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

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= m₀ϵ, the reduced Planck constant to equal ħ= ϵ and the cutoff frequency to equal Λ= E_Λ/ ϵ, where m₀ and E_Λ are positive constants, so that the particle’s de Broglie wavelength and the largest energy scale of the bath are fixed as ϵ→ 0. We study the limit as ϵ→ 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.