Dynamics of dissipative systems and computational physics

During the first year of research activity in the frame of this project there have been investigated two main topics: I. Dynamics of systems of fermions in complex dissipative media; II. Solitons with topologic charge in dissipative systems. An essential problem of the quantum information systems is the controllability and observability of the quantum states, generally described by Lindblad's master equation with phenomenological coefficients. In its usual form, this equation describes a decay of the mean-values, but not necessarily the expected decaying transitions. The basic and very difficult problem of a dissipative quantum theory is to project the evolution of the total system (the system of interest + the environment) on the space of the system of interest. In this case, one obtains a quantum master equation where the system evolution is described by two terms: 1) a Hamiltonian term for the processes with energy conservation, and 2) a non-Hamiltonian term with coefficients depending on the dissipative coupling. That means that a master equation is based on some approximations enabling the replacement of the operators of the dissipative environment with average value coefficients. It is often assumed that the evolution operators of the dissipative system define a semigroup, not a group as in the case of an isolated system. In this framework, Lindblad obtained a quantum master equation in agreement with all the quantum-mechanical principles. However, the Lindblad master equation was unable to secure a correct description of the decaying states. To do that, one has to take into account the transition operators between the system eigenstates with appropriate coefficients. Within this investigation, we have obtained an equation obeying to this requirement, giving the ρ(t) time derivative in terms of creation-annihilation operators of the single-particle states |i>, and λij, representing the dissipative coefficients, the microscopic expressions of which are given. These coefficients describe correlated transitions of the system and environment particles, depending on the dissipative two-body potential V, the populations f(εα), f(εβ) and the densities g(εα), g(εβ) of the environment states. Therefrom we infer that for a normal Fermi-Dirac distribution of the environment particles, the decay processes are favored in comparison with the excitation ones, while for a reversed distribution of the environment populations the excitations are favored. Concerning the second topics approached in the frame of this project one starts from admitting that the topologic charge of a soliton is an integer number 's' which arises in the axial symmetric solution of the local amplitude of the electromagnetic wave, A(z, x, y) = U(z, r) exp (isθ) of the (2+1)-dimensional Ginzburg-Landau equation. The 's' parameter is also called 'spin' or 'vorticity'. The investigation conducted within this topics has been directed along two main lines: (i) The study of fundamental phenomena concerning vortex solitons in dissipative (open) systems, and (ii) Comparison of the specific properties of the vortex type solitons in Hamiltonian (conservative) systems and in dissipative systems. The following fundamental results have been obtained: 1. Formulation of the relevant physical model and identification of the values of the physical parameters of the model. 2. Systematic analysis of the stable localized solutions of the (2+1)-dimensional Ginzburg-Landau equation in media characterized by cubic saturable nonlinearities. 3. Extensive numerical simulations of the (2+1)-dimensional Ginzburg-Landau equation in polar coordinates resulting in the demonstration of the occurrence of stable two-dimensional solutions characterized by axial symmetry both for non-vanishing 'spin' (annular, vortex type solitons) and vanishing 'spin' (fundamental solitons). The study of the propagation of these solitons under azimuthal perturbations demonstrates soliton stability under such perturbations. This result is in strong contrast with the behavior of the 'spin' (annular, vortex type) solitons defined in the corresponding conservative system. These vortex type solitons are found to be quite strong attractors of the system. They can be generated from rather arbitrary localized initial conditions characterized by the same vorticity. 4. The research has evidenced the phenomenon of coexistence of the 'spin' s = 0, 1, and 2 solitonic solutions for a given set of values of the system parameters, each soliton being an attractor within the class of pulses characterized by the same 'spin'. 5. Identification, for isolated values in the parameter space, of the existence of dynamic regimes characterized by internal quasi-periodic persistent vibrations. (authors)