Optimal control of a nitrogen-vacancy spin ensemble in diamond for sensing in the pulsed domain

At a Glance

Metadata	Details
Publication Date	2022-07-05
Journal	Physical review. B./Physical review. B
Authors	Andreas F. L. Poulsen, Joshua D. Clement, James L. Webb, Rasmus Høy Jensen, Luca Troise
Institutions	Technical University of Denmark
Citations	20
Analysis	Full AI Review Included

Technical Documentation & Analysis: Optimal Control of NV Spin Ensembles

This document analyzes the research paper “Optimal control of a nitrogen-vacancy spin ensemble in diamond for sensing in the pulsed domain” and outlines how 6CCVD’s advanced MPCVD diamond materials and fabrication services can directly support and extend this groundbreaking quantum sensing research.

Executive Summary

The research successfully demonstrates enhanced coherent control over a large, macroscopic ensemble of Nitrogen-Vacancy (NV) centers in bulk diamond, achieving significant improvements in quantum sensing sensitivity.

Core Achievement: Robust, coherent manipulation of a large ensemble of NV centers (estimated up to 4 x 10⁹ centers) using shaped microwave pulses derived from optimal control theory and Floquet analysis.
Sensitivity Gain: The optimized shaped pulses resulted in an 11% enhancement in the optically detected magnetic resonance (ODMR) slope (a direct measure of sensitivity) compared to the best conventional three-frequency flat pulses.
Performance Benchmark: This represents a 78% sensitivity improvement over standard single-frequency flat $\pi$-pulses commonly used for coherent control in the literature.
Methodology: The optimal control algorithm explicitly included the physics of the ¹⁴N hyperfine splitting, ensuring high fidelity state transfer ($|0\rangle$ to $|-1\rangle$) across the inhomogeneous ensemble.
Material Limitation: The experiment utilized off-the-shelf optical-grade diamond ($\sim 0.5$ ppb NV), indicating that the sensitivity is currently limited by the material quality, pointing to massive potential gains using 6CCVD’s high-purity, engineered SCD.
Application: The demonstrated technique is highly beneficial for DC/low-frequency quantum sensing applications, such as precision magnetometry and thermometry.

Technical Specifications

The following hard data points were extracted from the research paper, detailing the experimental parameters and key results:

Parameter	Value	Unit	Context
NV Center Ensemble Size (Estimated)	4 x 10⁹	centers	Minimum estimated size addressed by pump laser
Diamond Material Grade	Optical Grade (Element 6)	N/A	Standard, non-optimized material
Nitrogen-Vacancy Concentration	$\sim 0.5$	ppb	Used in the experiment
Diamond Dimensions	5 x 5 x 1.2	mm³	Sample size
Zero-Field Splitting (ZFS)	2.87	GHz	NV ground state splitting
¹⁴N Hyperfine Splitting ($\delta_I$)	2.16	MHz	Separation of $m_I$ levels
Maximum Achievable Rabi Frequency ($R_{max}$)	3.2	MHz	Experimental limit using flat pulses
Optimal Shaped Pulse Duration ($t_p$)	1.85	µs	Best performing pulse
ODMR Slope Improvement (vs. 3-Freq Flat)	11	%	Direct sensitivity gain
ODMR Slope Improvement (vs. Single-Freq Flat)	78	%	Sensitivity gain over standard method
Estimated Shot-Noise-Limited Sensitivity ($\eta$)	10	nT/$\sqrt{\text{Hz}}$	Based on current non-optimized setup
Laser Reinitialization Time Constant ($\tau_R$)	$\sim 1.4$	ms	Decay constant of the contrast

Key Methodologies

The experiment relied on advanced quantum control techniques and specialized hardware integration:

Optimal Control Pulse Design: Shaped microwave pulses were constructed using smooth optimal control theory, maximizing the state transfer fidelity ($|0\rangle$ to $|-1\rangle$) across the ensemble’s inhomogeneous broadening.
Floquet Theory Integration: Floquet theory was used to solve the time-periodic Hamiltonian, allowing the explicit inclusion of the ¹⁴N hyperfine interaction (three $m_I$ levels) in the optimization algorithm.
Microwave Generation: Pulses were generated via In-phase/Quadrature (IQ) modulation using an Arbitrary Waveform Generator (AWG) and an RF signal generator, amplified, and delivered via a near-field square split-ring resonator antenna (resonant at $\sim 2.8$ GHz).
Magnetic Bias Field: A static magnetic field of 2.9 mT was applied along the [111] crystallographic axis to split the $m_s = \pm 1$ states, isolating the $m_s=0 \rightarrow m_s=-1$ transition.
Pulsed ODMR Sequence: The sequence involved: (1) Laser reinitialization (3 ms pulse duration, $t_i$), (2) Microwave control pulse application ($t_p$), and (3) Readout via red fluorescence detection using an Avalanche Photodiode (APD).
Noise Reduction: Software lock-in detection and referencing the fluorescence signal against a fraction of the pump laser (V_ref) were used to reject DC and high-frequency common-mode noise.

