FLUCTUATIONSPECTRAOFMESOSCOPICVIBRATIONALSYSTEMSByYaxingZhangADISSERTATIONSubmittedtoMichiganStateUniversityinpartialoftherequirementsforthedegreeofPhysicsŒDoctorofPhilosophy2016ABSTRACTFLUCTUATIONSPECTRAOFMESOSCOPICVIBRATIONALSYSTEMSByYaxingZhangWestudythespectraofinlinearandnonlinearvibrationalsystems.Fluctuationsplayamajorroleinmesoscopicsystemsexploredinnanomechanics,cavityandcircuitquantumelec-trodynamics,andJosephsonjunctionbasedsystemstomentionbutafew.Wethatimportantinsightsintothenatureofthecanbegainedbyinvestigatingthesystemdynamicsinthepresenceofperiodicdriving.Thisisbecausetheinterplayofthedrivingandleadstopronouncedspectralfeatures.Ourpredictionsarecorrobaratedbymeasurementsonacarbonnanotuberesonatorwhichshowthatthetheoryallowsonebothtorevealandtocharacterizefrequencyinavibrationalsystem,aswellastodeterminethedecayratewithoutring-downmeasurements.Ourresultsbearonthegeneralareaofdecoherenceofmesoscopicoscillatorsandalsoontheclassicalproblemsofresonanceandlightscatteringbyoscillators.Animportantandpoorlyunderstoodmechanismofinmesoscopicsystemsisthedispersivemodecoupling.Thiscouplingisinherentessentiallytoallmesoscopicsystems.Itcomesfromthenonlinearinteractionbetweenvibrationalmodeswithnon-resonatingfrequencies.Weconsiderthepowerspectrumofoneofthesemodes.Thermalofthemodesnonlinearlycoupledtoitleadtoofthemodefrequencyandthustothebroadeningofitsspectrum.However,thecoupling-inducedbroadeningispartlymaskedbythespectralbroadeningduetothemodedecay.Weshowthattheeffectofthemodecouplingcanbeandcharacterizedusingthechangeofthespectrumbyresonantdriving.Thetheoreticalanalysisiscomplicatedbythefactthatthedispersive-couplinginducedarenon-Gaussian.Wedevelopapath-integralmethodofaveragingovertheandobtainthepowerspectruminanexplicitform.Theshapeofthespectrumdependsontheinterrelationbetweenthecouplingstrengthandthedecayratesofthemodesinvolved,providingameansofcharacterizingthesemodesevenwheretheycannotbedirectlyaccessed.Theanalysisisextendedtothecaseofcouplingtomanymodeswhich,becauseofthecumulativeeffect,canbecomeeffectivelystrong.Wealsothepowerspectrumofadrivenmodewherethemodehasinternalnonlinearity.Unexpectedly,foradrivenmode,thepowerspectradominatedbytheintra-andinter-modenonlinearitiesarequalitativelydifferent.Theanalyticalresultsareinexcellentagreementwiththenumericalsimulations.Ofinterestforphysicsandbiophysicsareoverdampedmesoscopicandmicroscopicsystems.Inertialeffectsplaynoroleintheirdynamics.Weshowthatwheresuchsystemsareperiodicallydriven,alongwiththeconventionaldelta-peakatthedrivingfrequencytheirpowerspectradisplayextrafeatures.Thesecanbepeaksordipswithheightquadraticinthedrivingamplitude,forweakdriving.Thepeaks/dipsaregenerallylocatedatzerofrequencyandatthedrivingfrequency.Theshapeandintensityofthespectrasensitivelydependontheparametersofthesystemdynamics.Toillustratethissensitivityandthegeneralityoftheeffect,westudythreetypesofsystems:anoverdampedBrownianparticle(e.g.,anopticallytrappedparticle),atwo-statesystemthatswitchesbetweenthestatesatrandom,andanoisythresholddetector.Theanalyticalresultsareinexcellentagreementwithnumericalsimulations.CopyrightbyYAXINGZHANG2016ACKNOWLEDGEMENTSFirstofall,Iwouldliketothankmyadvisorandteacher,Dr.MarkDykman.IwasluckyenoughtomeetMark,wholatertaughtmethefundamentalsofdoingscience,andledmetodiscoverandenjoythebeautyofphysicsworld.Inadditiontotheinvaluableknowledgethathehastaughtme,MarkalwaysencouragesmetogoaheadanddothecalculationwhenIwasintimidatedbytheproblemintheplace.Markalsotaughtmetheimportanceofinterpretingthemathematicalresultsinaphysicaltermandputtingtheminaphysicalpicture.Onthepersonalmatters,MarkhasalwaysbeenverycaringandconsiderateforwhichIgreatlyappreciate.IwouldalsoliketothankmyThesisCommitteemembers,Dr.NormanBirge,ScottPratt,CarlSchmidt,andSteveShaw,whohavebeenenthusiasticaboutmyworkandhelpfulwiththeircomments.Particularly,Normanhasalwaysbeencriticalandraisingquestionsduringmypresen-tationsforwhichIgreatlyappreciate.SpecialthanksgotoScottwhohasbeenverysupportiveonacademicandpersonalmattersduringmyyearsatEastLansing.Infact,withoutencouragementfromScott,Iprobablywouldnotstartmyadventureasatheorist.IwouldliketothankCarl,fromwhomIlearnedthebasicsofthequantumtheoryanddiagrammatictechniquewhichIfrominmyeverydayresearch.AsaclosecollaboratorofSteve,IenjoyedandlearnedalotonmechanicalresonatorsandnonlineardynamicswhichIwillcontinuetoworkoninmyfuturecareer.ManythanksgotomycolleaguesandofatMSU,DongLiu,JuanAtalaya,PavelPulunin,andKirillMoskovtsev,withwhomIenjoyeddiscussingphysicsandeverythingelse.SpecialthanksgotoJuan,whohasbeenreallypatientandhelpfulinansweringmyquestionsinresearch.IwouldalsoliketothankthestaffatthePhysicsandAstronomyDepartment,whomademystayatMSUsmoothandpleasant.Lastbutnotleast,Iwouldliketothankmyfamily.Icannotimaginespendingtheseveyearsvwithoutmywife,Jun,bymysidewhoisalwayspatient,supportive,andcaringtowardsme.Iwouldliketothankmyparentsandparents-in-lawwhoarealsoverypatienttomeandalwaysteachmehowtobeagoodperson.Iwouldalsoliketothankmydaugher,Audrey,whoteachesmebeingcuriousisimportant.viTABLEOFCONTENTSLISTOFFIGURES.......................................ixCHAPTER1INTRODUCTION...............................1CHAPTER2INTERPLAYOFDRIVINGANDFREQUENCYNOISEINTHESPEC-TRAOFVIBRATIONALSYSTEMS.....................62.1Introduction......................................62.2Powerspectrumofweaklydrivensystems......................82.2.1Generalexpression..............................82.2.2Spectrumofadrivenharmonicoscillatorwithfrequency....102.3Oscillatorpowerspectruminthelimitingcases...................122.3.1Weakfrequencynoise.............................122.3.2Narrow-bandfrequencynoise........................122.3.3Broadbandfrequencynoise..........................132.3.4Gaussianfrequencynoise...........................142.3.5Theweak-noisecondition...........................152.3.6Susceptibilitywithweaklyfrequency..............162.4Theareaofthedriving-inducedspectralpeak....................162.4.1Scalingofthedriving-inducedpowerspectrum...............182.5Experimentsoncarbonnanotubevibrationalsystem.................202.6Conclusion......................................23CHAPTER3SPECTRALEFFECTSOFDISPERSIVEMODECOUPLINGINMESO-SCOPICSYSTEMS..............................253.1Introduction......................................253.1.1Thestructureofthechapter..........................283.2Driving-inducedpartofthepowerspectrum.....................293.3Equationsofmotionfortheslowvariables......................303.3.1Stochasticequationsforslowvariables....................313.4Thedriving-inducedspectrumFF(w)fordispersivecoupling............323.5Averagingoverthefrequencynoisefordispersivecoupling.............343.5.1Findingthedeterminant............................363.5.2Theaveragesusceptibility..........................373.5.3Theaverageoftheproductofthesusceptibilities..............373.5.4Thetransfer-matrixtypeconstruction....................383.5.5Alternativepath-integralapproachtoaveragingoverfrequencynoise....393.6Discussionofresults.................................413.6.1ThespectrumFF(w)inthelimitingcases..................423.6.1.1Weakfrequencynoise.......................423.6.1.2Broad-bandfrequencynoise....................43vii3.6.1.3Narrow-bandfrequencynoise...................443.6.2EvolutionofFF(w)withthevaryingbandwidthandstrengthofthefre-quencynoise.................................453.6.3EffectonFF(w)ofthedetuningofthedrivingfrequency..........483.6.4Theareaofthedrivinginducedpowerspectrum...............493.7Dispersivecouplingtoseveralmodes.........................513.7.1Anintermediatenumberofmodes:weakandeffectivelystrongcoupling.533.7.1.1Thedriving-inducedspectrum...................543.7.1.2Driving-inducedspectrumforaneffectivelystrongdispersivecouplingtoalargenumberofmodes...............563.8Powerspectrumofadrivennonlinearoscillator...................573.8.1Weaknonlinearity...............................593.8.2Largedetuningofthedrivingfrequency................593.8.3Numericalsimulations............................603.9Conclusions......................................62CHAPTER4FLUCTUATIONSPECTRAOFDRIVENOVERDAMPEDNONLIN-EARSYSTEMS................................654.1Introduction......................................654.1.1Qualitativepicture..............................664.2Generalformulation..................................684.3PowerspectrumofadrivenBrownianparticle....................704.3.1MethodofMoments.............................704.3.2Powerspectrumforcomparativelylargedrivingfrequency.........724.4Powerspectrumofadriventwo-statesystem.....................744.4.1Themodel:modulatedswitchingrates....................744.4.2Kineticequationanditsgeneralsolution...................764.4.3Thedriving-inducedpartofthepowerspectrum...............774.5Thresholddetector...................................804.6Formulationintermsofsusceptiblities..................844.6.1Fluctuatingsusceptibilityofathresholddetector...............854.7Conclusions......................................86CHAPTER5CONCLUSIONS................................895.1Outlook........................................91BIBLIOGRAPHY........................................92viiiLISTOFFIGURESFigure2.1Top:sketchesofthepowerspectraofadrivenlinearoscillatorF(w).Panels(a)and(b)refertolargeandsmallcorrelationtimeofthefrequencynoisetccomparedtotheoscillatorrelaxationtimetr,respectively,i.e.,tonarrow-andbroad-bandfrequencynoise.Theblue(lower)lineshowsthespectrumofthermalintheabsenceofdriving;itiscenteredattheoscillatoreigenfrequencyw0=hwosc(t)i.Inthepresenceofdrivingthereisaddedad-peakatthedrivingfrequencywF.Thegreenareasshowthespectralfeaturesfromtheinterplayofthedrivingandofwosc(t).Bottompanels:wosc(t)fortc˛tr(a)andtc˝tr(b)........................7Figure2.2ThepowerspectrumoftheoscillatorwithaGaussianfrequencynoisewiththespectrumX(W)=2Dl2=(l2+W2).ThenoiseintensityisD=G=2.Panelsaandb:thefullspectrum.ThecolorcodingisthesameasinFig.2.1,F2=16G2=20kBT.Panelc:thedriving-inducedterm.Thesolidlinesanddotsshowtheanalytictheoryandsimulations;theconsecutivecurvesareshiftedby0.25alongtheordinate.........................15Figure2.3ThescaledareaŸSF=8G2w20SFofthedriving-inducedpeakintheoscilla-torpowerspectrumasafunctionofthefrequencynoiseparameters.ThedatarefertoGaussianfrequencynoisewiththepowerspectrumX(W)=2Dl2=(l2+W2)..................................18Figure2.4AFMimageofa4-mm-longnanotubebeforeremovingthesiliconoxide(top)andschematicofthedevice(bottom)........................21Figure2.5(a)ThepowerspectrumofthecurrentdI(t)throughadrivencar-bonnanotube.Themeasurementbandwidthis4.7Hz.Theeigenfrequencyofthestudiedxuralmodeis6.3MHz.Thedrivingfrequencyis100Hzbelowtheresonancefrequency.Thebluelinereferstothepowerspectrumwithoutdriving;thegreenareashowsthedriving-inducedspectralchange.Thischangeisseparatedintothebroadpeak(darkergreen),narrowpeak(lightergreen),andadelta-spikeatthemodulationfrequency.Thisspikelieswithin3bins,withinourexperimentalresolution,andisrepresentedbytheblackverticallines.Theseparationofthebroadandnarrowpeaksisdonebythestraightlinethatinterpolatesthebroadpeak.Showninthelowerpanelsisthedependenceofthelightergreenarea(b),thedarkergreenarea(c),andtheareaunderthed-peak(d)onthesquaredamplitudeofthemodulatinggatevoltage;asexpectedfromthetheory,itisclosetolinear...........22ixFigure2.6Narrowbandfrequencynoisespectrum.ItisobtainedbythebroadparthdI2ibroad(w)oftheexperimentalspectruminFig.2.5atoaLorentzian,andthenbysubtractingthisfromtheexperimentalspectrum.Theredlineisato1=f1=2,wheref=jwwFj=2p.....................23Figure3.1Thescaleddriving-inducedpartofthepowerspectrumofthedrivenmodedispersivelycoupledtoanothermode,whichwecallthed-mode.Thermalofthed-modeleadtofrequencyofthedrivenmode.Panels(a)to(d)showthechangeofthespectrumwiththevaryingratioGd=Gofthedecayratesofthed-modeandthedrivenmode.Thescaledstrength(standarddeviation)ofthefrequencynoiseisadGd=G=1.ThespectrumFF(w)isscaledusingthenoise-freesusceptibilityc0(wF),Eq.