Study of the dynamics of open quantum systems using Lie groups and algebras

Abstract

Abstract. The paper proposes approaches to the description of quantum system states, as well as the properties of continuity and symmetry, through the density matrix, and explores symmetries in systems, operators, and their commutators using Lie groups and Lie algebras. Specifically, by solving the Neumann equation (or quantum Liouville equation) for a stationary Hamiltonian, information was obtained from all processes occurring in the system, and the underlying causes of these processes were analysed through Lie algebra. By considering the transition from temporal dependencies to the dependence of observables, unitary equations were obtained that describe the dynamics of observables in the Heisenberg representation while preserving the structure of the Lie algebra. The dynamics of open quantum systems weakly interacting with the environment are described through solutions of the Lindblad equation for a stationary super operator. The quantum-to-classical transition, for a system interacting with its environment, was demonstrated under the condition of Bloch sphere contraction. The conditions for transitions from one algebra to another newly formed algebra was established. The Lindblad equation and its solution were expressed in the Gell-Mann matrix representation. In the course of the calculations, the relations between the bases of internal algebras and generators were reduced to the four-dimensional case. For large values of the time variable, a mathematical formulation of the density matrix contractivity was presented.