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We study in this paper the time evolution of $PT$-symmetric non-HermitianHamiltonian consisting of periodically driven $SU(1,1)$ generators. Anon-Hermitian invariant operator is adopted to solve the Schr\"{o}dingerequation, since the time-dependent Hamiltonian is no longer a conservedquantity. We propose a scheme to construct the non-Hermitian invariant with a$PT$-symmetric but non-unitary transformation operator. The eigenstates ofinvariant and its complex conjugate form a bi-orthogonal basis to formulate theexact solution. We obtain the non-adiabatic Berry phase, which reduces to theadiabatic one in the slow time-variation limit. A non-unitary time-evolutionoperator is found analytically. As an consequence of the non-unitarity the ket($|\psi (t)\rangle $) and bra ($\langle \psi (t)|$) states are not normalizedeach other. While the inner product of two states can be evaluated with thehelp of a metric operator. It is shown explicitly that the model can berealized by a periodically driven oscillator.

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