(3.40),ŸFF=4GFF=jc0(wF)j2.Thesolidlinesandthedotsshowtheanalyticaltheoryandthenumericalsimulations,respectively.......................47Figure3.2Theevolutionofthedriving-inducedpartofthepowerspectrumwiththevaryingstrengthofthefrequencynoiseduetodispersivecoupling.Curve1to3refertothescaledstandarddeviationofthenoiseadGd=G=0:5,2.5,and12.5,respectively.Theratioofthenoisebandwidthtothedecayrateofthedrivenmodeis2Gd=G=1.Thescaleddetuningofthedrivingfre-quencyfromtheeigenfrequencyofthedrivenmodeisdwF=G=5.Thespectrumisscaledusingthenoise-freesusceptibilityc0(wF),Eq.(3.40),ŸFF=4GFF=jc0(wF)j2.Thecurves1and3areadditionallyscaledbyfac-tors3.15and1.3,respectively,sothatthepeaksnearwFhavethesameheight.TheinsetshowsthespectrumF0(w)intheabsenceofdrivingforthesamevaluesofthefrequencynoisestrengthadGd=Gasinthemainpanel.Thesolidlinesandthedotsshowtheanalyticaltheoryandthesimulations,respectively.....................................49Figure3.3Theevolutionofthedriving-inducedpartofthepowerspectrumwiththevaryingdetuningofthedrivingfrequencywF.Thescaledstrengthofthefre-quencynoiseinducedbythedispersivecouplingisadGd=G=2:5.Theratioofthenoisebandwidthtothedecayrateofthedrivenmodeis2Gd=G=1.Thespectrumisscaledusingthenoise-freesusceptibilityc0(wF),Eq.(3.40),ŸFF=4GFF=jc0(wF)j2.Thesolidlinesandthedotsshowtheanalyticalthe-oryandthesimulations,respectively........................50Figure3.4Theareaofthedriving-inducedpartofthepowerspectrumasafunctionofadfordifferentratioofthefrequencynoisebandwidthtothedecayrateofthedrivenmode2Gd=G.Thered(solid),blue(dashed),andgreen(dotted)linesrefertoGd=G=10,2,and0.1,respectively.TherelativedetuningofthedrivingfrequencyisdwF=G=5.Inpanel(a),theareaSFisscaledusingthenoise-freesusceptibilityc0(wF),Eq.(3.40),ŸSF=2SF=pjc0(wF)j2.Inpanel(b),SFisscaledbytheareaofthedpeakinthepowerspectrumofthedrivenmode,Sd=pjc(wF)j2=2.........................52xFigure3.5Thedriving-inducedpartofthepowerspectrumofanonlinearoscillatorforlargedetuningofthedrivingfrequency,dwF=Dw=40.Thesolidcurvesandthedotsshowtheanalyticalexpressionsandtheresultsofsimulations,respectively.Thevaluesofthenonlinearityparameterandthescaleddrivingstrengthforthecurves1to3are,respectively,aDw=G=0.125,1.25,and5,andb3gF2=32w30dw3F=0.016,0.004,and0.004.TheinsetshowsthechangeofthepowerspectrumintheabsenceofdrivingwithvaryingDw=G...61Figure3.6Thedriving-inducedpartofthespectrumofanonlinearoscillatorforsmalldetuningofthedrivingfrequency.Thesolidcurve(red)showstheanalyticalresultsforFF(w)forsmallDw=Gforthesameparametersasthedottedcurve1.Thedotsshowtheresultsofsimulations.Thescaledvaluesofthenonlinearityparameter,thedetuning,andthedrivingstrengthonthecurves1and2are,respectively,aDw=G=0.05,and1.25,dwF=G=0.5and5,andb3gF2=32w30(dwF)3=0.64and0.01.Theinsetshowsthefullspectrumfortheparametersofcurve2(bluedots,simulations);thespectrumwithoutdrivingforthesameDw=Gisshownbythesolidline(analytical)and(green)dotsontopofthisline,whichareobtainedbysimulations............62Figure4.1Sketchofapotentialofanonlinearsystemnearthepotentialminimum.Be-causeoftheinterplayofnonlinearityandthecurvatureofthepotentialTheseareshownasthesmearingofthesolidline,whichrepresentsthepotentialintheabsenceof.....67Figure4.2ScaleddrivinginducedtermsinthepowerspectrumofanoverdampedBrow-nianparticlemovinginthequarticpotentialU(q)givenbyEq.(4.6),ŸFF(w)=102k2FF(w)=2D.Panels(a),(b),and(c)refertothescaledcubicnonlin-earityb2D=k3=0:002andquarticnonlinearitygD=k2=0.0006,0.00147,and0.002,respectively.TheblackdotsandredsolidcurvescorrespondtothenumericalsimulationsandEq.(4.12).ThescaleddrivingfrequencyiswF=k=5andthedrivingstrengthiskF2=w2FD=20.Forthisdrivingstrengthandthenoiseintensity,thesimulationresultsinpanels(b)and(c)de-viatefromthetheoreticalcurve.Thedeviationdecreasesforweakerdriving.Thisisseenfromthesimulationdatainpanel(b)thatrefertokF2=w2FD=5(bluetriangles)and1.25(greensquares).Thecorrespondingspectraarescaledupbyfactors4and16,respectively.....................73xiFigure4.3Thedrivinginducedtermsinthepowerspectrumofthetwo-statesystemfortheratiooftheswitchingratesW21=W12=7=3.ThescaleddrivingfrequencyandamplitudearewF=W+=5andFa12=W12=1:Onthethicksolid(red),dot-dashed(black),long-dashed(blue),short-dashed(green),andthinsolid(purple)linestheratioa21=a12is7/3,7/6,0,7=6,and7=3.TheverticallineatwFshowsthepositionofthed-peakatwF.Theareasofthed-peaksfordifferenta21=a12aregivenbytheheightsoftheverticalsegments.Theheightsarecountedofffromthelinestothesymbolsofthesamecolor,i.e.,tothecircle,triangle,andopenandfullsquare,intheorderofdecreasinga21=a12;thereisnosymbolfora21=a12=7=3asthereisnod-peakinthiscase.Theinsetshowsthefullspectrumwith(red)andwithout(black)drivingfora21=a12=7=3.Thecurvesandthedotsshowtheanalyticaltheoryandthesimulations,respectively............................79Figure4.4Powerspectrumofthethresholddetector.(a):Thefullpowerspectrum;thescaledfrequencyandtheintensityofthedrivingarewF=2pk=100andF2k=D=0:0025.Thescaledthresholdish(k=D)1=2=0:5.Inset:thespectrumnearthedrivingfrequency.Thedeltapeakhasbeensubtracted.Thecurvesandblackdotsrefertothetheoryandsimulations,respectively.(b):Thelow-frequencypartofthedriving-inducedterminthepowerspectrumforwF=k=50asgivenbyEq.(4.28).Thesolid(black),long-dashed(red),short-dashed(blue)anddot-dashed(green)curvescorrespondtothescaledvalueofthethresholdh(k=D)1=2=0:1;0:8;1:2,and2.Inset:thespectrumnearthedrivingfrequency,wF=k=50......................83xiiCHAPTER1INTRODUCTIONMesoscopicvibrationalsystems(oscillators)haveattractedmuchinterestinrecentyears,includingnanomechanicalresonators[1],optomechanicalsystems[2],andsuperconductingcavitymodes[3].Thesesystemsareusuallyweaklycoupledtotheenvironment,thereforetheyhaveverysmalldecayrate,muchsmallerthantheirvibrationfrequency.WhataccompaniesthecouplingtotheenvironmentisThemesoscopicnatureofthesesystemsistwo-fold:ontheonehand,duetotheirsmallsize(typicallyofnano/microscale),thesesystemsusuallyexperiencecompara-tivelylargequantumandclassicalOntheotherhand,thesystemscanbeindividuallyaccessedandmanipulated,thusallowingtostudytheirwithoutperformingensembleaveraging.Inaword,tostudythedynamicsofthesesystems,itiscrucialtounderstandtheationsinthesystems:howtomeasurethem,wheretheycomefromandhowtheyaffectthesystemdynamics.Anotherfeatureofmesoscopicvibrationalsystemsisthattheyhaverelativelystrongnonlin-earity[4].Asiswellknown,frequencyofanonlinearoscillatordependsonitsamplitude.Theeffectofnonlinearitybecomesalreadystrongwhenthenonlinearity-inducedfrequencyshiftiscomparabletothedecayrateofoscillators.Foroscillatorswithlowdecayratesthishappenswellbeforetheconventionalstronglynonlineareffects,suchasdynamicalchaos,forexample,comeintoplay.Therearevariousmechanismsofnonlinearity.Someareintrinsicinthesystems,andsomecomefromnonlinearcouplingtotheexternaldegreesoffreedom.Fornanomechanicalresonators,thenonlinearitycancomefromtheintrinsicphononnonlinearity,orphonon-phononscatteringprocess.Theyareparticularlyimportantforresonatorsofsmallsize,forexample,avibratingnanobeam.Thenonlinearitycanalsocomefromnonlinearcouplingtoexternalelectricthatisusedtodrivetheresonator.Forsuperconductingcircuits,thedynamicsoftheemployedJosephsonjunctionsisintrinsicallynonlinear.1Astandardtooltocharacterizemesoscopicvibrationalsystemsisspectroscopy.Itcanbetrans-spectrumforacavitymode,orthepowerspectrumofthedisplacementofme-chanicalresonators.Quiteoften,thespectrumismodeledasaLorentzianwhosewidthisthoughttobegivenbythedecayrateoftheoscillators.However,thereareothermechanismsofspectralbroadening.Oneofthemisinthevibrationalfrequencies,thatis,thesystemeigen-frequencyissubjecttoarandomperturbationintime.Forananomechanicalresonator,frequencycanresultfromattachmentordetachmentofmoleculesontheresonator,chargetuationsinthesubstrate,ordispersivecouplingbetweendifferentvibrationalmodes,etc..Foracavitymode,itcancomefrominthedielectricconstant.ThenonlinearityinthesystemsalsoleadstospectralbroadeningviaconvertingamplitudeofvibrationstofrequencyTheconvolutedeffectsofdifferentspectralbroadeningmechanismsmaketheshapeofspectrallinecomplicated,andgenerallynon-Lorentzianandasymmetric.Inordertoquantifydifferentsourcesofandnonlinearityinthesystem,itisimportanttobeabletoidentifyandcharacterizetheireffectsonspectralbroadening.Thisisthecentraltopicofthethesis.Theeffectsoffrequencynoiseonspectralbroadeninghavebeenobservedindifferentmeso-scopicvibrationalsystems.Tonameafew,Sansaetal.[5]showedthatfrequencyplayacrucialroleinsiliconnanoresonatorsbasedonaAllenvarianceanalysis,yetthesourceoftuationslargelyremainunknown.Barnardetal.[6]showedthatincarbonnanotuberesonator,thefrequencynoisethatcomesfrommode-modecouplingaccountsformostoftheobservedspectrallinewidthatroomtemperature;Miaoetal.[7]observedthesameeffectingrapheneresonator.Insuperconductivitycavity,thefrequencynoise/phasenoisewasmeasuredviahomodynedetection[8,9],andwasattributedtocouplingbetweenthecavityandtwo-levelinthecavitywallsorthesubstrate.Torevealandcharacterizefrequencyremainsachallenge.Thepreviouslymen-tionedAllenvarianceanalysisgivesinformaitonaboutfrequencynoiseinanarrowband,notthewholespectrum.Othermethodssuchashomodynedetection,anddirectobservationofthespec-2trallinedonotprovideadirectprobeoffrequencynoise,andareoftenmixedbyothersourcesofInthisthesis,weproposeamethodtoidentifyandcharacterizefrequency-noiseandnonlin-earity-inducedspectralbroadening.Themethodisbasedonapplyinganearresonantdrivingtotheoscillator,andanalyzingtheresultingchangetotheoscillatorpowerspectrumbecauseofthedriving.More,theoscillatorpowerspectrumwilldisplayspectralpeaksofcertainshapeandstrengthasaresultofdriving.Aswewillshow,,thesepeakswillnotoccurifthereisnofrequencynoise.Theyareconsequencesoftheinterplaybetweenfrequencynoiseandthedriving.Secondly,thecharacteristicsofthepeakssensitivelydependonthepropertiesoffre-quencynoisesuchasnoisestrengthandspectrum,thereforeallowingonetoextractinformationabouttheunderlyingorthenonlinearitymechanisms.Theideabehindtheproposedmethodisanalogoustoshiningelectromagneticwaveontoanoscillatingcharge(achargedharmonicoscillator),andmeasuretheluminescencespectrumofthecharge.Asastandardtextbookresult,aharmonicoscillatoronlyscatterslightelastically.There-fore,itsluminescencespectrumwillsimplybeasuperpositionofad-peakattheincidentlightfrequencyandthethermalspectrumduetothermaloftheoscillator[10].However,iftheoscillatorfrequencyisrandomlyperturbedbytheenvironment,theoscillatingchargecanscatterlightinelasticallyforwhichtheenergyoffsetisprovidedbyordumpedintotheenviron-ment.Asaresult,theluminescencespectrumwillshowextrastructureawayfromthefrequencyofincidentlight.Dependingontheenergystoredinthefrequencynoise(classicallyitrelatestothecorrelationtimeofthenoise)andtherelaxationrateoftheoscillatingcharge,oneexpectsthatthescatteredlightcanbeatfrequenciesdifferentfromtheincidentlightfrequency,andwidthofthespectralpeaksthebandwidthofthefrequencynoise.Toillustratethemethod,westudyinChap.2theoscillatorresponsetoanear-resonantdrivebasedonaphenomenologicalmodelofaharmonicoscillatorwithgenericfrequency[11].Weformulatetheproblemintermsofoscillatorsusceptibilitythatintimeduetofrequencynoise.Weshowthatindeeddependingontheinterrelationofthenoisecorrelationtime3andtheoscillatorrelaxationtime,thedriving-inducedpowerspectrum("luminescencespectrum")hasdifferentstructure.Wethenshowtheexperimentalresultsonacarbonnanotuberesonatorobtainedbyourexperimentalcollaborators,andapplythetheorytoquantitativelyextractthepropertiesoftheobservedfrequencynoiseinthesystem.Animportantsourceoffrequencynoiseisnonlineardispersivecouplingbetweenvibrationalmodes,asamplitudeofonemodeleadtofrequencyoftheothermode.Dispersivemodecouplingplaysacentralroleinquantumnon-demolitionmeasurements,inpar-ticularinsuperconductingcircuits[12,13]andoptomechanicalsystems[14].However,revealingandcharacterizingthecoupling-inducednoiseoftheoscillatorfrequencybecomeschallenginginthepresenceofdissipationwhenthestructureoftheoscillatorabsorptionspectrumcannotberesolved.InChap.3,wefocusonmode-coupling-inducedfrequencynoise.Westudyamicroscopicmodeloftwononlinearlycoupledharmonicoscillator(ortwomodes),bothofwhicharecoupledtoathermalreservior.Becauseofthenonlinearcouplingbetweenthetwooscillators,amplitudeofoneoscillatorbecomefrequencyoftheother.Weanalyticallyusingapath-integraltechniquetheresponseofoneoftheoscillatorstoanearresonantdriving,andthedriving-inducedpowerspectrum[15].Asinthegenericcase,thisspectrumthepropertiesofthefrequencynoise,inparticular,itdependssensitivelyontheinterrelationbetweenthecouplingstrength,anddecayratesofthetwooscillators.Wethengeneralizetheanalysisfromtwocoupledmodestomanycoupledmodes.Thedriving-inducedspectralpeaksofthepowerspectrumresultfrominthesystemsusceptibility.Thesepeaksaresensitivetoolstostudysystemdynamicsandirrespec-tiveoftheparticulartypeofthesystem.Withthisinmind,inChap.4,westudythedriving-inducedpowerspectraofseveraltypesofsystemsdifferentfromaoscillator,includinganover-dampedBrownianparticle(e.g.,anopticallytrappedparticle),atwo-statesystemthatswitchesbetweenthestatesatrandom,andanoisythresholddetector.Inallstudiedcasesweshowthatdrivingleadstotheonsetofspectralfeaturesnearthedrivingfrequencyandthecharacteristicfre-4quenciesofthesystems[16].Theshapeandintensityofthesefeaturesaresensitivetotheformofthesystemnonlinearityandthemechanism.InChap.5,weconcludebysummarizingthemajorresultsofthethesisanddiscussfuturedirections.5CHAPTER2INTERPLAYOFDRIVINGANDFREQUENCYNOISEINTHESPECTRAOFVIBRATIONALSYSTEMS2.1IntroductionThespectrumofresponseandthepowerspectrumofanoscillatorisatextbookproblemthatgoesbacktoLorentzandEinstein[17,18,10].Ithasattractedmuchattentionrecentlyinthecontextofnanomechanicalsystems.Here,thespectraareamajorsourceofinformationabouttheclassicalandquantumdynamics[19,20,21,22,23,24,25,26].Thisisthecasealsoformesoscopicoscillatorsofdifferentnature,suchassuperconductingcavitymodes[8,9,27,28]andoptomechanicalsystems[2].MesoscopicoscillatorsexperiencecomparativelylargeAlongwithdissipation,thesedeterminetheshapeofthevibrationalspectra.Awell-understoodandmostfrequentlyconsidered[10]sourceofisthermalnoisethatcomesfromthecouplingofanoscillator(vibrationalsystem)toathermalreservoirandisrelatedtodissipationbythetheorem.Dissipationleadstothebroadeningoftheoscillatorpowerspectrumandthespectrumoftheresponsetoexternaldriving.Spectralbroadeningcanalsocomefromoftheoscillatorfrequency,whichplayanimportantroleinmesoscopicoscillators.Fornanomechanicalresonators,frequencycanbecausedbytensionandmassofthechargeinthesubstrate,ordispersiveintermodecoupling[22,23,24,25,26,29,30,6,7],whereasforelectromagneticcavitymodestheycancomefromoftheeffectivedielectricconstant[8,9].Identifyingdifferentbroadeningmechanismsisadelicatetaskthathasbeenattractingmuchattention[24,26,8,6,31,32].Inthischapterwestudythecombinedeffectofperiodicdrivingandfrequencyonthepowerspectraofnanomechanicalvibrationalsystems.Foralinearoscillatorwithnofrequency6drivingleadstoad-likepeakatthedrivingfrequencywF[10,19],becauseheretheonlyeffectofthedrivingisforcedvibrationslinearlysuperimposedonthermalmotion.Frequencymakeforcedvibrationsrandom.Asweshow,thisqualitativelychangesthespectrumleadingtocharacteristicnewspectralfeatures.Weobservethesefeaturesinacarbon-nanotuberesonatorandusethemtoseparatetheenergyrelaxationratefromtheoverallbroadeningofthepowerspectrumintheabsenceofdriving,aswellasrevealandexplorethenarrow-bandfrequencynoise.Figure2.1Top:sketchesofthepowerspectraofadrivenlinearoscillatorF(w).Panels(a)and(b)refertolargeandsmallcorrelationtimeofthefrequencynoisetccomparedtotheoscillatorrelaxationtimetr,respectively,i.e.,tonarrow-andbroad-bandfrequencynoise.Theblue(lower)lineshowsthespectrumofthermalintheabsenceofdriving;itiscenteredattheoscillatoreigenfrequencyw0=hwosc(t)i.Inthepresenceofdrivingthereisaddedad-peakatthedrivingfrequencywF.Thegreenareasshowthespectralfeaturesfromtheinterplayofthedrivingandofwosc(t).Bottompanels:wosc(t)fortc˛tr(a)andtc˝tr(b).Foralinearoscillator,thespectralfeaturesresultingfromtheinterplayofdrivingandfrequencynoisearesketchedinFig.2.1.ThetwolimitingcasesshowninFig.2.1correspondtothelongandshortcorrelationtimeofthefrequencynoisetccomparedtotheoscillatorrelaxation(decay)timetr.Fortc˛tr(panela)theoscillatorfrequencywosc(t)slowlyaboutwhatcanbecalledtheeigenfrequencyw0=hwosc(t)i.Onecanthenthinkofslowoftheoscillatorsusceptibilityc,whichdependsonthedetuningofthedrivingfrequencywFfromwosc(t).TheassociatedslowoftheamplitudeandphaseofforcedvibrationsatfrequencywFleadtoaspectralpeakcenteredatwF.Thisisafrequency-domainanalogoftheEinstein7lightscatteringduetospatialsusceptibility[33].Fortc˝tr(panelb),driving-inducedrandomvibrationsquicklylosethememoryofthedriv-ingfrequency.Theybecomesimilartothermalvibrations.However,theiramplitudeisdeterminedbythedriving,notthetemperature.Thisleadstoaspectralpeakcenteredattheoscillatoreigen-frequencyw0,withtheheightquadraticinthedrivingamplitude.Inthequantumpicture,onecanthinkthat,asaresultofpumpingbyadrivingtheoscil-latoremitsenergyquanta.Forthefamiliarexampleofanoscillatingchargedrivenbyanelectro-magneticthesequantacanbephotons,andonecanspeakoflightscatteringandbyanoscillator.Aquantumisemittedovertimetraftertheabsorptionevent.Fortc˝trthefrequencyofthequantumisuncorrelatedwiththeexcitationfrequencywF.Thisisatypeprocess.Theenergydifference¯h(wFw0)comesfromthefrequencynoise.Fortc˛tremissionoccursatfrequenciesclosetowF.Inthebothcasesthespectrumisqualitativelydifferentfromjustad-likepeakintheabsenceoffrequency[10].2.2Powerspectrumofweaklydrivensystems2.2.1GeneralexpressionTodescribethepowerspectrumofanonlinearsystemonehastogobeyondtheapproximationimpliedabove,whereonlythelinearsusceptibilityisIfdrivingisdescribedbythetermqF(t)intheoscillatorHamiltonian,whereqistheoscillatorcoordinateandF(t)=FcoswFtisthedrivingforce,toobtaintermsµF2inthepowerspectrumoneshouldkeeptermsµFandµF2intheresponse,q(t)ˇq0(t)+Zt¥dt0c1(t;t0)F(t0)+ZZt¥dt0dt00c2(t;t0;t00)F(t0)F(t00):(2.1)Hereq0(t)isthermaldisplacementintheabsenceofdriving.Equation(2.1)doesnotincludeaveraging,c1andc2arethelinearandnonlinearsusceptibilities.Thestandardlinear8susceptibilityishc1(t;t0)i,itisafunctionoftt0.Foraharmonicoscillator,whichisthecentraltopicofthischapter,c2=0.TheconventionallymeasuredoscillatorpowerspectrumisF(w)=2ReZ¥0dteiwthhq(t+t0)q(t0)ii;wherehhiindicatesstatisticalaveragingandaveragingwithrespecttot0overthedrivingperiod2p=wF.ForweakdrivingF(w)ˇF0(w)+p2F2jc(wF)j2d(wwF)+F2FF(w):(2.2)ThisspectrumissketchedinFig.2.1.FunctionF0isthepowerspectrumintheabsenceofdriving,aresonantpeakassociatedwiththermalvibrationsoftheoscillator.Thed-peakatthedrivingfrequencyinEq.(2.2)andinFig.2.1describesaverageforcedoscillatorvibrations,c(w)istheFouriertransformofhc1(t;t0)iovertt0.OfprimaryinteresttousisthetermFF(w),shownbytheenvelopeofthegreenareainFig.2.1.Itdescribestheinterplayoffrequencyandthedriving.WeconsideritforwclosetowFassumingahighqualityfactor,w0tr˛1,typicalformesoscopicsystems,andresonantdriving,jwFw0j˝wF.Theexplicitexpressionforthedriving-inducedterminthepowerspectrumofoftheoscillatorreadsFF(w)=12ReZ¥0dtei(wwF)tZZ0¥dtdt0eiwF(t0t)hc1(t;t+t)c1(0;t0)hc1(0;t0)ii+F(2)F(w):(2.3)ThisexpressionfollowsfromEqs.(2.1)and(2.2).Thetermgivesthecontributionoftheofthelinearsusceptibility.Thesecondtermgivesthecontributionfromthenonlinearsusceptibility,F(2)F(w)=ReZ¥0dteiwtZZ0¥dtdt0cos[wF(tt0)]hc2(t;t+t;t+t0)q0(0)i+hq0(t)c2(0;t;t0)i:(2.4)9Thistermdescribesthecorrelationbetweenofthesecond-ordersusceptibilityandther-malintheabsenceofperiodicdriving.Weemphasizethat,foraresonantlymodulatedunderdampedoscillator,itispronouncedatfrequencieswclosetothedrivingfrequencywF,not2wF.Equation(2.4)describes,inparticular,thecontributiontothespectrumfromthenonlinearsusceptibilityofanonlinearoscillator.Itisespeciallyconvenientinthecaseofweaknonlinearity,wheretheoscillatorspectrumF0(w)isbroadenedprimarilybythedecayratherthanbyfrequencyduetotheinterplayofthenonlinearityandtheamplitudeInthiscasethetermF(2)FgivesthemaincontributiontoFF.ThetheoryofanonlinearoscillatorwillbediscussedinChap.3.2.2.2SpectrumofadrivenharmonicoscillatorwithfrequencyForaharmonicoscillatorwithfrequency,wosc(t)=w0+x(t),wherex(t)iszero-meannoise.Weassumethatthenoiseisweakcomparedtow0andthatitscorrelationtimetc˛w10.Thenoisethendoesnotcauseparametricexcitationoftheoscillator[34,35].Themostsimplemodeloftheoscillatordynamicsisdescribedbyequation¨q+2Gq+[w20+2w0x(t)]q=FcoswFt+f(t);(2.5)wheref(t)isthermalnoiseandG=t1ristherelaxationrate.Bothf(t)andthedirectfrequencynoisex(t)leadtooftheoscillatorphase.SeparatingtheircontributionsbymeasuringthecommonlyusedAllanvariance(cf.[19])iscomplicated.However,thesetwotypesofnoisehavedifferentphysicalorigin,andourresultsshowhowtheycanbeseparatedusingthepowerspectrum;adifferentapproach,whichhowevermaynotbeimplementedwithastandardspectrumanalyzer,wasproposedin[36].Thesusceptibilityofalinearunderdampedoscillatorwithfrequencycanbefoundinastandardwaybychangingfromthefastoscillatingvariablesq;qtoslowcomplexoscillatoramplitudeu(t)=[q(t)+(iwF)1q(t)]exp(iwFt)=2.FromEq.(2.5),theequationforu(t)inthe10rotatingwaveapproximationreadsu=[G+idwFix(t)]uiF4w0+fu(t):(2.6)Here,dwF=wFw0isthedetuningofthedrivingfrequencyfromtheoscillatoreigenfrequency;fu(t)=[f(t)=2iw0]exp(iw0t).Equation(2.6)appliesonthetimescalethatlargelyexceedsw10.Onthisscalefu(t)isd-correlatedevenwhereinthelabframetheoscillatordynamicsisnon-Markovian,cf[37].Solvingthelinearequation(2.6),oneimmediatelyobtainsthelinearsusceptibilityofadampedharmonicoscillator,c1(t;t0)=i2w0e(G+iw0)(tt0)iRtt0dt00x(t00)+c.c.(2.7)(c2=0).Equation(2.7)oftenappliesevenwheretheoscillatordynamicsinthelabframeisnon-Markovian.Wedisregardcorrections˘jdwFj=wF;inparticularinEq.(2.6)forconveniencewereplacedF=wFwithF=w0;similarly,intheexpressionforfuwereplacedf=wFwithf=w0.Wenotethatthenoisefu(t)dropsoutfromthemomentshun(t)i[36].Thiscanbeusedtochar-acterizethestatisticsofthefrequencynoise.Hereweconsiderthechangeoftheconventionallymeasuredcharacteristic,thepowerspectrum,andtheextraspectralfeaturesrelatedtotheinterplayofthedrivingandfrequencynoise.ItisconvenienttorewriteEq.(2.3)forthespectrumFF(w)nearitsmaximumintheformthatexplicitlytakesintoaccountthat,whentheexpressionforthesusceptibilityissubstitutedintoEq.(2.3),thefast-oscillatingtermsintheintegrandscanbedisregarded.ThisgivesFF(w)=(8w20)1ReZ¥0dtexp[i(wwF)t]Zt¥dt0Z0¥dt01csl(t;t0)[csl(0;t01)hcsl(0;t01)i];csl(t;t0)=e(GidwF)(tt0)expiZtt0dt00x(t00):(2.8)Here,functioncsl(t;t0)givestheslowlyvaryingfactorinthefast-oscillatingtime-dependentos-cillatorsusceptibilityc1(t;t0).Functionhcsl(0;t)ihcsl(t;0)igivesthestandard(average)sus-11ceptibilityc(wF)=Z¥0dteiwFthc1(t;0)i=i2w0Z¥0dthcsl(t;0)i:(2.9)Themeanforceddisplacementoftheoscillatorinthelinearresponsetheoryishq(t)i=12FeiwFtc(wF)+c.c.:2.3Oscillatorpowerspectruminthelimitingcases2.3.1WeakfrequencynoiseExplicitexpressionsforFF(w)canbeobtainedfromEq.