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REVIEW 1 major objections 5 minor 62 references

Amplitude and frequency sensing of microwave fields with a superconducting transmon qudit

T0 review · 1 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Ac Stark shifts of a transmon qudit reveal the amplitude and frequency of an on-chip microwave field.

desk verdict A useful, honest experimental follow-up: Ramsey-based ac-Stark sensing improves precision by an order of magnitude, but the amplitude channel lacks independent ground truth and should be tempered or cross-checked. read the letter →

arxiv 1908.09556 v3 pith:TZMJR2YH submitted 2019-08-26 quant-ph cond-mat.supr-con

Maximilian Kristen , Andre Schneider , Alexander Stehli , Tim Wolz , Sergey Danilin , Hsiang S. Ku , Junling Long , Xian Wu
show 4 more authors
Russell E. Lake David P. Pappas Alexey V. Ustinov Martin Weides
This is my paper · ORCID
classification quant-phcond-mat.supr-con
keywords transmonquditacStarkshiftRamseyinterferometrymicrowavesensingtransferfunctionsuperconductingcircuitslookuptableinversionquantum
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 shows that a single superconducting transmon, used as a multilevel sensor, can determine both the frequency and the amplitude of a microwave field arriving at the chip by measuring how that field shifts the energies of its first two transitions. The shifts are read out through Ramsey fringes, which raises the precision by an order of magnitude over the earlier spectroscopic scheme. Over a span of several hundred megahertz, the sensor reports field frequency and amplitude with uncertainties of 25 MHz and 3.4 MHz, respectively, at an energy sensitivity around $10^{-4}$. Such on-chip metrology matters because conventional electronics cannot probe signals at the millikelvin stage, and knowing the transfer function from source to sample would let experimenters correct distorted microwave pulses before they reach the qubit.

What carries the argument

The load-bearing object is a pair of lookup tables: numerical master-equation simulations of the full transmon Hamiltonian that map field frequency and amplitude onto the expected Stark shifts of the first two transitions. A Ramsey sequence on each transition supplies the measured shifts, and the tables are searched for the field parameters whose predicted shifts match both measurements. These numerical tables replace the analytic perturbation-theory version from the earlier sensing work, which is not accurate enough when frequency shifts are resolved at the few-kilohertz level.

What would settle it

Apply a known microwave tone at several amplitudes and frequencies, including a tone near the readout resonator at 6.878 GHz and amplitudes around A_F/2pi = 0.75 GHz where the simulation predicts 22.7% population of the |2> state, and compare the sensor's extracted amplitude to an independently calibrated cryostat attenuation model; agreement within the quoted uncertainties would confirm the lookup-table model, while systematic deviations beyond those uncertainties would falsify it.

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Extended reading notes

Core claim

The central claim is that the ac Stark shifts of the first two transmon transitions, measured via Ramsey interferometry and inverted through pre-computed lookup tables, uniquely determine the frequency and amplitude of an unknown on-chip microwave field. The paper verifies this by sending a known tone across a 450 MHz range and showing that the extracted frequencies track the applied frequencies, while the extracted amplitudes trace the frequency-dependent attenuation of the transfer function. Compared with the prior spectroscopic implementation, the time-resolved approach reduces the uncertainties to $\Delta$ omega_F/2pi = 25 MHz and $\Delta$ A_F/2pi = 3.4 MHz in about a minute of measurement time. The method is limited to fields above the first qudit transition and below the power where higher transmon levels become substantially populated.

Load-bearing premise

The sensor's calibration charts are computed by a simulation that ignores the readout resonator and uses only the two measured qudit transition frequencies, so if that model's predicted Stark shifts do not match what the real device feels, the extracted amplitude and frequency will be systematically wrong.

Editorial extensions

If this is right

  • The sensor measures the amplitude of the microwave transfer function from source to chip over a band of several hundred megahertz, exposing frequency-dependent distortion from the readout resonator and cable resonances.
  • Because both frequency and amplitude come from the same two shift measurements, the method can characterize arbitrary microwave fields, not just tones whose parameters are already known.
  • Combined with the phase-sensitive extension outlined in the supplemental material, the scheme could yield the full complex transfer function needed for digital pulse pre-distortion and higher gate fidelities.
  • The achieved energy sensitivity near 10^-4 makes weak fields detectable, with the ultimate floor in this device set by a noise offset of about 2.2 kHz on the second transition.
  • The one-minute measurement time can plausibly shrink to seconds using parametric amplifiers and active reset, both already demonstrated in similar circuit-QED setups.

