{"id":"d3db0d22-ce1a-46a4-b7bb-361338fa1f2a","arxiv_id":"2501.18521","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"low","formal_verification":"none","parameter_count":2,"one_line_summary":"The squared Rabi frequency of a weakly driven qubit gains a detuning-dependent correction equal to g^2 (k^2/2)(Delta/alpha), verified on fluxonium qubits with opposite anharmonicity signs.","lead":"This paper derives and measures how a qubit's anharmonicity changes its Rabi oscillation frequency when the drive is slightly detuned, adding a correction proportional to the ratio of detuning to anharmonicity. The result matters for calibrating microwave-activated two-qubit gates on fluxonium qubits, where ignoring this correction causes small leakage and phase errors.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Three-level truncation is the least secure pillar: the harmonic-mode separation is quoted relative to sweet-spot ν01, while the measured effect is a virtual 1–2 transition whose nearby modes could renormalize s; no truncation-error estimate is given.","rationale":"The analytic derivation is internally consistent: re-deriving the cubic from the RWA Hamiltonian in Eq. (1) yields the depressed coefficients that produce Eq. (2) to the stated order, so I found no algebra or sign error in the central formula. The experimental confirmation is also strong: the measured slopes at the two sweet spots (2.08 ± 0.18 ns and −2.16 ± 0.19 ns) agree with Eq. (3) at the 1σ level, with opposite signs tracking the sign of α. The residual worry is model truncation. The paper explicitly flags assumptions (i)–(v), but the justification for excluding higher levels is incomplete: the harmonic mode is guaranteed to be >1 GHz from ν01, yet the effect being measured lives on the 1–2 transition, and at the high sweet spot ν12 is only 0.403 GHz below ν01. A weakly coupled level near ν12 would contribute a second-order Stark shift comparable in structure to δ from Eq. (4), so the quoted mode separation alone does not settle the three-level approximation. This is not an observed discrepancy, and the existing data already set a tight empirical bound, so I do not regard it as invalidating the paper. A numerical multi-level check or a direct measurement of the residual 0–2 coupling would convert this from a plausible caveat into a settled one. The reader's weakest-assumption identification matches this concern, and the verdict does not need to change.","tokens_in":10403,"tokens_out":40983,"duration_ms":408496,"concrete_test":"Use the device parameters from Table I together with the actual harmonic-mode frequency and coupling (currently not reported) to numerically diagonalize the full fluxonium-plus-harmonic-mode Hamiltonian at both sweet spots, using the charge operator n for the low sweet spot and flux operator φ for the high sweet spot. For the same g and Δ ranges as in Fig. 2, extract Ω² from exact time evolution and fit s(Δ); if the resulting slope differs from Eq. (3) by more than ~0.2 ns, the three-level truncation is the controlling error. Alternatively, bound the omitted 0–2 matrix element and the 1–2 coupling to the harmonic mode directly by two-photon and two-tone spectroscopy.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central prediction, Eq. (2), rests on assumptions (i)–(v) of Section II. The least secure is the three-level truncation combined with the exact 0–2 forbiddance in assumption (v). Section III states that the additional harmonic mode is engineered to differ by more than 1 GHz from the sweet-spot frequencies, but the quantity being probed is the virtual 1–2 transition that produces the Stark shift δ = k²g²/(4α). At the high sweet spot ν01 = 4.66 GHz and α = −0.403 GHz, so ν12 = 4.257 GHz; a mode placed >1 GHz from 4.66 GHz can lie within roughly 0.7 GHz of ν12. Because the entire detuning-dependent correction is a second-order effect confined to the 1–2 subspace, any extra level with a non-negligible matrix element to |1⟩ near this frequency renormalizes k and α and changes the predicted slope s. The paper gives no quantitative truncation-error estimate and no measured bound on the residual 0–2 matrix element. The agreement at the ~0.1 ns level provides an empirical bound, but it does not by itself distinguish the three-level model from a four-level model that happens to produce the same slope over the explored Δ range.