1

Tunable, synchronized frequency down-conversion in

magnetic lattices with defects

Marc Serra-Garcia1, Miguel Moleron1 and Chiara Daraio*2

1Department of Mechanical and Process Engineering, Swiss Federal Institute of

Technology (ETH), Zürich, Switzerland

2Engineering and Applied Science, California Institute of Technology, Pasadena,

California 91125, USA

*email: daraio@caltech.edu

2

Abstract -We study frequency conversion in nonlinearmechanical lattices, focusingon achainofmagnets as amodel system.We show thatby insertingmassdefects at suitablelocations,wecanintroducelocalizedvibrationalmodesthatnonlinearlycoupletoextendedlattice modes. The nonlinear interaction introduces an energy transfer from the high-frequency localizedmodes to a low-frequency extendedmode. This system is capable ofautonomously converting energy between highly tunable input and output frequencies,whichneednotberelatedbyintegerharmonicorsubharmonicratios.Itisalsocapableofobtainingenergyfrommultiplesourcesatdifferentfrequencieswithatunableoutputphase,due to thedefect synchronizationprovidedby theextendedmode.Our lattice isapurelymechanicalanalogofanopto-mechanicalsystem,wherethelocalizedmodesplaytheroleoftheelectromagneticfield,andtheextendedmodeplaystheroleofthemechanicaldegreeoffreedom.Introduction-Frequencyconvertingprocesseshaveapplicationsinavarietyofproblems,forexample,inobtainingdifferentwavelengthsfromafixed-frequencylaser1,harvestingenergyfromvibrationsources2andgeneratingentangledphotons3.Typically,frequencyconversionis accomplished through wave mixing4 (which requires at least two input signals with aprescribedfrequencydifference),harmonicgeneration1(whichproducesanoutputthatisamultipleoftheinputsignal)andsubharmonicbifurcations5(whichproduceanoutputthatisanintegerfractionoftheoriginalsignal).Inaddition,thesefrequencyconversionmechanismsprescribe theoutput’s signal phase,whichhinders theprocess of harvesting energy frommultiplesources.Combinationresonances6,processesthatariseinthepresenceofmultiplenonlinearly-interactingmodes, can achieve frequency down-conversion between arbitraryinputandoutputsignalsnotrelatedbyaharmonicorsubharmonicratios.Theresultinginputandoutput frequenciescanbe tunedbyadjusting themodes’ frequencies.Combinationalresonancescanbefound,forexample,invibratingbeams7,membranesandplates8.In this paper, we show that nonlinear lattices have the potential to act as frequency-convertingdevices,duetothecombinationresonancesarisingfromthenonlinearinteractionbetweenthelattice’snormalmodes.Chainsofnonlinearlyinteractingelementshavebeenstudied for decades, beginning in the FPU problem9,10 . They present a wide variety ofphenomena, including solitons11,12, band-gaps13, energy trapping14, breathers15,16,unidirectional wave propagation17, negative or extreme stiffness18, localized modes withtunableprofile19,shocksandrarefactionwaves20.Thesephenomenacanbeusedtorealizeacoustic rectifiers17, logic gates21, lenses22, filters23, vibration-attenuation24 and energyharvesting systems16. Nonlinear lattices can be implemented using a broad range ofmaterials25,geometries26andinteractions27,allowingtotunethemasses,couplingstrengthsand damping values of the particles, to optimize the performance under the requiredoperating conditions. Because of this tunability, nonlinearmetamaterials are a promisingcandidateforenergyconvertingdevices.

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Figure 1. (Color Online) Frequency-converting metamaterial concept. (a) Metamaterial design,consistingofachainofnonlinearly-interactingmagnets.Thecentralparticleofthechainisadefect,which has a lowermass. Thismagnet acts as the high-frequency input to the system. The down-convertedenergycanbeextractedfarawayfromthedefect.Inourexperiments,thedefectisdrivenbyawirecarryinganelectricalcurrent(yellowarrow).(b)Croppedimageoftheexperimentalmagnetchain,obtainedusingthesamecomputervisioncamerathatisalsousedtotrackthemagnets.Eachmagnetisenclosedina3Dprintedcase,andhasarandomspecklepatterntofacilitateitstrackingbydigitalimagecorrelation.(c)Extendedmodeofvibration.Theredhollowcircleisthedefectparticle,whilethebluesoliddotsrepresenttheotherparticles.(d)Experimentalfrequencyresponseoftheextendedmode (blue dots) and Lorentzian fit (red solid line). (e) Localizedmodeof vibration. (d)Experimentalfrequencyresponseofthelocalizedmode(bluedots)andLorentzianfit(redsolidline).Experimentalsystem–Ourexperimentalsetupconsistsofachainofmagnets27floatingonanairtable(Fig.1(a)).Eachmagnetisembeddedina3Dprintedcasethataddsanadditionalmass,withdifferentcasedesignsresultingindifferentparticlemasses(! = 0.45' forthenon-defectparticles,!() = 0.197' forthefirstdefectand!() = 0.244' fortheseconddefect).Thepresenceofdefectsintroduceslocalizedmodesaroundeachdefectparticle(Fig1(b,c)).Whenthesemodesareexcited,theresultingmotionisexponentiallylocalizedaroundthedefect.Inourexperimentalsetup,thedefectsactasinputsforthefrequency-conversionsystem.Weexcitethembypassingcurrentthroughasmallconductivewirenormaltothelengthofthechain(Fig.1(a)).Thewireisdrivenharmonicallywithasignalgenerator(Agilent33220A) amplified by an audio amplifier (Topping TP22, class D). An extended mode ofvibration (Figs. 1(c) and (d)) interacts with the localized mode to introduce frequencyconversion.Wemonitorthemotionofthemagnetsusingacomputervisioncamera(PointGreyGS3-U3-41C6C-C),withaframeratebetween40and200fpsthatallowsustoresolveall particles’motion.Weuse the softwareVIC-2D fromCorrelated Solutions, to track theparticlesanddeterminetheirtrajectory.

