Behaviour of ohmic systems in cavities—a non-perturbative approach

Abstract

We consider a system consisting of a particle in the approximation of a harmonic oscillator, coupled to an ohmic environment inside a spherical reflecting cavity of radius R. Starting from the solution of the confined problem, we study the behaviour of the system in free space understood as the limit of an arbitrarily large radius in the quantities describing the confined solution. From a mathematical point of view we show that this method to approach the problem is not equivalent to consider the system a priori embedded in unbounded space. In particular, the matrix elements of the transformation turning the system to principal axis do not tend to distributions in the limit of an arbitrarily large cavity as it should be the case if the two procedures were mathematically equivalent. By introducing dressed coordinates we define dressed states which allow a non-perturbative unified description of the time evolution process for the system, in both cases of a small or an arbitrarily large cavity, for weak and strong coupling regimes. In particular we perform a study of the time energy distribution in a small cavity, with the initial condition that the particle is in the first excited state. We conclude for the quasi-stability of the excited particle in the weak coupling regime, which is a well-known experimental fact. Also our study of a superposition of dressed states in a large cavity concludes that it obeys a statistical distribution for very large times, in other words, we find a kind of “spontaneous” decoherence in free space.