Longitudinal relaxation of a nitrogen-vacancy center in a spin bath by generalized cluster-correlation expansion method

At a Glance

Metadata	Details
Publication Date	2020-01-07
Journal	Annals of Physics
Authors	Zhi-Sheng Yang, Yanxiang Wang, Ming‐Jie Tao, Wen Yang, Mei Zhang
Institutions	King Abdulaziz University, Beijing Computational Science Research Center
Citations	24
Analysis	Full AI Review Included

Longitudinal Relaxation of NV Centers in Diamond: A 6CCVD Technical Analysis

This document analyzes the theoretical study of longitudinal relaxation (T₁) in Nitrogen-Vacancy (NV) centers coupled to a $^{13}$C nuclear spin bath, utilizing the Generalized Cluster-Correlation Expansion (CCE) method. The findings underscore the critical need for high-purity, precisely engineered Single Crystal Diamond (SCD) substrates, a core capability of 6CCVD.

Executive Summary

Research Focus: Theoretical investigation of the longitudinal relaxation ($T_1$) dynamics of the NV electron spin in diamond, driven by coupling to the surrounding $^{13}$C nuclear spin bath.
Methodology: Generalized Cluster-Correlation Expansion (CCE) was employed, incorporating the central electron spin into the cluster definition to accurately model population dynamics in the nearly-resonant regime.
Key Finding (Resonance): $1/T_1$ increases dramatically (nearly one order of magnitude) when the external magnetic field ($B_z$) is tuned to the level anticrossing point (approximately 1025 G), confirming enhanced cross-relaxation.
Decay Dynamics: In the nearly-resonant case, the electron spin population decays irreversibly to zero around 10 µs, demonstrating the critical role of the spin bath in decoherence.
Material Requirement: The study implicitly requires high-purity diamond to isolate the NV- $^{13}$C interaction and minimize decoherence from other impurities (e.g., P1 centers).
6CCVD Value Proposition: 6CCVD provides the necessary high-quality, optical-grade SCD substrates with controlled impurity levels and superior surface finishing (Ra < 1 nm) required for advanced quantum sensing and computing applications based on NV centers.

Technical Specifications

The following hard data points were extracted from the theoretical simulation parameters and physical constants used in the study:

Parameter	Value	Unit	Context
NV Zero-Field Splitting ($D$)	2.87	GHz	Separation between $
Electronic Gyromagnetic Ratio ($\gamma_e$)	-1.76 x 10¹¹	rad·s^-1T^-1	NV electron spin constant
¹³C Nuclear Gyromagnetic Ratio ($\gamma_c$)	6.73 x 10⁷	rad·s^-1T^-1	Nuclear spin bath constant
¹³C Natural Abundance	1.1	%	Standard concentration in diamond
Resonance Magnetic Field ($B_z$)	1024.975	G	Field required for maximum $1/T_1$ increase
Typical Operating Field ($B_z$)	~1025	G	Required to approach level anticrossing
Simulated Bath Size ($N$)	50	Spins	Number of $^{13}$C spins used in 4-CCE simulation
Longitudinal Relaxation Time ($T_1$)	~10	µs	Decay time observed near resonance
Optical Initialization Wavelength	532	nm	Required for NV state preparation

Key Methodologies

The study employed the Generalized Cluster-Correlation Expansion (CCE) method to simulate the quantum dynamics of the NV center and the surrounding $^{13}$C spin bath.

System Hamiltonian Definition: The total system Hamiltonian ($H$) was constructed, including the NV center Hamiltonian ($H_{NV}$), the $^{13}$C bath Hamiltonian ($H_{bath}$), and the hyperfine interaction ($H_{int}$).
CCE Generalization: The traditional CCE method was generalized by explicitly including the central electron spin in the cluster definition (e.g., 2-CCE includes the NV center and one bath spin; 4-CCE includes the NV center and three bath spins).
Initial State Preparation: The NV electron spin was initialized to the $|0\rangle$ state ($\rho_{NV} = |0\rangle\langle 0|$), and the $^{13}$C bath was assumed to be in a thermal equilibrium state.
Numerical Simulation: The survival probability $P(t)$ of the initial state was calculated by partially tracing the time-evolved density matrix over the bath degrees of freedom.
Convergence and Truncation: Simulations were performed up to 5th-order CCE, confirming that 4th-order CCE (4-CCE) provided a reliable and convergent result for a bath size of $N=50$ nuclear spins.
Magnetic Field Tuning: Simulations were conducted by tuning the static magnetic field ($B_z$) along the NV axis to control the energy gap between the electron spin levels, specifically focusing on the nearly-resonant regime where cross-relaxation is maximized.

6CCVD Solutions & Capabilities

This research highlights the need for ultra-high-quality diamond materials to realize robust quantum devices. 6CCVD is uniquely positioned to supply the necessary substrates and engineering services to replicate and extend this critical research into practical applications.

Applicable Materials

To achieve the long coherence times and controlled spin bath environment required for this research, Optical Grade Single Crystal Diamond (SCD) is the optimal material solution.

6CCVD Material	Key Specification	Relevance to NV Research
Optical Grade SCD	Ultra-low Nitrogen (P1) Concentration	Minimizes parasitic decoherence sources, ensuring the $^{13}$C bath is the dominant relaxation mechanism, as modeled.
Isotopically Controlled SCD	Depleted or Enriched $^{13}$C	Allows researchers to precisely control the spin bath concentration (currently 1.1% natural abundance) to optimize $T_1$ and $T_2$ times, extending the scope of the CCE model.
Standard PCD	Plates up to 125 mm diameter	Suitable for large-area quantum sensing arrays where high crystal purity is less critical than scale and cost.

Customization Potential

The precise control of the NV environment requires highly customized substrates, which 6CCVD delivers through its advanced fabrication capabilities:

Custom Dimensions and Thickness:
- 6CCVD supplies SCD plates ranging from 0.1 µm to 500 µm thickness, and robust substrates up to 10 mm thick, allowing for optimal integration into high-field magnet setups (required for $B_z \approx 1025$ G).
- Custom laser cutting and shaping services ensure precise alignment of the NV axis with the applied magnetic field.
Surface Engineering:
- Achieving reliable optical initialization (532 nm laser) and readout requires pristine surfaces. 6CCVD guarantees atomic-scale polishing with roughness Ra < 1 nm for SCD, minimizing surface-related decoherence.
Integrated Metalization:
- While this study is theoretical, practical NV experiments require microwave control lines for dynamic decoupling and spin manipulation. 6CCVD offers in-house metalization using materials such as Ti/Pt/Au, W, or Cu for direct integration of control structures onto the diamond surface.

Engineering Support

The sensitive dependence of $T_1$ on the magnetic field near the level anticrossing point is crucial for quantum sensing applications.

Decoherence Optimization: 6CCVD’s in-house PhD team specializes in material science for quantum applications and can assist researchers in selecting the optimal diamond grade (e.g., $^{13}$C depletion level) to maximize $T_1$ and $T_2$ coherence times for similar NV-based Quantum Sensing and Quantum Computing projects.
Material Characterization: We provide detailed characterization data (e.g., nitrogen concentration, surface roughness) to ensure the supplied material meets the stringent requirements for high-fidelity spin dynamics simulations and experiments.

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

Tech Support

Original Source

DOI: https://doi.org/10.1016/j.aop.2019.168063