Detecting nonequilibrium phase transitions via continuous monitoring of space-time trajectories and autoencoder-based clustering

TL;DR

This work presents a machine-learning approach to detect nonequilibrium phase transitions from the measurement time-records of continuously-monitored quantum systems, and benchmarks it using the quantum contact process, a model featuring an absorbing-state phase transition.

Abstract

The characterization of collective behavior and nonequilibrium phase transitions in quantum systems is typically rooted in the analysis of suitable system observables, so-called order parameters. These observables might not be known a priori, but they may in principle be identified through analyzing the quantum state of the system. Experimentally, this can be particularly demanding as estimating quantum states and expectation values of quantum observables requires a large number of projective measurements. However, open quantum systems can be probed in situ by monitoring their output, e.g. via heterodyne-detection or photon-counting experiments, which provide space-time resolved information about their dynamics. Building on this, we present a machine-learning approach to detect nonequilibrium phase transitions from the measurement time-records of continuously-monitored quantum systems. We benchmark our method using the quantum contact process, a model featuring an absorbing-state phase transition, which constitutes a particularly challenging test case for the quantum simulation of nonequilibrium processes.

Figures (4)

• Figure 1: Quantum trajectories across the phase diagram of the quantum contact process. The top part of the figure sketches an experimental setting in which spatially resolved emissions from a many-body system (S) are monitored continuously in time. The sketched quantum trajectories labeled by S show qualitatively the dynamics of the expectation of a local order parameter (here the local density of active sites). The latter display clearly distinct structures in the subcritical, critical, and supercritical regimes of the quantum contact process. In contrast, space-time records O of the output from continuous monitoring [here the real part of the complex heterodyne current, Eq. (\ref{['eq:Jhet']})] appear noisy and structureless throughout all dynamical regimes. While trajectories of the order parameter are usually inaccessible due to postselection overheads, trajectories of the output signal are directly available in experiments. As we show, they can provide useful information to detect nonequilibrium phase transitions.
• Figure 2: Classification of quantum trajectories. Individual trajectories of the one-dimensional quantum contact process with $N=30$ sites are classified using an autoencoder. The trajectories are encoded by two latent variables $z_1$, $z_2$, which span the latent space where trajectories from the active and absorbing phases separate. As mentioned in the main text, the latent variables are assumed to follow a bimodal multivariate Gaussian (sketched in in the central panel), which allows to assign to each encoded trajectory a probability of belonging to the component associated with the active phase. (a) Latent space representation of $1000$ trajectories constructed from the local density of active sites $S_k(t)$. Note that such trajectories are not practically accessible in experiment albeit simple to compute in numerical calculations. The probability of belonging to the active phase exhibits a sharp crossover as a function of the control parameter $\Omega/\gamma$. (b) Corresponding analysis based on the absolute value of the time-averaged heterodyne current: $|\overline{O_k(t)}|$. These quantum trajectories can be in principle accessed in an experiment. For both analyses the density of data points in the interval $\Omega/\gamma \in [4.0,\,7.0]$ is increased to better resolve the transition region.
• Figure 3: Neural network architecture and training. (a) Architecture of the autoencoder used to compress the full space-time trajectories into a two-dimensional latent representation $z_1,z_2$. The network consists of layers with dimensions $6000 \times 1000 \times 2 \times 1000 \times 6000$. For the time-averaged heterodyne current the model size is reduced to $5730 \times 1000 \times 2 \times 1000 \times 5730$ nodes. During training, trajectories $\mathbf{x}=(x_1,x_2,\dots)$ are processed by the autoencoder, and the network parameters $\theta$ are updated at each iteration to minimize the reconstruction loss $L(\mathbf{x},\hat{\mathbf{x}})$ between the input $\mathbf{x}$ and its reconstruction $\hat{\mathbf{x}}$. The learning rate $\eta$ controls the step size of parameter updates $\theta\to\theta^\prime$ during optimization. (b) Summary of the training parameters.
• Figure 4: Critical scaling at the quantum contact process transition. (a) Probability of being associated with the active phase, extracted from trajectories of the local density of active sites $S_k(t)$ [see Fig. \ref{['fig:Clustering']}(a)]. A power-law behavior is fitted to the data in the interval $\omega\in[6.0,7.0]$. (b) Probability of being associated with the active phase, extracted from trajectories of the absolute value of the time-averaged heterodyne current $|\overline{O_k(t)}|$ [see Fig. \ref{['fig:Clustering']}(b)]. A power-law fit is performed over the interval $\omega\in[5.5,7.0]$. In both cases, the critical exponent $\beta^\text{AE}$ and the critical point $\omega^\text{AE}_c$ are treated as free parameters.