Transition from non-ergodic to ergodic dynamics in an autonomous discrete time crystal

TL;DR

This paper investigates a transition in an autonomous discrete time crystal system, showing how increasing coupling strength shifts the system from non-ergodic, time-crystalline behavior to ergodic dynamics, with observable signatures at the transition.

Contribution

It introduces a new class of autonomous discrete time crystals and characterizes the transition from non-ergodic to ergodic behavior through Loschmidt echo dynamics.

Findings

Periodic Loschmidt echo oscillations indicate non-ergodic, time-crystalline order.

Transition to exponential decay of Loschmidt echo marks ergodic behavior.

Maximum variance of photon number signals the transition point.

Abstract

We consider an autonomous system of two coupled single-mode cavities, one of which interacts with a multimode resonator. We demonstrate that for small coupling strengths between single-mode cavities, the Loschmidt echo oscillates periodically in time and spontaneous breaking of time translation symmetry takes place. The Loschmidt echo behavior is an indication of the non-ergodic nature of the system when its evolution is time-reversible and the system retains a memory of the initial state under the action of small perturbations. This behavior reveals the presence of a time crystalline order in the autonomous system. In this regime, the system is a new class of time crystals - autonomous discrete time crystals. An increase in the coupling strength leads to a transition from periodic oscillations to an exponential decay in time of the Loschmidt echo. This corresponds to the transition from non-ergodic behavior to ergodic one in the system, and is accompanied by the disappearance of time crystalline order. We demonstrate that at the transition point the time-averaged variance of the number of photons reaches a maximum, which serves as a signature of the transition. We show that such a transition can also be observed when changing the number of degrees of freedom in the resonator, which is achieved by changing its length.