SpinDefectSim — Physical Models and Mathematical Reference¶

This document describes every physical model and formula implemented in SpinDefectSim. It is intended as a compact reference for researchers using or extending the library; full derivations can be found in the cited literature.

Table of Contents¶

Spin Hamiltonian
ODMR Transition Frequencies
Lineshape and CW ODMR Spectrum
Sensing Protocols
4.1 Ramsey Free-Induction Decay
4.2 Hahn-Echo
4.3 XY8 Dynamical Decoupling
4.4 Lock-in Differential Signal
Shot-Noise SNR Model
Electric-Field Models
6.1 Gate Bias Field
6.2 Screened Coulomb Field — Bare
6.3 Yukawa Screening
6.4 Dual-Gate Image-Charge Model
6.5 Dielectric Transmission
Magnetic-Field Models
7.1 Finite Wire Segment (Analytic)
7.2 Edge Currents from Magnetization
7.3 Bulk Currents from Non-Uniform Magnetization
Ensemble Modelling
Model Assumptions and Limitations
Nuclear-Spin Hyperfine Interaction
- 10.1 Hyperfine Hamiltonian
- 10.2 Nuclear Quadrupole Coupling
- 10.3 Nuclear Zeeman Term
- 10.4 Tensor-Product Hilbert Space
- 10.5 Isotope Constants
Rate-Equation Model for CW Contrast
- 11.1 Level Structure
- 11.2 Rate Matrix and Steady State
- 11.3 CW ODMR Contrast
- 11.4 Pre-built Defect Parameters
References

1. Spin Hamiltonian¶

The library supports spin-$S$ defects with an arbitrary quantization axis. For a defect with quantization axis $\hat{z}'$ in the lab frame, all applied fields are first rotated into the defect local frame before the Hamiltonian is constructed.

The full Hamiltonian (in frequency units, $H/h$, Hz) is:

\[ \frac{H}{h} = D_0 \left(S_{z'}^2 - \frac{S(S+1)}{3}\,I\right) + E_0 \left(S_{x'}^2 - S_{y'}^2\right) + d_\parallel E_{z'}\left(S_{z'}^2 - \frac{S(S+1)}{3}\,I\right) + d_\perp\left[E_{y'}(S_{x'}^2 - S_{y'}^2) + E_{x'}\{S_{x'},S_{y'}\}\right] + \gamma_e\left(B_{x'} S_{x'} + B_{y'} S_{y'} + B_{z'} S_{z'}\right) \]

Symbols

Symbol	Meaning	Typical value (VB⁻ in hBN)
$S$	spin quantum number	1
$D_0$	axial zero-field splitting (Hz)	3.46 GHz
$E_0$	transverse ZFS strain (Hz)	50 MHz
$d_\perp$	transverse E-field coupling (Hz V⁻¹ m)	0.35 Hz/(V/m)
$d_\parallel$	axial E-field coupling (Hz V⁻¹ m)	≈ 0 (usually neglected)
$\gamma_e$	gyromagnetic ratio (Hz T⁻¹)	28 GHz/T
$\vec{B}$	total magnetic field in local frame (T)	—
$\vec{E}$	total electric field in local frame (V/m)	—

Frame rotation. If the defect's quantization axis is $\hat{z}' \ne \hat{z}_\text{lab}$, a rotation matrix $R$ (constructed via Gram–Schmidt from $\hat{z}'$) is applied:

\[ \vec{B}_\text{local} = R\,\vec{B}_\text{lab}, \qquad \vec{E}_\text{local} = R\,\vec{E}_\text{lab} \]

Spin operators for arbitrary $S$ are built from the standard ladder operators and the Wigner–Eckart theorem. For $S = 1$ (e.g. VB⁻, NV⁻) the $3\times 3$ matrix representation is used; for $S = 1/2$ (e.g. P1 centre) a $2\times 2$ representation.

Total B field entering the Hamiltonian is the sum of the bias field $\vec{B}_0$ (from Defaults.B_mT) and any stray signal field $\vec{B}_\text{extra}$:

\[ \vec{B} = \vec{B}_0 + \vec{B}_\text{extra} \]

References: [1] Gottscholl et al., Nature Materials 2020; [2] Dolde et al., Nature Physics 2011.

