# Scaling Theory of Quantum Ratchet

## TL;DR

This paper investigates the quantum ratchet model to understand how dissipation influences nonreciprocal responses, revealing temperature-dependent nonlinear mobility behaviors and a crossover from classical to quantum regimes.

## Contribution

It demonstrates the critical role of dissipation in quantum nonreciprocal responses and derives the temperature scaling of nonlinear mobility near the localization transition.

## Key findings

- Nonlinear mobility $rac{T^{6/lpha -4}}$ for $lpha<1$
- Nonlinear mobility $rac{T^{2(lpha -1)}}$ for $1alpha>1$
- High-temperature behavior $rac{1}{T^{11/4}}$

## Abstract

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.