Continual Measurements in Quantum Mechanics and Quantum Stochastic Calculus

We speak of continuous measurements in the case in which one ore more observables of a quantum system are followed with continuity in time. Traditional presentations of quantum mechanics consider only instantaneous measurements, but continuous measurements on quantum systems are a common experimental practice; typical cases are the various forms of photon detection. The statements of a quantum theory about an observable are of probabilistic nature; so, it is natural that a quantum theory of continuous measurements gives rise to stochastic processes. Moreover, a continuously observed system is certainly open. All these things show that the development and the applications of a quantum theory of continuous measurements needs quantum measurement theory, open system theory, quantum optics, operator theory, quantum probability, quantum and classical stochastic processes... There are essentially three approaches to continuous measurements. These approaches have received various degrees of development, any one of them has its own merits and range of applicability, but “morally” all the three approaches are equivalent and one can go from one to the other and this feature is certainly at the bases of the flexibility and interest of the theory. The first approach is the operational one, which is based on positive operator valued measures or (generalized) observables and operation valued measures or instruments. A variant of this approach is based on the Feynman integral. The second approach is based on quantum stochastic calculus and quantum stochastic differential equations and it is connected to quantum Langevin equations and the notion of input and output fields in quantum optics. The last approach is based on (classical) stochastic differential equations and the notion of a posteriori states and it is related to some notions appeared in quantum optics: quantum trajectories, Monte-Carlo wave function method, unravelling of master equation. This report is concerned mainly with the second approach, the one based on quantum stochastic calculus. Both the general mathematical theory and some physical applications are developed.