2. ODMR Transition Frequencies¶

The Hamiltonian is diagonalised numerically:

\[ H\,|\psi_n\rangle = E_n\,|\psi_n\rangle, \qquad E_n / h \equiv \varepsilon_n \text{ (Hz)} \]

The eigenstate with the highest overlap with the bare $|m_S = 0\rangle$ Zeeman state is identified as the "reference" state (index ms0_index). ODMR-active transition frequencies are defined as the energy differences between this reference and all other eigenstates:

\[ f_i = |\varepsilon_i - \varepsilon_{m_S=0}|, \qquad i \ne m_S=0 \]

For a spin-1 system this gives two transitions $f_1 < f_2$. For spin-1/2 it gives one transition $f_1$.

3. Lineshape and CW ODMR Spectrum¶

Each transition is modelled with a Lorentzian lineshape:

\[ \mathcal{L}(f;\, f_0,\, \Delta f) = \frac{(\Delta f/2)^2}{(f - f_0)^2 + (\Delta f/2)^2} \]

where $\Delta f = (\pi T_2^*)^{-1}$ is the FWHM determined by the inhomogeneous dephasing time $T_2^*$.

The normalised PL contrast for a single defect is:

\[ \text{PL}(f) = 1 - C \sum_i \mathcal{L}(f;\, f_i,\, \Delta f) \]

where $C$ is the optical spin contrast (dimensionless; 0.02 for VB⁻ in hBN).

Ensemble CW ODMR. For an ensemble of $N$ defects each with its own transition frequencies $\{f_i^{(n)}\}$ (derived from individual local E-fields), the spectrum is averaged incoherently:

\[ \overline{\text{PL}}(f) = \frac{1}{N}\sum_{n=1}^{N} \text{PL}^{(n)}(f) \]

Reference: [3] Haykal et al., Nature Communications 2022.

4. Sensing Protocols¶

4.1 Ramsey Free-Induction Decay¶

Sequence: $\frac{\pi}{2} - \tau - \frac{\pi}{2}$ — readout

The Ramsey sequence is sensitive to quasi-static frequency shifts (long correlation times) and decays at rate $1/T_2^*$. For a defect with ODMR transition frequency $f_0$ (in the presence of the signal field) vs. $f_\text{ref}$ (without), the accumulated phase during free precession is:

\[ \phi(\tau) = 2\pi\,(f_0 - f_\text{ref})\,\tau \]

The PL signal (ensemble-averaged) is:

\[ S_\text{Ramsey}(\tau) = \frac{1}{N}\sum_{n=1}^{N} \left[1 - C \cos\left(2\pi\,\delta f^{(n)}\,\tau\right)\right] e^{-\tau/T_2^*} \]

where $\delta f^{(n)} = f_0^{(n)} - f_\text{ref}^{(n)}$ is the local frequency shift, and $T_2^*$ is the inhomogeneous dephasing time.

Sensitivity (single shot):

\[ \eta_\text{Ramsey} \sim \frac{1}{C\sqrt{N_\text{ph}}}\,\frac{1}{\gamma_e\,\tau_\text{opt}} \]

with optimal $\tau_\text{opt} \approx T_2^* / \sqrt{2}$.

Reference: [4] Degen, Reinhard & Cappellaro, Rev. Mod. Phys. 2017.

4.2 Hahn-Echo¶

Sequence: $\frac{\pi}{2} - \tau - \pi - \tau - \frac{\pi}{2}$ — readout

The $\pi$ refocusing pulse cancels quasi-static inhomogeneity; the echo is sensitive to fluctuations at frequency $\sim 1/(2\tau)$ and decays at the slower rate $1/T_2$.

\[ S_\text{echo}(\tau) = \frac{1}{N}\sum_{n=1}^{N} \cos\left(2\pi\,\delta f^{(n)}\,\tau\right) e^{-\tau/T_2} \]

The decay envelope $e^{-\tau/T_2}$ is a phenomenological simple exponential parameterised by the coherence time $T_2$. (The physically exact envelope under a nuclear-spin-bath noise spectral density is a stretched exponential $e^{-(2\tau/T_2)^3}$; the library uses the simpler form for efficiency.)

Optimal τ: $\tau_\text{opt} = T_2$, the maximum of the signal envelope $\tau\,e^{-\tau/T_2}$.

