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We analyze the microscopic model of quantum Brownian motion, describing aBrownian particle interacting with a bosonic bath through a coupling which islinear in the creation and annihilation operators of the bath, but may be anonlinear function of the position of the particle. Physically, thiscorresponds to a configuration in which damping and diffusion are spatiallyinhomogeneous. We derive systematically the quantum master equation for theBrownian particle in the Born-Markov approximation and we discuss theappearance of novel terms, for various polynomials forms of the coupling. Wediscuss the cases of linear and quadratic coupling in great detail and wederive, using Wigner function techniques, the stationary solutions of themaster equation for a Brownian particle in a harmonic trapping potential. Wepredict quite generally Gaussian stationary states, and we compute the aspectratio and the spread of the distributions. In particular, we find that thesesolutions may be squeezed (super-localized) with respect to the position of theBrownian particle. We analyze various restrictions to the validity of ourtheory posed by non-Markovian effects and by the Heisenberg principle. Wefurther study the dynamical stability of the system, by applying a Gaussianapproximation to the time dependent Wigner function, and we compute thedecoherence rates of coherent quantum superpositions in position space.Finally, we propose a possible experimental realization of the physicsdiscussed here, by considering an impurity particle embedded in a degeneratequantum gas.

Quantum Brownian Motion with Inhomogeneous Damping and Diffusion