Quantum squeezing amplification with a weak Kerr nonlinear oscillator

Domain: physics
Published: 2025-12-18
Editorial outcome: major_revision

Abstract
Quantumsqueezedstates,withbiasedquantumnoise,havebeenwidelyuti-lizedinquantumsensingandquantumerrorcorrectionapplications.How-ever,generatingandmanipulatingthesenonclassicalstateswithalargesqueezingdegreetypicallyrequiresstrongnonlinearity,whichinevitablyinducesadditionaldecoherencethatdiminishestheoverallperformance.Here,wedemonstratethegenerationandamplificationofsqueezedstatesina
superconductingmicrowavecavitywithweakKerrnonlinearity.Bysubtlyengineeringanoff-resonantmicrowavedrive,weobservecyclicdynamicsofthequantumsqueezingevolutioninadisplacedframeofthecavity.Further-more,wedeterministicallyrealizequantumsqueezingamplificationbyalternatelydisplacingtheKerroscillatorusingtheTrotterizationtechnique,achievingamaximumsqueezingdegreeof14.6dBandasqueezingrateof0.28MHz.Ourdemonstrateddisplacement-enhancedsqueezingoperationoffersahardware-efficientapproachforgeneratinglargesqueezedstates,promisingpotentialapplicationsinquantum-enhancedsensingandquantuminformationprocessing.

Introduction
Thelandscapeofquantuminformationsciencehasbeensubstantiallyenrichedbytheutilizationofnonclassicalbosonicstates,whichareessentialforcontinuousvariablequantuminformationprocessing1,2.Amongthesestates,squeezedstates3,whichexhibitreducednoiseinonequadraturebutincreasednoiseintheconjugateone,standoutfortheir remarkable ability to enhance the measurement sensitivity beyondthe standardquantum limit4,5.This enhancementis particularlycriticalinsensingapplications6,7,suchasthesearchforelusive darkmatterparticles8andtheobservationofgravitationalwaves9,10.In addition, these nonclassical squeezed states serve as essential resources to generate Gottesman-Kitaev-Preskill states11,12, catstates13,14,andsqueezedFockstates15forquantumerrorcorrectionprotocols16–19.

Overthepastfewdecades,thegenerationandmanipulationofsqueezedstateshavebeensuccessfullydemonstratedonavarietyofphysicalplatforms20,includingopticalandmicrowavephotons21–28,as well as mechanical and acoustic phonons29–33. The underlying mechanismforgeneratingthesenonclassicalstatesgenerallyinvolves engineeringanonlinearinteractionwithinthesebosons.Thepursuitof alargesqueezingtypicallynecessitatesastrongnonlinearity,which, however,inevitablyintroducesadditionaldecoherenceandlimitsthe accessibility of the large Hilbertspace of an oscillator with a linear displacementdrive.

Intherealmofquantumopticswithsuperconductingmicrowave circuits34,KerrnonlinearitycanbeachievedusingJosephsonjunctions, whichareessentialelementsfordevelopingparametricamplifiers35,36, generating itinerant squeezed microwave fields21,27, and realizing squeezed superconducting qubits37. By coupling to a strong nonlinear superconducting qubit, a harmonic oscillator in the single-photon Kerr regimehasbeendemonstratedtorealizethecollapseandrevivalof coherent states38 and to autonomously generate microwave Fock states39,40.In addition, the use of weak Kerr nonlinearitieshasrecently beenproposedformanipulatingphoton-blockadedstates41,42,withthe operationspeedenhancedbyalargedisplacement.Asimilarideawas also demonstrated to implement fast conditional operations26,43. However,despitethesetheoreticalandexperimentaladvancements, demonstratingthefastgenerationoflargesqueezedstateswithaweak Kerrnonlinearoscillatorremainsaformidableexperimentalchallenge.

Inthiswork,weexperimentallydemonstratethegenerationandamplification of squeezed states in a superconducting microwave cavity with weak Kerr nonlinearity. By subtly engineering an off-resonant microwave drive on the Kerroscillator,weexperimentallyobservethecyclicdynamicsofthesqueezingevolutioninadisplacedframe without state collapse during the evolution. In addition, we employ the Trotterization technique to realize quantum squeezing amplificationbyalternatelydisplacingtheKerroscillator,achievingamaximumsqueezingdegreeof14.6dB,which,toourknowledge,isthe largestvalueofmicrowavephotonsqueezingreportedinpreviousstudies.This technique offers an efficient approach for realizing displacement-enhanced squeezing operations to generate large squeezed states. Furthermore, we demonstrate the metrological advantages of these nonclassical states for sensing small displacements, promising potential applications in quantum-enhanced metrology.

Results
Alinearharmonicoscillatorhasaquadraticpotentialwell,whichcanbequantizedintodiscreteandequallyspacedenergylevels,asdepictedinFig.1a.Theoscillatorwouldinduceaself-Kerrnonlinearitywhencouplingtoanonlinearsuperconductingqubitincircuitquantumelectrodynamics37. This nonlinear Kerr oscillator has slightly unequally spaced energy levels with the Hamiltonian expressed as H = \frac{K}{2}a^y2a^2(assuming\hbar=1)intherotatingframeoftheoscillatorfrequency\omega_c.Here,a(a^\dagger)istheannihilation(creation)operator,andKisthestrengthoftheKerrnonlinearityoftheoscillator.UnderthisKerrHamiltonian,acoherentstatewillfirstevolveintoaslightlycrookedsqueezedstateandthenintoacompletephasecollapsestate38, asindicatedinFig.1b.

To avoid the state collapse and further improve the squeezing degree, we engineer an off-resonant microwave drive on the Kerr oscillator with a drive frequency of \omega_d, resulting in a driven Kerr oscillator

H = \Delta_d a^y a - \frac{K}{2} a^y2 a^2 + \Omega_d (a + a^y), (1)

where \Delta_d = \omega_d - \omega_c represents the frequency detuning between the oscillatorandthedrive,and\Omega_d isthestrengthofthedrive(ignoring the drive phase for simplicity without any loss of generality). This Hamiltonian represents the well-known prototypical model of a quantum Duffing oscillator, exhibiting nonequilibrium phase transitiondynamics44andnonclassicalWignernegativity45.

The quantum dynamics of the driven Kerr Hamiltonian with engineered parameters can be employed to generate two-photon squeezing(Fig.1c),whichcanbefurtheramplifiedthroughtheTrotterizationtechnique(Fig.1d).Tointuitivelyunderstandthesqueezingmechanism,wemoveintoadisplacedframewithaunitaryoperationofD(\beta)=exp(\beta a^y - \beta^* a). Then, the aforementioned driven Kerr Hamiltoniancanbetransformedinto

H_\beta = \Delta_0 a^y a - \frac{K}{2} a^y2 a^2 - \frac{K}{2} (\beta^2 a^y2 + \beta^{*2} a^2) - K\beta^y (a^y a - r) + H.C. (2)

with \Delta_0 = \Delta_d - 2K|\beta|^2 and r = \Delta_d/K + \Omega_d/K\beta - |\beta|^2 representing the photon-blockade parameter. The first line of the above equation correspondstotheHamiltonianofaKerrparametricoscillator,which promises potentialapplications inquantum computation46–49, quantum metrology50, quantum phase transition51,52, and quantum tunneling53,54.

Here, we first discuss the driven squeezing dynamics. By properly engineeringthedetuneddriveontheKerroscillator,wefindthata coherentstatenolongercollapsesduringtheevolutionandundergoes acyclicsqueezingevolution,asshowninFig.1c.Intuitively,thephoton blockade term in the displaced frame arises from the interplay between the detuned drive and the Kerr nonlinearity. The detuned drive acts as a driving force pulling the coherent state along the negativeimaginaryaxisinphasespace,whiletheKerrtermrotatesthe state anticlockwise at a rate proportional to the average photon number.Carefullybalancingthesetwoeffectspreventsstatecollapse andenablesperiodiccyclingtrajectorydynamics(seeSupplementaryFig.2).Themirror-symmetrytrajectoryaroundtherealaxisinphasespaceeffectivelycancelsthephotonblockadeeffectattheendofeach cycle,achievingthedisplacement-enhancedsqueezingoperationwith asqueezingrateK|\beta|^2,asindicatedbythetwo-photonsqueezingtermin Eq. (2). The optimal parameters of the detuned drive are extracted from numerical simulations by evaluating the fidelity between the evolvedsqueezedstateandanidealsqueezedstate|\xi>=S(\xi)|0>,where S(\xi)=exp(\xi^* a^2 - \xi a^y2) representsasqueezingoperator.Here,\xi isthesqueezingparameter.Thedetails of the numericalsimulationsarepresentedinSupplementaryNote2.

Inourexperiment,werealizeaKerrnonlinearoscillatorbydispersivelycouplingahigh-qualitysuperconductingmicrowavecavity(resonance frequency of \omega_c/2\pi=6.60GHz) to an ancillary superconductingqubit(transitionfrequencyof\omega_q/2\pi=5.28GHz).ThecavitymodehasaweakKerrnonlinearityofK/2\pi=5.83kHz(K/\omega_c <10^{-6}) andasingle-photonlifetimeof395\mu s(correspondingtoadecayrate \kappa_c/2\pi=0.40kHz), serving as the storage cavity for storing and manipulating the multiphoton squeezed states. The ancillary qubit mode,withanenergyrelaxationtimeofabout38\mu s anda Ramsey (echo)decoherencetimeof50(58)\mu s,isutilizedforcharacterizingthe generatedsqueezedstatesinthestoragecavity.Detailedinformation regardingthedeviceparametersandtheexperimentalsetupisprovidedinSupplementaryNote1.Notethat,ourdisplacement-enhanced squeezing method is also applicable even with an ultra-weak Kerr nonlinearity K < \kappa_c (seeSupplementaryFig.6).

