  We analyze the microscopic model of quantum Brownian motion, describing a
Brownian particle interacting with a bosonic bath through a coupling which is
linear in the creation and annihilation operators of the bath, but may be a
nonlinear function of the position of the particle. Physically, this
corresponds to a configuration in which damping and diffusion are spatially
inhomogeneous. We derive systematically the quantum master equation for the
Brownian particle in the Born-Markov approximation and we discuss the
appearance of novel terms, for various polynomials forms of the coupling. We
discuss the cases of linear and quadratic coupling in great detail and we
derive, using Wigner function techniques, the stationary solutions of the
master equation for a Brownian particle in a harmonic trapping potential. We
predict quite generally Gaussian stationary states, and we compute the aspect
ratio and the spread of the distributions. In particular, we find that these
solutions may be squeezed (super-localized) with respect to the position of the
Brownian particle. We analyze various restrictions to the validity of our
theory posed by non-Markovian effects and by the Heisenberg principle. We
further study the dynamical stability of the system, by applying a Gaussian
approximation to the time dependent Wigner function, and we compute the
decoherence rates of coherent quantum superpositions in position space.
Finally, we propose a possible experimental realization of the physics
discussed here, by considering an impurity particle embedded in a degenerate
quantum gas.
