Quantum Phase Transitions Revealed By Fisher Information’s Long-Time Average

Recent advances in controlling isolated quantum systems facilitate the investigation of many-body quantum systems as they evolve away from equilibrium, generating increased interest in dynamical quantum phase transitions (DQPTs). These transitions describe abrupt changes in the long-term behaviour of a quantum system as external parameters are altered, offering a unique perspective on non-equilibrium quantum dynamics that mirrors equilibrium phase transitions but occurs in time rather than in static properties. Understanding DQPTs requires examining how quantum systems respond to sudden changes, often termed ‘quenches’, and researchers typically probe these transitions by monitoring the long-time average of an observable to identify a point where the system’s behaviour qualitatively shifts.

The quantum Fisher information (QFI) emerges as a valuable tool for characterising these non-equilibrium dynamics, quantifying the sensitivity of a quantum state to changes in a parameter and establishing a fundamental limit on the precision of parameter estimation. Beyond its role in quantum metrology, the QFI links to out-of-time-ordered correlators, central to understanding phenomena like chaos, thermalisation, and information scrambling, broadening its applicability as a diagnostic for DQPTs and a probe of underlying quantum dynamics.

Researchers currently investigate DQPTs in systems exhibiting well-defined classical behaviours, with a particular focus on spinor Bose-Einstein condensates and the Lipkin-Meshkov-Glick model. These transitions differ from traditional phase transitions by manifesting in systems driven by time-dependent forces. The investigation centres on diagnosing these transitions using the long-time average of the QFI, allowing researchers to pinpoint the critical points where the system’s behaviour undergoes a qualitative shift.

By employing both mean-field and semiclassical approximations, researchers aim to establish universal behaviour of the QFI across different systems, independent of specific details. Mean-field theory simplifies the many-body problem by replacing interactions between particles with an average field experienced by each particle, reducing computational complexity. Semiclassical approximations bridge the gap between quantum and classical mechanics by treating certain quantum variables classically.

The Lipkin-Meshkov-Glick model serves as a complementary system to the spinor Bose-Einstein condensate, providing a tractable theoretical framework for validating findings obtained from the more complex condensate system. This strengthens the claim of universality and increases confidence in the robustness of the results.

This research focuses on DQPTs in systems exhibiting clear classical analogues, utilising spinor Bose-Einstein condensates (BECs) and the Lipkin-Meshkov-Glick model as primary investigative tools. Researchers diagnose these transitions by monitoring the long-time average of the QFI, serving as a robust indicator of the transition’s occurrence as the QFI exhibits an abrupt change precisely at the DQPT point.

The study demonstrates that the long-time average of the QFI reveals universal behaviour, persisting across diverse physical systems. Spinor Bose-Einstein condensates are a state of matter formed when bosons—particles with integer spin—are cooled to temperatures near absolute zero, exhibiting unique properties due to the collective behaviour of the atoms, with the ‘spinor’ aspect referring to the internal angular momentum, or spin, of the atoms. The Lipkin-Meshkov-Glick model serves as a theoretical benchmark for understanding the behaviour of interacting quantum systems, providing a consistent methodology for identifying DQPTs using the QFI, independent of the specific system under investigation.

This approach provides a valuable tool for studying non-equilibrium quantum phenomena in various contexts, including quantum simulation, quantum information processing, and condensed matter physics, contributing to a deeper understanding of the fundamental principles governing DQPTs and their potential applications. This research establishes a robust connection between DQPTs and the long-time average of the QFI, demonstrating that the QFI exhibits an abrupt change precisely at the DQPT point in both the spinor Bose-Einstein condensate and the Lipkin-Meshkov-Glick model, providing a clear diagnostic for these transitions. The investigation successfully employs both mean-field and semiclassical approximations to reveal universal behaviour in the long-time average of the QFI, suggesting that the observed relationship between the QFI and DQPTs transcends specific system details, strengthening the validity of the QFI as a reliable indicator of DQPTs.

Specifically, the research highlights the utility of the QFI in diagnosing DQPTs in systems possessing well-defined classical limits, which is crucial because many real-world quantum systems exhibit behaviour that can be approximated by classical physics under certain conditions. This enables the identification of DQPTs in these systems, facilitating an understanding of their behaviour and potential applications. Future work should focus on extending this analysis to more complex many-body systems and exploring the impact of environmental noise on the observed relationship between the QFI and DQPTs, investigating the potential for utilising this diagnostic tool in experimental settings, particularly in the context of quantum sensing and metrology, and exploring the connection between the QFI and other measures of quantum coherence for a more complete understanding of the dynamics underlying DQPTs. The findings contribute to a growing body of knowledge concerning non-equilibrium quantum dynamics and the emergence of collective behaviour in many-body systems, providing valuable insights into the behaviour of complex quantum systems and opening up new possibilities for their control and manipulation by establishing a clear link between a measurable quantity, the QFI, and a fundamental quantum phenomenon, the DQPT.