TheexperimentalsequenceisshowninFig.2a,wheretheinitial cavityvacuumstateisfirsttransformedintoadisplacedframeusinga displacementoperationD(\beta),thenevolvesunderthedetuneddriven KerrHamiltonian,andfinallyrevertstotheoriginalframeviaareversedisplacementoperationD(-\beta).Toachievealargersqueezingdegree, we choose an optimal displacement amplitude \beta=2, and drive frequencydetuning\Delta_d/2\pi=56kHzandamplitude\Omega_d/2\pi=2.01MHz,as determinedfromnumericalsimulations(seeSupplementaryFig.3).At the end of the sequence, we employ Wigner tomography to characterizethecavitystates.Duringthefirstevolutioncyclewitha periodof2250ns,wemeasuretheWignerfunctionsofthecavitystates at various moments, which are in good agreement with numerical simulations,asindicatedinFig.2b.Themeasurementresultsreveal thatthecoherentstaterotateswithoutcollapsinginphasespaceunder the detuned driven Kerr Hamiltonian, and is compressed into an ellipticalstructureinphasespaceattheendofeachevolutioncycle, indicatingthesuccessfulgenerationofsqueezedstates.Wedisplacethecompressedcoherentstatetotheorigininphasespace,performa virtualphaserotationoperationtoeliminatetherotationangle(see Supplementary Note 4), and measure the Wigner functions of the generatedsqueezedstatesatthefinalmomentofdifferentcycles.The experimental results displayed in Fig. 2c, clearly show that as the numberofcyclesincreases,thecavitystateisgraduallycompressedin onedirectionandstretchedintheorthogonaldirection.Thisstructural feature indicates that the squeezing level of the cavity state is progressivelyamplifiedwithincreasingcycles,demonstratingtheeffectiveness of the displacement-enhanced squeezing approach. In addition,thedrivensqueezingapproachmayofferpotentialapplicabilityandflexibilityinengineeringcomplexsqueezingevolution.For example,weemploythisapproachtogeneratesqueezedmultiphoton Fock states S(\xi)|N> with N up to 6 (see Supplementary Fig. 7). The generated squeezed Fock states, with a large squeezing level and photon number, could potentially achieve higher sensitivities than either squeezed vacuum states or multiphoton-number Fock states alone,offeringsignificantadvantagesinquantum-enhancedprecision metrology.

Tofurtherenhancethesqueezinglevelswithoutrelyingonbrute-force optimizations,weproposeutilizingtheTrotterizationtechnique55to eliminatetheresidualphoton-blockadeeffectinHamiltonianEq.(2). ThisTrotterizationstrategyisbasedontheKerrsqueezingdynamics mentioned above, enabling a digital squeezing protocol that relies solelyonsimple,nativelyavailablegatesanddynamics,asshownin Fig.1d.Thekeypointofthisapproachliesintheadditionofthedriven KerrHamiltonianintwooppositedisplacementframes,H_\beta and H_{-\beta}, whichcouldcompletelyeliminatethephoton-blockadeterm,resulting inaKerrparametricoscillatorHamiltonian:

H_{KPO} = \frac{H_\beta + H_{-\beta}}{2} = \Delta_0 a^y a - \frac{K}{2} a^y2 a^2 - \frac{K}{2} (\beta^2 a^y2 + \beta^{*2} a^2). (3)

BasedontheTrotterizationformula,thetimeevolutionoftheKerr parametricoscillatorHamiltoniancanbeexpressedas

e^{-iH_{KPO} \delta t} = e^{-iH_{-\beta} \delta t/2} e^{-iH_\beta \delta t/2} + O[(\delta t)^2], (4)

inadiscretizedevolutionarytimestep\delta twithTrottererrorssuppressed toanorderofO[(\delta t)^2].ThisTrotterizationmethodalternatesthephases of the displacement frame, which effectively cancels the undesired nonlinear photon-blockade phase errors while reinforcing the two-photonsqueezingterm.Byincreasingthedisplacementamplitude\beta, thetwo-photonsqueezingtermwoulddominateintheKerrparametric oscillator Hamiltonian of Eq. (3) for achieving large squeezed states. Notethatinthisscenario,therapidswitchingbetweenopposite-signed displaced states prevents any single step of Kerr evolution from going far beyond the squeezing stages, rendering the detuned driveunnecessaryduringtheKerrevolution.Wethereforeset\Omega_d=0 to restrict the average photon numbers of the intermediate states and engineer a virtual phase shift after each evolution cycle to mitigate the Kerr-induced phase on the squeezed state (see SupplementaryNote4).

The experimental sequence to demonstrate the quantum squeezing amplification with the Trotterization method is shownin Fig.3a.Inthissequence,wealternatelytransformtheKerroscillator into different displacement frames with opposite phases. This is similartoanechoeddisplacedKerrevolutionthatwouldaverageaway theundesiredphoton-blockadetermandsuppresscavitydephasing errors (see Supplementary Note 7). To evaluate the squeezing performance of the generated squeezed states, we measure one-dimensional(1D)Wignerfunctioncutsalongtherealaxis(Im(\alpha)=0) andtheimaginaryaxis(Re(\alpha)=0),respectively,withtheexperimental resultsshowninFig.3b.AstheTrotterstepsincrease,thelinewidthof onetraceiscompressedbelow1/2,whiletheotheroneisexpanded beyond 1/2, indicating an effective squeezing ofthe quantum state. Fromaglobalfittobothtraces,weextractthesqueezingparameters|\xi| of the generated squeezed states as a function of the number of Trotter steps for various displacement amplitudes |\beta|, with the experimentalresultsshowninFig.3c.

The experimental results reveal that as the number of Trotter steps increases, the squeezing level is amplified towards a large squeezed state. TheeventualsaturationinsqueezinglevelsisattributedtoKerrnonlinearity,verifiedbynumericalsimulations.Byperformingalinearfittotheexperimentaldatawithinthelinearregion,we extractthesqueezingrateasafunctionof|\beta|^2,asshownintheinsetof Fig.3c.Asthedisplacementamplitude|\beta|increases,thesqueezingrate isenhancedaccordingly,butdeviatesfromtheidealrelationshipK|\beta|^2. ThisdeviationisprimarilyduetotheneglectofKerrevolutionduring the displacement operation and the influence of higher-order Kerr nonlinearities. When these aspects are taken into account, the numericalsimulationresultsalignwellwiththeexperimentaldata.

Furthermore,wehavemeasuredthe2DWignerfunctionsofthe generatedstatesatvariousTrottersteps,asshowninFig.3d.The2D Wignerfittingresultsdemonstrateamaximumsqueezingdegreeof 14.6dB,which is larger thanprevious demonstrations of intracavity squeezing24–26,28. Despite the gradual twisting and stretching of the squeezedstatesinphasespaceduetoKerrnonlinearity,thesestates continuetoexhibitlargeFisherinformation(FI),whichquantifiesthe maximuminformationthatcanbeextractedaboutaparameterfrom thequantumstate.

Toassessthepracticalutilityofthesegeneratedsqueezedstates,we furtherconductaquantummetrologyexperimenttomeasuresmall displacementsalongthepositionquadratureusingthesestates.Inthis experiment,weemployaparitymeasurementtomaptheinformation aboutthedisplacementontothequbitstate,withtheexperimental sequenceillustratedinFig.4a.Fromthemeasuredqubitgroundstate population P_g, the classical FI is calculated with the formula I_F(\alpha) = \frac{1}{P_g(1-P_g)} (\frac{dP_g}{d\alpha})^2 andisboundedbyquantumFII_Q=4e^{2|\xi|}foranideal squeezedstate|\xi>.TheseFIsareexplicitlylinkedtothevarianceofthe estimateddisplacementamplitudeviatheCramér-Raoinequality56

\Delta\alpha \ge \sqrt{\frac{1}{N_{meas} I_F}} \ge \sqrt{\frac{1}{N_{meas} I_Q}}, (5)

where N_{meas} representsthenumberofindependentmeasurements.

UsingtheparitymeasurementresultsfromN_{meas} experiments,we estimate a sample of displacement amplitude \alpha by leveraging the Bayesianinference(seeSupplementaryNote5).Thevarianceofthe estimateddisplacementamplitudeiscalculatedbytheAllandeviation formula \sigma_{\alpha} = \sqrt{\frac{1}{2(l-1)} \sum_{k=1}^{l-1} (\alpha_{k+1} - \alpha_k)^2} withl=10samples.InFig.4b, weplottheAllandeviationfordisplacementestimationversusN_{meas} usingthesqueezedstatesgeneratedwithdifferentTrottersteps.In addition,Fig.4cdepictsthemeasuredAllandeviationasafunctionof the number of Trotter steps with a fixed N_{meas}=3.2\times 10^4.Theresults indicate that as the number of Trotter steps increases, the Allan deviation for displacement measurement significantly decreases, highlighting the large metrological advantage of these nonclassical statesinsurpassingthestandardquantumlimit.

