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 \(C_{3v}\) symmetry with the symmetry axis along the \(\langle 111 \rangle\) crystallographic direction.

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 \({}^3A_2\)

A spin triplet with \(S=1\). The orbital singlet has \(A_2\) symmetry. Zero-field splitting \(D_{\text{gs}} = 2.87\,\text{GHz}\) at room temperature.

Excited State \({}^3E\)

A spin triplet with orbital doublet symmetry. Zero-field splitting \(D_{\text{es}} = 1.42\,\text{GHz}\).

Singlet States \({}^1A_1\) and \({}^1E\)

Intermediate states enabling intersystem crossing (ISC) and spin polarization.

The optical transition between \({}^3A_2\) and \({}^3E\) occurs at:

\[\lambda_{\text{ZPL}} = 637\,\text{nm} \quad (1.945\,\text{eV})\]

Hilbert Space

The full Hilbert space of the NV center in this simulator is:

\[\mathcal{H} = \mathcal{H}_{\text{electronic}} \otimes \mathcal{H}_{\text{spin}} \otimes \mathcal{H}_{\text{nuclear}}\]

with dimensions:

• \(\mathcal{H}_{\text{electronic}}\): 2-dimensional (ground \(|g\rangle\), excited \(|e\rangle\))

• \(\mathcal{H}_{\text{spin}}\): 3-dimensional (\(|m_s = +1\rangle, |0\rangle, |-1\rangle\))

• \(\mathcal{H}_{\text{nuclear}}\): 3-dimensional (\(|m_I = +1\rangle, |0\rangle, |-1\rangle\) for N14)

Total dimension: \(2 \times 3 \times 3 = 18\)

Basis States

The computational basis is ordered as:

\[|n\rangle = |g/e\rangle \otimes |m_s\rangle \otimes |m_I\rangle\]

with the index mapping:

\[n = 9 \cdot \delta_{e} + 3 \cdot (1 - m_s) + (1 - m_I)\]

where \(\delta_e = 1\) for excited state, \(\delta_e = 0\) for ground state.

Explicitly:

Basis state ordering

Index	State	Description
0	\(|g, +1, +1\rangle\)	Ground, ms=+1, mI=+1
1	\(|g, +1, 0\rangle\)	Ground, ms=+1, mI=0
2	\(|g, +1, -1\rangle\)	Ground, ms=+1, mI=-1
3	\(|g, 0, +1\rangle\)	Ground, ms=0, mI=+1
4	\(|g, 0, 0\rangle\)	Ground, ms=0, mI=0
5	\(|g, 0, -1\rangle\)	Ground, ms=0, mI=-1
6	\(|g, -1, +1\rangle\)	Ground, ms=-1, mI=+1
7	\(|g, -1, 0\rangle\)	Ground, ms=-1, mI=0
8	\(|g, -1, -1\rangle\)	Ground, ms=-1, mI=-1
9-17	\(|e, m_s, m_I\rangle\)	Excited state analogues

Spin-1 Operators

The electron spin operators for \(S=1\) in the \(\{|+1\rangle, |0\rangle, |-1\rangle\}\) basis:

\[\begin{split}\hat{S}_x = \frac{1}{\sqrt{2}} \begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix}\end{split}\]

\[\begin{split}\hat{S}_y = \frac{1}{\sqrt{2}} \begin{pmatrix} 0 & -i & 0 \\ i & 0 & -i \\ 0 & i & 0 \end{pmatrix}\end{split}\]

\[\begin{split}\hat{S}_z = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \end{pmatrix}\end{split}\]

These satisfy the angular momentum commutation relations:

\[[\hat{S}_i, \hat{S}_j] = i\epsilon_{ijk}\hat{S}_k\]

and the Casimir relation:

\[\hat{S}^2 = \hat{S}_x^2 + \hat{S}_y^2 + \hat{S}_z^2 = S(S+1)\mathbb{1} = 2\mathbb{1}\]

Raising and lowering operators:

\[\hat{S}_+ = \hat{S}_x + i\hat{S}_y, \quad \hat{S}_- = \hat{S}_x - i\hat{S}_y\]

with:

\[\hat{S}_\pm |m_s\rangle = \sqrt{S(S+1) - m_s(m_s \pm 1)} |m_s \pm 1\rangle\]

Physical Constants

NV Center Physical Constants

Parameter	Value	Unit
Ground state ZFS \(D_{\text{gs}}\)	2.87	GHz
Excited state ZFS \(D_{\text{es}}\)	1.42	GHz
Electron g-factor \(g_e\)	2.0028	—
Gyromagnetic ratio \(\gamma_e\)	\(1.76 \times 10^{11}\)	rad/(s·T)
N14 parallel hyperfine \(A_\parallel\)	-2.14	MHz
N14 perpendicular hyperfine \(A_\perp\)	-2.70	MHz
N14 quadrupole \(P\)	-5.0	MHz
Optical lifetime \(\tau_{\text{rad}}\)	~12	ns
ZPL wavelength	637	nm
Temperature coefficient \(dD/dT\)	-74	kHz/K

Optical Transitions

Spin-Conserving Transitions

The primary optical transitions are spin-conserving:

\[|g, m_s\rangle \xrightarrow{637\,\text{nm}} |e, m_s\rangle\]

These transitions are used for:

• Initialization: Optical pumping into \(|g, m_s=0\rangle\)

• Readout: Spin-dependent fluorescence

Intersystem Crossing (ISC)

Non-radiative decay through singlet states enables spin polarization:

\[|e, m_s = \pm 1\rangle \xrightarrow{\text{ISC}} |{}^1A_1\rangle \rightarrow |{}^1E\rangle \xrightarrow{\text{ISC}} |g, m_s = 0\rangle\]

The ISC rate is spin-dependent:

• \(m_s = \pm 1\): High ISC probability (\(\sim 30-50\%\))

• \(m_s = 0\): Low ISC probability (\(\sim 0\%\))

This spin-dependent shelving enables:

1. Optical spin polarization: After ~1 μs of illumination, >90% population in \(|m_s=0\rangle\)

2. Spin readout: \(m_s=0\) is brighter than \(m_s=\pm 1\)

Contrast

The fluorescence contrast between spin states is:

\[C = \frac{I_{m_s=0} - I_{m_s=\pm 1}}{I_{m_s=0}} \approx 20-30\%\]

This contrast is the basis for optically detected magnetic resonance (ODMR).