Nonreciprocal Synchronization of Active Quantum Spins

Active agents are capable of exerting nonreciprocal forces upon one another. For instance, one agent, say $A$, may attract another agent $B$ while $B$ repels $A$. These antagonistic nonreciprocal interactions have been extensively studied in classical systems, revealing a wealth of exciting phenomena such as novel phase transitions and traveling-wave states. Whether these phenomena can originate in quantum many-body systems is an open issue, and proposals for their realization are lacking. In this work, we present a model of two species of quantum spins that interact in an antagonistic nonreciprocal way of the attraction-repulsion type. We propose an implementation based on two atomic ensembles coupled via chiral waveguides featuring both braided and non-braided geometries. The spins are active due to the presence of local gain, which allows them to synchronize. In the thermodynamic limit, we show that nonreciprocal interactions result in a nonreciprocal phase transition to time-crystalline traveling-wave states, associated with spontaneous breaking of parity-time symmetry. We establish how this symmetry emerges from the microscopic quantum model. For a finite number of spins, signatures of the time-crystal phase can still be identified by inspecting equal-time or two-time correlation functions. Remarkably, continuous monitoring of the output field of the waveguides induces a quantum traveling-wave state: a time-crystalline state of a finite-size quantum system, in which parity-time symmetry is spontaneously broken. Our work lays the foundation to explore nonreciprocal interactions in active quantum matter.