Some remarks on dissipative Ermakov systems and damping in Bose–Einstein condensates

Abstract

For physical systems where the Hamiltonian is no longer a constant of motion, like oscillators with time-dependent frequency or dissipative systems, a dynamical invariant of the Ermakov type might still exist. This invariant depends on the classical position and velocity and an auxiliary variable that is proportional to position uncertainty. It is possible to construct this invariant via an algebraic method based on the classical Poisson bracket. A modified version for dissipative systems that also employs anti-Poisson brackets and is related to a description in an expanding coordinate system will be presented. The resulting invariant is identical to that obtained from a logarithmic nonlinear Schrödinger equation. The uncertainties of position and momentum, expressed in terms of the auxiliary variable, can be compared with the so-called moment-method for the description of the time-evolution of Bose–Einstein condensates, corresponding to a description in terms of the cubic nonlinear Gross–Pitaevskii equation. It will be shown how this method can be extended to also include dissipative damping effects into the Bose–Einstein dynamics.