[Paper Review] Interplay between charge and spin noise in the near-surface theory of decoherence and relaxation of $C_{3v}$ symmetry qutrit spin-1 centers

Denis R. Candido, Michael E. Flatté|arXiv (Cornell University)|Mar 23, 2023

Diamond and Carbon-based Materials ResearchMaterials Science3 citations

TL;DR

This paper presents a complete Lindblad-based theory for decoherence and relaxation in $C_{3v}$ symmetry spin-1 qutrits (e.g., NV$^-$ centers) near crystal surfaces, accounting for both charge and spin noise. It reveals that charge noise—previously underestimated—significantly contributes to relaxation via the $d' E_±$ dipole term, and explains experimental $T_2$ depth dependencies through distinct noise mechanisms (dipole vs. point-like fluctuations).

ABSTRACT

Decoherence and relaxation of solid-state defect qutrits near a crystal surface, where they are commonly used as quantum sensors, originates from charge and magnetic field noise. A complete theory requires a formalism for decoherence and relaxation that includes all Hamiltonian terms allowed by the defect's point-group symmetry. This formalism, presented here for the $C_{3v}$ symmetry of a spin-1 defect in a diamond, silicon cardide, or similar host, relies on a Lindblad dynamical equation and clarifies the relative contributions of charge and spin noise to relaxation and decoherence, along with their dependence on the defect spin's depth and resonant frequencies. The calculations agree with the experimental measurements of Sangtawesin $ extit{et al.}$, Phys. Rev. X $ extbf{9}$, 031052 (2019) and point to an unexpected importance of charge noise.

Motivation & Objective

• To develop a complete theoretical framework for decoherence and relaxation in $C_{3v}$ symmetry spin-1 qutrits under both charge and magnetic noise.
• To clarify the relative contributions of charge and spin noise to relaxation and dephasing, especially near crystal surfaces where quantum sensors operate.
• To explain the experimentally observed depth dependence of decoherence times ($T_2$) in nitrogen-vacancy centers in diamond.
• To identify how surface treatments (e.g., annealing) alter dominant noise sources by suppressing dipole fluctuations.

Proposed method

• Derives a Lindblad dynamical equation with eight distinct Lindblad operators to describe all symmetry-allowed charge and spin noise couplings in $C_{3v}$ spin-1 systems.
• Models charge noise from both fluctuating dipoles (e.g., surface defects) and point-like charges (e.g., 2D hole/electron gases) using analytical expressions for electric field fluctuations.
• Analyzes magnetic noise from fluctuating magnetic moments and moving charges in surface-confined 2D systems, including spin-orbit coupling effects.
• Calculates relaxation and dephasing rates as functions of magnetic field $B_z$, energy splitting $ω_{+0}$, and defect depth $z_{\text{def}}$, using spectral noise density models.
• Applies the theory to experimental data from Sangtawesin *et al.* (2019), comparing theoretical $T_2$ depth dependencies with measured values.
• Distinguishes between $T_2 ∝ z_{\text{def}}^{-4}$ (dipole fluctuations) and $T_2 ∝ z_{\text{def}}^{-2}$ (point-like fluctuations) to interpret sample-specific behavior.

Figure 1: Schematic plot of a spin-1 negatively-charged nitrogen-vacancy (NV - ) center within diamond in the presence of fluctuating surface charges (gray spheres) and magnetic moments (ruby arrows). (Lower right) NV - center’s spin levels and response to the magnetic field. Dephasing and relaxatio

Experimental results

Research questions

• RQ1 What is the relative contribution of charge noise versus magnetic noise to relaxation and dephasing in $C_{3v}$ spin-1 qutrits near surfaces?
• RQ2 How do the relaxation rates $1/T_{1\pm}$ depend on the magnetic field $B_z$ and energy level splitting $\omega_{+0}$?
• RQ3 Why do different diamond samples exhibit distinct depth dependencies of $T_2$ decoherence times?
• RQ4 What role does surface treatment (e.g., high-temperature and oxygen annealing) play in suppressing specific noise sources?
• RQ5 How do point-like charge fluctuations and dipole fluctuations differ in their $z_{\text{def}}$-dependence and impact on $T_2$?

Key findings

• Charge noise contributes significantly to relaxation between $|T_\pm\rangle$ and $|T_0\rangle$ states via the $d' E_\pm$ dipole term, challenging the assumption that such transitions are dominated solely by magnetic noise.
• The $T_2$ decoherence time for shallow NV centers follows $T_2 \propto z_{\text{def}}^{-4}$ in samples with rough surfaces, consistent with dipole fluctuation noise.
• In contrast, $T_2 \propto z_{\text{def}}^{-2}$ is observed in samples with smooth, annealed surfaces, indicating suppression of dipole noise and dominance of point-like charge fluctuations.
• Sample B, which underwent high-temperature and oxygen annealing, exhibits the longest coherence times due to reduced dipole fluctuations, confirming surface treatment efficacy.
• The $B_z$-dependence of $1/T_{1+}$ shows a non-monotonic behavior with a maximum at $\omega_{+0} = \Delta\omega_\mu$, indicating resonant enhancement of relaxation rates.
• Theoretical predictions for $1/T_{1\pm}$ and $1/T_2$ match experimental data from Sangtawesin *et al.* (2019), validating the model’s accuracy across different sample conditions.

Figure 2: Population dynamics for the spin-defect energy levels as a function of time for $\gamma\gg\Omega_{\pm}$ , $\gamma=\Omega_{\pm}$ and $\gamma\ll\Omega_{\pm}$ , respectively. (a) solid (dot-dashed) lines: $\gamma=20\times 10^{3}$ s -1 with $\Omega_{+}=\Omega_{-}=0.1\times 10^{3}$ s -1 ( $\Ome