Statistical inference with quantum measurements - methodologies for nitrogen vacancy centers in diamond

At a Glance

Metadata	Details
Publication Date	2017-11-27
Journal	New Journal of Physics
Authors	Ian Hincks, Christopher Granade, David G Cory
Citations	17
Analysis	Full AI Review Included

6CCVD Technical Analysis: Statistical Inference in NV Diamond Metrology

Reference Paper: Statistical Inference with Quantum Measurements: Methodologies for Nitrogen Vacancy Centers in Diamond (Hincks, Granade, Cory, et al., 2017)

Executive Summary

This paper presents a rigorous statistical framework for analyzing photon count data from Nitrogen Vacancy (NV^-) centers in diamond, addressing a critical need for accurate uncertainty quantification in quantum sensing (metrology) and characterization (tomography, Hamiltonian learning).

Rigorous NV Measurement Model: The research establishes a detailed physical model of the NV optical process using Lindblad jump operators, defining photon emission as an inhomogeneous Poisson process.
Addressing Experimental Imperfections: The model explicitly incorporates real-world limitations crucial for high-fidelity devices, including finite visibility, dark counts, imperfect state preparation (pseudo-pure states), and temporal reference drift.
Superior Statistical Methods: Comparison between Maximum Likelihood Estimation (MLE) and Bayesian estimators (Sequential Monte Carlo, SMC) demonstrates the superior risk performance and seamless error propagation offered by the Bayesian approach, especially near physical boundaries (p=0, p=1).
Quantified Uncertainty: Utilizes the Cramér-Rao bound to derive practical formulas for estimating the required experimental repetition count ($N$) needed to achieve a specified uncertainty ($\Delta p$).
Successful Quantum Hamiltonian Learning (QHL): Validates the Bayesian methodology by applying SMC inference to experimental Rabi and Ramsey data, accurately characterizing key Hamiltonian parameters ($\Omega, \omega_e, \Delta_g, A_N, T_{2}^{-1}$).
Relevance to High-Fidelity Diamond Components: The findings underscore that achieving high-contrast (low $\alpha/\beta$ ratio) and high-count (large $\alpha$) NV systems—which rely heavily on the quality and purity of the SCD material—is the primary bottleneck for quantum error reduction.

Technical Specifications

Parameter	Value	Unit	Context
Zero-Field Splitting (Ground State)	$\Delta_g \approx 2.87$	GHz	Hamiltonian Parameter
Zero-Field Splitting (Excited State)	$\Delta_e \approx 1.4$	GHz	Hamiltonian Parameter
Spontaneous Emission Rate ($\gamma_{eg}$)	$77$	MHz	Optical decay rate (Lifetime $\approx 13$ ns)
Spin-Selective Decay Rate ($\gamma_{es}$)	$30$	MHz	Decay rate to singlet state (ISC path)
Singlet Lifetime Decay Rate ($\gamma_{sg}$)	$3$	MHz	Dominates optimal measurement time
Non-Spin Conserving Rate ($\gamma_{01}$)	$1$	MHz	Sets maximum achievable polarization purity
Excitation Wavelength	$532$	nm	Green laser source used for initialization/measurement
Emission Wavelength	$600 - 800$	nm	Red/near-infrared photons collected
QHL Reference Bright Counts ($\alpha$)	$0.006$	Photons/Shot	Average detected bright photons (single repetition)
QHL Reference Dark Counts ($\beta$)	$0.004$	Photons/Shot	Average detected dark photons (single repetition)
QHL Contrast (C)	$0.2$	Dimensionless	Low visibility experimental setup (C = ($\alpha-\beta$)/($\alpha+\beta$))
Data Required for $\Delta p = \pm 0.01$	$\approx 170,000$	Experiments (N)	Worst-case estimate via Cramér-Rao bound

Key QHL Parameter Estimates (SMC Fit, All Data, Table I):

Parameter	Estimate (E)	Standard Deviation ($\sigma$)	Unit	Context
Nutation Frequency ($\Omega$)	$5555$	$1.1$	kHz	Microwave pulse strength
External Field Projection ($\omega_e$)	$1432$	$0.5$	kHz	Static magnetic field along z-axis
ZFS Mismatch ($\delta\Delta$)	$597$	$13.9$	kHz	Mismatch vs. applied MW frequency
¹⁴N Hyperfine Splitting ($A_N$)	$2171$	$0.8$	kHz	Nitrogen atom coupling
Dephasing Rate ($T_{2}^{-1}$)	$35$	$0.5$	kHz	Relaxation rate

Key Methodologies

The NV measurement process is modeled using a 7-level system (3 ground, 3 excited, 1 singlet) and described by the Lindblad master equation, which is simplified into a $5 \times 5$ rate equation for tractability.

