We study monitored quantum dynamics of infinite-range interacting bosonic systems in the thermodynamic limit. We show that under semiclassical assumptions, the quantum fluctuations along single monitored trajectories adopt a deterministic limit for both quantum-jump and state-diffusion unravelings, and they can be exactly solved. In particular, the hierarchical structure of the equations of motion explains the coincidence of entanglement criticalities and dissipative phase transitions found in previous finite-size numerical studies. We illustrate the findings on a Bose-Hubbard dimer and a collective spin system.
We study monitored quantum dynamics of infinite-range interacting bosonic systems in the thermodynamic limit. We show that under semiclassical assumptions, the quantum fluctuations along single monitored trajectories adopt a deterministic limit for both quantum-jump and state-diffusion unravelings, and they can be exactly solved. In particular, the hierarchical structure of the equations of motion explains the coincidence of entanglement criticalities and dissipative phase transitions found in previous finite-size numerical studies. We illustrate the findings on a Bose-Hubbard dimer and a collective spin system.
A key challenge in observing measurement-induced phase transitions is the mitigation of the post-selection barrier, which causes the reproducibility of specific sequences of measurement readouts—the trajectory—to be exponentially small in system size. Recent studies suggest that some classes of monitored infinite-range systems alleviate this problem by exhibiting a fast saturation of entanglement, resulting in only a polynomial post-selection overhead. This paper answers whether this feature is inherent in infinite-range systems, due to their underlying semiclassical dynamics. We consider three experimentally relevant monitored models: a Tavis-Cummings model, a Superradiance model, and a Bose-Hubbard dimer, each exhibiting nontrivial monitored dynamics. We unveil the occurrence of entanglement phase transitions in these models, showing how the saturation time is strongly affected by bistability regions, which also prevent the mitigation of the post-selection barrier. Finally, we propose experimental realizations of these models, providing a discussion of post selection from an experimental perspective.
The competition between unitary quantum dynamics and dissipative stochastic effects, as emerging from continuous-monitoring processes, can culminate in measurement-induced phase transitions. Here, a many-body system abruptly passes, when exceeding a critical measurement rate, from a highly entangled phase to a low-entanglement one. We consider a different perspective on entanglement phase transitions and explore whether these can emerge when the measurement process itself is modified, while keeping the measurement rate fixed. To illustrate this idea, we consider a noninteracting fermionic system and focus on diffusive detection processes. Through extensive numerical simulations, we show that, upon varying a suitable unraveling parameter---interpolating between measurements of different quadrature operators---the system displays a transition from a phase with area-law entanglement to one where entanglement scales logarithmically with the system size. Our findings may be relevant for tailoring quantum correlations in noisy quantum devices and for conceiving optimal classical simulation strategies.
Dissipation generally leads to the decoherence of a quantum state. In contrast, numerous recent proposals have illustrated that dissipation can also be tailored to stabilize many-body entangled quantum states. 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. Specifically, we analyze the competition between two different dissipation channels arising from two incompatible continuous monitoring protocols. 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. 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. 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.
In this paper we consider a bosonic Josephson junction described by a two-mode Bose-Hubbard model, and we thoroughly analyze a quantum phase transition occurring in the system in the limit of infinite bosonic population. We discuss the relation between this quantum phase transition and the dynamical bifurcation occurring in the spectrum of the discrete self-trapping equations describing the system at the semiclassical level. In particular, we identify five regimes depending on the strength of the effective interaction among bosons, and study the finite-size effects arising from the finiteness of the bosonic population. We devote special attention to the critical regime which reduces to the dynamical bifurcation point in the thermodynamic limit of infinite bosonic population. Specifically, we highlight an anomalous scaling in the population imbalance between the two wells of the trapping potential, as well as in two quantities borrowed from quantum information theory, i.e., the entropy of entanglement and the ground-state fidelity. Our analysis is not limited to the zero-temperature case, but considers thermal effects as well.
Equilibrium phase transitions usually emerge from the microscopic behavior of many-body systems and are associated to interesting phenomena such as the generation of long-range order and spontaneous symmetry breaking. They can be defined through the non-analytic behavior of thermodynamic potentials in the thermodynamic limit. This limit is obtained when the number of available configurations of the system approaches infinity, which is conventionally associated to spatially-extended systems formed by an infinite number of degrees of freedom (infinite number of particles or modes). Taking previous ideas to the extreme, we argue that such a limit can be defined even in non-extended systems, providing a specific example in the simplest form of a single-mode bosonic Hamiltonian. In contrast to previous non-extended models, the simplicity of our model allows us to find approximate analytical expressions that can be confronted with precise numerical simulations in all the parameter space, particularly as close to the thermodynamic limit as we want. We are thus able to show that the system undergoes a change displaying all the characteristics of a second-order phase transition as a function of a control parameter. We derive critical exponents and scaling laws revealing the universality class of the model, which coincide with that of more elaborate non-extended models such as the quantum Rabi or Lipkin-Meshkov-Glick models. Analyzing our model, we are also able to offer insights into the features of this type of phase transitions, by showing that the thermodynamic and classical limits coincide. In other words, quantum fluctuations must be tamed in order for the system to undergo a true phase transition.
We propose a variational framework for the efficient simulation of stochastic quantum dynamics in interacting bosonic systems. Our method generalizes the time-dependent variational principle to accommodate quantum state diffusion processes and is tailored for describing quantum trajectories unraveled from a master equation. We adopt a wavefunction ansatz composed of a coherent superposition of Gaussian wavepackets, which allows the fully analytical assembly of the variational equations of motion while capturing both the quantum trajectory dynamics and the Lindblad evolution beyond semiclassical approximations. The method is carefully benchmarked on a Bose-Hubbard dimer, and applied to extended Bose-Hubbard lattices in both 1D and 2D. In particular, we show the emergence of a symmetry-breaking phase transition in 2D lattices, which is absent in 1D chains.