Single nitrogen-vacancy ($\mathrm{N}$-$V$) centers in diamond are excellent nano-NMR sensors with the ability to function in ambient conditions. However, the fluorescence signal detected from single $\mathrm{N}$-$V$ centers is weak and often requires measurement durations on the order of several hours to acquire enough signal for the conventional curve fit estimation (CFE) technique. In this work, we explore maximum-likelihood estimation (MLE) as an alternative method to process the fluorescence signal from single $\mathrm{N}$-$V$ centers in XY8-$k$ measurements, particularly for online and real-time applications. There are three key points of interest in this work. First, we present a closed-form analytical expression for $\mathrm{N}$-$V$ center electron phase accumulation in the time domain. Second, we provide an explicit derivation of the Fisher information and Cramér-Rao lower bounds for parameter estimators in XY8-$k$ data. Third, we study the use of MLE for direct parameter estimation in XY8-$k$ fluorescence data with single $\mathrm{N}$-$V$ centers in different environmental settings through simulations, as well as providing a demonstration of performance on real data. Our results show that MLE converges faster and is more robust than CFE in most conditions.

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• Nuclear & electron resonance
• Quantum parameter estimation
• Quantum sensing
• Nitrogen vacancy centers in diamond
• Fluorescence spectroscopy
• Optically detected magnetic resonance
• Photon counting
• Resonance fluorescence