Reading between the lines

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

  • Because the lookup tables are generated numerically, the same sensor concept could be retuned to other transmon frequencies or higher qudit levels simply by regenerating the tables, extending the measurable band without changing the measurement logic.
  • A practical phase sensor built on the supplement's two-detuned-pulse idea would likely need a clearer readout signal than this sample's noise floor permits, suggesting that readout improvement is the natural next step.
  • Temporal fluctuations of the qudit frequencies from two-level-system noise could be the dominant systematic error in field extraction; continuously tracking omega_1 and omega_2 during a sensor run should tighten the reported uncertainties.
  • One could test whether the method extends beyond continuous tones to pulsed or multi-tone microwave fields, by comparing the Ramsey-derived amplitudes against an independent, calibrated power measurement in the time domain.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 5 minor

Summary. The paper reports a time-resolved microwave-field sensor based on a superconducting transmon qudit. A continuous microwave tone detuned from the first two qudit transitions produces ac Stark shifts Δ1 and Δ2 that are measured via Ramsey interferometry. The measured shifts are inverted using lookup tables generated by QuTiP master-equation simulations of Eq. (1) to extract the on-chip field frequency ωF and amplitude AF. The authors validate the frequency channel by comparing extracted and applied source frequencies over a 450 MHz range, and they use the extracted amplitude to display the transfer function of the microwave line. They report an order-of-magnitude precision improvement over the earlier spectroscopic implementation, with ΔAF/2π = 3.4 MHz and ΔωF/2π = 25 MHz for about one-minute measurement times, and they discuss sensor limits, the Ramsey noise floor, TLS fluctuations, and a phase-measurement scheme.

Significance. This is a useful and timely contribution to in-situ calibration of microwave control lines in circuit QED. The paper's strengths are the direct frequency validation over 450 MHz, the quantitative error analysis (Ramsey fit standard errors, shot-noise scaling, TLS frequency-fluctuation estimates, and simulation uncertainty for input parameters), and the concrete Ramsey-based protocol with clear sensor limits. If the amplitude channel can be independently validated, the demonstrated order-of-magnitude precision gain over Ref. [18] and the broadband transfer-function measurement would justify publication in a good quantum-engineering journal.

major comments (1)
  1. [Lookup table calculations; Fig. 3c; Supplementary 'Simulation Uncertainty'] The amplitude channel is not independently calibrated, so the headline amplitude uncertainty ΔAF/2π = 3.4 MHz is not yet established. The applied source power PF,apl is not a ground truth for the on-chip amplitude AF, and the transfer function shown in Fig. 3c is precisely the quantity the sensor is meant to measure; therefore Fig. 3c cannot serve as a self-calibration. The lookup tables are generated by master-equation simulations of Eq. (1) that explicitly neglect the readout resonator, and the Supplementary 'Simulation Uncertainty' analysis varies only ω1 and ω2, concluding that the error is at most 2σR,i. That argument does not bound model-form error from the neglected resonator, the truncated level basis, or the modeling of the gate pulses. A systematic bias in the lookup tables would shift AF,ex without affecting the statistical shaded error bars, and the successful frequency validation in Fig. 3a does not rule this out because the frequency contours of the lookup table can remain accurate even if the amplitude scaling is wrong. I request an independent amplitude check, for example resonant Rabi oscillations at a known transition frequency to calibrate the on-chip drive amplitude, or a full-system simulation including the readout resonator that quantifies the model-form error.
minor comments (5)
  1. [Results, Example extraction] The expression PF,ex = AF,ex ℏωF,ex appears dimensionally inconsistent as written; please specify the unit conversion used to obtain a value in dBm.
  2. [Abstract] The claim of an 'energy sensitivity on the order of 10^-4' is never defined; please state which measured quantity this refers to and how it is derived.
  3. [Supplementary Information, Unprocessed sensor data] The sentence 'the qudit ground state is not effected' contains a typo; 'effected' should be 'affected'.
  4. [Methods, Lookup table calculations] The text refers to 'the full system Hamiltonian in Eq. (1)', but Eq. (1) neglects the readout resonator; this wording is misleading and should be revised to 'the qudit Hamiltonian' or similar.
  5. [Results, Eq. (3)] The distinction between upper and lower sensor limits in Eq. (3) would be clearer if the text explicitly stated that the first inequality is an upper bound set by the sampling rate and the second is a lower bound set by the maximum Ramsey delay, and why the lower bound is controlled by Δ2.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the field extraction is a forward-model lookup-table inversion with independently measured qudit frequencies; the acknowledged model-uncertainty gaps are validation concerns, not circular reductions.