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Experimentalresultsforthesystemwithasingledefect–Westartbystudyingalatticeof21magnetscontainingasingedefect(!() = 0.197')inthemiddleposition(i=11).Thefirstandlastmagnetsarefixed.Wesettheexcitationfrequencytoapproximatelythesumofthedefect’sfrequency(Fig.1(f))andtheextendedmode’sfrequency(Fig.1(d)),withthegoalofexcitingacombinationalresonance(./ + .1)betweentheextendedandlocalizedmodes6.Weslowlyincreasetheexcitationamplitudeuntilathresholdisreachedandself-sustainingoscillationsdevelopfarawayfromthedefect,indicatingthetransferofenergybetweenthelocalized mode and an extended mode (Fig. 2(a)). In this regime, the defect motion ismodulated by the extended mode (Fig. 2(b)). Due to the exponential localization of thedefect’smotion,theFouriertransformofafarfromthedefectparticle’sdisplacement(Fig.2(c))doesnotrevealsignificantmotionattheinputfrequency,andconsistsalmostexclusivelyof down-converted energy. The frequency conversion efficiency, defined as the energydissipatedintheextendedmodeincomparisonwiththeenergyinputintothesystem2 =

3456578

34:6:78;<:<=, equals 10.8 ± 0.9%. This efficiency arises from our particular system

parametersandisnotanabsolutelimit.

Figure2.(ColorOnline)Experimentalresponseofthesystemunderharmonicexcitation.(a)Positionofthemagnetsasafunctionoftime.Thereddottedlinecorrespondstothedefectmagnet,whichactsastheinputtothefrequency-convertingsystem.Thegreendashedlineistakenastheoutputofthesystem.(b)Fouriertransformofthedefectmagnet’sposition,whichismodulatedattheextendedmode’sfrequency.Theverticaldottedlinerepresentstheexcitationfrequency.(c)Fouriertransformof theoutputmagnet’sposition. Thismagnet’smotionhappensprimarily at the secondextendedmode’sfrequency.

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Theoreticalmodel–Ourtheoreticalmodeldescribesthemagnetsaspointmasses.Wemodelthe interactionbetweenparticlesusinganempiricalpower-lawmodel,A B = CBD,withC = 3.378 ∙ 10G)HI!J.K)LandM = −4.316determinedexperimentally(SeeSupplementaryInformation for the fitted force-displacement curves). This model does not have astraightforwardphysicaljustificationintermsofthematerialpropertiesandthegeometryofthemagnets,butitischosenbecauseitreproducestheexperimentalforcelawwithveryhighprecisionandlowcomplexity.Usingthisforce-displacementlawwecanwritetheequationofmotionforthesystem(theindicesinparenthesesindicatethatnosummationisperformedoverthem):!(Q)S(Q) + T(Q)S(Q) − C BU V − W + SQ − SX

D

)YX3Q

+ C BU W − V + SX − SQD

Q3XYZ

= AQ [ (1)

Where!(Q)andT(Q)arethemassanddampingcoefficientoftheV-thparticle,CandMarethemagnetic force law parameters,AQ [ is the external driving force acting on the V-thparticle (whichmaybezero if theparticle isnotexternallydriven),andBU is thedistancebetweenmagnetsatrest.Whenperformingthereduced-orderanalysis,wewillassumethatBU isthesameforallmagnets.Thisisanapproximation,sincemagnetsthatarenotinthecenterofthelatticearesubjecttoasymmetriclong-rangeforces.However,wehavefoundthisapproximationtoyieldacceptableresults.Weemphasizethatourtheoreticalmodelisnot limited to nearest-neighbor interactions and takes into account the magnetic forcebetweenallpairsofmagnets.Allnumericalintegrationinthispaperisdoneusinga4thorderRunge-Kuttaalgorithmwithatimestepof1!\.Reduced modal description and frequency conversion mechanism – The mechanismresponsibleforthefrequencyconversioninourlatticebecomesmuchclearerwhenwelookattheevolutionofthesystemintermsofthenormalmodesofthelinearizedsystem.Wecanobtainthisdescriptionbyapproximatingtheforce-displacementrelationbyasecondorderTaylorseries.Whenwedothisapproximation,thesystembecomes:

