Hamiltonian Terms¶

The total NV center Hamiltonian is composed of several terms:

\[\hat{H} = \hat{H}_{\text{ZFS}} + \hat{H}_{\text{Zeeman}} + \hat{H}_{\text{HF}} + \hat{H}_{\text{MW}} + \hat{H}_{\text{optical}} + \hat{H}_{\text{Stark}} + \hat{H}_{\text{strain}} + \hat{H}_{\text{JT}}\]

Each term is implemented as an 18×18 matrix in the full Hilbert space.

Zero-Field Splitting (ZFS)¶

Physical Origin¶

The zero-field splitting arises from spin-spin dipolar interaction between the two unpaired electrons in the NV center. This creates an energy splitting even in the absence of external magnetic fields.

Mathematical Form¶

The ZFS Hamiltonian is:

\[\hat{H}_{\text{ZFS}} = D \hat{S}_z^2 + E(\hat{S}_x^2 - \hat{S}_y^2)\]

where:

\(D\): Axial ZFS parameter (splitting between \(m_s=0\) and \(m_s=\pm 1\))
\(E\): Transverse ZFS parameter (strain-induced splitting of \(m_s=\pm 1\))

Typical values:

\(D_{\text{gs}} = 2.87\,\text{GHz}\) (ground state at 300 K)
\(D_{\text{es}} = 1.42\,\text{GHz}\) (excited state)
\(E = 0\) to \(\sim 10\,\text{MHz}\) (depends on local strain)

Energy Levels¶

For \(E = 0\):

\[\begin{split}E(m_s = 0) &= 0 \\ E(m_s = \pm 1) &= D\end{split}\]

For \(E \neq 0\):

\[\begin{split}E(m_s = 0) &= 0 \\ E(m_s = +1) &= D + E \\ E(m_s = -1) &= D - E\end{split}\]

The \(E\) term lifts the degeneracy between \(m_s = +1\) and \(m_s = -1\).

Matrix Representation¶

In the \(\{|+1\rangle, |0\rangle, |-1\rangle\}\) basis:

\[\begin{split}\hat{S}_z^2 = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}\end{split}\]

\[\begin{split}\hat{S}_x^2 - \hat{S}_y^2 = \begin{pmatrix} 0 & 0 & 1 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end{pmatrix}\end{split}\]

Therefore:

\[\begin{split}\hat{H}_{\text{ZFS}} = \begin{pmatrix} D & 0 & E \\ 0 & 0 & 0 \\ E & 0 & D \end{pmatrix}\end{split}\]

Temperature Dependence¶

The D parameter has a temperature dependence:

\[D(T) \approx D_0 + \alpha (T - 300\,\text{K})\]

with \(\alpha \approx -74\,\text{kHz/K}\). This enables NV-based thermometry.

Zeeman Effect¶

Physical Origin¶

The Zeeman effect describes the interaction between the electron spin magnetic moment and an external magnetic field.

Mathematical Form¶

\[\hat{H}_{\text{Zeeman}} = \gamma_e \vec{B} \cdot \hat{\vec{S}} = \gamma_e (B_x \hat{S}_x + B_y \hat{S}_y + B_z \hat{S}_z)\]

where \(\gamma_e = g_e \mu_B / \hbar \approx 1.76 \times 10^{11}\,\text{rad/(s·T)}\).

Energy Shifts¶

For a field \(\vec{B} = B_z \hat{z}\) along the NV axis:

\[E(m_s) = \gamma_e B_z m_s\]

The splitting between adjacent \(m_s\) levels:

\[\Delta E = \gamma_e B_z \approx 28\,\text{MHz/mT} \cdot B[\text{mT}]\]

The splitting between \(m_s = +1\) and \(m_s = -1\):

\[\Delta E_{\pm 1} = 2\gamma_e B_z \approx 56\,\text{MHz/mT} \cdot B[\text{mT}]\]

Combined with ZFS¶

The full energy levels with ZFS and Zeeman:

\[\begin{split}E(m_s = 0) &= 0 \\ E(m_s = +1) &= D + \gamma_e B_z \\ E(m_s = -1) &= D - \gamma_e B_z\end{split}\]

This creates two distinct ODMR frequencies:

\[\begin{split}f_+ &= D + \gamma_e B_z / (2\pi) \\ f_- &= D - \gamma_e B_z / (2\pi)\end{split}\]