Methods
Characterizationofsqueezingparameters
The generated squeezed states are characterized bycomparing measuredWignerfunctionstothoseofidealsqueezedstates|\xi>.Thetheoretical 2D Wigner function for an ideal squeezed state is given by W(\alpha)= \frac{2}{\pi}exp(-2|\nu|^2), where \nu=cosh(|\xi|)\alpha^*+sinh(|\xi|)\alpha and \alpha is the complexphase-spacevariable.The measured 2DWignerfunctionsin Figs.3dand2b,carefitteddirectlywiththisexpressiontoextractthe squeezingparameter.However,measuringthefull2DWignerfunctionis time-consuming.Therefore,weemployanefficientapproachbymeasuringonly1DWignercutsalongthesqueezingquadrature(Im(\alpha)=0) andtheanti-squeezingquadrature(Re(\alpha)=0),thenperformingaglobal fittothesetwotracestoefficientlyextractthesqueezingparameter|\xi| (seeFig.3b).SeetheSupplementaryNote3fordetaileddescriptions.

ExtractionofFisherinformation
Inquantummetrologyexperiments,weemploytheFItoquantifythe precisionachievableinestimatingthedisplacementparameter\alpha using nonclassical squeezed states. The classical FI is defined as I_F(\alpha)= \sum_{\mu} \frac{1}{P_{\mu}} (\frac{\partial P_{\mu}}{\partial \alpha})^2 56, where P_{\mu} denotes the probability distribution from a set of measurement projections M_{\mu}. In our experiment, the measurementperformedontheancillaryqubityieldsonlytwooutcomes, \mu={g,e},corresponding tothe qubitinground and excited states. Consequently,theclassicalFIsimplifiestoI_F(\alpha)= \frac{1}{P_g(1-P_g)} (\frac{dP_g}{d\alpha})^2. For an ideal squeezed state |\xi>, the quantum FI is calculated by I_Q = 4( <\xi|h_{\alpha}^2 |\xi> - <\xi|h_{\alpha} |\xi>^2 ) = 4e^{2|\xi|}, with h_{\alpha}=i(a - a^\dagger) representing the interrogation Hamiltonian that encodes the displacement parameter via the unitary transformation U_{\alpha}=exp(-i\alpha h_{\alpha}) (assuming \alpha is real). These FI quantities are directly related to the variance of the estimated displacement amplitude through the Cramér-Rao inequality56: \Delta\alpha \ge \sqrt{\frac{1}{N_{meas} I_F}} \ge \sqrt{\frac{1}{N_{meas} I_Q}}.

Bayesianinference
To estimate the displacement parameter \alpha from the measured qubit population,weemploytheBayesianinferenceapproachbymaximizing theposteriorprobabilityP(\alpha|M) \propto P(\alpha) \times P(M|\alpha), where P(M|\alpha) isthe likelihoodofobservingtheancillainthegroundstate(M=g)orexcited state(M=e)under\alpha.AssumingauniformpriorprobabilityP(\alpha)over \alpha \in [0, 0.5], Bayesian estimation reduces to maximum likelihood estimation.Thedisplacementamplitudeisestimatedbymaximizingthe log-likelihood function L(M|\alpha)=ln[P(M|\alpha)]=P_g^{exp}ln[P_g(\alpha)]+(1-P_g^{exp})ln[1-P_g(\alpha)] via \alpha_{est} = argmax_{\alpha} L(M|\alpha), incorporating the measuredqubitgroundstatepopulation P_g^{exp} andfittedqubitgroundstate probabilitydistribution P_g(\alpha)(seeSupplementaryNote5).

Dataavailability
Sourcedataareprovidedwiththispaper,andmoredetaileddataare availablefromthecorrespondingauthorsuponrequest.Sourcedata areprovidedinthispaper.