]QXSX + MQXSX + Q̂XSX + _QX`SXS` = AQ [ (2)

Here,theindicesWandaaresummedoveralldegreesoffreedom,],Mdenotethemassanddampingmatrices defined conventionally, and the terms^ and_ areobtainedby Taylorexpansionoftheforcelaw:

Q̂X =BBSX

C BU V − ! + SQ − Sb D

)Yb3Q

+ C BU ! − V + Sb − SQ D

Q3bYZ

(3)

_QX` =12

BH

BSXBS`C BU V − ! + SQ − Sb D

)Yb3Q

+ C BU ! − V + Sb − SQ D

Q3bYZ

(4)

Since]issymmetricandpositive-definiteand^symmetric,wecanfindaninvertiblematrixcsuchthatcd]candcd^carebothdiagonal.Forsimplicity,weassumethatdampingisproportionaltothemassmatrixandthereforealsodiagonalizable.Inthisbasis,theequationsofmotionbecome:

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cQb]QXcXZeZ + cQbMQXcXZeZ + cQb Q̂XcXZeZ + _QX`cQbcXZc̀ feZef = Ab [ (5)

ThediagonalizedsysteminEq.(5)doesnotprovideanysignificantnumericaladvantage,since_QX`ishighlynon-localinthemodalbasis(i.e.,modesfarapartinteractasstronglyasnearbymodes).However,wecanseethemotivationforthisapproachifwelookattheexperimentalresultsinthemodalbasis(Fig.3(a)).Inthisbasis,mostofthemotionoccursinthesecondextendedmode and in the localizedmode. In fact, thesemodes hold around 90%of thesystem’senergy(Fig.3(b)).Therefore,wecanrestrictourdescriptiontothesetwomodeswithoutincurringasignificanterror.Thisreduced-orderdescriptionis:

!1e1 + T1e1 + a1 − 2ge/ S1 = Ah cos 2lmh[ (6)

!/e/ + T/e/ + a/e/ − ge1H = 0 (7)

In this description, e1 and e/ are the displacements of the localized and extended modesrespectively,Ah istheinputforce,mh istheinputfrequency,!1,T1anda1aretheeffectivemass,dampingandstiffnessofthelocalizeddefectmode,!/,T/ anda/ aretheeffectivemass,dampingandstiffnessoftheextendedmode.Thetermg = _QX`cQ/cX1c̀ 1 = _QX`cQ1cX/c̀ 1 =_QX`cQ1cX1c̀ / denotes the quadratic interaction between modes. This term can bedeterminedbyperformingaTaylorexpansionofthe interactionforce,orbyanalyzingthefrequencyresponseofthelocalandextendedmodes(SeeSupplementaryInformation).Wenote that the reduced equations of motion present an asymmetry: There is no termproportionaltoS/HinEq.(6)andthereisnotermproportionaltoS/S1inEq.(7).Thesetermsdonotappearinourlatticeduetothelocationofthedefect,buttheyarenotgenerallyzero(SeeSupplementary Information forastudyontherelationbetweennonlinear termsanddefect location).The interactionbetweenmodescanbeunderstood in the followingway:Duetononlinearity,thevibrationofthedefectmodepushesagainstitsneighbors,inawaythat is analogous to thermal expansion of a crystal18 or the optical pressure in an opto-mechanical system (Fig. 3(c)). For small amplitudes, this expansion is proportional to thesquareofthevibrationamplitude,resultinginthetermgS1Hintheextendedmodeequation.Inaddition,themotionoftheextendedmodemodulatesthedistancebetweenthedefectparticleanditsneighbors(Fig.3(d)).Thisaffectsthelocalizedmode’seffectivestiffnessandintroducestheparametrictermna1 = gS/,analogoustothemodulationoftheopticalcavitywavelength inanopto-mechanical system.This typeof interactionappears inavarietyofsystems,suchasphononmodesinsuperconductors28,andcanresultinstochasticheatengineopearation29. This reduced-order model can reproduce the experimentally-observedbehavior (Figs. 3(e) and (f)) with remarkable accuracy.We highlight that the only fittingparameterusedistheexcitationamplitude.Theparticlemass,modequalityfactorandinter-particleforcelawhaveallbeenmeasuredexperimentally.

