15 mK-Level Accuracy in All-Optical NV Diamond Thermometry via Interpretable Deep Learning

Technical Summary: 15 mK-Level Accuracy in All-Optical NV Diamond Thermometry via Interpretable Deep Learning

Problem Statement
Precise temperature measurement is critical for applications ranging from quantum information systems to industrial process control. While nitrogen-vacancy (NV) diamond thermometry has emerged as a promising technique, all-optical sensing methods (which rely on photoluminescence, PL, rather than optically detected magnetic resonance, ODMR) have historically suffered from limited accuracy, typically restricted to ±3% or ±9 K around 300 K. This limitation stems from the nature of spectral changes at elevated temperatures: as temperature increases, the zero-phonon line (ZPL) at 637 nm broadens and redshifts, while the phonon sideband (PSB) becomes more pronounced. These changes introduce noise and complexity that traditional expert-derived linear models, which rely on isolated features like ZPL amplitude or width, struggle to interpret accurately. Consequently, the utility of all-optical NV sensors in high-precision applications remains constrained.

Methodology
The study systematically evaluates machine learning (ML) and deep learning models to predict temperature from the full PL spectra of three fiber-coupled NV sensors (Sensors A, B, and C) across a temperature range of 243 K to 343 K. The experimental setup involved acquiring spectra under varying conditions, including different acquisition rates, spectral ranges, and sensor configurations (bulk diamond particles vs. chips).

The methodology compares three categories of approaches:

1. Expert Knowledge (EK) Models: Traditional linear regression models based on physically motivated features (ZPL amplitude, peak center, width, Debye-Waller factor, and KL divergence).
2. Unsupervised Dimensionality Reduction: Principal Component Analysis (PCA) and Autoencoders (AE), which learn low-dimensional representations without explicit temperature labels.
3. Supervised Learning Models: Linear Discriminant Analysis (LDA), Multi-Layer Perceptrons (MLP), Convolutional Neural Networks (CNN), and an "Optimized MLP" (a 6-layer network with sigmoidal activation functions).

A novel interpretability framework, termed "explainable modes of deviation" (xMoDs), was introduced to visualize the latent space of these models. xMoDs project the activations of hidden layers or dimensionality reduction outputs back onto the original spectral space, allowing researchers to identify which spectral components (e.g., ZPL shifts, PSB changes, or noise artifacts) drive the model's predictions. Additionally, sparse regression and distance computations were used to compare the features learned by ML models against those derived from expert knowledge.

Key Results
The study demonstrates that deep learning models significantly outperform both expert-derived linear models and simpler ML techniques:

• Accuracy Improvement: Expert-derived univariate and multivariate regression models yielded testing errors between 4.1 K and 19.6 K. In contrast, the optimal 6-layer MLP with sigmoidal activation achieved an average testing error of ≈0.015 K (15 mK) for Sensor A. This represents a ≥200-fold improvement over the best multivariate regression models and up to a 600-fold improvement over previous all-optical results.
• Model Performance Hierarchy: The Optimized MLP (sigmoidal) outperformed ReLU-based MLPs, which in turn outperformed CNNs, LDA, PCA, and Autoencoders. Notably, unsupervised Autoencoders underperformed compared to PCA, suggesting that the reconstruction objective of AEs retains "dark variables" (non-thermal dynamic information) detrimental to temperature regression.
• Sensor Variability: While Sensor A achieved the highest accuracy (15 mK), Sensors B and C showed higher errors (0.28 K and 0.80 K, respectively), attributed to differences in experimental conditions such as spectral range (Sensor B excluded longer PSB wavelengths) and photon flux (Sensor C had lower collection efficiency).

Interpretability Findings (xMoDs)
The xMoD analysis revealed distinct differences in how models process data:

• Unsupervised vs. Supervised: Unsupervised methods (PCA, AE) produced modes highly correlated with expert knowledge features (e.g., ZPL amplitude), indicating they capture general spectral variance. Supervised methods (LDA, MLP) identified distinct latent subspaces optimized specifically for temperature, often ignoring parts of the spectrum (like the long-wavelength PSB) that unsupervised methods deemed important for reconstruction.
• Noise Isolation: The superior performance of the Optimized MLP was attributed to its ability to isolate non-thermal artifacts. Its xMoDs revealed the detection of "cosmic ray" spikes and stray light events (e.g., a unique spike past 775 nm) that other models averaged out or failed to distinguish. By isolating these non-thermal events, the model avoided their interference with temperature inference.

Significance and Claims
The paper claims that deep neural networks, when paired with interpretable frameworks like xMoDs, can unlock the full potential of all-optical NV thermometry. The primary significance lies in:

1. Breaking the Accuracy Barrier: Achieving 15 mK accuracy places ML-enhanced all-optical thermometry on competitive footing with legacy technologies (like thin-film resistance thermometers) and other emerging quantum sensing methods.
2. Mechanistic Insight: The xMoD framework provides a "spectroscopist-friendly" lens to understand why deep models succeed, moving beyond the "black box" perception. It demonstrates that these models learn to filter out non-thermal noise and focus on specific, temperature-sensitive spectral components that linear models miss.
3. Data-Driven Optimization: The results suggest that the entire NV-PL spectrum contains dynamic information relevant to temperature that is inaccessible to feature-engineered approaches, validating the shift toward end-to-end deep learning for spectroscopic regression tasks.

The authors conclude that while model complexity is not a guarantee of performance (as seen with the underperforming Autoencoders), the combination of appropriate architecture (sigmoidal MLPs), rigorous preprocessing, and interpretability tools (xMoDs) is essential for realizing high-precision, all-optical temperature sensing.