References: [4] Degen et al. 2017; [5] de Lange et al., Science 2010.

4.3 XY8 Dynamical Decoupling¶

Sequence: $\frac{\pi}{2}_X - \left[\frac{\tau}{2} - \pi_X - \tau - \pi_Y - \tau - \pi_X - \tau - \pi_Y - \tau - \pi_Y - \tau - \pi_X - \tau - \pi_Y - \tau - \pi_X - \frac{\tau}{2}\right] - \frac{\pi}{2}$

XY8 applies 8 $\pi$-pulses with alternating X/Y axes, extending coherence by suppressing higher-order noise terms via the filter function:

\[ F_{XY8}(\omega) = 8\tan^2\left(\frac{\omega\tau}{2}\right)\,\frac{\cos^2(4\omega\tau)}{\omega^2} \]

The filter peak at $\omega = \pi/(2\tau)$ selects AC signals at $f_\text{AC} = 1/(2\tau)$ while rejecting DC and low-frequency noise. Sensitivity scales as $\sim 1/(8^{1/2})$ relative to the Hahn-echo at the same total time, because the $\sqrt{8}$ more phase accumulations overcome the $\sqrt{8}$ slower rep rate.

Reference: [6] Yan et al., Nature Communications 2013.

4.4 Lock-in Differential Signal¶

All protocols are evaluated in a lock-in differential scheme: the experiment is alternately run with ("with") and without ("no") the signal field (E or B), and the difference is recorded:

\[ \Delta S(\tau) = S_\text{with}(\tau) - S_\text{no}(\tau) \]

This cancels common-mode noise (laser intensity fluctuations, constant background PL) and makes the signal zero-mean in the absence of the analyte.

The peak signal is:

\[ \Delta S_\text{peak} = \max_\tau|\Delta S(\tau)| \]

achieved at the optimal free-precession time $\tau_\text{opt}$.

5. Shot-Noise SNR Model¶

The fundamental sensitivity limit in photon-counting ODMR is shot noise.

Single-shot noise floor:

\[ \sigma = \frac{1}{C\sqrt{N_\text{ph}}} \]

where $C$ is the spin-state contrast and $N_\text{ph}$ is the number of photons collected per shot.

SNR after $N_\text{avg}$ averages:

\[ \text{SNR} = \frac{|\Delta S|}{\sigma}\sqrt{N_\text{avg}} = |\Delta S|\, C\sqrt{N_\text{ph}\, N_\text{avg}} \]

Averages required to reach threshold SNR $\rho$:

\[ N_\text{avg} = \left(\frac{\rho\,\sigma}{\Delta S}\right)^2 = \frac{\rho^2}{C^2 N_\text{ph}\,\Delta S^2} \]

Integration time for a given sequence with repetition rate $R(\tau)$:

\[ T_\text{int} = \frac{N_\text{avg}}{R(\tau)}, \qquad R(\tau) = \frac{1}{T_\text{init} + T_\text{pulses} + n_\text{fp}\,\tau + T_\text{ro}} \]

where $T_\text{init}$, $T_\text{pulses}$, $T_\text{ro}$ are the initialisation, pulse, and readout gate times, and $n_\text{fp}$ is the number of free-precession intervals.

Reference: [4] Degen et al. 2017, Section IV.

6. Electric-Field Models¶

6.1 Gate Bias Field¶

A uniform background field with an optional linear spatial gradient:

\[ \vec{E}_\text{gate}(\vec{r}) = \vec{E}_0 + \begin{pmatrix} G_{xx}\,x + G_{xy}\,y \\ G_{yx}\,x + G_{yy}\,y \\ 0 \end{pmatrix} \]

where $\mathbf{G} \in \mathbb{R}^{2\times 2}$ is the field-gradient matrix (V m⁻²). Implemented in electrometry.efield.E_gate_bias.

6.2 Screened Coulomb Field — Bare¶

For a point charge $q$ at position $\vec{r}_s$ and an observation point $\vec{r}_\text{obs}$:

\[ \vec{E}(\vec{r}_\text{obs}) = \frac{q}{4\pi\varepsilon_0\,\varepsilon_\text{eff}} \frac{\vec{r}_\text{obs} - \vec{r}_s}{|\vec{r}_\text{obs} - \vec{r}_s|^3} \]

$\varepsilon_\text{eff}$ is an effective relative permittivity that accounts for the dielectric screening of the host material and any encapsulation layers. Implemented in electrometry.efield.E_disorder_point_charges with screening_model=None.