(2.8)inthelimitingcases.Forweakfrequencynoise,onecanexpandcslinx(t).Totheleadingorder,thespectrumFFisproportionaltothenoisepowerspectrumX(W)=R¥¥dthx(t)x(0)iexp(iWt),FF(w)ˇ116w20[G2+(wFw0)2]X(wwF)G2+(ww0)2:(2.10)Thisexpressionprovidesadirectmeansformeasuringthefrequencynoisespectrum.Italreadyshowsthepeculiarfeaturesqualitativelydiscussedabove.IfX(W)peaksatzerofrequencyandisnarrowonthescaleG(asfor1=f-typenoise,forexample),FF(w)hasapeakatwF,cf.Fig.2.1a.TheshapeofthispeakcoincideswiththatofX(W).If,ontheotherhand,X(W)isalmostonthefrequencyscaleG;jwFw0j(broad-bandnoise),FF(w)hasaLorentzianpeakatw0,cf.Fig.2.1b.2.3.2Narrow-bandfrequencynoiseTodescribetheeffectofanarrow-band,butnotnecessarilyweakfrequencynoise,onecanreplacex(t00)inEq.(2.8)withx(t).ThenitfollowsfromEq.(2.9)thatthesusceptibilityreadsc(wF)=i2w0hX(t)i;X(t)=[GidwF+ix(t)]1:(2.11)12whereastheexpressionforthedriving-inducedterminthepowerspectrumreadsFF(w)ˇ18w20ReZ¥0dtei(wwF)tX(t)[X(0)hX(0)i]:(2.12)ThequantityiX(t)=2w0correspondstothefiinstantaneous"slowlysusceptibility.ThenarrowspectrumFF(w)isdeterminedbythespectrumandstatisticsofthefrequencynoise.Theseexpressionscanbeusedfornumericalcalculationsifthestatisticsofthenoisex(t)isknown.Thesimplerelation(2.10)betweenFF(w)andX(w)followsfromthisanalysisforhx2i˝G2+(wFw0)2.Importantly,thisconditioncanbeachievedbytuningwFsomewhatawayfromw0.2.3.3BroadbandfrequencynoiseThecaseofX(W),i.e.,ofx(t)beingd-correlatedontimescaletr,canbeanalyzedforanarbitrarynoisestrengthusingthecharacteristicfunctionalofad-correlatednoiseisP[k(t)]=hexp[iZdtk(t)x(t)]i=exp[Zm(k(t))dt];wherefunctionm(k)isdeterminedbythenoisestatistics.AsseenfromEq.(2.8),functionhcsl(t;t0)iisdeterminedbyP[k(t00)]withk(t00)=1ift0<t00<tandk(t)=0otherwise.Ford-correlatednoise,whereP[k(t)]=exp[Rdtm(k(t))],takingintoaccountthatm(0)=(dm=dk)k=0=0andm(k)=m(k),weobtainhcsl(t;t0)i=exp(GidwF+m(1)])(tt0);c(wF)=(i=2w0)ŸGi(wFŸw0)1(2.13)withŸG=G+Rem(1)andŸw0=w0Imm(1).Thus,frequencynoiseleadstothebroadeningoftheconventionalsusceptibilityRem(1)andtheeffectiveshiftoftheoscillatoreigenfrequencybyImm(1).Wenotethatthenoisecanbeconsideredd-correlatedwhenitsspectrumisnotjustonthescale&G,butonthescale&G+Rem(1),whichitselfdependsonthenoiseintensity.Atthesametime,thenoisespectrumisassumedtobemuchnarrowerthanw0.AsseenfromEq.(2.8)thenoisecomponentsoscillatingatfrequenciesmuchhigherthanG+Rem(1);jdwFjareaveraged13out;frequencynoisewithfrequencies˘w0wasdisregardedinEq.(2.6).WhenwritingEq.(2.6)wealsoassumedthatnoiseatfrequenciescloseto2w0ˇ2wFisveryweakandcanbedisregarded.Ifthiswerenotthecase,onewouldhavetotakeintoaccounttheeffectsofnonlinearfrictionthatcomefromthecouplingtothesourceofthenoise,cf.[37].Averagingthetermhcsl(t;t0)csl(0;t01)iinEq.(2.8)comestocalculating*exp"iZtt0dt00x(t00)+iZ0t01dt001x(t001)#+˝expiZ¥¥dt2k(t2)x(t2):(2.14)Heret>0and¥<t0t;¥<t010.Clearly,inthisequationk(t2)=0;1.Fort0<0wehavek(t2)=sgn(t0t01)ifmin(t0;t01)<t2<max(t1;t01)andk(t2)=1if0<t2<t;fort0>0wehavek(t2)=1,ift01<t2<0andk(t2)=1,ift0<t2<t;otherwisek(t2)=0.Forad-correlatednoisetheaveragingusingtheexplicitformofP[k(t)]andintegrationovert0;t01;tgivesFF(w)=[Rem(1)]=G8w20[ŸG2+(wFŸw0)2]ŸGŸG2+(wŸw0)2:(2.15)Thespectrum(2.15)andthespectrumF0(w)intheabsenceofperiodicdrivinghavethesameshapegivenbythelastfactorin(2.15):aLorentziancenteredatthenoise-renormalizedoscilla-toreigenfrequencyŸw0=w0Imm(1)withhalfwidthŸG=G+Rem(1).However,incontrasttoF0(w),theareaofF2FF(w)isindependentoftheintensity(µkBT)ofthedissipation-relatednoise.Insteaditisproportionaltothefrequency-noisecharacteristicRem(1).Equation(2.15)suggestshowtoseparatethenoise-inducedbroadeningoftheoscillatorspectrumfromthedecay-inducedbroadening,seebelow.2.3.4GaussianfrequencynoiseForastationaryGaussiannoisethecharacteristicfunctionalisexpressedintermsofthenoisecorrelator[38],P[k(t)]=exp12Zdtdt0hx(t)x(t0)ik(t)k(t0):14Figure2.2ThepowerspectrumoftheoscillatorwithaGaussianfrequencynoisewiththespectrumX(W)=2Dl2=(l2+W2).ThenoiseintensityisD=G=2.Panelsaandb:thefullspectrum.ThecolorcodingisthesameasinFig.2.1,F2=16G2=20kBT.Panelc:thedriving-inducedterm.Thesolidlinesanddotsshowtheanalytictheoryandsimulations;theconsecutivecurvesareshiftedby0.25alongtheordinate.Ifthecorrelatorhx(t)x(t0)iorequivalently,thepowerspectrumX(W),areknown,usingthevaluesofk(t)givenbelowEq.(2.14)onecanperformtheaveraginginEq.(2.8)andthenperformintegra-tionovertimetothepowerspectrumFF.TheresultsareshowninFig.2.2forthenoisepowerspectrumwithbandwidthl,X(W)=2Dl2=(l2+W2).TheyillustratehowtheshapeofFF(w)changesfromapeakatwFforanarrow-bandnoise(l˝G)toapeakatw0forabroadbandnoise(l˛G).TheoverallareaofthespectrumFFnonmonotonicallydependsonthefrequencynoiseintensity:itislinearinthenoiseintensityforweaknoise,cf.Eq.(2.10),butforalargenoiseintensityitdecreases,sincethedecoherencerateoftheoscillatorincreases.2.3.5Theweak-noiseconditionInthelimitofweakslownoise,hx2(t)i˝jGidwFj2,Eq.(2.12)goesoverintotheresultforsuchnoiseobtainedabove;notethatinEq.(2.10)oneshouldreplaceww0withwFw0intheslow-noiselimit,sincefunctionX(W)isconcentratedintherangeofsmallW˝G.Forthebroad-15bandnoise,ontheotherhand,theweak-noiselimitdiscussedabovecorrespondstojm(1)j˝G.InthiscasethenoisepowerspectrumisandX(W)=(d2m=dk2)k=0˘jm(1)j˝G.Generally,theweaknoiseconditionusedtoobtainEq.(2.10)certainlyholdsformaxX(W)˝G.Itisimportantthat,forslownoise,theconditionislessstringentandcanbemetbyincreasingthedetuningjdwFj,allowingonetoreadtheslow-noisepowerspectrumdirectlyofftheoscillatorpowerspectrum.2.3.6SusceptibilitywithweaklyfrequencyBoththestandardsusceptibilityc(w)andthepowerspectrumintheabsenceofdrivingF0(w)areaffectedbyfrequencynoise.Intheconsideredcasetheyarerelatedbytherelation,F0(w)=(2kBT=w)Imc(w).Foranon-whitefrequencynoisethespectrumF0(w)becomesnon-Lorentzian.Theexplicitexpressionsforthesusceptibilityinthelimitingcasesoffastandslowfrequencynoiseweregivenabove,Eqs.(2.13)and(2.11).Asimpleexplicitexpressionforc(w)followsfromEqs.(2.8)and(2.9)alsointhecaseofweaknoise.Here,thesusceptibilitybecomesc(w)ˇi2w0(Gidw)1ZdW2p(Gidw)X(W)GidwiW;(2.16)whereX(W)isthefrequencynoisepowerspectrumanddw=ww0.Importantly,thenoise-inducedcorrectionjustslightlydistortsthesusceptibility.Forexample,asharplow-frequencypeakofX(W)doesnotleadtoanarrowpeakinc(w)and,respectively,inthepowerspectrumF0(w).ThisshouldbecontrastedwiththenarrowpeakinFF(w),whichemergesinthiscase.2.4Theareaofthedriving-inducedspectralpeakWenowconsidertheareaSFofthedrivinginducedspectralpeakforwclosetow0;wF;notethatthispeakmayhaveseveralmaxima,asseenfromFig.2.2.Wetheareaasanintegraloverpositivefrequencies,SF=R¥0dwFF(w).KeepinginmindthatFF(w)issmallforlarge16jwwFj˘wF[infact,Eq.(2.8)doesnotapplyforsuchw],weobtainSF=p8w20ZZ0¥dtdt0hcsl(0;t)csl(0;t0)ip8w20Z0¥dthcsl(0;t)i2:(2.17)Thisexpressiondescribesthedependenceoftheareaofthedriving-inducedspectrumonthepa-rametersandstatisticsofthefrequencynoise.FromEq.(2.17),theareaSFbecomeszerointheabsenceoffrequencynoise,sincehcsl(t;t0)i=csl(t;t0)inthiscase.TheareaSFlinearlyincreaseswiththefrequencynoiseintensityforweaknoise,asseenfromEq.(2.10).AnexplicitexpressionforSFcanbeobtainedforwhitefrequencynoise.FromEq.(2.15),SF=p8Gw20Rem(1)jG+i(wFw0)+m(1)j2:(2.18)FromEq.(2.18),SFlinearlyincreaseswiththecharacteristicnoisestrengthRem(1)whereitissmall,butoncethenoisebecomesstrong,SFdecreaseswithincreasingjm(1)j,withSFµRem(1)=jm(1)j2forjm(1)j˛G;jwFw0j.Forweaknarrow-bandfrequencynoise,fromEq.(2.10)oneobtainsSFintermsofthenoisevariancehx2(t)iasSF=p8w20hx2(t)i[G2+(wFw0)2]2:AnexplicitexpressionforSFcanbeobtainedalsoforastrongGaussiannoise.Wewillassumethatthenoisecorrelatorhx(t)x(0)iisnotfastoscillatingand,respectively,thenoisespectrumX(W)doesnothavenarrowpeaksordips.Forthenoisevariancehx2(t)imuchlargerthanG2;dw2F,andthesquaredreciprocalnoisecorrelationtimet2c,fromEq.(2.17)SFˇp28Gw20[2phx2(t)i]1=2:(2.19)ThevariationofSFwiththevaryingfrequency-noiseintensityandbandwidthisshowninFig.2.3,whichreferstotheexponentiallycorrelatedGaussiannoise.AsseenfromthisSFdisplaysamaximumasafunctionofthenoiseintensityD.Thedependenceonthenoisebandwidthlismorecomplicated;SFcanhavetwomaximaasafunctionoflforsufstrongnoiseintensity.17Figure2.3ThescaledareaŸSF=8G2w20SFofthedriving-inducedpeakintheoscillatorpowerspectrumasafunctionofthefrequencynoiseparameters.ThedatarefertoGaussianfrequencynoisewiththepowerspectrumX(W)=2Dl2=(l2+W2).2.4.1Scalingofthedriving-inducedpowerspectrumAconvenientscalingfactorforthedistributionFF(w)andfortheareaSFisprovidedbytheareaSdofthed-peakintheoscillatorpowerspectrumatthedrivingfrequency.AsseenfromEq.2.2,Sd=(p=2)jc(wF)j2.Ifc(wF)isknownfromthemeasuredpowerspectrumintheabsenceofdrivingwiththeinvokedtheorem,scalingbySdallowsonetoavoidtheactualmeasurementoftheforceF,whichrequiresknowledgeofthecouplingtothedrivingTheexpressionforSdifthefrequencynoisecanbethoughtofasasumofaweaknarrow-bandnoisexnb(t)andabroad-band(d-correlatedinslowtime)noisexbb(t),whichisnotweak,generally,andisstatisticallyindependentfromthenarrrow-bandnoise.Inthiscase,combiningEqs.(2.13)and(2.11)andexpandingtotheleadingorderintheweaknarrow-bandnoise,weobtain,Sdˇp8w20[ŸG2+(wFŸw0)2]1"1ŸG2(wFŸw0)2p[ŸG2+(wFŸw0)2]2ZdwXnb(w)#:(2.20)Here,Xnb(w)isthepowerspectrumofthenarrow-bandnoise;thevarianceofthenarrow-bandnoiseishx2nb(t)i=(2p)1RdwXnb(w).18Fornotweaknarrowbandfrequencynoise,onehasgenerally,Sd=p8w20jh[ŸG+i(wFŸw0xnb(t))]1inbj2(2.21)whereh::inbindicatesaveragingoverthenarrow-bandfrequencynoise.ThecorrectionthatcontainsXnbcanbedirectlyreadofftheareaofthenarrowpeakF(nb)F(w)inthespectrumFF(w),whichisduetothenarrow-bandnoise.Forweaknarrow-bandnoise,thispeakisdescribedbyEq.(2.10)ifonereplacesinthisequationGwithŸG,w0withŸw0,andX(w)withXnb(w).ThiscanbeseenfromEq.(2.8).Indeed,intheexpressionforcsl(t;t0)inEq.(2.8)onecanwriteRtt0dt00xnb(t00)ˇxnb(t)(tt0).ToF(nb)F(w),oneshouldintegrateovertherangeoftgivenbythereciprocalbandwidthofthenarrow-bandnoise.Sinceitlargelyexceeds1=G,thecontributionsofthebroad-bandnoisetocsl(t;t0)andcsl(0;t01)arestatisticallyindependent.Thereforetheaveragingoverthebroad-bandnoiseinthesesusceptibilitiescanbedoneindependently.Ifthisaveragingisdenotedbyibb,hcsl(t;t0)csl(0;t01)ibbˇhcsl(t;t0)ibbhcsl(0;t01)ibbforGt˛1.Herehcsl(t;t0)ibbˇehŸGiwFŸw0xnb(t)i(tt0):SincefunctionXnb(w)quicklyfallsoffwithincreasingjwj,inthedenominatorofEq.(2.10)onecanreplacewwithwF.Onethenseesthat,totheleadingorderinthenarrow-bandnoisestrength,theratiooftheareaSnbofthenarrowpeakF(nb)F(w)totheareaofthed-peakinthespectrumisSnbSdˇ12p1ŸG2+(wFŸw0)2ZdwXnb(w):(2.22)Fornotweaknarrow-bandnoise,itfollowsfromEq.(2.17)that,Snb=p8w20h[ŸG2+(wFŸw0xnb(t))2]1inbSd(2.23)whereSdisgivenbyEq.(2.21).19Equations(2.20)and(2.22)canbeusedtoscaletheareaofthebroaderpeakofFF(w)bySdwithaccounttakenoftheeffectofthenarrow-bandfrequencynoise.AlongwiththeonsetofanarrowpeakinFF,thisnoiseleadstothechangeoftheshapeandareaofthebroadpeak.TotheareaofthebroadpeakF(bb)F(w),oneneedtointegrateovertherangeoftimetgivenby1=ŸGinEq.(2.8).Duringthisperiodoftime,thenarrowbandfrequencynoisestaysalmostconstant,thatis,xnb(t)ˇxnb(0).ThentheonecansimplyreplacedwFwithdwFxnb(t)inEq.(2.8)forcsl(t;t0).Theresultisparticularlysimpleintheconsideredcaseherewherethebroad-bandfrequencynoiseisd-correlatedinslowtimeandthebroadpeakofFF(w)isdescribedbyEq.(2.15).OnejusthastoreplaceinthisequationsŸw0withŸw0+xnb(t),thenaveragingtheexpressionoverthedistributionofxnb(t).Tothesecondorderinthexnb(t),theareaofthebroaderpeakisgivenby,Sbbˇp8w20Rem(1)=G[ŸG+(wFŸw0)]2"1ŸG23(wFŸw0)2)2p[ŸG2+(wFŸw0)2]2ZdwXnb(w)#:(2.24)Fornotweaknarrowbandnoise,onehasSbb=p8Gw20hRem(1)jG+i(wFw0xnb(t))+m(1)j2inb:(2.25)OneimmediatelyseesfromEq.