Where Spin Defects Lose Energy: Calculating the Hardest Step

Inside a diamond sits a flaw of a few atoms. A laser pulse hits it, the flaw lights up, and within a tiny fraction of a second it gives the energy back, partly as light and partly as something you cannot see at all. Where the energy goes decides how a quantum sensor built from that flaw can be read out, and for years part of the answer could not be calculated.

Physicists who simulate such flaws have been able to account for part of that story and not the rest. A team at the University of Chicago has now closed much of the gap with a method that follows the whole cycle, including the step that was hardest to calculate. Their paper in Physical Review Letters, published on 14 October 2026, also turns up a surprise in silicon carbide, a material that is attracting growing attention as a platform for quantum devices.

The work comes from the group of Giulia Galli at the Pritzker School of Molecular Engineering, with first author Stefano Villani and Yu Jin, who is now at the Flatiron Institute in New York. The article below is based on the journal paper, on the authors' preprint (arXiv:2606.14545, submitted in June 2026), whose numbers we quote, and on the university's press release. The published version may differ in details from the preprint.

A flaw that behaves like an atom

Real crystals are never perfect. A diamond is a lattice of carbon atoms, and sometimes one carbon is replaced by a nitrogen atom sitting next to an empty site, a vacancy. This pairing is called a nitrogen-vacancy center, or NV center. In its negatively charged form, written NV⁻, it has become one of the most studied defects in physics. A cousin of it lives in silicon carbide (SiC), a hard compound of silicon and carbon: the neutral divacancy, written VV⁰, in which two neighboring atoms are missing from the lattice.

Both belong to a family called optically active spin defects. "Optically active" means they absorb and emit light. "Spin" refers to a quantum property of electrons, a tiny magnetic orientation, that can be prepared, manipulated and read. The authors of the paper describe these defects as foundational for quantum technologies, and the press release says they can function as stable qubits whose states can be read, which is why researchers want to understand exactly how they behave.

The potential uses the paper cites are concrete. Its reference list includes a demonstration of a spin in silicon carbide that stayed coherent for five seconds, with single-shot readout; a report of room-temperature coherent readout of single spins in silicon carbide using photocurrent; and a non-invasive, bioinert quantum sensor made from silicon carbide qubits. Each of these depends on one thing: knowing, in detail, what happens to the defect between the moment the laser hits it and the moment the signal is read.

The optical cycle in plain words

Physicists describe that sequence as the optical cycle. In both the NV⁻ center and the divacancy, the defect has a lowest-energy "ground" state and a higher "excited" state, both of which are triplets, meaning the electron spins add up to a total spin of one. Between them sit two singlet states, with total spin zero, which the paper calls shelving states because the defect can park there for a while. The paper labels them ³A₂ (triplet ground), ³E (triplet excited), and ¹A₁ and ¹E (the two singlets). The labels come from the symmetry of the defect.

The cycle runs like this. A laser lifts the defect from the ground triplet to the excited triplet. From there the defect has two main options. It can fall straight back to the ground state, emitting a photon, which is the light you see. Or it can cross over into the upper singlet state through a process called inter-system crossing, in which the spin orientation changes. From the upper singlet, the defect relaxes to the lower singlet without any change of spin, then crosses a second time back to the triplet ground state.

The paper notes that the magnetic sublevels of the ground state act as the effective qubit. Which route the defect takes, and how long it spends on each, sets how efficiently the qubit can be read out.

That is why the details matter. To work out how efficient the readout is, and to interpret the spectra that experiments record by sweeping microwave frequency while watching the fluorescence, you need to know the lifetimes of each state and how the defect splits its decay among the available paths. The authors say a detailed description of the lifetimes, and of the competition between radiative and non-radiative decay, is necessary to determine the efficiency of the readout.

The step nobody could calculate well

Three kinds of transition appear in that cycle. The defect can release a photon. It can change spin, the inter-system crossing. Or it can dump its energy into the vibrations of the crystal around it without emitting light. That last route is called internal conversion, and it is the one the Chicago team set out to model.

