Pound-Drever-Hall Feedforward for Trapped-Ion Optical Qubits

Jianwei Lee; Jia-Yang Gao; Manas Mukherjee; Morteza Ahmadi

REVIEW 3 major objections 5 minor 42 references

Pound-Drever-Hall Feedforward for Trapped-Ion Optical Qubits

T0 review · 3 major / 5 minor · reviewed 2026-07-31 · deepseek-v4-flash

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Pith's one-line read PDH feedforward suppresses servo-bump noise by 15 dB and lifts optical-qubit Rabi coherence sixfold.

desk verdict First demonstration of PDH feedforward on a trapped-ion optical-qubit laser: the 15 dB bump suppression and 6x T2 improvement are direct and credible; the ΔF≈0.22 fidelity estimate is model-bound and not yet supported. read the letter →

arxiv 2607.24050 v1 pith:XQUIOQTR submitted 2026-07-27 quant-ph physics.app-phphysics.atom-phphysics.optics

Jia-Yang Gao , Jianwei Lee , Morteza Ahmadi , Manas Mukherjee This is my paper · ORCID

classification quant-phphysics.app-phphysics.atom-phphysics.optics

keywords Pound-Drever-Hallfeedforwardlaserphasenoiseservobumpopticalqubitcoherencedelayedself-heterodyneinterferometrytrapped-ionquantumgatesbarium-138quadrupoletransition

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

This paper establishes that the residual Pound-Drever-Hall error signal of a cavity-locked laser can be recycled as a feedforward phase correction, canceling the servo-bump phase noise that standard PDH feedback cannot remove. Demonstrated on the 1762 nm quadrupole-transition laser of a trapped 138Ba+ optical qubit, the method suppresses bump noise by about 15 dB near the 132 kHz bump peak, leaves the laser linewidth unchanged, and improves the measured Rabi coherence time from 3.69±0.35 µs to 21.73±0.87 µs when the Rabi frequency lies inside the bump band. A finite-time noise-response model connects the measured phase-noise spectrum to a phase-noise-limited average π-gate infidelity, predicting a gain of about ΔF≈0.22. If correct, the technique extends the usable Rabi-frequency range of optical-qubit gates without the optical-power penalty of cavity filtering.

What carries the argument

The load-bearing object is the PDH error signal, which after tight locking is proportional to the residual instantaneous phase noise ϕ(t), including components beyond feedback bandwidth. Recycling it with inverted gain and matched delay into a fiber EOM gives ϕ_out≈ϕ(t)+G_ff ϕ(t−τ); with G_ff=−1 and τ tuned to the 132 kHz bump, the bump noise cancels. The setup uses a 325 m BNC delay line and 20 m fiber; DSHI with 15 m fiber resolves the bump spectrum and a 5 km fiber measures the linewidth. The qubit model is a second-order time-convolutionless expansion of the toggling-frame Bloch equation, yielding a finite-time Pauli-transfer matrix M_I(t), the infidelity ϵ=(3−Tr[M_I(t)])/6, and a scalar

What would settle it

Run a positivity-preserving Monte-Carlo simulation of the stochastic Bloch equation (rather than the second-order truncation) using the measured DSHI phase-noise spectrum, and compare its predicted π-pulse infidelity versus Rabi frequency to the paper's Fig. 5 curves; disagreement near f_R≈63 kHz, where the second-order estimate reaches the unital-channel ceiling of about 0.5, would show the calibrated infidelity model is unreliable in that regime.

Watch

Extended reading notes

Core claim

The central claim is that PDH feedforward—applying the locked laser's own residual error signal, after matched delay and inverted gain, to a fast fiber EOM—cancels the servo-bump phase noise of the PDH lock beyond feedback bandwidth. The authors verify this on a 138Ba+ S1/2↔D5/2 optical qubit: DSHI spectra show up to 15 dB suppression at the bump peak with no linewidth broadening, Rabi oscillations driven near the bump show a roughly sixfold increase in T2, and a calibrated finite-time dynamical map translates the measured spectrum into a phase-noise-limited π-gate fidelity improvement of about 0.22. They present this as the first experimental demonstration that PDH feedforward is compatible

Load-bearing premise

The quantitative bridge from spectra to qubit claims—specifically the ΔF≈0.22 fidelity gain—assumes the second-order time-convolutionless expansion is valid at the strong-noise operating points, even though the paper's Appendix A warns that the truncated map may lose complete positivity for stronger noise and the spectral scale factor is fitted to the very decay it explains.

