Quasi-parity-time symmetric dynamics in periodically driventwo-level non-Hermitian system

In recent years, there have been intensive studies of non-Hermitian physics and parity-time (PT) symmetrydue to their fundamental importance in theory and outstanding applications. A distinctive character in PT-symmetric system is phase transition (spontaneous PT-symmetry breaking), i.e. an all-real energy spectrumchanges into an all-complex one when the non-Hermitian parameter exceeds a certain threshold. However, theconditions for PT-symmetric system with real energy spectrum to occur are rather restrictive. Thegeneralization of PT-symmetric potentials to wider classes of non-PT-symmetric complex potentials with all-realenergy spectra is a currently important endeavor. A simple PT-symmetric two-level Floquet quantum system isnow being actively explored, because it holds potential for the realization of non-unitary single-qubit quantumgate. However, studies of the evolution dynamics of non-PT-symmetric two-level non-Hermitian Floquetquantum system are still relatively rare. In this paper, we investigate the non-Hermitian physics of a periodically driven non-PT-symmetric two-level quantum system. By phase-space analysis, we find that there exist so-called pseudo fixed points in phasespace representing the Floquet solutions with fixed population difference and a time-dependent relative phasebetween the two levels. According to these pseudo fixed points, we analytically construct a non-unitaryevolution operator and then explore the dynamic behaviors of the non-PT-symmetric two-level quantum systemin different parameter regions. We confirm both analytically and numerically that the two-level non-HermitianFloquet quantum system, although it is non-parity-time-symmetric, still features a phase transition with thequasienergy spectrum changing from all-real to all-complex energy spectrum, just like the PT symmetricsystem. Furthermore, we reveal that a novel phenomenon called quasi-PT symmetric dynamics occurs in thetime evolution process. The quasi-PT symmetric dynamics is so named in our paper, in the sense that the time-evolution of population probabilities in the non-PT-symmetric two-level system satisfies fully the time-spacesymmetry (PT symmetry), while time-evolution of the quantum state (containing the phase) does not meetsuch a PT symmetry, due to the fact that time-evolution of the phases of the probability amplitudes on the twolevels violates the requirement for the PT symmetry.