System Outputs Reveal Hidden Shifts in Dynamics

Scientists are developing new methods to identify critical changes in complex systems, moving beyond traditional analysis of system observables. Erik Fitzner from the Institut f ur Theoretische Physik, Universit at T ubingen, and Francesco Carnazza from Universit e Paris Cit e, CNRS, Mat eriaux et Ph enom`enes Quantiques, led a study demonstrating a technique for detecting nonequilibrium phase transitions by continuously monitoring the space-time trajectories of a system, in collaboration with Federico Carollo from Centre for Fluid and Complex Systems, Coventry University, and Igor Lesanovsky from School of Physics and Astronomy and Centre for the Mathematics and Theoretical Physics of Quantum nonequilibrium Systems, The University of Nottingham. This research is significant because it offers a way to characterise these transitions without needing prior knowledge of key system properties, instead relying on analysing dynamic output data, and represents a powerful tool for understanding complex behaviours in diverse physical systems.

Understanding how systems change state just became far simpler. Upon observing a system’s continuous evolution, rather than snapshots, reveals transitions previously hidden within noise. This new technique offers a powerful way to chart complex behaviour and identify critical points in active processes. Scientists are increasingly focused on understanding collective behaviours and phase transitions in systems that are not at equilibrium.

By characterising these phenomena traditionally relies on identifying suitable system observables, termed order parameters, which reveal changes in the system’s state as it approaches a critical point. To determine these order parameters experimentally, however, presents a significant challenge, often requiring numerous precise measurements of the framework’s quantum state.

Open quantum systems offer a different avenue for investigation, as their dynamics can be monitored by observing their output via methods like heterodyne-detection or photon-counting, providing information about how the system evolves in both space and time. Scientists have developed a new approach to detect nonequilibrium phase transitions by analysing the time-records generated from continuously-monitored systems.

This method moves beyond the need to predefine or even know the relevant order parameter, instead learning directly from the system’s output — the technique was initially tested using the contact process, a well-known model exhibiting an absorbing-state phase transition. A particularly difficult scenario for simulating systems far from equilibrium, and a machine-learning framework has been introduced that can discern collective properties of open quantum systems directly from the structure of quantum trajectories. The paths the system takes through its possible states.

Unlike conventional methods, this approach does not require a direct link between the measured output and any pre-defined system observable. The machine learning algorithm identifies patterns within the complex, space-time resolved data, effectively creating an emergent order parameter. Once trained, the system maps these trajectories onto a lower-dimensional space, clustering them according to the dynamical phase from which they originate.

Here, this dimensionality reduction allows for the identification of critical points and phase transitions from complex datasets. Without relying on prior knowledge of the system’s behaviour. Simulations using the quantum contact process and synthetic heterodyne measurements demonstrate the method’s ability to distinguish between absorbing and active phases — even in one dimension where such transitions are notoriously difficult to observe. In turn, this advancement promises new ways to analyse quantum systems and potentially unlock insights into a wide range of physical phenomena.

Unsupervised learning of quantum trajectories reveals nonequilibrium phase transitions

Heterodyne detection and The assessment of resulting quantum trajectories underpin this effort’s methodology for identifying nonequilibrium phase transitions. Meanwhile, scientists generated synthetic quantum trajectories using tensor networks, a computational technique for efficiently simulating quantum many-body systems. These trajectories mimicked the continuous monitoring of an open quantum system, specifically the quantum contact process, a model exhibiting an absorbing-state phase transition.

At the same time, the quantum contact process presents a demanding test case, even in one dimension, serving as a benchmark for evaluating the method’s performance. Here, the core of this approach lies in an unsupervised machine-learning framework, learning collective properties directly from the structure of the generated quantum trajectories, rather than relying on pre-defined system observables.

High-dimensional space-time resolved data from these trajectories were mapped onto a low-dimensional latent space, reducing the complexity of the data. Within this latent space, trajectories originating from the same dynamical phase clustered together. It to discern patterns indicative of phase transitions. The technique diverges from traditional approaches requiring repeated state preparation and projective measurements.

