Equilibrium and oscillations in a turbulent Bose–Einstein condensate

Abstract

A general approach to equilibrium and oscillations in a turbulent Bose–Einstein condensate, valid in the mean-field approximation, is proposed. Two different wave equations, describing the quasi-static equilibrium and the elementary excitations of the medium are derived. In particular, it is shown that a modified Thomas–Fermi equilibrium can be defined in the presence of an arbitrary spectrum of oscillations.The oscillating modes are also determined using Bogoliubov–de Gennes (BdG) equations, valid in different geometrical configurations, such as infinite, cylindrical and toroidal. The case of a non-homogenous density is considered, which includes the modified equilibrium profiles. A general form of dispersion relation for the elementary excitations is derived. Twisted phonons, carrying a finite amount of angular momentum, are shown to be a particular class of BdG modes. They were previously derived from fluid equations. Nonlinear analysis also shows that envelope dark solitons can exist, and correspond to a rarefaction of twisted phonons.