  We investigate both the quantum and classical dynamics of a non-Hermitian
system via a kicked rotor model with $\mathcal{PT}$ symmetry. For the quantum
dynamics, both the mean momentum and mean square of momentum exhibits the
staircase growth with time when the system parameter is in the neighborhood of
the $\mathcal{PT}$ symmetry breaking point. If the system parameter is very
larger than the $\mathcal{PT}$ symmetry breaking point, the accelerator mode
results in the directed spreading of the wavepackets as well as the ballistic
diffusion in momentum space. For the classical dynamics, the non-Hermitian
kicking potential leads to exponentially-fast increase of classical complex
trajectories. As a consequence, the imaginary part of trajectories
exponentially diffuses with time, while the real part exhibits the normal
diffusion. Our analytical prediction of the exponential diffusion of imaginary
momentum and its breakdown time is in good agreement with numerical results.
The quantum signature of the chaotic diffusion of the complex trajectories is
reflected by the dynamics of the out-of-time-order correlators (OTOC). In the
semiclassical regime, the rate the exponential increase of the OTOC is equal
with that of the exponential diffusion of complex trajectories.