Codeavailability
The code used for this study is available from the corresponding authorsuponrequest.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "Response to Reviewers #1 & #4’s Comments:\nComment 1:\nIn this article, the authors presented a scheme to squeeze vacuum and Fock states in a\nsuperconducting microwave cavity. They use the weak Kerr non-linearity of the cavity with an\noff-resonant drive. The resulting action resembles a squeezing operation, whose effective\nstrength is enhanced during the evolution by the strong displacement in phase space. To\nmitigate the effect of the unwanted term in the Hamiltonian (second line in Eq.2), the authors\nemployed Trotterization method. The experimental results in the paper show states that\nresemble highly squeezed vacuum and Fock states.\nThe positive aspects of this papers are:\n1. The method does not use the qubit in the evolution, which makes it not prone to qubit\ndecoherence.\n2. The experimental technique is simple and only uses natively available gates and dynamics.\nThus, it is easily implemented on typical bosonic cQED setups.", "reviewer_2": "I enjoyed this manuscript’s clever use of a weak Kerr nonlinearity plus an off-resonant drive to generate and amplify squeezing inside a superconducting microwave cavity, reaching 14.6 dB of intracavity squeezing. The idea of cycling a coherent state in a displaced frame to avoid collapse, then using a Trotterized echo of alternating displacement phases to reinforce two-photon squeezing while canceling unwanted Kerr rotations, is both elegant and original. It’s worth adding a brief, intuitive note on “photon blockade” (the Kerr-induced photon–photon interaction) and how the drive detuning neutralizes it at cycle endpoints, and a one-sentence plain-English description of the Trotterization, alternating displacement phases so that nonlinear phase errors cancel while squeezing terms accumulate. It would also be more informative to give T₂* or T₂ echo times in the main text instead of only the pure dephasing time. Speaking of limits, the text notes that “the squeezing degree can be further enhanced by engineering a weaker Kerr nonlinearity and a larger displacement amplitude,” but it’s unclear what the practical ceiling is; a short discussion of realistic bounds—perhaps set by calibration fidelity, drive amplitude constraints, or decoherence rates—would be a nice addition. A sentence contrasting this intracavity approach with standard JPAs or other methods would help to contrast with other squeezing work. Despite these minor points, the experiments are convincing and I recommend publication in Nature Communications.", "reviewer_3": "Referee report for “Quantum squeezing amplification with a weak Kerr nonlinear oscillator”\nThis work outlines two different approaches for generating squeezed states, both of which utilize the inherent non-linearity associated with a bosonic mode coupled to a non-linear element, a transmon qubit in this case. Squeezed states have many applications, especially in sensing due to their asymmetry of sensitivity along the two different quadratures. The generation of highly squeezed states can be difficult, and this manuscript proposed and implemented a novel protocol to accomplish record degree of squeezing that leverages a very commonplace device technology/architecture to the field of circuit QED, making the work broadly applicable to the field. Thus, I believe the experiments presented are clever and useful along with being of interest to a relatively large reader base in the field. However, I have some issues with the presentation of the work as written.\nThe first major issue is the lack of intuitive/heuristic explanation of the experiment/results. The layout of the manuscript is to propose the model in the form of a Hamiltonian, show the results and then show agreement with a numerical simulation of the model. This does not really provide a nice intuitive/heuristic picture of the underlying physics for the reader to understand. This issue is most prominent in the discussion of the driven case. The oblong trajectory shown in the cartoon in fig. 1c) and the data in fig. 2 b) is rather unique/interesting. It is not immediately clear what the heuristic reason for the shape of this trajectory is and the importance of the drive parameters. I believe providing some more intuitive explanation for this would be very useful. It seems that there should be a clear intuitive picture looking at the drive as pulling the state along the negative im(α) axis while the Kerr term is rotating the state around with a frequency proportional to the average photon number. This should also explain how the drive prevents a full rotation in phase space, leading to the squeezing effect from the Kerr before it fully evolves to the point where the squeezing effect is ruined.\nThe second major issue is the unclear relationship between the two techniques presented, the driven approach and the Trotterization technique. Currently the manuscript introduces the driven approach as the main focus, while the Trotterization technique appears to be an extension. However, the Trotterization approach appears superior as it achieves stronger and faster squeezing. Furthermore, it does not have to deal with a brute-force-optimized external drive to perform and is generally easier to understand. The manuscript would be better served by highlighting the Trotterization technique and treating the driven method as an extension instead. If there are some advantages/disadvantages to the two approaches that make them more or less useful in different applications, this should be discussed to give clear motivation for including the different approaches. This explanation/comparison can then be incorporated into a more cohesive transition between the sections of the two methods which would tie the paper together much more nicely and give clearer motivation for what is presented.\nFurthermore, there are some areas where further discussions or revisions are needed:\nIn the discussion on page 4 of the Trotterization technique, the manuscript mentions that the detuned drive is unnecessary so Ω_d=0 but then it is mentioned that the this allows one to “engineer the frequency detuning to mitigate Kerr-induced phases”. This is confusing as I don’t understand what a frequency detuning from a drive means if the drive has 0 amplitude. Again, a more intuitive explanation of the drive would be helpful to understand the differences in this technique versus the first one and would allow for more of a smooth segue in the paper between topics. It seems that the drive in the first technique is what prevents the full Kerr rotation, only allowing a part of the evolution to occur where the squeezing happens whereas in the Trotterization method, the rapid switching between opposite signed displaced states prevents any single step of Kerr evolution from going beyond just the squeezing stages but this should be thought out and more explicitly explained in the paper.\nSince all the squeezed states generated are warped to a degree, or deviated from an ideal squeezed state, it would be nice to add a note about how this would affect their use in detection applications regarding potential detriments this effect has.\nThe fit of K^(-1/3) Ω^(-2/3) is in very good agreement with the data in the supplementary, it seems that there should be a clear derivation for this. I think some more work to derive or understand this relation would be useful. At the very least, simulations with a broader parameter spectrum should be performed so the limits of applicability of this expression can be confidently explained.\nIn section IV of the supplementary, the detuning of the drive is again discussed as a tuned parameter but here it is in relation to the Trotterization technique where there is no drive and I don’t understand what the meaning of detuning has with respect to a drive with no amplitude that does not play a role in the process. Please explain this in the text.\nIn the abstract, the first line states, “Quantum squeezed states, with reduced quantum noise…”. The word “reduced” here does not seem accurate and “biased noise” seems to be a more appropriate term as the noise will be reduced in one quadrature but enhanced in another for a squeezed state.\nExcessively self-congratulatory adjectives like “ingenious” (near the end of the introduction section) is not appropriate for a scientific publication.\nIn the concluding paragraph, the authors made a strong claim without evidence that “the squeezing degree can be further enhanced by engineering a weaker Kerr nonlinearity and a larger displacement amplitude”. Weaker single-photon nonlinearity but a larger displacement does not necessarily mean weaker spurious higher-order terms. Furthermore, as the displacement amplitude increases and the average photon number becomes larger, would the increased photon loss rate cause problems for this scheme? The authors must provide much more in-depth analysis or even experimental tests to back this claim up. Otherwise, this claim should be removed.\nFinal recommendation:\nOverall, the experiments performed and the approach to squeezed state generation are very nice and innovative and would find useful applications in the field. However, the presentation of the results, as written, needs some work. At the moment I cannot recommend publication but if the above-mentioned problems are fixed, I would recommend publication.", "reviewer_4": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_5": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts."}, "author_response": "Response to Reviewers #1 & #4’s Comments:\nComment 1:\nIn this article, the authors presented a scheme to squeeze vacuum and Fock states in a\nsuperconducting microwave cavity. They use the weak Kerr non-linearity of the cavity with an\noff-resonant drive. The resulting action resembles a squeezing operation, whose effective\nstrength is enhanced during the evolution by the strong displacement in phase space. To\nmitigate the effect of the unwanted term in the Hamiltonian (second line in Eq.2), the authors\nemployed Trotterization method. The experimental results in the paper show states that\nresemble highly squeezed vacuum and Fock states.\nThe positive aspects of this papers are:\n1. The method does not use the qubit in the evolution, which makes it not prone to qubit\ndecoherence.\n2. The experimental technique is simple and only uses natively available gates and dynamics.\nThus, it is easily implemented on typical bosonic cQED setups.\nResponse:\nWe thank the Reviewer for the nice summary and positive assessments of our method,\nincluding its robustness against qubit decoherence and experimental simplicity. These novel\nfeatures have also been recognized and highlighted by other reviewers. Following the\nReviewer’s suggestions, we have included additional experiments and clarifications\nthroughout the revised manuscript. We believe these major revisions would significantly\nstrengthen our manuscript and are confident that it now meets the high standards for\npublication in Nature Communications.\nComment 2:\nThe authors only presented the degree of squeezing as a figure of merit. To us, this does not\ncapture the full picture. The states visually look at squeezed states but have many artefacts,\nas the Kerr dynamics is not the same as squeezing. So to claim that strongly squeezed states\nare produced seems to oversimplify what is going on. What about the overall quality of the\ngenerated states, i.e., the fidelity of the generated state to the target squeezed states?