  The asymmetric responses of the system between the external force of right
and left directions are called "nonreciprocal". There are many examples of
nonreciprocal responses such as the rectification by p-n junction. However, the
quantum mechanical wave does not distinguish between the right and left
directions as long as the time-reversal symmetry is intact, and it is a highly
nontrivial issue how the nonreciprocal nature originates in quantum systems.
Here we demonstrate by the quantum ratchet model, i.e., a quantum particle in
an asymmetric periodic potential, that the dissipation characterized by a
dimensionless coupling constant $\alpha$ plays an essential role for nonlinear
nonreciprocal response. The temperature ($T$) dependence of the second order
nonlinear mobility $\mu_2$ is found to be $\mu_2 \sim T^{6/\alpha -4 }$ for
$\alpha<1$, and $\mu_2 \sim T^{2(\alpha -1)}$ for $\alpha>1$, respectively,
where $\alpha_c=1$ is the critical point of the localization-delocalization
transition, i.e., Schmid transition. On the other hand, $\mu_2$ shows the
behavior $\mu_2 \sim T^{-11/4}$ in the high temperature limit. Therefore,
$\mu_2$ shows the nonmonotonous temperature dependence corresponding to the
classical-quantum crossover. The generic scaling form of the velocity $v$ as a
function of the external field $F$ and temperature $T$ is also discussed. These
findings are relevant to the heavy atoms in metals, resistive superconductors
with vortices and Josephson junction system, and will pave a way to control the
nonreciprocal responses.
