Formatted abstract

We investigate the dynamics of formation of topological defects in ultracold quantum gases. Using classical field methods, such as the truncated Wigner approximation and the stochastic GrossPitaevskii equation, we explore the connection between the rate of a phase transition and the number of topological defects formed as a consequence of it. We show that in various systems such defects form in number comparable with the predictions of the KibbleZurek theory, known for thirty years but as yet unverified experimentally. We also consider the spontaneous production of entangled vortices following a quantum phase transition in a multicomponent spin1 BoseEinstein condensate (BEC).
To begin, we provide a literature review and describe some of the background theory of multicomponent condensates, stochastic methods and KibbleZurek theory.
Part of this thesis focus on a two component BoseEinstein condensates of the same atomic species. In particular on the quantum critical point separating the miscible from the immiscible phase of such systems. We show that such phase transition cannot only be achieved with the tuning of the interaction constants between the components, but also through changes in the intensity of the coupling between atoms in different hyperfine states. We first determine the value of the critical coupling, and also the critical exponents defining the universality class of the coupling—induced phase transition. We then proceed to simulate the dynamics of the transition in effective 1D ring and harmonically trapped condensates, and test the results against predictions of the KibbleZurek (KZ) mechanism. We find good agreement between the KZ theory and the numerical results in the ring geometry, and we motivate the origin of the discrepancies between the two in harmonically trapped BECs. We use the ”equivalence” of inhomogeneous phase transitions in anisotropic systems and spatially inhomogeneous quenches to engineer coupling quenches and systematically study key aspects of the inhomogeneous KZ mechanism. The coupling induced miscible—immiscible transition results in the formation of a stable pattern of domains that easily be detected and counted in current ultracold gases experiment, and the scheme is a viable candidate for the first experimental testing of the KZ predictions.
We then examine the spontaneous formation of a superflow as result of the thermal BoseEinstein phase transition in ring traps. According to the KibbleZurek theory, the fast production of a condensate in a ring trap should result in the formation of solitons, with the number depending on the rate of the temperature quench. Solitons are shortliving topological defects and their observation in ultracold gases experiments remains challenging. We show here that even after they decay, solitons leave a signature in the variance of the final winding number of the condensate. Solitons fragment a 1D condensate into regions of random phase, hence the larger the number of solitons, the higher the probability that the resulting phase wraps around the ring by a multiple of 2π. We numerically simulate the BEC transition in this system with the stochastic GrossPitaevskii equation, and observe that the winding number scales with the quenching time as predicted by the KZ mechanism. We examine the effect of inhomogeneity on the expectation value of the winding number, and find that the tilting of the ring greatly suppress the spontaneous formation of vortices. This has implications for the necessary horizontal alignment of any experiment on this system.
Finally, we consider the polar to antiferromagnetic phase transition in magnetically quenched spin1 BEC. In this quantum phase transition a pure condensate of atoms in the m_{F} = 0 Zeeman sublevel is converted into a three component BEC with atoms in m_{F} = 0, ±1 through spin changing collisions. An analogous mechanism, four—wave mixing, in quantum optics produces entangled photons, and it has been proposed in ultracold gases as a method to obtain EinsteinPodolskiRosen entanglement of macroscopic atomic ensembles. We begin with a rotating 1D spinor condensate of atoms in the m_{F} = 0 component, and perform a quench of the magnetic field through the critical point. We show how the transition produces a pair of entangled components with different winding numbers, and quantify the entanglement as a function of the modes considered and of the type of magnetic quench, sudden or instantaneous. Lastly, we propose a method to use the original spin zero component as coherent source to perform homodyne detection and consider the two rotating entangled components as a resource for the sensing of rotations.