Radiative transitions and inter-system crossing have been studied with a range of first-principles methods, meaning calculations that start from the equations of quantum mechanics rather than from fitted parameters. The authors say internal conversion has remained much less explored. In their words, this has hampered a full description of the optical cycle from first principles.

It is useful to picture what internal conversion is. A crystal is not a rigid scaffold. Its atoms vibrate, and the quanta of those vibrations are called phonons. When a defect sits in the lattice, the electrons that carry its excitation are coupled to the nearby atoms: if the electrons rearrange, the atoms feel it and shift, and if the atoms jiggle, the electronic energy levels wobble with them. This coupling between electrons and phonons can let the defect hand its excitation over to many phonons at once, without a photon leaving. Because the process involves thermal motion, it depends on temperature. The authors point out that it is key to understanding how temperature affects the optical cycle and to predicting excited-state lifetimes.

Villani summarized the difficulty in the press release: "Although the equations are known, it's hard to describe the process accurately because of computational difficulties. You need to account for complex interactions between the electrons and surrounding atoms, and that's challenging in realistic solids represented by models with hundreds or thousands of atoms."

Why earlier shortcuts fell short

The paper names two obstacles. The first is describing the excited states themselves. Many standard calculations use Kohn-Sham orbitals, single-electron wavefunctions that arise in density functional theory. They work well for many problems, but the excited states of these defects are not well captured by a single electron configuration. They mix several arrangements of the electrons, which the authors call multi-configurational character. A description built from single orbitals misses that.

The second obstacle is the coupling between electrons and phonons. A solid with hundreds of atoms has hundreds of vibrational modes, and computing how each one couples to the electronic transition is expensive. To save effort, earlier methods often used shortcuts. One keeps only a single vibrational mode, the single-mode approximation. Another, the accepting-mode approximation, builds one effective mode from the overall displacement of atoms between the initial and final geometries. The authors note that approaches based on one effective mode and Kohn-Sham orbitals have worked for capturing charge carriers at defects in the strong-coupling regime, but when applied to neutral excitations in spin defects they gave inaccurate lifetimes.

The result was a long-running mismatch between simulation and experiment, in which internal conversion rates came out orders of magnitude too low. The paper states that common approximations underestimate these rates by orders of magnitude.

Building a calculation that sees every vibration

The Chicago group's answer has two ingredients, and the paper calls both central.

First, they describe the defect's states as many-body wavefunctions, computed with linear-response time-dependent density functional theory, using hybrid functionals. A functional is the recipe that sets how electrons interact in density functional theory; a hybrid functional mixes in an exact piece of the electron interaction and is generally more reliable for localized defect states. The method is a way of obtaining excited states, and the "linear-response" part means they are found by studying how the system reacts to a small perturbation. The group used a flavor of the approach that can handle spin-flip excitations, and that captures the multi-configurational character that single orbitals cannot.

Second, they worked out how strongly each excited state couples to atomic motion by deriving the non-adiabatic coupling, a quantity that measures how much the electronic state changes when the atoms move, and doing it analytically. Previous work typically estimated such derivatives by nudging atoms by small amounts and recomputing everything, a procedure called finite differences, which becomes impractical with hundreds of atoms. Yu Jin said in the press release: "In the past, a clear mathematical derivation and efficient algorithms were not available. The new derivation came together after years of work on the optical cycle of spin defects."

With analytic derivatives, the team can include every phonon mode of the supercell, the block of crystal used in the simulation. They then plug the couplings into Fermi's golden rule, a standard formula from quantum mechanics for the rate at which a system jumps between two states under a weak interaction. The rate is a sum over all vibrational modes and depends on temperature. To keep the sum tractable they use the generating-function approach, a mathematical technique long used for this type of problem, and the displaced harmonic oscillator approximation, which treats each vibration as an ideal spring whose equilibrium position shifts when the defect changes state.

The framework is implemented in WEST, an open-source code optimized for graphics processors that Galli's group has developed over the past decade within the Midwest Integrated Center for Computational Materials, funded by the U.S. Department of Energy since 2015. Because it is already part of WEST, other laboratories can start using it, although the press release cautions that it needs substantial computing resources.

The test: a defect with known answers

A new method needs a defect where the answer is known. The team chose the NV⁻ center in diamond, which has been measured many times, and began with the singlet transition from ¹A₁ to ¹E.