\nResponse:\nWe thank the Reviewer for raising this point. In our manuscript, we have shown the squeezing\ndegree as a figure of merit to quantify the quantum sensing capability of the generated\nsqueezed states. Actually, we have presented the fidelities of these states to ideal squeezed\nstates in Fig. S13 in the Supplementary Material. These fidelities are extracted from the\nreconstructed density matrices via Wigner tomography, achieving a maximum squeezing\ndegree of 14.6 dB with a fidelity exceeding 0.5. To provide a clearer picture of the generated\nstates’ quality, we have now included the state fidelity alongside the squeezing degree in Fig.\n4(d) in the main text of the revised manuscript.\nComment 3:\nThe description of the analysis is also quite vague. They mention that they perform fit to extract\nthe squeezing level. What about the fitting errors? This is particularly a good thing to show as\none can achieve high squeezing level but the state smears out in phase space (like in Fig. 4d\nM=13). What about the ani-squeezing level on these states? Do they make sense? It is not\ncomplete to put maximum squeezing 14.6 dB in the abstract without also presenting how good\nthe quality of the state is.\nResponse:\nWe appreciate the Review’s insightful feedback. In our manuscript, the fitting errors for the\nsqueezing level, obtained from both the 1D and 2D Wigner function fits, are evaluated using\n95% confidence intervals and are represented by the error bars in Fig. 3(d) and Fig. 4(c) in the\nmain text. These fits are performed globally to account for both the squeezing and anti-\nsqueezing quadratures equally.\nTo separately quantify the anti-squeezing level, we also fit individual orthogonal quadrature\ncuts of the measured Wigner function. The results, shown in the following Figure 1 [see Fig.\nS8(d) in the revised Supplementary Material], are consistent with those obtained from the\nglobal 2D Wigner fits, validating the effectiveness and reliability of the fitting methodology.\nFigure 1: Extracted squeezing and anti-squeezing parameters from individual fits as a function of\nthe number of Trotter steps. Dashed lines correspond to the squeezing parameters obtained from\nglobal 2D Wigner fits.\nThe quality of the generated states is quantified by calculating the state fidelities between the\nreconstructed density matrices and ideal squeezed states. These values are now included in\nFig. 4(d) in the main text and Fig. S13 in the Supplementary Material of the revised manuscript.\nComment 4:\nThe authors mentioned that at the end they displaces back the compressed coherent state\nback to origin and perform a virtual phase rotation to eliminate rotation angle. What is then the\nrole of the cycle, as one can stop at any point in Fig. 2b and performed the mentioned\ncorrections?\nResponse:\nWe agree with the Reviewer that the phase correction operation could, in principle, be applied\nat any point during the evolution to generate squeezed states. This approach was indeed used\nto track the state trajectory dynamics during the cyclic evolution, as shown in Fig. 2b in the\nmain text. However, to achieve optimal state fidelity, we specifically select the evolved state at\nthe end of each evolution cycle. To clarify this point, we have performed numerical simulations\nof the cyclic evolution dynamics and examined both the state fidelity and squeezing parameter\nas a function of the evolution time, as shown in the following Figure 2 [see Fig. S2(d) in the\nrevised Supplementary Material]. The results indicate that the fidelity reaches a maximum at\nthe end of each cycle, and the squeezing level is enhanced by increasing the number of cycles.\nTherefore, we choose to perform phase correction rotation on the evolved state at the end of\neach cycle to maximize the fidelity of the generated squeezed states. In response, we have\nadded an initiative explanation in the fourth paragraph on page 2 in the main text and Fig. S2(d)\nin the Supplementary Material of the revised manuscript.\nFigure 2: Numerically simulated state fidelity and squeezing parameter as a function of the\nevolution time. Dashed vertical lines correspond to the final moment of each evolution cycle.\nComment 5:\nIdeally, squeezed vacuum does not have Wigner negativity, it is simply a vacuum state with\npositive Wigner function squeezed in one quadrature. The Wigner logarithmic negativity that\nthe authors computed are caused by imperfections and does not show the main non-classical\nfeature of the squeezed state.\nResponse:\nWe thank the Reviewer for raising this point. We agree that an ideal squeezed vacuum state\nis a Gaussian state and exhibits no Wigner negativity. In our experiment, the observed Wigner\nnegativity of the generated states arises from the evolution of the Kerr parameter oscillator\nHamiltonian, as also evidenced in Ref. [Nat. Phys. 20, 1448 (2024)]. The coexistence of\nsqueezing and non-Gaussian features may be beneficial for certain quantum information\napplications. To avoid any misunderstanding, we have therefore removed the discussions of\nWigner negativity throughout the revised manuscript and focused instead on the squeezing\namplification and quantum metrology application using the generated nonclassical states.\nComment 6:\nFrom what is presented in the manuscript, there is little evidence that the states created as\nclose to ideal squeezed states. Given this, are the reported states actually useful for anything\nthat a typical squeezed vacuum is suited for? The last figure computes the FI, but it is not in\nthe context of any real tasks that uses the state (such as metrology). It does not seem\ncompelling or obvious to us that such states are useful given their pronounced artefacts and\ndeviations from ideal squeezing. To show us, the authors would need to perform a real\nexperiment using this state, which I think would require a rather major revision.\nResponse:\nWe appreciate the Reviewer’s constructive suggestion and make a major revision to perform\na quantum sensing experiment using the generated nonclassical states.\nTo evaluate how closely our generated states resemble ideal squeezed states, we have\ncalculated the state fidelities using reconstructed density matrices, as presented in Fig. S13 of\nthe Supplementary Material.\nFollowing the Reviewer’s suggestion, we have performed additional experiments to\ndemonstrate the practical utility of these generated states in quantum metrology. Specifically,\nwe have conducted a displacement sensing task in which the generated squeezed states are\nused to estimate small displacements along the position quadrature. The experimental details\nof the procedure, analysis, and results are presented in the following Figure 3 [see the newly\nadded Fig. 5 in the main text] and Section V in the Supplementary Material of the revised\nmanuscript.\nIn this experiment, we first generate the squeezed states with varying Trotter steps ( ), then\nimprint a small displacement parameter, and finally perform a parity measurement. Using\n𝑀𝑀\nBayesian inference, we extract the Allan deviations for displacement estimation as a function\nof both the number of independent measurements ( ) and Trotter steps ( ). The results\nindicate that the displacement estimation precision im\n𝑁𝑁\npmreoavses as the number of\n𝑀𝑀\nTrotter steps or\nthe number of experiments increases. We achieve a maximum metrological gain of 9 dB\nbeyond the standard quantum limit at . These experimental results confirm that our\ngenerated squeezed states remain highly useful for practical metrology applications, despite\n𝑀𝑀 = 12\nphase-space twisting and deviations from ideal squeezing. We believe these major revisions\nwould significantly strengthen our manuscript and meet the high standards for publication in\nNature Communications.\nFigure 3: Quantum metrology for measuring displacements using generated squeezed states. (a)\nExperimental sequence. (b) Extracted Allan deviation for displacement measurements as a\nfunction of the number of experiments. (c) Extracted Allan deviation as a function of the number\nof Trotter steps for generating the squeezed states.\nResponse to Reviewer #2's Comments:\nComment 1:\nI enjoyed this manuscript’s clever use of a weak Kerr nonlinearity plus an off-resonant drive to\ngenerate and amplify squeezing inside a superconducting microwave cavity, reaching 14.6 dB\nof intracavity squeezing. The idea of cycling a coherent state in a displaced frame to avoid\ncollapse, then using a Trotterized echo of alternating displacement phases to reinforce two-\nphoton squeezing while canceling unwanted Kerr rotations, is both elegant and original.\nResponse:\nWe sincerely appreciate Reviewer #2's thorough summary of our key results, including cyclic\nsqueezing dynamics and Trotterized echo squeezing amplification. We are particularly gratified\nby the recognition of our approach as “clever” and \"elegant and original\".\nComment 2:\nIt’s worth adding a brief, intuitive note on “photon blockade” (the Kerr-induced photon–photon\ninteraction) and how the drive detuning neutralizes it at cycle endpoints, and a one-sentence\nplain-English description of the Trotterization, alternating displacement phases so that\nnonlinear phase errors cancel while squeezing terms accumulate.\nResponse:\nWe thank the Reviewer for this valuable suggestion. In the displacement frame of the driven\nHamiltonian, the photon blockade term comes from the combination of the Kerr nonlinearity\nand the detuned drive, as also evidenced in Ref. [Sci. Adv. 7, eabj1916 (2021)]. By choosing\nappropriate drive strength and detuning, these two effects result in cyclic state trajectory\ndynamics, achieving a maximum squeezing fidelity at the endpoint of each cycle. Following\nthe Reviewer’s suggestion, we have added an intuitive note: “Intuitively, the photon blockade\nterm in the displaced frame arises from the interplay between the detuned drive and the Kerr\nnonlinearity. The detuned drive acts as a driving force pulling the coherent state along the\nnegative imaginary axis in phase space, while the Kerr term rotates the state anticlockwise at\na rate proportional to the average photon number. Carefully balancing these two effects\nprevents state collapse and enables periodic cycling trajectory dynamics (see Supplement).\nThe mirror-symmetry trajectory around the real axis in phase space effectively cancels the\nphoton blockade effect at the end of each cycle, achieving the displacement-enhanced\nsqueezing operation with a squeezing rate , as indicated by the two-photon squeezing\n2\nterm in Eq. (2).” in the fourth paragraph on page 2 in the main text of the revised manuscript.\n𝐾𝐾𝛽𝛽\nWe have also provided a more detailed discussion and derivation of the cyclic trajectory\ndynamics in Sections II-B and II-C of the revised Supplementary Material.\nAccording to the Reviewer's suggestion regarding the Trotterization method, we have also\nadded a one-sentence description: “This Trotterization method alternates the phases of the\ndisplacement frame, which effectively cancels the undesired nonlinear photon-blockade phase\nerrors while reinforcing the two-photon squeezing term.” in the first paragraph on page 4 in the\nmain text of the revised manuscript.\nComment 3:\nIt would also be more informative to give T * or T echo times in the main text instead of only\nthe pure dephasing time.\n₂ ₂\nResponse:\nThanks for this suggestion. We have added the Ramsey and echo decoherence times in the\nmain text of the revised manuscript.