Identifying optimal magnetic field configurations for decoherence mitigation of boron vacancies in hexagonal boron nitride

TL;DR

The paper addresses decoherence in the negatively charged boron vacancy (V_B-) in hexagonal boron nitride (hBN) arising from a nuclear-spin bath. It combines detailed numerical simulations of the VB- plus three 14N spins with an analytical perturbation-theory model for spin-1 systems to map static magnetic-field configurations that minimize transition-energy gradients. Key contributions include identifying low-gradient anti-crossings in both parallel and transverse field geometries, deriving a gradient-minimization condition that balances Zeeman and hyperfine terms, performing curvature analysis to capture second-order sensitivities, and providing T2 estimates that indicate substantial decoherence suppression in selected low-field regions. The results offer a practical framework for mitigating decoherence in low-field sensing with 2D spin defects and are extendable to other spin-1 systems coupled to nuclear baths, guiding experimental validation and future theoretical work.

Abstract

The negatively charged boron vacancy center in 2D hexagonal boron nitride has emerged as a promising quantum sensor. However, its sensitivity is constrained due to ubiquitous nuclear spins in the environment. The nuclear spins, hyperfine coupled with the central electron spin, effectively behave as magnetic field fluctuators, leading to rapid decoherence. Here, we explore the effectiveness of static magnetic field strength and orientation in realizing peculiar subspaces that can lead to enhanced spin coherence. Specifically, using detailed numerical simulations of the spin Hamiltonian, we identify specific field configurations that minimize energy gradients and, consequently, are expected to facilitate decoherence suppression. We also develop an approximate analytical model based on the perturbation theory that accurately predicts these low-gradient subspaces for magnetic fields aligned with the electron spin quantization axis, applicable not only to boron vacancies but to any spin-1 electronic system coupled to nearby nuclear spins. Furthermore, to stimulate experimental validation, we estimate coherence lifetimes as a function of various bias field configurations and demonstrate that significant decoherence suppression can indeed be achieved in certain regions. These findings and the developed methodology offer valuable insights for mitigating decoherence in a low-field regime.

Figures (7)

• Figure 1: Structure of the $V_B^-$ defect in hBN, showing the three nearest-neighbor $^{14}\mathrm{N}$ nuclear spins. $\theta$ and $\varphi$ represent the polar and azimuthal angles associated with the applied magnetic field $\bm{B}$ of magnitude $B_0$. Pink and blue spheres denote boron and nitrogen atoms, respectively, while the red circle marks the boron vacancy site. The coordinate axes and magnetic field orientation are indicated for reference.
• Figure 2: (a) Energy levels of the system as a function of the magnetic field applied parallel to the quantization axis of the electron spin. The black lines correspond to $m_s = 0$, and orange and blue are for $m_s = \pm{1}$ electronic sub-level. These sub-levels split into 27 $(3\times 3\times 3)$ hyperfine levels. The red arrows indicate the chosen magnetic field values. (b), (c) and (d) Energy levels as a function of the polar angle of the applied magnetic field with respect to the quantization axis for 1.7, 3.4, and 5.1 mT magnetic field, respectively. (e), (f) and (g) The magnitude of energy gradient (in MHz/T), as defined in \ref{['gradient']}, corresponding to the spin transition exhibiting the maximum transition probability at the aforementioned fields. The black arrows represent the $\theta$-range depicted in the figures (b), (c), and (d). The spherical shell radius, in principle, represents the bias magnetic field magnitude $B_0$ (in mT), however, for visualization purposes, all spheres are shown with the same size, and the radii do not scale with the field strength. The colorbar indicates the magnitude of the energy gradient.
• Figure 3: (a) Energy levels as a function of the transverse magnetic field. The black lines correspond to $m_s = 0$, and orange and blue are for $m_s = \pm{1}$ electronic sub-level. The red arrows indicate the chosen magnetic field values. Energy levels as a function of the polar angle of the applied magnetic field with respect to the quantization axis for 7.5, 15, and 23.5 mT magnetic field, respectively. (e), (f) and (g) The magnitude of energy gradient (in MHz/T), as defined in \ref{['gradient']}, corresponding to the maximum transition probability at the aforementioned fields. The black arrows represent the $\theta$-range depicted in the figures (b), (c), and (d). The spherical shell radius, in principle, represents the bias magnetic field magnitude $B_0$ (in mT), however, for visualization purposes, all spheres are shown with the same size, meaning the radii do not scale with the field strength. The colorbar indicates the magnitude of the energy gradient.
• Figure 4: The magnitude of transition energy gradient (as defined in \ref{['gradient']}) and curvature (as defined in \ref{['curvature']}) associated with all $(2\times27\times27)$ transitions between $m_s = 0$ and $m_s = \pm{1}$ as a function of the magnetic field applied parallel (a) and (c), perpendicular (b) and (d) to the quantization axis. The color bar corresponds to the magnitude of the gradient and the curvature; In gradient plots values beyond an arbitrary threshold are whitened for visibility purposes.
• Figure 5: Estimated coherence lifetime as a function of bias magnetic field for (a) parallel field configuration, and (b) transverse field configuration. Here, the upper bound of the colorbar has been restricted to 0.5 $\mu$s for enhancing the visibility of the high $T_2$ regions. Dependence of $T_2$ as a function of the applied field corresponding to select transition energies for (c) parallel field configuration, and (d) transverse field configuration. The red arrows in (a) and (b) indicate the chosen energies, and the gray lines in (c) indicate the magnetic field values that satisfy Eq. \ref{['dfs_condition']}.