The reason this transition is a good test is that it is dominated by non-radiative decay. Its energy gap is small, with a zero-phonon line (the energy of the pure electronic jump with no vibrations involved) in the infrared at about 1.19 electron volts. Measured infrared emission from this transition is three orders of magnitude weaker than the visible light the defect gives off when falling from the excited triplet. In other words, almost all of the energy leaves as heat in the lattice.

The team modeled the NV⁻ center in a 215-atom supercell. Across temperatures from zero to 400 kelvin, they found internal conversion rates of the order of 10 gigahertz, meaning roughly ten billion transitions per second. For comparison, the predicted rate at which this transition would emit a photon is about 0.53 megahertz, more than ten thousand times smaller. Internal conversion wins by a very wide margin.

The authors then compared the implied lifetime of the upper singlet state with two experiments. A 2010 study by Acosta and colleagues measured the infrared fluorescence with a phase-shift technique and estimated a lifetime of 0.9 nanoseconds at 4 kelvin, with a large uncertainty of 0.5 nanoseconds. That corresponds to a rate of about 1.1 gigahertz, roughly eight times lower than the 8.5 gigahertz the calculation gives. The authors point out the large relative error of that measurement as a possible reason for the gap.

The second experiment is the more precise one. Ulbricht and Loh used femtosecond transient absorption: an ultrashort laser pulse starts the optical cycle and a second pulse, tuned to the singlet transition, probes how the population evolves over time. In two different diamond samples at 78 kelvin they inferred lifetimes of 102.3 and 92.1 picoseconds, which correspond to rates of about 9.8 and 10.9 gigahertz. The new calculation gives 117.4 picoseconds, or 8.52 gigahertz, at the same temperature. The authors describe this as being in accord with experiment and call it the first time a calculation reaches excellent agreement for this defect.

Which vibrations do the work?

One strength of the method is that it tells you which vibrations matter. The authors computed a quantity called the electron-phonon spectral density, which shows how much each vibrational energy contributes to the decay.

For the singlet transition in the NV⁻ center, the spectrum has a peak at about 170 millielectron volts, from a vibrational mode localized around the defect itself, shown in the paper as an inset. It is the strongest candidate to be the main driver. One might therefore expect that a calculation with only this mode would be good enough. The paper tests that expectation, and it fails: with only the local mode, the rate at 78 kelvin comes out at 4.37 gigahertz, roughly half the 8.52 gigahertz obtained with all modes included. The rest comes from vibrations spread across the whole crystal.

The same table lists the accepting-mode result of 3.27 gigahertz. Each shortcut therefore captures only part of the total. The authors note that the single-mode approximation describes the tail of the line shape better, while the accepting-mode approximation performs better at lower energies, so neither is a safe replacement for the full sum.

The most striking comparison is with the earlier calculation based on Kohn-Sham orbitals. Using the same accepting-mode approximation at 300 kelvin, the new many-body approach gives 2.49 gigahertz; a previous study using Kohn-Sham orbitals gave 0.048 gigahertz. The factor of about fifty comes from how the electrons are described, and the authors trace it to the electron-phonon coupling term: for this transition it is 239.2 in the new calculation and 10.5 with Kohn-Sham orbitals, in units of millielectron volts per ångström per square root of atomic mass unit. Describing the electrons as a many-body state raises the coupling more than twentyfold.

Where internal conversion should not matter

A good method should also report "no effect" when there is none. The team checked the triplet transition from ³E down to ³A₂ in the NV⁻ center. This is the bright transition, with a zero-phonon line at about 1.95 electron volts, and it is known to be predominantly radiative. The energy gap is large, so many phonons are needed to bridge it, which makes internal conversion there a rare event.

The calculation agrees: the non-radiative rate stays below about 5 megahertz from zero to 400 kelvin, negligible compared with the radiative rate of 83.3 megahertz that earlier work had computed. Earlier first-principles estimates of the triplet lifetime that included radiative decay and inter-system crossing already matched the measurements, and this result confirms internal conversion plays no significant part there.

