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arXiv:2604.09528

Machine Learning Phase Field Reconstruction in a Bose-Einstein Condensate

Jackson Lee, Andrew J. Millis

Published Apr 10, 2026• 0 citations
arXivHTMLPDF

Abstract

A basic challenge in experimental physics is the extraction of information related to variables that are not directly measured. The challenge is particularly severe in quantum systems where one may be interested in correlations of operators that are not diagonal in the measurement basis. In this paper we take a step towards addressing this issue in the context of Boson superfluids, where standard in-situ imaging yields only the spatially resolved density, leaving the phase field - crucial for identifying topological defects such as vortices and confirming superfluidity - indirectly encoded. Previous work has shown that the location of vortices in the phase field may be detected, but has not solved the problems of fully reconstructing the phase or identifying the charge (vortex vs. antivortex). This paper shows that a combination of a deep machine learning (ML) model and classical computer vision post-processing steps can address this gap. We use realistic snapshots of the thermal state of a two-dimensional BEC in a harmonic trap using synthetic data obtained from projected Gross-Pitaevskii equation simulations to train a U-Net-based architecture to infer the absolute values of the phase field gradients from an observed density field, and then employ a separate ML model to locate the positions of the vortex cores and a post-processing graphical analysis to determine with high accuracy the phase field, including the quantized charge of each vortex.