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper reports an experimental and theoretical study of Rabi oscillations in a fluxonium qubit under weak near-resonant driving, motivated by microwave-activated coupler CZ gates. The authors reduce the system to a three-level model with the 0–2 transition forbidden, solve the resulting Hamiltonian exactly in Appendix A, and obtain the approximation Ω_Rabi² ≈ Δ² + g²(1 + (k²/2)(Δ/α)), so that the slope s of Ω_Rabi² versus g² is predicted to vary linearly with the detuning-to-anharmonicity ratio. They measure s near both fluxonium sweet spots, where the anharmonicity has opposite signs, obtaining slopes 2.08 ± 0.18 ns and −2.16 ± 0.19 ns against predicted values 2.23 and −2.26 ns. They then estimate leakage and phase errors in a coupler-activated CZ gate if this correction is neglected.","tokens_in":10684,"tokens_out":16557,"duration_ms":168565,"significance":"The result is a clean, externally checked prediction: α and k are extracted from spectroscopy rather than fitted to the Rabi data, and the predicted sign reversal of the slope with the sign of α is confirmed. The exact diagonalization in Appendix A makes the expansion checkable, and the CZ error analysis gives a concrete practical motivation. If the truncation concerns are addressed, this is a useful calibration correction for microwave-activated gates on anharmonic qubits. The main limitation is that the three-level truncation and the exactly forbidden 0–2 transition are assumed rather than quantitatively bounded, so the theoretical prediction currently lacks a systematic error estimate.","major_comments":[{"comment":"The three-level truncation is the least quantitatively supported pillar of the derivation. The manuscript states that the harmonic mode is engineered to differ by more than 1 GHz from the sweet-spot frequencies, but the probed correction is a virtual 1–2 transition: at the low sweet spot ν01 = 0.75 GHz and α = 1.335 GHz, ν12 = 2.085 GHz, so a mode placed more than 1 GHz from ν01 can lie within roughly 0.3 GHz of ν12. Such a mode can renormalize k and α and hence the predicted slope s. No quantitative estimate is given for the error from truncating to three levels, and no measured bound is given for the residual 0–2 matrix element assumed to vanish in assumption (v). Please add a truncation-error estimate, for example a four-level Schrieffer–Wolff calculation or a numerical diagonalization including the harmonic mode, and report the harmonic-mode frequency relative to ν12 together with a bound on the 0–2 coupling. The empirical agreement provides a posteriori support, but it does not by itself distinguish the three-level model from a four-level model with a renormalized slope.","section":"Section II, assumptions (i) and (v); Section III"},{"comment":"As written, δ = k²g²/(4α) has the wrong sign to be the Stark shift responsible for Eq. (2) under the stated convention α = ν12 − ν01. For α > 0 the virtual 1–2 coupling shifts level |1> downward; using δ with the opposite sign in the effective two-level detuning gives Ω² ≈ Δ² + g²(1 − k²Δ/(2α)), which contradicts both Eq. (2) and the measured positive slope at the low sweet spot. To be consistent with Eq. (2), the shift entering the effective detuning must be δ = −k²g²/(4α), or the text should clearly state a different sign convention. Please correct Eq. (4) and any related text in Fig. 1.","section":"Section II, Eq. (4)"},{"comment":"The agreement statement would be quantitative if the predicted slopes carried uncertainties propagated from the spectroscopy-derived parameters k and α from Appendix C. Currently the predictions are point values (2.23 and −2.26 ns) while the fitted slopes have ±0.18 and ±0.19 ns errors. Please propagate the uncertainties in k and α, including correlations, into s, and state whether the residuals are within the combined experimental and theoretical uncertainty.","section":"Section III, Fig. 2(e)"}],"minor_comments":[{"comment":"The expansion is written with O(g⁴/α⁴), but the slope s in Eq. (3) is valid only after also neglecting terms of order Δ²/α² in the coefficient of g²; please state explicitly