(2.23)andEq.(2.25)that,ŸG=G=1+Sbb=(Snb+Sd):(2.26)Whenthenarrow-bandfrequencynoiseisweak,thewidthofthebroadpeakisdeterminedbyprimarilythebroadbandnoise,thatis,itisclosetoaLorentzianwithhalfwdithŸG.Onecanthenusetheaboveequationtoextractthespectralbroadeningduetobroadbandfrequencynoise.2.5ExperimentsoncarbonnanotubevibrationalsystemTocorroboratethetheory,Moseretal.[11]measuredthespectrumofamodulatedcarbonnanotuberesonator.Thedeviceconsistsofacarbonnanotubecontactedbysourceanddrainelectrodesand20suspendedoveragateelectrode,seeFig.2.4.ThevibrationalmodeunderstudyisaxuralmodeasindicatedinthelowerpaneloftheFig.2.4.DetailsofthefabricationandthegeometryofthedevicecanbefoundinRef.[39].ThemeasurementwasperformedatT=1.2K.ForsuchTandweakdriving,low-lyingxuralmodesoftheresonatorarewelldescribedbyharmonicoscillators.ThedrivingwasappliedasanacvoltagedVgonthegateelectrode,andthepowerspectrumofthemechanicalvibrationswasprobedbymeasuringthenoiseinthecurrentwingthroughthenanotube.Figure2.4AFMimageofa4-mm-longnanotubebeforeremovingthesiliconoxide(top)andschematicofthedevice(bottom).Fig.2.5comparesthepowerspectraofthenanotubevibrationsobtainedwithandwithouttheoscillatingforce.Thespectrumwithoutmodulation(bluetrace)isclosetoaLorentzian,asexpectedforthermalvibrations.Thespectrumwithmodulation(greentrace)displaysanarrowpeakcenteredatthemodulationfrequencyandamuchbroaderpeakofthesameshapeasthespectrumwithoutmodulation.Theareasofthebothmodulation-inducedpartsofthespectrumscaleasdV2g(Fig.2.5b,c),inagreementwithEq.(2).TheseparationbetweenthepartsissubjecttosomeuncertaintybecauseofthemeasurementnoiseinFig.2.5a.TheresultinguncertaintyintheslopesinFig.2.5b,cis.10%.Thechangeofthespectrumisnotaheatingeffectassociatedwiththemodulation,sinceweestimatetemperatureincreasesas.108K.ThespectralfeatureatwFisnotrelatedtothephasenoiseofthesourceusedforthemodulation;indeed,thephasenoiseofoursourceˇ10HzawayfromwFcouldonlyaccountforˇ0:01%ofthemeasuredpowerspectrum.21Figure2.5(a)ThepowerspectrumofthecurrentdI(t)throughadrivencarbonnan-otube.Themeasurementbandwidthis4.7Hz.Theeigenfrequencyofthestudiedxuralmodeis6.3MHz.Thedrivingfrequencyis100Hzbelowtheresonancefrequency.Thebluelinereferstothepowerspectrumwithoutdriving;thegreenareashowsthedriving-inducedspectralchange.Thischangeisseparatedintothebroadpeak(darkergreen),narrowpeak(lightergreen),andadelta-spikeatthemodulationfrequency.Thisspikelieswithin3bins,withinourexperimentalresolution,andisrepresentedbytheblackverticallines.Theseparationofthebroadandnarrowpeaksisdonebythestraightlinethatinterpolatesthebroadpeak.Showninthelowerpanelsisthedependenceofthelightergreenarea(b),thedarkergreenarea(c),andtheareaunderthed-peak(d)onthesquaredamplitudeofthemodulatinggatevoltage;asexpectedfromthetheory,itisclosetolinear.Thedriving-inducedspectralchangeprovidesasimplemeansforestimatingtheintrinsicrelax-ationrateoftheresonatorGfromourexperimentaldata.ThiscanbedoneusingtheareasF2SnbandF2SbbofthenarrowandbroadpeaksinFig.2.5,respectively.Comparingthemtotheareaunderthedriving-inducedd-peakF2Sd,weeliminateFandobtainfromEq.(2.26)ŸG=Gˇ1+(Sbb=Sd)[1(Snb=Sd)];(2.27)weusedSnb˝Sd.WithŸG=(2p)'230Hzreadoutfromthecollectedspectra(suchastheoneinFig.2.5a),alongwithSbb=SdandSnb=SdmeasuredfromFigs.2.5b-d,weobtainŸG=G'2:1,which22givesG=(2p)'110Hz.Therefore,thebroad-bandoftheresonantfrequencyaccountfor&50%ofthemeasuredmechanicallinewidth.BecauseofthenoiseinthemeasurementinFig.2.5a,theuncertaintyinGis.10%.Thenarrow-bandfrequencynoisecanalsobecharacterizedfromthemeasurements.Sincethenarrow-bandfrequencynoiseinthenanotubeiscomparativelyweak,onecaninterprettheresultsusingtheweak-noiseexpressionforthespectrumEq.(2.10).Thentheshapeoftheresonatorspec-trumgivestheshapeofthenoisepowerspectrum.Figure2.6showsthenarrowbandfrequencynoisespectrumobtainedbysubtractingthebroadpartoftheresonatorspectrumfromwholespec-trumasshowninFig.2.5a.Thespectrumisof1=fatype.Thedataindicatethataiscloseto1/2.ObtainingthepowerspectraFFforseveralvaluesofwFw0shouldallowseparatingthelow-frequencypartofthefrequencynoisespectrumX(w)evenwhereitisnotweakerthanthebroad-bandpart,andreadingitdirectlyoffthedataonthepowerspectrumusingEq.(2.10).Figure2.6Narrowbandfrequencynoisespectrum.ItisobtainedbythebroadparthdI2ibroad(w)oftheexperimentalspectruminFig.2.5atoaLorentzian,andthenbysubtractingthisfromtheexperimentalspectrum.Theredlineisato1=f1=2,wheref=jwwFj=2p.2.6ConclusionTheaboveresultsshowthattheinterplayofdrivingandfrequencynoisequalitativelychangesoscillatorspectracomparedtothespectrawithnofrequency[10].Thechangesensi-tivelydependsonthefrequencynoiseintensityandpowerspectrum.Thepossibilitytoseparate23contributionsfromdifferentpartsofthefrequency-noisespectrumisofinterest,astheymaycomefromphysicallydifferentsources,liketwo-levelanddispersivecouplingtoothermodes,tomentionbutafew.Theresultssuggestawayofdiscriminatingbetweentwomajorfactorsofthebroadeningoftheoscillatorspectra:decay(energyrelaxation),frequencyinduceddirectlybythenoisethatmodulatestheeigenfrequency.Forlinearoscillators,oursimpleprocedureyieldsthedecayratewithouttheneedofanactualring-downmeasurementthatisoftendiftoimplement,inparticularfornanotubemechanicalresonators.Theanalysisofdrivenlinearoscillatorswithafrequencyimmediatelyextendstothequantumregime,whichisattractingmuchinterestinnano-andoptomechanics[2,40,41,42,43,32].Fornonlinearoscillators,thenonequidistanceoftheenergylevelscanbringinnewfeaturescomparedtotheclassicallimit.24CHAPTER3SPECTRALEFFECTSOFDISPERSIVEMODECOUPLINGINMESOSCOPICSYSTEMS3.1IntroductionMesoscopicvibrationalsystemstypicallyhaveseveralnonlinearlycoupledmodes.Thesecanbexural,torsional,oracousticmodesinthecaseofnanomechanicalresonators[6,44,45,46,47,48,49,7],photonandphononmodesinoptomechanics[50,51,2,52,53,54],ormodesofmicrowavecavitiesincircuitquantumelectrodynamicalsystems[13].Oftendifferentmodeshavedifferentfrequencies,sothattheinteractionbetweenthemisprimarilydispersive.Amajoreffectofsuchinteractionisthatthefrequencyofonemodedependsontheamplitudeoftheothermode.Therelatedshiftofthemodefrequencyprovidesameansofcharacterizingthecouplingstrengthwherebothmodescanbeaccessed,cf.Refs.[45,55,49,56]andcanbeusedforquantumnondemolitionmeasurementsoftheoscillatorFockstates.[57,58]Inmanymesoscopicsystemsonlyoneorafewmodescanbedirectlyaccessedandcontrolled.Thepresenceofdispersivecouplingtoothermodesandthestrengthofthiscouplinghavetobeinferredfromthedataontheaccessiblemodes.Tothebestofourknowledge,therehavebeennogenerallyacceptedmeansofdoingthis.Animportantconsequenceofdispersivecouplingisthatamplitudeofonemodeleadtofrequencyoftheothermode[37].Theamplitudecomefromthecouplingtoathermalreservoir,buttheycanalsobeofnonthermalorigin.Themode-couplinginducedfrequencybroadenthespectrumoftheresponsetoanexternalforceandthepowerspectrum.Suchbroadeninghasbeensuggestedasamajorbroadeningmechanismforx-uralmodesincarbonnanotubes[6],graphenesheets[7],doublyclampedbeams[55,49]aswellasmicrocantilevers.[56]25Separatingthemode-couplinginducedfromotherspectralbroadeningmechanismsisnontrivial,seeRef.[5]forarecentreviewofthebroadeningmechanisms.Themostfamiliarbroadeningmechanismisvibrationdecayduetoenergydissipation.Anothermechanismofin-terestforthepresentchapterisinternalvibrationnonlinearity.Becauseofsuchnonlinearity,thefrequencyofavibrationalmodedependsonthemodeamplitudeandthermaloftheamplitudeleadtofrequencyThisisreminiscentofthemode-couplingeffect,yetthenonlinearityisdifferent.Weshowbelowhowthetwononlinearitymechanismscanbeclearlydistinguished.Inthischapter,weproposeameansforidentifyingandcharacterizingthemode-couplingin-ducedfrequencyTheapproachisbasedonstudyingthechangeofthepowerspectrumoftheconsideredmode,whichisduetoanadditionallyappliedperiodicdriving.Theapproachdoesnotrequireaccesstoothermodes,incontrasttoRefs.[45,55,49,56],forexample.Itreliesonthefactthat,quitegenerally,frequencyleadtothefeaturesinthepowerspectrumofaperiodicallydrivenmode,whichdonotoccurwithoutsuchseeChap.2.Ifonethinksofthedrivenmodeasachargedoscillatorinastationaryradiationthesefeaturescor-respondtoandquasi-elasticlightscattering.Theabsenceoftheseeffectsinthecaseofaperiodicallydrivenlinearoscillatorwithconstantfrequencyisatextbookresult.[17,18,10]TheanalysisofthepowerspectraofdrivenmodeswithfrequencyinChap.2wasphenomenological.TheresultswereobtainedinseverallimitingcasesandinthecaseofGaussianThedispersivemodecouplingstudiedhereleadstostronglynon-GaussianfrequencyThesimplesttypeofsuchcouplingcorrespondstothecouplingenergyµq2q2d,whereqisthecoordinateoftheconsidereddrivenmodeandqdisthecoordinateofthemodetowhichitisdispersivelycoupledandwhichwecallthed-mode.Wherethemodesarefarfromresonance,thefrequencychangeoftheconsideredmodeisproportionaltotheperiod-averagevalueofq2d,i.e.,tothesquaredamplitudeofqd(t).Evenwhereqd(t)isGaussian,asinthecaseofthermaldisplacementofalinearmode,[18]thesquareddisplacementq2d(t)isnot.Thefeaturesofthespectrarelatedtothedispersive-couplinginducedfrequency26dependontheinterrelationbetweenthreeparameters.ThesearethetypicalmagnitudeofthefrequencyDw,theirreciprocalcorrelationtimeGd,andthedecayrateoftheconsideredmodeG.Weassumethatalltheseparametersaresmallcomparedtothemodeeigenfrequenciesandtheirdifference.Intheabsenceofdriving,thepowerspectrumandthelinearresponsespectrumoftheconsid-eredmodehavetheformofaconvolutionofthespectrumcalculatedwithoutdispersivecouplingandafunctionthatdependsontheparameterad=Dw=Gd.[59]Wecalladthemotionalnarrow-ingparametertodrawthesimilarity(althoughsomewhatindirect)withthemotionalnarrowingeffectinnuclearmagneticresonance(NMR).[60,61]Forad˝1thecorrelationtimeofthefre-quencyiscomparativelysmall.Thentheareaveragedoutandtheireffectissmall,asinthecaseoffastdecayofcorrelationsinNMR.Ontheotherhand,forad˛1,thespectrumcanbethoughtofasasuperpositionofpartialspectra,eachforagivenvalueoftheam-plitudeofqd(t).Theweightofthepartialspectrumisdeterminedbytheprobabilitydistributionofthisamplitude.Inthepresenceofdriving,thesituationisdifferent.Thedistinctionisthat,withoutthedispersivecoupling,thereisnodriving-inducedpartofthepowerspectrumatall,exceptforthetriviald-peakatthedrivingfrequency.BasedonthepreviousresultsinChap.2,weexpectthatthedriving-inducedpartofthespectrumwillstronglydependontheinterrelationbetweentheratesGandGd.Itisclearthatitwillalsostronglydependonad,butthisdependenceisnotknowninadvance.Theformulationbelowisintheclassicalterms.However,asweexplain,theresultsfullyapplyalsotothecasewheretheconsidereddrivenmodeisquantum,i.e.,itsenergylevelspacingiscomparableorexceedsthetemperature.Atthesametime,itisessentialthatthed-modeisclassical.Thecaseofthedeeplyquantumregimeofthed-modeisinawaysimpler.Inthiscase,thepowerspectrumoftheconsideredmodewithoutdrivingcanhaveapronouncedstructure,withdifferentlinescorrespondingtodifferentoccupationnumbersofthed-mode.[37]Thisissimilartothespectrumofatwo-levelsystemdispersivelycoupledtoaquantumvibrational27mode[62,63],wherethestructurehasbeenseenintheexperiment.