6.3 Yukawa Screening¶

A phenomenological Yukawa (screened Coulomb) potential accounts for free-carrier screening at finite carrier density:

\[ V(r) = \frac{q}{4\pi\varepsilon_0\,\varepsilon_\text{eff}} \frac{e^{-r/\lambda}}{r}, \qquad \vec{E} = -\nabla V = \frac{q}{4\pi\varepsilon_0\,\varepsilon_\text{eff}} \frac{(1 + r/\lambda)}{r^3}\,e^{-r/\lambda}\,(\vec{r}_\text{obs}-\vec{r}_s) \]

where $\lambda$ is the Thomas–Fermi screening length. Activated via screening_model="yukawa" with parameter lambda_screen.

6.4 Dual-Gate Image-Charge Model¶

For a defect layer sandwiched between two grounded metallic gates at $z = 0$ and $z = d_\text{gate}$, the electrostatic boundary conditions require zero potential at both plates. This is satisfied by an infinite series of image charges:

\[ \vec{E}_\text{total}(\vec{r}_\text{obs}) = \frac{1}{4\pi\varepsilon_0\,\varepsilon_\text{eff}} \sum_{n=-N_\text{im}}^{N_\text{im}}(-1)^n\,q\, \frac{\vec{r}_\text{obs} - \vec{r}_s^{(n)}}{|\vec{r}_\text{obs} - \vec{r}_s^{(n)}|^3} \]

where the image charge positions are:

\[ z_s^{(n)} = z_s + 2n\,d_\text{gate} \]

$N_\text{im}$ controls the number of image-charge pairs included (truncation). The series converges rapidly; $N_\text{im} \ge 10$ is typically sufficient. Activated via screening_model="dual_gate".

Reference: [7] Jackson, J. D. Classical Electrodynamics, 3rd ed. (Wiley, 1999), §2.9.

6.5 Dielectric Transmission¶

When the observation layer has a dielectric constant $\varepsilon_\text{layer}$ different from the host (e.g. hBN flake encapsulated in air), the quasi-static planar transmission factor is:

\[ \eta = \frac{2\varepsilon_\text{layer}}{\varepsilon_\text{layer} + \varepsilon_\text{host}} \]

and the transmitted field is $\vec{E}_\text{transmitted} = \eta\,\vec{E}_\text{source}$.

7. Magnetic-Field Models¶

All B-field calculations use the Biot–Savart law. The global prefactor is $\mu_0/(4\pi) = 10^{-7}$ T m A⁻¹.

7.1 Finite Wire Segment (Analytic)¶

For a straight wire carrying current $I$ from $\vec{A}$ to $\vec{B}$, the analytic Biot–Savart result at observation point $\vec{P}$ is:

\[ \vec{B} = \frac{\mu_0 I}{4\pi R}(\sin\theta_1 - \sin\theta_2)\;(\hat{d} \times \hat{r}_\perp) \]

where $R = |\vec{P} - \vec{A} - [(\vec{P} - \vec{A})\cdot\hat{d}]\,\hat{d}|$ is the perpendicular distance from $\vec{P}$ to the wire line, $\hat{d}$ is the unit vector along the wire, and:

\[ \sin\theta_i = \frac{s_i}{\sqrt{s_i^2 + R^2}}, \qquad s_1 = (\vec{P} - \vec{A})\cdot\hat{d}, \quad s_2 = (\vec{P} - \vec{B})\cdot\hat{d} \]

This formula has no quadrature error and is used for all edge-current segments. Implemented in magnetometry.bfield.B_from_wire_segment and magnetometry.bfield.B_from_edge_segments.

7.2 Edge Currents from Magnetization¶

A 2-D ferromagnetic or ferrimagnetic layer with out-of-plane magnetization $M_z(\vec{r})$ (units: A, i.e. magnetic moment per unit area in SI) carries edge currents along its boundary:

\[ I_\text{edge} = M_z \cdot \hat{t} \]

where $\hat{t}$ is the tangent to the boundary. This is the Amperian current picture: a region of uniform $M_z$ is equivalent to a closed loop current $I = M_z$ (in amperes) flowing along its perimeter.

