Solver

Tools for solving the NV center hamiltonian.

Zeeman Hamiltonian

$\frac{H}{\hbar} = D_{gs}\left(S_z^2 - \frac{1}{3}S(S+1)\right) + E(S_x^2 - S_y^2) + \gamma_{\text{NV}}\vec{B} \cdot \vec{S}$

from nvtools import unit
from nvtools.solver import Solver

solver = Solver()

# calculate absolute field value and angle to NV axis from two resonances
b_abs, theta = self.solver.field_from_resonances(2865 * unit.MHz, 2875 * unit.MHz)

# calculate B-Field vector from eight resonance frequencies
b_field = self.solver.field_vector_from_resonances([<list of eight resonance frequencies>])

Solver for Zeeman Hamiltonian.

class nvtools.solver.zeeman.SolverZeeman[source]

B-field Solver.

static field_from_resonances(fl, fu)[source]

Calculate absolute field value and angle to NV-axis.

$\mathcal{B}^2 = \frac{1}{3} \left(f_{\text{u}}^2 + f_{\text{l}}^2 - f_{\text{u}}f_{\text{l}} - D^2 - 3E^2\right)$

$\cos^2\theta = \frac{2f_{\text{l}}^3-3f_{\text{l}}^2f_{\text{u}}-3f_{\text{l}}f_{\text{u}}^2+2f_{\text{u}}^3}{27(D-E/2)\mathcal{B}^2} + \frac{2D^3 - 18DE^2}{27(D-E/2)\mathcal{B}^2} + \frac{2D-3E}{6(D-E/2)}$

Parameters:

fl – lower resonance frequency
fu – upper resonance frequency

field_from_spectrum(mw_frequency, odmr, p0_frequency=None)[source]

Calculate field from odmr spectrum.

Parameters:

mw_frequency – MW frequency
odmr – ODMR
p0_frequency (list | None) – estimates for resonance frequencies

field_vector_from_resonances(resonances: list[float]) → BField[source]: Field vector from list of eight resonances.

static resonances_from_field(b_abs, theta, units=True)[source]

Calculate resonance frequencies.

$p = -\left(\frac{1}{3}D^2 + E^2 + \mathcal{B}^2\right)$

$q = - \frac{1}{2}D\mathcal{B}^2\cos(2\theta) - E\mathcal{B}^2\sin^2\theta - \frac{1}{6}D\mathcal{B}^2 + \frac{2}{27}D^3 - \frac{2}{3}DE^2$

$\lambda_k = \frac{2}{\sqrt{3}}\sqrt{-p}\cos\left(\frac{1}{3}\arccos\left(\frac{3\sqrt{3}}{2}\frac{q}{\sqrt{-p^3}}\right)-k\frac{2\pi}{3}\right), \enspace k = 0,1,2$

$f_{\text{u}} = \lambda_2 - \lambda_0 \enspace \text{and} \enspace f_{\text{l}} = \lambda_1 - \lambda_0$

Parameters:

b_abs – absolute B-field value
theta – angle between B-field and NV-axis
units – apply / ignore units

static vector_from_field(b_abs: list[float], theta: list[float], theta_var: None | list[float] = None) → Vector[source]

Calculate B-field vector from angles to NV axes.

$\vec{B} = B \frac{\sqrt{3}}{4} \begin{pmatrix} 1 & -1 & -1 & 1 \\ 1 & -1 & 1 & -1 \\ 1 & 1 & -1 & -1 \\ \end{pmatrix} \cdot \begin{pmatrix} \pm\cos\theta_1 \\ \pm\cos\theta_2 \\ \pm\cos\theta_3 \\ \pm\cos\theta_4 \\ \end{pmatrix}$

Parameters:

b_abs – absolute B-field value
theta – angle between B-field and NV-axis
theta_var – variances of theta, if known

Hyperfine Hamiltonian

$\frac{H}{\hbar} = D_{gs}S_z^2 + E(S_x^2 - S_y^2) + \gamma_{\text{NV}}\vec{B} \cdot \vec{S} + \vec{S}\cdot\hat{A}\cdot\vec{I} + P I_z^2$

from nvtools import unit
from nvtools.solver import Solver

# create a solver for a given nitrogen isotope
solver = Solver("14N")  # or Solver("15N")

# get absolute field value and angle to NV axis from two resonances
b_abs, theta = self.solver.field_from_resonances(2865 * unit.MHz, 2875 * unit.MHz)

Solver for Hyperfine Hamiltonian.

class nvtools.solver.hyperfine.SolverHyperfine(isotope: Literal['14N', '15N'])[source]

B-field Solver.

full_resonances_from_field_vector(b_field: Vector)[source]: Compute full resonance spectrum for all four NV axes.

resonances_from_field(b_field: Vector)[source]: Compute spin resonance frequencies from Hamiltonian.