Observation of a quantum phase from classical rotation of a single spin

Alexander A. Wood; Andy M. Martin; Lloyd C. L. Hollenberg; Robert E. Scholten

Observation of a quantum phase from classical rotation of a single spin

T0 review · 0 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Physical rotation of a diamond directly shifts the quantum phase of a single NV electron spin, accumulating nonlinearly and measurable by spin-echo interferometry.

desk verdict First direct measurement of a rotation-induced quantum phase on a single spin, with a clean model and strong internal controls; the main gap is missing raw data and a wobble estimate. read the letter →

arxiv 1908.08682 v1 pith:A4QY4POG submitted 2019-08-23 quant-ph

Alexander A. Wood , Lloyd C. L. Hollenberg , Robert E. Scholten , Andy M. Martin This is my paper · ORCID

classification quant-ph

keywords nitrogen-vacancycenterspinqubitrotationsensingquantumphaseechorotatingframeeffectivediamond

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper reports a measurement in which the quantum phase of a single nitrogen-vacancy (NV) electron spin in diamond is set directly by the diamond's mechanical rotation, without transduction through magnetic fields or ancillary spins. The diamond is spun at 200,000 rpm, and spin-echo interferometry reads out a rotation-induced phase that accumulates nonlinearly because the effective microwave drive axis rotates with the NV. If correct, this demonstrates the fundamental angular-momentum link between physical rotation and spin phase in a room-temperature solid, and it gives a readout principle for single-spin rotation sensors and for trapped nanoparticles containing spins.

What carries the argument

The central object is the effective microwave phase $\varphi_{\mathrm{eff}}$, the azimuthal angle of the microwave drive in the rotating frame of the NV spin, defined as the argument of the complex off-diagonal matrix element of the interaction Hamiltonian after the rotating-wave approximation. It carries the physical rotation angle $\varphi = \omega_{\mathrm{rot}} t$ into each quantum gate: every microwave pulse rotates the spin about an axis whose azimuth is $\varphi_{\mathrm{eff}}$. When the microwave polarization is tilted by $\theta_{\mathrm{mw}} \neq 0$, $\varphi_{\mathrm{eff}}$ is a nonlinear function of $\varphi$, and a spin-echo sequence cancels the linear part exactly as it cancels a static magnetic field; the residual $\delta\varphi$ is what the experiment measures.

What would settle it

Measure spin-echo fringes over a full range of starting rotation angles and microwave tilt angles and compare the fitted phase shift $\delta\varphi$ with the rigid-rotation model reconstructed from stationary Rabi measurements; a systematic mismatch, or an independent stroboscopic measurement of the diamond orientation resolving 3.33 kHz motion, would reveal wobble or slip that breaks the model.

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Extended reading notes

Core claim

The central claim is that physical rotation of the host diamond changes the phase of the NV electron spin directly, through the phase of the microwave drive in the rotating NV frame, and that this phase can be separated from magnetic shifts because it accumulates nonlinearly. After transforming into the NV frame and applying the rotating-wave approximation, the off-diagonal coupling has the form $H_I^{(i,j)} = \Omega_0 e^{-i\varphi_0}(\cos\theta_{\mathrm{NV}}\cos\varphi\sin\theta_{\mathrm{mw}} - \cos\theta_{\mathrm{mw}}\sin\theta_{\mathrm{NV}} + i\sin\theta_{\mathrm{mw}}\sin\varphi)/2$, so the microwave drive direction has azimuth $\varphi_{\mathrm{eff}} = \mathrm{Arg}(H_I^{(i,j)})$. For a microwave tilt angle $\theta_{\mathrm{mw}} \neq 0$, $\varphi_{\mathrm{eff}}$ advances nonlinearly with rotation angle $\varphi = \omega_{\mathrm{rot}} t$. In a spin-echo sequence the linear part of this advance cancels, leaving the spin population proportional to $\cos 2\delta\varphi$, where $\delta\varphi = \varphi_{\mathrm{eff}}(\tau)/2 - \varphi_{\mathrm{eff}}(\tau/2)$. The measured spin-echo fringe phase shift as a function of $\theta_{\mathrm{mw}}$ matches the model built from stationary Rabi-frequency measurements, supporting the claim that the observed phase is set by physical rotation alone.

Load-bearing premise

The measurement assumes the diamond rotates rigidly about the fixed motor axis at the stated rate, with no wobble, slip, or precession, so the NV's instantaneous orientation always matches the model used to reconstruct the effective phase.