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Figure3.(ColorOnline)Reduced-orderdescriptionofthefrequencyconversionprocess.(a)Projectionof the experimental time evolution (Fig. 2(a)) in the linear modal basis. (b) Average energy as afunctionof themodenumber.Thesystem’senergy ishighlyconcentrated inthesecondextendedmodeandthe localizeddefectmode.(c)Dynamicexpansionofthedefectmode.Whenthedefectvibrates,thenonlinearmagneticinteractionresultsinanon-zeroaverageforceactingonthedefect’sneighbors.(d)Themotionofthesecondextendedmodemodulatesthedistancebetweenthedefectparticleanditsneighbors,dynamicallytuningthedefectmodefrequency.(e)Detailoftheextendedmode and localized mode evolution, measured experimentally. (f) Theoretical prediction for theextendedandlocalizedmodeevolution,obtainedfromareduced-ordermodelconsideringonlytwomodes (Eq. (6) and (7)). The numerical simulation in panel f corresponds to a systemwith!/ =0.45', !1 = 0.232', m/ = 0.5664op, m1 = 3.913op, mh = 4.38op, Ah = 45qI, r/ =4.518,r1 = 66.62andg = 1.801 I !H,whereas = !s 2lms HandTs = !s2lms rs.Thetwo-modesystem,describedinEq.6andEq.7,isapurely-mechanicalanalogofanopto-mechanicalsystem30-32.Theextendedmodeplaystheroleofthelow-frequencymechanicalmotion,whilethelocalizedmodeplaystheroleofthehigh-frequencyelectromagneticfield.ThetermgS1Hactingontheextendedmodeplaystheroleoftheopticalpressure,whiletheterm2gS/S1 actingon the localizedmode reproduces themodulationof theFabry-Perotresonancebythemechanicaldegreeoffreedominanopto-mechanicalsystem.Thisanalogycan be made explicit by expressing the motion of the defect mode as e1 [ =1 2 [u [ vQwx + u∗ [ vGQwx]andassumingthatu [ changesslowlyandthat1 r1 ≪ 1.With these assumptions, we arrive at the following equation (a detailed derivation andcomparisonwiththefullmodelareprovidedintheSupplementaryInformation):

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!e/ + T/e/ + a/e/ = gu H

2

(8)

u + u.U2r1

− Vn e/ = Ah (9)

Here,.U is the natural frequency of the localized mode and the detuningn e/ is thedifference between the localized mode’s natural frequency and the defect’s excitationfrequency,asafunctionoftheextendedmode’sposition.Allotherparametershavethesamemeaningthantheydid inEq. (6)andEq. (7).Whilebeinganapproximation,this formhasnumericaladvantagesbynotcontainingrapidlychangingcomponentsatthefrequencyofthelocalizedmode,andnotrequiringtheevaluationoftrigonometricfunctionsfortheexcitation.Besidesnumericalreasons,thedescriptionprovidedinEq.(8)andEq.(9)isidenticaltothemodel of an opto-mechanical system30,32,33, for which there is extensive analyticalliterature32,34. This analogy provides a lucid interpretation of the frequency convertingmechanism,wherebytheself-sustainingoscillationsoftheextendedmorearetheresultofafeedback mechanism between the extended mode’s motion and the localized modeamplitude.Inthispicture,thelocalizedmodeamplitudeudependsontheextendedmodedisplacementthroughthetermn e/ .Equation(9)imposesaretardationbetweenuande/ and,asaconsequence,thetermg u Hhasaquadraturecomponent(shifted90degreesfrome/([))thatresultsinnegativedamping31.WhenthisnegativedampingexceedsthevalueofT/,thesystemdevelopsself-sustainingoscillations,whichsaturateatafinitevalueduetonon-linearity31.Multiple-defectsynchronizedfrequencyconversion–Systemscontainingmultipledefectscan present synchronized frequency conversion, where themotion ofmultiple defects isdetermined by the same extendedmode, thereby synchronizing the defect’smodulationenvelopes and resulting in the conversionof energy frommultiple input frequencies to asingle output frequency. The use of an extendedmode to synchronizemultiple resonantelementsappearsinthecontextofJosephsonjunctionarrays35,andhereweuseittoextractenergyfrommultiplesourcesofmechanicalvibrations.Ourexperimentalsystemconsistsof20magnets,withdefectparticlesinpositions7(0.244')and14(0.197').Theinitialandfinalparticlesarefixed.Asinthecasewithasingledefect,wesetanexcitationfrequencyforeachdefectequaltothedefect’sfrequencyplusanextendedmode’sfrequency.Thistimeweusethethirdextendedmodeinsteadofthesecond,becauseitpresentstworegionsofmaximumstrainat the twodefect’spositons.Aswedid in the singledefect case,we increasebothdefect’sexcitationamplitudessimultaneously,untilweobserveself-sustainingoscillationsfarfromthedefect.Figure4ashowsthetrajectoriesofthemagnetsintheself-sustainingregime.Wecalculatetheenergytransferbetweenbothdefectsandtheextendedmode,byutilizingtheempiricalforce-displacementrelationandthedefect’strajectories,andweobservethatbothdefectsarecontributingenergytotheextendedmodewithapower(c =< AS/ >=<gS1HS/ >)of16.9 ± 1.5~�and25.8 ± 4.0~�respectively,indicatingsuccessfulextractionofenergyfrommultiplesources.Thefrequencyconversionefficiencyis20.5 ± 10.4%.Asinthepreviouscase,themotionofthedefectspresentssidebandsindicatingamodulationbytheextendedmode(Fig.4(b)).Farfromthedefect,themotiontakesplaceexclusivelyattheextendedmode’sfrequency,asrequiredforsuccessfulfrequencyconversionoperation.

