Understanding Decoherence of the Boron Vacancy Center in Hexagonal Boron Nitride

TL;DR

This study provides a detailed, field-resolved picture of decoherence for the V$_\mathrm{B}^-$ center in h$^{11}$B$^{15}$N by applying the generalized cluster-correlation expansion to a large nuclear-spin bath. It uncovers five distinct magnetic-field regions with different dominant noise channels, including zero-field multi-spin correlations, low-field boron- and nitrogen-spin dynamics, and a transition region where electron-spin–mediated processes and nuclear-spin precession compete, culminating in a high-field regime limited by nuclear dipolar flip-flops. A key finding is that operating in the moderate field window of $180$–$350$ mT can boost $T_2$ to $\mathcal{O}(10\ \mu$s), dramatically improving coherence relative to the low-field regime. The results deliver actionable guidance for optimizing V$_\mathrm{B}^-$–based sensing in chemically pure hBN and illuminate the roles of hyperfine interactions, ESEEM, and multi-spin correlations in 2D spin bath decoherence.

Abstract

Hexagonal boron nitride (hBN) has emerged as a significant material for quantum sensing, particularly due to its ability to host spin active defects, such as the negatively charged boron vacancy (V$_\mathrm{B}^-$ center). The optical addressability of the V$_\mathrm{B}^-$ center and hBN's 2D structure enable high spatial resolution and integration into various platforms. However, decoherence due to the strong magnetic noise in hBN imposes fundamental limitations on the sensitivity of V$_\mathrm{B}^-$ center-based applications. Understanding the phenomena behind decoherence and identifying parameter settings that provide the highest performance are essential for advancing V$_\mathrm{B}^-$ sensors. This study employs state-of-the-art computational methods to investigate the decoherence of the V$_\mathrm{B}^-$ center in hexagonal boron nitride across a wide range of magnetic field values from 0 T up to 3 T. The provided in-depth numerical and analytical analysis reveals an intricate interplay of various decoherence mechanisms. This study identifies five distinct magnetic field regions governed by different types of magnetic interactions with and within the abundant nuclear spin bath. In addition to magnetic field, the effects of zero-field splitting, nuclear polarization, and different hyperfine coupling terms are studied, representing an important step forward in utilizing V$_\mathrm{B}^-$ ensembles in sensing. In particular, this study proposes operation in the moderate $180-350$ mT magnetic field range in chemically pure h$^{11}$B$^{15}$N samples, where the coherence time can reach $1-20$ $μ$s, significantly exceeding the $\mathcal{O}( 100~\text{ns})$ low-field $T_2$ values.

Figures (5)

• Figure 1: The V$_\mathrm{B}^-$ center in hexagonal boron nitride.a Structure and spin density of the V$_\mathrm{B}^-$ center in hexagonal boron nitride. Orange lobs depict the spin density distribution of the triplet electron spin localized on the dangling bonds of the neighboring nitrogen atoms. b Frobenius-norm of the hyperfine coupling tensors of different species of nuclear spins in the spin bath. c Ground state energy level structure of the V$_\mathrm{B}^-$ center electronic spin coupled to the three first-neighbor $^{15}$N nuclear spins. The inset shows the energy level structure close to zero-field. d Schematic representation of the Hahn-echo pulse sequence used to measure decoherence due to dynamical magnetic noise. Red rectangles represent laser pulses, while blue and green rectangles represent microwave $\pi$-half and $\pi$ pulses, respectively. e Illustration of first, second, and third-order clusters of the spin bath centered around the V$_\mathrm{B}^-$ electron spin. f Comparison of measured and simulated coherence decay function at $B_z=0$ mT. Experimental data is reproduced from Ref. VBmeas-liu2022 by rescaling the contrast values to the values of the coherence function $L$.
• Figure 2: Magnetic field dependence of the Hahn-echo coherence time ($T_2$) as a function of the external magnetic field.a Calculated coherence time from 1 mT to 3000 mT on log-log scale (blue). The orange curve serves as a guide to the eye. The results obtained in the pseudo-secular approximation of the hyperfine couplings are shown in red. Dashed lines indicate approximate boundaries of different regions dominated by different decoherence mechanisms. Green and yellow markers indicate experimental and simulation data available in the literature (see also Table \ref{['tab:prev']}). The inset shows the coherence times at zero-field. b, c$T_2(B_z$) on a linear scale close to zero-field and in the transition region. d, e, f, g Density plot of the calculated coherence function $L$ as a function of the spin echo time $t$ and the external magnetic field $B_z$ in different regions and on relevant timescales. Gold regions are indicators of coherent V$_\mathrm{B}^-$ centers, while dark blue regions indicate the loss of coherence. Transition and periodic modulation of the coloring function represent decay and electron spin echo envelop modulation, respectively.
• Figure 3: Hahn-echo coherence time close to zero-field for different configurations and interaction parameters.a Coherence functions in different orders of gCCE at zero-field. b Coherence functions obtained by considering pure nitrogen ($^{15}$N), pure boron ($^{11}$B), and the combined boron-nitrogen spin bath. a and b The dashed line shows the fitted decay curve to the convergent $L(t)$ curve. c$T_2$ as a function of magnetic field in a hypothetical system, where first-neighbor nitrogen spins are removed. d, e Coherence time as a function of the transverse zero-field splitting ($E$) and the external magnetic field. f The coherence function in different orders of gCCE at zero-field in the case of ideal threefold rotational symmetry of the defect.
• Figure 4: Analysis of coherence decay in the low-field regime.a Coherence time as a function of magnetic field in the low-field regime. b Coherence functions in different orders of gCCE, the coherence function obtained by neglecting the role of all three first-neighbor nitrogen spins, and the first-order coherence decay derived from ESEEM theory. The dashed line shows the fitted decay. c$T_2$ against the initial population of $\ket{\downarrow}$ states of first-neighbor nitrogen nuclear spins. d, e Coherence functions in different orders of gCCE and analytical results obtained from ESEEM theory for spin baths consisting of solely boron and nitrogen nuclei, respectively. f Coherence function obtained from simulations compared to the result of ESEEM and pseudo-spin theory.
• Figure 5: Coherence time in the transition region.a Coherence time as a function of magnetic field in the transition region. Lines indicate the sampled magnetic fields of the subsequent Figure. b The coherence as a function of the spin-echo time and the external magnetic field. c, d, e, f, g, h, i Coherence functions in different orders of gCCE. The magnetic field values indicated in the top-right corner of each subfigure were sampled from the transition region and the high-field regime. Dashed lines show the fitted coherence decays.