{"id":"92c4d07f-5f1b-4eff-9b69-bfb82923dc89","arxiv_id":"2412.20971","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"Researchers prepared phonon Fock states up to |6> in a mechanical resonator, proved genuine quantum non-Gaussianity at that level, and showed the states beat an ideal |3> Fock state for displacement sensing under loss.","lead":"This paper prepares mechanical Fock states up to |6> in a vibrating crystal resonator by coupling it to a superconducting qubit and shaping microwave pulses with optimal control. The states show genuine quantum non-Gaussianity and better displacement sensitivity than an ideal three-phonon Fock state under realistic losses, which matters for quantum force sensing.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"RPN measurement model bias is the load-bearing risk: both QNG n=6 and the Fisher-information comparison are computed from fitted Pn, so an unmodeled readout artifact could overturn both central claims.","rationale":"The reader's weakest assumption is that the RPN measurement yields unbiased estimates of the phonon-number distribution. My independent stress-test reaches the same point: the two central claims (genuine QNG compatible with Fock |6> and metrological sensitivity beyond ideal Fock |3>) are both computed from Pn values extracted via model-dependent fits. This is the single most load-bearing link because a systematic error in the basis functions would propagate directly to both threshold crossings. The paper has internal consistency checks—the measured τn follow τ1/n and agree with the independently measured T1, and the Wigner function matches simulation—but these validate the loss and pulse models rather than the absolute scale of the extracted Pn. The check I propose is feasible because the same state was independently characterized via displaced-parity measurements; comparing the two reconstructions would settle whether the RPN model bias is real. I do not see a need to change the reader's conditional verdict: the concern is concrete but not yet demonstrated to be fatal, and the suggested test is the decisive next step.","tokens_in":13930,"tokens_out":6587,"duration_ms":73495,"concrete_test":"Use the displaced-parity Wigner data that the authors already acquired for the n=6 state (Fig. 1(c)): perform a maximum-likelihood reconstruction of the density matrix from the raw parity measurement, extract the phonon-number distribution, and compute the QNG functional (P6, P≥7) and the Fisher information from this independent distribution. If the independently reconstructed point lies below the n=6 QNG threshold—or if its FI does not exceed the lossy Fock |3> bound—then the RPN model bias is real and the central claims fail; if the two reconstructions agree within combined uncertainties, the concern is settled.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Both headline claims are downstream of the resonant-interaction phonon-number (RPN) reconstruction: the QNG violation for n=6 (Fig. 2(a)) and the metrological comparison against ideal Fock |3> (Fig. 3(a)) are evaluated from the fitted phonon-number distribution {Pn}. The RPN fit is model-dependent: the basis functions are generated by a master-equation simulation of the qubit-phonon interaction, and any unmodeled imperfection—qubit leakage outside {g,e}, a detuning error in the resonant interaction, a second near-degenerate HBAR mode, or a measurement-induced dephasing channel—will bias the extracted Pn. A small shift in P6 or P≥7 could move the n=6 point across the QNG threshold, because the threshold curve is tight for high n. Similarly, the FI advantage over the lossy ideal |3> state in Fig. 3(a) is not a large margin, so a redistribution of Pn could flip the conclusion. The paper notes that Wigner tomography is 'delicate' and uses RPN instead; this makes the RPN model the single point of failure. The internal consistency checks (τn=τ1/n vs. T1, and agreement of the Wigner function with simulation) validate the overall decoherence rates but do not independently verify the absolute scale of Pn, the quantity that sets both thresholds.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This manuscript reports the preparation of Fock states up to |6> in a high-overtone bulk acoustic wave resonator coupled to a superconducting qubit, using quantum optimal control. The states are characterized through resonant-interaction phonon-number (RPN) measurements, and the resulting phonon-number distributions are used to evaluate a robust genuine quantum non-Gaussianity criterion from the literature. The measured {P_n, P_{>= n+1}} points are reported to violate the QNG thresholds for n=1,...,6, and the loss tolerance of the violations is quantified. The same measured distributions are used to compute the classical Fisher information for displacement-amplitude