6CCVD Solutions & Capabilities

The research highlights the critical role of material quality in achieving high sensitivity. The authors note that sensitivity is limited by the standard optical-grade diamond used and suggest future improvements using isotopically purified diamond. 6CCVD is uniquely positioned to supply the advanced materials and fabrication services required to replicate and significantly extend this research.

Applicable Materials

To achieve the ultimate $T_2$-limited sensitivity suggested by the authors, researchers must move beyond standard optical-grade diamond.

Research Requirement	6CCVD Material Recommendation	Technical Advantage
High Sensitivity / Extended Coherence Time ($T_2$): Requires minimizing spin bath noise (¹³C and residual N).	High-Purity Single Crystal Diamond (SCD): Isotopically purified (low ¹³C) and ultra-low residual nitrogen content.	Maximizes $T_2$ coherence time, directly improving the shot-noise-limited sensitivity ($\eta \propto 1/C’T_2$).
Ensemble Control Scaling: Need large, uniform material for macroscopic sensing volumes.	Optical Grade SCD or Large Area PCD: For bulk sensing applications requiring large, homogeneous ensembles.	SCD wafers up to 500 µm thick. PCD plates up to 125 mm diameter, polished to Ra < 5 nm.
Boron Doping (BDD): Potential for integrated microwave delivery or alternative sensing platforms (e.g., electrochemical).	Boron-Doped Diamond (BDD): Available in both SCD and PCD formats.	Custom doping levels for conductive layers, enabling integrated on-chip microwave transmission lines or electrodes.

Customization Potential

The experiment relies on precise geometry and integration of microwave components. 6CCVD offers comprehensive fabrication services to meet these demands:

Custom Dimensions: We provide plates and wafers in custom dimensions, including the ability to supply thick substrates (up to 10 mm) for bulk sensing geometries, exceeding the 1.2 mm thickness used in the paper.
Precision Polishing: Achieving uniform microwave coupling and minimizing optical scattering losses is crucial. We offer ultra-low roughness polishing:
- SCD: Surface roughness (Ra) < 1 nm.
- Inch-size PCD: Surface roughness (Ra) < 5 nm.
Custom Metalization: The integration of near-field antennas or transmission lines (as required for the $2.8$ GHz drive) demands high-quality metal deposition. 6CCVD offers in-house metalization services, including: Au, Pt, Pd, Ti, W, and Cu stacks, tailored for specific microwave or ohmic contact requirements.
Laser Cutting and Shaping: We provide precise laser cutting services to shape diamond samples to fit custom antenna designs (like the split-ring resonator used) or specialized experimental setups (e.g., integration into portable sensor devices or cryostats).

Engineering Support

6CCVD’s in-house team of PhD material scientists and quantum engineers can assist researchers in optimizing material selection for similar NV Center Quantum Sensing projects. We specialize in tailoring CVD growth parameters to control NV density, isotopic purity, and surface termination, ensuring the material perfectly matches the requirements of advanced optimal control protocols.

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

View Original Abstract

Defects in solid state materials provide an ideal, robust platform for quantum sensing. To deliver maximum sensitivity, a large ensemble of non-interacting defects hosting coherent quantum states are required. Control of such an ensemble is challenging due to the spatial variation in both the defect energy levels and in any control field across a macroscopic sample. In this work we experimentally demonstrate that we can overcome these challenges using Floquet theory and optimal control optimization methods to efficiently and coherently control a large defect ensemble, suitable for sensing. We apply our methods experimentally to a spin ensemble of up to 4 $\times$ 10$^9$ nitrogen vacancy (NV) centers in diamond. By considering the physics of the system and explicitly including the hyperfine interaction in the optimization, we design shaped microwave control pulses that can outperform conventional ($\pi$-) pulses when applied to sensing of temperature or magnetic field, with a potential sensitivity improvement between 11 and 78%. Through dynamical modelling of the behaviour of the ensemble, we shed light on the physical behaviour of the ensemble system and propose new routes for further improvement.