Optimal control theory techniques for nitrogen vacancy ensembles in single crystal diamond

At a Glance

Metadata	Details
Publication Date	2023-09-27
Journal	Quantum Information Processing
Authors	Madelaine S. Z. Liddy, Troy W. Borneman, Peter Sprenger, David G. Cory
Institutions	University of Waterloo
Citations	1
Analysis	Full AI Review Included

Technical Documentation & Analysis: Optimal Control for NV Ensembles

Executive Summary

This research successfully demonstrates the application of Optimal Control Theory (OCT) to achieve orientation-selective and transition-selective control of Nitrogen Vacancy (NV) center ensembles in single crystal diamond (SCD) without an applied static magnetic field.

Core Achievement: Arbitrary simultaneous control over all four Principal Axis Systems (P.A.S.) of NV ensembles was achieved using circularly polarized microwaves generated by dual microstrip resonators.
Application Focus: The techniques are critical for realizing high-sensitivity NV-based quantum sensing devices, particularly for vector magnetometry and collective control applications.
Material Limitation Identified: The use of a thick 500 µm SCD substrate resulted in the NV focal volume being far (70 µm) from the microstrips, limiting the Rabi drive strength (max 2.64 MHz).
Performance Impact: The limited Rabi drive necessitated long pulse lengths (up to 32 µs), increasing the vulnerability of the quantum operation to decoherence effects, RF inhomogeneities, and hyperfine splittings.
Path to Improvement: The authors explicitly recommend using thinner SCD substrates (< 300 µm) to increase the Rabi drive strength, shorten pulse times, and improve pulse robustness.
6CCVD Value Proposition: 6CCVD specializes in providing custom, ultra-thin, high-purity SCD substrates and integrated metalization solutions required to overcome these specific hardware limitations and scale the research.

Technical Specifications

The following parameters were extracted from the experimental setup and results:

Parameter	Value	Unit	Context
Diamond Crystal Orientation (Used)	(100)	N/A	Chosen for symmetry and collective control demonstration.
Diamond Thickness (Used)	500	µm	DNV-B1 sample used in experiments.
Diamond Thickness (Recommended)	< 300	µm	Recommended by authors to increase Rabi drive strength.
Zero Field Splitting (ZFS, Delta)	$\approx 2.87$	GHz	Quantization frequency of the NV$^-$ center.
Central Control Frequency ($\omega_{\tau}$)	2.87	GHz	Frequency provided by the synthesizer.
High-Frequency Sub-Ensemble Rabi Drive ($\Omega_{NVA}$)	2.64	MHz	Measured Rabi frequency for sub-ensemble A.
Low-Frequency Sub-Ensemble Rabi Drive ($\Omega_{NVB}$)	1.03	MHz	Measured Rabi frequency for sub-ensemble B.
Short Pulse Length (Identity/Selective $\pi$)	10	µs	Achieved in 250 steps of 40 ns.
Long Pulse Length (Complete Selective $\pi^+$ map)	32	µs	Required for full map under experimental conditions (800 steps).
Experimental Field Orthogonality ($\eta$)	115	°	Measured phase value (Ideal is 90°).
Microstrip Width	127	µm	Planar microstrip resonator dimension.
Microstrip Thickness	17.5	µm	Planar microstrip resonator dimension.
Focal Volume Distance from Microstrips	$\approx 70$	µm	Result of 160 µm working distance objective on 300 µm diamond.
Estimated NV Centers in Focal Volume	$\approx 16,000$	N/A	Contained within 0.59 µm diameter beam spot.

Key Methodologies

The experimental control relied on a highly integrated optical and RF system utilizing dual-channel control and advanced pulse shaping.