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Figure4.(ColorOnline)Synchronizedfrequencyconversion.(a)Positionofthemagnetsasafunctionof time.Theyellowdotted line (particle7)andthereddotted-dashed line (particle14)aredefectmagnets that act as the high-frequency inputs of the system. The green dashed line is the lowfrequency output. (b) Fourier transform of the defects’ positions, which are modulated at theextendedmode’sfrequency.Theverticaldottedlinerepresentstheexcitationfrequency.(c)Fouriertransform of the output magnet’s position. This magnet’s motion happens primarily at the thirdextendedmode’sfrequency.As in the case with a single defect, expressing themagnet’s trajectories in terms of thelattice’slinearnormalmodesrevealsthatthemotion(Fig.5(a))andtheenergy(Fig.5(b))areprimarilyconcentratedinanextendedmode(Fig.5(c),left)andinthetwolocalizedmodes,theprofilesofwhicharedepictedinFig.5(c).Thisenergyconcentrationallowsustoformulatea reduced-order description following the sameprocedure as in the systemwith a singledefect.Theresultingsystemofequationshastheform:

!1)S1) + T1)S1) + a1) − 2g)S/ S1) = Ah) cos 2lmhH[ (10)

!1HS1H + T1HS1H + a1H − 2gHS/ S1H = AhH cos 2lmhH[ (11)

!/S/ + T/S/ + a/S/ − g)S1)H − gHS1HH = 0 (12)

ThemodelinEqs.(10)-(12)iscapableofqualitativelypredictingtheevolutionofthemodes(Fig.5(d)and5(e)),butunder-estimatestheoutputamplituderelativetotheexperiments.Weattributethisdifferencetouncertaintyinthesystem’sresonancefrequenciesandqualityfactors.Thisissuggestedbythedifferencebetweentheoryandexperimentintheextendedmode’s frequencies (See Supplementary Information) and in the phase of the localizedmode’svibration.

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Figure5.(ColorOnline)Reduced-orderdescriptionofthesynchronizedfrequencyconversion.(a)Timeevolutionofthemagnetsintermsofthelineareigenmodebasis.(b)Energydistributionineachnormalmode.Theenergyisconcentratedinthethirdextendedmodeandinthetwolocalizeddefectmodes.(c)Modeprofilesofthethreemostrelevanteigenmodes.(d)Experimentaltimeevolutionofthethirdextendedmodee/ andthetwolocalizedmodese1)ande1Hasafunctionoftime.(e)Theoreticalpredictionforthetimeevolutionoftheeigenmodes.Thetheoreticalpredictionshavebeenobtainedusinga3-DOFreducedordermodel.Thenumericalparametersusedinpanel(e)are:!/ = 0.45',!1) = 0.2318',!1H = 0.2915',m/ = 0.7494op,m1) = 3.404op,m1H = 3.063op,mh) = 4.1op,mhH = 3.81op, Ah) = AhH = 42qI, r/ = 12.27, r1) = 39.3, r1H = 60.27, g) = −2.4293I !H,gH = 2.5761 I !H.Outputphasetunability– Inourlattice,theoutputsignal’sphaseisnotprescribedbytheinputsandcanbedynamicallytunedwhilethesystemisoperating.Thisoffersanopportunityto synchronize multiple devices, create passive and tunable phased arrays or transferinformation by modulating the output signal’s phase. We theoretically demonstrate thisoutputphasetunabilityinFig.6(a)-(c).Thephasemodulationisaccomplishedbyperturbing

thelastparticlefollowingaGaussianprofilegivenbySZ [ = CÄvG ÅÇÅÉ 7

7Ñ7 (Fig.6(a)),whereCÄdenotesthemaximumperturbationamplitude,[Uisthemomentwheretheperturbationis applied andÖ represents thewidth of the perturbation.We choose aGaussian profile

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becauseitishighlylocalizedinbothtimeandfrequencydomains.Applyingthisperturbationresultsinachangeintheoutputsignalphase,thatpersistslongaftertheperturbationhasvanished. Figure 6(b) shows the extended mode’s displacement 1790 seconds after aperturbation, for different perturbation amplitudes. In this calculation, the perturbationwidthÖ = 30\ ismuchsmallerthanthewaittime,ensuringthatnoeffectremainsbythetime the results are obtained. Two remarkable observations shall bemade regarding thephasetunabilityshowninFig.6(c):Firstly,thistunabilitycoversthewholerange(0° − 360°).Secondly, the perturbation-induced phase shift persists for a period of time that ismuchlongerthananyofthesystem’stimeconstants,sincethephasedoesnotchangesignificantlyif we wait an additional 1000\ until [ = 2790\. In the experimental system, this phasestabilitywouldbelimitedbythepresenceofexternalnoisesourcesandBrownianmotion.

