We study the properties of a monitored ensemble of atoms driven by a laser field and in the presence of collective decay. The properties of the quantum trajectories describing the atomic cloud drastically depend on the monitoring protocol and are distinct from those of the average density matrix. By varying the strength of the external drive, a measurement-induced phase transition occurs separating two phases with entanglement entropy scaling subextensively with the system size. Incidentally, the critical point coincides with the superradiance transition of the trajectory-averaged dynamics. Our setup is implementable in current light-matter interaction devices, and most notably, the monitored dynamics is free from the postselection measurement problem, even in the case of imperfect monitoring.
We study the properties of a monitored ensemble of atoms driven by a laser field and in the presence of collective decay. The properties of the quantum trajectories describing the atomic cloud drastically depend on the monitoring protocol and are distinct from those of the average density matrix. By varying the strength of the external drive, a measurement-induced phase transition occurs separating two phases with entanglement entropy scaling subextensively with the system size. Incidentally, the critical point coincides with the superradiance transition of the trajectory-averaged dynamics. Our setup is implementable in current light-matter interaction devices, and most notably, the monitored dynamics is free from the postselection measurement problem, even in the case of imperfect monitoring.
We study the localization properties of continuously monitored dynamics and associated measurement-induced phase transitions in disordered quantum many-body systems on the basis of the quantum trajectory approach. By calculating the fidelity between random quantum trajectories, we demonstrate that the disorder and the measurement can lead to dynamical properties distinct from each other, although both have a power to suppress the entanglement spreading. In particular, in the large-disorder regime with weak measurement, we elucidate that the fidelity exhibits an anomalous power-law decay before saturating to the steady-state value. Furthermore, we propose a general method to access physical quantities for quantum trajectories in continuously monitored dynamics without postselection. It is demonstrated that this scheme drastically reduces the cost of experiments. Our results can be tested in ultracold atoms subject to continuous measurement and open the avenue to study the dynamical properties of localization, which cannot be understood from the stationary properties of the entanglement entropy.
We study the competing effects of collective generalized measurements and interaction-induced scrambling in the dynamics of an ensemble of spin-1/2 particles at the level of quantum trajectories. This setup can be considered as analogous to the one leading to measurement-induced transitions in quantum circuits. We show that the interplay between collective unitary dynamics and measurements leads to three regimes of the average Quantum Fisher Information (QFI), which is a witness of multipartite entanglement, as a function of the monitoring strength. While both weak and strong measurements lead to extensive QFI density (i.e., individual quantum trajectories yield states displaying Heisenberg scaling), an intermediate regime of classical-like states emerges for all system sizes where the measurement effectively competes with the scrambling dynamics and precludes the development of quantum correlations, leading to sub-Heisenberg-limited states. We characterize these regimes and the crossovers between them using numerical and analytical tools, and discuss the connections between our findings, entanglement phases in monitored many-body systems, and the quantum-to-classical transition.
The dynamics of a quantum-many body system subject to measurements is naturally described by an ensemble of quantum trajectories, which can feature measurement-induced phase transitions (MIPTs). This phenomenon cannot be revealed through ensemble-averaged observables, but it requires the ability to discriminate each trajectory separately, making its experimental observation extremely challenging. We explore the fate of MIPTs under an observer's reduced ability to discriminate each measurement outcome. This introduces uncertainty in the state of the system, causing observables to probe a restricted subset of trajectories rather than a single one. By introducing an exactly-solvable Liouvillian model, we examine how long-time spatial correlations are influenced by varying degrees of trajectory averaging. We compute exactly the correlation matrix, Liouvillian gap, and entanglement negativity to demonstrate that averaging over multiple realizations introduces an effective finite lengthscale, beyond which long-range correlations are suppressed. This suggests that partial averaging over trajectories conceals the critical features of individual realizations, thereby blurring away the signatures of distinct measurement-induced phases.
The interplay of unitary evolution and local measurements in many-body systems gives rise to a stochastic state evolution and to measurement-induced phase transitions in the pure state entanglement. In realistic settings, however, this dynamics may be spoiled by decoherence, e.g., dephasing, due to coupling to an environment or measurement imperfections. We investigate the impact of dephasing and the inevitable evolution into a non-Gaussian, mixed state, on the dynamics of monitored fermions. We approach it from three complementary perspectives: (i) the exact solution of the conditional master equation for small systems, (ii) quantum trajectory simulations of Gaussian states for large systems, and (iii) a renormalization group analysis of a bosonic replica field theory. For weak dephasing, constant monitoring preserves a weakly mixed state, which displays a robust measurement-induced phase transition between a critical and a pinned phase, as in the decoherence-free case. At strong dephasing, we observe the emergence of a new scale describing an effective temperature, which is accompanied with an increased mixedness of the fermion density matrix. Remarkably, observables such as density-density correlation functions or the subsystem parity still display scale invariant behavior even in this strongly mixed phase. We interpret this as a signature of gapless, classical diffusion, which is stabilized by the balanced interplay of Hamiltonian dynamics, measurements, and decoherence.
Monitored quantum systems undergo measurement-induced phase transitions (MiPTs) stemming from the interplay between measurements and unitary dynamics. When the detector readout is postselected to match a given value, the dynamics is generated by a non-Hermitian Hamiltonian with MiPTs characterized by different universal features. Here, we derive a stochastic Schrödinger equation based on a microscopic description of continuous weak measurement. This formalism connects the monitored and postselected dynamics to a broader family of stochastic evolution. We apply the formalism to a chain of free fermions subject to partial postselected monitoring of local fermion parities. Within a two-replica approach, we obtain an effective bosonized Hamiltonian in the strong postselected limit. Using a renormalization group analysis, we find that the universality of the non-Hermitian MiPT is stable against a finite (weak) amount of stochasticity. We further show that the passage to the monitored universality occurs abruptly at finite partial postselection, which we confirm from the numerical finite size scaling of the MiPT. Our approach establishes a way to study MiPTs for arbitrary subsets of quantum trajectories and provides a potential route to tackle the experimental postselected problem.
Entanglement phase transitions driven by quantum measurements have emerged as a central paradigm in open quantum many-body physics. Such phase transitions are well established for systems with finite local Hilbert-space dimensions, such as qubits and fermions, while their realization in bosonic systems with unbounded local occupation numbers remains poorly understood. Even in the absence of interactions, number states of bosons are intrinsically non-Gaussian, preventing the use of standard correlation-matrix approaches. To address this problem, we develop a replica-free Keldysh field-theoretic framework that expresses the Renyi entropy of bosonic systems initialized in on-site Fock states in terms of permanents of matrices constructed from single-particle Green's functions. Applying this framework to a continuously monitored one-dimensional cross-stitch lattice conditioned on the no-click trajectory, we uncover a transition from volume-law to logarithmic entanglement scaling. We show that the transition is controlled by a restructuring of the non-Hermitian spectrum that changes the number of long-lived modes from extensive to finite. In the strongly monitored regime, bosons dynamically condense into a microscopic number of slowest-decaying modes, producing logarithmic entanglement scaling, whereas an extensive manifold of long-lived modes at weak monitoring gives rise to volume-law entanglement. Our results establish a distinct mechanism for measurement-induced entanglement transitions in free bosonic systems and provide a computationally efficient diagnostic of the measurement-induced bosonic condensation.