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We investigate both the quantum and classical dynamics of a non-Hermitiansystem via a kicked rotor model with $\mathcal{PT}$ symmetry. For the quantumdynamics, both the mean momentum and mean square of momentum exhibits thestaircase growth with time when the system parameter is in the neighborhood ofthe $\mathcal{PT}$ symmetry breaking point. If the system parameter is verylarger than the $\mathcal{PT}$ symmetry breaking point, the accelerator moderesults in the directed spreading of the wavepackets as well as the ballisticdiffusion in momentum space. For the classical dynamics, the non-Hermitiankicking potential leads to exponentially-fast increase of classical complextrajectories. As a consequence, the imaginary part of trajectoriesexponentially diffuses with time, while the real part exhibits the normaldiffusion. Our analytical prediction of the exponential diffusion of imaginarymomentum and its breakdown time is in good agreement with numerical results.The quantum signature of the chaotic diffusion of the complex trajectories isreflected by the dynamics of the out-of-time-order correlators (OTOC). In thesemiclassical regime, the rate the exponential increase of the OTOC is equalwith that of the exponential diffusion of complex trajectories.

Chaotic diffusion of complex trajectory and its quantum signature