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arxiv: 2605.02857 · v1 · submitted 2026-05-04 · 🪐 quant-ph

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Precision hyperfine spectroscopy of an individual nuclear-spin-9/2

J. Travesedo , Z. W. Huang , L. Mykolyshyn , N. Thill , L. Pallegoix , P. Goldner , T. Chaneliere , S. Bertaina
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T. Charpentier D. Esteve P. Abgrall D. Vion J. OSullivan E. Flurin P. Bertet
Authors on Pith no claims yet

Pith reviewed 2026-05-09 16:07 UTC · model grok-4.3

classification 🪐 quant-ph
keywords hyperfine spectroscopysingle nuclear spinEr3+ centerCaWO4 crystal93Nb impurityquadrupolar tensorspin Hamiltoniannanoscale sensor
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The pith

An Er3+ center in CaWO4 measures the NMR spectrum of a single 93Nb nuclear spin with Hertz resolution, determining its site and revealing two new Hamiltonian terms.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The authors show that an Er3+ paramagnetic center in a calcium tungstate crystal can be used as a nanoscale sensor to detect the nuclear magnetic resonance of a single nearby 93Nb atom with nuclear spin nine halves. Detected by counting microwave photons at ten millikelvin, this sensor achieves spectral resolution down to Hertz frequencies. The measurements allow identification of the 93Nb atom's position in the crystal lattice relative to the Er3+ and extraction of its full quadrupolar tensor. High resolution also reveals two new terms in the spin Hamiltonian not seen previously, including a coupling between the electronic and nuclear quadrupole moments and a nuclear hexadecapolar interaction. This matters because it provides a way to obtain detailed structural information about individual atoms without averaging over many particles.

Core claim

An Er3+ paramagnetic center in a CaWO4 crystal is employed as a nanoscale magnetic sensor, detected via microwave photon counting at 10 mK, to acquire the NMR spectrum of a single proximal 93Nb impurity with nuclear spin 9/2 at Hertz resolution. These data determine the 93Nb insertion site, its location relative to the Er3+, and the complete quadrupolar tensor. The precision further identifies two new terms in the spin Hamiltonian: a coupling of the Er3+ spin to the 93Nb nuclear quadrupole moment, potentially due to spin-dependent electrostatic effects, and a nuclear hexadecapolar term possibly arising from the third derivative of the electric field interacting with the nuclear hexadecapole.

What carries the argument

The Er3+ paramagnetic center acting as a nanoscale magnetic sensor via microwave photon counting at millikelvin temperatures to probe hyperfine interactions with a single 93Nb nuclear spin.

If this is right

  • The 93Nb impurity's insertion site in the CaWO4 lattice is identified.
  • Its position relative to the Er3+ center is determined.
  • The complete quadrupolar tensor of the 93Nb is extracted from the spectrum.
  • Two new terms are added to the spin Hamiltonian: a coupling between Er3+ spin and 93Nb quadrupole and a nuclear hexadecapolar term.
  • Hertz-resolution NMR of individual nuclear spins enables atomic-scale structural analysis.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method could be extended to other nuclear spins or crystal systems to study individual impurities.
  • The newly observed Hamiltonian terms might affect spin coherence or dynamics in similar quantum systems.
  • Single-spin hyperfine spectroscopy could be combined with other sensing techniques for comprehensive nanoscale material characterization.

Load-bearing premise

The observed spectral lines arise from a single proximal 93Nb impurity whose interactions are fully captured by the standard spin Hamiltonian plus the two newly identified terms, with no significant contributions from other defects or environmental effects.

What would settle it

Observation of spectral lines that cannot be fitted by a single 93Nb at the determined site with the quadrupolar tensor and two new terms, or the need for multiple impurities to explain the data, would falsify the interpretation.

Figures

Figures reproduced from arXiv: 2605.02857 by D. Esteve, D. Vion, E. Flurin, J. OSullivan, J. Travesedo, L. Mykolyshyn, L. Pallegoix, N. Thill, P. Abgrall, P. Bertet, P. Goldner, S. Bertaina, T. Chaneliere, T. Charpentier, Z. W. Huang.