[12]3.1.1ThestructureofthechapterAnimportantpartofthechapteristheaveragingovertheofthed-mode.Thisisaninterestingtheoreticalproblem,withdirectrelevancetotheexperiment.Itinvolvesanexplicitcalculationoftheappropriatepathintegral.ThecalculationispresentedinSecs.3.4and3.5.ThesesectionsaswellasSec.3.3.1canbeskippedifoneisinterestedprimarilyinthepredictionsfortheexperiment.BelowinSec.3.2wegiveageneralexpressionforthepowerspectrumofamodedrivenbyanexternalwhichapplieswheretheiscomparativelyweak,sothattheinternalnonlinearityofthemoderemainssmall.Theeffectoftheismostpronouncedwhereitisclosetoresonance.InSec.3.3wederiveequationsofmotionforweaklydampeddispersivelycoupledmodesintherotatingwaveapproximation.Section3.6providestheexplicitanalyticalexpressionsforthedriving-inducedpartofthepowerspectruminthelimitingcases,whichrefertothefastorslowrelaxationofthed-modecomparedtotherelaxationofthedrivenmodeandalsotothelargeorsmallfrequencyshiftduetothedispersivecouplingcomparedtotherelaxationrateofthed-mode.Thissectionalsopresentsresultsofnumericalcalculationsofthespectra,whicharecomparedwiththeresultsofsimulations.Thelastpartofthesectiondescribesthedependenceoftheareaofthedriving-inducedpeakonthedispersivecouplingparameters.Section3.7extendstheresultstodispersivecouplingtoseveralmodes.Inparticular,weconsiderthecumulativeeffectofdispersivecouplingtomany,butnottoomanymodes,whichmaybecomestrongevenwherethecouplingtoeachofthemodesisweak.Section3.8describesthedriving-inducedpartofthepowerspectrumforanonlinearoscillator,wheretheoftheoscillatorfrequencyareduenottodispersivecouplingbuttotheinternalnonlinearity.Thelastsectionprovidesasummaryoftheresults.Thetransfer-matrixmethodusedtoperformtheaveragingoverthedispersive-couplinginducedandanoutlineofanalternativederivationofthemajorresultaregivenintheappendices.283.2Driving-inducedpartofthepowerspectrumFrequencyrenderthemoderesponsetodrivingrandom.Generally,theresponseisnonlinearinthedrivingstrength.Respectively,evenforsinusoidaldriving,themodepowerspec-trumF(w)isacomplicatedfunctionofthedrivingamplitudeFandfrequencywF.Theconven-tionalwayofmeasuringthepowerspectrumofadrivensystemwithcoordinateqcorrespondstotheF(w)=2ReZ¥0dteiwthhq(t+t0)q(t0)ii;wherehhiindicatesstatisticalaveragingandaveragingwithrespecttot0overthedrivingperiod2p=wF(sometimesthepowerspectrumisalsoasF(w)=2p).Ifthedrivingisweak,onecankeepinF(w)onlythetermsquadraticinF,F(w)ˇF0(w)+p2F2jc(wF)j2d(wwF)+F2FF(w):(3.1)Here,F0isthepowerspectrumintheabsenceofdriving.Fortheconsideredunderdampedmode,itisaresonantpeakatthemodeeigenfrequencyw0,whichisduetothermalvibrations;thewidthofthepeakissmallcomparedtow0.Functionc(w)isthemodesusceptibility,F0(w)=(2kBT=w0)Imc(w)intheclassicallimit.Thetermµjc(wF)j2describesthecontributionofstationaryforcedvibrationsatfrequencywF.OfutmostinteresttousisthelastterminEq.(3.1),FF(w).Foralinearoscillatorwithfrequency,thistermisequaltozero.Indeed,themotionofsuchoscillatorisasuperpositionofthermalvibrationsandforcedvibrationsatfrequencywF,whichareuncoupled.Frequencyaffectboththermalvibrations,leadingtospectralbroadening,andforcedvibrations.Thelattereffectisparticularlyeasytounderstandinthecaseofslowfrequencytuations.Here,onecanthinkoftheamplitudeandphaseofforcedvibrationsasbeingdeterminedbythedetuningoftheinstantaneousmodefrequencyfromthedrivingfrequency.Thereforetheyintime,whichleadstotheonsetofafipedestal"ofthed-peakatw=wFinEq.(3.1).SomeotherlimitingcaseshavebeenconsideredearlierinChap.2.293.3EquationsofmotionfortheslowvariablesWewillconsiderthepowerspectrumF(w)inthecasewherethemodeofinterestisdispersivelycoupledtoanothermode(moded).Themodesareweaklycoupledtoathermalreservoir,sothattheirdecayratesaresmall.ThemodeofinterestisdrivenbyaperiodicFcoswFt.TheHamiltonianofthesystemisH=H0+Hb+Hi;H0=12(p2+w20q2)+14gq4+12(p2d+w2dq2d)+34gdq2q2dqFcoswFt:(3.2)Here,pandpdarethemomentaoftheconsideredmodeandthed-mode,respectively,w0andwdarethemodeeigenfrequencies,gistheparameteroftheintrinsicnonlinearityoftheconsideredmode,andgdisthedispersivecouplingparameter.Wedonotincorporatetheintrinsicnonlinearityofthed-mode,asitwillnotaffecttheresultsifitissmall,seebelow.ThetermHbistheHamiltonianofthethermalbath(eachmodecanhaveitsownbath,butweassumethatinthiscasethebathshavethesametemperature),whereasHidescribesthemodes-to-bathcoupling.Weassumethecouplingtobelinearinthemodescoordinates,Hi=qh+qdhd,wherehandhdarefunctionsofthedynamicalvariablesofthebath.Suchcouplingisthedominantmechanismofrelaxationforsmallmodedisplacementsandvelocities.ForHiofthisform,thedecayrateoftheconsideredmodeis[64,65]GG(w0)=¯h2ReZ¥0dth[h(0)(t);h(0)(0)]ieiw0t;(3.3)whereh(0)isfunctionhcalculatedintheneglectofthemode-bathcoupling.TheexpressionforthedecayrateGdofthed-modeissimilartoEq.(3.3),withh(0)replacedwithh(0)dandw0replacedwithwd.ParametersG;Gdcorrespondtofrictioncoefinthephenomenologicaldescriptionofthemodedynamics¨q+¶qH0=2Gqh(0)(t):(3.4)Theanalysisbelowreferstoslowlyvaryingamplitudesandphasesofthemodesandappliesevenwheretheabovephenomenologicaldescriptiondoesnotapply.[64,65,37]30Inwhatfollowsweassumethatthemodesareunderdamped,thenonlinearityisweak,andthedrivingfrequencyisclosetoresonance,G;Gd;jgjhq2iw0;jgdjhq2diw0;jw0wFj˝w0;wd;jw0wdj:(3.5)Theconditionong;gdmeansthatthechangeofthemodefrequenciesduetothenonlinearityissmallcomparedtotheeigenfrequency.However,itdoesnotmeanthattheeffectofthenonlinearityissmall,asthefrequencychangehastobecomparedwiththefrequencyuncertaintyduetothedecayG;Gd.Weassumethatthenonlinearchangeofthed-modefrequency,whichcomesfromthedispersivecouplingandtheinternalnonlinearityofthed-mode,isalsosmallcomparedtowd.3.3.1StochasticequationsforslowvariablesWhereconditions(3.5)apply,themotionoftheunderdampedmodespresentsalmostsinusoidalvibrationswithslowlyvaryingamplitudesandphases.Itcanbedescribedbythestandardmethodofaveraging,whichissimilartotherotatingwaveapproximationinquantumoptics.Tothisend,wechangetocomplexvariablesu(t)=12[q(t)+(iwF)1q(t)]exp(iwFt)(3.6)andsimilarlyud(t)=12[qd(t)+(iwd)1qd(t)]exp(iwdt).Disregardingfastoscillatingtermsintheequationforu(t),weobtainu=[G+idwFix(t)]uiF4w0+f(t);dwF=wFw0;x(t)=3g2w0ju(t)j2+3gd2w0jud(t)j2;(3.7)wheref(t)=(2iw0)1h(0)(t)exp(iwFt).Similarly,theequationforud(t)readsud=Gdi3gd2wdju(t)j2)ud+fd(t);(3.8)withfd(t)=(2iwd)1h(0)d(t)exp(iwdt).31Functionsf;fddescribetheforcesonthemodesfromthethermalbath.Theforcesarerandom,andonecanalwayschoosehfi=hfdi=0.Asymptotically,theyaredelta-correlatedGaussiannoises,hf(t)f(t0)i=(GkBT=w20)d(tt0);hfd(t)fd(t0)i=(GdkBT=w2d)d(tt0);(3.9)allothercorrelatorsvanish.Thed-functionshereared-functionsinfislow"timecomparedtow10;w1d,andthecorrelationtimeofthethermalbath.Thestochasticdifferentialequations(3.7)and(3.8)areunderstoodintheStratonovichsense:hu(t)f(t)i=GkBT=2w20;hud(t)fd(t)i=GdkBT=2w2d.Theseequationswereobtainedandtheirrangeofapplicabilityestablishedforahar-monicoscillatorcoupledtoabath,[64,65]buttheyalsoholdforaweaklyanharmonicoscillator.[37]Asseenfromtheofthecomplexamplitudeu(t),functionx(t)inEq.(3.7)describesachangeofthefrequencyoftheconsideredmodeduetotheintrinsicnonlinearityandthedispersivecoupling.Becauseofthenoisetermsf;fd,thefrequencybecomesarandomfunctionoftime.Therelatedfrequencynoiseisofprimaryinterestforthischapter.3.4Thedriving-inducedspectrumFF(w)fordispersivecouplingInthisandthetwofollowingsectionswewillstudythespectrumFF(w)inthecasewheretheinternalnonlinearityofthemodecanbedisregarded,i.e.,onecansetg=0.Inthiscasefrequencyx(t)µjud(t)j2inEq.(3.7)aredueonlytothedispersivenonlinearmodecoupling.Animportantfeatureofthiscouplingisthatitdoesnotaffectthefrequencynoisex(t)itself,asessentiallywasnoticedearlier.[59]Inotherwords,ofjud(t)j2arethesameasifthed-modewerejustalinearoscillatoruncoupledfromtheconsideredmode.Thesimplestwaytoseethisistochangeintheequationofmotion(3.8)fromud(t)toŸud(t)=K(t)ud(t)withK(t)=exp[3i(gd=2wd)Rtdt0ju(t0)j2].TheLangevinequationforŸudisŸud=GdŸud+Ÿfd(t)withŸfd(t)=K(t)fd(t).FromEq.(3.9),thenoiseŸfd(t)hasthesamecorrelation32functionsasfd(t).ThereforethetermµgddropsoutoftheequationforŸud.Sincejudj2=jŸudj2,thetermµgddoesnotaffectjud(t)j2either[eventhoughitdoesaffectud(t)].Itisconvenienttowritex(t)intermsofthescaledrealandimaginarypartofŸud=(2w0=3jgdj)1=2(Qd+iPd):FunctionsQd;Pdarethescaledquadraturesofadampedharmonicoscillator.TheyaredescribedbytheindependentexponentiallycorrelatedGaussiannoises(theOrnstein-Uhlenbecknoises)[66].UsingtheLangevinequationforŸud,weobtainhQd(t)Qd(0)i=hPd(t)Pd(0)i=adGdexp(Gdjtj);ad=3jgdjkBT=8w0w2dGd;x(t)=[Q2d(t)+P2d(t)]sgngd:(3.10)Thefrequencynoiseofthedrivenmodex(t)isnon-Gaussian.Parameteradcharacterizestheratioofthestandarddeviationofthenoise,whichisequaltothenoisemeanvaluehx(t)i=3gdkBT=4w0w2d,toitscorrelationrateGd.Sincex(t)isindependentofu(t),theLangevinequationforu(t)(3.7)islinear.Itssolutionreadsu(t)=Zt¥dt0csl(t;t0)[(iF=4w0)+f(t0)];csl(t;t0)=e(GidwF)(tt0)expiZtt0dt00x(t00):(3.11)Functioncsl(t;t0)describestheresponseoftheconsidereddrivenmodetoaresonantperturbation.ThecoefatFaveragedoverrealizationsofx(t)givestheresonantsusceptibilityofthemode,c(wF)=i2w0Z¥0dthcsl(t;0)i:(3.12)Itdeterminesthed-peakinthepowerspectrum(3.1).FromEq.(3.11),thetermFF(w)inEq.(3.1)33forthepowerspectrumhastheformFF(w)=(8w20)1ÂZ¥0dtexp[i(wwF)t]Zt¥dt0Z0¥dt01hcsl(t;t0)csl(0;t01)i4w20jc(wF)j2(3.13)(intheintegralovertitisimpliedthatImw!+0).Thetermµf(t0)inEq.(3.11)determinesthepowerspectrumofthemodeF0(w)nearitsmaximum,jww0j˝w0,intheabsenceofdriving.Thus,Eqs.(3.11)-(3.13)giveageneralexpressionforthepowerspectrumofanunderdampedmodewithfrequencyintheabsenceofinternalnonlinearity.3.5AveragingoverthefrequencynoisefordispersivecouplingTothedriving-inducedpartofthepowerspectrumF2FF(w),onehastoperformaveragingoverofx(t)inEq.(3.13).Thiscalculationisthecentraltheoreticalpartofthechapter.Inwhatfollowsweoutlinethecriticalstepsthatareinvolved.TheintegrandintheexpressionforFF(w)canbewrittenashcsl(t;t0)csl(0;t01)i=e(GidwF)(tt0)+(G+idwF)t01G2(t;t0;t01);G2(t;t0;t01)=heifx(t;t0)+ifx(0;t01)i;fx(t;t0)=Ztt0dt00x(t00):(3.14)Here,fx(t;t0)istheincrementofthephaseoftheoscillatoroverthetimeinterval(t0;t)duetothefrequencynoisex(t).FunctionGdescribestheresultoftheaveragingoverthenoise.Expression(3.14)canbeslightlyusingtherelation(3.10)betweenx(t)andthequadraturesQdandPd.Thesequadraturesarestatisticallyindependent,andthereforeaveragingoverthemcanbedoneindependently,sothatG(t;t0;t01)=˝expiZtt0dt00k(t00)Q2d(t00)˛;(3.15)34wheret0=min(t0;t01)andintheintervalt0<t00<tfunctionk(t00)isequalto0;1,k(t00)=8>>>><>>>>:sgngd;tt00t;t=max(t0;0)0;t0<t00<t;t0=maxhmin(0;t0);t01isgn(t0t01)sgngd;t0=min(t0;t01)t00t0:(3.16)TheaveraginginEq.(3.15)canbeconvenientlydoneusingthepath-integraltechnique.Atheoryofthepowerspectrumbasedonthistechniquewaspreviouslydevelopedforthecasewherethereisnodriving.[59]Theapproach[59]canbeextendedtothepresentproblem,asdiscussedintheSec.3.5.5.However,thecalculationiscumbersome.Hereweuseadifferentmethod,whichisbasedonthetechnique[67]forcalculatingdeterminantsinpathintegrals.Intermsofapathintegral,themeanvalueofafunctionalL[Qd(t)]ofQd(t)canbewrittenasRDQd(t)L[Qd(t)]P[Qd(t)].FortheconsideredexponentiallycorrelatednoiseQd(t),theproba-bilitydensityfunctionalis(cf.Ref.[38])P[Qd(t)]=exp(4adG2d)1Zdt(Qd+GdQd)2:ToG(t;t0;t01)weneedtoperformaveragingoverthevaluesofQd(t00)intheinterval(t0;t).Inastandardway,wediscretizethetimeastn=t0+ne;n=0;:::;N,wheree=(tt0)=NandN˛1.ThepathintegralisthenreducedtointegratingoverthevaluesofQnQd(tn)=(4adG2de)1=2with1nNfollowedbyaveragingoverQ0Qd(t0)=(4adG2de)1=2withtheBoltzmannweight-ingfactorexp[2eGdQ20].Inthemid-pointdiscretization,intheintegralsovertimeoneusesQd(t)![Qd(t)Qd(te)]=eandfQd(t)![fQd(t)+fQd(te)]=2foranarbitraryf(Qd).[68]ThentheexponentinP[Qd(t)]becomes(4adG2d)1Zdt(Qd+GdQd)2!Nån=1h(QnQn1)2+e2G2dQ2nieGd(Q2NQ20):(3.17)35Theintegralovert00inEq.(3.15)issimilarlydiscretizedandgoesoverinto4adG2de2ånknQ2nwithknk(tn).ThentheexpressionforfunctionGbecomesG(t;t0;t01)=I[k]=I[0];I[k]=ZdQ0eQ20(1+eGd)ZNÕn=1dQnexphQƒ‹L[k]Q+2Q0Q1i:(3.18)Here,QisavectorwithcomponentsQ1;:::;QN.FromEq.