Pseudo PT-symmetry in time periodic non-Hermitian Hamiltonians systems

We investigate the concept of the pseudo-parity-time (pseudo-PT) symmetry in periodic quantum systems. This pseudo parity-time symmetry manifests itself dynamically in the framework of the non-unitary evolution (Floquet) operator U(τ)=e−iLτ, over a period τ, which shows that the stability of the dynamics occurs when the PT-symmetry (or pseudo-PT) of the time-independent non-Hermitian Hamiltonian L is unbroken i.e. its quasienergies En are real. Nevertheless, when the PT-symmetry of the non-Hermitian Hamiltonian L is broken, which corresponds to the complex conjugate quasienergies En, an instable dynamics arises. We investigate in greater detail a harmonic oscillator with imaginary time-dependent periodic driving term linear in x. The Floquet operator for the modulated system is pseudo-PT symmetric if the relative phase ϕ of the applied mode is not 0 or π.