Resource-efficient adaptive Bayesian tracking of magnetic fields with a quantum sensor

At a Glance

Metadata	Details
Publication Date	2021-02-05
Journal	Journal of Physics Condensed Matter
Authors	K Craigie, E M Gauger, Y. Altmann, C Bonato, K Craigie
Institutions	Heriot-Watt University
Citations	12
Analysis	Full AI Review Included

Resource-Efficient Quantum Sensing: Accelerating Real-Time Magnetic Field Tracking with 6CCVD Diamond

This technical documentation analyzes the research paper “Resource-efficient adaptive Bayesian tracking of magnetic fields with a quantum sensor,” focusing on the material requirements and computational advancements relevant to quantum metrology applications utilizing Nitrogen-Vacancy (NV) centers in diamond.

Executive Summary

The research demonstrates a highly efficient adaptive Bayesian protocol for real-time magnetic field tracking using NV centers in diamond, achieving significant computational speed improvements critical for practical quantum sensing deployment.

Core Application: Nanoscale magnetic field tracking and quantum metrology using diamond NV centers.
Computational Breakthrough: Implementation of a Gaussian mixture approximation within the Bayesian tracking protocol, drastically reducing computational complexity.
Speed Increase: The approximation yielded a computation speed increase ranging from 1.3x to 13.5x (typically an order of magnitude) compared to non-approximate methods.
Real-Time Capability: The protocol enables computation times comparable to or faster than the required real-time threshold of < 10 µs for single-shot spin readout.
Performance in Low Coherence: The Gaussian protocol demonstrated superior tracking accuracy (lower Mean Squared Error, MSE) in regimes characterized by short coherence times (T₂ = 1 µs), where the original method failed.
Material Requirement: Successful implementation relies on high-quality Single Crystal Diamond (SCD) substrates to host NV centers, particularly for achieving long coherence times (T₂ up to 100 µs).

Technical Specifications

The following data points summarize the performance metrics and experimental parameters used in the simulation and direct comparison tests.

Parameter	Value	Unit	Context
Computation Speed Increase	1.3 to 13.5	Factor	Gaussian vs. Original Protocol
Target Real-Time Computation	< 10	µs	Required for adaptive optimization
Coherence Time (T₂) Tested	100, 10, 1	µs	Material quality parameter
Overhead Time (t_oh) Tested	10, 6, 2	µs	Time between Ramsey measurements
Prediction Coefficient (κ)	10	MHz Hz^1/2	Rate of magnetic field change
Typical MSE (Successful Run)	~0.09	MHz/ms	Mean Squared Error
Fail Rate (F.R.) Threshold	> 0.15	MHz/ms	Definition of tracking failure
Average Parameters Used (Gaussian)	8 to 9	Count	Computational complexity indicator

Direct Comparison Results (T₂ vs. Speed Increase)

T₂ (µs)	Overhead (µs)	Original F.R. (%)	Gaussian F.R. (%)	Speed Increase (Factor)
100	10	0.5	1	8.1
100	6	0	5.5	10.5
100	2	0.5	3	13.5
10	10	0.25	1.25	9.4
10	2	0.75	6.5	10.8
1	10	99.75	91.25	2.1
1	2	99.5	21	1.3

Key Methodologies

The resource-efficient tracking protocol is based on an adaptive Bayesian filter implemented via Ramsey measurements on NV centers.

Ramsey Measurement Sequence: The core sensing mechanism involves initializing the NV spin, allowing free precession under the magnetic field for sensing time $\tau_n$, and then performing optical readout.
Bayesian Estimation with Approximation: The likelihood function and posterior distribution of the Larmor frequency ($f_B$) are approximated as a finite sum of Gaussian functions (Gaussian mixtures). This approximation reduces the number of parameters required to describe the distribution from thousands (in the original method) to typically 8 or 9.
Adaptive Optimization: Between measurements, the protocol determines the optimal experimental settings:
- Adaptive Sensing Time ($\tau_n$): Chosen based on the standard deviation of the posterior distribution to minimize uncertainty.
- Adaptive Control Phase ($\theta_n$): Set to maximize the information gained from the next measurement, calculated from the prior probability distribution in Fourier space.
Computational Pruning: To prevent the number of Gaussian terms from increasing exponentially, two steps are implemented:
- Amplitude Thresholding: Gaussians with amplitudes below a threshold ($A_{th} = 0.04$) are discarded.
- Kullback-Leibler (KL) Divergence Merging: Similar Gaussians (KL divergence below $KL_{th} = 0.001$) are merged into a single term.
Tracking Fluctuation Model: Magnetic field fluctuations are modeled as a Wiener process characterized by a diffusion coefficient ($\kappa$).