\nComment 4:\nSpeaking of limits, the text notes that “the squeezing degree can be further enhanced by\nengineering a weaker Kerr nonlinearity and a larger displacement amplitude,” but it’s unclear\nwhat the practical ceiling is; a short discussion of realistic bounds—perhaps set by calibration\nfidelity, drive amplitude constraints, or decoherence rates—would be a nice addition\nResponse:\nWe appreciate Reviewer #2's valuable suggestion. The demonstrated Trotterized squeezing\napproach indicates that the squeezing degree is enhanced by increasing the displacement\namplitudes. However, in our realistic experiment, the achievable displacement amplitudes are\nprimarily limited due to electronic hardware constraints and the breakdown of the dispersive\napproximation at high photon numbers, which would induce parasitic qubit excitations during\nthe evolution. The intrinsic Kerr nonlinearity of the cavity would also restrict the achievable\nphoton number and introduce phase-space twisting. To generate a larger squeezing degree\nwith high fidelity, one could engineer a weaker Kerr nonlinearity, allowing larger displacements\nwithout disrupting the dispersive approximation.\nIn order to clarify this point, we have performed numerical simulations, and the results are\npresented in Section VII and Fig. S13 in the revised Supplementary Material. The simulation\nresults indicate that employing a weaker Kerr nonlinearity kHz would yield a state\nfidelity exceeding 0.99 with a squeezing parameter . In response to the Reviewer’s\n𝐾𝐾/2𝜋𝜋 = 1\nsuggestion, we have revised the relevant sentence in the last paragraph to: “The squeezing\n|𝜉𝜉| ≈ 1.5\ndegree is primarily limited by the displacement drive amplitude due to electronic hardware\nconstraints and the breakdown of the dispersive approximation.” to discuss the realistic bounds\nof our squeezing approach. A more detailed error analysis is also provided in Section VII of\nthe revised Supplementary Material.\nComment 5:\nA sentence contrasting this intracavity approach with standard JPAs or other methods would\nhelp to contrast with other squeezing work.\nResponse:\nWe thank Reviewer #2 for this suggestion. In the revised manuscript, we have added a brief\ndescription, “This method distinguishes itself from previous displacement-enhanced\noperations and other squeezing approaches by eliminating decoherence errors induced by the\nancillary qubit and suppressing cavity dephasing errors through echoed displacements during\nthe evolution.” in the concluding paragraph in the main text to contrast our approach to other\nsqueezing works. A more detailed comparison and analysis is provided in Section VIII of the\nrevised Supplementary Material.\nComment 6:\nDespite these minor points, the experiments are convincing and I recommend publication in\nNature Communications.\nResponse:\nWe appreciate Reviewer #2’s recognition of our convincing experiments and are highly\nencouraged by the recommendation for publication in Nature Communications. We have\nclearly addressed all the minor points raised by the Reviewer in the revised manuscript.\nResponse to Reviewers #3 & #5's Comments:\nComment 1:\nThis work outlines two different approaches for generating squeezed states, both of which\nutilize the inherent non-linearity associated with a bosonic mode coupled to a non-linear\nelement, a transmon qubit in this case. Squeezed states have many applications, especially in\nsensing due to their asymmetry of sensitivity along the two different quadratures. The\ngeneration of highly squeezed states can be difficult, and this manuscript proposed and\nimplemented a novel protocol to accomplish record degree of squeezing that leverages a very\ncommonplace device technology/architecture to the field of circuit QED, making the work\nbroadly applicable to the field. Thus, I believe the experiments presented are clever and useful\nalong with being of interest to a relatively large reader base in the field. However, I have some\nissues with the presentation of the work as written.\nResponse:\nWe sincerely thank the Reviewer for their thorough summary and positive assessment of our\nwork. We are particularly encouraged by their recognition of the novelty and practical relevance\nof our approach, as well as its potential interest to the circuit QED community. We also\nappreciate the constructive suggestions aimed at improving the clarity and presentation of the\nmanuscript. In the following, we address each comment individually.\nComment 2:\nThe first major issue is the lack of intuitive/heuristic explanation of the experiment/results. The\nlayout of the manuscript is to propose the model in the form of a Hamiltonian, show the results\nand then show agreement with a numerical simulation of the model. This does not really\nprovide a nice intuitive/heuristic picture of the underlying physics for the reader to understand.\nThis issue is most prominent in the discussion of the driven case. The oblong trajectory shown\nin the cartoon in fig. 1c) and the data in fig. 2 b) is rather unique/interesting. It is not immediately\nclear what the heuristic reason for the shape of this trajectory is and the importance of the drive\nparameters. I believe providing some more intuitive explanation for this would be very useful.\nIt seems that there should be a clear intuitive picture looking at the drive as pulling the state\nalong the negative im(α) axis while the Kerr term is rotating the state around with a frequency\nproportional to the average photon number. This should also explain how the drive prevents a\nfull rotation in phase space, leading to the squeezing effect from the Kerr before it fully evolves\nto the point where the squeezing effect is ruined.\nResponse:\nWe appreciate the Reviewer for pointing out this issue. We agree that the cyclic squeezing\ntrajectory of our experimental results is rather unique and interesting, and providing a clearer\nintuitive picture significantly strengthens the manuscript. As suggested by the Reviewer, we\nhave added an intuitive explanation: “Intuitively, the photon blockade term in the displaced\nframe arises from the interplay between the detuned drive and the Kerr nonlinearity. The\ndetuned drive acts as a driving force pulling the coherent state along the negative imaginary\naxis in phase space, while the Kerr term rotates the state anticlockwise at a rate proportional\nto the average photon number. Carefully balancing these two effects prevents state collapse\nand enables periodic cycling trajectory dynamics (see Supplement). The mirror-symmetry\ntrajectory around the real axis in phase space effectively cancels the photon blockade effect\nat the end of each cycle, achieving the displacement-enhanced squeezing operation with a\nsqueezing rate , as indicated by the two-photon squeezing term in Eq. (2).” in the fourth\n2\nparagraph on page 2 in the main text of the revised manuscript. We have also provided a more\n𝐾𝐾𝛽𝛽\ndetailed discussion and derivation of the cyclic trajectory dynamics in Sections II-B and II-C of\nthe revised Supplementary Material.\nComment 3:\nThe second major issue is the unclear relationship between the two techniques presented, the\ndriven approach and the Trotterization technique. Currently the manuscript introduces the\ndriven approach as the main focus, while the Trotterization technique appears to be an\nextension. However, the Trotterization approach appears superior as it achieves stronger and\nfaster squeezing. Furthermore, it does not have to deal with a brute-force-optimized external\ndrive to perform and is generally easier to understand. The manuscript would be better served\nby highlighting the Trotterization technique and treating the driven method as an extension\ninstead. If there are some advantages/disadvantages to the two approaches that make them\nmore or less useful in different applications, this should be discussed to give clear motivation\nfor including the different approaches. This explanation/comparison can then be incorporated\ninto a more cohesive transition between the sections of the two methods which would tie the\npaper together much more nicely and give clearer motivation for what is presented.\nResponse:\nWe thank the Reviewer for this insightful comment. We agree with the Reviewer that the\nTrotterization approach offers significant advantages, including stronger and faster squeezing,\nand simpler and easier implementation without the need for brute-force optimization. In our\nmanuscript, the driven approach serves to illustrate the fundamental cyclic squeezing\ndynamics under the detuned driven Kerr oscillator Hamiltonian, which forms the basis for\nintroducing the digital Trotterization squeezing approach. To better motivate and contrast these\ntwo methods, we have added sentences: “The quantum dynamics of the driven Kerr\nHamiltonian with engineered parameters can be employed to generate two-photon squeezing,\nwhich can be further amplified through the Trotterization technique.” in the third paragraph on\npage 2 and “To further enhance the squeezing levels without relying on brute-force\noptimizations, we propose utilizing the Trotterization technique to eliminate the residual\nphoton-blockade effect in Hamiltonian Eq. (2). This Trotterization strategy is based on the Kerr\nsqueezing dynamics mentioned above, enabling a digital squeezing protocol that relies solely\non simple, natively available gates and dynamics.” in the third paragraph on page 3 in the\nrevised main text. These revisions provide a clearer comparison of the two strategies and\nemphasizes the practical benefits of the Trotterization technique, thus improving the transition\nbetween the two sections to enhance the overall cohesion of the manuscript.\nComment 4:\nFurthermore, there are some areas where further discussions or revisions are needed:\nIn the discussion on page 4 of the Trotterization technique, the manuscript mentions that the\ndetuned drive is unnecessary so Ω_d=0 but then it is mentioned that the this allows one to\n“engineer the frequency detuning to mitigate Kerr-induced phases”. This is confusing as I don’t\nunderstand what a frequency detuning from a drive means if the drive has 0 amplitude. Again,\na more intuitive explanation of the drive would be helpful to understand the differences in this\ntechnique versus the first one and would allow for more of a smooth segue in the paper\nbetween topics. It seems that the drive in the first technique is what prevents the full Kerr\nrotation, only allowing a part of the evolution to occur where the squeezing happens whereas\nin the Trotterization method, the rapid switching between opposite signed displaced states\nprevents any single step of Kerr evolution from going beyond just the squeezing stages but\nthis should be thought out and more explicitly explained in the paper.\nResponse:\nWe thank the Reviewer for pointing out this issue and apologize for the confusion. We agree\nwith the Reviewer that for the driven squeezing approach, the detuned drive serves as a driving\nforce to prevent the full Kerr rotation and only allows a part of evolution to occur, while for the\nTrotterization technique, the rapid switching between opposite-signed displaced states\nprevents any single step of Kerr evolution from going beyond the squeezing stages, thus\nrendering the detuned drive unnecessary during the evolution. Therefore, in our Trotter\nexperiment, we set and apply a virtual phase rotation (effectively emulating a\nfrequency detuning du\n𝛺𝛺\nrin𝑑𝑑g\n=\nth\n0\ne evolution) on the cavity after each Kerr evolution to compensate\nfor Kerr-induced phase accumulation. To avoid any confusion and clarify this point, we have\nrevised the relevant sentence in the first paragraph on page 4 to “Note that in this scenario,\nthe rapid switching between opposite-signed displaced states prevents any single step of Kerr\nevolution from going far beyond the squeezing stages, rendering the detuned drive\nunnecessary during the Kerr evolution. We therefore set to restrict the average photon\nnumbers of the intermediate states and engineer a virtual\n𝛺𝛺\np𝑑𝑑ha\n=\nse\n0\nshift after each evolution cycle\nto mitigate the Kerr-induced phase on the squeezed state (see Supplement).” in the revised\nmanuscript.