Dynamics of a Discrete Quantum Nonlinear Schrödinger Equation

Abstract

Numerical solution of the density matrix formulation of a discrete quantum non-linear Schrödinger equation, which has been used to describe transport in one-dimensional molecular chains, has been carried out in the case of a trimer. Results show that the occupation probabilities tend to their initial values in the limit of large nonlinearity. How the asymptotic values are approached depends strongly on the initial conditions. It has been seen that the occupation probabilities behave quasiperiodically or chaotically with varying nonlinearity, and periodically if there is no nonlinearity. Amplitudes of oscillation of the site occupation probabilities are larger in chaotic intervals than in quasiperiodic intervals. Transition to localisation of the excitation is also marked by a sharp change in the probability amplitudes, if this transiton is preceeded by a chaotic interval. The degree of localisation is never bigger than initially. If the excitation is well, but not necessarily completely, localised initially, transition to localisation is preceeded by one or more prelocalised intervals. In a prelocalised interval excitation oscillates mainly between two sites.