The formation of a Bose-Einstein condensate has been a widely-debated topic already in the 1990’s, even before its first experimental observation in weakly-interacting systems, led by trapped ultracold atoms in 1995 (and followed by similar observations in other physical systems: exciton-polaritons, photons and magnons), as reviewed in our recent topical review . Our lates joint theoretical-experimental analysis, addresses various open questions on the formation of a phase-coherent Bose-Einstein condensate across a phase transition, when induced by an external quench, which is an important issue across different areas of physics, and of relevance to potential ultracold atom quantum technologies.
Early ultracold atom experiments  (1998 – 2007) tackled this issue by studying initially the growth of the condensate fraction, followed by dynamical studies of the growth of coherence after a sudden evaporation quench. Open questions arising from such studies included the emergence of a so-called “quasicondensate” phase as the system dynamically transitions from a thermal to a phase-coherent system  and the related interplay of timescales  between number and coherence growth. The next phase of controlled experiments (2008 – now) addressed condensate formation induced by a gradual (linear) external parameter quench, focusing on the important question of how a dynamical system can cross the (second-order) phase transition by spontaneously breaking a symmetry of its Hamiltonian, through the spontaneous emergence of defects in its density. This topic was first raised by Tom Kibble in the cosmological context, and subsequently brought to the condensed-matter realm by Wojciech Zurek, with the study of such a Kibble-Zurek universal scaling law also becoming a key area of study in ultracold gases in which defect number [5,6,7] and associated critical exponents  have been experimentally studied. Open questions here relate to the role of the inhomogeneous confining potential and the interplay of spontaneous defect generation with the underlying defect dynamics occurring in the system, which challenges the validity of this universal Kibble-Zurek mechanism and leads to defect relaxation and final equilibration.
Based on experiments conducted by our Trento collaborators (with whom we have a joint QuantERA Cofund grant) , our recently-published work Dynamical equilibation across a quenched phase transition in a trapped quantum gas (Comms. Phys (Nature), 1, 24 (2018)) makes a unified advance on the above concepts and questions by means of state-of-the-art numerical simulations. In addition to demonstrating agreement with experiments, the main achievements of the current work include:
(i) A unique visualization of the entire dynamical process (including experimentally-inaccessible regimes) of how an equilibrium thermal gas relaxes to a phase-coherent Bose-Einstein condensate;
(ii) A clear demonstration of spontaneous emergence of symmetry-breaking as the system crosses the critical region;
(iii) An analysis of the role of defect interactions and decay which transform the initial turbulent-like state to a state of few interacting defects that are experimentally detectable.
(iv) The main focus of this work is to make advances on the largely unexplored study of the re-equilibration dynamics, i.e. the process by which a system relaxes, after crossing the phase transition dynamically, to its final (equilibrium) phase-coherent state by the ejection, or decay, of its defects.
We convincingly demonstrate (for gradual quenches in an inhomogeneous system) a decoupling of timescales for number and coherence growth, which we discuss in the context of the emergent “quasi-condensation” during the phase transition, argued here to be determined by the interplay of the spontaneously-generated defect and their subsequent dynamics; in this manner, we also address the important, yet still unresolved, interplay between Kibble-Zurek and coarse-graining dynamics during quenched condensate formation.
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