\nComment 5:\nSince all the squeezed states generated are warped to a degree, or deviated from an ideal\nsqueezed state, it would be nice to add a note about how this would affect their use in detection\napplications regarding potential detriments this effect has.\nResponse:\nWe appreciate the Reviewer for this valuable suggestion. Although the generated squeezed\nstates are warped and deviated from an ideal squeezed state, they still provide promising\nmetrological advantages in quantum sensing applications.\nTo clarify this point and directly assess the practical utility of these warped squeezed states,\nwe have conducted additional experiments to sense small displacements along the position\nquadrature. The experimental details of the procedure, analysis, and results are presented in\nFig. 5 in the main text and Section V in the Supplementary Material of the revised manuscript.\nIn this experiment, we first generate the squeezed states with varying Trotter steps ( ), then\nimprint a small displacement parameter, and finally perform a parity measurement. Using\n𝑀𝑀\nBayesian inference, we extract the Allan deviations for displacement estimation as a function\nof both the number of independent measurements ( ) and the number of Trotter steps ( ).\nThe results indicate that the displacement estima\n𝑁𝑁\ntiomne apsrecision improves as the number\n𝑀𝑀\nof\nTrotter steps or the number of experiments increases. We achieve a maximum metrological\ngain of 9 dB beyond the standard quantum limit at . The difference between the\nmeasurement results and that from ideal squeezed states arises from the infidelity of the\n𝑀𝑀 = 12\ngenerated squeezed state. Despite the deviations from ideal squeezing, these experimental\nresults confirm that our generated squeezed states remain highly useful for practical metrology\napplications.\nComment 6:\nThe fit of K^(-1/3) Ω^(-2/3) is in very good agreement with the data in the supplementary, it\nseems that there should be a clear derivation for this. I think some more work to derive or\nunderstand this relation would be useful. At the very least, simulations with a broader\nparameter spectrum should be performed so the limits of applicability of this expression can\nbe confidently explained.\nResponse:\nWe thank the Reviewer for this suggestion. In response, we have added a new subsection II-\nC in the revised Supplementary Material that provides a detailed analytical deviation of the\nrelation using a semiclassical approximation of the Heisenberg\nequations of motion during the evolution. Additionally, we perform systematic numerical\n𝐾𝐾^(−1/3) 𝛺𝛺^(−2/3)\nsimulations across a broader parameter range to further verify the applicability of the\nexpression. The results, presented in Fig. S4 in the revised Supplementary Material, confirm\nthe scaling behavior and offer a comprehensive explanation of the cyclic trajectory dynamics.\nComment 7:\nIn section IV of the supplementary, the detuning of the drive is again discussed as a tuned\nparameter but here it is in relation to the Trotterization technique where there is no drive and I\ndon’t understand what the meaning of detuning has with respect to a drive with no amplitude\nthat does not play a role in the process. Please explain this in the text.\nResponse:\nWe thank the Reviewer for pointing out this and apologize for the confusion. As noted in our\nresponse to comment 4, the Trotterization approach in our experiment does not need to apply\nthe detuned drive, so we set and apply a virtual phase rotation after each Kerr\nevolution step to compensate fo\n𝛺𝛺\nr 𝑑𝑑K\n=\nerr\n0\n-induced phase accumulation. To avoid any confusion\nand clarify this point, we have removed the description of frequency detuning and changed to\nvirtual phases in the revised manuscript.\nComment 8:\nIn the abstract, the first line states, “Quantum squeezed states, with reduced quantum noise…”.\nThe word “reduced” here does not seem accurate and “biased noise” seems to be a more\nappropriate term as the noise will be reduced in one quadrature but enhanced in another for a\nsqueezed state.\nResponse:\nWe thank the Reviewer for catching this inaccuracy. As suggested, we have revised the word\n“reduced” to “biased” in the abstract of the revised manuscript.\nComment 9:\nExcessively self-congratulatory adjectives like “ingenious” (near the end of the introduction\nsection) is not appropriate for a scientific publication.\nResponse:\nWe have removed these adjectives in the revised manuscript following the Reviewer’s\nsuggestion.\nComment 10:\nIn the concluding paragraph, the authors made a strong claim without evidence that “the\nsqueezing degree can be further enhanced by engineering a weaker Kerr nonlinearity and a\nlarger displacement amplitude”. Weaker single-photon nonlinearity but a larger displacement\ndoes not necessarily mean weaker spurious higher-order terms. Furthermore, as the\ndisplacement amplitude increases and the average photon number becomes larger, would the\nincreased photon loss rate cause problems for this scheme? The authors must provide much\nmore in-depth analysis or even experimental tests to back this claim up. Otherwise, this claim\nshould be removed.\nResponse:\nWe appreciate the Reviewer's valuable comment and apologize for this confusion. In response,\nwe have removed this statement from the main text and discussed in more detail in the revised\nSupplementary Material. To further enhance the squeezing degree in our experiment, the\nprimary limitation is the maximum achievable displacement amplitude. The amplitude is\nconstrained by the electronic hardware and the breakdown of the dispersive approximation at\nhigh photon numbers, which would induce parasitic qubit excitations during the evolution. The\nintrinsic Kerr nonlinearity of the cavity would also restrict the achievable photon number and\nintroduce phase-space twisting.\nTo support the statement that a weaker Kerr nonlinearity could enable higher squeezing, we\nhave performed numerical simulations, and the results are presented in Section VII and Fig.\nS13 in the revised Supplementary Material. The simulation results indicate that employing a\nweaker Kerr nonlinearity kHz would yield a state fidelity exceeding 0.99 with a\nsqueezing parameter . In addition, we have discussed the limitations of the\n𝐾𝐾/2𝜋𝜋 = 1\nTrotterized squeezing approach in our experiment, and performed numerical simulations\n|𝜉𝜉|≈ 1.5\nincorporating the single-photon loss errors, dephasing errors, and higher-order Kerr\nnonlinearities to evaluate their influences on the state fidelity and squeezing parameter of the\ngenerated nonclassical states. A more detailed error analysis is also provided in Section VII of\nthe revised Supplementary Material.\nComment 11:\nOverall, the experiments performed and the approach to squeezed state generation are very\nnice and innovative and would find useful applications in the field. However, the presentation\nof the results, as written, needs some work. At the moment I cannot recommend publication\nbut if the above-mentioned problems are fixed, I would recommend publication.\nResponse:\nWe greatly appreciate the Reviewer for their positive assessment of our work as “very nice and\ninnovative”. We have thoroughly addressed all the comments raised by the Reviewer in the\nrevised manuscript and believe that the clarifications and additions made have significantly\nimproved the presentation. We hope the revised version now meets the high standards\nrequired for publication in Nature Communications."}, {"round": 2, "reviewer_comments": {"reviewer_2": "I think the paper can now be published", "reviewer_3": "The additions made to the text were very good overall and the manuscript has really been elevated from the first draft. The\naddition of an intuitive picture to the results was very helpful in understanding the experiment. The addition of the metrology\nexperiment gave much better motivation for the experimental technique and put the claims on much more solid footing. The\nderivation of the power law in the supplementary was very useful. A lot of wording and confusing explanations were cleared\nup, making the paper much more cohesive and understandable.\nThere are still three points we think should be addressed:\n1. As we noted earlier, the Trotterization approach seemed easier to implement, achieved better results faster and was more\nintuitive to understand, while the driven approach seemed like more of a side-show and has no real advantage. We were\ntherefore confused about the strong emphasis on the driven approach in the writing. This does not impact the overall\nsignificance/correctness of the work, but it just seems an awkward way to present the results. The authors added some\nsentences in the revision to segue these results a bit nicer, but didn’t really justify this approach of presentation. If there is\nsome distinct advantage to the driven approach, it should be made more clear.\n2. There still seems to be insufficient evidence to support the claim that the squeezing degree can be increased with a larger\ndisplacement amplitude and weaker Kerr. The authors added an explanation for this statement with a simulation showing that\nwith a larger displacement and smaller Kerr, the simulated fidelity is greater than 99%. However, it is stated that a\npredominant source of errors is the higher order Kerr terms such as 6th an 8th order terms. It would seem that these higher\norder terms scale such that they would have an even larger contribution as photon numbers get larger, making them a\ndominant error pathway in this discussion, so using a simulation with just the 4th order Kerr terms and single photon loss\nseems to be neglecting a very large error avenue for the statement that is trying to be made here. Either the higher order\nterms should be taken into account and shown concretely to still allow for higher fidelities, or this statement should be taken\nout of the text.\n3. The additional experiments and writeup on the newly added Quantum metrology section are a great addition to the paper\nand make the message much more convincing. The only point to add here is an explanation for the data point in Fig. 5b) for\nthe m=13 point at the highest N_meas would be nice, it is odd that this point is so far off the trend line and outside of any\nstatistical fluctuation based on the size of the error bars.\nWith resolution of these points, we would recommend publication.", "reviewer_4": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature\nCommunications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career\nResearchers who co-review manuscripts.", "reviewer_5": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature\nCommunications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career\nResearchers who co-review manuscripts."}, "author_response": "Response to Reviewer #1 & # 4’s Comments\nComment 1:\nWe appreciate the thorough revisions made the authors and the detailed responses to our\nprevious questions. Overall, we believe that the manuscript is significantly more compelling\ncompared to the earlier version. In particular, the new experimental results on amplitude\nestimation are very useful in supporting the claims of utility for the states they created\ndespite the imperfections/deviations from ideal squeezed states. With these revisions, we\nbelieve that the manuscript meets the standards of Nature Communications and\nrecommend publication after addressing a few additional questions listed below.\nResponse:\nWe thank the Reviewer for their positive assessment of our revised manuscript and for\nacknowledging that the new experimental results strengthen the metrological utility of the\ngenerated squeezed states. We are pleased that they find our manuscript now meets the\nstandards of Nature Communications and recommend its publication. Below, we provide\npoint-by-point responses to their additional questions.