Editorial extensions

If this is right

Single-qubit optical gates can be operated at Rabi frequencies inside the servo-bump band, removing a constraint on gate-speed choice imposed by PDH lock noise.
Because the technique recycles the already-available error signal, it adds no optical loss; it is a power-preserving alternative to high-finesse cleanup cavities when transmitted cavity power is only microwatts.
The measured log-linear relation (≈1.21× in T2 per dB of bump suppression) gives a quantitative rule for predicting coherence gain from spectral noise reduction.
The method is presented as readily transferable to other optical qubits and transitions, with the main hardware caveat being the power handling of the phase actuator at the operating wavelength.
For two-qubit gates, suppressing servo bumps near motional frequencies should mitigate fidelity loss in Mølmer–Sørensen gates, which the authors identify as a motivation for future work.

Reading between the lines

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

One could use the same feedforward loop to suppress bumps at multiple or higher frequencies by choosing delay lines matched to several bump harmonics, effectively widening the usable gate-speed window beyond the single peak demonstrated here.
The finite-time Rabi-filter formalism suggests a design heuristic: to predict the coherence impact of a given locking servo, evaluate the overlap between the phase-noise PSD and the Rabi filter for the target pulse time, rather than simply reading the PSD at the Rabi frequency.
A direct extension would be to apply PDH feedforward to Mølmer–Sørensen two-qubit gates and measure the entangling-gate fidelity as a function of motional frequency; the single-qubit results imply a measurable improvement when a motional sideband coincides with the bump.
The model's reliance on a single spectral calibration factor could be sharpened by measuring Rabi decay at several Rabi frequencies and checking whether the predicted log-linear T2 scaling across the whole bump region is borne out independently of the fitted decay.

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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

First, the part that's solid: this is the first experimental demonstration of PDH feedforward applied to a trapped-ion optical-qubit laser. They measure ~15 dB suppression of the servo bump at ~132 kHz with DSHI, show the linewidth doesn't broaden (160±15 Hz vs 156±16 Hz from ion spectroscopy), and see a real coherence improvement — T2 goes from 3.69±0.35 µs to 21.73±0.87 µs when the Rabi frequency is in the bump band. The 300 m cable control showing incomplete cancellation is a good experimental touch. These direct measurements are credible and cross-checked.

The soft spot is the model that converts spectra to gate fidelity. The headline ΔF_avg ≈ 0.22 comes from a second-order time-convolutionless expansion that App. A itself warns can lose complete positivity at stronger noise — and their own un-stabilized estimate reaches 1−F ≈ 0.498 at 63 kHz, exactly that regime. The scale factor is calibrated using the feedforward-off Rabi decay, so the model is being fit and validated on the same data. There's also a 16% discrepancy between the measured log-linear T2 slope (0.084 per dB) and the model's predicted 0.100 per dB, and the fidelity gain is quoted without uncertainty. These don't undermine the direct measurements, but they mean the quantitative gate-fidelity claim is not yet supported.

The authors are open about several of these limitations, which is to their credit. The fix is straightforward: publish the spectra and T2 data with ΔdB coordinates, reconcile or explain the slope discrepancy, and re-quote fidelity gains with uncertainties.

I'd send this to peer review — the experimental core is valuable and the technique is transferable to other optical qubits. I'd also bring it to a reading group; it's a good example of where direct measurements outrun the theory used to interpret them.

Referee Report

3 major / 5 minor

Summary. The manuscript reports an experimental demonstration of Pound-Drever-Hall (PDH) feedforward phase-noise suppression for the 1762 nm laser driving the S1/2↔D5/2 optical-qubit transition in 138Ba+. The authors recycle the residual PDH error signal through a delay-matched electrical path to a fiber EOM, and measure, via delayed self-heterodyne interferometry, a ~15 dB suppression of the servo-bump phase noise near 132 kHz without measurable linewidth broadening (160±15 Hz by DSHI versus 156±16 Hz by ion spectroscopy, App. H). They then present a theoretical model of a resonantly driven qubit under stochastic detuning noise, based on a second-order time-convolutionless expansion and a Pauli-transfer-matrix average gate infidelity, and use DSHI spectra plus a spectral calibration factor to estimate a phase-noise-limited π-gate fidelity improvement ΔF_avg ≈ 0.22 near the servo-bump region. Trapped-ion Rabi measurements show that enabling feedforward increases the extracted coherence time from 3.69±0.35 µs to 21.73±0.87 µs for Rabi frequencies near the bump, and the D-state population at the first π time increases from 0.641±0.014 to 0.870±0.009. The paper also reports a log-linear relation between T2 and servo-bump suppression in dB.

