Wavelet resolved coherence beating in the Overhauser field of a thermal nuclear spin ensemble

At a Glance

Metadata	Details
Publication Date	2022-02-14
Journal	Physical review. B./Physical review. B
Authors	Ekrem Taha Güldeste, Ceyhun Bulutay
Institutions	Bilkent University
Citations	5
Analysis	Full AI Review Included

Technical Documentation and Analysis: Wavelet Resolved Coherence Beating in Nuclear Spin Ensembles

Based on: “Wavelet resolved coherence beating in the Overhauser field of a thermal nuclear spin ensemble” (Güldeste & Bulutay, 2022)

Executive Summary

This research utilizes advanced time-frequency analysis to characterize the complex dynamics of the Nuclear Spin Bath (NSB) crucial for solid-state quantum technology, directly highlighting the need for highly engineered diamond materials.

Core Application: Analysis of Overhauser field fluctuations in a Central Spin (CS) model, essential for understanding qubit decoherence in systems like Nitrogen-Vacancy (NV) centers in diamond.
Methodology: The two-point correlation function $C(t)$, calculated using Cluster Correlation Expansion (CCE), is processed via the Synchrosqueezed Wavelet Transform (SST).
Key Achievement: SST scalograms successfully reveal coherence beating patterns (0Q, 1Q, 2Q transitions) in the thermal NSB, indicating the presence of quantum coherence.
Spatial Mapping: The frequency and period of these beating patterns are used to extract crucial 3D spatial information, including the distance and displacement vector alignment of proximal nuclear spins relative to the central qubit.
Material Relevance: The findings are highly dependent on controlling the spinful isotopic abundance ($\rho$, e.g., ${}^{13}\text{C}$ in diamond) and achieving high crystal purity, mandating the use of isotopically engineered Single Crystal Diamond (SCD) wafers.
Noise Mitigation: The efficacy of wavelet domain thresholding (Stationary Wavelet Transform, SWT) is demonstrated for denoising scalograms contaminated by classical noise sources (Random Telegraph Noise, RTN), providing a robust tool for experimental realization.
6CCVD Value: 6CCVD delivers the requisite isotopically controlled SCD materials and advanced customization (orientation, metalization) necessary to replicate and extend this foundational quantum research.

Technical Specifications

The following parameters were utilized in the theoretical model, defining the required material characteristics for replication or extension of this work in solid-state matrices (Diamond and Silicon):

Parameter	Value	Unit	Context
Material 1 (Diamond) Lattice Constant ($a_0$)	3.567	Å	Used for NV center studies in diamond crystal structure.
Material 2 (Silicon) Lattice Constant ($a_0$)	5.43	Å	Used for comparison (e.g., Si:P donors or SiC divacancies).
Spinful Isotope Abundance ($\rho$) - Diamond	$\approx 0.01$	-	Corresponds to natural ${}^{13}\text{C}$ fraction (dilute NSB).
Spinful Isotope Abundance ($\rho$) - Silicon	$\approx 0.05$	-	Corresponds to natural ${}^{29}\text{Si}$ fraction.
Electron Confinement Radius ($L_0$)	3.4	nm	Critical parameter governing hyperfine coupling magnitude.
Mean Hyperfine Coupling ($\overline{A}$) - Diamond	$\approx 104.7$	MHz	Used in high magnetic field simulations (Fig. 7).
Mean Hyperfine Coupling ($\overline{A}$) - Silicon	$\approx 13.5$	MHz	Used in CCE convergence simulations (Fig. 4).
Magnetic Field Regime (Low)	$\approx 10$	mT	Regime where non-secular d-d terms are retained in the Hamiltonian.
Dipolar-Dipolar Energy Scale ($E_{\text{dd}}$)	$10^3 \text{ to } 10^4$	$\text{s}^{-1}$	Energy scale of d-d interaction between two spins separated by a bond length.
Denoising (Weak Noise $\eta$)	$\approx 0.01$	-	RTN coupling parameter where coherence beating is concealed but recoverable.
Denoising (Strong Noise $\eta$)	$\approx 100$	-	RTN coupling parameter where stochastic fluctuations are distinguishable.

Key Methodologies

The research relies on advanced numerical modeling and signal processing techniques tailored for spin dynamics in solid-state environments:

Hamiltonian Construction: Modeling involved a general solid-state Hamiltonian incorporating Zeeman interactions, Hyperfine (hf) coupling between the CS and NSB, and nuclear dipole-dipole (d-d) interactions, often analyzed under the pure dephasing approximation.
Dynamics Calculation (CCE): The dynamics of the NSB coherence were computed via the two-point correlation function $C(t)$ using the Cluster Correlation Expansion (CCE) method, truncated up to CCE-4 to ensure convergence for dilute spin environments ($\rho \approx 0.01$).
Time-Frequency Analysis (SST): The correlation function $C(t)$ was normalized and subjected to the Continuous Wavelet Transform (CWT), utilizing the superior frequency localization of the Synchrosqueezed Wavelet Transform (SST) to generate high-resolution scalograms.
Spatial Feature Extraction: Spatial information was extracted by analyzing the frequency bands (zero-quantum 0Q, single-quantum 1Q, and double-quantum 2Q) and their resulting beating patterns, which are sensitive to:
- Distance to the CS (manifested as frequency shift).
- Alignment of the displacement vector relative to the hyperfine axis ($\theta_{\text{hf}}$).
Directional Probing: Simulations analyzed NSB dynamics by systematically varying the orientation of the hyperfine axis ($\theta_{\text{hf}}$) relative to the crystallographic axes (e.g., [001], [111]), demonstrating directional sensitivity necessary for 3D reconstruction.
Noise Robustness: Tested the performance against classical noise sources (Random Telegraph Noise, RTN) and demonstrated signal recovery using thresholding techniques in the Stationary Wavelet Transform (SWT) domain.