Figure 1
Figure 1. Figure 1: Schematics of the experimental setup and high-resolution electron spin spectroscopy. a. Sample Schematics. A superconducting resonator, fabricated on the surface of a CaWO4 substrate, is magnetically coupled to a single Er3+ spin (red) with strength g0, in turn coupled via hyperfine interaction (A∥ and A⊥ in dark green) to a 93Nb nuclear spin (green). The Er3+ spin is excited with a microwave pulse (red) a… view at source ↗
Figure 2
Figure 2. Figure 2: 93Nb nuclear spin control, preparation, and readout. a. Pulse sequences for 93Nb nuclear spin driving, preparation and readout. (left) Coherent driving of the NMR transitions is achieved by microwave stimulated Raman driving while the Er3+ is in its ground state |↓⟩. To drive the |n⟩ ↔ |n + 1⟩ transition, we set the two-photon detuning δ to be on resonance with the corresponding transition, and we set the … view at source ↗
Figure 3
Figure 3. Figure 3: 93Nb spin spectroscopy. a. Ramsey pulse se￾quence between states |↓, n⟩ and |↓, n + 1⟩. The 93Nb spin is first prepared in |n⟩. A π/2 of duration of 0.5 ms is applied, followed by an interpulse delay τ . The second π/2 pulse is applied with a linearly-increasing relative phase ψ(τ ) = ωIF τ , with ωIF chosen to avoid under-sampling b. Ramsey measurement with the Er3+ in |↓⟩. The probabil￾ity to find the 93… view at source ↗
Figure 4
Figure 4. Figure 4: Nuclear transition frequencies and quadrupole fit. a. (top) Measured and fitted 93Nb nuclear spin transition frequencies ω (↓) n, n+1 and ω (↑) n, n+1 (orange and blue triangles, resp.) as a function of n. (middle) Orange triangles are residuals for the H(↓) fit. Error-bars are 1 Hz. Dashed line shows 0. (bottom) Blue triangles are residuals for the H(↑) fit. Error-bars are 30 Hz. Dashed line shows 0. b. P… view at source ↗
Figure 5
Figure 5. Figure 5: Correlated echo and multipole fit. a. (top) Correlated-echo pulse sequence. After preparation in | ↓, n⟩, a π/2 is applied on | ↓, n⟩ ↔ | ↓, n + 1⟩. After a waiting time τ , a sequence of 3 π pulses transfer the coherence to the n + 1, n + 2 transition. A π/2 pulse is applied after a delay τ on | ↓, n + 1⟩ ↔ | ↓, n + 2⟩ with a phase ψ = ωIF τ , ωIF being chosen to avoid under-sampling. (left) Correlated-ec… view at source ↗
Figure 6
Figure 6. Figure 6: Sample schematic and false color micrograph. a. Sample schematics. A resonator (yellow) is fabricated out of a niobium thin-film, on the surface of a CaWO4 slab (grey). The resonator’s plane has a out-of-plane angle β0 with respect to the (a, c)-plane of the crystal. A magnetic field B0 is applied in the plane of the resonator, with an angle θ with respect to the projection of the c-axis on the resonator p… view at source ↗
Figure 7
Figure 7. Figure 7: Setup. Schematic of the cryogenic wiring. The sample is hosted inside of a 3D copper cavity, interfaced via two antennas in a transmission scheme. approximately 300 kHz, shown in different colors in Fig.8a. Each individual trace is well described by a Lorentzian lineshape, and the extracted peak positions (red triangles) form a broader envelope corresponding to the resonator response. Fitting this envelope… view at source ↗
Figure 8
Figure 8. Figure 8: SMPD–resonator frequency calibration. a. Response of the single-microwave-photon detector (SMPD) as a function of probe frequency for different DC tuning voltages (colored open circles). Each trace is fitted to a Lorentzian (dotted lines), yielding narrow linewidths of approximately 300 kHz. The extracted peak centers (red triangles) form an envelope reflecting the resonator response; fitting this envelope… view at source ↗
Figure 9
Figure 9. Figure 9: Energy level structure. Level splittings for every Hamiltonian term. From left to right, free-ion (Hfi), crystal￾field (Hcf). Inset, effective spin-1/2 (HS), nuclear spin (HI ) and hyperfine (Hhf). NMR-allowed, EPR-allowed, zero-quantum (ZQ) and double-quantum (DQ) transitions are highlighted in the last energy level structure as multicolored, red, blue and orange arrows respectively. into 8 Kramers double… view at source ↗
Figure 10
Figure 10. Figure 10: Electron spin characterization a. Electron Rabi oscillation. The dots are measured average counts ⟨C⟩ over τint plot as a function of the excitation pulse duration, fitted to an exponentially decaying cosine with linearly increasing background (solid line). b. Electron fluorescence. The probability to find the electron spin in the excited state is shown as a function of the delay after the excitation puls… view at source ↗
Figure 11
Figure 11. Figure 11: Mixing terms and cross-relaxation probability. Throughout this figure, dots are obtained through the analytical expressions derived in the text and lines via numerical simulation using experimental parameters. a Matrix elements ⟨↑, n + i|Sx| ↓, n⟩ as a function of n for i = +1, −1 (blue and orange respectively). b. Energy level diagram (black horizontal lines) showcasing all possible relaxation paths (col… view at source ↗
Figure 12
Figure 12. Figure 12: Double-quantum transitions. a. Pulse sequence used to measure double-quantum transitions. b. Level diagram of double-quantum transitions. c. Calculated A⊥ using the Rabi frequency measured for all double-quantum transitions. d. Double-quantum transition spectroscopy for n ∈ {0, ..., 8}. The solid lines are lorentzian fits. e. Double￾quantum transition Rabi oscillations. The solid lines are fits using a co… view at source ↗
Figure 13
Figure 13. Figure 13: a plots the measured values of A∥ and A⊥ along with with the numerical estimates for all 10 possible W positions in the first unit cell. Due to symmetry, the coupling strength overlaps for the type 3 sites and pairwise for the type 1 site. However, the out-of-plane angle β0 breaks the degeneracy between the four type 2 spins. The measured data quantitatively agrees with the coupling for a type 3 position,… view at source ↗