System Initialization:
- NV center is illuminated with a 532 nm (green) laser pulse, driving the system to a steady state (pseudo-pure state $\rho_0$).
- The selective decay path via the Inter-System Crossing (ISC) state $|s\rangle$ preferentially depopulates the $m_s=\pm 1$ states, resulting in high spin state polarization (near $|g, 0\rangle$).
Spin State Manipulation:
- Qubit state is prepared (e.g., $|g, 0\rangle$) or manipulated using microwave pulses (Rabi and Ramsey sequences) prior to measurement.
Photon Counting Measurement:
- Measurement consists of a short laser pulse (~500 ns) and counting spontaneously emitted photons (600 nm-800 nm, red).
- The $m_s=0$ state appears ‘bright’ (higher photon emission rate $\mu_0$), while the $m_s=\pm 1$ states appear ‘dark’ (lower rate $\mu_1$).
Statistical Modeling (Poisson Process):
- The total detected photon count ($n_d$) is modeled as a Poisson distribution derived from the expected number of true NV emissions ($\eta \mu$) plus dark counts ($\Gamma \Delta t$).
- $n_d \sim \text{Poisson}(\alpha p + \beta (1-p))$, where $p$ is the survival probability of the spin state, $\alpha$ is the bright reference count, and $\beta$ is the dark reference count.
Drift and Uncertainty Handling (Bayesian SMC):
- References $\alpha$ and $\beta$ are treated as stochastic variables (Gaussian or Gamma processes) to account for temporal drift (e.g., changes in laser power or confocal alignment).
- Sequential Monte Carlo (SMC) algorithm is employed for Bayesian inference, allowing for robust updating of Hamiltonian parameters ($\vec{\chi}$) while seamlessly propagating the uncertainty and noise from the measurement process.

6CCVD Solutions & Capabilities

6CCVD is uniquely positioned to supply the advanced diamond materials and engineering services required to replicate and extend this foundational NV research into deployable quantum technologies.

Applicable Materials

The rigorous statistical modeling demonstrates that high NV contrast ($C = \frac{\alpha-\beta}{\alpha+\beta}$) and high count rates ($\alpha$) are paramount for reducing uncertainty. These factors are critically dependent on the quality of the starting diamond material.

Required Material Property	6CCVD Material Recommendation	Engineering Justification
Purity & NV Formation	Optical Grade Single Crystal Diamond (SCD)	Low nitrogen incorporation (Type IIa) is essential for isolated, high-quality NV^- centers with long coherence times ($T_2$).
Thermal Management	High Purity SCD Substrates (up to 10mm thickness)	Necessary to manage heat load from continuous green laser excitation (532 nm) and mitigate thermal drift effects (Section V), ensuring stable reference counts ($\alpha, \beta$).
Electronic/RF Applications	Boron-Doped Diamond (BDD)	While not the primary NV material, BDD is available for integrated electrochemistry or high-speed detection elements adjacent to the NV layer.

Customization Potential

The experimental setup described in the paper—involving precise microwave delivery for Rabi and Ramsey sequences—is greatly enhanced by integrated microfabrication capabilities. 6CCVD offers end-to-end processing for quantum device construction.

Paper Requirement / Implication	6CCVD Customization Service	Value Proposition
Integrated Microwave Control	Custom Metalization (Au, Pt, Pd, Ti, W, Cu)	Fabrication of on-chip RF strip lines and electrodes directly onto the SCD surface, allowing for precise control of the nutation frequency ($\Omega$).
Optical Interface Quality	Ultra-Low Surface Roughness (Ra < 1 nm for SCD)	Essential for maximizing collection efficiency ($\eta$) in the confocal microscope setup and maintaining high measurement visibility.
Unique Wafer Geometry	Custom Dimensions (Plates/wafers up to 125 mm)	Provision of wafers/plates in specific dimensions required for integration into proprietary cryogenic or optical alignment systems.
NV Layer Engineering	Epitaxial Thickness Control (SCD: 0.1 µm - 500 µm)	Precise growth control of the NV layer depth allows optimization for either near-surface sensing (magnetometry) or bulk sensing (coherent control), depending on application.

Engineering Support

This research validates the use of highly sophisticated, computationally intensive Bayesian methods (SMC) for accurately characterizing NV systems. Replicating and advancing this work requires materials designed to minimize nuisance parameters.

6CCVD’s in-house PhD-level engineering team can assist researchers in selecting optimal diamond material specifications—such as nitrogen concentration, epitaxial thickness, and surface preparation (Ra < 1 nm)—to maximize NV quantum yield and minimize the drift and noise characterized in this statistical model.

We specialize in optimizing material parameters to achieve the high contrast and high count rates ($\alpha$) necessary to push the limits of the Cramér-Rao bound, ensuring that our customers’ experimental risk (R) is minimized.

For custom specifications or material consultation, visit 6ccvd.com or contact our engineering team directly.

View Original Abstract

The analysis of photon count data from the standard nitrogen vacancy (NV)\nmeasurement process is treated as a statistical inference problem. This has\napplications toward gaining better and more rigorous error bars for tasks such\nas parameter estimation (eg. magnetometry), tomography, and randomized\nbenchmarking. We start by providing a summary of the standard phenomenological\nmodel of the NV optical process in terms of Lindblad jump operators. This model\nis used to derive random variables describing emitted photons during\nmeasurement, to which finite visibility, dark counts, and imperfect state\npreparation are added. NV spin-state measurement is then stated as an abstract\nstatistical inference problem consisting of an underlying biased coin\nobstructed by three Poisson rates. Relevant frequentist and Bayesian estimators\nare provided, discussed, and quantitatively compared. We show numerically that\nthe risk of the maximum likelihood estimator is well approximated by the\nCramer-Rao bound, for which we provide a simple formula. Of the estimators, we\nin particular promote the Bayes estimator, owing to its slightly better risk\nperformance, and straight-forward error propagation into more complex\nexperiments. This is illustrated on experimental data, where Quantum\nHamiltonian Learning is performed and cross-validated in a fully Bayesian\nsetting, and compared to a more traditional weighted least squares fit.\n

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Original Source

DOI: https://doi.org/10.1088/1367-2630/aa9c9f