that Eq. (3) is the leading-order slope in both Δ/α and g/α.","section":"Section II, Eq. (2)"},{"comment":"The fitting procedure is not fully specified; please state how many repetitions were averaged, how the Rabi frequencies were extracted, and how the slope uncertainties were obtained.","section":"Section III, Fig. 2(d)–(e)"},{"comment":"The gate parameters g = √(5/3) Δ, νd = (ν00 + ν10)/2, and τ = √(3/2) π/Δ are introduced without derivation; a brief derivation or reference would help the reader assess the error estimates.","section":"Section IV, Eq. (6)"},{"comment":"Reference [22] contains a formatting artifact and should be cleaned before publication.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the journal's scope and the central experimental result appears sound. I am recommending major revision primarily to obtain a quantitative bound on the three-level truncation and to fix the sign in Eq. (4). The novelty relative to prior strong-drive multilevel Rabi work is modest but sufficient: the detuning-dependent linear correction and its experimental confirmation are new. The self-citation to the authors' prior coupler-gate work is contextual and not circular."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this paper has a real, small result and verifies it cleanly. The formula Ω² ≈ Δ² + g²(1 + (k²/2)(Δ/α)) — the s coefficient depends linearly on Δ/α — is new relative to the strong-resonant and high-anharmonicity literature, and the two-sweet-spot measurement with opposite signs of α makes the effect convincing. The authors are not overselling; the claimed magnitude is a small correction that matters for coupler-activated CZ calibration.\n\nWhat's done well: the derivation in Appendix A is straightforward and the exact diagonalization checks out. The experiment is careful: they measure the signal-line transfer function to avoid amplitude systematic errors, and they extract α and k from independent spectroscopy rather than from the Rabi data. The predicted slopes (2.23 and -2.26 ns) agree with the measured ones (2.08±0.18 and -2.16±0.19 ns) within the statistical errors. The two-sign anharmonicity test is a nice discriminating check.\n\nSoft spots, in proportion: the main one is the three-level truncation. The stress-test note is right: the harmonic-mode separation is quoted relative to ν01, but the relevant virtual transition is 1-2, and at the high sweet spot ν12 sits only ~0.4 GHz below ν01, so a mode >1 GHz away from ν01 could still be within ~0.7 GHz of ν12. That could renormalize k and α. The paper gives no quantitative truncation-error estimate and no bound on the 0-2 residual coupling. This isn't fatal — the agreement with experiment provides an empirical bound — but a referee should ask for a numerical estimate of the next-level contribution over the explored detuning range. Two minor things: raw data and code aren't provided, and the predicted slopes don't include propagated systematic uncertainties from α and k. Both are easy to fix.\n\nWho it's for: quantum-control people working on microwave-activated two-qubit gates, especially with fluxonium or low-anharmonicity couplers. It's a practical calibration note with a clean analytic core, not a paradigm shift. The paper deserves a serious referee and, with minor revisions, publication.\n\nRecommendation: send to peer review. I'd accept it after the authors add a truncation-error estimate and address the systematic uncertainty propagation.","headline":"A clean, small but real result: the Rabi frequency carries a detuning-anharmonicity correction that matters for coupler-activated CZ gates, and the two-sweet-spot verification gives it credibility.","tokens_in":11259,"tokens_out":2966,"would_cite":true,"duration_ms":28523,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Near-resonant Rabi oscillations in a three-level superconducting qubit acquire a correction to the squared Rabi frequency that is linear in the detuning-to-anharmonicity ratio, and fluxonium measurements confirm the predicted slope at…","keywords":["qubit anharmonicity","Rabi oscillations","fluxonium","three-level model","Stark shift","microwave-activated CZ gate","near-resonant drive","leakage error"],"falsifier":"Measure the squared Rabi frequency versus drive