  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.