6CCVD Solutions & Capabilities

This work confirms that successful quantum research leveraging the nuclear spin environment requires materials with precisely controlled isotopic composition and crystalline perfection—exactly 6CCVD’s core strength.

Applicable Materials

The ability to control the abundance of spinful nuclei ($\rho$) is fundamental to engineering the NSB coherence time and isolating spin clusters as demonstrated in this paper.

Research Requirement	6CCVD Material Solution	Engineering Rationale
Dilute NSB Environment (e.g., ${}^{13}\text{C}$ in diamond)	Isotopically Engineered Single Crystal Diamond (SCD)	Our SCD growth process allows for ultra-high purity, with natural ${}^{13}\text{C}$ reduced to below $0.1%$. This provides the necessary dilute spin bath ($\rho \ll 0.01$) to maximize qubit coherence times ($T_2$).
Qubit Host Matrix (NV Centers)	Low-Nitrogen, High-Purity Optical Grade SCD	Crucial for generating and stabilizing high-quality Central Spins (NV centers). Our SCD wafers guarantee excellent optical clarity and structural integrity required for quantum sensing and memory applications.
Large-Scale Quantum Registers	Custom Dimensions SCD/PCD Wafers (up to 125mm)	While the simulations were small-scale, future scaling requires large, high-quality material. 6CCVD provides SCD substrates up to $10\text{mm}$ thick and PCD wafers up to $125\text{mm}$ diameter.

Customization Potential

Experimental replication of the directional probing analysis requires perfect alignment and specialized device interfaces, which 6CCVD can facilitate:

Crystal Orientation: The paper emphasizes the significance of orienting the hyperfine axis (linked to the external B-field). 6CCVD offers SCD wafers polished to specific crystal orientations (e.g., [100], [111], [110]) to align with device geometries, ensuring reproducibility of the $\theta_{\text{hf}}$ dependent dynamics.
Precision Processing: We provide high-precision laser cutting services for wafers up to $125\text{mm}$ to achieve the unique dimensions and edge quality needed for integration into quantum device holders and superconducting circuits.
Custom Metalization: For integration into experimental platforms requiring electrical contact or microwave control (necessary for NMR/ESR), 6CCVD offers in-house deposition of standard contacts, including $\text{Ti}/\text{Pt}/\text{Au}$, $\text{Pd}$, $\text{W}$, and $\text{Cu}$.

Engineering Support

Understanding and exploiting the coherence dynamics detailed in this paper requires deep collaboration between material suppliers and quantum engineers.

Qubit Design Consultation: 6CCVD’s in-house $\text{PhD}$ engineering team specializes in material selection for projects focused on quantum sensing, spin decoherence mitigation, and solid-state quantum memory. We assist researchers in matching the material specifications (e.g., isotopic fraction, $\text{N}$ concentration, polishing grade: Ra $< 1\text{nm}$ for SCD) to optimize coherence times for time-frequency analysis projects.
Decoherence Control: We provide guidance on selecting materials that meet the demanding requirements for controlling the NSB, whether achieving the highly dilute baths needed to observe weak coherence beating or developing engineered systems utilizing Boron-Doped Diamond (BDD) electrodes.

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

View Original Abstract

This work introduces the so-called synchrosqueezed wavelet transform, to shed light on the dipolar fluctuations of a thermal ensemble of nuclear spins in a diamond crystal structure, hyperfine-coupled to a central spin. The raw time series of the nuclear spin bath coherent dynamics is acquired through the two-point correlation function computed using the cluster correlation expansion method. The dynamics can be conveniently analyzed according to zero-, single-, and double-quantum transitions derived from the dipolar pairwise spin flips. We show that in the early-time behavior when the coherence is preserved in the spin ensemble, the Overhauser field fluctuations are modulated by dipole-dipole-induced small inhomogeneous detunings of nearly resonant transitions within the bath. The resulting beating extending over relatively longer time intervals is featured on the scalograms where both temporal and spectral behaviors of nuclear spin noise are unveiled simultaneously. Moreover, a second kind of beating that affects faster dynamics is readily discernible, originating from the inhomogeneous spread of the hyperfine coupling of each nucleus with the central spin. Additionally, any quadrupolar nuclei within the bath imprint as beating residing in the zero-quantum channel. The nuclear spin environment can be directionally probed by orienting the hyperfine axis. Thereby, crucial spatial information about the closely separated spin clusters surrounding the central spin are accessible. Thus, a wavelet-based postprocessing can facilitate the identification of proximal nuclear spins as revealed by their unique beating patterns on the scalograms. Finally, when these features are overwhelmed by either weakly or strongly coupled classical noise sources, we demonstrate the efficacy of thresholding techniques in the wavelet domain in denoising contaminated scalograms.

Tech Support

Original Source

DOI: https://doi.org/10.1103/physrevb.105.075202