For a discretised boundary (polygon with vertices $\vec{v}_i$), each segment $(\vec{v}_i, \vec{v}_{i+1})$ carries a current equal to the interpolated mean $M_z$ at its endpoints:

\[ I_i = \frac{M_z(\vec{v}_i) + M_z(\vec{v}_{i+1})}{2} \]

The total edge contribution to $\vec{B}$ is the vectorial sum over all segments, each computed with the analytic formula from §7.1.

7.3 Bulk Currents from Non-Uniform Magnetization¶

Spatial gradients in $M_z$ produce bulk Amperian currents:

\[ \vec{K}_\text{bulk} = \nabla \times (M_z\,\hat{z}) = \left(\frac{\partial M_z}{\partial y},\; -\frac{\partial M_z}{\partial x},\; 0\right) \quad [\text{A m}^{-1}] \]

These are computed numerically on the grid via central finite differences and integrated by the Biot–Savart surface-current formula. For a current element $\vec{K}\,dA$ in the $z = 0$ plane:

\[ d\vec{B} = \frac{\mu_0}{4\pi}\frac{\vec{K} \times (\vec{r}_\text{obs} - \vec{r}_s)}{|\vec{r}_\text{obs} - \vec{r}_s|^3}\,dA \]

Summed over all grid cells inside the sample mask. The observation height $z_s = 0$ for all source points.

Edge / bulk split. The implementation follows the convention that both the edge and bulk terms must be included with include_edge=True and include_bulk=True (default) to obtain the correct total field for a step-function magnetization profile. The outermost grid ring then contributes half to the edge integral and half to the bulk gradient term; the two together reproduce the exact $\delta$-function boundary current.

Reference: [8] Lima & Weiss, Physical Review B 2009; [9] Thiel et al., Science 2019.

8. Ensemble Modelling¶

Defect positions¶

Defect centres are placed uniformly at random inside a circular patch of radius $R_\text{patch}$ at height $z_\text{defect}$ above the sample surface:

\[ (x_i, y_i) \sim \mathcal{U}(\text{disk}, R_\text{patch}) \]

Alternatively a Gaussian (laser-beam) profile can be used:

\[ (x_i, y_i) \sim \mathcal{N}(0, w_0^2/2) \]

where $w_0$ is the $1/e^2$ beam waist.

Per-defect field evaluation¶

For $N$ defect positions $\{(x_i, y_i)\}$:

\[ \vec{E}_i = \vec{E}(x_i, y_i, z_\text{defect}), \qquad \vec{B}^\text{extra}_i = \vec{B}(x_i, y_i, z_\text{defect}) \]

Each defect then has its own Hamiltonian:

\[ H^{(i)} = H[\vec{B}_0 + \vec{B}^\text{extra}_i,\; \vec{E}_i] \]

Ensemble averaging¶

Transition frequencies $\{f_1^{(i)}, f_2^{(i)}\}$ are computed for each defect. The CW spectrum and echo/Ramsey lock-in signals are incoherently averaged over the ensemble:

\[ \overline{S}(\tau) = \frac{1}{N}\sum_{i=1}^{N} S^{(i)}(\tau) \]

This is valid when inter-defect correlations are negligible (separations $\gg$ spin-spin dipolar coupling length) and when the measurement repetition time exceeds the fluctuation correlation time of the environment.

Quantization axes¶

Each defect can have its own quantization axis $\hat{z}'_i$, sampled from: - Fixed axis — all defects share the same axis (e.g. all perpendicular to the substrate). - Random / powder — axes drawn uniformly from the unit sphere ($\hat{z}' \sim \mathrm{Uniform}(\mathbb{S}^2)$). - User-supplied — a $(N, 3)$ array of pre-computed axes.