Significance. If the direct measurements stand, the work has clear practical value: it provides a power-efficient alternative to cavity filtering that extends the usable Rabi-frequency range of optical-qubit gates without sacrificing optical throughput, and it is the first experimental integration of PDH feedforward with trapped-ion optical-qubit control. The 15 dB suppression, the unchanged linewidth, and the ~6x improvement in Rabi coherence time are direct, cross-checked measurements and are the strongest parts of the paper. However, the specific quantitative gate-fidelity gain (ΔF_avg ≈ 0.22) and the Fig. 5 infidelity curves depend on a perturbative finite-time model that the paper itself warns may lose complete positivity, and the model is calibrated using the very Rabi-decay observable it is used to explain. These issues do not invalidate the direct measurements, but they mean the gate-fidelity headline is not yet quantitatively supported. The paper includes detailed appendices and a clear discussion of the model's limitations, which is helpful, but the limitations are not reflected in the strength of the central fidelity claim.

major comments (3)

[App. A, App. F, Fig. 5, Sec. V.B] The phase-noise-limited gate-fidelity estimate ΔF_avg ≈ 0.22 is produced by the second-order time-convolutionless generator of App. A. Appendix A explicitly warns (after Eq. A13) that for stronger noise or longer evolution times the truncated map 'may lose complete positivity' and that Monte-Carlo propagation should be used as the reference. The paper's own Fig. 5/App. F shows an un-stabilized infidelity of 1−F_avg ≈ 0.498 at f_R ≈ 63 kHz, which is essentially the maximum possible average infidelity for a unital single-qubit channel. No error bound, no comparison with Monte-Carlo simulation, and no finite-size check is given to establish that the second-order expansion is controlled in this regime. Therefore the quantitative fidelity improvement and the Fig. 5 curves are not reliably supported. The direct 15 dB and T2 measurements stand independently, but the gate-fidelity claim should b
[Sec. V.B, App. F] The model is calibrated using the measured Rabi decay: App. F states that 'a single spectral calibration factor, determined from the feedforward-off Rabi-decay measurement, is used for all Rabi frequencies and both feedforward conditions.' The same observable (Rabi decay / T2) is thus used both to fix the noise scale and to validate the model's predictions. This makes the ΔF_avg ≈ 0.22 estimate, and the per-dB T2 scaling in Fig. 4, model-dependent rather than independent predictions. I would like to see an out-of-sample check — e.g., calibrating on one Rabi frequency and predicting another, or predicting the feedforward-on data using only feedforward-off calibration — or a sensitivity analysis showing how the inferred fidelity gain changes with the calibration factor.
[App. G, Fig. 4] There is an unexplained quantitative discrepancy in the log-linear relation. Appendix G derives that if the dephasing rate is proportional to the phase-noise PSD at the Rabi frequency, then log10(T2) should increase with slope 0.100 per dB of suppression. The measured fit in Fig. 4 gives a slope of 0.084 per dB with R² = 0.992 — a 16% deviation. This discrepancy is not discussed in the text, yet the abstract and discussion quote the empirical '1.21x per dB' factor. Since the per-dB improvement is a central quantitative claim, the origin of this deviation (finite-time filter effects, non-Lorentzian noise shape, non-exponential decay, or fitting choices) should be addressed, or the theoretical slope should not be presented as the expected behavior without comment.

minor comments (5)