This case also shows the weakness of shortcuts. In the spectral density, no single mode stands out, so single-mode approximations are not valid. The accepting-mode result underestimates the rate by several orders of magnitude when compared with the full sum over phonons. The shortcut matters little for the answer, because the answer is nearly zero in both calculations, but it shows how unreliable the approximation can be.

The surprise in silicon carbide

The divacancy in silicon carbide is the case where the new method changes the textbook picture. The authors studied the kk configuration of the neutral divacancy in the 4H polytype of silicon carbide, one of the crystal structures of the material, which has been used in experiments including work on controlling spin qubits by choice of polytype. They used a larger supercell of 398 atoms.

The question was the lifetime of the excited triplet state. The gap between the triplet ground and excited states is large (the zero-phonon line is about 1.096 electron volts), and it had often been assumed that this separation makes non-radiative contributions from internal conversion negligible. First-principles estimates that counted only radiative decay did not match experiment, however. The measured excited-state lifetime is about 15 nanoseconds, while calculations including only radiative decay gave about 36.8 nanoseconds with standard density functional theory and about 28 nanoseconds with a constrained version of the theory. The simulated state lived too long, which means that something besides light emission was draining it.

The Chicago calculation supplies that something. When the authors add internal conversion to the radiative rate, using the relation that the inverse of the lifetime equals the sum of the radiative and non-radiative rates, the computed lifetime moves toward experiment. The paper shows this in a figure that compares the radiative-only lifetime with the total lifetime and with two sets of measurements, from work published in 2019 in Science and in 2014 in Physical Review Letters. The text does not give a single number for the new lifetime, so we do not quote one here.

As in diamond, no individual vibrational mode dominates, and an effective-mode approximation severely underestimates the rate. The conclusion from the paper is that internal conversion is a significant, previously overlooked contribution to the lifetime of the triplet excited state of the divacancy, which resolves a long-standing discrepancy between computed and measured lifetimes. Galli put it plainly in the press release: "We were expecting to see an effect, but of a smaller magnitude. What we found is that this internal conversion process turns out to be more important than we thought."

The authors also computed the singlet transition in the divacancy. No experiment exists for it, and neither the transition energy nor the singlet lifetime has been measured, so the paper presents its numbers as predictions. The rates are again of the order of 10 gigahertz, similar to those in the NV⁻ center, against a radiative rate of 0.32 megahertz, so internal conversion should dominate there too.

A caution about the functional

The supplementary material of the preprint includes a check that deserves attention from anyone who uses these tools. The group repeated its calculations with PBE, a simpler and cheaper approximation widely used in density functional theory, and compared the results with those from the hybrid functional, called DDH.

For the NV⁻ singlet transition the two differ by a factor of about two. The authors trace this mostly to the energy gap: PBE gives 0.871 electron volts, DDH gives 1.397. Because the rate depends steeply on the gap, a small shift in the gap changes the result a lot. When the experimental zero-phonon line of 1.195 electron volts is used instead, the ratio shrinks to about 1.5.

For the SiC triplet transition the difference is larger, and the cause is different. The coupling vectors themselves disagree between the two functionals. The authors find that PBE places the energy of one of the defect's orbitals, called a1, below the top of the valence band, while the hybrid functional puts it in the band gap, where a defect state should be. The PBE orbital is also spread out over the crystal rather than localized at the defect. They attribute this to known inaccuracies of PBE in describing charge-transfer processes in spin-defective systems. The cheaper functional can get the physics of this defect qualitatively wrong, which is a reason the authors used the hybrid functional for their headline results.

What the calculation costs and what it assumes

The method is demanding, and the supplementary material is frank about the compromises. For the main calculations the group used Quantum ESPRESSO for the electronic structure, the WEST code for the excited states and couplings, and phonopy to generate the vibrational properties. All the dynamical matrices, the tables that give vibrational modes, were computed with the cheaper PBE functional, a practice the authors justify with earlier work. They assumed no mixing between the vibrational modes of the initial and final states and used the displaced harmonic oscillator approximation throughout.

