-Hermitian invariant operator is adopted to solve the Schrödinger equation, since the time-dependent Hamiltonian is no longer a conserved quantity. We propose a PT-symmetric but non-unitary transformation operator in the construction of the non-Hermitian invariant. The eigenstates of invariant and its complex conjugate form a bi-orthogonal basis to formulate the exact solution. We obtain the non-adiabatic Berry phase, which reduces to the adiabatic one in the slow time-variation limit. A non-unitary time- evolution operator is found analytically. As a consequence of the non-unitarity the ket (|ψ(t)〉) and bra (〈ψ(t)|) states are not normalized each other. While the inner product of two states can be evaluated with the help of a metric operator. It is shown explicitly that the model can be realized by a periodically driven oscillator.