A Fourier Neural Operator Approach for Modelling Exciton-Polariton Condensate Systems

TL;DR

The paper addresses the computational bottleneck of predicting exciton-polariton condensate properties described by the GPE coupled to reservoir dynamics. It introduces a Fourier Neural Operator (FNO) surrogate, motivated by the Split-Step Fourier Method (SSFM), to learn mappings between input pump profiles and steady-state density distributions using both numerical and experimental data. Theoretical results establish universal approximation capabilities and a formal SSFM-FNO correspondence, enabling the FNO to emulate first-order SSFM steps with a Fourier kernel operator. The approach delivers about a three-orders-of-magnitude speedup over CUDA-based solvers while maintaining high accuracy, enabling scalable design of large-scale all-optical polariton devices and potentially impacting quantum computing and neuromorphic photonics.

Abstract

A plethora of next-generation all-optical devices based on exciton-polaritons have been proposed in latest years, including prototypes of transistors, switches, analogue quantum simulators and others. However, for such systems consisting of multiple polariton condensates, it is still challenging to predict their properties in a fast and accurate manner. The condensate physics is conventionally described by Gross-Pitaevskii equations (GPEs). While GPU-based solvers currently exist, we propose a significantly more efficient machine-learning-based Fourier neural operator approach to find the solution to the GPE coupled with exciton rate equations, trained on both numerical and experimental datasets. The proposed method predicts solutions almost three orders of magnitude faster than CUDA-based solvers in numerical studies, maintaining the high degree of accuracy. Our method not only accelerates simulations but also opens the door to faster, more scalable designs for all-optical chips and devices, offering profound implications for quantum computing, neuromorphic systems, and beyond.

Figures (10)

• Figure 1: Comparison of pump profiles and wavefunction density with scattering process illustration.a The upper layer shows the nonresonant pump profile featuring three Gaussian spots, while the lower one shows the wavefunction density of the condensates at the final time. Three white dashed lines indicate the central positions of the pump regions and align with their corresponding locations on the condensate density map. b Depiction of the scattering process, tracing the transition from the hot electron-hole plasma phase, through the reservoir cooling phase, to the scattering in the condensates. Only the lower polariton branch of the polariton energy mode is shown here.
• Figure 2: Comparison of the prediction with different short-distanced-spots pump configurations using theoretical datasets and the Fourier Neural Operator approach.a-d From left to right, the different pump configurations are $P=0.85, 0.9, 1.2, 1.4\,P_{\mathrm{th}}$. e-h Corresponding condensate solutions $|\Psi_p|$ with pump profiles, each featuring a distinct spatial profile and intensity from the prediction datasets. i-l Corresponding numerical steady-state solutions $|\Psi_g|$ from the ground truth. m-p Corresponding absolute errors between prediction and ground truth $||\Psi_p|-|\Psi_g||$. The white bar on all panels is $10\,\mathrm{\mu m}$. The corresponding percentage errors of the number of condensate particles, taken from Fig. \ref{['S_curve']}, are $1.12\%$, $4.07\%$, $0.04\%$, $0.24\%$. $P_{\mathrm{th}}$ is the threshold of the power density per single Gaussian spot.
• Figure 3: S-curve of the condensate particles as a function of the pumping density. The logarithmic scale of number of particles for a ground truths denoted as $\mathrm{log}(N_{g})$,b predictions denoted as $\mathrm{log}(N_{p})$, and c the relative error of the condensate particles in the prediction $N_{p}$ with respect to the ground truth $N_{g}$ as a function of pumping density in the unit of $P_{\mathrm{th}}$, where $P_{\mathrm{th}}$ is the threshold power of a single Gaussian spot.
• Figure 4: Comparison of the prediction with different normalized pump configurations using the preprocessed experimental datasets and Fourier Neural Operator approach.a-d From left to right, the different pump configurations. e-h Corresponding predictions from the pump profiles.i-l Corresponding post-processed photoluminescence from the experiment. The number of fringes on e-h is $3$, $5$, $6$, $8$, respectively, which is the same as those on i-l. The white bar on all panels is $10\,\mathrm{\mu m}$. The pump density for the whole experiment is $3.6$ times the threshold value.
• Figure 5: Architecture of the Fourier Neural Operator. The process begins with the input $a(x)$ which undergoes a lifting operation, denoted as $\mathcal{P}$. This is followed by $4$ consecutive Fourier layers. Subsequently, a projector $\mathcal{Q}$ transforms the data to the desired target dimension, resulting in the output $u(x)$. The inset provides a detailed view of the structure of a Fourier layer. Data initially flow to the layer as $\nu(x)$ and are bifurcated into two branches: one undergoes a linear transformation $W$, and the other first experiences a Fourier transformation, from which the 128 lowest Fourier modes are kept, and the other higher modes are filtered out by undergoing a transformation $R$, and ends with an inverse Fourier transformation with these left modes. The two data streams then converge, followed by the application of an activation function $\sigma$.

Theorems & Definitions (4)

• Theorem 1
• Theorem B1
• Theorem B2
• Theorem B3