Non-Resonant Boundary Time Crystals from Quantum Synchronization Breakdown

TL;DR

This paper introduces a Liouvillian framework to classify driven-dissipative quantum systems, revealing a phase transition from quantum synchronization to boundary time crystals, with distinct behaviors depending on the system's background attractor.

Contribution

It provides a novel classification of quantum synchronization breakdown as a Hopf-type transition into boundary time crystals using a background-based Liouvillian approach.

Findings

Quantum synchronization breaks down via a Hopf transition.

Robust non-resonant boundary time crystals occur with self-sustained oscillators.

Time crystals are supported only at resonance with polar fixed points.

Abstract

Quantum synchronization (QS) in dissipative systems is often inferred from smooth phase locking, leaving open whether its breakdown constitutes a genuine nonequilibrium transition. Here we introduce a Liouvillian framework that classifies driven-dissipative dynamics by the structure of the undriven dissipative background and show that QS breaks down via a Hopf-type dynamical phase transition into a boundary time crystal (BTC). The character of this transition is determined by the background attractor: systems with a self-sustained oscillator (SSO) support robust non-resonant BTCs, whereas those with a polar fixed point (PFP) sustain BTCs only at resonance and lose them under detuning. We identify sharp dynamical and spectral signatures of the QS-BTC transition and thereby establish, within U(1)-symmetric collective-spin Lindbladians driven by a single coherent tone, a background-based allowed/forbidden criterion that unifies QS, its breakdown, and time-crystalline order within a single Liouvillian framework.