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 particles 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 particles 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) particles 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.