From Single Spins to Ensembles: A Calibrated Symbolic Regression Pipeline for NV Spectroscopy

Abstract

Characterizing nitrogen-vacancy (NV) ensembles requires connecting their collective spectra to individual centers that experience different local environments. We develop a symbolic-regression pipeline that carries single-center spin structure into a calibrated ensemble model by generating individual spectral responses and combining them with explicit weights. A constrained search selects Hermitian operator terms from single-center optically detected magnetic resonance spectra, fits their coefficients, and retains the selected expressions as source models for the ensemble. Ensemble calibration then accounts for resonance shifts, local disorder, orientation, and optical response while keeping the individual spectral contributions accessible. On public single-center and ensemble data, the zero-field ensemble held-frequency root-mean-square error decreases from 0.008835 to 0.003872 in normalized photoluminescence, a 56.2% improvement over matched uncorrected composition. In controlled QuTiP ensembles of 128 sites spanning four NV orientations, models calibrated at 0 and 3 mT predict an independently held-out 5.17 mT spectrum with errors of 0.000207-0.000228 across three random seeds, comparable to the 0.000200 noise standard deviation. Resolving the simulated response into its individual-center contributions shows how local spectral diversity combines into the ensemble line shape. By retaining the source Hamiltonians while calibrating sample-dependent spectral differences, the pipeline connects local spin physics to collective optical readout and makes the contributions to an ensemble line shape available for physical interpretation.

The paper addresses how to connect individual nitrogen-vacancy (NV) center spin models to ensemble optically detected magnetic resonance spectra when centers have different local environments. It proposes a symbolic-regression pipeline that selects Hermitian operator terms from single-center spectra, fits their coefficients, and retains the resulting Hamiltonians as source models. Individual spectral responses are combined with explicit weights, followed by ensemble calibration for resonance shifts, local disorder, orientation, and optical response. The construction keeps individual-center spectral contributions accessible rather than representing only the aggregate line shape. Validation uses public single-center and ensemble data, together with controlled QuTiP simulations. For a zero-field experimental ensemble spectrum, the root-mean-square error on held-out frequencies decreases from $0.008835$ to $0.003872$ in normalized photoluminescence, a reported $56.2\%$ reduction relative to matched uncorrected composition. In simulated ensembles containing $128$ sites across four NV orientations, calibration at $0$ and $3\,\mathrm{mT}$ yields prediction errors of $0.000207$–$0.000228$ at an independently held-out field of $5.17\,\mathrm{mT}$ across three random seeds, compared with a noise standard deviation of $0.000200$. The simulated response is also decomposed into individual-center contributions to examine how local spectral diversity produces the ensemble line shape. The abstract does not state explicit limitations. Its reported held-out-field validation is simulation-based; the reported experimental test concerns held-out frequencies at zero field. Further methodological details and limitations cannot be assessed from the abstract alone.