Material Selection: A 500 µm-thick DNV-B1 (100) single crystal diamond was used, containing an estimated 16,000 NV centers.
Optical Setup: NVs were excited using off-resonant green light (532 nm) and red light emission was collected via an Avalanche Photo-Diode (APD). A Half Wave Plate (HWP) was rotated to equalize fluorescence across all NV orientations (optimal value found at 42°).
Microwave Generation: Dual parallel microstrip resonators were fabricated (150 µm spacing, 127 µm width, 17.5 µm height) on a PCB to create independently controllable microwave fields.
RF Control System: A frequency synthesizer provided the 2.87 GHz central frequency, which was split into two channels. Each channel used an IQ mixer to combine the central frequency with amplitude and phase control envelopes generated by a four-channel Arbitrary Waveform Generator (AWG).
Hamiltonian Modeling: A phenomenological control Hamiltonian was used, incorporating experimentally measured parameters: Rabi drive strength ($\Omega_{NVA/B}$) and field orthogonality ($\eta$).
Optimal Control Theory (OCT): The Gradient Ascent Pulse Engineering (GRAPE) algorithm was used to optimize amplitude-only controls ($\Omega_{1/2}$) to achieve target state-to-state transfers (identity and selective $\pi$ operations) with a target fidelity of 0.99.

6CCVD Solutions & Capabilities

The research highlights a critical need for optimized diamond substrates and integrated RF components to advance NV quantum sensing. 6CCVD is uniquely positioned to supply the next generation of materials required for robust, high-fidelity OCT pulse implementation.

Applicable Materials for Replication and Extension

Research Requirement	6CCVD Solution	Technical Specification	Sales Advantage
Thinner Substrates	Optical Grade SCD	Thickness: 0.1 µm to 300 µm (or custom up to 10 mm).	Directly addresses the need for < 300 µm diamond to increase Rabi drive strength and shorten pulse times.
Crystal Orientation	Custom SCD Substrates	Orientation: (100) or (110) available.	Enables replication of the (100) experiments or extension to the four non-degenerate P.A.S. control demonstrated in the (110) simulation.
High Purity/Low Strain	High-Quality MPCVD SCD	Polishing: Ra < 1 nm (essential for minimizing crystal strain $E(S_x^2 - S_y^2)$).	Low strain is critical for simplifying the NV Hamiltonian and achieving high-fidelity quantum control in zero magnetic field.

Customization Potential for Integrated Devices

The authors’ use of microstrip resonators on a PCB necessitates precise material integration, a core strength of 6CCVD:

Custom Dimensions: While the paper used a 4 x 4 mm sample, 6CCVD offers SCD and PCD plates/wafers in custom sizes, enabling scaling of the NV ensemble area for larger, high-throughput quantum sensors.
Integrated Metalization: The microstrip design requires high-quality conductive layers. 6CCVD offers in-house metalization services (Au, Pt, Pd, Ti, W, Cu) directly applied to the diamond substrate. This capability allows researchers to integrate RF structures directly onto the diamond surface, minimizing interface losses and maximizing field uniformity.
Precision Fabrication: 6CCVD provides laser cutting and shaping services to match the exact geometry required for complex RF structures (like the 127 µm wide microstrips) or for integration into existing cryogenic or vacuum systems.

Engineering Support

6CCVD’s in-house team of PhD material scientists and quantum engineers can provide authoritative support to researchers seeking to replicate or extend this work:

Material Selection Consultation: Assistance in selecting the optimal SCD thickness and nitrogen concentration (DNV equivalent) to balance signal-to-noise ratio (ensemble size) with Rabi drive strength (proximity to microstrips).
Geometry Optimization: Support in designing diamond-based RF structures, leveraging our expertise in thin-film metalization and high-precision polishing (Ra < 1 nm) to ensure high-Q resonators and optimal field orthogonality ($\eta \rightarrow 90^\circ$).
Global Logistics: We offer global shipping (DDU default, DDP available) to ensure rapid delivery of custom quantum-grade diamond materials worldwide, accelerating research timelines.

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

View Original Abstract

Abstract Nitrogen vacancy centre ensembles are excellent candidates for quantum sensors due to their vector magnetometry capabilities, deployability at room temperature and simple optical initialization and readout. This work describes the engineering and characterization methods required to control all four principle axis systems (P.A.S.) of NV ensembles in a single crystal diamond without an applied static magnetic field. Circularly polarized microwaves enable arbitrary simultaneous control with spin-locking experiments and collective control using optimal control theory (OCT) in a (100) diamond. These techniques may be further improved and integrated to realize high-sensitivity NV-based quantum sensing devices using all four P.A.S. systems.

Tech Support

Original Source

DOI: https://doi.org/10.1007/s11128-023-04106-x