One compromise is specific to silicon carbide. The excited triplet state of the divacancy relaxes into a lower-symmetry, broken-symmetry geometry. Computing its vibrations in a 398-atom supercell would need thousands of excited-state force calculations, which the authors call prohibitively expensive even with PBE. For that transition they used ground-state phonons instead of excited-state ones. The result is therefore a calculation with a controlled, stated approximation, not an exact solution.

Other inputs come from experiment. The parameter that sets the width of the electronic levels was estimated from low-temperature photoluminescence linewidth measurements of each transition. For the SiC singlet transition, where no data exist, the authors reused the value from the NV⁻ center. As the press release notes, the simulations still demand considerable computing power.

What it means for quantum sensors

Together with earlier first-principles methods for inter-system crossing, including a 2025 Physical Review Letters study of inter-system crossing rates that the paper cites, the new framework lets researchers calculate a full optical cycle without fitting parameters. The authors say this enables parameter-free predictions of complete optical cycles and of the optically detected magnetic resonance spectra that experiments record. Galli described the result as "the missing piece of the puzzle needed to completely describe the optical cycle of spin defects."

For someone designing a quantum sensor, this changes the process from trial and error toward screening. A researcher considering a new defect or a new host crystal can calculate how it loses energy before growing a single sample. Yu Jin said the work "can now be used to make more accurate predictions about many materials of interest for quantum technologies." The silicon carbide result is a first example: if a lifetime in the lab disagrees with the calculation, there is now a trustworthy source of the discrepancy to examine.

The method is not limited to quantum defects. The authors write that it is broadly applicable to non-adiabatic processes in heterogeneous solids and molecules, and that the analytical couplings pave the way for ab initio non-adiabatic molecular dynamics in extended systems and for studies of polaron and exciton dynamics in solids. Galli noted in the press release that internal conversion is "also related to processes of interest to energy technologies," which is one reason the team is extending the framework to materials used in energy conversion and storage.

Limits and open questions

Several things remain unsettled. The agreement with experiment is best-documented for a single defect, the NV⁻ center, and the one quantity that is directly compared there is the lifetime of the upper singlet state, measured in two samples at one temperature. The measurement at 4 kelvin by an older technique disagrees with the calculation by about a factor of eight, though the authors argue that experiment has a large uncertainty.

For the silicon carbide divacancy, the agreement is described in terms of a trend: including internal conversion brings the computed triplet lifetime closer to measured values. The paper does not claim a quantitative match, and it relies on ground-state phonons for that transition. The singlet transition there has no data at all.

The results also depend on the choice of functional. The authors show that the hybrid functional and PBE give different rates, in some cases by large factors, and that PBE describes one key orbital incorrectly in silicon carbide. That is a reminder that the numbers are only as good as the electronic-structure theory underneath.

Finally, the method is costly. The group says it is working on machine-learned dynamical matrices and surrogate models for the non-adiabatic couplings to reduce the cost. Villani said in the press release: "We are working on ways to integrate machine learning into excited states calculations to further improve the efficiency of our method." Whether those surrogates keep the accuracy of the full calculation is a question for future work.

The bottom line

The result is a story about a missing piece in a model. For a decade, the optical cycle of a quantum defect could be drawn but not fully computed: light emission and spin-flipping crossings could be calculated, while the quiet, heat-releasing step through lattice vibrations was either approximated crudely or ignored. By treating the defect's electrons as a many-body system and including every vibration of the surrounding crystal, the Chicago team can now compute that step, reproduce a precise measurement in diamond, and show that in silicon carbide it is more important than assumed.

For readers following quantum technology, the practical point is that this is a tool, not a device. It does not make a better qubit by itself. What it offers is a way to predict, before an experiment, how a candidate defect will behave under light, which is where a sensor's usefulness begins. How much it speeds up the search for better materials will depend on how far the cost of the calculation can be brought down. For earlier coverage of the result, see our news report on the silicon carbide finding, and for related stories browse the qubits tag.

Sources

• S. P. Villani, Y. Jin, G. Galli, "First-Principles Calculations of Internal Conversion Processes in Spin Defects", Physical Review Letters, 14 October 2026
• Preprint of the same work, arXiv:2606.14545 (12 June 2026)
• University of Chicago Pritzker School of Molecular Engineering, "Quantum calculations reveal how quantum defects lose energy"