[Fig. 3b] The caption states 'pi time used: 22.1 us,' which corresponds to a Rabi frequency of approximately 22.6 kHz, not the 132 kHz servo-bump frequency discussed elsewhere in the text. Please clarify whether the frequency scan was performed at a Rabi frequency away from the bump, and state explicitly how the off-resonant excitation shoulder at ±132 kHz is produced in that case.
[Sec. V.B / Conclusion] The improvement in D-state probability at the first π time (0.641→0.870) is a single-state transfer measurement, not an average gate fidelity. The authors acknowledge this distinction in Sec. V.B, but the Conclusion states that the results show 'clear improvements in the coherence time and gate fidelity.' I recommend reserving 'gate fidelity' language for actual randomized or state-tomography-based fidelity measurements, or for the model-based estimate after it has been properly validated.
[Abstract / Introduction] There are several typographical errors and slightly awkward phrasings, e.g., 'supressing' in the Conclusion, 'an qubit' in the Introduction, and 'without limited by the transmission optical power' in Sec. II.A. A careful proofread would improve clarity.
[Data Availability] The statement 'available from the corresponding author upon reasonable request' is weaker than current best practice. Since the Figs. 2–5 and the calibration procedure are central to the quantitative claims, I encourage depositing the data and analysis scripts in a permanent repository (e.g., Zenodo or figshare) so that the DSHI spectra, Rabi fits, and calibration factor can be independently checked.
[Fig. 4] The text says the red points at ≈345 kHz are 'shown for comparison' and are not included in the fit. It would be helpful to state this directly in the figure caption and to report the confidence interval on the fitted slope, not only R².

Circularity Check

0 steps flagged · score 2.0 of 10

Direct 15 dB suppression and T2 improvement are measured; the model-based ΔF≈0.22 fidelity estimate is transparently calibrated and not used to support the central experimental claims.

full rationale

The principal claims—15 dB servo-bump suppression by DSHI and the T2 improvement from 3.69±0.35 µs to 21.73±0.87 µs from Rabi oscillations—are direct interferometric and qubit measurements and do not depend on the cumulant model. The model in App. A is used to convert measured phase-noise spectra and a calibrated noise scale into an estimated phase-noise-limited π-gate fidelity improvement (Sec. V.B, App. F). This is a transparent calibration: 'A single spectral calibration factor, determined from the feedforward-off Rabi-decay measurement, is used for all Rabi frequencies and both feedforward conditions.' Fitting a noise scale to one measured decay and then using the measured on/off spectral ratio to estimate a different metric (fidelity) is a normal model-calibration procedure, not a derivation where the input is renamed as the output. App. A's warning that the truncated map 'may lose complete positivity' for stronger noise is a validity caveat relevant to the absolute infidelity numbers, but it is a correctness risk, not circularity. There is no load-bearing self-citation chain: references to the authors' earlier work are for ion-trap setup details and amplifier characterization. Thus no circular step meets the required standard.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The ledger shows a modest, mostly measurable axiom load. The free parameters are concentrated in the quantitative bridge between spectra and qubit metrics: one global calibration factor fitted to the feedforward-off decay, empirical log-linear fit coefficients, and two experimental tunables (cable length, gain setting). No invented entities are introduced. The main fragility is the combined validity of the Gaussian, weak-noise, TCL2 truncation and phase-noise-dominance axioms in the un-stabilized regime, where the model's own estimates approach maximal qubit-channel infidelity.

free parameters (4)

Spectral calibration factor = not stated
Single scale factor fitted so the finite-time model reproduces the feedforward-off Rabi decay (T2 = 3.69 µs at f_R ≈ 132 kHz); reused for all Rabi frequencies and both feedforward conditions (App. F). The ΔF_avg ≈ 0.22 estimate and Fig. 5 curves inherit this fit.
Log-linear T2 fit parameters = slope 0.084, intercept 0.741
Empirical least-squares fit to T2 vs ΔdB (Fig. 4, R² = 0.992). The fitted slope deviates from the model prediction 10^(ΔdB/10) (slope 0.1) derived in App. G, and the intercept corresponds to T2(0) ≈ 5.5 µs, above the measured 3.69 µs.
Feedforward delay cable length = 325 m RG-58 (optimized)
Chosen by fine-tuning around a first estimate to flatten the DSHI bump region (Sec. II B); a 300-m control shows incomplete suppression. A hand-optimized experimental parameter, not a physics constant.
Servo gain setting = bump at ≈132 kHz
The bump frequency "appears at approximately 132 kHz dependent of the gain setting"; the feedforward delay and the T2 measurements are matched to this operating point. The 15 dB claim holds at this setting only.

