Dynamics of spontaneous symmetry breaking in a space-time crystal

TL;DR

This paper develops a comprehensive theoretical framework for understanding spontaneous symmetry breaking in space-time crystals formed by atomic Bose-Einstein condensates, covering both classical and quantum regimes and aligning with experimental observations.

Contribution

It introduces a unified approach to analyze the non-equilibrium dynamics and symmetry breaking in space-time crystals, incorporating Langevin and Fokker-Planck formalisms for classical and quantum cases.

Findings

Derived Langevin and Fokker-Planck equations for the system

Analyzed the formation and dynamics of space-time crystals

Compared theoretical results with experimental data

Abstract

We present the theory of spontaneous symmetry breaking (SSB) of discrete time translations as recently realized in the space-time crystals of an atomic Bose-Einstein condensate. The non-equilibrium physics related to such a driven-dissipative system is discussed in both the Langevin as well as the Fokker-Planck formulation. We consider a semi-classical and a fully quantum approach, depending on the dissipation being either frequency independent or linearly dependent on frequency, respectively. For both cases, the Langevin equation and Fokker-Planck equation are derived, and the resulting equilibrium distribution is studied. We also study the time evolution of the space-time crystal and focus in particular on its formation and the associated dynamics of the spontaneous breaking of a Z2 symmetry out of the symmetry unbroken phase, i.e., the equilibrium Bose-Einstein condensate before the periodic drive is turned on. Finally, we compare our results with experiments and conclude that our theory provides a solid foundation for the observations.