=============================== The Nitrogen-Vacancy Center =============================== Physical Structure ================== The nitrogen-vacancy (NV) center is a point defect in diamond consisting of a substitutional nitrogen atom adjacent to a lattice vacancy. The NV center possesses :math:`C_{3v}` symmetry with the symmetry axis along the :math:`\langle 111 \rangle` crystallographic direction. .. figure:: /_static/nv_structure.png :align: center :width: 400px :alt: NV center structure Schematic of the NV center in the diamond lattice. The nitrogen atom (blue) is adjacent to a vacancy (white), surrounded by three carbon atoms. Electronic Structure ==================== The negatively charged NV⁻ center has six electrons: three from dangling bonds of carbon atoms, two from the nitrogen, and one captured from the environment. These electrons form a spin-1 system. Energy Levels ------------- The electronic structure consists of: **Ground State** :math:`{}^3A_2` A spin triplet with :math:`S=1`. The orbital singlet has :math:`A_2` symmetry. Zero-field splitting :math:`D_{\text{gs}} = 2.87\,\text{GHz}` at room temperature. **Excited State** :math:`{}^3E` A spin triplet with orbital doublet symmetry. Zero-field splitting :math:`D_{\text{es}} = 1.42\,\text{GHz}`. **Singlet States** :math:`{}^1A_1` and :math:`{}^1E` Intermediate states enabling intersystem crossing (ISC) and spin polarization. The optical transition between :math:`{}^3A_2` and :math:`{}^3E` occurs at: .. math:: \lambda_{\text{ZPL}} = 637\,\text{nm} \quad (1.945\,\text{eV}) Hilbert Space ============= The full Hilbert space of the NV center in this simulator is: .. math:: \mathcal{H} = \mathcal{H}_{\text{electronic}} \otimes \mathcal{H}_{\text{spin}} \otimes \mathcal{H}_{\text{nuclear}} with dimensions: - :math:`\mathcal{H}_{\text{electronic}}`: 2-dimensional (ground :math:`|g\rangle`, excited :math:`|e\rangle`) - :math:`\mathcal{H}_{\text{spin}}`: 3-dimensional (:math:`|m_s = +1\rangle, |0\rangle, |-1\rangle`) - :math:`\mathcal{H}_{\text{nuclear}}`: 3-dimensional (:math:`|m_I = +1\rangle, |0\rangle, |-1\rangle` for N14) **Total dimension**: :math:`2 \times 3 \times 3 = 18` Basis States ------------ The computational basis is ordered as: .. math:: |n\rangle = |g/e\rangle \otimes |m_s\rangle \otimes |m_I\rangle with the index mapping: .. math:: n = 9 \cdot \delta_{e} + 3 \cdot (1 - m_s) + (1 - m_I) where :math:`\delta_e = 1` for excited state, :math:`\delta_e = 0` for ground state. Explicitly: .. list-table:: Basis state ordering :widths: 10 30 30 :header-rows: 1 * - Index - State - Description * - 0 - :math:`|g, +1, +1\rangle` - Ground, ms=+1, mI=+1 * - 1 - :math:`|g, +1, 0\rangle` - Ground, ms=+1, mI=0 * - 2 - :math:`|g, +1, -1\rangle` - Ground, ms=+1, mI=-1 * - 3 - :math:`|g, 0, +1\rangle` - Ground, ms=0, mI=+1 * - 4 - :math:`|g, 0, 0\rangle` - Ground, ms=0, mI=0 * - 5 - :math:`|g, 0, -1\rangle` - Ground, ms=0, mI=-1 * - 6 - :math:`|g, -1, +1\rangle` - Ground, ms=-1, mI=+1 * - 7 - :math:`|g, -1, 0\rangle` - Ground, ms=-1, mI=0 * - 8 - :math:`|g, -1, -1\rangle` - Ground, ms=-1, mI=-1 * - 9-17 - :math:`|e, m_s, m_I\rangle` - Excited state analogues Spin-1 Operators ================ The electron spin operators for :math:`S=1` in the :math:`\{|+1\rangle, |0\rangle, |-1\rangle\}` basis: .. math:: \hat{S}_x = \frac{1}{\sqrt{2}} \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix} .. math:: \hat{S}_y = \frac{1}{\sqrt{2}} \begin{pmatrix} 0 & -i & 0 \\ i & 0 & -i \\ 0 & i & 0 \end{pmatrix} .. math:: \hat{S}_z = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \end{pmatrix} These satisfy the angular momentum commutation relations: .. math:: [\hat{S}_i, \hat{S}_j] = i\epsilon_{ijk}\hat{S}_k and the Casimir relation: .. math:: \hat{S}^2 = \hat{S}_x^2 + \hat{S}_y^2 + \hat{S}_z^2 = S(S+1)\mathbb{1} = 2\mathbb{1} Raising and lowering operators: .. math:: \hat{S}_+ = \hat{S}_x + i\hat{S}_y, \quad \hat{S}_- = \hat{S}_x - i\hat{S}_y with: .. math:: \hat{S}_\pm |m_s\rangle = \sqrt{S(S+1) - m_s(m_s \pm 1)} |m_s \pm 1\rangle Physical Constants ================== .. list-table:: NV Center Physical Constants :widths: 40 30 30 :header-rows: 1 * - Parameter - Value - Unit * - Ground state ZFS :math:`D_{\text{gs}}` - 2.87 - GHz * - Excited state ZFS :math:`D_{\text{es}}` - 1.42 - GHz * - Electron g-factor :math:`g_e` - 2.0028 - — * - Gyromagnetic ratio :math:`\gamma_e` - :math:`1.76 \times 10^{11}` - rad/(s·T) * - N14 parallel hyperfine :math:`A_\parallel` - -2.14 - MHz * - N14 perpendicular hyperfine :math:`A_\perp` - -2.70 - MHz * - N14 quadrupole :math:`P` - -5.0 - MHz * - Optical lifetime :math:`\tau_{\text{rad}}` - ~12 - ns * - ZPL wavelength - 637 - nm * - Temperature coefficient :math:`dD/dT` - -74 - kHz/K Optical Transitions =================== Spin-Conserving Transitions --------------------------- The primary optical transitions are spin-conserving: .. math:: |g, m_s\rangle \xrightarrow{637\,\text{nm}} |e, m_s\rangle These transitions are used for: - **Initialization**: Optical pumping into :math:`|g, m_s=0\rangle` - **Readout**: Spin-dependent fluorescence Intersystem Crossing (ISC) -------------------------- Non-radiative decay through singlet states enables spin polarization: .. math:: |e, m_s = \pm 1\rangle \xrightarrow{\text{ISC}} |{}^1A_1\rangle \rightarrow |{}^1E\rangle \xrightarrow{\text{ISC}} |g, m_s = 0\rangle The ISC rate is spin-dependent: - :math:`m_s = \pm 1`: High ISC probability (:math:`\sim 30-50\%`) - :math:`m_s = 0`: Low ISC probability (:math:`\sim 0\%`) This spin-dependent shelving enables: 1. **Optical spin polarization**: After ~1 μs of illumination, >90% population in :math:`|m_s=0\rangle` 2. **Spin readout**: :math:`m_s=0` is brighter than :math:`m_s=\pm 1` Contrast -------- The fluorescence contrast between spin states is: .. math:: C = \frac{I_{m_s=0} - I_{m_s=\pm 1}}{I_{m_s=0}} \approx 20-30\% This contrast is the basis for optically detected magnetic resonance (ODMR).