Figure6.(ColorOnline)Theoreticalinvestigationofphaseandfrequencytunability.(a)Phasetuningscheme. The output signal’s phase is tuned bymoving the last particle (SZ) following a Gaussianprofile. (b) Extended mode signal 2790 seconds after the phase-shifting perturbation has beeneffected.ThelinescorrespondtoperturbationswithCUequalto0mm(blue,solid),6.2562mm(red,dotted) and 6.5917 mm (yellow, dashed). (c) Output phase as a function of the maximumdisplacementofthephase-adjustingperturbation.Thebluesolidlineismeasured1790secondsaftertheperturbation,while the circles aremeasured1000 seconds after the firstmeasurement, 2790seconds after the perturbation’s peak. Panelsb and c have been obtained by integrating the fullequation of motion (Eq. (1)) with BU = 16.3!!, !Q,Qà)) = 0.45', !)) = 0.197', TQ,Qà)) =306.83 qI\ !, T)) = 42.62 qI\ !, AQ,Qà)) = 0I,A)) = Ah sin 2lmQ[, Ah = 48.45qI and mh =4.38op.Theforce-lawparametersareasdescribedinthetheoreticalmodelsection.(d)Frequencydown-conversionratio(top)andinputandoutputfrequencies(bottom)asafunctionofthemassratiobetweenthedefectandextendedmodes.Theseplotshavebeenobtainedbykeepingtheextendedmode’smassconstantandmodifyingthedefect’smass.(e)Frequencydown-conversionratio(top)andinputandoutputfrequencies(bottom)asafunctionoftheextendedmodemass,whilekeeping

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themodalmassratio!/ !1constant.Inthissection,allparametersexceptthemassesareidenticaltothoseinFig.3(f).Tunability–A remarkable featureofour frequency-convertingsystem is thepossibilityoftuningtheinputandoutputfrequenciesoverabroadrange,bothduringthedesignphaseanddynamicallyoncethesystemhasalreadybeenbuilt.Figure(6)theoreticallyexplorestherelationship between the input and output frequencies and the modal masses. We firstexploretheeffectofthemassratiobyalteringthemassofthedefectmodewithoutalteringthatoftheextendedmode(Fig.6(a)).Thisresultsinachangeintheoptimalinputfrequencywithoutasignificanteffectontheoutputfrequency.Wethenproceedtoalterthemassesofthe localized and extended mode simultaneously (Fig. 6(b)). This affects both input andoutputfrequencies,whilemaintainingthedown-conversionratioconstant.Intheconversionratiocalculation,weidentifytheoptimalinputfrequencybysweepingtheinputbetweentheresonance frequencyof the localizedmode and the resonance frequencyof the localizedmode plus twice the resonance frequency of the extended mode, and finding the inputfrequencythatresultsinthehighestenergytransfer.Inadditiontotheparticle’smass,therearemanyunexploredavenuesfortuningthefrequencyconversionratio.Examplesincludethestaticcompressionappliedonthechain,themagnets’strengthandgeometryandtheapplicationofanexternalmagneticfields36.Inaddition,modern3Dprintedmaterialsallowus to engineer nonlinear inter-particle interactions37 beyond these offered by magneticsystems.Conclusionsandoutlook–Wehavedemonstratedthat latticescomposedofmagneticallyinteractingparticleswithdefectsarecapableofconvertingenergyfromhighfrequenciestolower frequencies, which need not be linked by harmonic/subharmonic relations. This ispossiblethroughthenonlinearcouplingbetweenextendedandlocalizednormalmodes.Suchfrequency-convertinglattice,analogoustoopto-mechanicalsystems,ishighlytunableinbothfrequencyandphase,andcanextractenergyfrommultiplesignalsatdifferentfrequenciestogenerate a single-component output. This work may motivate the design of innovativenonlinearmetamaterialsanddeviceswithtunableenergyconversioncapabilities.References1. P.A.Franken,A.E.Hill,C.W.PetersandG.Weinreich,PhysicalReviewLetters7(4),118-119(1961).2. S.-M.JungandK.-S.Yun,AppliedPhysicsLetters96(11),111906(2010).3. J.G.Rarity,P.R.Tapster,E.Jakeman,T.Larchuk,R.A.Campos,M.C.TeichandB.E.A.Saleh,PhysicalReviewLetters65(11),1348-1351(1990).4. R.H.Stolen, J. E.BjorkholmandA.Ashkin,AppliedPhysics Letters24 (7),308-310(1974).5. Y.C.Chen,H.G.WinfulandJ.M.Liu,AppliedPhysicsLetters47(3),208-210(1985).6. A.H.Nayfeh,Nonlinearoscillations/AliHasanNayfeh,DeanT.Mook. (Wiley,NewYork,1979).7. P.MalatkarandA.H.Nayfeh,NonlinearDynamics31(2),225-242(2003).8. K.OhandA.H.Nayfeh,NonlinearDynamics11(2),143-169(1996).9. J.P.E.Fermi,S.Ulam,M.Tsingou,1955.10. J.Ford,PhysicsReports213(5),271-310(1992).

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Supplementary Information: Tunable, synchronized

frequency down-conversion in magnetic lattices with

defects

Marc Serra-Garcia1, Miguel Moleron1 and Chiara Daraio*2

1Department of Mechanical and Process Engineering, Swiss Federal Institute of

Technology (ETH), Zürich, Switzerland

2Engineering and Applied Science, California Institute of Technology, Pasadena,

California 91125, USA

*email: daraio@caltech.edu

Force-displacementrelationWeexperimentallydeterminetheparametersfortheinteractionforcebymeasuringtheforce-displacementrelationandfittingitusingapowerlawfunctionoftheform! = #$% . Themeasurements and fitted curve are plotted in Fig. S1. Through thisprocedure, we obtain values of# = 3.378 ± 0.751/012.345 and 6 = −4.316 ±0.0460.