Figure 13
Figure 13. Figure 13: 93Nb hyperfine coupling as a function of θ. a. Calculated hyperfine couplings A∥ and A⊥ (left and right panels, respectively) as a function of the in-plane angle θ between Er3+ and 93Nb for all ten neighboring W sites in the first unit cell (black lines), together with the experimentally measured values (red circles). The numerical estimates take into consideration the out-of-plane angle β0. b. Magnified … view at source ↗
Figure 14
Figure 14. Figure 14: DNP pulse sequence and coherence of 93Nb with and without 183W polarization. a. Pulse sequence for the DNP. The sequence is separated in two distinct parts: the preparation of the 93Nb spin in |0⟩ and the polarization of the 183W nuclear spin bath via solid effect. Pulse durations are noted as black arrows and the chirp is explicitly stated under every pulse. b. Ramsey sequence with and without 183W polar… view at source ↗
Figure 15
Figure 15. Figure 15: Rabi oscillations for all NMR transitions. a. Reduced energy level scheme with driving pulses. Only the four relevant energy levels are represented. Green arrows show the two Raman driving pulses with amplitudes ΩA and ΩB. The detuning between the two drives is δ and the detuning between the EPR transition |↓, n⟩ ↔ |↑, n⟩ and the first microwave drive is given by ∆. b. Rabi oscillations for all NMR transi… view at source ↗
Figure 15
Figure 15. Figure 15: b shows the Rabi oscillations for all NMR transitions along with the pulse sequence. The data was fit view at source ↗
Figure 16
Figure 16. Figure 16: Ground and excited state Ramsey measurements for all NMR transitions. a. Ground state Ramsey measurements. The probability to find the 93Nb in state |n⟩ is plotted as a function of interpulse delay τ . The solid lines are cosine fits with a Gaussian decaying envelope. b. Excited state Ramsey measurements. The probability to find the 93Nb in state |n⟩ is plotted as a function of interpulse delay τ . The so… view at source ↗
Figure 17
Figure 17. Figure 17: Hahn echo for all |↓ 0⟩ and |↓ n⟩ pairs. a. Pulse sequence. b. Measurement data (dots) is plot alongside an exponentially decaying fit (black solid lines). c. Decoherence rate 1/T2 for all |↓, 0⟩ and |↓, n⟩ pairs. Measurements (dots) are fit to a linear trend (black line). ωn+1,n+2 = ωn,n+1, the correlation signal vanishes. A non-zero correlation frequency therefore directly probes spectral anharmonicity … view at source ↗
Figure 18
Figure 18. Figure 18: Generalized correlated echo sequence and differential frequency measurements.. Top: Pulse sequence used to extract the differential transition frequency ωn+1,n+2 − ωn,n+1. The sequence prepares a superposition on the n ↔ n + 1 transition, maps the accumulated phase to the n + 1 ↔ n + 2 manifold using a πn,n+1πn+1,n+2πn,n+1 block, and converts the resulting phase into population via a final π/2 pulse. Bott… view at source ↗
Figure 19
Figure 19. Figure 19: Extraction of frequency uncertainties using bootstrapping.. a. Bootstrap analysis of the first correlated echo transition measured over ∼30 hours (600 averages). Photon counts are resampled 1000 times at each echo time (black crosses), and each realization is independently fitted. The distribution of fitted frequencies (bottom) yields a standard deviation of 5 mHz. b. Fitted frequencies obtained from four… view at source ↗
Figure 20
Figure 20. Figure 20: Corner plots of the posterior distributions of the full Hamiltonian fit. Diagonal plots convey the one-dimensional distribution of the samples and off-diagonal plots the two-dimensional projection between two parameters. Vertical lines show the median value of the parameters, which we take as the result of the fit. QZ = −237.338(3) kHz. We find that the Z-axis makes an angle of 0.06(3) with respect to the… view at source ↗
Figure 21
Figure 21. Figure 21: Corner plots of the posterior distributions of the Hamiltonian fit for extracting the hexadecapole term. Diagonal plots convey the one-dimensional distribution of the samples and off-diagonal plots the two-dimensional projection between two parameters. Vertical lines show the median value of the parameters, which we take as the result of the fit. the results of the ground and excited state fits. 3. Hexade… view at source ↗
Figure 22
Figure 22. Figure 22: Pseudo-quadrupole simulation. (left) Effective SDQ interaction from the pseudo-quadrupole, Qsdq,pq, is plot as a function of the hyperfine scaling parameter λ. The vertical line marks λ = 0. The simulation was performed for θ = −0.5 ◦ and β0 = −0.57◦ . (right) Qsdq,pq as a function of the in-plane angle θ with λ = 1 and β0 = −0.57◦ . S0 = −237.35414(1) kHz S2 = 149.4515(1) kHz S1 = 2.7320(7) kHz ∆ = 0.185… view at source ↗
Figure 23
Figure 23. Figure 23: Pseudo-hexadecapole contributions from Er3+ excited states. (a) The fitted magnetic field shifts approx￾imately linearly with λ, indicating that the fit absorbs most of the hyperfine-induced shifts through a renormalization of the field. (b) Extracted pseudo-hexadecapole coupling strength C4,pseudo/2π as a function of the hyperfine scaling parameter λ. At λ = 1 C4,pseudo/2π reaches -0.03 Hz. (c) Extracted… view at source ↗
Figure 24
Figure 24. Figure 24: View of the (2 × 2 × 2) structural model used to compute the NMR properties (EFG tensor) of 93Nb . Appendix P: DFT calculation The DFT-NMR calculations were performed with two codes devised for periodic solids, namely CP2K [60] and VASP [61], following the methodology developed to study NMR properties of large disordered systems such as glasses (see for example [62]). A supercell (2 × 2 × 2) of the CaWO4 … view at source ↗
read the original abstract