amplitude on a device whose 0-2 transition is not parity-forbidden, or at detunings approaching the anharmonicity, and check whether the slope departs from $1 + (k^2/2)(\\Delta/\\alpha)$ by more than the experimental uncertainty.","tokens_in":10227,"feed_emoji":"⚛️","tokens_out":13269,"duration_ms":104315,"temperature":0.7,"pith_summary":"This paper argues that anharmonicity leaves a small but measurable imprint on the Rabi frequency of a weakly driven, near-resonant superconducting qubit. The squared Rabi frequency of the 0-1 transition is predicted to be $\\Delta^2 + g^2(1 + (k^2/2)(\\Delta/\\alpha))$, where $\\Delta$ is detuning, $g$ is drive amplitude, $\\alpha$ is anharmonicity, and $k$ is the ratio of 1-2 to 0-1 transition matrix elements. The authors verify this on fluxonium qubits near both sweet spots, measuring slopes $2.08 \\pm 0.18$ ns and $-2.16 \\pm 0.19$ ns against predicted $2.23$ ns and $-2.26$ ns. If correct, the result means that calibrations of microwave-activated two-qubit gates on couplers must include this correction, especially for low-anharmonicity couplers where ignoring it causes leakage and phase errors of about $0.02\\%$ and $0.016$ rad in the analyzed CZ implementation.","feed_headline":"Rabi frequency gains a detuning-anharmonicity correction","feed_subtitle":"Fluxonium data confirm it and ignoring it degrades microwave-activated CZ gates on low-anharmonicity couplers.","key_machinery":"The load-bearing object is the rotating-wave three-level Hamiltonian of Eq. (1), restricted to the states $|0\\rangle$, $|1\\rangle$, $|2\\rangle$ with the 0-2 transition forbidden by parity and with drive amplitude $g$, detuning $\\Delta$, anharmonicity $\\alpha$, and matrix-element ratio $k = m_{12}/m_{01}$. The argument goes by diagonalizing this Hamiltonian exactly, then expanding the 0-1 Rabi frequency in $g/\\alpha$ under the assumptions $g \\ll \\alpha$ and $\\Delta \\ll \\alpha$, producing Eq. (2) with the slope $s = 1 + (k^2/2)(\\Delta/\\alpha)$. The physical mechanism is the ac Stark shift $\\delta = (k^2/4)(g^2/\\alpha)$ of the 1-2 transition, which shifts the 0-1 frequency and thereby changes the Rabi frequency. This machinery carries the argument by converting a multi-level drive problem into a single three-level diagonalization whose leading correction is directly measurable in the slope of $\\Omega_{\\mathrm{Rabi}}^2$ versus $g^2$.","core_discovery":"The paper's central claim is that the Rabi frequency of the 0-1 transition of a weakly driven, near-resonant three-level system is not $\\sqrt{\\Delta^2 + g^2}$ but rather, to leading order in $g/\\alpha$, $$\\Omega_{\\mathrm{Rabi}}^2 \\approx \\$\\Delta$^2 + $g^{2}$\\left(1 + \\frac{$k^{2}$}{2}\\frac{\\$\\Delta$}{\\$\\alpha$}\\right),$$ where $\\Delta$ is the drive detuning, $g$ the drive amplitude, $\\alpha$ the anharmonicity, and $k = m_{12}/m_{01}$ the ratio of 1-2 to 0-1 transition matrix elements. The coefficient $s = \\Omega_{\\mathrm{Rabi}}^2/g^2$ therefore depends on $\\Delta/\\alpha$. The paper derives this from the rotating-wave three-level Hamiltonian with the 0-2 transition forbidden, identifies the physical origin as the ac Stark shift of the 1-2 transition, and verifies it experimentally on fluxonium qubits near both sweet spots, obtaining slopes $2.08 \\pm 0.18$ ns and $-2.16 \\pm 0.19$ ns against predictions of $2.23$ ns and $-2.26$ ns.","pith_inferences":["The same correction should appear in any three-level system driven near resonance, not just fluxonium, so transmon-based couplers with smaller anharmonicity would show a larger relative effect.","The measured slope $s$ is an independent handle on the matrix-element ratio $k$, so the same Rabi experiment could double as a parameter-extraction tool.","A natural calibration protocol follows from the formula: measure $\\Omega_{\\mathrm{Rabi}}^2$ versus $g^2$ at two detunings and use the slope difference to set the drive parameters for the gate.","Pushing the drive amplitude up until $g$ approaches $\\alpha$ would test the limits of the leading-order expansion, since the exact cubic solution predicts deviations that the linearized formula does not."],"forward_implications":["The slope $s$ relating $\\Omega_{\\mathrm{Rabi}}^2$ to $g^2$ carries the sign of the anharmonicity, so