  Phase transitions are ubiquitous in nature, ranging from protein folding and
denaturisation, to the superconductor-insulator quantum phase transition, to
the decoupling of forces in the early universe. Remarkably, phase transitions
can be arranged into universality classes, where systems having unrelated
microscopic physics exhibit identical scaling behaviour near the critical
point. Here we present an experimental and theoretical study of the
Bose-Einstein condensation phase transition of an atomic gas, focusing on one
prominent universal element of phase transition dynamics: the spontaneous
formation of topological defects during a quench through the transition. While
the microscopic dynamics of defect formation in phase transitions are generally
difficult to investigate, particularly for superfluid phase transitions,
Bose-Einstein condensates (BECs) offer unique experimental and theoretical
opportunities for probing such details. Although spontaneously formed vortices
in the condensation transition have been previously predicted to occur, our
results encompass the first experimental observations and statistical
characterisation of spontaneous vortex formation in the condensation
transition. Using microscopic theories that incorporate atomic interactions and
quantum and thermal fluctuations of a finite-temperature Bose gas, we simulate
condensation and observe vortex formation in close quantitative agreement with
our experimental results. Our studies provide further understanding of the
development of coherence in superfluids, and may allow for direct investigation
of universal phase-transition dynamics.

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