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$PT$-symmetric non-Hermitian Hamiltonian and invariant operator in periodically driven $SU(1,1)$ system

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 scheme to construct the non-Hermitian invariant with a $PT$-symmetric but non-unitary transformation operator. 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 an consequence of the non-unitarity the ket ($|ψ(t)\rangle $) and bra ($\langle 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.