The stochastic approach for non-Hamiltonian systems

Sergio A Albeverio, Ph. Combe, M. Sirugue-Collin

V. P. Belavkin, P. Staszewski

A semiclassical non-Hamiltonian model of a spontaneous collapse of unstable quantum system is given. The time evolution of the system becomes non-Hamiltonian at random instants of transition of pure states to reduced ones, η⟼Cη, given by a contraction C. The counting trajectories are assumed to satisfy the Poisson law. A unitary dilation of the concractive stochastic dynamics is found. In particular, in the limit of frequent detection corresponding to the large number limit we obtain the Itô–Schrödinger stochastic unitary evolution for the pure state of unstable quantum system providing a new stochastic version of the quantum Zeno effect.

Ashat M. Basharov

Abstract We show that a number of particle decay problems, which are described by the non-Hermitian Hamiltonian (correspondingly, the nonunitary dynamics is spoken of), can be correctly and consistently reformulated in terms of stochastic differential equations upon application of the algebraic perturbation theory. In this formulation, kinetic equations are obtained in the standard scheme of the theory of open quantum systems. The parameters of the kinetic equation coincide with similar parameters describing the decay of a particle when the decay is considered on the basis of the non-Hermitian Hamiltonian.

S.S. Sannikov, I. I. Uvarov

Luis de la Peña, Ana Marı́a Cetto

Y. B. Band

We study the stochastic dynamics of a two-level quantum system interacting with a stochastic magnetic field, and a single frequency electromagnetic field, with and without making the rotating wave approximation (RWA). The transformation to the rotating frame does not commute with the stochastic Hamiltonian if the stochastic field has nonvanishing components in the transverse direction. Hence, making the RWA modifies the stochastic terms in the Hamiltonian. Modification of the decay terms is also required in a master equation approach (i.e., the Liouville--von Neumann density matrix equation) for describing the dynamics. For isotropic Gaussian white noise, the RWA dynamics remains Markovian, although the Lindblad terms in the master equation for the density matrix become time-dependent when the non-commutation of the RWA transformation and the noise Hamiltonian is properly accounted for. We also treat Ornstein--Uhlenbeck noise, and find, in contra-distinction to the white noise case, a significant difference in the dynamics calculated with the RWA when the non-commutation of the RWA transformation and the noise Hamiltonian is taken into account. These findings are applicable to the modeling of any open quantum system coupled to an electromagentic field.