Elucidating the Inter-system Crossing of the Nitrogen-Vacancy Center up to Megabar Pressures

Abstract

The integration of Nitrogen-Vacancy color centers into diamond anvil cells has opened the door to quantum sensing at megabar pressures. Despite a multitude of experimental demonstrations and applications ranging from quantum materials to geophysics, a detailed microscopic understanding of how stress affects the NV center remains lacking. In this work, using a combination of first principles calculations as well as high-pressure NV experiments, we develop a complete description of the NV's optical properties under general stress conditions. In particular, our ab initio calculations reveal the complex behavior of the NV's inter-system crossing rates under stresses that both preserve and break the defect's symmetry. Crucially, our proposed framework immediately resolves a number of open questions in the field, including: (i) the microscopic origin of the observed contrast-enhancement in (111)-oriented anvils, and (ii) the surprising observation of NV contrast-inversion in certain high-pressure regimes. Our work lays the foundation for optimizing the performance of NV high-pressure sensors by controlling the local stress environment, and more generally, suggests that symmetry-breaking stresses can be utilized as a novel tuning knob for generic solid-state spin defects.

Figures (5)

• Figure 1: (a) Schematic of the diamond anvil cell (DAC) geometry. The DAC sample chamber is defined by the gasket-anvil assembly; it is loaded with the sample of interest, pressure-transmitting medium, and a ruby microsphere. A $\sim$50-nm layer of NV centers (about 1 ppm density) is embedded into the diamond anvil directly below the sample chamber. For ODMR measurements, a platinum wire is placed on the bottom culet to deliver microwaves. (b) Major quantum sensing applications using the NV center include magnetometry rondin2014magnetometry and sensing normal and shear (depicted) stresses in the sample broadway2019microscopicho2021recentkehayias2019imagingsuda2025gpa. (c) Continuous-wave ODMR measurements of NV centers in the (100)-cut anvil exhibit a drastic reduction in contrast with increasing pressure. The dominant culet stresses have symmetry-preserving and breaking projections on all NV subgroups, thereby inducing both a blue shift, $\Pi_z$, and a splitting, $2\Pi_\perp$ in the ODMR peaks. Notably, a surprising inversion of contrast is observed on the left peak around 60 GPa, as shown in the inset.
• Figure 2: The negatively charged NV center's energy level diagram and its optical cycle under (a) symmetry-preserving and (b) symmetry-breaking stress (here $\Pi_x$ stress, defined in Supplemental Materials). The NV center's low-lying electronic states contain two spin triplets ${}^3\!A_2$ and ${}^3\!E$, and two spin singlets ${}^1\!A_1$ and ${}^1\!E$, with three energy gaps defined as $\Delta, \Lambda, \Sigma$ in (a). The spin-1 basis adopted here is $\left|m_s=0\right\rangle$, and $\left|m_s=\pm\right\rangle = \frac{1}{\sqrt{2}}\left(\left|m_s=1\right\rangle \pm \left|m_s=-1\right\rangle\right)$, which are the spin eigenstates under $\Pi_x$ stress. Notably, for symmetry-preserving stress, $\sigma_\perp=\frac{1}{2}\left(\sigma_{xx} + \sigma_{yy}\right)$ and $\sigma_\parallel=\sigma_{zz}$ play qualitatively different roles in shifting energy gaps. Symmetry-breaking stress, however, breaks the defect's point group symmetry and allows every spin state to participate in the optical cycle. The ISC rates are color coded for their spins, with blue, dark red, and grey for $\ket{m_s=-, +, 0}$, respectively in (b). The line styles denote the microscopic origin for these ISCs (see Supplemental Materials for details).
• Figure 3: (a) First principles calculations of the intermediate components of the upper ISC, i.e., transverse spin-orbit coupling $\lambda_\perp$ (red) and vibrational overlap $F(\Delta)$ between the ${}^3\!E$ and ${}^1\!A_1$ manifolds (gray) versus hydrostatic (circle) and uniaxial [111] strain (triangle). The inset shows a cluster model of the NV center, which we use as a basis for our ab initio calculations. (b) Upper ISC rate $\Gamma_\text{ave}$ assembled from $\lambda_\perp$ and $F(\Delta)$ (dashed), and comparison of the contrast (relative to that at the ambient condition) between simulation (solid) and experiments hilberer2023enablingwang2024imaging (dots), where the color codes the hydrostaticity $\alpha$. Notably, the ISC rate exhibits a strong correlation with the relative contrast. (c) Schematic of a zoom-in DAC with (111)-cut diamond, and the embedded NV centers. (d) Rabi oscillations of the [111] NV at $\sigma_{ZZ}=18, 32, 56$ GPa respectively, from which contrast is extracted. The experimental data are fitted by damped sine waves and plotted by the orange, purple and blue dashed lines.
• Figure 4: First principles calculations of the ISC rates and ODMR contrast of NV centers in the (100)-cut diamond under stress. (a) Upper ISC rates versus stress with color codes for the three spins. (b) Lower ISC rates versus stress, with $\Gamma_z^\text{lower}$ exhibiting non-monotonic trend coming from negative interference between different ISC mechanisms (see main text and SM). (c) Ground state population distribution among the three spins. A gradual transfer from $n_0$ to $n_-$ begins around $25$ GPa and $n_-$ dominates the population from $65$ GPa. (d) Simulated ODMR contrast (solid) obtained by solving the rate model defined in the main text, with ISC rates acting as inputs. Notably, the contrast inversion in the left peak (representing transitions $\left|m_s=0\right\rangle \leftrightarrow \left|m_s=-\right\rangle$ driven by the MW) observed from experiments bhattacharyya2024imaging (discrete) is reproduced, as shown in the inset. The 'predicted' onset of positive contrast occurs slightly later compared to experiments, and the magnitude is also smaller, with possible reasons for this discrepancy discussed in detail in the Supplemental Materials.
• Figure 5: Positive contrast observed from ODMR measurements performed on NV centers in (a) (110)-cut anvil at 300 K, 25 GPa, $B_Z=85$ G hsieh2019imaging, and (b) (111)-cut anvil at 30 K, 28 GPa, $B_Z=150$ G. For the latter, the positive contrast originates from the non-[111]-oriented NV centers.