Quantum system in contact with a thermal environment: Rigorous treatment of a simple model

Abstract

We study the quantum dynamics of a particle of massM in an external potentialV(Q), interacting with a simple model environment—a harmonic chain of 2N particles with massm and spring constantk. The classical version of this model was studied by Rubin and is equivalent to standard models of a particle interacting with a phonon bath. Settingm=m*/L andk=k*L, we prove that for a suitable class of potentialsV and initial statesω ₀, the time evolution of the massM particle converges, whenN → ∞ andL → ∞, to the time evolution governed by the Quantum Langevin Equation (QLE) which has been found by Ford, Kac and Mazur. Furthermore we show that, for this class of potentials, the QLE has a unique solution for all positive times, such solution can be expressed as a convergent expansion in the deviation ofV(Q) from a harmonic potential. The equilibrium properties of the particle with massM can be expressed in terms of an integral, over path space, with a Gaussian measure which has mean zero and covariance proportional to$[ - \Delta + \eta h/M\sqrt { - \Delta } ]^{ - 1}$; where$\eta = 2\sqrt {km}$ is the friction constant, andh is the Plancks' constant (divided by 2π).