Editorial extensions

If this is right

A single electron spin can detect physical rotation at 3.33 kHz through a phase shift, with no magnetic-field gradient and no auxiliary spin involved.
Spin-echo interferometry cancels linear phase drift from magnetic fields and temperature, so the rotation-induced nonlinear phase can be read out in a noisy room-temperature environment.
The strength and sign of the rotation-induced phase are set by the microwave tilt angle $\theta_{\mathrm{mw}}$, giving a control knob for the effective microwave phase during a pulse sequence.
The same readout principle applies to any spin whose quantization axis is not fixed in the lab, including proposed spin-based rotation sensors and trapped nanoparticles containing spins.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

An implication the authors leave implicit: the nonlinearity of $\varphi_{\mathrm{eff}}$ should be generic, so any spin with a fixed crystal axis driven by a tilted oscillating field while being rotated should show an analogous nonlinearly accumulating phase; other color centers or donor-bound electron spins are natural search targets.
A neighbouring problem this connects to is multi-axis rotation: in a levitated nanodiamond tumbling in a fluid or trap, the effective phase trajectory would be more complex than $\varphi_{\mathrm{eff}}(t)$, but the same spin-echo contrast mechanism could in principle track rotational diffusion on quantum timescales.
A testable extension would be to vary the spin-echo duration $\tau$ at fixed rotation speed and check that the measured $\delta\varphi$ follows the predicted $\varphi_{\mathrm{eff}}(\tau)$; this would distinguish rotational phase from any residual magnetic or temperature effect that also depends on total interrogation time.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Worth reading and worth refereeing. This is the first measurement I know of where a single spin's quantum phase is set by physical rotation of its host, with no mediating magnetic field or ancilla. The experiment is a rotating-diamond NV spin-echo, and the key idea is elegant: when the microwave drive is tilted relative to both the NV axis and the rotation axis, the effective microwave phase phi_eff sampled by each pulse becomes a nonlinear function of rotation angle. Spin-echo cancels the linear part and leaves a delta-phi that depends on theta_mw. The internal control is convincing: for a starting angle where the model predicts delta-phi ≈ 0, the fitted fringe phase stays flat across all theta_mw; for the nonlinear starting angle it tracks the model's growth.

The model work is the clean part. Eq. (2) follows from a standard frame transformation under the RWA, and the authors do not fit the spin-echo data to get it. They calibrate the microwave coupling amplitude at stationary 'park' angles, across a range of wire positions, with a single free azimuthal offset between the NV axis and the motor's park angle. From that independent calibration they reconstruct phi_eff and predict the rotating spin-echo shifts. The agreement in Fig. 3 is good, and the linear-vs-nonlinear comparison rules out the mundane artifacts, such as pulse-area errors or a theta_mw-dependent microwave phase, that would shift both datasets equally.

Soft spots, in proportion. The rigid-rotation assumption (no wobble, no slip, fixed axis) is the biggest one. I don't think it is load-bearing: a static misalignment is constrained by the stationary Rabi mapping, and a dynamic wobble big enough to explain tens of degrees of delta-phi would have to reproduce Eq. (2) for seven tilt angles and two starting angles. Still, the paper should state the motor's wobble limits or provide a stability check. More practically, v1 omits the raw traces and the supplementary error analysis; the fringes show only statistical error bars, and drift, especially temperature-driven magnetic-field drift, is discussed only briefly. The published version needs the supplement, a clear account of systematic uncertainties, and ideally the raw data. The microwave-field model as a simple azimuthal wire field is adequate to fit the stationary data, but a finite-element check would silence nitpickers.

My verdict: send it to review. It is a genuine experimental result, the theory is textbook but the realization is not, and the internal controls are strong. A serious referee will ask for the supplement and a wobble estimate, not for a new experiment. This paper is for the NV-and-quantum-sensing community and for anyone working on rotating quantum systems or trapped nanodiamonds. I would bring it to reading group.

Referee Report

0 major / 6 minor

Summary. The authors report an experiment on a single nitrogen-vacancy (NV) center in a diamond spinning at 3.33 kHz, and they measure a phase shift in spin-echo fringes that depends on the tilt angle of the microwave field. They derive an effective microwave phase φ_eff from a rotating-frame transformation under the rotating-wave approximation, calibrate the model against stationary Rabi-oscillation data using one free azimuthal offset, and compare the rotating spin-echo fringe phases with a parameter-free prediction of δφ = φ_eff(τ)/2 − φ_eff(τ/2). The central claim is that the electron-spin quantum phase is set directly by physical rotation, without transduction through magnetic fields or ancillary spins, and that the nonlinear accumulation of this phase is detected via spin-echo interferometry.

