(a) We analyze the stochastic evolution of the system under continuous monitoring with two competing monitoring protocols, characterized by the registering of jump operators d j and c j with rates Λ and Γ, respectively. Our work provides an alternative perspective on the measurement-induced phase transition: the measurement can be viewed as monitoring and registering quantum jumps, offering an intriguing extension of these phase transitions through the long-established realm of quantum optics.Ī schematic illustration of our setup and the scaling of trajectory entanglement entropy. By studying the unraveling of stochastic quantum trajectories associated with the continuous monitoring protocols, we present a transition for the scaling of the averaged trajectory entanglement entropies, from critical scaling to area-law behavior. The first protocol locks the phase of neighboring sites upon registering a quantum jump, thereby generating a long-range entanglement through the system, while the second destroys the coherence via a dephasing mechanism. Specifically, we analyze the competition between two different dissipation channels arising from two incompatible continuous monitoring protocols. While the focus of these works has been primarily on engineering the nonequilibrium steady state, we investigate the buildup of entanglement in the quantum trajectories. In contrast, numerous recent proposals have illustrated that dissipation can also be tailored to stabilize many-body entangled quantum states. Dissipation generally leads to the decoherence of a quantum state.
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