estimation, which is compared with ideal Fock states and with ideal Fock states after an amplitude-damping channel representing measurement loss; from this comparison the authors conclude that the prepared n=6 state has displacement sensitivity better than an ideal Fock |3> state under the same lossy readout. The theoretical tools are previously established, and the experimental platform is described in detail.","tokens_in":14190,"tokens_out":23309,"duration_ms":228590,"significance":"If the central claims are fully supported, this is a significant experimental result: it demonstrates high-order Fock states in a mechanical oscillator, certified by externally defined QNG thresholds rather than by Wigner negativity alone, and it provides a Fisher-information-based metrological benchmark that connects non-Gaussianity to sensing performance. Strengths of the manuscript include the use of parameter-free QNG thresholds, an independently measured phonon T1 in the loss model, the internal consistency of the measured decay times tau_n = tau_1/n with T1, and the explicit inclusion of readout-induced relaxation in the simulations. The principal risk is that both headline claims—the n=6 QNG violation and the metrological advantage over an ideal lossy Fock |3>—are computed from the same model-dependent RPN reconstruction of the phonon-number distribution; the report therefore focuses on validating that reconstruction.","major_comments":[{"comment":"The QNG violation for n=6 in Fig. 2(a) and the displacement-sensitivity comparison in Fig. 3(a) are both computed from the phonon-number distribution {P_n} extracted by fitting RPN traces to master-equation basis functions. The manuscript does not provide a systematic-error analysis of this fit: no residuals, no calibration of the coupling g, no explicit test for qubit leakage, spurious near-degenerate modes, detuning errors, or readout assignment errors. Because the n=6 point sits close to a high-n threshold and the FI advantage over the ideal lossy |3> state is a modest margin, an unmodeled readout imperfection could shift the extracted Pn enough to affect both headline conclusions. Please add a systematic-error budget for the fitted {P_n} and an independent cross-check of the absolute scale of the distribution—for example, comparing RPN results with Wigner-tomography-based populations for at least the n=5 and n=6 states—so that the model dependence is demonstrated to be under control.","section":"Methods: RPN measurement; Supplement Fig. 8 and Sec. V"},{"comment":"The claimed QNG violations are presented graphically without the corresponding numerical values and confidence levels. The supplement states that the error bars come from fitting-function uncertainties, which are statistical only, and the threshold curves become increasingly tight at higher n. Please report the measured {P_n, P_{>= n+1}} values with their statistical and systematic uncertainties for each n and state explicitly the significance (e.g., number of standard deviations) of each threshold violation, in particular for n=6. This information is necessary to establish that the n=6 violation is not an artifact of the fit uncertainties.","section":"Fig. 2(a); Supplement Fig. 8"}],"minor_comments":[{"comment":"Notation: P_{n+1} is used both for the single Fock probability and for the tail probability P_{>= n+1}; please introduce a distinct symbol, e.g., P_{>= n+1}, and use it in the text and in the axes of Fig. 2(a).","section":"Notation throughout; Fig. 2(a)"},{"comment":"Supplement Eq. (25): the left-hand side should be Delta F_0 sqrt(T) (or an equivalent statement of the sensitivity in N/sqrt(Hz)), not Delta F_0 sqrt(1/t_cycle); as printed the equation is dimensionally inconsistent.","section":"Supplement Eq. (25)"},{"comment":"The two methods used in Fig. 2(d) to quantify QNG depth are the same amplitude-damping model expressed as a wait time and as a beamsplitter transmittance; calling them 'independent' is inaccurate.","section":"Fig. 2(d)"},{"comment":"The abstract and Fig. 3(a) should clarify that the post-loss hierarchy is the classical Fisher information for Fock-basis population measurements, not the quantum Fisher information of the lossy states; Fock-basis measurements are optimal only for the ideal pure Fock states.","section":"Abstract; Fig. 3(a)"},{"comment":"The caption of Fig. 1(a) should define what is meant by 'fidelity' for each of the four curves (optimizer target, simulated prepared state, simulated measured state, and experimentally measured fidelity).","section":"Fig. 1(a) caption"},{"comment":"Please include a numerical table of the experimental Pn values and the maximum FI values in the main text or supplement, to allow readers to reproduce Figs. 2 and 3 without reading values off the plots.","section":"Data availability in main text"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is a good fit for a high-profile quantum physics journal if the RPN-related validation is added. I do not see a fundamental flaw in the theoretical framework, but the paper's two headline claims rest on a single, model-dependent readout pipeline; the systematic-error and significance analysis are therefore essential. I would be willing to accept after a major revision that addresses these points."