\nComment 2:\nWhy is Allan deviation used? I would think that the variance of the amplitude estimation\ncan be directly inferred from the posterior distribution from the Bayesian method. Is the\nAllan deviation a more sensitive or robust measure? Could they directly provide the\nvariances as well?\nResponse:\nWe thank the Reviewer for raising this point. In quantum sensing experiments, the Allan\ndeviation serves as a well-established metric for characterizing measurement precision,\nparticularly useful for identifying and quantifying different types of noise. While the\nBayesian posterior distribution indeed allows for the direct estimation of the variance for a\nfixed set of measurements, the Allan deviation is a more sensitive measure for\ncharacterizing low-frequency correlated noise than conventional variance, which quantifies\nonly Gaussian white noise across the entire dataset and fails to capture temporal noise\ncorrelations.\nIn response to the Reviewer’s suggestion, we have now included the standard deviation\n(square root of the variance) extracted from the Bayesian posterior distributions as a\nfunction of the number of measurements in the following Figure 1 [see Supplementary\nFigure 13].\nFigure 1: Extracted standard deviation (triangles) from the Bayesian posterior distributions for\nestimating the displacement amplitude using squeezed states generated with different Trotter\nsteps (red: M=1, green: M=13) versus the number of independent measurements .\nDashed lines represent the theoretical Cramér-Rao bound. 𝑁𝑁𝑚𝑚𝑚𝑚𝑚𝑚𝑚𝑚\nComment 3:\nSome treatments of the data are not presented, which are important for reproducibility. For\nexample, the maximum operational point alpha_opt (this deviates from zero?); the fitting\nthe Authors used to get a smooth pg for the calculation of Fisher information; Fisher vs\namplitude. A few exemplary plots would be nice.\nResponse:\nWe appreciate the Reviewer for this valuable suggestion aimed at improving reproducibility.\nTo address this, we have added detailed descriptions and exemplary plots in\nSupplementary Note 5-A of the revised manuscript.\nAs shown in Figure 2 [see Supplementary Figure 12], we now include representative\nresults for the measured qubit ground state population and the corresponding\nextracted classical Fisher information as a function of 𝑃𝑃t𝑔𝑔h(e𝛼𝛼 d)isplacement amplitude α\n(for M = 6). The data are fitted us𝐼𝐼i 𝐹𝐹 n (g𝛼𝛼 ) , which are used\n2|𝐵𝐵| 2\nfor calculating th𝑃𝑃𝑔𝑔e( 𝛼𝛼F)isher information 𝑃𝑃𝑔𝑔 (t𝛼𝛼o )e=xtr𝐴𝐴a(cetx tph(e− 2o𝑒𝑒ptima𝛼𝛼l )o)p+er𝐷𝐷ating point\nas clearly indicated 𝐼𝐼i𝐹𝐹n (𝛼𝛼th)e figure. Notably, deviates from 𝛼𝛼zopetro=,\nainrdgi cmaatixn𝛼𝛼g 𝐼𝐼 𝐹𝐹th (e𝛼𝛼 d)isplacement required to maximize the metrologic𝛼𝛼aolp stensitivity. Additionally,\nwe provide a plot of the experimentally determined as a function of the number of\nTrotter steps M. 𝛼𝛼opt\nFigure 2: Extracting the optimal displacement amplitude for quantum sensing using the\ngenerated squeezed states. (a) Measured qubit ground state populations (blue symbols),\ncorresponding fit (blue line), and extracted Fisher information (yellow line) as a function of the\ndisplacement amplitude with M=6. The optimal operating point with maximum FI is\nindicated. (b) Extracted optimal displacements as a function of the number of Trotter steps M.\n𝛼𝛼\nComment 4:\nThe pg(alpha) used Eq. S18 is theoretical or experimental fit?\nResponse:\nThe used in Eq. S18 refers to the experimental fit using the expression\n𝑃𝑃𝑔𝑔(𝛼𝛼) , where fitting parameters A and D account for exper𝑃𝑃im𝑔𝑔(e𝛼𝛼n)ta=l\n2|𝐵𝐵| 2\n𝐴𝐴impeexrpfe−ct2io𝑒𝑒ns, 𝛼𝛼and +B𝐷𝐷 corresponds to the extracted squeezing parameter (see\nSupplementary Note 5-A).\nComment 5:\nIn SM section V. A. Might be typo. Should the pg in pg(alpha)=0.5(pi/2...) be pe?\nResponse:\nWe thank the Reviewer for their careful reading. This expression is indeed correct as\nwritten. In our displacement sensing experiment, we employ the parity measurement\nthrough a sequence involving a controlled-phase gate between and\n𝑖𝑖𝑖𝑖𝜋𝜋|𝑚𝑚⟩⟨𝑚𝑚|\nqubit rotations. This sequence transforms the𝐶𝐶 𝜋𝜋in=itia𝑒𝑒l joint cavity-qub𝑋𝑋iπt /2state\n−𝑋𝑋π/2 into a final state , from which\n𝑖𝑖𝑖𝑖𝜋𝜋\n∑th𝑖𝑖e𝑐𝑐 𝑖𝑖q|u𝑛𝑛b⟩i|t𝑔𝑔 ⟩ground state popu∑l𝑖𝑖a[t𝑐𝑐io𝑖𝑖n|𝑛𝑛 i⟩s( |𝑔𝑔d⟩e+riv𝑖𝑖e|𝑒𝑒d⟩ )a/s2 −𝑖𝑖𝑒𝑒 𝑐𝑐𝑖𝑖|𝑛𝑛⟩(𝑖𝑖|𝑔𝑔⟩+ . |𝑒𝑒S⟩i)n/c2e] the Wigner\n𝑖𝑖𝑖𝑖𝑖𝑖\n1+𝑚𝑚 2\nfunction of the initial cavity state can be calculated𝑃𝑃 𝑔𝑔 a=s ∑𝑖𝑖 2 |𝑐𝑐𝑖𝑖| , we thus derive\n2 2 𝑖𝑖𝑖𝑖𝜋𝜋\n. Therefore, the expression presented in𝑊𝑊 th=e m𝜋𝜋∑an 𝑖𝑖 u|𝑐𝑐s 𝑖𝑖 c|ri𝑒𝑒pt is accurate.\n1 𝜋𝜋\n𝑃𝑃𝑔𝑔 =2[2𝑊𝑊+1]\nResponse to Reviewer #3 & # 5’s Comments\nComment 1:\nThe additions made to the text were very good overall and the manuscript has really been\nelevated from the first draft. The addition of an intuitive picture to the results was very\nhelpful in understanding the experiment. The addition of the metrology experiment gave\nmuch better motivation for the experimental technique and put the claims on much more\nsolid footing. The derivation of the power law in the supplementary was very useful. A lot\nof wording and confusing explanations were cleared up, making the paper much more\ncohesive and understandable.\nThere are still three points we think should be addressed:\nResponse:\nWe sincerely thank the Reviewer for their positive assessment of our revised manuscript\nand for recognizing the added intuitive explanations, the new metrology experiments, and\nthe improved theoretical derivations, all of which have significantly strengthened our work.\nWe have now carefully addressed the three remaining points below.\nComment 2:\nAs we noted earlier, the Trotterization approach seemed easier to implement, achieved\nbetter results faster and was more intuitive to understand, while the driven approach\nseemed like more of a side-show and has no real advantage. We were therefore confused\nabout the strong emphasis on the driven approach in the writing. This does not impact the\noverall significance/correctness of the work, but it just seems an awkward way to present\nthe results. The authors added some sentences in the revision to segue these results a bit\nnicer, but didn’t really justify this approach of presentation. If there is some distinct\nadvantage to the driven approach, it should be made more clear.\nResponse:\nWe appreciate the Reviewer’s insightful suggestion. In response, we have reorganized the\npresentation to better justify the driven approach and highlight the Trotterization method.\nThe driven approach serves as the conceptual foundation, demonstrating the fundamental\ncyclic squeezing dynamics of a Kerr nonlinear oscillator and offering potential applicability\nand flexibility in engineering complex squeezing evolution. Meanwhile, the Trotterization\nprotocol builds upon this foundation to achieve enhanced squeezing performance with\nexperimental simplicity for generating large squeezed vacuum states.\nTo better contextualize this in the manuscript, we have added a schematic diagram (Fig.\n1d) to intuitively illustrate and highlight the Trotterized squeezing technique. In addition, we\nhave removed the original Fig. 3 and corresponding discussions to the Supplementary to\navoid overemphasis on the driven squeezing approach, and added a sentence: “In addition,\nthe driven squeezing approach may offer potential applicability and flexibility in engineering\ncomplex squeezing evolution. For example, we employ this approach to generate\nsqueezed multiphoton Fock states with up to 6 (see Supplementary Figure 7).”\nin the last paragraph on page 3 in the main text of the revised manuscript.\n𝑆𝑆(𝜉𝜉)|𝑁𝑁⟩ 𝑁𝑁\nThese revisions better justify our presentation while highlighting the Trotterization\nsqueezing technique as suggested by the Reviewer.\nComment 3:\nThere still seems to be insufficient evidence to support the claim that the squeezing degree\ncan be increased with a larger displacement amplitude and weaker Kerr. The authors\nadded an explanation for this statement with a simulation showing that with a larger\ndisplacement and smaller Kerr, the simulated fidelity is greater than 99%. However, it is\nstated that a predominant source of errors is the higher order Kerr terms such as 6th an\n8th order terms. It would seem that these higher order terms scale such that they would\nhave an even larger contribution as photon numbers get larger, making them a\ndominant error pathway in this discussion, so using a simulation with just the 4th order Kerr terms\nand single photon loss seems to be neglecting a very large error avenue for the statement\nthat is trying to be made here. Either the higher order terms should be taken into account\nand shown concretely to still allow for higher fidelities, or this statement should be taken\nout of the text.\nResponse:\nWe thank the Reviewer for raising this point. We agree that higher-order Kerr nonlinearities\n(e.g., 6th- and 8th-order terms) would distort the squeezing evolution when using large\ndisplacements, and should be included in the simulation to avoid overestimating the\nachievable squeezing performance. In our simulation, we directly compare the squeezing\nfidelity considering only the Kerr term ( kHz) and including the higher-\n𝐾𝐾 †2 2\norder Kerr terms and (2𝑎𝑎 𝑎𝑎 𝐾𝐾H/2z,𝜋𝜋 =1 Hz), yielding a fidelity\n𝐾𝐾2 †3 3 𝐾𝐾3 †4 4\nof 0.99 and 0.926, 𝑎𝑎resp𝑎𝑎ectivel2y4, 𝑎𝑎afte𝑎𝑎r fi𝐾𝐾ve2/ 2T𝜋𝜋ro=tte5rizati𝐾𝐾on3/ 2st𝜋𝜋ep=s.0 .T0h2e5 results indicate that\nhigher-order Kerr nonlinearities indeed introduce a certain degree of fidelity loss. However,\nit is challenging to accurately characterize these small strengths of high-order Kerr\nnonlinearities in our experiment. Therefore, in accordance with the Reviewer's suggestion,\nwe have removed the corresponding description from the Supplementary Information to\navoid potential overclaim.\nComment 4:\nThe additional experiments and writeup on the newly added Quantum metrology section\nare a great addition to the paper and make the message much more convincing. The only\npoint to add here is an explanation for the data point in Fig. 5b) for the m=13 point at the\nhighest N_meas would be nice, it is odd that this point is so far off the trend line and outside\nof any statistical fluctuation based on the size of the error bars.\nResponse:\nWe appreciate the Reviewer for recognizing the value of the newly added metrology\nexperiments. The increase observed in the last data point of the Allan deviation for\nindicates the presence of low-frequency correlated noise in the system, as Allan\n𝑀𝑀 =\ndeviation is a well-established sensitive metric for identifying and quantifying such low-\n13\nfrequency fluctuations. This deviation from the trend line may be primarily attributed to low-\nfrequency displacement drifts in the experimental apparatus. As suggested, we have\nadded a brief note: “The increase of the last point for M = 13 may primarily be attributed to\nlow-frequency displacement drifts in the experimental apparatus.” in the caption of Fig. 4b\nin the revised manuscript."}, {"round": 3, "reviewer_comments": {"reviewer_1": "We thank the authors for their response to our remaining technical questions. All our questions are fully resolved now and we recommend publication.", "reviewer_3": "My co-reviewer and I have read the revised manuscript, and we agree that the remaining issues of the manuscript have been address adequately. We recommend publication of the manuscript in Nature Communications.", "reviewer_4": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts.", "reviewer_5": "I co-reviewed this manuscript with one of the reviewers who provided the listed reports. This is part of the Nature Communications initiative to facilitate training in peer review and to provide appropriate recognition for Early Career Researchers who co-review manuscripts."}, "author_response": ""}]

Decision letter