the correction appears with opposite trends at the two fluxonium sweet spots.","For the coupler-activated CZ gate with $\\Delta = 14$ MHz, $\\alpha = -550$ MHz, and $k = 1.29$, neglecting the correction yields a leakage of about $0.02\\%$ and a phase error of about $0.016$ rad.","The error from neglecting the correction grows as the coupler anharmonicity shrinks, making the effect important for low-anharmonicity couplers.","A frequency calibration that accounts for the multilevel Stark shift removes these leakage and phase errors entirely."],"supporting_citations":[{"why":"Introduces the coupler microwave-activated controlled-phase gate on fluxonium qubits whose calibration the paper takes as its motivating application.","marker":"[1]"},{"why":"Earlier analytic treatment of multilevel effects on Rabi oscillations of a phase qubit, which the paper extends by including detuned drive.","marker":"[20]"},{"why":"Introduces the fluxonium qubit, the device used in the experiments.","marker":"[30]"},{"why":"Establishes the ac Stark shift of a superconducting qubit, the mechanism the paper attributes to the 1-2 transition's influence on the 0-1 Rabi frequency.","marker":"[32]"},{"why":"Describes the extra harmonic mode in fluxonium, whose engineered >1 GHz separation justifies truncating the model to three levels.","marker":"[33]"}],"fun_headline_variants":["Anharmonicity corrects Rabi frequency near resonance","Rabi frequency gains anharmonicity term","Anharmonicity shifts Rabi oscillations","Microwave Rabi frequency feels anharmonicity","Beyond Rabi: anharmonicity adds a detuning term"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The prediction stands or falls on the three-level truncation with the 0-2 transition exactly forbidden and the rotating-wave approximation; if higher levels or the 0-2 coupling participate, the simple linear correction changes.","fun_headline_variants_meta":{"raw":{"variants":["Anharmonicity corrects Rabi frequency near resonance","Rabi frequency gains anharmonicity term","Anharmonicity shifts Rabi oscillations","Microwave Rabi frequency feels anharmonicity","Beyond Rabi: anharmonicity adds a detuning term"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000328,"raw_usage":{"total_tokens":1828,"prompt_tokens":934,"completion_tokens":894,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":550,"completion_tokens_details":{"reasoning_tokens":822}},"tokens_in":550,"tokens_out":894,"duration_ms":8769,"temperature":1.0,"reasoning_tokens":822,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T23:09:28.915848+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Measure the squared Rabi frequency versus drive amplitude on a device whose 0-2 transition is not parity-forbidden, or at detunings approaching the anharmonicity, and check whether the slope departs from $1 + (k^2/2)(\\Delta/\\alpha)$ by more than the experimental uncertainty.","supporting_citations":[{"cited_title":"Coupler microwave-activated controlled-phase gate on fluxonium qubits","cited_arxiv_id":null,"evidence_quote":"Introduces the coupler microwave-activated controlled-phase gate on fluxonium qubits whose calibration the paper takes as its motivating application."},{"cited_title":"Strong-field effects in the rabi oscillations of the superconducting phase qubit","cited_arxiv_id":null,"evidence_quote":"Earlier analytic treatment of multilevel effects on Rabi oscillations of a phase qubit, which the paper extends by including detuned drive."},{"cited_title":"Fluxonium: Single cooper-pair circuit free of charge offsets","cited_arxiv_id":null,"evidence_quote":"Introduces the fluxonium qubit, the device used in the experiments."},{"cited_title":"ac stark shift and dephasing of a superconducting qubit strongly coupled to a cavity field","cited_arxiv_id":null,"evidence_quote":"Establishes the ac Stark shift of a superconducting qubit, the mechanism the paper attributes to the 1-2 transition's influence on the 0-1 Rabi frequency."},{"cited_title":"Tunable coupling scheme for implementing two- qubit gates on fluxonium qubits","cited_arxiv_id":null,"evidence_quote":"Describes the extra harmonic mode in fluxonium, whose engineered >1 GHz separation justifies truncating the model to three levels."}],"review_version":1}