DOI
10.48550/arxiv.2604.09528

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A basic challenge in experimental physics is the extraction of information related to variables that are not directly measured. The challenge is particularly severe in quantum systems where one may be interested in correlations of operators that are not diagonal in the measurement basis. In this paper we take a step towards addressing this issue in the context of Boson superfluids, where standard in-situ imaging yields only the spatially resolved density, leaving the phase field - crucial for identifying topological defects such as vortices and confirming superfluidity - indirectly encoded. Previous work has shown that the location of vortices in the phase field may be detected, but has not solved the problems of fully reconstructing the phase or identifying the charge (vortex vs. antivortex). This paper shows that a combination of a deep machine learning (ML) model and classical computer vision post-processing steps can address this gap. We use realistic snapshots of the thermal state of a two-dimensional BEC in a harmonic trap using synthetic data obtained from projected Gross-Pitaevskii equation simulations to train a U-Net-based architecture to infer the absolute values of the phase field gradients from an observed density field, and then employ a separate ML model to locate the positions of the vortex cores and a post-processing graphical analysis to determine with high accuracy the phase field, including the quantized charge of each vortex.
Abstract Quantum vortices in atomic Bose–Einstein condensates (BECs) are topological defects characterized by quantized circulation of particles around them. In experimental studies, vortices are commonly detected by time-of-flight imaging, where their density-depleted cores are enlarged. In this work, we describe a machine learning-based method for detecting vortices in experimental BEC images, particularly focusing on turbulent condensates containing irregularly distributed vortices. Our approach employs a convolutional neural network (CNN) trained solely on synthetic simulated images, eliminating the need for manual labeling of the vortex positions as ground truth. We find that the CNN achieves accurate vortex detection in real experimental images, thereby facilitating analysis of large experimental datasets without being constrained by specific experimental conditions. This novel approach represents a significant advancement in studying quantum vortex dynamics and streamlines the analysis process in the investigation of turbulent BECs.
Abstract Quantum vortices naturally emerge in rotating Bose–Einstein condensates (BECs) and, similarly to their classical counterparts, allow the study of a range of interesting out-of-equilibrium phenomena, such as turbulence and chaos. However, the study of such phenomena requires the determination of the precise location of each vortex within a BEC, which becomes challenging when either only the density of the condensate is available or sources of noise are present, as is typically the case in experimental settings. Here, we introduce a machine-learning-based vortex detector motivated by state-of-the-art object detection methods that can accurately locate vortices in simulated BEC density images. Our model allows for robust and real-time detection in noisy and non-equilibrium configurations. Furthermore, the network can distinguish between vortices and anti-vortices if the phase profile of the condensate is also available. We anticipate that our vortex detector will be advantageous for both experimental and theoretical studies of the static and dynamic properties of vortex configurations in BECs.
Quantum vortices in Bose-Einstein condensates (BECs) are essential phenomena in condensed matter physics, and precisely locating their positions, especially the vortex core, is a precondition for studying their properties. With the rise of machine learning, there is a possibility to expedite the localization process and provide accurate predictions. However, traditional machine learning requires particular considerable amount of manual data annotation, leading to uncontrollable accuracy. In this paper, we utilize the U-Net method to detect vortex positions accurately at the pixel level and propose an Automatic Correction Labeling (ACL) approach to optimize the acquisition of data sets for vortex localization in BECs. This approach addresses inaccuracies in the labeled vortex positions and improves the accuracy of vortex localization, especially the vortex core positions, while enhancing the tolerance for human mislabeling. The main process involves Rough Labeling [Formula: see text] Machine Learning [Formula: see text] Probability Region Search [Formula: see text] Data Relabeling [Formula: see text] Machine Learning again. The objective of ACL is to secure more accurate labeled data for model retraining. Through vortex localization experiments conducted in a two-dimensional Bose-Einstein condensate, our results establish the following: 1. Even under conditions of biased and missing manual annotations, U-Net can still accurately locate vortex positions; 2. Vortices exhibit certain regularities, and training U-Net with a small number of samples yields excellent predictive consequences; 3. The machine learning vortex locator based on the ACL method effectively corrects errors in manually annotated data, significantly improving the model's performance metrics, thus enhancing the precision and metrics of vortex localization. This substantial advancement in the application of machine learning in vortex localization provides an effective way for vortex dynamics localization. Furthermore, this method of obtaining more accurate positions of approximate human labels through machine learning offers new insights for machine learning in other types of image recognition problems.
Abstract Quantum vortices play a crucial role in both equilibrium and dynamical phenomena in two-dimensional (2D) superfluid systems. Experimental detection of these excitations in 2D ultracold atomic gases typically involves examining density depletions in absorption images, however the presence of a significant phononic background renders the problem challenging, beyond the capability of simple algorithms or the human eye. Here, we utilize a convolutional neural network to detect vortices in the presence of strong long- and intermediate-length scale density modulations in finite-temperature 2D Bose gases. We train the model on datasets obtained from ab initio Monte Carlo simulations using the classical-field method for density and phase fluctuations, and Gross–Pitaevskii simulation of realistic expansion dynamics. We use the model to analyze experimental images and benchmark its performance by comparing the results to the matter-wave interferometric detection of vortices, confirming the observed scaling of vortex density across the Berezinskii–Kosterlitz–Thouless critical point. The combination of a relevant simulation pipeline with machine-learning methods is a key development towards the comprehensive understanding of complex vortex-phonon dynamics in out-of-equilibrium 2D quantum systems.
The phase of the mean-field wave function of a Bose-Einstein condensate can be recovered from a time series of images. We adapt an iterative retrieval method that has been successful in linear electron and optical imaging systems to solve the nonlinear Gross-Pitaevskii equation. We address a number of issues related to the successful application of this method to the nonlinear system, including the retrieval of wave functions with nonzero net topological charge (i.e., containing vortices) and the effects of repulsive and attractive interactions in the condensate on the convergence properties of the method. Recovering the phase from the continuity-of-density equation is also investigated. An understanding of these issues is of importance for the practical implementation of phase retrieval to Bose-Einstein condensates.
We present an application of dark-field imaging that enables in situ detection of two-dimensional vortex distributions in single-component Bose-Einstein condensates (BECs). By rotating a 87Rb^{87}\mathrm{Rb}87Rb BEC in a magnetic trap, we generate a triangular lattice of vortex cores in the BEC, with core diameters on the order of 400 nm and cores separated by approximately $9\ensuremath{\mu}\mathrm{m}$. We have experimentally confirmed that the positions of the vortex cores near the BEC center can be determined without the need for ballistic expansion of the BEC. Our imaging method should allow for the determination of arbitrary distributions of vortices and other superfluid density defects in cases where expansion of the BEC is either impractical or would significantly alter the physical characteristics and appearance of vortices or defects. Our method is also a step toward real-time measurements of complex two-dimensional vortex dynamics within a single BEC.