FIG S1: Magnetic force-displacement relation. The blue circles are the experimentalmeasurementsandtheredlinerepresentsthepower-lawfit.

DerivationoftheoptomechanicalequationsofmotionHere we show that the equations governing the reduced-order dynamics of thefrequency converting lattice (Eq. 6-7 in themain paper) can be approximated bythose describing an optomechanical system, under realistic assumptions. Theequationsthatwewanttoapproximateare:

1:;: + =:;: + >: − 2@;A ;: = !B cos FG Eq.S1

1A;A + =A;A + >A;A − @;:H = 0 Eq.S2

Westartourapproximationbyexpressingthedisplacementofthelocalizedmodeasa harmonic function, with a slowly-changing amplitude and phase given by thecomplexfunctionI G :

;: =12I G JKLM + IJNKLM Eq.S3

ThisallowsustorewriteEq.S1as:

1:OP

OMP4HI G JKLM + I G JNKLM + QRLSR

TR

OOM

4H I G JKLM + I G JNKLM + Eq.S4

>: − 2@;A4H[I G JKLM + I G JNKLM] = 4

H[!WJKLM + !WJNKLM]

ForequationS4tohold,termsmultipliedbyJKLMmustbeidenticalonbothsidesoftheequation(TakingtermsmultipliedbyJNKLMwouldyieldan identicalcondition.Thisidentityresultsin:

1:XH

XGH I G JKLM +

1:FW:Y:

XXG I G JKLM + >: − 2@;A I G JKLM = !WJKLM Eq.S5

ByevaluatingthederivativeandbothsidesbyJKLMweobtain:

1: −FHI G + 2ZFI G + I G +1:FW:Y:

ZFI G + I G

+ >: − 2@$A I G = !W

Eq.S6

Nowwemakeour firstassumption:That the localizedmodeamplitudeandphaseareslowlychanging.ThisallowsustoneglectthesecondderivativeofI inEq.S6,resultingin:

1: −FHI G + 2ZFI G +1:FW:Y:

ZFI G + I G + >: − 2@$A I G = !B Eq.S7

Whichcanberegroupedas:

FW:Y:

+ 2ZF I G +>: − 2@;A

1:− FH +

FW:Y:

ZF I G =!B1: Eq.S8

SinceY: isa largevalue(40-60inourexperimentalsetup)andFW: isofthesameorderthanF,wecanneglectthecontributionofFW: Y:intherighthandside.Thisresultsin:

I G +−Z2F

>: − 2@;A1:

− FH +FW:2Y:

I G =!B

2F1: Eq.S9

Wenotethattheterm >: − 2@$A 1:isthesquareofthelocalizedmode’sangularfrequency,asafunctionofthedisplacementoftheextendedmode.Wecallthisterm

FW: $AHtodistinguishitfromthenaturalfrequencyofthelocalizedmodewhen

the extended mode is at rest ($A = 0), which we termed FW: . By defining thedetuning [ $A = FW: $A − F as the difference between the natural frequencyFW: $A andtheexcitationfrequency,weobtain:

I G +−Z

2(FW: ;A − [ ;A )FW: ;A

H− FW: ;A − [ ;A

H

+FW:2Y:

I G =!B

2F1:

Eq.S10

ExpandingthesquaresinthelefthandsideofEquation10,weobtain:

I G +Z([ ;A − 2FW: ;A )[ ;A

2(FW: ;A − [ ;A )+FW:2Y:

I G =!B

2F1: Eq.S11

Since[ ≪ FW: $A we can neglect the additive term[ $A in the numerator anddenominator:

I G +Z(−2FW: ;A )[ ;A

2(FW: ;A )+FW:2Y:

I G =!B

2F1: Eq.S12

Whichcanbesimplifiedtotheequationforaclassicaloptomechancialsystem:

I G +FW:2Y:

− Z[ ;A I G =!B

2F1: Eq.S13

Wenowexamine theequation for theextendedmode,byreplacing;: G (Eq.S3)intoEq.S2.:

1A;A + =A;A + >A;A −@4IHJKHLM + IJNKHLM + 2II = 0 Eq.S14

By neglecting the rapidly varying degrees of freedom at 2ω, we arrive at theequation:

1A;A + =A;A + >A;A −@2I H = 0 Eq.S15

EquationsS13andS15correspond to theoptomechanicalmodelpresented in themainpaper.

