Scaling theory of a quantum ratchet

The asymmetric responses of the system between the external force of the right and left directions are called ``nonreciprocal.'' There are many examples of nonreciprocal responses, such as the rectification by the $p\text{\ensuremath{-}}n$ junction. However, the quantum-mechanical wave does not distinguish between the right and the 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 $\ensuremath{\alpha}$ plays an essential role for nonlinear nonreciprocal response. The temperature ($T$) dependence of the second-order nonlinear mobility ${\ensuremath{\mu}}₂$ is found to be ${\ensuremath{\mu}}₂\ensuremath{\sim}{T}^{(6/\ensuremath{\alpha})\ensuremath{-}4}$ for $\ensuremath{\alpha}<1$, and ${\ensuremath{\mu}}₂\ensuremath{\sim}{T}^{2(\ensuremath{\alpha}\ensuremath{-}1)}$ for $\ensuremath{\alpha}>1$, respectively, where ${\ensuremath{\alpha}}_c=1$ is the critical point of the localization-delocalization transition, i.e., Schmid transition. On the other hand, ${\ensuremath{\mu}}₂$ shows the behavior ${\ensuremath{\mu}}₂\ensuremath{\sim}{T}^{\ensuremath{-}11/4}$ in the high-temperature limit. Therefore, ${\ensuremath{\mu}}₂$ 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.