Dynamical excitations in low-dimensional condensates: sound, vortices and quenched dynamics

The dynamics of systems out of equilibrium, such as the phase transition process, are very rich, and related to largely scalable problems, from very small ultracold gases to large expanding galaxies. Quantum low-dimensional systems show interesting features, notably different from the canonical three-dimensional case. Bose-Einstein condensates are very good platforms to study macroscopic quantum phenomena. These three points describe well the motivation behind the study presented in this work. In this thesis, some dynamical problems of trapped and uniform condensates are studied, both at zero and finite temperature. In particular, we focus on the analysis of the propagation of linear and nonlinear excitations in a quasi-1D and in quasi-2D systems. In the first case, we are able to correctly describe the dynamics of a solitonic vortex in an elongated condensate, as measured by Serafini et al. [Phys. Rev. Lett. 115, 170402 (2015)]. In the second case, we reproduce the decay rate of a phase-imprinted soliton (collaboration with Birmingham), and assess its dependence on the temperature. We also replicate the propagation speed of sound waves over a wide range of temperatures as in Ville et al. [arXiv:1804.04037] (collaboration with Collège de France). The result of this analysis is included in Ota et al. [arXiv:1804.04032], which is currently under revision. In uniform low-dimensional systems Bose-Einstein condensation is technically not possible, and in two dimensions it is replaced by the Berezinskii-Kosterlitz-Thouless superfluid phase transition. We study its critical properties by analysing the spontaneous generation of vortices during a quench, produced via the Kibble-Zurek mechanism. This procedure predicts, for any dimension, the scaling for the density of defects formed during a fast transition, when the system is not adiabatically following the control parameter, and regions of phase inhomogeneity are formed. We address the role of reduced dimensionality on this process. All finite temperature simulations are performed by means of the stochastic (projected) Gross-Pitaevskii equation, a model fully incorporating density and phase fluctuations for weakly interacting Bose gases.