6CCVD Solutions & Capabilities

The successful implementation of this high-speed quantum tracking protocol is fundamentally dependent on the quality and engineering of the diamond material hosting the NV centers. 6CCVD is uniquely positioned to supply the necessary high-purity materials and custom fabrication services required to replicate and advance this research.

Applicable Materials

To achieve the long coherence times (T₂ up to 100 µs) necessary for robust tracking performance, researchers require ultra-high purity diamond.

Single Crystal Diamond (SCD): 6CCVD recommends Optical Grade SCD for quantum sensing applications. Our SCD material offers extremely low nitrogen and defect concentrations, minimizing environmental noise and maximizing the NV center T₂ coherence time, which is critical for high-fidelity Ramsey measurements.
Polycrystalline Diamond (PCD): For applications requiring larger area coverage or integration into complex micro-electromechanical systems (MEMS), our High-Purity PCD offers excellent thermal and mechanical properties, suitable for dense NV ensembles.
Boron-Doped Diamond (BDD): While not the primary sensor material here, our BDD films are available for integrated electrode structures or high-speed electronic components required for the FPGA-based real-time computation and control systems mentioned in the paper.

Customization Potential

The integration of NV sensors into practical devices, such as those used in levitated nanodiamond experiments or magnetic traps (as discussed in the Conclusion), often requires precise dimensions and specialized surface preparation.

Research Requirement	6CCVD Capability	Technical Advantage
Large Area Sensing	Plates/wafers up to 125mm (PCD)	Enables scaling up of quantum metrology arrays.
High-Fidelity Integration	Custom thicknesses: SCD (0.1µm - 500µm)	Tailored material thickness for optimal NV depth and integration into optical setups.
Surface Quality	Polishing: Ra < 1nm (SCD)	Essential for minimizing surface noise and maximizing optical coupling efficiency for spin readout.
Device Integration	Custom Metalization: Au, Pt, Pd, Ti, W, Cu	We offer in-house deposition of metal contacts (e.g., Ti/Pt/Au stacks) necessary for microwave delivery, electrodes, or integration with 3D Helmholtz coils used in trapping experiments.

Engineering Support

The computational efficiency demonstrated in this paper opens the door for real-time tracking of complex physical phenomena, such as temperature drift in living cells or rotational dynamics of trapped nanodiamonds.

6CCVD’s in-house team of PhD material scientists and engineers specializes in optimizing diamond growth parameters for quantum applications. We provide consultation on:

Material Selection: Choosing the optimal diamond grade (SCD vs. PCD) and NV creation method (implantation vs. in-situ growth) to meet specific T₂ and NV density requirements for high-speed magnetic field tracking projects.
Surface Termination: Customizing surface chemistry to ensure stable NV performance and compatibility with subsequent processing steps (e.g., metalization or integration into fluidic systems).

For custom specifications or material consultation regarding high-performance diamond substrates for quantum metrology, visit 6ccvd.com or contact our engineering team directly.

View Original Abstract

Abstract Single-spin quantum sensors, for example based on nitrogen-vacancy centres in diamond, provide nanoscale mapping of magnetic fields. In applications where the magnetic field may be changing rapidly, total sensing time is crucial and must be minimised. Bayesian estimation and adaptive experiment optimisation can speed up the sensing process by reducing the number of measurements required. These protocols consist of computing and updating the probability distribution of the magnetic field based on measurement outcomes and of determining optimized acquisition settings for the next measurement. However, the computational steps feeding into the measurement settings of the next iteration must be performed quickly enough to allow real-time updates. This article addresses the issue of computational speed by implementing an approximate Bayesian estimation technique, where probability distributions are approximated by a finite sum of Gaussian functions. Given that only three parameters are required to fully describe a Gaussian density, we find that in many cases, the magnetic field probability distribution can be described by fewer than ten parameters, achieving a reduction in computation time by factor 10 compared to existing approaches. For <mml:math xmlns:mml=“http://www.w3.org/1998/Math/MathML” display=“inline” overflow=“scroll”> <mml:msubsup> <mml:mrow> <mml:mi>T</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>*</mml:mo> </mml:mrow> </mml:msubsup> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> <mml:mspace class=“nbsp” width=“0.3333em”/> <mml:mi>μ</mml:mi> <mml:mi mathvariant=“normal”>s</mml:mi> </mml:math> , only a small decrease in computation time is achieved. However, in these regimes, the proposed Gaussian protocol outperforms the existing one in tracking accuracy.

Tech Support

Original Source

DOI: https://doi.org/10.1088/1361-648x/abe34f