We investigate the directed momentum current in the quantum kicked rotor model with PT symmetric deriving potential. For the quantum non-resonance case, the values of quasi-energy become to be complex when the strength of imaginary part of the kicking potential exceeds a threshold value, which demonstrates the appearance of the spontaneous PT symmetry breaking. In the vicinity of the transition point, the momentum current exhibits a staircase growth with time. Each platform of the momentum current corresponds to the mean momentum of some eigenstates of the Floquet operator whose imaginary parts of the quasi-energy are significantly large. Above the transition point, the momentum current increases linearly with time. Interestingly, its acceleration rate exhibits a kind of "quantized" increment with the kicking strength. We propose a modified classical acceleration mode of the kicked rotor model to explain such an intriguing phenomenon. Our theoretical prediction is in good agreement with numerical results.

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We investigate both the classical and quantum dynamics of a kicked particle with P T symmetry. In chaotic situation, the mean energy of the real parts of momentum linearly increases with time, and that of the imaginary momentum exponentially increases. There exists a breakdown time for chaotic diffusion, which is obtained both analytically and numerically. The quantum diffusion of this non-Hermitian system follows the classically chaotic diffusion of Hermitian case during the Ehrenfest time, after which it is completely suppressed. Interestingly, the Ehrenfest time decreases with the decrease of effective Planck constant or the increase of the strength of the non-Hermitian kicking potential. The exponential growth of the quantum out-of-time-order correlators (OTOC) during the initially short time interval characterizes the feature of the exponential diffusion of imaginary trajectories. The long time behavior of OTOC reflects the dynamical localization of quantum diffusion. The dynamical behavior of inverse participation ratio can quantify the P T symmetry breaking, for which the rule of the phase transition points is numerically obtained.

We investigate the quantum-classical correspondence of the Bose–Einstein condensate (BEC) which is described by the Gross–Pitaevski equation with the periodic, both temporally and spatially, modulation of nonlinear interaction. It is found that this system is equivalent to a generalized kicked rotor (GKR) model. Interestingly, the classical dynamics of the GKR system exhibits the regular-to-chaotic transition as time evolves, and the corresponding energy diffusion is exponentially fast. For the BEC system, the energy diffusion can be predicted by the classical mapping equation of the GKR model. However, the ratio of the energy between the BEC system and the classical GKR model is not unity, even if the effective Planck constant is very small, which is contradictory to our conventional understanding of quantum-classical correspondence. We find that the BEC system is not exactly a quantum counterpart of the classical GKR system. We construct a better quantum counterpart and derive an analytical expression for the exponentially-fast diffusion of energy, which is in perfect agreement with numerical results.