9. Model Assumptions and Limitations¶

Assumption	Scope	Comment
Weak driving	All protocols	Microwave pulses are assumed ideal (hard pulses); pulse imperfections are not modelled.
Classical fields	E and B sources	Quantum fluctuations of the electromagnetic field are neglected.
Static disorder	Electrometry	Disorder charges are fixed during a measurement shot; charge hopping is not modelled.
No Lindblad dynamics	Relaxation	Decoherence enters only through phenomenological $T_2^*$ and $T_2$ times; the full Lindblad master equation is not solved.
Planar geometry	Electrometry	The gate and defect layer are assumed flat and infinite in $x$–$y$; edge effects of the gate are neglected.
2-D magnetization	Magnetometry	$M_z$ is assumed independent of $z$ (thin-film limit).
Incoherent ensemble	Signals	No inter-defect correlations or collective effects.
Shot-noise limited	SNR	Background fluorescence contributions beyond shot noise (e.g. substrate PL) are not included.
Diagonal hyperfine tensor	Hyperfine	The hyperfine tensor $A$ is supplied in the defect local frame; off-diagonal elements are supported but typically assumed to vanish for axially symmetric sites.
Single metastable state	Rate model	A single collective shelving state pools all ISC population; multiple distinct singlets are not resolved.

10. Nuclear-Spin Hyperfine Interaction¶

10.1 Hyperfine Hamiltonian¶

Coupling between the electron spin $\vec{S}$ and a nuclear spin $\vec{I}_k$ is described by the hyperfine interaction:

\[ \frac{H_\text{hf}}{h} = \vec{S} \cdot A_k \cdot \vec{I}_k = \sum_{i,j} A_k^{ij}\, S_i \otimes I_{kj} \]

where $A_k$ is the $3\times 3$ hyperfine coupling tensor (Hz) of nucleus $k$ in the defect local frame ($z' =$ electron spin quantization axis).

For an axially symmetric site the tensor is diagonal:

\[ A_k = \operatorname{diag}(A_\perp,\, A_\perp,\, A_\parallel) \]

where $A_\parallel = A_{zz}$ is the longitudinal component (along the defect quantization axis) and $A_\perp = A_{xx} = A_{yy}$ is the transverse component. Built with axial_A_tensor(A_zz_Hz, A_perp_Hz).

For an isotropic (Fermi contact) coupling:

\[ A_k = A_\text{iso}\,\mathbf{1}_3 \]

Built with isotropic_A_tensor(A_iso_Hz).

Typical values:

Defect	Nucleus	$A_\parallel$ (MHz)	$A_\perp$ (MHz)
NV⁻ (diamond)	¹⁴N on-site	−2.14	−2.70
VB⁻ (hBN)	¹⁴N ×3 in-plane	≈0	+47.8

10.2 Nuclear Quadrupole Coupling¶

For nuclei with spin $I \ge 1$, the non-spherical nuclear charge distribution couples to the electric field gradient at the nuclear site:

\[ \frac{H_Q}{h} = P_k\left(I_{z,k}^2 - \frac{I_k(I_k+1)}{3}\,\mathbf{1}\right) \]

where $P_k$ (Hz) is the quadrupole coupling constant. For NV⁻ ¹⁴N: $P \approx -4.95$ MHz.

Reference: [10] Felton et al., Physical Review B 2009.

10.3 Nuclear Zeeman Term¶

The nuclear spin also precesses in the applied magnetic field:

\[ \frac{H_{nZ}}{h} = \gamma_{n,k}\,\bigl(B_{x'} I_{x,k} + B_{y'} I_{y,k} + B_{z'} I_{z,k}\bigr) \]

where $\gamma_{n,k}$ (Hz T⁻¹) is the nuclear gyromagnetic ratio for isotope $k$. At typical laboratory fields ($\lesssim 10$ mT) the nuclear Zeeman splitting ($\lesssim 100$ kHz for ¹⁴N) is much smaller than the hyperfine splitting and acts as a small perturbation.

10.4 Tensor-Product Hilbert Space¶

The full electron + nuclear Hamiltonian is assembled in the tensor-product space:

\[ \mathcal{H} = \mathcal{H}_e \otimes \mathcal{H}_{n_1} \otimes \mathcal{H}_{n_2} \otimes \cdots \]

with total dimension $(2S+1)\prod_k(2I_k+1)$. Each term in $H$ is embedded as a tensor-product operator acting on its own subspace and as the identity on all others:

\[ \frac{H}{h} = \underbrace{H_e \otimes \mathbf{1}_{n_1} \otimes \cdots}_{\text{electron}} + \sum_k\underbrace{\mathbf{1}_e \otimes \cdots \otimes H_{n_k} \otimes \cdots}_{\text{nuclear }k} + \sum_k \sum_{i,j} A_k^{ij}\,S_i \otimes I_{kj} \]

Implemented in full_hyperfine_hamiltonian_Hz.