Tuningthenonlinearparametersbymodifyingthedefect’slocationWecantunethenonlinearparametersinourreduced-ordermodelbychangingthedefect location.Themost general equationofmotion for the system, truncated tocontainonlysecond-orderterms,isgivenby:

1:;: + =:;: + >:;: − _:;:H − 2@A;:;A − @:;AH = !B cos FG Eq.S16

1A;A + =A;A + >A;A − _A;AH − 2@:;:;A − @A;:H = 0 Eq.S17

FigureS2presentsthenonlinearparametersasafunctionofthedefectlocation.Theselected defect locations maximize the @A coupling, while ensuring that all othernonlinearparametersaresmall.Thepointofmaximal@A correspondstotheregionwherethemode’sstrain`K = $Ka4 − $K ismaximal,resultinginthehighestchangeinthedefect-neighbordistanceduringthemotionoftheextendedmode.

FIG S2:Nonlinear parameters as a function of the defect location. (a) Here, the extendedmodeisthesecondextendedmodeofthelattice,correspondingtothesingle-defectsystemin the main paper. (b) The extended mode is the third extended mode of the lattice,correspondingtothetwo-defectssysteminthemainpaper.Inbothpanels,thedottedlinerepresentstheexperimentallightdefectlocation.Inpanelb,thedashedlinerepresentstheheavydefectlocation.

Comparisonbetweenfull,reducedandoptomechanicalsystemHerewepresentacomparisonbetweenthesystem’sevolutionpredictedbythefullmodel(Eq.XXinthemainpaper),thetwo-modereducedordermodel(Eq.1inthemainpaper)andtheoptomechanicalmodel(Eq.6andEq.7inthemainpaper).

FIG S3: Comparison of full and reduced models. a Time evolution of the localized andextended modes calculated using the full system simulation. The modal description hasbeenobtainedbyprojecting the trajectories into themodalbasis.bModal timeevolutioncalculated using the two-mode reduced order model. cModal time evolution calculatedusingtheoptomechanicalmodel.TheG = 0pointhasbeenselectedindependently ineachsimulation,inordertopresentaconsistentphase.We observe that the threemodels produce similar predictions. This allows us toconcludethatareduced-ordermodellingapproachprovidesagoodapproximation,andthatournonlinearlatticeaccuratelymimicsthedynamicsofanoptomechanicalsystem.

Determinationofthenaturalfrequenciesinthetwo-defectsystemHerewediscussthedeterminationoftheresonancefrequenciesandqualityfactorsfor the third extended mode and the two localized modes, used in the sectionMultiple-defectsyncronizedfrequencyconversionofthemainpaper.Thefrequencyisdetermined by exciting themodes using a variable frequency signal.Wemonitoreach particle’smotion and project it into the theoretically-predictedmodal basis.TheamplitudeisdeterminedbycalculatingtheRMSvalueofthemodalcoordinateaftersubtractingtheaverage.WethenfitthefrequencyresponseusingaLorentzianfunctiontoobtainthemode’sfrequencyandqualityfactor.

FIG S4: Fitting of the two-defect system parameters. a Frequency response of the thirdextended mode. b Frequency response of the first localized mode (Centered around thedefect with mass 1b4 = 0.197d. c Frequency response of the second localized mode(Centeredaroundthedefectwithmass1b4 = 0.244d

ExtendedmodeFrequency 0.7494 ± 0.0197efQualityfactor 12.27 ± 6.24

FirstLocalizedmodeFrequency 3.404 ± 0.004efQualityfactor 39.30 ± 3.35

SecondlocalizedmodeFrequency 3.063 ± 0.004efQualityfactor 60.27 ± 10.34

Table1:Two-defectsystemmodelparameters.

DeterminationofthenonlinearconstantfromthefrequencyresponseInallofourpaper’ssimulations,thenonlinearparameter@hasbeendeterminedbyperforming a Taylor expansion of the magnetic force-displacement relationpresentedinFig.S1.Insomecircumstances(Forexample,inmicroscopicsystems)itmay not be possible to accurately measure the interaction potential. Here wecalculatethenonlinearparameter@fromthefrequencyresponsecurves(Fig.S5a),by simultaneously monitoring the displacement of the extended mode (Fig. S5b)duringthefrequencyresponsecharacterization.Theequationofmotionfortheextendedmodeisgivenby:

1A;A + =A;A + >A;A − @;:H = 0 Eq.S18

Forexcitationamplitudesbelowtheself-oscillationthreshold,;:followsaharmonicmotionwithconstantamplitude.Undertheseconditions,;A cannotfollowtherapidchanges of ;:H and reacts only to its average value. Since the displacement of ;A duringthefrequencyresponsemeasurementisquasistatic,wecanneglecttheterms1A;A and=A;A .Thisresultsintheequation:

< ;A >=@>A

< ;:H >

Eq.S19

FigureS5cpresentstheextendedmodedisplacementasafunctionofthelocalizedmodeamplitude.By fitting this relationusingaquadraticpolynomial,weobtainanonlinear coefficient @ = 1.79 ± 0.56 0 1H which compares extremely well withthe value @ = 1.81 ± 0.42 0 1H obtained from the experimental force-displacementrelation.

FIGS5:Experimentaldeterminationofthenonlinearparameter@.aFrequencyresponseofthe localized mode. b Displacement of the extended mode as a function of the localizedmode excitation frequency, measured simultaneously with panel a. c Experimentalrelationship between localized mode amplitude and extended mode static displacement(Crosses),andpolynomialfit(Redline).