On the Small Mass Limit of Quantum Brownian Motion with Inhomogeneous Damping and Diffusion

TL;DR

This paper investigates the small mass limit of quantum Brownian motion with inhomogeneous damping, deriving a limiting equation that includes quantum noise-induced drift terms, revealing quantum effects absent in classical models.

Contribution

The paper derives a quantum stochastic limit for inhomogeneous quantum Brownian motion, revealing quantum noise-induced drifts in the small mass limit.

Findings

Derived the Heisenberg-Langevin equations for the model.

Established the limit as mass and Planck constant tend to zero with fixed ratios.

Identified quantum noise-induced drift terms in the limiting equation.

Abstract

We study the small mass limit (or: the Smoluchowski-Kramers limit) of a class of quantum Brownian motions with inhomogeneous damping and diffusion. For Ohmic bath spectral density with a Lorentz-Drude cutoff, we derive the Heisenberg-Langevin equations for the particle's observables using a quantum stochastic calculus approach. We set the mass of the particle to equal $m = m_{0} \epsilon$, the reduced Planck constant to equal $\hbar = \epsilon$ and the cutoff frequency to equal $\Lambda = E_{\Lambda}/\epsilon$, where $m_0$ and $E_{\Lambda}$ are positive constants, so that the particle's de Broglie wavelength and the largest energy scale of the bath are fixed as $\epsilon \to 0$. We study the limit as $\epsilon \to 0$ of the rescaled model and derive a limiting equation for the (slow) particle's position variable. We find that the limiting equation contains several drift correction terms, the quantum noise-induced drifts, including terms of purely quantum nature, with no classical counterparts.