Time Crystal in the Nonlinear Phonon Mode of the Trapped Ions

TL;DR

This paper proposes a practical scheme to realize a phonon time crystal in trapped ions, demonstrating continuous-time symmetry breaking through controlled nonlinear phonon modes and robust dissipative dynamics.

Contribution

It introduces a new method to generate and observe a phonon time crystal in a two-ion system using laser control and nonlinear damping.

Findings

Successful numerical simulation with experimental parameters

Robustness to thermal states and control errors

Observation of discrete time-translation symmetry breaking

Abstract

Time crystals constitute a novel phase of matter defined by the spontaneous breaking of timetranslation symmetry. Here we present a scheme to realize a continuous-time crystal of the vibrational phonon in the normal mode of two coupled ultra-cold ions. By utilizing two addressable standing-wave lasers and adiabatic elimination method, we generate a controllable nonlinear phonon mode with the well-designed efficient linear gain and nonlinear damping. By controlling these parameters to satisfy the phase transition conditions of Hopf bifurcation and limit cycle phase, it behaves as a stable dissipative dynamics over timescales significantly longer than the oscillation period, indicating the emergence of discrete time-translation symmetry breaking in the phonon mode, i.e., a phonon time crystal. We further numerically simulate this phonon time crystal by using accessible experimental parameters and also demonstrate a robustness to the initial thermal state and thermalization of phonon mode, spin dephasing, and the control errors of Rabi frequencies. These results provide a practical scheme for observing a time crystal in a nonlinear phonon mode and will advance the research of time crystals.