Continuous monitoring provides a stream of dynamical information encoded in the quantum trajectories, accessible without ensemble averaging or postselection. By exploiting the intrinsic dimensionality reduction capabilities of machine learning, The project bypasses the need for identifying a suitable order parameter a priori. At the same time, the real part of the complex heterodyne current was used as the output signal from the continuous monitoring process. Providing the raw data for the machine-learning algorithm.

The choice of the quantum contact process was deliberate, as its absorbing phase presents a significant challenge for both theoretical and experimental investigations. Once generated, these trajectories were fed into the unsupervised learning framework. The identification of critical signatures from complex datasets without dependence on predefined observables. At this stage, the algorithm effectively extracts effective order parameters directly from the quantum trajectories themselves.

Autoencoder latent space characterises phase transitions and critical behaviour

Analysis of trajectories from the quantum contact process revealed a clear separation of phases within the latent space generated by the autoencoder0.1000 trajectories exhibited a distinct colour gradient, demonstrating the model’s ability to differentiate between active and absorbing states. Classification based on these trajectories placed the critical point at (Ω/γ)AE c within the range of 5.5 to 6.5, closely aligned with previously estimated values of 5.9 ≤(Ω/γ)c ≤7.

Further investigation into the scaling behaviour near this critical point yielded an estimated critical exponent of βAE = 0.33 ±0.13, calculated from a power-law fit to the transition region. By applying the autoencoder to experimentally accessible heterodyne trajectories, after mitigating noise through a sliding average, again produced a sharp crossover between phases.

This analysis pinpointed the critical point to a range of (Ω/γ)AE c between 5.0 and 6.5 — demonstrating comparable accuracy to The assessment of the local order parameter. The power-law fit applied to these heterodyne trajectories resulted in a refined critical value of (Ω/γ)AE c = 5.4 ±0.1 and a critical exponent of βAE = 0.28 ±0.05. These the autoencoder-based clustering algorithm effectively identifies nonequilibrium phase transitions and accurately estimates critical parameters using readily obtainable quantum trajectories, and the consistency between analyses using both the local order parameter and the output signal highlights the method’s versatility and potential for broader application.

Inferring systemic shifts from output data in non-equilibrium dynamics

For years, detecting when a system undergoes a fundamental shift, a phase transition, has demanded painstaking measurement of its internal state. This effort offers a bypass, inferring these transitions simply by watching what the system outputs, rather than probing its inner workings directly. Instead of needing to know what to look for. Researchers have devised a method to identify these critical points from the continuous stream of data a framework naturally emits.

This is a considerable step forward, particularly for complex systems where direct observation is impractical or even impossible. Identifying phase transitions isn’t merely a technical challenge. Many real-world systems, from ecological networks to financial markets, exist far from equilibrium, making traditional analytical approaches difficult to apply.

These systems don’t settle into neat, predictable states, pinpointing the moment of change requires techniques capable of handling noisy, active data. By employing machine learning, specifically autoencoders, scientists can extract meaningful signals from these ongoing outputs, revealing hidden order within apparent chaos. While the demonstrated accuracy with the ‘contact process’ model is encouraging, extending this approach to genuinely complex, high-dimensional systems will be a significant test.

Real-world data is often incomplete, corrupted by noise, and subject to unforeseen influences, unlike the simulated environment used here. Also, the reliance on autoencoders, while effective, introduces a degree of ‘black box’ interpretation. By understanding why the algorithm identifies a particular transition. Whether that aligns with underlying physical mechanisms, remains an open question.

Once further refinements are made, this technique could find applications in diverse fields — beyond physics, monitoring the health of power grids, predicting outbreaks of disease. Or even tracking the stability of social networks could all benefit from a method that can detect subtle shifts in system behaviour from external observations, and the focus is on improving the algorithm’s ability to discern critical points and exponents. But future work might explore combining this approach with other analytical tools, creating a more complete and interpretable picture of complex system dynamics.