🔬 atomic physics

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

This paper presents exact, closed-form expressions for the hyperfine energy eigenvalues and eigenstates of the nitrogen-vacancy center in diamond for both nitrogen-14 and nitrogen-15 isotopes under an axial magnetic field, eliminating the need for the numerical methods and analytic approximations previously required.

Original authors: Alex Tritt, Lincoln D. Turner, Michael S. J. Barson

Published 2026-09-15

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CC BY 4.0

Original authors: Alex Tritt, Lincoln D. Turner, Michael S. J. Barson

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Deep within the crystal lattice of a diamond, hidden among the carbon atoms, sits a tiny defect known as the nitrogen-vacancy center. This is not a flaw in the usual sense, but a specific arrangement where a nitrogen atom has replaced a carbon atom, and a neighboring spot in the grid is empty. This vacancy acts like a trap for electrons, creating a system that behaves with the precision of an atomic clock. Scientists have long been fascinated by these centers because they can sense magnetic fields with incredible sensitivity, making them powerful tools for quantum computing and for mapping the magnetic fields of living cells or geological samples. To use them effectively, researchers must understand exactly how the energy levels of the electrons inside the defect shift when exposed to a magnetic field. This shifting is complicated by the fact that the nitrogen atom itself has a magnetic "spin," which interacts with the electron, creating a complex web of energy states that change as the magnetic field strength varies.

For decades, calculating these energy states has been a difficult task. When the magnetic field is very weak or very strong, physicists have reliable formulas to predict the behavior. However, in the middle ground—where the magnetic field is neither weak nor strong, but somewhere in between—the mathematics becomes notoriously difficult. In this region, the energy levels can come very close to each other without touching, a phenomenon known as an avoided crossing. To navigate this complex terrain, researchers have traditionally relied on computer simulations or rough approximations that work well enough for some purposes but fail to capture the full, exact picture. These approximations are like using a map with missing details; they get you close, but they cannot tell you the precise location of every turn.

A team of researchers at Monash University in Australia has now solved this long-standing puzzle. They have derived a set of exact, closed-form mathematical expressions that describe the energy levels and the specific states of the nitrogen-vacancy center for both of the stable forms of nitrogen found in nature. One form has a nucleus that spins like a top with three distinct orientations, while the other has a nucleus that spins with only two. The researchers found that by breaking the problem down into smaller, manageable pieces, they could write down exact formulas for the energy levels without needing to rely on computer crunching or approximations. Their work provides a complete and precise map of how these quantum systems behave under any axial magnetic field, filling in the gaps that previous methods left open.

The significance of this achievement lies in the precision it offers. The researchers showed that their new formulas work perfectly across the entire range of magnetic fields, from the very weak fields used in ultra-sensitive magnetometers to the stronger fields where specific quantum effects occur. They demonstrated that their exact solutions match the results of high-precision computer simulations down to the smallest decimal places, confirming that the new formulas are not just theoretical curiosities but accurate descriptions of reality. This is particularly important at specific magnetic field strengths where the energy levels come closest together, known as level anticrossings. At these points, the system becomes highly sensitive, and even tiny errors in calculation can lead to incorrect predictions about how the diamond will respond to a magnetic field. The team calculated that for the nitrogen-15 isotope, this critical point occurs at approximately 102.3 millitesla for the ground state and around 51.3 millitesla for the excited state. For the more common nitrogen-14 isotope, they identified two distinct crossing points near 102.5 millitesla and 102.2 millitesla, depending on the specific quantum state involved.

By providing these exact formulas, the researchers have removed the need for guesswork in designing experiments that rely on these diamond defects. Whether the goal is to build a quantum computer that processes information using the spin of these electrons or to build a sensor that can detect the faint magnetic fields of a single neuron, having an exact description of the system's behavior is essential. The new formulas allow scientists to predict exactly where the energy levels will be and how the system will respond to changes in the magnetic field, ensuring that their instruments are tuned to the right frequencies. This work bridges the gap between the simple rules that apply at extreme magnetic fields and the complex reality found in the middle, offering a clear and complete understanding of one of the most promising tools in modern quantum technology.

Technical Summary: Exact Breit-Rabi Formulae for the Nitrogen-Vacancy Center in Diamond

Problem Statement
The hyperfine structure of the nitrogen-vacancy (NV) center in diamond, specifically its interaction with the nitrogen nucleus, is critical for quantum sensing and computing applications. While the Breit-Rabi formula provides an exact analytic solution for the hyperfine ground state of alkali atoms (electron spin $S=1/2$ ), it cannot be applied to the NV center, which possesses an electron spin of $S=1$ . Consequently, determining the hyperfine Zeeman energy eigenvalues and eigenstates for the NV center under an axial magnetic field has historically relied on numerical methods or analytic approximations. These approximations are often insufficient near level anticrossings (LACs) or in regimes where simple linear expressions break down, such as in zero and ultra-low field (ZULF) applications.