Transverse Fields¶

For non-axial fields, \(B_x, B_y \neq 0\), there is state mixing. The off-diagonal elements couple \(m_s = 0\) to \(m_s = \pm 1\):

\[\begin{split}\hat{H}_{\text{Zeeman}}^{\perp} = \frac{\gamma_e}{\sqrt{2}} \begin{pmatrix} 0 & B_x - iB_y & 0 \\ B_x + iB_y & 0 & B_x - iB_y \\ 0 & B_x + iB_y & 0 \end{pmatrix}\end{split}\]

N14 Hyperfine Coupling¶

Physical Origin¶

The N14 nucleus (\(I = 1\)) at the vacancy site couples to the electron spin through:

Fermi contact interaction: Isotropic coupling from electron density at nucleus
Dipolar interaction: Anisotropic coupling from magnetic dipole-dipole interaction

Additionally, N14 has a nuclear electric quadrupole moment that interacts with the local electric field gradient.

Mathematical Form¶

\[\hat{H}_{\text{HF}} = A_\parallel \hat{S}_z \hat{I}_z + A_\perp (\hat{S}_x \hat{I}_x + \hat{S}_y \hat{I}_y) + P \left(\hat{I}_z^2 - \frac{2}{3}\right)\]

where:

\(A_\parallel = -2.14\,\text{MHz}\): Parallel hyperfine coupling
\(A_\perp = -2.70\,\text{MHz}\): Perpendicular hyperfine coupling
\(P = -5.0\,\text{MHz}\): Nuclear quadrupole parameter

Hyperfine Tensor¶

The hyperfine interaction can be written as:

\[\hat{H}_{\text{HF}} = \hat{\vec{S}} \cdot \mathbf{A} \cdot \hat{\vec{I}}\]

where the hyperfine tensor (in the NV frame) is:

\[\begin{split}\mathbf{A} = \begin{pmatrix} A_\perp & 0 & 0 \\ 0 & A_\perp & 0 \\ 0 & 0 & A_\parallel \end{pmatrix}\end{split}\]

Energy Shifts¶

For \(m_s = 0\) (only quadrupole contributes):

\[E(m_s=0, m_I) = P \left(m_I^2 - \frac{2}{3}\right)\]

For \(m_s = \pm 1\) (hyperfine + quadrupole):

\[E(m_s, m_I) = A_\parallel m_s m_I + P \left(m_I^2 - \frac{2}{3}\right)\]

This creates a characteristic triplet structure in ODMR spectra.

9×9 Matrix¶

In the \(|m_s\rangle \otimes |m_I\rangle\) basis (9-dimensional):

\[\hat{H}_{\text{HF}} = A_\parallel (\hat{S}_z \otimes \hat{I}_z) + A_\perp (\hat{S}_x \otimes \hat{I}_x + \hat{S}_y \otimes \hat{I}_y) + P (\mathbb{1}_3 \otimes \hat{I}_z^2) - \frac{2P}{3}\mathbb{1}_9\]

C13 Hyperfine Coupling¶

Physical Origin¶

Carbon-13 nuclei (\(I = 1/2\), 1.1% natural abundance) in the diamond lattice couple to the NV electron spin through dipolar interaction. The coupling strength depends on the distance and orientation to the NV center.

Mathematical Form¶

For a C13 nucleus at position \(\vec{r}\) from the NV:

\[\hat{H}_{\text{C13}} = \hat{\vec{S}} \cdot \mathbf{A}_{\text{dip}} \cdot \hat{\vec{I}}\]

The dipolar tensor:

\[A_{\text{dip}}^{ij} = \frac{\mu_0}{4\pi} \frac{\gamma_e \gamma_{\text{C13}} \hbar}{r^3} \left(\delta_{ij} - 3\frac{r_i r_j}{r^2}\right)\]

Secular Approximation¶

In the high-field limit (\(B \gg A/\gamma_e\)), only the secular terms survive:

\[\hat{H}_{\text{C13}}^{\text{sec}} = A_{zz} \hat{S}_z \hat{I}_z\]

where:

\[A_{zz} = \frac{\mu_0}{4\pi} \frac{\gamma_e \gamma_{\text{C13}} \hbar}{r^3} (1 - 3\cos^2\theta)\]

with \(\theta\) the angle between \(\vec{r}\) and the NV axis.

