Accelerating Quantum Emitter Characterization with Latent Neural Ordinary Differential Equations

TL;DR

A latent neural ordinary differential equation model is demonstrated that can forecast a complete and noise-free PCFS experiment from a small subset of noisy correlation functions, enabling up to a 20-fold speedup in experimental acquisition time from $\sim$3 hours to 10 minutes.

Abstract

Deep neural network models can be used to learn complex dynamics from data and reconstruct sparse or noisy signals, thereby accelerating and augmenting experimental measurements. Evaluating the quantum optical properties of solid-state single-photon emitters is a time-consuming task that typically requires interferometric photon correlation experiments, such as Photon correlation Fourier spectroscopy (PCFS) which measures time-resolved single emitter lineshapes. Here, we demonstrate a latent neural ordinary differential equation model that can forecast a complete and noise-free PCFS experiment from a small subset of noisy correlation functions. By encoding measured photon correlations into an initial value problem, the NODE can be propagated to an arbitrary number of interferometer delay times. We demonstrate this with 10 noisy photon correlation functions that are used to extrapolate an entire de-noised interferograms of up to 200 stage positions, enabling up to a 20-fold speedup in experimental acquisition time from $\sim$3 hours to 10 minutes. Our work presents a new approach to greatly accelerate the experimental characterization of novel quantum emitter materials using deep learning.

Figures (6)

• Figure 1: ($\boldsymbol{\mathrm{a}}$). Schematic of a PCFS optical setup. BS: beamsplitter, M: mirror, RR: retroreflector, SPD: single-photon detector. ($\boldsymbol{\mathrm{b}}$) Ten $g^{(2)}(\tau, t_{i})$ curves, drawn at values of $t$ shown in the legend, are used as inputs to our model to predict ($\boldsymbol{\mathrm{c}}$) the full $g^{(2)}(\tau, t)$ map. ($\boldsymbol{\mathrm{d}}$) Fourier transform of the PCFS interferogram (1 - $g^{(2)}(\tau, t)$) along the $t$ axis, resulting in the spectral correlation $p(\tau, \zeta)$ that shows how the single emitter lineshape evolves over time.
• Figure 2: ($\boldsymbol{\mathrm{a}}$). Input $g^{(2)}(\tau, t_{i})$ functions at slices of optical delay $t$, ($\boldsymbol{\mathrm{b}}$) $\hat{g}^{(2)}(\tau, t)$ predicted by LSTM-ODE using data in ($\mathbf{a}$), and ($\boldsymbol{\mathrm{c}}$) the true $g^{(2)}(\tau, t$).
• Figure S1: (a) Static homogeneous spectrum, (b) spectrum convolved with a diffusing Gaussian that broadens with $\tau$ (using a Wiener diffusion mechanism) to form $\rho(\tau, \zeta)$, (c) Fourier transform of $\rho(\tau, \zeta)$ along $\zeta$ to form the interferogram $I(\tau, \delta)$, (d) $g^{(2)}(\tau, t)$ resulting from the interferogram via eq. \ref{['eq:pcfs_g2']}, (e) representative individual $g^{(2)}(\tau, t)$ functions along $\tau$ for different values of $\delta$, and (f) an example of a $g^{(2)}(\tau, t)$ curve with and without noise added, compared to a real experimental correlation function.
• Figure S3: ($\boldsymbol{\mathrm{a}}$). Input $g^{(2)}(\tau, t_{i})$ functions at slices of optical delay $t$, ($\boldsymbol{\mathrm{b}}$) True ${g}^{(2)}(\tau, t)$ map, ($\boldsymbol{\mathrm{c}}$) Predicted $\hat{g}^{(2)}(\tau, t)$ from the LSTM-ODE, ($\boldsymbol{\mathrm{d}}$) Conv-ODE and ($\boldsymbol{\mathrm{e}}$) 1D ResNet models.
• Figure S4: Four examples of true ${g}^{(2)}(\tau, t)$ (left panels) and predicted $\hat{g}^{(2)}(\tau, t)$ (right panels) for LSTM-ODE models trained without the Fourier loss term $\mathcal{L}_{\mathcal{F}}$ (eq. \ref{['eq:fourier_loss']}). In practice, we found it more memory efficient to only use the Fourier transformed data at the first, middle, and last index of the $\tau$ axis.