NV ODMR Spectrum Lab (2D)

Spin-1 ground state of the NV centre

The ground state is a spin-1 with zero-field splitting D = 2.870 GHz along the NV axis. In a magnetic field B at angle theta to that axis the Hamiltonian is H = D S_z^2 + gamma B (S_z cos theta + S_x sin theta) with gamma = 28.024 GHz/T. The page diagonalises this 3x3 matrix exactly, so the two resonances f- , f+ are the true transition frequencies from the lowest level. On axis they are D -/+ gamma B; a transverse component shifts them only at second order.

Lineshape, power broadening and sensitivity

Each dip is a Lorentzian with half-width Gamma = Gamma0 sqrt(1+s) and contrast C = C0 s/(1+s), where s is the microwave saturation parameter. With photon rate R the shot-noise limited sensitivity at maximum slope is eta = (4/(3 sqrt 3)) Gamma / (gamma C sqrt R), so eta ~ (1+s)^1.5 / s: too little power gives no contrast, too much only broadens the line; the minimum is at s = 2.

Noise and the fit

Every frequency point collects N = R t photons, so the normalised signal has relative noise 1/sqrt N (Gaussian approximation of Poisson noise). Each sweep is averaged into the running spectrum, and the field is estimated by least squares of the two-dip model (known angle, width and contrast) over B, with a golden-style refinement. The error plot shows the estimate improving about as 1/sqrt(sweeps).

Numerical checks on this page

(1) The exact diagonalisation reproduces D -/+ gamma B on axis. (2) A scan finds the sensitivity minimum at s = 2. (3) A seeded noisy spectrum is fitted and the error must be below 30 microtesla.

Honest simplifications

No 14N hyperfine triplet (about 2.2 MHz spacing) and no strain or electric-field term; one NV orientation only (a real diamond has four, giving four dip pairs); a phenomenological saturation law instead of full optical-Bloch dynamics; the photon rate does not depend on power; Poisson noise is approximated as Gaussian; room temperature, constant D (it drifts about -74 kHz/K).

• Load Optimal power s=2 and note the sensitivity readout; then Too little power and Power broadened are both worse.
• Tilt the field to 50 degrees: the dips move toward each other because only the axial component splits them.
• Lower the photon rate or the time per point: the spectrum gets noisy and the fit error grows.
• Watch the error graph while sweeps accumulate: it falls roughly as one over the square root of the number of sweeps.