Semiclassical quantum trajectories in the monitored Lipkin-Meshkov-Glick model

Monitored quantum system have sparked great interest in recent years due to the possibility of observing measurement-induced phase transitions (MIPTs) in the full-counting statistics of quantum trajectories. Here, we investigate the dynamics of the Lipkin-Meshkov-Glick model, composed of 𝑁 long-range interacting spins, under a weak external monitoring. In the thermodynamic limit, we derive a set of semiclassical stochastic equations describing the evolution of the expectation values of global spin observables. Our results show that the limit 𝑁→∞ does not commute with the long-time limit: while for any finite 𝑁 the average over trajectories is expected to converge towards a trivial steady state, in the thermodynamic limit a transition appears. The phase transition is not affected by postselection issues, as it is already visible at the level of ensemble averages. Within a semiclassical approach we derive a quantitative theoretical picture explaining the nature of the transition, and, remarkably, show that only the quantum trajectories of this model have a semiclassical limit, while the dynamics of the full many-body state does not.