(3.17),thediagonalmatrixelementsof‹L[k]areLnn[k]=2+e2G2d(14iadkn)for1nN1,LNN[k]=1+e2G2d(14iadkN)+eGd.Theonlynon-zerooff-diagonalmatrixelementsof‹LareLnn1[k]=1.3.5.1FindingthedeterminantTheintegralsinEq.(3.18)areGaussian.ThereforethecalculationofGrequiresthedeter-minantsofthematrices‹L[k]and‹L[0].Thiscanbedonefollowingtheapproach[67].WeconsiderthedeterminantDnDn[k]ofthesquaresubmatrixof‹L[k],whichislocatedatthelowerrightcornerandhasrankNn+1.Forexample,D1isthedeterminantofthewholematrix‹L,whereasDNisthematrixelementLNN.TheresultofintegrationoverQ1;:::;QNinEq.(3.18)forI[k],besidestheQ0-dependentfactordiscussedbelow,ispN=2=pD1[k].ItisstraightforwardtoseethatDntherecurrencerelationDn=[2+e2G2d(14iadkn)]Dn+1Dn+2;1nN2:(3.19)Inthelimite!0,Dn[k]goesoverintoD(tn;k)andEq.(3.19)reducestoadifferentialequationforD(t00)D(t00;k),¨D(t00)G2d[14iadk(t00)]D(t00)=0(3.20)TheobviousboundaryconditionsforfunctionDareD(t)=lime!0DN=1andD(t)=lime!0(DNDN1)=e=Gd.ThequantityofinterestisD1[k]ˇD(t0;k);wewillalsoneedD(t0;k),seebelow.IntegratingthelinearinQ1termintheexponentinEq.(3.18)givesthefactorexp[Q20(‹L1)11].Itfollowsfromtheaboveanalysisthat(‹L1)11=D(t0+2e)=D(t0+e)ˇ1+eD(t0)=D(t0).ThentheresultofintegrationoverQ0inEq.(3.18)isthefactorfp=e[GdD(t0;k)=D(t0;k)]g1=2inI[k].36Fork=0,wehavefromEq.(3.20)D(t00;0)=exp[Gd(tt00)].Withthis,theexpression(3.18)forfunctionGbecomesG(t;t0;t01)=n2GdeGd(tt0)=[GdD(t0;k)D(t0;k)]o1=2(3.21)Thisexpressionisthecentralresultofthesection.Itreducestheproblemofcalculatingthedriving-inducedpartofthepowerspectrumtosolvinganordinarydifferentialequation(3.20).3.5.2TheaveragesusceptibilityWestartthediscussionoftheapplicationsofthegeneralresult(3.21)withtheanalysisofthefactorhcsl(t;0)i,whichgivestheaveragemodesusceptibility,Eq.(3.12).Functionhcsl(t;0)iisgivenbyhexp[iRt0dt00x(t00)]i,whichinturnisgivenbyEq.(3.14)withGoftheformofEq.(3.15)inwhicht0=t01=0andk(t00)=sgngd.SolvingEq.(3.20)withk(t00)=constisstraightforward,asisalsothenG(t;0;0)fromEq.(3.21).Theresultreadshcsl(t;0)i=exp[(GidwF)t]Ÿc(t);Ÿc(t)=eGdt[coshadt+(Gd=ad)(1+2iadsgngd)sinhadt]1;ad=Gd(1+4iadsgngd)1=2:(3.22)Equation(3.22)expressestheaveragesusceptibilityinelementaryfunctions.Itagreeswiththeresult[59]forthecorrelationfunctionofthemodedispersivelycoupledtoamode.3.5.3TheaverageoftheproductofthesusceptibilitiesSolvingthefullEq.(3.20)withdiscontinuousk(t00)ismorecomplicated.ItcanbedonebyfunctionD(t00)piecewisewherek(t00)=constasasumofflincident"andwavesandthenmatchingthesolutions.Thecorrespondingmethodremindsthetransfermatrixmethod.Itis37describedinSec.3.5.4.Theresultreads[G(t;t0;t01)]2=[Ÿc(t1)Ÿc0(t3)]1(G2da2d)(G2da02d)4G2dada0dsinh(adt1)sinh(a0dt3)exp[Gd(2t2+t1+t3)](3.23)wheret1=tt;t2=tt0;t3=t0t0;parameterst0,t,andt0areexpressedintermsoft;t0;t01inEq.(3.16),andŸc(t)isgivenbyEq.(3.22).FunctionŸc0(t)=Ÿc(t)anda0d=adfort0<t01,whereasŸc0(t)=Ÿc(t)anda0d=adfort0>t01.Thisexpressionisunexpectedlysimple.Weuseitbelowforanalyticalcalculations,inpartic-ularforcalculatingthedriving-inducedpartofthepowerspectruminthelimitingcases.3.5.4Thetransfer-matrixtypeconstructionThecentralpartofthecalculationofthedriving-inducedpowerspectrumistheaveragingoverthefrequencynoiseduetodispersivecoupling.Equations(3.14)and(3.21)reducethisaveragingtosolvinganordinarydifferentialequation(3.20)withthecoefthatvarieswithtimestepwise.Thesolutioncanbebytakingadvantageofthistimedependence.FromEq.(3.16),theinterval(t0;t)inEq.(3.20)isseparatedintothreeregionsm=1;2;3withinwhichthetime-dependentcoefk(t00)=¯kmisconstant.Theboundariesbetweentheregionstandt0andthevaluesof¯kmareinEq.(3.16).Weenumeratetheregionsintheorderofdecreasingtime,thatis,theregiont<t00<tcorrespondstom=1,etc.IneachregionD(t00;k)=M11(t00t0;m)Am+M12(t00t0;m)Bm:(3.24)Here,Mijarethematrixelementsofthematrix‹M(t00;m)=0B@coshamt00sinhamt00amsinhamt00amcoshamt001CA;ama(¯km)=Gd(14iad¯km)1=2(3.25)38Wenotethat,fromEq.(3.16),a1=Gd(1+4iadsgngd)1=2,whereasa2=Gd;a3isequaltoeithera1ora1dependingonwhethert0<t01ort0>t01intheargumentoftheG-functionin(3.21).ThevaluesofA1;B1inEq.(3.24)aredeterminedbytheconditionsD(t;k)=1;D(t;k)=Gd.ThevaluesofAm;Bmform=2;3arefoundfromthecontinuityofD(t00;k);D(t00;k)atthebound-ariest00=t;t0.FunctionG(t;t0;t01)inEq.(3.21)isdeterminedbyD(t0;k)=A3andD(t0;k)=a3B3.FromEqs.(3.24)and(3.25)wehave0B@A3B31CA=‹M1(t0t0;3)‹M(t0t0;2)‹M1(tt0;2)‹M(tt0;1)‹M1(tt0;1)0B@1Gd1CA;(3.26)ThissimplerelationcombinedwithEq.(3.21)givetheintegrandintheexpressionforthepowerspectrumFF(w)inasimpleform,whichisconvenientfornumericalintegration.Theexpression(3.26)canbeevaluatedintheexplicitform.TheresultisgiveninSec.3.5.3.ItisadvantageouswhenonelooksfortheasymptoticexpressionsforthespectrumFF(w).3.5.5Alternativepath-integralapproachtoaveragingoverfrequencynoiseHereweprovideanalternativeapproachtoevaluatingfunctionG(t;t0;t01),whichisbyEq.(3.15)anddescribestheoutcomeofaveragingoverthefrequencynoise.Themethodisrelated,albeitfairlyremotely,tothemethoddevelopedforcalculatingthepowerspectrumofanonlinearoscillatorintheabsenceofdriving.[59,37]WestartwithwritingtheprobabilitydensityfunctionaloftheGaussianprocessQd(t)onthewholetimeaxes,¥<t<¥,intermsofthecorrelationfunctionA(t2;t3)=hQd(t2)Qd(t3)ianditsinverseA1(t2;t3),P[Qd(t)]=exp12ZZdt1dt2Qd(t1)A1(t1;t2)Qd(t2);Zdt2A1(t1;t2)A(t2;t3)=d(t1t3);(3.27)39cf.Ref.[38]).FromEq.(3.15),functionGanditsderivative¶G=¶tcanbewrittenasG(t;t0;t01)=RD[Qd]ŸP[Qd]RD[Qd]P[Qd];¶G¶t=isgn(gd)RD[Qd]Q2d(t)ŸP[Qd]RD[Qd]P[Qd];(3.28)wherefunctionalŸPhastheformŸP[Qd]=exp12ZZdt1dt2Qd(t1)ŸA1(t1;t2)Qd(t2);ŸA1(t1;t2)=A1(t1;t2)2ik(t2)d(t1t2):(3.29)Herek(t2)isastepwisefunction,whichisequalto0or1inthetimeinterval(t0;t),wheret0min(t0;t01)isinEq.(3.16).Thishastobeextendedinthepresentformulation,k(t2)=0fort2>tandt2<t0.AkeyobservationisthatfunctionalŸP[Qd]isalsoGaussian.OnecanintroduceanoperatorŸA(t;t1)reciprocaltoŸA1,Zdt1ŸA(t;t1)ŸA1(t1;t2)=d(tt2):(3.30)Intermsofthisoperator,ŸA(t;t)=RD[Qd]Q2d(t)ŸP[Qd]RD[Qd]ŸP[Qd]:(3.31)Multiplyingequation(3.29)forŸA1byA(t2;t3)ŸA(t1;t)andintegratingwithrespecttot1;t2,weobtainanintegralequationforŸA(t;t3),ŸA(t;t3)2iZk(t1)ŸA(t;t1)A(t1;t3)dt1=A(t;t3):(3.32)Thisequationcanbereducedtoadifferentialequationbydifferentiatingtwicewithrespecttot3,¶2ŸA(t;t3)¶t23G2d[14iadk(t3)]ŸA(t;t3)=2adG2dd(t3t):(3.33)Interestingly,Eq.(3.33)hasthesamestructureasthedifferentialequationforthe"time-dependent"determinantfoundintheothermethod,seeEq.(3.20).Thusitcanbesolvedinasimilarfashionas40inSec.(3.5.4).TheboundaryconditionsareŸA(t;¥)=0.ItfollowsfromthedecayofcorrelationsofQd(t).Atthevaluesoft3wherek(t3)changesstepwise,seeEq.(3.16),ŸAand¶ŸA=¶t3remaincontinuous,exceptt3=t,where¶ŸA=¶t3changesby2adG2d,asseenfromEq.(3.33).WecannowwriteEq.(3.29)forfunctionG(t;t0;t01),intermsoffunctionŸA,¶tG(t;t0;t01)=isgn(gd)ŸA(t;t)G:(3.34)TheboundaryconditionforthisequationisG(t;t;0)=1.FromtheexplicitexpressionforGwealsohave¶t0G(t;t0;t01)=isgn(gd)ŸA(t0;t0)G(t;t0;t01);¶t01G(t;t0;t01)=isgn(gd)ŸA(t01;t01)G(t;t0;t01):(3.35)ThesolutionoftheseequationsreadsG(t;t0;t01)=exp(isgn(gd)"Ztt0ŸA(t00;t00)dt00Z0t01ŸA(t00;t00)dt00#):(3.36)WehavecheckedthattheexpressionforfunctionGthatfollowsfromEqs.(3.33)and(3.36)coincideswiththeresultobtainedusingthetransfermatrixmethod.3.6DiscussionofresultsInthissectionweusetheaboveresultstodiscusstheformofthedriving-inducedpartF2FF(w)ofthepowerspectruminthecaseofdispersivecoupling.Wegiveexplicitexpressionsforthespec-truminthelimitingcases.Wealsopresenttheresultsofthenumericalcalculationsofthespectrumbasedonthegeneralexpressions(3.13),(3.14),and(3.23).Theseresultsarecomparedwiththesimulations.Thesimulationswereperformedinastandardwaybyintegratingthestochasticequa-tionsofmotion(3.7)-(3.9)usingtheHeunscheme[69].Asweshow,theshapeandthemagnitudeofFF(w)sensitivelydependontwofactors.OneistheinterrelationbetweenthemagnitudeoffrequencyDwandtheirbandwidth2Gd,41themotional-narrowingparameteradinEq.(3.10).TheotheristheinterrelationbetweenGdandthewidthofthemodespectrumintheabsenceofdrivingF0(w).ThewidthofthemodespectrumisnotgivenjustbythemodedecayrateG.Itisaffectedbythefrequencynoiseanddependsonad.ThisisseenfromEqs.(3.12)and(3.22)forthesusceptibilityImc(w),cf.Ref.[59].ThehalfwidthofthepeakofImc(w)varieswiththefrequencynoisestrengthadGdfromG+2a2dGd,forad˝1,to˘G+2adGdforad˛1(thepeakofImc(w)isprofoundlynon-Lorentzianforad˛1andadGd&G).AnimportantoutcomeoftheanalysisintheprevioussectionsisthatthespectrumFF(w)allowsonetomeasureboththestrengthandthecorrelationtimeofthefrequencynoiseduetodispersivecoupling.Intheplace,itallowsonetodirectlyidentifytheverypresenceofthisnoise.Weemphasizethatthefulldriving-inducedterminthepowerspectrumF2FF(w)canbeseenevenwherethermalofthedrivenmodeareweakandthepeakinthepowerspectrumF0(w)istoosmalltoberesolved.3.6.1ThespectrumFF(w)inthelimitingcases3.6.1.1WeakfrequencynoiseThegeneralexpressionforthespectruminthelimitingcaseswherethefrequencynoiseisweakoritsbandwidthislargeorsmallcomparedtothewidthofthedrivenmodespectrumF0(w).Inthecaseofdispersivecoupling,thelimitofweakfrequencynoiseisrealizedwhereadGd˝G.AgeneralexpressionforFF(w)forweakfrequencynoisewasobtainedearlierEq.(2.10).ItrelatesFF(w)tothepowerspectrumofthefrequencynoisex(t).FromEq.(3.10),inthepresentcasewehavehx(t)i=2hQ2d(t)isgngd=2adGdsgngd,whereasthecorrelationfunc-tionofthenoisedeviationfromtheaveragedx(t)=x(t)hx(t)iis4a2dG2dexp(2Gdjtj).Then,42extendingtheresults(2.10)tonoisewithnon-zeroaverage,weobtainFF(w)=(a2dG3d=w20)[(wFŸw0)2+G2]1f[(wwF)2+4G2d][(wŸw0)2+G2]g1;(3.37)whereŸw0=w0+hx(t)i=w0+2adGdsgngd.ThisexpressioncanbealsodirectlyobtainedfromEq.(3.23)inthecorrespondinglimit.FromEq.(3.37),forweakdispersive-couplinginducednoise,theintensityofthespectrumFF(w)isproportionaltothesquareofthecouplingparameter.Ifthedetuningofthedrivingfrequencyfromtheeigenfrequencyofthedrivenoscillatorlargelyexceedsthehalfwidthsofthepowerspectraofthebothoscillatorsintheabsenceofdriving,jwFŸw0j˛G;Gd,thepowerspectrumFF(w)hastwodistinctpeaks.OneislocatedattheoscillatorfrequencyŸw0andhashalfwidthG.TheotherislocatedatthedrivingfrequencywFandhashalfwidth2Gd.Wenotethatthenoisevariance4a2dG2disindependentofGd.Forconstanta2dG2d,theareasofthepeaksatŸw0andwFareµGd=Gdw4Fanddw4F,respectively(theratioGd=Gaffectsonlytheareaofthepeakatw0).AsjdwFjdecreases,thepeaksstartoverlapping.ForsmalljdwFjtheycannotberesolved.Thisbehaviorisgeneralandoccursalsowherethefrequencynoiseisnotweak,asweshowbelow.3.6.1.2Broad-bandfrequencynoiseWenowconsiderthecaseofthebroad-bandfrequencynoise,wherethedecayrateofthed-modeGdlargelyexceedsthewidthofthedrivenmodespectrum.Thisconditionrequiresthatthemotional-narrowingparameterbesmall,ad˝1.Atthesametime,thecontributionofthefrequencynoisetothespectrumwidthoftheconsideredmodedoesnothavetobesmallcomparedtoitsdecayrate,i.e.,theratioa2dGd=Gcanbearbitrary.Inotherwords,thebroadeningofthespectrumoftheconsideredmodecanstilllargelycomefromthedispersivecoupling.TodescribethemostpronouncedpeakinthespectrumFF(w)forlargeGdandsmallad,onecansolveEq.(3.20)intheWKBapproximation,D(t00)ˇexpGdRt00tdt2[14iadk(t2)]1=2g.CombinedwithEq.(3.21),thissolutionimmediatelygivestheaveragingfactorGinEq.(3.14)43andthusthespectrumFF.Alternatively,onecanusetheexplicitexpression(3.23)forfunctionG.Onlythetermhastobekeptinthisexpressionforad˝1andG˝Gd.IntegrationovertimeinEq.(3.13)givesFF(w)ˇ2a2dGd=G4w0[ŸG2+(wFŸw0)2]Imc(w);c(w)=(i=2w0)[ŸGi(wŸw0)]1:(3.38)ParametersŸGandŸw0arethehalfwidthofthespectrumandtheeigenfrequencyofthedrivenmoderenormalizedduetothedispersivecoupling,ŸG=G+2a2dGdandŸw0isthesameasinEq.