Semiclassical wave-packets emerging from interaction with an environment

We study the quantum evolution in dimension three of a system composed by a test particle interacting with an environment made of $N$ harmonic oscillators. At time zero the test particle is described by a spherical wave, i.e. a highly correlated continuous superposition of states with well localized position and momentum, and the oscillators are in the ground state. Under suitable assumptions on the physical parameters characterizing the model, we give an asymptotic expression of the solution of the Schr\"odinger equation of the system with an explicit control of the error. The result shows that the approximate expression of the wave function is the sum of two terms, orthogonal in $L^2(\erre^{3(N+1)})$ and describing rather different situations. In the first one all the oscillators remain in their ground state and the test particle is described by the free evolution of a slightly deformed spherical wave. The second one consists of a sum of $N$ terms where in each term there is only one excited oscillator and the test particle is correspondingly described by the free evolution of a wave packet, well concentrated in position and momentum. Moreover the wave packet emerges from the excited oscillator with an average momentum parallel to the line joining the oscillator with the center of the initial spherical wave. Such wave packet represents a semiclassical state for the test particle, propagating along the corresponding classical trajectory. The main result of our analysis is to show how such a semiclassical state can be produced, starting from the original spherical wave, as a result of the interaction with the environment.