We consider a free fermionic chain with monitoring of the particle density on a single site of the chain and study the entanglement dynamics of quantum jump trajectories. We show that the entanglement entropy grows in time towards a stationary state which display volume law scaling of the entropy, in stark contrast with both the unitary dynamics after a local quench and the no-click limit corresponding to full post-selection. We explain the extensive entanglement growth as a consequence of the peculiar distribution of quantum jumps in time, which display superpoissonian waiting time distribution characterised by a bunching of quantum jumps followed by long dark intervals where no-clicks are detected, akin to the distribution of fluorescence light in a driven atom. We show that the presence of dark intervals is the key feature to explain the effect and that by increasing the number of sites which are monitored the volume law scaling gives away to the Zeno effect and its associated area law.
We consider a free fermionic chain with monitoring of the particle density on a single site of the chain and study the entanglement dynamics of quantum jump trajectories. We show that the entanglement entropy grows in time towards a stationary state which display volume law scaling of the entropy, in stark contrast with both the unitary dynamics after a local quench and the no-click limit corresponding to full post-selection. We explain the extensive entanglement growth as a consequence of the peculiar distribution of quantum jumps in time, which display superpoissonian waiting time distribution characterised by a bunching of quantum jumps followed by long dark intervals where no-clicks are detected, akin to the distribution of fluorescence light in a driven atom. We show that the presence of dark intervals is the key feature to explain the effect and that by increasing the number of sites which are monitored the volume law scaling gives away to the Zeno effect and its associated area law.
Recently, there has been interest in the dynamics of monitored quantum systems using linear jump operators related to the creation or annihilation of particles. Here, we study the dynamics of the entanglement entropy under quantum jumps that induce local particle losses in a model of free fermions with hopping and Z2 pairing. We solve the nonunitary dynamics using the recently developed Faber polynomial method and explore the different steady-state entanglement regimes by tuning the pairing strength, thus interpolating between monitored free fermions coherently driven by a particle-number-conserving Hamiltonian and a parity-conserving one. In the absence of pairing, all quantum trajectories approach the vacuum at long times, with the entanglement entropy showing nonmonotonic behavior over time that we capture with a phenomenological quasiparticle ansatz. In this regime, quantum jumps play a key role, and we highlight this by exactly computing their waiting-time distribution. On the other hand, the interplay between losses and pairing gives rise to quantum trajectories with entangled steady states. We show that by tuning the system parameters, a measurement-induced entanglement transition occurs, where the entanglement entropy scaling changes from logarithmic to area law. We compare this transition with the one derived in the no-click limit and observe qualitative agreement in most of the phase diagram. Furthermore, the statistics of entanglement gain and loss are analyzed to better understand the impact of the linear jump operators.
We study the entanglement dynamics of a one-dimensional chain of monitored non-interacting complex fermions with random power-law hopping characterized by a decay exponent α. For α≲1, in stark contrast with the case of hopping to nearest neighbors, the scaling of the entanglement entropy (EE) of the steady state with system size L is faster than logarithmic for any monitoring or disorder strength and it tends towards a linear (volume-law) scaling for sufficiently small α≲1/2. For α>3/2, the EE is in the area-law phase, namely, no scaling with L, for any monitoring strength. For 1<α≲3/2, we identify an α-dependent measurement-induced phase transition (MIPT) at a critical value of the monitoring strength separating the mentioned area-law and sub-volume-law phases. At this critical point, the EE scales logarithmically with system size, and the density-density correlation function, closely related to the EE, exhibits multifractal features. These results highlight the importance of superdiffusive classical hopping in the entanglement dynamic of quantum many-body systems and also help differentiate its role with respect to conventional sources of entanglement such as genuine quantum non-locality.
We compute the entanglement dynamics across a monitored quantum point contact, where particle losses are recorded on a given site, and demonstrate how this single-site local monitoring substantially reshapes the entanglement production. Contrary to the unitary case, where entanglement entropy grows logarithmically in time, here we find first a linear growth, up to a maximum value displaying volume-law scaling, and then a slow decay to zero, as the system empties out. We capture this crossover using a quasiparticle picture, where the first linear growth arises due to an emergent bias voltage established by the losses, which eventually decays away as the system depletes. We connect our results to studies of the Page curve and to experimentally relevant probes, via full counting statistics of charge transfer across a subregion, with only a single channel to unravel leading to a favorable scaling of the postselection overhead. Natural platforms for this setting include mesoscopic systems and ultracold atoms.
Recent years have witnessed major progress in the understanding of monitored quantum dynamics, an open quantum setting where the environment is a measurement apparatus that probes the system locally. For the resulting quantum trajectory, unitary evolution and local measurements have competing tendencies in the building of quantum correlations: while the coherent interactions in the former increase the entanglement, the resolving nature of the local measurement collapses the state of the system. Here, the authors study the interplay of monitoring and system dynamics by focusing on the monitored quantum Ising chain as the measurement strength is varied, going from the regime of weak detection, corresponding to the limit of infinite small jumps, to the case of post-selected no-jump case. A sharp transition is found between a logarithmic law phase to an area law phase for the entanglement entropy, which occurs at the same value of the measurement rate in the two limiting situations.
For one-dimensional non-interacting complex fermions, we compute numerically the probability distribution of the change in the entanglement entropy (EE) after saturation, resulting from a single measurement of the occupation number by using different measurements protocols. In the thermodynamic limit, the system is in the area-law phase for any monitoring strength, however we can also study the distribution in the critical phase characterized by a logarithmic scaling of the EE with system size by considering a sufficiently weak monitoring strength. For a quantum state diffusion protocol, where the change in EE is defined between two consecutive time steps, the distribution, Gaussian for weak monitoring, gradually develops symmetric exponential tails. For strong monitoring, the core turns from Gaussian to strongly peaked at zero suggesting the dominance of quantum Zeno effect. For the quantum jump and the projective measurement protocols, we observe clear deviations from Gaussianity characterized by broader and asymmetric tails, exponential for positive values of the change, and a peak at zero, likely a precursor of Zeno effect, that increases with the system size and the monitoring strength. Intriguingly, the distribution is spatially inhomogeneous. For sites around the boundary separating the subsystems defining the EE, the distribution is close to Gaussian with a broad support while for the rest of sites has asymmetric exponential tails and a much narrower support. As the monitoring strength increases, the full distribution is controlled by the boundary sites.
We study the entanglement entropy of the quantum trajectories of a free fermion chain under continuous monitoring of local occupation numbers. We propose a simple theory for entanglement entropy evolution from disentangled and highly excited initial states. It is based on generalized hydrodynamics and the quasi-particle pair approach to entanglement in integrable systems. We test several quantitative predictions of the theory against extensive numerics and find good agreement. In particular, the volume law entanglement is destroyed by the presence of arbitrarily weak measurement.