# Emergent limit cycles and time crystal dynamics in an atom-cavity system

## TL;DR

This paper proposes an experimental setup with an atomic Bose-Einstein condensate in an optical cavity to realize a time crystal, demonstrating stable limit cycles and breaking of continuous time translation symmetry.

## Contribution

It introduces a novel experimental realization of a time crystal using atom-cavity systems and maps out the dynamical phase diagram showing stable limit cycles.

## Key findings

- Identification of parameter regions with stable limit cycles
- Demonstration of robustness of limit cycles against perturbations
- Evidence of continuous time translation symmetry breaking

## Abstract

We propose an experimental realization of a time crystal using an atomic Bose-Einstein condensate in a high finesse optical cavity pumped with laser light detuned to the blue side of the relevant atomic resonance. By mapping out the dynamical phase diagram, we identify regions in parameter space showing stable limit cycle dynamics. Since the model describing the system is time-independent, the emergence of a limit cycle phase indicates the breaking of continuous time translation symmetry. Employing a semiclassical analysis to demonstrate the robustness of the limit cycles against perturbations and quantum fluctuations, we establish the emergence of a time crystal.