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arXiv:2004.09642 · Physical Review Research

Kibble-Zurek dynamics in a trapped ultracold Bose gas

I-Kang Liu, Jacek Dziarmaga, Shih-Chuan Gou, F. Dalfovo, Nick P. Proukakis

Published Aug 3, 2020• 35 citations
arXivHTMLPDF

Abstract

This work examines the growth of a Bose-Einstein condensate in a trapped ultracold bosonic gas quenched across the phase transition. The validity of the Kibble-Zurek framework in the early stages of the symmetry-breaking dynamics is assessed, clarifying the role of different time and spatial scales in the quench and filling in the gap between the early evolution and the corresponding long-time evolution observed in the experiments.

DOI
10.1103/physrevresearch.2.033183

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This work examines the growth of a Bose-Einstein condensate in a trapped ultracold bosonic gas quenched across the phase transition. The validity of the Kibble-Zurek framework in the early stages of the symmetry-breaking dynamics is assessed, clarifying the role of different time and spatial scales in the quench and filling in the gap between the early evolution and the corresponding long-time evolution observed in the experiments.
The dynamical evolution of an inhomogeneous ultracold atomic gas quenched at different controllable rates through the Bose-Einstein condensation phase transition is studied numerically in the premise of a recent experiment in an anisotropic harmonic trap. Our findings based on the stochastic (projected) Gross-Pitaevskii equation are shown to be consistent at early times with the predictions of the homogeneous Kibble-Zurek mechanism. This is demonstrated by collapsing the early dynamical evolution of densities, spectral functions and correlation lengths for different quench rates, based on an appropriate characterization of the distance to criticality felt by the quenched system. The subsequent long-time evolution, beyond the identified dynamical critical region, is also investigated by looking at the behavior of the density wavefront evolution and the corresponding phase ordering dynamics.
Universal dynamics of spontaneous symmetry breaking is central to understanding the universal behavior of spontaneous defect formation in various systems from the early universe, condensed-matter systems to ultracold atomic systems. We explore the universal real-time dynamics in an array of coupled binary atomic Bose-Einstein condensates in optical lattices, which undergo a spontaneous symmetry breaking from the symmetric Rabi oscillation to the broken-symmetry self-trapping. In addition to Goldstone modes, there exist gapped Higgs modes whose excitation gap vanishes at the critical point. In the slow passage through the critical point, we analytically find that the symmetry-breaking dynamics obeys the Kibble-Zurek mechanism. From the scalings of bifurcation delay and domain formation, we numerically extract two Kibble-Zurek exponents, b 1 = ν / ( 1 + ν z ) and b 2 = 1 / ( 1 + ν z ) , which give the static correlation-length critical exponent ν and the dynamic critical exponent z . Our approach provides an efficient way for the simultaneous determination of the critical exponents ν and z for a continuous phase transition.
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 problem of understanding how a coherent, macroscopic Bose- Einstein condensate (BEC) emerges from the cooling of a thermal Bose gas has attracted significant theoretical and experimental interest over several decades. The pioneering achievement of BEC in weakly interacting dilute atomic gases in 1995 was followed by a number of experimental studies examining the growth of the BEC number, as well as the development of its coherence. More recently, there has been interest in connecting such experiments to universal aspects of nonequilibrium phase transitions, in terms of both static and dynamical critical exponents. Here, the spontaneous formation of topological structures such as vortices and solitons in quenched cold-atom experiments has enabled the verification of the Kibble-Zurek mechanism predicting the density of topological defects in continuous phase transitions, first proposed in the context of the evolution of the early universe. This chapter reviews progress in the understanding of BEC formation and discusses open questions and future research directions in the dynamics of phase transitions in quantum gases. Introduction The equilibrium phase diagram of the dilute Bose gas exhibits a continuous phase transition between condensed and noncondensed phases. The order parameter characteristic of the condensed phase vanishes above some critical temperature T c and grows continuously with decreasing temperature below this critical point. However, the dynamical process of condensate formation has proved to be a challenging phenomenon to address both theoretically and experimentally. This formation process is a crucial aspect of Bose systems and of direct relevance to all condensates discussed in this book, despite their evident system-specific properties. Important questions leading to intense discussions in the early literature include the time scale for condensate formation and the role of inhomogeneities and finite-size effects in “closed” systems. These issues are related to the concept of spontaneous symmetry breaking, its causes, and implications for physical systems (see, for example, Chapter 5 by Snoke and Daley). In this chapter, we give an overview of the dynamics of condensate formation and describe the present understanding provided by increasingly well-controlled cold-atom experiments and corresponding theoretical advances over the past twenty years. We focus on the growth of BECs in cooled Bose gases, which, from a theoretical standpoint, requires a suitable nonequilibrium formalism.
Abstract Out-of-equilibrium phenomena are a subject of considerable interest in many fields of physics. Ultracold quantum gases, which are extremely clean, well-isolated and highly controllable systems, offer ideal platforms to investigate this topic. The recent progress in tailoring trapping potentials now allows the experimental production of homogeneous samples in custom geometries, which is a key advance for studies of the emergence of coherence in interacting quantum systems. Here we review recent experiments in which temperature quenches have been performed across the Bose–Einstein condensation phase transition in an annular geometry and in homogeneous 3D and quasi-2D gases. Combined, these experiments comprehensively explore and validate the Kibble–Zurek (KZ) scenario through complementary measurements of correlation functions and density of topological defects. They allow the measurement of KZ scaling laws, the direct confirmation of the ‘freeze-out’ hypothesis that underlies the KZ theory, and the extraction of critical exponents of the Bose–Einstein condensation transition.
Atomic Bose-Einstein condensates confined to a dual-ring trap support Josephson vortices as topologically stable defects in the relative phase. We propose a test of the scaling laws for defect formation by quenching a Bose gas to degeneracy in this geometry. Stochastic Gross-Pitaevskii simulations reveal a -1/4 power-law scaling of defect number with quench time for fast quenches, consistent with the Kibble-Zurek mechanism. Slow quenches show stronger quench-time dependence that is explained by the stability properties of Josephson vortices, revealing the boundary of the Kibble-Zurek regime. Interference of the two atomic fields enables clear long-time measurement of stable defects and a direct test of the Kibble-Zurek mechanism in Bose-Einstein condensation.