Quantum Trajectories I

Abstract

We are now able to begin the enterprise towards which the previous six lectures have been heading. We axe going to develop a new way of thinking about and analyzing the master equation for a photoemissive source. The character of the new approach can be appreciated by considering an analogy with classical statistical physics. In classical statistical physics there are two ways of approaching the dynamical evolution of a system. In the first the system is described by a probability distribution and a Fokker-Planck equation, or its equivalent, generates the evolution in time. In the second the system is describe by an ensemble of noisy trajectories and a set of stochastic differential equations is used to generate the trajectories. The quantum-classical correspondence (Sects. 4.1–4.3) allows both of these methods to be used to analyze a source master equation. But the usefulness of this method is limited. It is limited ultimately by the fact that at a fundamental level, quantum dynamics does not fit the classical statistics mold. It is actually rare that an operator master equation is converted into a Fokker-Planck equation under the quantum-classical correspondence. Most often the system size expansion (small quantum noise assumption) is used to “shoehorn” the quantum dynamics into a classical form. If this cannot be done, then we always have the operator master equation itself, which might be solved directly, using a computer if necessary. The master equation is an equation for the density operator — the quantum mechanical version of a probability distribution.