Exact Breit-Rabi formulae for the nitrogen-vacancy center in diamond

Calculating the hyperfine structure of the nitrogen-vacancy (NV) center in diamond is critical for quantum sensing and computing applications, particularly near level anticrossings (LACs). Historically, researchers have relied on numerical simulations or approximate analytic models to determine these energy levels and states. This paper seeks to provide an exact, closed-form solution—a Breit-Rabi formula—for the NV center, analogous to the well-known solutions for alkali atoms.

The authors treat the NV center as a spin-one electronic system coupled to a nuclear spin (spin-1/2 for 15N and spin-one for 14N). By constructing the Hamiltonian in matrix form and utilizing the block-diagonal structure inherent to systems with conserved total magnetic quantum numbers, the authors reduce the complex nine-level (for 14N) or six-level (for 15N) systems into smaller, solvable subsystems. They apply the algebraic solutions for cubic and quadratic equations to derive exact trigonometric expressions for the energy eigenvalues and eigenstates.

The paper presents complete, closed-form analytic expressions for the hyperfine Zeeman energy eigenvalues and eigenstates for both nitrogen isotopes. These formulas are valid across all magnetic field regimes, including low-field and high-field limits. The authors demonstrate that their exact solutions match numerical evaluations to float64 precision. Furthermore, they show how these exact results can be used to derive and refine the quadratic approximations previously used in the literature, providing a more rigorous basis for understanding the NV center's behavior near level anticrossings.

These exact formulas provide a powerful tool for researchers designing NV-based quantum technologies. By providing a closed-form solution, the paper enables faster and more accurate modeling of spin dynamics, which is essential for applications like zero-field magnetometry, nuclear magnetic resonance (NMR), and hyperpolarization protocols. It removes the computational overhead of numerical methods and provides deeper physical insight into the spin structure of the NV center.