Thermometry of simulated Bose--Einstein condensates using machine learning

TL;DR

This work presents a proof-of-principle non-destructive thermometry method for ultracold Bose gases that estimates the chemical potential $μ$ and temperature $T$ from a single in situ density image using a CNN trained on SGPE simulated quasi-2D condensates. The approach processes 2D density profiles through a three-layer convolutional feature extractor and a small fully connected predictor to output $μ$ and $T$ in nanokelvin with millisecond inference times. It demonstrates robust performance on equilibrium harmonic traps and partial zero-shot generalisation to toroidal geometries and during thermalisation, suggesting potential for real-time, geometry-agnostic thermometry in quantum gas experiments. The work bridges finite-temperature quantum fluid dynamics with machine learning and provides open-source codes for replication and extension to broader geometries and parameter spaces.

Abstract

Precise determination of thermodynamic parameters in ultracold Bose gases remains challenging due to the destructive nature of conventional measurement techniques and inherent experimental uncertainties. We demonstrate a machine learning approach for rapid, non-destructive estimation of the chemical potential and temperature from a single image of an \emph{in situ} imaged density profiles of finite-temperature Bose gases. Our convolutional neural network is trained exclusively on quasi-2D `pancake' condensates in harmonic trap configurations. It achieves parameter extraction within fractions of a second. The model also demonstrates {some} zero-shot generalisation across both trap geometry and thermalisation dynamics, successfully estimating the temperature (although not the chemical potential) for toroidally trapped condensates with errors of only a few nanokelvin despite no prior exposure to such geometries during training, and maintaining predictive accuracy during dynamic thermalisation processes after a relatively brief evolution without explicit training on non-equilibrium states. These results suggest that supervised learning can overcome traditional limitations in ultracold atom thermometry, with extension to broader geometric configurations, temperature ranges, and additional parameters potentially enabling comprehensive real-time analysis of quantum gas experiments. Such capabilities could significantly streamline experimental workflows whilst improving measurement precision across a range of quantum fluid systems.

Figures (19)

• Figure 1: Architecture of the convolutional neural network designed to take the density profile of an atomic Bose--Einstein condensate as an input, and from it determine the condensate's chemical potential, $\mu$, and temperature, $T$. The network processes 2D density profiles through three convolutional layers (extracting 12, 24, and 48 features, respectively), each followed by a maximum pooling layer with stride 2 (see \ref{['pooling']} for details). We apply global pooling to each feature before passing through two fully connected layers (FC1 and FC2), and use ReLU activation functions throughout.
• Figure 2: The prediction pipeline architecture, showing in more detail the final part of the network architecture depicted in figure \ref{['ch5:architecture']}. Three fully connected, feedforward layers convert the spatial information from the feature extraction pipeline to a pair of scalar values associated with the chemical potential and temperature, highlighting the weights $w_{j,k}^{\ell}$ between layers and the biases $b_{j}^{\ell}$ associated with the neurons in a layer, for $\ell=5$.
• Figure 3: The first convolutional layer Conv1, elaborating in more detail the beginning of the image processing and feature extraction pipeline as depicted in figure \ref{['ch5:architecture']}. a) Input of the atomic density, $\rho$. b) Cross-correlation of the atomic density with the weights matrices [see equation (\ref{['eq:layer1_zeta']})]. c) Application of the ReLU activation function and a bias [see equation (\ref{['eq:layer1_xi']})]. d) Carrying out maximum pooling (resulting in the halving of the image dimensions); this is one of $j=12$ outputs of the first convolutional layer, as per equation (\ref{['eq:layer1_Xi_pool']}). To the right we show zoomed-in sections of each step of the first convolutional layer as we construct the first layer feature maps.
• Figure 4: Schematic showing the production of the final 48 feature maps from an initial density profile, via the image processing and feature extraction pipeline, which then feed into the prediction pipeline (see also figure \ref{['fig:prediction_pipeline']}) to determine estimated values of the chemical potential $\mu$ and temperature $T$. The initial density profile is the same as that of figure \ref{['fig:feature_progression']}(a), the (1 of 12) pre-activation image is the same as in figure \ref{['fig:feature_progression']}(b), the following post-activation image is the same as in figure \ref{['fig:feature_progression']}(c), and the first of the following sample from 12 feature maps is the same as in figure \ref{['fig:feature_progression']}(d). Relative to figure \ref{['ch5:architecture']}, this schematic depicts detailed progress, a single channel of each layer at a time, through the image processing and feature extraction pipeline (see section \ref{['sec:batching']} for details). The end-of-layer outputs from all channels (individual "slices" in figure \ref{['ch5:architecture']}) are inputs to the subsequent layer; the 48 feature maps output by layer 3 are individually globally maximally pooled, as per section \ref{['sec:prediction_pipeline']}. The complete set of pre-activation images, post-activation images, and feature maps, for each layer, can be seen in in \ref{['appendix:conv_layers']}.
• Figure 5: The training and validation cost metrics for batch sizes 16, 32, 64, and 128 from a model extracting 12, 24 and 48 features in the first, second, and third convolutional layers. A smaller batch size results in models with a lower overall training and validation cost.