Quantum System Monitoring Reveals Deterministic Limits of Bosonic Dynamics.

Monitored bosonic systems exhibiting infinite-range interactions demonstrate deterministic fluctuation limits along individual trajectories under semiclassical conditions, allowing exact solutions. This explains the observed concurrence of entanglement and dissipative phase transitions, as verified through the analysis of a Bose-Hubbard dimer and a collective spin system.

The behaviour of quantum systems under continuous observation presents a significant challenge to conventional understanding, often leading to the expectation of increased randomness. However, recent research suggests a surprising degree of predictability can emerge even within these seemingly chaotic processes. Zejian Li, from The Abdus Salam International Center for Theoretical Physics, alongside Anna Delmonte of SISSA, and Rosario Fazio, affiliated with both The Abdus Salam International Center for Theoretical Physics and Dipartimento di Fisica “E. Pancini”, Universitá di Napoli “Federico II”, investigate this phenomenon in a study titled “Emergent deterministic entanglement dynamics in monitored infinite-range bosonic systems”. Their work focuses on infinite-range interacting bosonic systems – systems where particles interact with all others regardless of distance – and demonstrates that, under specific conditions, the fluctuations observed along individual quantum trajectories become predictable, exhibiting deterministic behaviour.

This predictability arises from a hierarchical structure within the equations governing the system’s evolution, offering a theoretical explanation for previously observed connections between entanglement transitions – changes in the quantum correlation between particles – and dissipative phase transitions, where energy is lost from the system. The researchers illustrate these findings using models of a Bose-Hubbard dimer, a simple system of two interacting bosons, and a collective spin system.

Recent investigations into the behaviour of interacting bosonic systems, where particles exhibit wave-like properties and occupy the same quantum state, reveal a predictable limit to the fluctuations observed when tracking individual quantum trajectories. This offers a powerful analytical tool for understanding complex quantum phenomena, particularly in systems with infinite-range interactions, meaning particles interact regardless of distance. Researchers demonstrate this deterministic limit applies to both quantum jump and state-diffusion unravelings, computational methods used to simulate quantum dynamics by tracing numerous possible evolutionary paths.

Quantum jumps and state diffusions are stochastic processes, meaning they involve inherent randomness. However, as the number of particles in the system increases, the collective behaviour becomes increasingly predictable, converging towards a deterministic limit described by classical-like equations of motion. This allows researchers to bypass computationally expensive simulations of individual quantum trajectories and instead focus on analysing the average behaviour of the system. The work establishes a clear connection between quantum entanglement, a correlation between particles that persists even when separated, and dissipative phase transitions, changes in the system’s macroscopic properties due to energy loss.

Investigations utilise both the Bose-Hubbard model, a standard framework for describing interacting bosons in a lattice, and a collective spin system as test cases, confirming the broad applicability of this approach. The hierarchical structure inherent in the derived equations of motion explains why entanglement and dissipative phase transitions coincide in numerical simulations of finite-sized systems, offering a deeper theoretical understanding of the interplay between quantum coherence and energy dissipation. This suggests that the loss of quantum coherence, measured by entanglement, drives the system towards a new phase with different macroscopic properties.

Researchers demonstrate the derivation and validation of deterministic equations of motion for collective spin systems, which originate from a quantum master equation that describes the time evolution of a quantum system, and employ a mean-field approximation within a large-N limit. The mean-field approximation simplifies the many-body problem by replacing interactions between individual particles with an average interaction field. The large-N limit assumes the number of particles (N) is very large, further simplifying the calculations. This approach successfully replaces quantum operators, mathematical entities representing physical observables, with their expectation values, which represent the average value of the observable. This yields a closed-form description of the system’s dynamics, specifically tracking the angular coordinates and variances, which characterise the orientation and spread of the spins. The resulting equations incorporate angle-dependent coefficients crucial for accurately modelling the system’s behaviour given specific Hamiltonian and dissipation parameters. The Hamiltonian defines the total energy of the system, while dissipation parameters describe the rate at which energy is lost.

Benchmarking confirms the accuracy of these deterministic equations through comparison with numerical simulations utilising the quantum jump unraveling method. The entanglement entropy, a measure of the degree of entanglement between particles, serves as a key metric for validating the agreement between the two approaches. Results indicate strong agreement, particularly as the number of spins (N) increases. This suggests the validity of the approximations employed and confirms the deterministic equations accurately capture the qualitative behaviour of the system, including the emergence of the time-crystal phase, a state of matter exhibiting periodic behaviour in time without any external driving force.

The successful derivation and validation of these deterministic equations provide an efficient and accurate method for studying the dynamics of collective spin systems, facilitating the prediction of system behaviour across various parameter regimes and offering valuable insights into phenomena such as time-crystal formation and dissipative phase transitions. Future work should focus on extending this framework to explore more complex system interactions and geometries, investigating the impact of noise and imperfections on the deterministic behaviour, and applying this methodology to many other body systems to broaden our understanding of collective quantum phenomena.