Methodology
The authors derive exact, closed-form expressions for the NV hyperfine energy eigenvalues and eigenstates for both nitrogen-14 ( $^{14}\text{N}$ , nuclear spin $I=1$ ) and nitrogen-15 ( $^{15}\text{N}$ , nuclear spin $I=1/2$ ) isotopes.

Fine Structure Derivation: The authors first address the fine structure of the NV center under arbitrary magnetic fields. By rotating the coordinate system to align the transverse magnetic field with the $x$ -axis, they reduce the $3 \times 3$ fine-structure Hamiltonian to a real, symmetric matrix. They demonstrate that the characteristic polynomial of this matrix is cubic. Using trigonometric solutions for cubic equations (specifically Cardano's method adapted for real roots), they derive exact expressions for the eigenvalues and eigenvectors in terms of the magnetic field components.
Hyperfine Derivation ( $^{15}\text{N}$ ): For the $^{15}\text{N}$ case ( $S=1, I=1/2$ ), the total Hamiltonian is a $6 \times 6$ matrix. The authors exploit the conservation of the total magnetic quantum number $m_F = m_S + m_I$ to decompose the Hamiltonian into block-diagonal form. This reduces the problem to solving independent $1 \times 1$ and $2 \times 2$ blocks, allowing for standard analytic solutions.
Hyperfine Derivation ( $^{14}\text{N}$ ): For the $^{14}\text{N}$ case ( $S=1, I=1$ ), the Hamiltonian is a $9 \times 9$ matrix. Unlike the $^{15}\text{N}$ case, the initial matrix is not block-diagonal. The authors apply a specific permutation matrix to reorder the basis states, revealing a block-diagonal structure consisting of $1 \times 1$ , $2 \times 2$ , and $3 \times 3$ blocks. The $3 \times 3$ block (corresponding to $m_F=0$ ) is mathematically identical to the fine-structure matrix derived earlier, allowing the reuse of the cubic trigonometric solution.
Approximation Recovery: The authors perform a Taylor and Puiseux expansion of their exact expressions near the level anticrossing points. This allows them to analytically recover the quadratic approximations previously proposed by Auzinsh et al., providing a rigorous mathematical link between the new exact formulae and established approximate models.

Key Contributions and Results

Exact Closed-Form Solutions: The paper provides the first exact, closed-form analytic expressions for the hyperfine energy eigenvalues and eigenstates of the NV center for both $^{14}\text{N}$ and $^{15}\text{N}$ isotopes. These expressions are valid across all magnetic field regimes, from the low-field Zeeman effect to the high-field Paschen-Back regime.
Level Anticrossing (LAC) Predictions: The authors derive precise conditions for the ground-state (GSLAC) and excited-state (ESLAC) level anticrossings.
- For $^{15}\text{N}$ , the GSLAC is predicted at $B \approx 102.318$ mT.
- For $^{14}\text{N}$ , the authors identify distinct GSLACs for different sublevels: one for the $m_F = \pm 1$ levels at $B \approx 102.522$ mT and another for the $m_F = 0$ levels at $B \approx 102.247$ mT.
- The derived formulae reveal that the LAC position is shifted by terms dependent on the transverse hyperfine coupling ( $A_\perp$ ), correcting the assumption that LACs occur exactly where the Zeeman splitting equals the zero-field splitting.
Validation: The derived analytic curves are compared against point-wise numerical evaluations. The authors confirm that their exact formulae match numerical results to float64 precision.
Refinement of Approximations: By expanding the exact solution, the authors derive a corrected quadratic approximation that accounts for second-order shifts in the magnetic field and energy levels, demonstrating superior accuracy over previous approximations near the LACs.

Significance
The paper claims that these exact formulae are essential for the design and interpretation of NV-based quantum technologies, particularly in low-field applications such as ZULF NMR and magnetometry. The exact solutions eliminate the need for numerical diagonalization in simulations and provide a rigorous foundation for measuring coupling parameters (such as the excited-state hyperfine tensor and ground-state pseudo-Stark tensor) at level anticrossings. Furthermore, the work clarifies the mathematical structure of the NV hyperfine system, demonstrating that despite the complexity of a 9-level system, the specific symmetries of the NV center allow for exact analytic solutions similar to those found in alkali atoms.

Technical Summary: Exact Breit-Rabi Formulae for the Nitrogen-Vacancy Center in Diamond

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