10.5 Isotope Constants¶

Gyromagnetic ratios (Hz T⁻¹) shipped with the library:

Constant	Isotope	$I$	$\gamma_n$ (MHz T⁻¹)
`GAMMA_14N`	¹⁴N	1	+3.077
`GAMMA_15N`	¹⁵N	1/2	−4.316
`GAMMA_11B`	¹¹B	3/2	+13.660
`GAMMA_10B`	¹⁰B	3	+4.575
`GAMMA_13C`	¹³C	1/2	+10.708
`GAMMA_29Si`	²⁹Si	1/2	−5.319

Reference: [11] Harris et al. (IUPAC recommendations), Pure and Applied Chemistry 2001.

11. Rate-Equation Model for CW Contrast¶

11.1 Level Structure¶

For a spin-$S$ defect with $N = 2S+1$ magnetic sublevels, the level structure has three sub-manifolds:

\[ \underbrace{|g, +S\rangle, \ldots, |g, -S\rangle}_{N\text{ ground states}} \quad \underbrace{|e, +S\rangle, \ldots, |e, -S\rangle}_{N\text{ excited states}} \quad \underbrace{|\text{shelf}\rangle}_{1\text{ metastable state (if ISC active)}} \]

Level indices: ground $= 0\ldots N-1$, excited $= N\ldots 2N-1$, shelving $= 2N$.

11.2 Rate Matrix and Steady State¶

The population vector $\vec{P}$ evolves as $\dot{\vec{P}} = R\,\vec{P}$, where the rate matrix $R$ encodes the following processes:

Process	Rate	Comment
Laser excitation	$k_\text{opt}$	Spin-blind; same for all $m_S$
Radiative decay	$k_\text{rad}$	Spin-preserving: $
ISC to shelving	$k_\text{ISC}(m_S)$	Spin-selective; faster for $
Return from shelf	$k_\text{shelf}(m_S)$	Spin-polarising: predominant return to $m_S=0$
Non-radiative	$k_\text{nr}$	Spin-blind; contributes to lifetime, not contrast

Steady state is the null vector of $R$ normalised to $\sum_i P_i = 1$:

\[ R\,\vec{P}_\text{ss} = 0, \qquad \sum_i P_i = 1 \]

11.3 CW ODMR Contrast¶

PL in the reference state (spin polarised, far from MW resonance):

\[ \text{PL}_\text{off} = k_\text{rad} \sum_{n=N}^{2N-1} P_n^\text{(ss,off)} \]

PL when a saturating MW drives transition $(m_{S,0} \leftrightarrow m_{S,1})$:

\[ \text{PL}_\text{on} = k_\text{rad} \sum_{n=N}^{2N-1} P_n^\text{(ss,on)} \]

CW ODMR contrast:

\[ C = \frac{\text{PL}_\text{off} - \text{PL}_\text{on}}{\text{PL}_\text{off}} \]

The sign convention is positive for dips (PL decreases on resonance, as for NV⁻ and VB⁻).

11.4 Pre-built Defect Parameters¶

Defect	Spin	Host	$k_\text{rad}$ (MHz)	$\tau$	Predicted $C$
NV⁻	1	diamond	77	13 ns	≈23 %
VB⁻	1	hBN	294	3.4 ns	≈2 %
V_SiC	1	4H-SiC	167	6 ns	≈15 %
P1	1/2	diamond	100	—	0 % (no ISC)
Cr/GaN	3/2	GaN	100	—	0 % (ISC unknown)

Values from: Tetienne et al., NJP 14, 103033 (2012) [NV⁻]; Haykal et al., npj Quantum Inf. 8, 16 (2022) [VB⁻]; Christle et al., Nat. Mater. 14, 160 (2015) [V_SiC].

The contrast is auto-computed via Defaults.get_contrast() when contrast=None.

References: [12], [13], [14], [15].

