We study the vortex nucleation dynamics in inhomogeneous atomic Bose gases quenched into a superfluid phase and investigate the dependence of the Kibble-Zurek (KZ) scaling exponent on the underlying trap configuration. For samples in a number of different inhomogeneous traps, we observe the characteristic power-law scaling of the vortex number with the thermal quench rate, as well as an enhanced vortex suppression in the outer regions with lower particle density, in agreement with the causality effect as encapsulated in the inhomogeneous Kibble-Zurek mechanism (IKZM). However, the measured KZ scaling exponents show significant differences from the theoretical estimates, and furthermore their trends as a function of the underlying trap configuration deviate from the IKZM prediction. We also investigate the early-time coarsening effect using a two-step quench protocol as proposed in a recent study and show that the interpretation of the measurement results without including the causality effect might be misleading. This paper provides a comprehensive study of vortex formation dynamics in quenched Bose gases confined in inhomogeneous trapping potentials and calls for a refined theoretical framework for quantitative understanding of the phase transition and defect formation processes in such inhomogeneous systems.
We study the vortex nucleation dynamics in inhomogeneous atomic Bose gases quenched into a superfluid phase and investigate the dependence of the Kibble-Zurek (KZ) scaling exponent on the underlying trap configuration. For samples in a number of different inhomogeneous traps, we observe the characteristic power-law scaling of the vortex number with the thermal quench rate, as well as an enhanced vortex suppression in the outer regions with lower particle density, in agreement with the causality effect as encapsulated in the inhomogeneous Kibble-Zurek mechanism (IKZM). However, the measured KZ scaling exponents show significant differences from the theoretical estimates, and furthermore their trends as a function of the underlying trap configuration deviate from the IKZM prediction. We also investigate the early-time coarsening effect using a two-step quench protocol as proposed in a recent study and show that the interpretation of the measurement results without including the causality effect might be misleading. This paper provides a comprehensive study of vortex formation dynamics in quenched Bose gases confined in inhomogeneous trapping potentials and calls for a refined theoretical framework for quantitative understanding of the phase transition and defect formation processes in such inhomogeneous systems.
The Kibble–Zurek mechanism is applied to the spontaneous formation of vortices in a harmonically trapped thermal gas following a temperature quench through the critical value for Bose–Einstein condensation. Whereas in the homogeneous scenario, vortex nucleation is always expected, we show that it can be completely suppressed in the presence of the confinement potential whenever the speed of the spatial front undergoing condensation is lower than a threshold velocity. Otherwise, the interplay between the geometry and the causality leads to different scaling laws for the density of vortices as a function of the quench rate, as we also illustrate for the case of a toroidal trapping potential.
Continuous phase transitions occur in a wide range of physical systems, and provide a context for the study of non-equilibrium dynamics and the formation of topological defects.The Kibble-Zurek (KZ) mechanism predicts the scaling of the resulting density of defects as a function of the quench rate through a critical point, and this can provide an estimate of the critical exponents of a phase transition.In this work we extend our previous study of the miscible-immiscible phase transition of a binary Bose-Einstein condensate (BEC) composed of two hyperfine states in which the spin dynamics are confined to one dimension [J.Sabbatini et al., Phys.Rev. Lett.107, 230402 (2011)].The transition is engineered by controlling a Hamiltonian quench of the coupling amplitude of the two hyperfine states, and results in the formation of a random pattern of spatial domains.Using the numerical truncated Wigner phase space method, we show that in a ring BEC the number of domains formed in the phase transitions scales as predicted by the KZ theory.We also consider the same experiment performed with a harmonically trapped BEC, and investigate how the density inhomogeneity modifies the dynamics of the phase transition and the KZ scaling law for the number of domains.We then make use of the symmetry between inhomogeneous phase transitions in anisotropic systems, and an inhomogeneous quench in a homogeneous system, to engineer coupling quenches that allow us to quantify several aspects of inhomogeneous phase transitions.In particular, we quantify the effect of causality in the propagation of the phase transition front on the resulting formation of domain walls, and find indications that the density of defects is determined during the impulse to adiabatic transition after the crossing of the critical point.
We investigate the dynamics of formation of topological defects in ultra-cold quantum gases. Using classical field methods, such as the truncated Wigner approximation and the stochastic Gross-Pitaevskii 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 Kibble-Zurek 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 multi-component spin-1 Bose-Einstein condensate (BEC). To begin, we provide a literature review and describe some of the background theory of multi-component condensates, stochastic methods and Kibble-Zurek theory. Part of this thesis focus on a two component Bose-Einstein 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 Kibble-Zurek (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 ultra-cold 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 Bose-Einstein phase transition in ring traps. According to the Kibble-Zurek 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 short-living topological defects and their observation in ultra-cold 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 Gross-Pitaevskii 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 anti-ferromagnetic phase transition in magnetically quenched spin-1 BEC. In this quantum phase transition a pure condensate of atoms in the mF = 0 Zeeman sublevel is converted into a three component BEC with atoms in mF = 0, ±1 through spin changing collisions. An analogous mechanism, four—wave mixing, in quantum optics produces entangled photons, and it has been proposed in ultra-cold gases as a method to obtain Einstein-Podolski-Rosen entanglement of macroscopic atomic ensembles. We begin with a rotating 1D spinor condensate of atoms in the mF = 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.
Continuous phase transitions occur in a wide range of physical systems, and provide a context for the study of non-equilibrium dynamics and the formation of topological defects. The Kibble–Zurek (KZ) mechanism predicts the scaling of the resulting density of defects as a function of the quench rate through a critical point, and this can provide an estimate of the critical exponents of a phase transition. In this work, we extend our previous study of the miscible–immiscible phase transition of a binary Bose–Einstein condensate (BEC) composed of two hyperfine states in which the spin dynamics are confined to one dimension (Sabbatini et al 2011 Phys. Rev. Lett. 107 230402). The transition is engineered by controlling a Hamiltonian quench of the coupling amplitude of the two hyperfine states and results in the formation of a random pattern of spatial domains. Using the numerical truncated Wigner phase-space method, we show that in a ring BEC the number of domains formed in the phase transitions scales as predicted by the KZ theory. We also consider the same experiment performed with a harmonically trapped BEC, and investigate how the density inhomogeneity modifies the dynamics of the phase transition and the KZ scaling law for the number of domains. We then make use of the symmetry between inhomogeneous phase transitions in anisotropic systems, and an inhomogeneous quench in a homogeneous system, to engineer coupling quenches that allow us to quantify several aspects of inhomogeneous phase transitions. In particular, we quantify the effect of causality in the propagation of the phase transition front on the resulting formation of domain walls and find indications that the density of defects is determined during the impulse to adiabatic transition after the crossing of the critical point.
We investigate the saturation of defect density in an atomic Bose gas rapidly cooled into a superfluid phase. The number of quantum vortices, which are spontaneously created in the quenched gas, exhibits a Poissonian distribution not only for a slow quench in the Kibble-Zurek (KZ) scaling regime but also for a fast quench, in which case the mean vortex number is saturated. This shows that the saturation is not caused by destructive vortex collisions, but by the early-time coarsening in an emerging condensate, which is further supported by the observation that the condensate growth lags the quenching in the saturation regime. Our results demonstrate that the defect saturation is an effect beyond the KZ mechanism, opening a path for studying critical phase transition dynamics using the defect number distribution.