Semiquantum chaos

Abstract

%e consider a system in which a classical oscillator is interacting with a purely quantum mechanical oscillator, described by the Lagrangian L = 2x + 2A -2 (m +e A )x, where A is a classical variable and x is a quantum operator. With (x(t)) = 0, the relevant variable for the quantum oscillator is (x(t)x(t)) = G(t). The classical Hsmiltonian dynamics governing the variables A(t), II&(t), G(t), and IIa(t) is chaotic so that the results of making measurements on the quantum system at later times are sensitive to initial conditions. This system arises as the zero momentum part of the problem of pair production of charged scalar particles by a strong external electric 6eld.

Semiquantum chaos and the large N expansion

“…We now summarize some previous results from the numerical solution of the Hamiltonian equations obtained in the Gaussian approximation [3]. For calculational purposes, it turns out to be convenient to scale out the mass via the transformations…”

“…However the proper definition of chaos for quantum systems and its experimental manifestations are still unclear [2]. Here we first study a simple model of two coupled systems which displays semiquantum chaos [3] when one of the systems can be treated "semiclassically." We then study a purely quantum system of N + 1 degrees of fredom which has the identical dynamics in the large N limit as our original system.…”

“…The commutation relation (9) then follows automatically. It is easy to show using (3) and (10) that f (t) satisfies the equation of motion…”

“…In a classical chaotic system, such as the weather, we are accustomed to situations where there is lack of long time forecasting because of the sensitivity of the system to initial conditions. The simple model system considered here has the unusual feature that one has to give up long term forecasting even for the quantum mechanical probabilities, as exemplified by the average number of quanta at later times [3].…”

Characteristics of the wave function of coupled oscillators in semiquantum chaos

“…This system was first considered by Cooper et.al. [6], which can be described by a classical effective Hamiltonian, the expectation value of the quantum Hamiltonian. Faccioli et.al.…”