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nWhat you should know first: this is a real experimental step forward. The group prepared Fock states in an HBAR using optimal control, kept measured fidelity above 75% up to n=6, and certified genuine quantum non-Gaussianity at n=6 against an external, parameter-free threshold built on Pn + a Pn+1. They also go beyond the usual ideal-QFI comparison: they model the measurement as amplitude damping with an independently measured T1 and show the prepared n=6 state beats an ideal |3> Fock state under that same lossy channel. That is a more honest metrological benchmark than what most papers in this area offer.\n\nThe QNG criterion extension is useful, the tau_n = tau_1/n consistency check is reassuring, and the Wigner reconstruction agreeing with simulation is a good sanity check. I also appreciate that the QNG thresholds come from prior literature rather than being fit to the data.\n\nThe soft spot is exactly what the stress-test note flags. Both headline claims are computed from the phonon-number distribution Pn extracted by fitting RPN traces to master-equation basis functions. An unmodeled qubit leakage channel, a detuning error, a second near-degenerate mode, or a measurement-induced dephasing term could bias Pn. The n=6 QNG violation and the FI edge over the lossy |3> state are not huge margins, so a modest redistribution of Pn could move either result. The paper gives internal consistency checks, but those validate the decoherence rates, not the absolute Pn scale that sets both thresholds. There is also no raw data or code released, and the uncertainty bars on {Pn, Pn+1} appear to be fit-only, so systematic effects are not quantified. Minor point: the abstract's fidelity sentence reads as if all measured states exceed 75%, which Fig. 1(a) presumably clarifies, but it would be cleaner to state the number as measured and not simulated.\n\nThat said, I do not think these concerns are fatal. The thresholds are external, T1 is independent, and the internal physics checks behave as expected. The right fix is a frank systematic-error analysis of the RPN readout, plus data release. This paper deserves a serious referee, and I would send it out without hesitation.\n\nFor you: worth reading for the QNG criterion and the loss-adjusted metrology, but keep an eye on the RPN model as the single point of failure.","headline":"A solid cQAD advance: genuine QNG up to n=6 and a loss-adjusted metrological benchmark, but both claims sit on one model-dependent readout that deserves scrutiny.","tokens_in":14780,"tokens_out":1779,"would_cite":true,"duration_ms":20287,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper reports preparation of mechanical Fock states up to |6> with genuine quantum non-Gaussianity and a displacement sensitivity exceeding that of an ideal Fock |3> state.","keywords":["Fock states","mechanical resonator","quantum non-Gaussianity","quantum Fisher information","displacement sensing","quantum optimal control","HBAR","circuit quantum acoustodynamics"],"falsifier":"A direct check is to reconstruct the same prepared states with an independent measurement, such as displaced-parity Wigner tomography, and compare the resulting phonon number distribution and Fisher information with the RPN-based values; if the independent reconstruction disagrees with the RPN fits beyond the reported uncertainties, or if calibrating the RPN basis on known coherent states changes the extracted $P_n$ values significantly, the central claim would be falsified. A second check would be to prepare a known state (e.g., a coherent state) and verify that the RPN analysis recovers its expected distribution and Fisher information.","tokens_in":13735,"feed_emoji":"🎯","tokens_out":8397,"duration_ms":76036,"temperature":0.7,"pith_summary":"This paper reports the preparation of high-number Fock states, phonon number states of a mechanical resonator, in a high-overtone bulk acoustic wave resonator coupled to a superconducting qubit. The states are produced by microwave pulses optimized with quantum optimal control, and the paper's central claim is that the resulting states show genuine quantum non-Gaussianity up to the six-phonon level: they cannot be written as Gaussian operations (displacements and squeezing) applied to superpositions with at most five phonons. The paper further introduces a loss-adjusted metrological hierarchy