  We investigate the directed momentum current in the quantum kicked rotor
model with $\mathcal{PT}$ 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 \textbf{a}
threshold value, which demonstrates the appearance of the spontaneous
$\mathcal{PT}$ symmetry breaking. In the vicinity of the phase-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 phase-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.
