Projection operator techniques in stochastic processes

S. Chaturvedi

Heinz‐Peter Breuer, Francesco Petruccione

A generalization of the stochastic wave-function method to open quantum systems under the influence of strong laser fields is derived. To this end, quantum-statistical ensembles of reduced state vectors are represented in terms of probability distributions on the projective Hilbert space of the open system. By employing the Floquet expansion of the quantum time evolution, this representation is shown to lead in the Markov approximation to a piecewise deterministic stochastic process for the reduced wave function. The realizations of the stochastic process consist of smooth deterministic parts that are interrupted by sudden jumps. The jump operators are the eigenoperators of the Floquet Hamiltonian and describe instantaneous transitions between Floquet states. Two examples are given that serve to illustrate the method and the stochastic simulation technique of the process.

Alberto Barchielli, G. Lupieri

The physical idea of a continual observation on a quantum system has been recently formalized by means of the concept of operation valued stochastic process (OVSP). In this article, it is shown how the formalism of quantum stochastic calculus of Hudson and Parthasarathy allows, in a simple way, for constructing a large class of OVSP’s that in particular contains the quantum counting processes of Davies and Srinivas and continual ‘‘Gaussian’’ measurements. This result is obtained by means of a stochastic dilation of the OVSP’s: at the level of the enlarged system probabilities turn out to be expressed in terms of projection valued measures associated with certain time-dependent, commuting, self-adjoint operators.

Nicolas Gisin

A nonlinear stochastic process is presented that, for each realization and for large times, reproduces L\"uder's projection postulate. The corresponding density operator undergoes a linear evolution reproducing von Neumann's projection postulate. The violation of the Bell inequality, for instance, is described with the two apparatus acting independently on the composed system.

Yuri A. Rylov