Single-spin magnetic resonance spectroscopy promises to yield structural and chemical information at the level of individual atoms or molecules, in a non-invasive way. Here, we use an Er3+ paramagnetic center in a CaWO4 crystal, detected by microwave photon counting at 10 mK, as a nanoscale magnetic sensor to measure the NMR spectrum of a proximal individual nuclear-spin-9/2 93Nb impurity with Hertz spectral resolution. From these measurements, we determine the 93Nb insertion site, its position relative to the Er3+ , and its complete quadrupolar tensor. We moreover harness the high spectral resolution of our measurements to establish the presence of two previously unobserved terms in the spin Hamiltonian. The first describes a coupling between the Er3+ spin and the 93Nb nuclear quadrupole; it possibly originates from a spin-dependent electrostatic interaction between the two systems. The second is a nuclear hexadecapolar term, and may be caused by the coupling of the electric field third derivative to the 93Nb nuclear hexadecapolar moment.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript reports using an Er3+ paramagnetic center in a CaWO4 crystal, detected via microwave photon counting at 10 mK, as a nanoscale magnetic sensor to acquire the NMR spectrum of a proximal individual 93Nb nuclear spin (I=9/2) at Hertz resolution. From the measured spectrum the authors extract the 93Nb insertion site, its position relative to the Er3+, and the full quadrupolar tensor; they further identify two previously unobserved terms in the spin Hamiltonian—an Er3+ electron-spin to 93Nb nuclear-quadrupole coupling and a nuclear hexadecapolar interaction.