assumptions (5)

domain assumption Residual locked-laser phase noise is small enough that sin φ(t) ≈ φ(t) in the PDH error signal (Eq. 4)
The entire feedforward principle — gain −1 and delay matching give near-complete cancellation — relies on linearity of the error signal at the servo-bump frequency and beyond.
domain assumption Detuning noise β(t) is zero-mean, stationary, and Gaussian (Eq. 18)
The second-order TCL cumulant truncation and the scalar secular envelope (Apps. A, D) assume Gaussianity and stationarity; the paper does not test departures from either.
domain assumption Second-order time-convolutionless expansion is adequate (O(β⁴) terms dropped, Eq. A6)
App. A itself warns that the truncated map "may lose complete positivity" for stronger noise, yet the un-stabilized estimate reaches near-maximal infidelity ≈ 0.498, where the expansion is most strained.
domain assumption TDFA amplification preserves the seed's phase-noise structure (adds intensity, not phase, noise)
DSHI phase-noise and ion data are taken after TDFA amplification; linewidth checks (160±15 Hz vs 156±16 Hz) support this for linewidth, but the 132-kHz phase-noise attribution to the seed lock loop is inferred rather than separately measured pre/post amplifier.
domain assumption Dephasing during Rabi drive is dominated by laser phase noise; SPAM, intensity noise, magnetic-field fluctuations, and pulse-area errors are negligible
Explicitly stated in App. F: the estimates exclude these terms. The T2-per-dB model and gate-fidelity claims are phase-noise-only, yet App. H itself attributes low-frequency T2 differences (97 vs 125 µs) to intensity noise.

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Cite this review

Pith. "Pith review of Pound-Drever-Hall Feedforward for Trapped-Ion Optical Qubits." pith.science (2026). https://pith.science/paper/XQUIOQTR

@misc{pith2026260724050,
  author       = {Pith},
  title        = {Pith review of: Pound-Drever-Hall Feedforward for Trapped-Ion Optical Qubits},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XQUIOQTR}},
  note         = {Machine review of arXiv:2607.24050}
}

read the original abstract

Laser phase noise is one of the limiting factors that dictate gate fidelities and coherence times in trapped-ion quantum systems. Previous studies have reported that when the Rabi frequency of an ion qubit is close to the servo-bump phase-noise frequency, the driving laser limits the fidelity and coherence time. This issue has typically been mitigated by choosing a Rabi frequency outside the servo-bump region. However, this constrains the usable range of gate speeds and can limit the achievable fidelity. To address this issue, we developed an active phase-noise stabilization system for a barium-ion optical-qubit laser at 1762 nm, employing a fiber electro-optic modulator (EOM) with an electrical feedforward servo. Our results demonstrate that this setup, based on the Pound-Drever-Hall (PDH) feedforward method, can suppress servo-bump phase noise by 15 dB near the bump peak frequency in our locking system. The laser phase noise is analyzed using delayed self-heterodyne interferometry (DSHI). We further tested the stabilized laser on an optical qubit and observed a clear improvement in coherence time based on the measured amplitude decay of Rabi oscillations, even when the Rabi frequency lies within the servo-bump bandwidth. This technique can be readily adapted to other optical qubits with minimal modifications.

Figures

Figures reproduced from arXiv: 2607.24050 by the authors.

**Figure 2.** FIG. 2 [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** b compare the frequency scan excitation spectra with and without phase noise stabilization (pi time used: 22.1us). The additional off-resonant excitation is clearly observable (circled by a dashed green line), with its shape determined by the servo noise and locking bandwidth. We also summarize the relationship between the coherence time and the servo-bump phase-noise reduction in [PITH_FULL_IMAGE:figures/full_fig… view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: The data shows a log-linear dependence, fitted by log10(T2) = 0.084 × |∆dB| + 0.741, with a coefficient of determination R2 = 0.992. This corresponds to an improvement factor of approximately 1.21 times in T2 per dB of servo-bump noise suppression. The blue data points…

**Figure 5.** Figure 5: FIG. 5: Phase-noise spectrum and simulated phase-noise-limited average [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]

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Reviewed July 31, 2026 · model on record in the stance chip above.

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