Typical Coupling Strengths¶

C13 Hyperfine Couplings¶
Position	Distance	Coupling
Nearest neighbor	1.54 Å	~130 MHz
2nd shell	2.5 Å	~15 MHz
3rd shell	3.5 Å	~3 MHz

Microwave Drive¶

Physical Origin¶

Microwave radiation at frequencies near \(D \pm \gamma_e B\) drives transitions between \(m_s = 0\) and \(m_s = \pm 1\).

Mathematical Form (Rotating Frame)¶

In the rotating wave approximation:

\[\hat{H}_{\text{MW}} = \frac{\Omega}{2} \left(\hat{S}_+ e^{-i\phi} + \hat{S}_- e^{i\phi}\right) + \Delta \hat{S}_z\]

where:

\(\Omega\): Rabi frequency (proportional to MW amplitude)
\(\phi\): MW phase
\(\Delta = \omega_{\text{MW}} - \omega_0\): Detuning from resonance

On Resonance (\(\Delta = 0\))¶

\[\hat{H}_{\text{MW}} = \frac{\Omega}{2} \left(\hat{S}_+ e^{-i\phi} + \hat{S}_- e^{i\phi}\right)\]

For \(\phi = 0\) (X rotation):

\[\hat{H}_{\text{MW}} = \Omega \hat{S}_x\]

For \(\phi = \pi/2\) (Y rotation):

\[\hat{H}_{\text{MW}} = \Omega \hat{S}_y\]

Rabi Oscillations¶

Starting from \(|m_s = 0\rangle\), the population in \(|m_s = +1\rangle\):

\[P_{+1}(t) = \sin^2\left(\frac{\Omega t}{2}\right)\]

For a \(\pi\)-pulse (\(\Omega t = \pi\)), full population transfer occurs.

Pulse Durations¶

For a given Rabi frequency \(\Omega\):

\[\begin{split}t_{\pi/2} &= \frac{\pi}{2\Omega} \\ t_{\pi} &= \frac{\pi}{\Omega}\end{split}\]

Example: For \(\Omega = 2\pi \times 10\,\text{MHz}\):

\(t_{\pi/2} = 25\,\text{ns}\)
\(t_{\pi} = 50\,\text{ns}\)

Optical Coupling¶

Mathematical Form¶

The optical drive couples ground and excited states:

\[\hat{H}_{\text{opt}} = \frac{\Omega_{\text{opt}}}{2} \sum_{m_s, m_I} \left(|e, m_s, m_I\rangle \langle g, m_s, m_I| + \text{h.c.}\right)\]

For spin-conserving transitions. Spin-selective transitions can be achieved with polarization control.

Stark Effect¶

DC Stark Effect¶

Electric fields shift the ZFS through:

\[\hat{H}_{\text{Stark}}^{\text{DC}} = d_\perp E_\perp (\hat{S}_x^2 - \hat{S}_y^2) + d_\parallel E_z \hat{S}_z^2\]

where:

\(d_\perp \approx 17\,\text{Hz/(V/m)}\)
\(d_\parallel \approx 3.5\,\text{Hz/(V/m)}\)

AC Stark Effect¶

Off-resonant optical fields create effective spin Hamiltonians through the AC Stark shift.

Strain¶

Physical Origin¶

Lattice strain from defects or external stress perturbs the NV electronic structure.

Mathematical Form¶

\[\hat{H}_{\text{strain}} = \sum_{i,j} \varepsilon_{ij} \frac{\{\hat{S}_i, \hat{S}_j\}}{2}\]

where \(\varepsilon_{ij}\) is the strain tensor and \(\{A, B\} = AB + BA\).

The primary effect is through transverse strain \(E_1, E_2\):

\[\hat{H}_{\text{strain}} \approx E_1 (\hat{S}_x^2 - \hat{S}_y^2) + E_2 (\hat{S}_x \hat{S}_y + \hat{S}_y \hat{S}_x)\]

Jahn-Teller Effect¶

In the excited state, the orbital degeneracy leads to vibronic coupling:

\[\hat{H}_{\text{JT}} = \lambda_{\text{JT}} \hat{Q} \cdot \hat{\sigma}\]

where \(\hat{Q}\) are phonon coordinates and \(\hat{\sigma}\) are pseudo-spin operators in the orbital space.

This leads to dynamic averaging of the excited state ZFS at room temperature.