12. References¶

[1] Gottscholl, A. et al. "Initialization and read-out of intrinsic spin defects in a van der Waals crystal at room temperature." Nature Materials 19, 540–545 (2020). https://doi.org/10.1038/s41563-020-0619-6

[2] Dolde, F. et al. "Electric-field sensing using single diamond spins." Nature Physics 7, 459–463 (2011). https://doi.org/10.1038/nphys1969

[3] Haykal, A. et al. "Decoherence of VB− spin defects in hexagonal boron nitride." Nature Communications 13, 4347 (2022). https://doi.org/10.1038/s41467-022-31743-0

[4] Degen, C. L., Reinhard, F. & Cappellaro, P. "Quantum sensing." Reviews of Modern Physics 89, 035002 (2017). https://doi.org/10.1103/RevModPhys.89.035002

[5] de Lange, G. et al. "Universal dynamical decoupling of a single solid-state spin from a spin bath." Science 330, 60–63 (2010). https://doi.org/10.1126/science.1192739

[6] Yan, F. et al. "Rotating-frame relaxation as a noise spectrum analyser of a superconducting qubit undergoing driven evolution." Nature Communications 4, 2337 (2013). https://doi.org/10.1038/ncomms3337

[7] Jackson, J. D. Classical Electrodynamics, 3rd ed. (Wiley, 1999), §2.9 — Method of Images for parallel conducting planes.

[8] Lima, E. A. & Weiss, B. P. "Obtaining vector magnetic field maps from single-component measurements of geological samples." Journal of Geophysical Research 114, B06102 (2009). https://doi.org/10.1029/2008JB006006

[9] Thiel, L. et al. "Probing magnetism in 2D materials at the nanoscale with single-spin microscopy." Science 364, 973–976 (2019). https://doi.org/10.1126/science.aav6926

[10] Felton, S. et al. "Hyperfine interaction in the ground state of the negatively charged nitrogen vacancy center in diamond." Physical Review B 79, 075203 (2009). https://doi.org/10.1103/PhysRevB.79.075203

[11] Harris, R. K. et al. "NMR nomenclature: nuclear spin properties and conventions for chemical shifts." Pure and Applied Chemistry 73, 1795–1818 (2001). https://doi.org/10.1351/pac200173111795

[12] Tetienne, J.-P. et al. "Magnetic-field-dependent photodynamics of single NV defects in diamond." New Journal of Physics 14, 103033 (2012). https://doi.org/10.1088/1367-2630/14/10/103033

[13] Christle, D. J. et al. "Isolated electron spins in silicon carbide with millisecond coherence times." Nature Materials 14, 160–163 (2015). https://doi.org/10.1038/nmat4144

[14] Robledo, L. et al. "Spin dynamics in the optical cycle of single nitrogen-vacancy centres in diamond." New Journal of Physics 13, 025013 (2011). https://doi.org/10.1088/1367-2630/13/2/025013

[15] Haykal, A. et al. "Decoherence properties and optical lifetime of VB⁻ spin defects in hexagonal boron nitride." npj Quantum Information 8, 16 (2022). https://doi.org/10.1038/s41534-022-00528-0

Symbol	Meaning	Typical value (VB⁻ in hBN)
\(S\)	spin quantum number	1
\(D_0\)	axial zero-field splitting (Hz)	3.46 GHz
\(E_0\)	transverse ZFS strain (Hz)	50 MHz
\(d_\perp\)	transverse E-field coupling (Hz V⁻¹ m)	0.35 Hz/(V/m)
\(d_\parallel\)	axial E-field coupling (Hz V⁻¹ m)	≈ 0 (usually neglected)
\(\gamma_e\)	gyromagnetic ratio (Hz T⁻¹)	28 GHz/T
\(\vec{B}\)	total magnetic field in local frame (T)	—
\(\vec{E}\)	total electric field in local frame (V/m)	—

Process	Rate	Comment
Laser excitation	\(k_\text{opt}\)	Spin-blind; same for all \(m_S\)
Radiative decay	\(k_\text{rad}\)	Spin-preserving: $
ISC to shelving	\(k_\text{ISC}(m_S)\)	Spin-selective; faster for $
Return from shelf	\(k_\text{shelf}(m_S)\)	Spin-polarising: predominant return to \(m_S=0\)
Non-radiative	\(k_\text{nr}\)	Spin-blind; contributes to lifetime, not contrast