based on the quantum Fisher information and shows that the experimentally realized $|6\\rangle$ state, even after energy relaxation and readout imperfections, has a displacement sensitivity exceeding that of an ideal Fock $|3\\rangle$ state. If correct, this establishes optimal control as a practical route to multiphonon non-Gaussian mechanical states and connects their non-Gaussian character directly to a sensing advantage.","feed_headline":"Six-phonon mechanical states beat ideal Fock-3 in sensing","feed_subtitle":"Optimal-control pulses create acoustic Fock states that pass non-Gaussianity tests and beat ideal Fock-3 in sensing.","key_machinery":"The argument runs on three mechanisms. The first is the resonant-interaction phonon number (RPN) measurement, which initializes the qubit in its excited state, lets it interact with the phonon mode for a variable time, and fits the resulting qubit population oscillations to master-equation basis functions to extract the phonon number distribution $P_n$. The second is the genuine quantum non-Gaussianity criterion $F_{a,n}(\\rho) = P_n + a P_{n+1}$, whose threshold $F_n(a)$ is the maximum of this functional over all states of the form a displaced and squeezed mixture of Fock states up to $|n-1\\rangle$; exceeding the threshold for any $a$ certifies genuine $n$-phonon QNG, and the use of $P_{n+1}$ makes the test robust to loss. The third is the quantum Fisher information for displacement sensing: an ideal Fock state $|n\\rangle$ gives $F_Q = 4(2n+1)$, and the paper computes the classical Fisher information from Fock-basis populations after an amplitude-damping channel, producing a loss-adjusted hierarchy that ranks the prepared states against ideal Fock states under the same decoherence.","core_discovery":"On its own terms, the paper establishes three linked results. First, quantum-optimal-control pulses of a few microseconds prepare Fock states $|1\\rangle$ through $|7\\rangle$ in the acoustic resonator, with measured fidelities above 75% for states up to $|6\\rangle$. Second, a hierarchical criterion for genuine quantum non-Gaussianity—built from the probabilities $P_n$ and $P_{n+1}$ of detecting $n$ and at least $n+1$ phonons, with thresholds optimized over displaced, squeezed mixtures of states with at most $n-1$ phonons—is violated for $n = 1,\\dots,6$, so the prepared states cannot be generated by any Gaussian transformation on states containing up to five phonon contributions. Third, when the amplitude-damping channel of the measurement is included in the comparison, the classical Fisher information of the prepared $|6\\rangle$ state for estimating a displacement amplitude exceeds that of an ideal Fock $|3\\rangle$ state subjected to the same loss, implying a quantum-enhanced force sensitivity beyond the classical limit of $63.2\\ \\mathrm{fN}/\\sqrt{\\mathrm{Hz}}$ for the device.","pith_inferences":["The loss-adjusted Fisher hierarchy is a platform-independent benchmark: any bosonic mode with a known $T_1$ can be ranked against ideal Fock states under the same damping channel, so the comparison method should transfer to microwave cavities, trapped ions, and optical systems without modification.","Because the $P_n/P_{n+1}$ QNG criterion can be violated even for states with modest fidelity and purity, it offers a practical acceptance test for non-Gaussian resources in hybrid devices, complementing Wigner-function negativity measurements.","A testable extension suggested by the method: shortening the RPN readout time (through stronger coupling or a quantum non-demolition measurement) should increase both the highest certifiable Fock number and the metrological advantage, since less amplitude damping would occur during readout.","The same optimal-control preparation could be directed at superposition states such as $|0\\rangle + |n\\rangle$ or at two-mode number-difference states, where the Fisher hierarchy would need to be recomputed for the corresponding multi-mode displacement generator."],"forward_implications":["Fock states up to $|6\\rangle$ can be prepared in a mechanical resonator with fidelity exceeding 75% using optimal-control pulses shorter than the decoherence time.","Genuine quantum non-Gaussianity is certified for $n = 1$ to $6$, so the prepared states cannot be produced by Gaussian operations on states with fewer phonons, even when the criterion allows for losses.","The prepared $|6\\rangle$ state, measured through the device's lossy readout, has a higher Fisher information for displacement amplitude than an ideal Fock $|3\\rangle$ state under the same amplitude-damping channel.","Each prepared state can tolerate a quantifiable amount of loss—measured both by simulated free evolution and by an equivalent beamsplitter—before it stops violating its QNG threshold, with higher Fock states losing certification