Analyzing Nonequilibrium Quantum States through Snapshots with Artificial Neural Networks

Current quantum simulation experiments are starting to explore nonequilibrium many-body dynamics in previously inaccessible regimes in terms of system sizes and timescales. Therefore, the question emerges as to which observables are best suited to study the dynamics in such quantum many-body systems. Using machine learning techniques, we investigate the dynamics and, in particular, the thermalization behavior of an interacting quantum system that undergoes a nonequilibrium phase transition from an ergodic to a many-body localized phase. We employ supervised and unsupervised training methods to distinguish nonequilibrium from equilibrium data, using the network performance as a probe for the thermalization behavior of the system. We test our methods with experimental snapshots of ultracold atoms taken with a quantum gas microscope. Our results provide a path to analyze highly entangled large-scale quantum states for system sizes where numerical calculations of conventional observables become challenging.

Figure 1

Machine learning many-body localization. The Bose-Hubbard model with a quasiperiodic disorder potential exhibits a MBL phase, where thermalization breaks down, as the disorder strength is increased beyond a critical value. (a) We study the dynamics of the system after a quench for different disorder strengths by evaluating snapshots from a quantum gas microscope with neural networks. (b) A neural network is trained to distinguish exact diagonalization snapshots at $W/J=0.3$ and $W/J=11$ for $U/J=2.9$ and a system with 8 and 12 sites at time $tJ=100$ after a global quench. After the training process is finished, snapshots at intermediate values of the disorder strength are used as input. The plot shows the resulting classification for numerical data (shaded band) as well as experimental snapshots (symbols). As the system size is increased, the fraction of snapshots classified as MBL begins to increase at larger values of $W$, indicating the transition in the finite-size system. The accuracies are averaged over two independent runs and the errors denote one standard error of the mean (SEM).

Figure 2

Learning thermalization. A system with eight sites and $U/J=2.9$ is initialized in a Mott-insulating state of one particle per site and the ensuing time evolution is investigated. In each time step, the neural network is trained to distinguish snapshots from the current time step from snapshots from a thermal state with the same energy density, both obtained from exact diagonalization. A high accuracy indicates that the current time step can be easily distinguished from the thermal state. (a) The resulting classification as “dynamics” versus “equilibrium” for $W/J=1.0$ and $W/J=7.3$, averaged over 12 different disorder realizations (shaded line). Experimental data from the dynamics after the quench is used as input at selected time steps (symbols). (b) Exact diagonalization results for disorder strengths between $W/J=1$ and $W/J=10$ for the full dynamics. (c) Classification as dynamics versus equilibrium at time $tJ=100$ for disorder strengths between $W/J=1$ and $W/J=10$. The results are averaged over ten independent runs and the error bars correspond to the SEM.

Figure 3

Confusion learning. Snapshots of the many-body quantum state of a system with 12 sites, $U/J=2.9$, and various disorder strengths $W/J$ are analyzed using the confusion learning scheme. A neural network is trained to label all snapshots with $W<W^{*}$ as “phase A” and the remainder as “phase B.” If a qualitative change in the data occurs, the accuracy will peak at an intermediate value of $W^{*}$. (a) The resulting accuracy at time $tJ=100$ after the global quench for training on numerically simulated data (shaded line) and sorting experimental data (symbols). Inset: same data after subtracting the accuracy for randomly labeled data. (b) The accuracy for repeating the training process for different time points during the dynamics after the quench using numerically simulated data. The results are averaged over ten independent runs and the error bars correspond to the error based on one SEM.