  Dynamical behavior of vortices plays central roles in the quantum phenomena
of two-dimensional (2D) superconductors. Quantum metallic state, for example,
showing an anomalous temperature-independent resistive state down to
low-temperatures, has been a common subject in recently developed 2D
crystalline superconductors, whose microscopic origin is still under debate.
Here, we unveil a new aspect of the vortex dynamics in a noncentrosymmetric 2D
crystalline superconductor of MoS$_{2}$ through the nonreciprocal transport
measurement. The second harmonic resistance $R^{2w}$ at low temperature with
high current indicates the classical vortex flow accompanying the ratchet
motion. Furthermore, we found that $R^{2w}$ is substantially suppressed in the
quantum metallic state with low current region, allowing identification of the
quantum and classical ratchet motions of vortices by the magnitude of the
second harmonic generation. This suggests that nonreciprocal transport
measurement can be a powerful tool to probe the vortex dynamics in
noncentrosymmetric 2D superconductors.
