Irreversible evolution of quantum chaos

Abstract

The pendulum is the simplest system having all the basic properties inherent in dynamic stochastic systems. In the present paper we investigate the pendulum with the aim to reveal the properties of a quantum analogue of dynamic stochasticity or, in other words, to obtain the basic properties of quantum chaos. It is shown that a periodic perturbation of the quantum pendulum (similarly to the classical one) in the neighborhood of the separatrix can bring about irreversible phenomena. As a result of recurrent passages between degenerate states, the system gets self-chaotized and passes from the pure state to the mixed one. Chaotization involves the states, the branch points of whose levels participate in a slow "drift" of the system along the Mathieu characteristics this "drift" being caused by a slowly changing variable field. Recurrent relations are obtained for populations of levels participating in the irreversible evolution process. It is shown that the entropy of the system first grows and, after reaching the equilibrium state, acquires a constant value.

Paul Trap and the Problem of Quantum Stability

“…When there is a small dispersion in the values of parameter δq 0 , as was shown in work [22], mixed states are formed in the degenerate areas. As a result of this, the system may be with equal probability of 1/2 in two states ψ + 2n and ψ − 2n .…”

“…In this case, the problem of ion's motion inside Paul trap is reduced to the problem studied in work [22]. In the mentioned work stochastic absorption of energy by non stationary chaotic quantum system was studied.…”

“…In the mentioned work stochastic absorption of energy by non stationary chaotic quantum system was studied. Not going into details that can be found by interested reader in [22], here we only present the main results related to given problem.…”

“…At the same time we shall use the results obtained by us from the study of the problem of quantum chaos [20,21,22,23].…”

Quantum chaos and its kinetic stage of evolution

“…If ψ(ϕ) ≡ G(ϕ) means either ce n (ϕ) or se n (ϕ), then G(ϕ) and G(π − ϕ) satisfy the same equation (9) and the same boundary conditions (11). Therefore these functions differ from each other only by the constants.…”