sooner.","For this device, the sensitivity advantage translates into a quantum-enhanced force sensitivity beyond the classical limit of $63.2\\ \\mathrm{fN}/\\sqrt{\\mathrm{Hz}}$."],"supporting_citations":[{"why":"introduced the resonant-interaction phonon number measurement and prior mechanical Fock-state preparation that this work builds on.","marker":"[9]"},{"why":"founded the quantum non-Gaussianity concept used to define genuine multiphonon criteria.","marker":"[16]"},{"why":"supplied the hierarchical QNG criteria that the paper extends and applies.","marker":"[17]"},{"why":"provided the experimental threshold values for genuine n-phonon QNG based on Fock-state probabilities.","marker":"[18]"},{"why":"the GRAPE algorithm used to optimize the state-preparation pulses.","marker":"[19]"},{"why":"demonstrated the resonant-interaction phonon number measurement technique on which the RPN readout is based.","marker":"[22]"},{"why":"introduced the $P_n + a P_{n+1}$ functional that the paper adapts into its loss-robust QNG criterion.","marker":"[26]"},{"why":"the continuous-variable metrology framework showing Fock states are optimal for unknown displacement directions.","marker":"[13]"},{"why":"the supplementary material supplying the analytic amplitude-damping model and Fisher-information calculations for the loss-adjusted hierarchy.","marker":"[23]"}],"fun_headline_variants":["Six-phonon states beat ideal Fock-3 in sensing","Acoustic Fock-6 out-senses ideal Fock-3 under loss","Phonon Fock states pass non-Gaussianity, beat Fock-3","Optimal pulses create Fock-6 that surpasses ideal Fock-3","Mechanical resonator achieves genuine non-Gaussian Fock-6"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The conclusions rest on the assumption that the resonant-interaction phonon number measurement returns unbiased estimates of the phonon number distribution $P_n$; if the master-equation basis functions mis-model the readout—through an uncalibrated coupling strength, qubit leakage, or a decay channel that is not simple amplitude damping—the extracted $P_n$ values could shift enough to change the QNG certification or the comparison with ideal Fock states.","fun_headline_variants_meta":{"raw":{"variants":["Six-phonon states beat ideal Fock-3 in sensing","Acoustic Fock-6 out-senses ideal Fock-3 under loss","Phonon Fock states pass non-Gaussianity, beat Fock-3","Optimal pulses create Fock-6 that surpasses ideal Fock-3","Mechanical resonator achieves genuine non-Gaussian Fock-6"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000234,"raw_usage":{"total_tokens":1529,"prompt_tokens":1008,"completion_tokens":521,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":624,"completion_tokens_details":{"reasoning_tokens":421}},"tokens_in":624,"tokens_out":521,"duration_ms":5178,"temperature":1.0,"reasoning_tokens":421,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T23:05:41.747454+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A direct check is to reconstruct the same prepared states with an independent measurement, such as displaced-parity Wigner tomography, and compare the resulting phonon number distribution and Fisher information with the RPN-based values; if the independent reconstruction disagrees with the RPN fits beyond the reported uncertainties, or if calibrating the RPN basis on known coherent states changes the extracted $P_n$ values significantly, the central claim would be falsified. A second check would be to prepare a known state (e.g., a coherent state) and verify that the RPN analysis recovers its expected distribution and Fisher information.","supporting_citations":[{"cited_title":"Filip and L","cited_arxiv_id":null,"evidence_quote":"founded the quantum non-Gaussianity concept used to define genuine multiphonon criteria."},{"cited_title":"Lachman, I","cited_arxiv_id":null,"evidence_quote":"supplied the hierarchical QNG criteria that the paper extends and applies."},{"cited_title":"Podhora, L","cited_arxiv_id":null,"evidence_quote":"provided the experimental threshold values for genuine n-phonon QNG based on Fock-state probabilities."},{"cited_title":"Khaneja, T","cited_arxiv_id":null,"evidence_quote":"the GRAPE algorithm used to optimize the state-preparation pulses."},{"cited_title":"Hofheinz, E","cited_arxiv_id":null,"evidence_quote":"demonstrated the resonant-interaction phonon number measurement technique on which the RPN readout is based."},{"cited_title":"Provazn´ık, L","cited_arxiv_id":null,"evidence_quote":"introduced the $P_n + a P_{n+1}$ functional that the paper adapts into its loss-robust QNG criterion."},{"cited_title":"Quantum metrology with a continuous-variable system,","cited_arxiv_id":null,"evidence_quote":"the continuous-variable metrology framework showing Fock states are optimal for unknown displacement directions."}],"review_version":1}