Significance. If the result holds, it demonstrates a fundamental connection between classical rotation and quantum phase in a single spin, and it provides a measurement strategy for rotation sensing with spin qubits. The paper's strongest feature is the internal control: a 'linear' starting configuration in which the model predicts δφ ≈ 0 for all microwave tilt angles, and a 'nonlinear' configuration at the same tilt angles where a large phase shift appears. This comparison rules out several mundane artifacts such as a tilt-dependent microwave phase or pulse-area errors. The model is independently calibrated from stationary Rabi data, with only one free azimuthal calibration angle, so the rotating-data comparison is not a fit to the effect being claimed. The main limitations are presentation-related: the referenced Supplementary Material is not included in the arXiv version, and some key formulas are asserted without derivation.

minor comments (6)

[Section 2, Eq. (3)] The derivation of the spin-echo signal formula δφ = φ_eff(τ)/2 − φ_eff(τ/2) is not shown, and the statement that a linearly varying φ_eff is cancelled by spin echo would benefit from a short explicit calculation. The factor of 2 and the sign are not obvious from the text, and this relation is central to the comparison between data and model.
[Section 3] The text repeatedly references a Supplementary Material for details on magnetic-field drift and error analysis, but the arXiv v1 does not contain this material. Since the error bars in Fig. 3 are essential for the claim of agreement, the supplementary analysis should be included or a summary should be given in the main text.
[Setup, Fig. 1] The assumption of rigid rotation about a fixed axis with no wobble or slip is not discussed. A sentence justifying this assumption using the motor mount and the stationary Rabi calibration would help the reader assess whether mechanical imperfections could mimic the observed phase shifts.
[Section 2, Eq. (2)] The notation B_mw = φ_hat uses the same symbol φ for the azimuthal coordinate, which is confusing. A different symbol for the unit vector, e.g., e_φ, would improve clarity.
[Fig. 2] The claim that the model 'reproduces' the stationary Rabi data would be strengthened by reporting a goodness-of-fit statistic or showing residuals. The current figure shows agreement by eye only.
[Section 3, fitting] The fit function is written as cos^2(2π f0 − δφ) with f0 called the average fringe frequency, but f0 appears to have units of inverse magnetic-field current rather than frequency. Clarifying the notation would avoid confusion for readers who expect f0 in Hz.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the rotation-induced phase is a parameter-free prediction from Eq. (2), calibrated by stationary Rabi data rather than by the rotating fringe phases.

full rationale

The derivation chain is self-contained. Equation (2) is a standard rotating-frame calculation for a tilted microwave field in the rotating NV frame, and phi_eff is defined as the argument of the off-diagonal matrix element, not as an input. The only fitted parameter, the calibration between the absolute NV azimuthal orientation and the arbitrary motor park angle, is obtained from stationary Rabi-frequency measurements (Fig. 2, top) and does not encode the rotating spin-echo fringe phases shown in Fig. 3. The predicted delta_phi versus theta_mw is therefore not forced by the measured fringe phase. Moreover, the fringe-phase extraction uses a common average fringe frequency f0 that contains no theta_mw dependence, so the theta_mw trend in delta_phi is a genuine prediction rather than a renamed fit. The linear-regime configuration serves as an internal control: the model predicts delta_phi approximately zero at all theta_mw, while the nonlinear configuration shows the predicted growth, ruling out a trivial calibration artifact. Prior same-group references are cited for experimental configuration and for magnetic-field up-conversion effects, but the central phase claim does not rest on those citations; it follows from Eq. (2) and the stationary calibration. No circular step can be exhibited, so the score is 0.

Assumptions & free parameters 2 free parameters · 3 assumptions · 0 invented entities

The central model rests on standard angular-momentum transformation and the rotating-wave approximation; the experiment-specific assumptions are the rigid-rotation idealization, the purely azimuthal wire-field model, and a single azimuthal calibration. No new physical entities are introduced. The extracted phase shift is not used to set model parameters, so the circularity burden is low.

free parameters (2)

Azimuthal calibration between NV axis and motor park angle = Not stated numerically; fixed by fitting stationary Rabi data
The only free parameter in the phi_eff model of Fig. 2; it aligns the arbitrary motor trigger angle with the NV azimuthal orientation and is not independently measured.
Average fringe frequency f0 = Not stated numerically; fitted to cosine spin-echo fringes
Used in the fits cos^2(2 pi f0 - delta phi) to extract the phase shift; it is a shared calibration across the dataset, not a per-angle model parameter.