full rationale

The claimed derivation chain is: Ramsey measurements give oscillation frequencies, which give ac Stark shifts via Eq. (2); the shifts are then inverted through numerically generated lookup tables produced by master-equation simulations of Eq. (1) with independently measured qudit frequencies ω1 and ω2. The lookup tables contain no adjustable parameters tuned to the extracted fields, and the extracted frequency ωF,ex is compared with the externally applied source frequency ωF,apl as an independent validation. The only fitted constants in the paper, the noise-floor coefficients ai and ci in σR,i/2π = ai/√Navg + ci, are used for sensitivity and averaging characterization, not to produce the reported AF,ex and ωF,ex values. Citations to the authors' prior work, notably Ref. [18], provide background and the original sensor concept, but the load-bearing numerical inversion is implemented and validated in the present paper against an external frequency reference, so the self-citations are not carrying the argument. The supplementary 'Simulation Uncertainty' section explicitly bounds only the effect of uncertainty in ω1 and ω2, and the text acknowledges that details of the noise floor 'require further investigation' and that a general formulation for high drive powers is 'beyond the scope of this work.' These are honest statements of model uncertainty and unvalidated amplitude calibration, not circular reductions: they do not define the extracted amplitude in terms of the measured output or fit a parameter that is later renamed as a prediction. The lack of an independent on-chip amplitude calibration would be a benchmarking and correctness concern, but it does not make the forward-model inversion circular. No equation in the paper is equivalent to its own input by construction, and no fitted parameter is presented as a predicted result. Therefore no specific circular step can be exhibited, and the appropriate score is 0.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

The central extraction uses no fitted free parameters: the lookup tables are forward-model simulations from the independently measured qudit frequencies. The only fitted constants are the noise-floor characterization, which is not part of the sensing loop. The main axioms are the validity of the driven transmon Hamiltonian (with the readout resonator neglected) and the uniqueness of the two-table inversion.

free parameters (1)
  • Noise floor fit constants (a1, a2, c1, c2) = a1=355 kHz, a2=537 kHz, c1=0.12 kHz, c2=2.20 kHz
    Fitted to the measured standard error sigma_R,i versus the number of averages Navg (Supplementary Fig. 3e). Used only to report sensor limits and the noise floor, not in the central extraction of amplitude and frequency.
assumptions (3)
  • domain assumption The transmon Hamiltonian model in Eq. (1) with exact eigenstates and a drive term describes the sensor dynamics, and the readout resonator can be neglected.
    The lookup tables are generated from this Hamiltonian using QuTiP master-equation simulations (Methods, Lookup table calculations), so the inversion inherits all assumptions of this model.
  • domain assumption The observed Ramsey oscillation frequency shifts are equal to the ac Stark shifts Delta_i of the first two qudit transitions.
    Used in Eq. (2) and throughout; the sequential pi-pulse calibration depends on this relation.
  • domain assumption The pair of lookup tables provides a unique inversion from (Delta_1, Delta_2) to (A_F, omega_F) over the probed range.
    The paper states that searching both tables simultaneously yields an unambiguous result; uniqueness is assumed for the transfer function measurement.

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Pith. "Pith review of Amplitude and frequency sensing of microwave fields with a superconducting transmon qudit." pith.science (2026). https://pith.science/paper/TZMJR2YH

@misc{pith2026190809556,
  author       = {Pith},
  title        = {Pith review of: Amplitude and frequency sensing of microwave fields with a superconducting transmon qudit},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TZMJR2YH}},
  note         = {Machine review of arXiv:1908.09556}
}
abstract

Experiments with superconducting circuits require careful calibration of the applied pulses and fields over a large frequency range. This remains an ongoing challenge as commercial semiconductor electronics are not able to probe signals arriving at the chip due to its cryogenic environment. Here, we demonstrate how the on-chip amplitude and frequency of a microwave signal can be inferred from the ac Stark shifts of higher transmon levels. In our time-resolved measurements we employ Ramsey fringes, allowing us to detect the amplitude of the systems transfer function over a range of several hundreds of MHz with an energy sensitivity on the order of $10^{-4}$. Combined with similar measurements for the phase of the transfer function, our sensing method can facilitate pulse correction for high fidelity quantum gates in superconducting circuits. Additionally, the potential to characterize arbitrary microwave fields promotes applications in related areas of research, such as quantum optics or hybrid microwave systems including photonic, mechanical or magnonic subsystems.

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