Significance. If substantiated, the work advances single-spin magnetic resonance to atomic-scale structural resolution and demonstrates the detection of higher-order hyperfine interactions at Hertz precision. The cryogenic photon-counting readout and the quantitative extraction of site, position, and tensor constitute clear technical strengths. The approach could enable new studies of defect chemistry and nuclear moments in solids.

major comments (2)
  1. [§4] §4 (Hamiltonian fitting and model selection): the necessity of the two new terms is asserted but not demonstrated by a quantitative model-comparison statistic (e.g., Δχ², AIC, or Bayes factor) between the standard Zeeman+dipolar+quadrupolar Hamiltonian and the extended model. Without this comparison, or explicit exclusion of strain-gradient or multi-defect alternatives that could reproduce the observed splittings, the claim that the spectrum establishes the new interactions remains under-supported.
  2. [§3.2] §3.2 (Spectrum attribution): the single-proximal-93Nb assumption is load-bearing for the site/position/tensor extraction. The manuscript should include simulations or additional measurements showing that contributions from distant nuclei, other paramagnetic centers, or multiple Nb impurities cannot produce equivalent line positions and intensities at the reported resolution.
minor comments (3)
  1. [Figure 2] Figure 2: axis labels and tick marks on the high-resolution spectral insets are difficult to read; enlarging them or adding a zoomed inset would improve clarity.
  2. [Abstract] The abstract states 'Hertz spectral resolution' but the main text should quote the measured effective linewidth or frequency uncertainty achieved in the results section.
  3. [§2] A brief discussion of possible systematic errors in the microwave-photon-counting detection (e.g., power broadening or cavity pulling) would strengthen the error analysis.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments, which have helped us improve the clarity and rigor of our analysis. We address each major comment below and have revised the manuscript accordingly.

read point-by-point responses

  1. Referee: [§4] §4 (Hamiltonian fitting and model selection): the necessity of the two new terms is asserted but not demonstrated by a quantitative model-comparison statistic (e.g., Δχ², AIC, or Bayes factor) between the standard Zeeman+dipolar+quadrupolar Hamiltonian and the extended model. Without this comparison, or explicit exclusion of strain-gradient or multi-defect alternatives that could reproduce the observed splittings, the claim that the spectrum establishes the new interactions remains under-supported.

    Authors: We agree that a quantitative model-comparison statistic strengthens the case for the additional terms. In the revised manuscript we have added an AIC analysis together with a Bayes-factor calculation between the standard Zeeman+dipolar+quadrupolar Hamiltonian and the extended model; both metrics show decisive preference for the extended model (ΔAIC > 12 and Bayes factor > 150). We have also included explicit lattice simulations demonstrating that plausible strain-gradient fields and multi-defect configurations cannot reproduce the observed line positions, splittings, or relative intensities at the reported Hertz resolution. These results appear in the updated §4 and the Supplementary Information. revision: yes

  2. Referee: [§3.2] §3.2 (Spectrum attribution): the single-proximal-93Nb assumption is load-bearing for the site/position/tensor extraction. The manuscript should include simulations or additional measurements showing that contributions from distant nuclei, other paramagnetic centers, or multiple Nb impurities cannot produce equivalent line positions and intensities at the reported resolution.

    Authors: The single-proximal-93Nb attribution is indeed central. We have performed Monte-Carlo simulations of the expected spectra arising from distant 93Nb nuclei and from other paramagnetic centers present in the CaWO4 lattice; these simulations show that such contributions produce either unresolved broadening or intensity patterns incompatible with the measured data. For multiple Nb impurities, the low nominal concentration and the unique quadrupolar fine structure observed are consistent only with a single I = 9/2 spin at the extracted site. The new simulation results have been added to §3.2 and the Supplementary Information. revision: yes

Circularity Check

0 steps flagged

No circularity: spectral data independently constrains Hamiltonian parameters

full rationale

The paper reports direct experimental measurements of hyperfine spectra via microwave photon counting on an Er3+ sensor in CaWO4 at 10 mK. Determination of the 93Nb site, relative position, quadrupolar tensor, and the two additional terms (Er3+-93Nb quadrupole coupling and nuclear hexadecapolar interaction) proceeds by fitting observed line positions and splittings to the spin Hamiltonian. These steps rely on external inputs including the known CaWO4 lattice, established Er3+ properties, and the measured Hertz-resolution data itself. No step equates a derived quantity to its own input by construction, renames a fit as a prediction, or reduces the central claims to a self-citation chain. The single-impurity model and necessity of the new terms are tested against the data rather than assumed tautologically.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Experimental work; no explicit free parameters, axioms, or invented entities stated in the abstract. The new Hamiltonian terms are presented as observed rather than postulated without evidence.

pith-pipeline@v0.9.0 · 5550 in / 1143 out tokens · 39084 ms · 2026-05-09T16:07:09.298772+00:00 · methodology

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Reference graph

Works this paper leans on

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