assumptions (3)

domain assumption The rotating-wave approximation is valid: only slowly varying off-diagonal terms of the interaction Hamiltonian are kept in Eq. (1), and counter-rotating terms are neglected.
Used to obtain Eq. (2) and the azimuthal microwave vector. The driving frequency is 2.846 GHz, Rabi rates are 1-6 MHz, and rotation is 3.33 kHz, so the RWA is standard but still an assumed separation of timescales.
domain assumption The diamond and NV axis rotate rigidly about a fixed lab axis z, with phi = omega_rot t and no wobble or slip; the NV tilt is theta_NV = 54.7 degrees.
Introduced in the setup paragraph describing the mount; the entire phase reconstruction maps motor angle directly to NV orientation, and no wobble characterization is presented in this version.
ad hoc to paper The microwave field from the straight wire is purely azimuthal, B_mw = phi_hat, so the tilt angle theta_mw is set by the wire position relative to the rotation axis.
Adopted to reproduce the stationary Rabi angular dependence with one free azimuthal calibration. The paper states this approximation is sufficient, but it remains a simplified field model.

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Cite this review

Pith. "Pith review of Observation of a quantum phase from classical rotation of a single spin." pith.science (2026). https://pith.science/paper/A4QY4POG

@misc{pith2026190808682,
  author       = {Pith},
  title        = {Pith review of: Observation of a quantum phase from classical rotation of a single spin},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A4QY4POG}},
  note         = {Machine review of arXiv:1908.08682}
}

read the original abstract

The theory of angular momentum connects physical rotations and quantum spins together at a fundamental level. Physical rotation of a quantum system will therefore affect fundamental quantum operations, such as spin rotations in projective Hilbert space, but these effects are subtle and experimentally challenging to observe due to the fragility of quantum coherence. Here we report a measurement of a single-electron-spin phase shift arising directly from physical rotation, without transduction through magnetic fields or ancillary spins. This phase shift is observed by measuring the phase difference between a microwave driving field and a rotating two-level electron spin system, and can accumulate nonlinearly in time. We detect the nonlinear phase using spin-echo interferometry of a single nitrogen-vacancy qubit in a diamond rotating at 200,000rpm. Our measurements demonstrate the fundamental connections between spin, physical rotation and quantum phase, and will be applicable in schemes where the rotational degree of freedom of a quantum system is not fixed, such as spin-based rotation sensors and trapped nanoparticles containing spins.

Figures

Figures reproduced from arXiv: 1908.08682 by the authors.

**Figure 1.** FIG. 1. (a) Experimental setup, depicting NV center in a (100)-diamond mounted to an electric motor that rotates about an [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** FIG. 2. Reconstruction of effective phase accumulation from [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** FIG. 3. Detection of electron-spin phase shifts from physical [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

M. W. Doherty, N. B. Manson, P. Delaney, F. Jelezko, J. Wrachtrup, and L. C. L. Hollenberg, Physics Reports The nitrogen-vacancy colour centre in diamond, 528, 1 (2013)

work page 2013

[2] [2]

Schirhagl, K

R. Schirhagl, K. Chang, M. Loretz, and C. L. Degen, Annual Review of Physical Chemistry 65, 83 (2014)

2014

[3] [3]

A. A. Wood, E. Lilette, Y. Y. Fein, N. Tomek, L. P. McGuinness, L. C. L. Hollenberg, R. E. Scholten, and A. M. Martin, Science Advances 4, eaar7691 (2018)

work page 2018

[4] [4]

Balasubramanian, P

G. Balasubramanian, P. Neumann, D. Twitchen, M. Markham, R. Kolesov, N. Mizuochi, J. Isoya, J. Achard, J. Beck, J. Tissler, V. Jacques, P. R. Hem- mer, F. Jelezko, and J. Wrachtrup, Nat Mater 8, 383 (2009)

work page 2009

[5] [5]

A. A. Wood, E. Lilette, Y. Y. Fein, V. S. Perunicic, L. C. L. Hollenberg, R. E. Scholten, and A. M. Mar- tin, Nature Physics 13, nphys4221 (2017)

work page 2017

[6] [6]

A. A. Wood, A. G. Aeppli, E. Lilette, Y. Y. Fein, A. Stacey, L. C. L. Hollenberg, R. E. Scholten, and A. M. Martin, Phys. Rev. B 98, 174114 (2018)

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[7] [7]

Zhang, H

C. Zhang, H. Yuan, Z. Tang, W. Quan, and J. C. Fang, Applied Physics Reviews 3, 041305 (2016)

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[8] [8]

Degen, F

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work page 2017