Wigner-Weisskopf atom and quantum stochastic dilations

Abstract

Quantum stochastic processes are operator processes in Fock space adapted in a natural way with respect to the tensor product factorization of the space. One can describe the usual classical stochastic processes (e.g., Brownian, Poisson) as well as new variety of non-commuting processes in this language. In this framework, quantum stochastic calculus (like Ito calculus in the classical case) has been developed by Hudson and Parthasarathy. Using this calculus, one can rewrite the simple Wigner-Weisskopf model of a decaying atom. In this model, the system (atom) is coupled linearly to the environment (a bath of infinite degrees of freedom represented by a reducible bosonic quantum stochastic process at non-zero temperature). The quantum stochastic differential equation of evolution is solved, and the bath variables are averaged out by taking vacuum expectation values leading to a law of relaxation of the state of the system.