(3.37).Equation(3.38)showsthat,forabroad-bandfrequencynoise,thespectrumFF(w)hasthesameshapeasthespectrumintheabsenceofdriving,FF(w)µF0(w)µImc(w).However,fromthespectrumF0(w),whichisLorentzianinthiscase,onecannottellwhetherthespectrumhalfwidthŸGisduetodecayortofrequencyIncontrast,FFµa2disproportionaltothevarianceofthefrequencynoiseandhasacharacteristictemperaturedependence(a2dµT2ifGdisT-independent).Itenablesidentifyingthefrequencynoisecontributiontothespectralbroadening,asitwasearlierdemonstratedforad-correlatedfrequencynoise,seeChap.2.Alongwiththecomparativelynarrowpeak(3.38),fora2dGd&GthespectrumFF(w)hasabroadbackgroundnearw0,withthetypicalfrequencyscaleGd.Todescribeitonehastotakeintoaccountcorrectionsµa2dtotheleading-orderterminfunctionG(3.23).ForlargejdwFj˛ŸG,thespectrumalsohasapeaknearwFwithwidth˘2Gd.3.6.1.3Narrow-bandfrequencynoiseThespectrumFF(w)hasacharacteristicshapealsointheoppositelimitwherethebandwidthofthefrequencynoise2GdissmallcomparedtothewidthofthespectrumintheabsenceofdrivingF0(w).InthiscaseFF(w)displaysacharacteristicpeakatthedrivingfrequencywF,ascanbealreadyinferredfromtheweak-noiseexpression(3.37).IntheoverallspectrumF(w)itlookslikeapedestalofthed-peakatwF.ThetypicalhalfwidthofthepedestalisgivenbyGd.Thisallowsonetoreadoffthedecayrateofthed-modefromthespectrumFF(w)withoutaccessing44thed-modedirectly.Toresolvethepedestalforlargead,whereadGd&GsothatthewidthofthespectrumF0(w)isprimarilydeterminedbythedispersivecoupling,weneedacomparativelylargedetuningofthedrivingfrequencyfromthemaximumofF0(w).ThesimplestwaytoFF(w)nearwFforsmallGdisbasedonEq.(3.23).OnenoticesfromEq.(3.13)thatthemajorcontributiontoFF(w)inthiscasecomesfromthetimeranget˘G1d,whereastt0;jt01j.jGidwFj1.ThereforeinEq.(3.23)oneisinterestedinthelimitoflargetbutcomparativelysmalltt0;jt01j.Inthelimitt!¥butforedtt0,functionG2(t;t0;t01)!1=Ÿc(tt0)Ÿc(t01)withŸc(t)givenbyEq.(3.22).TheremainingterminG2isµexp(2Gdt).Onecanthenwritetheintegrandintheexpression(3.13)forFF(w)asaseriesinexp(2Gdt).Thenextcomesfromthefactthatjad(tt0)j;jadt01j˝1forjGidwFj=jadj˛1.ThereforeonecanexpandŸc(tt0)ˇ[1+2iadGd(tt0)sgngd]1andsimilarlyforŸc(t01).Ultimately,theresultofintegrationoverthetimest;t0;t01readsFF(w)=¥ån=11n!and¶n¶andc(wF)2nGd(wwF)2+(2nGd)2:(3.39)ThisexpressiondescribesthespectralpeakatsmalljwwFjintermsofthederivativesofthesusceptibilityc(w)calculatedasafunctionofthemotional-narrowingparameterad.Thewidthofthespectralpeak(3.39)isgivenbythebandwidthofthefrequencynoise2Gd.Thepeakisgenerallynon-Lorentzian.3.6.2EvolutionofFF(w)withthevaryingbandwidthandstrengthofthefrequencynoiseTovisualizethedependenceofthespectrumFF(w)ontheparametersofthesystem,wenowpresenttheresultsofnumericalevaluationofthegeneralexpressions(3.13),(3.14),and(3.23).Fig.3.1showstheevolutionofthedriving-inducedpowerspectrumFF(w)withthevaryingratioofthedecayratesGd=G,i.e.,thevaryingratioofthebandwidthofthefrequencynoiseandthedecayrateofthedrivenmode.Weuseasascalingfactorthesusceptibilityc0ofthedrivenmodeintheabsenceofdispersivecoupling,c0(wF)=i[2w0(GidwF)]1:(3.40)45InFig.3.1(a),thefrequencynoisebandwidthismuchlargerthanthewidthofthespectrumF0(w)intheabsenceofdriving.ThespectrumFF(w)isclosetoaLorentziancenteredneartheshiftedeigenfrequencyofthedrivenmode,seeEq.(3.38);forsmalladtheshiftshouldbe2adGd,whereasthehalfwidthshouldbeclosetoG+2a2dGd,[59]whichagreeswiththenumerics.InFig.3.1(d),ontheotherhand,thenoisebandwidthissmall.ThespectrumFF(w)isanarrowpeaknearthedrivingfrequencywF,seeEq.(3.39),withhalfwidthˇGd.InFigs.3.1(b)and(c)thefrequencynoisebandwidthiscomparabletothewidthofthespectrumF0(w).InthiscasethespectrumFF(w)displaystwopartlyoverlappingpeaks.TheoverlappingcanbereducedbytuningthedrivingfrequencywFfurtherawayfromtheresonance,seebelow.46Figure3.1Thescaleddriving-inducedpartofthepowerspectrumofthedrivenmodedispersivelycoupledtoanothermode,whichwecallthed-mode.Thermalofthed-modeleadtofrequencyofthedrivenmode.Panels(a)to(d)showthechangeofthespectrumwiththevaryingratioGd=Gofthedecayratesofthed-modeandthedrivenmode.Thescaledstrength(standarddeviation)ofthefrequencynoiseisadGd=G=1.ThespectrumFF(w)isscaledusingthenoise-freesusceptibilityc0(wF),Eq.(3.40),ŸFF=4GFF=jc0(wF)j2.Thesolidlinesandthedotsshowtheanalyticaltheoryandthenumericalsimulations,respectively.47Fig.3.2showstheevolutionofthespectrumFF(w)withthevaryingstrength(standarddevia-tion)2adGdofthefrequencynoise.Thefrequencynoisebandwidth2GdischosentobeclosetothedecayrateofthedrivenmodeG.ThedrivingfrequencywFistunedawayfromresonancesothatthetwopeaksofFF(w)arewellseparated.AninsightintotheshapeofthepeakscanbegainedfromtheaforementionedsimilarityofthespectrumFF(w)withthespectrumofandquasi-elasticlightscatteringbyaperiodicallydrivenoscillatingcharge.Forweakfrequencynoise,curve1inFig.3.2,thepeaksarelocatednearwF(quasi-elasticscattering)andw0cf.Eq.(3.37).Asthenoisestrengthincreases,thepeaknearw0becomesbroaderandthepositionofitsmaximumshiftstohigherfrequency(ifgd>0,asassumedintheThisresemblestheevolutionofthespectrumF0(w)intheabsenceofdrivingwithincreasingad;thisevolutionisshownintheinsetofFig.3.2.Forad>1thepeakbecomesnon-Lorentzianandasymmetric.Incontrast,theshapeofthepeaklocatednearwFstaysalmostthesamewithvaryingnoisestrength.Thisisconsistentwiththepictureofquasi-elasticscattering,wherethewidthofthepeakisdeterminedbythefrequencynoisebandwidth.Toillustratehowpersistentthisbehavioris,wescaledthespectrainFig.3.2sothatattheirmaximaatwFthespectrahavethesameheightfordifferentad.3.6.3EffectonFF(w)ofthedetuningofthedrivingfrequencyToprovidemoreinsightintothenatureofthedouble-peakstructureofthespectrumFF(w)forG˘Gd,weshowinFig.3.3theeffectofdetuningofthedrivingfrequencywFfromresonance.Panels(a),(b),and(c)refertothedrivingfrequencybeingreddetuned,equalto,andbluedetunedfromthethemaximumofthespectrumF0(w)intheabsenceofdriving,respectively.Theresultsweshowrefertothedispersivecouplingconstantgd>0.Forgd<0,theplotsshouldbemirror-withrespecttoww0,andwFw0shouldbereplacedwithw0wF.ThepeaklocatednearthefrequencywFiswellresolvedinFig.3.3(a).ItmovesalongwithwFasthelattervaries.InFig.3.3(a)onecanalsoseeabroaderpeak,whichislocatedclosetow048Figure3.2Theevolutionofthedriving-inducedpartofthepowerspectrumwiththevaryingstrengthofthefrequencynoiseduetodispersivecoupling.Curve1to3refertothescaledstandarddeviationofthenoiseadGd=G=0:5,2.5,and12.5,respectively.Theratioofthenoisebandwidthtothedecayrateofthedrivenmodeis2Gd=G=1.ThescaleddetuningofthedrivingfrequencyfromtheeigenfrequencyofthedrivenmodeisdwF=G=5.Thespectrumisscaledusingthenoise-freesusceptibilityc0(wF),Eq.(3.40),ŸFF=4GFF=jc0(wF)j2.Thecurves1and3aread-ditionallyscaledbyfactors3.15and1.3,respectively,sothatthepeaksnearwFhavethesameheight.TheinsetshowsthespectrumF0(w)intheabsenceofdrivingforthesamevaluesofthefrequencynoisestrengthadGd=Gasinthemainpanel.Thesolidlinesandthedotsshowtheanalyticaltheoryandthesimulations,respectively.andessentiallydoesnotchangeitspositionaswFchanges.Forsmallfrequency-noisebandwidth,thepeakatwFbecomesnarrowandisdescribedbyEq.(3.39).However,itiswell-resolvedforlargefrequencydetuningevenwherethenoisebandwidthandthewidthofthespectrumF0(w)areofthesameorderofmagnitude.IfthewidthsarecloseandwFisclosetoresonance,thepeaksoverlapandcannotbeasseeninpanel(b).Theareasofthepeaksaredramaticallydifferentforredandbluedetuning.ThisisduetotheasymmetryofthespectrumF0(w)inthepresenceofthefrequencynoiseinducedbydispersivecoupling,seetheinsetofFig.(3.2).AsseenfromFig.3.1,forverysmallGd=Gthepeakneartheoscillatoreigenfrequencydisappears;thiswasdiscussedearlierinthecaseofweaknoise,butisalsotrueinageneralcase.3.6.4TheareaofthedrivinginducedpowerspectrumTheareaSFofthedrivinginducedpowerspectrumFF(w)isasSF=R¥0dwFF(w).ThemajorcontributiontotheintegralcomesfromthefrequencyrangewherejwwFj;jww0j˝wF.49Figure3.3Theevolutionofthedriving-inducedpartofthepowerspectrumwiththevaryingde-tuningofthedrivingfrequencywF.ThescaledstrengthofthefrequencynoiseinducedbythedispersivecouplingisadGd=G=2:5.Theratioofthenoisebandwidthtothedecayrateofthedrivenmodeis2Gd=G=1.Thespectrumisscaledusingthenoise-freesusceptibilityc0(wF),Eq.(3.40),ŸFF=4GFF=jc0(wF)j2.Thesolidlinesandthedotsshowtheanalyticaltheoryandthesimulations,respectively.ThenintegrationoverwinEq.(3.13)givesafactor2pd(t).Furthercomesfromchangingfromintegratingovert0andt01tointegratingovert0andt0t01andusingEq.(3.12)forthesusceptibilityofthemode.TheresultreadsSF=p4w0GImc(wF)p2jc(wF)j2:(3.41)ThisreducesthecalculationoftheareaSFjusttothesusceptibilityc(wF)ofthemode.ThissusceptibilitywithaccounttakenofthedispersivecouplingisgivenbyEqs.(3.12)and(3.22).ThebehavioroftheareaSFcanbefoundexplicitlyforsmallandlargead.Inthelimitofsmallad,wherethefrequencynoiseisweak,fromEq.(3.37)SFµa2d.Forlargead,itisconvenient50towriteŸc(t)inEq.(3.22)asŸc(t)ˇ(2=piad)å¥n=0exp[2n(iad)1=2(2n+1)adt],whereweassumedgd>0;theultimateresultisindependentofthesignofgd.Thesusceptibilityc(wF)isgivenbytheintegralofŸc(t)overt,Eq.(3.12).InthelimitGda1=2d˛jGidwFjfromEq.(3.12)c(wF)ˇ(2w0adGd)1åexp[2n(iad)1=2]=(2n+1).Totheleadingorderin1=adthisgivesc(wF)ˇ12ln(4ad)+ip4=4w0Gdad:(3.42)WeseefromEqs.(3.41)and(3.42)thatSFµa1dfallsdownwithincreasingadforlargead.ThenonmonotonicdependenceoftheareaSFontheparameterad,whichisexpectedfromtheaboveasymptoticexpressions,isindeedseeninFig.3.4(a).ThisshowstheareaSFasafunctionofthemotionalnarrowingparameteradfordifferentGd=G.ThepositionofthemaximumofSFsensitivelydependsonGd=G.Intermsofacomparisonwithexperiment,itisadvantageoustoscalethespectrumFF,andinparticulartheareaSF,bytheareaofthed-peakinthepowerspectrumofthedrivenmode.ThisareaisgivenbytheexpressionSd=(p=2)jc(wF)j2,cf.Eq.(3.1).ThequantitiesmeasuredintheexperimentareF2SFandF2Sd.TheunknownscaledintensityF2dropsoutfromtheirratio.FromEqs.(3.41)and(3.42)SF=Sdµad=ln2adincreaseswithadforlargead.Forsmallad,SF=Sdµa2dalsoincreaseswithad.Onthewhole,wefoundthatSF=Sdmonotonicallyincreaseswithad.ThisincreaseisseeninFig.3.4(b).3.7DispersivecouplingtoseveralmodesTheresultscanbeeasilyextendedtothecaseofdispersivecouplingtoseveralmodesratherthanasingled-mode.Weenumeratethemodesbythesubscript{=1;2;:::.Themodeseigenfrequen-ciesanddecayratesarew{andG{.Theenergyofthedispersivecouplingis(3=4)å{g{q2q2{.Thecontributionsofdifferentmodes{tothefrequencyofthestudiedmodeandthere-foretotherandomaccumulationofitsphaseareadditiveandmutuallyindependent.Todescribethedriving-inducedspectrum,onecanuseEqs.(3.13)and(3.14)andaverageoverthephaseac-51Figure3.4Theareaofthedriving-inducedpartofthepowerspectrumasafunctionofadfordifferentratioofthefrequencynoisebandwidthtothedecayrateofthedrivenmode2Gd=G.Thered(solid),blue(dashed),andgreen(dotted)linesrefertoGd=G=10,2,and0.1,respectively.TherelativedetuningofthedrivingfrequencyisdwF=G=5.Inpanel(a),theareaSFisscaledusingthenoise-freesusceptibilityc0(wF),Eq.(3.40),ŸSF=2SF=pjc0(wF)j2.Inpanel(b),SFisscaledbytheareaofthedpeakinthepowerspectrumofthedrivenmode,Sd=pjc(wF)j2=2.cumulationinEq.(3.14)independentlyforeachmode{.Theresultistheproductoftheaverages[functionsG(t;t0;t01)]calculatedforeachmodetakenseparately.Perhapsofutmostphysicalinterestarethecaseswhereofiseitherdispersivecouplingtooneorveryfewmodes,asforexampleinsomeoptomechanicalsystemsinwhicharadiationmodeisdispersivelycoupledtoamechanicalmode[50,58],orwherethereisdispersivecouplingtomanymodes,asmaybethecaseincarbonnanotubesorgraphenemembranes[6,7].Thepresentchapterisfocusedonthecase.Thesecondcasemaybesimpler,sincetheparametersg{ofcouplingtoindividualmodesaresmall;inparticular,innanomechanicalsystemsthisisaconsequenceofthedifferenceofthespatialstructureofthestudiedmodeandthemodes{.Ifthenumberofthe{-modesisN,inthethermodynamiclimit,N!¥,wewouldhaveg{µ1=N.Inthislimitthespectrumofthemodes{isalmostcontinuous,andfrequencyofthestudiedmodecomefromthecouplingµå{;{0g{{0q2q{q{0.Thiscouplingleadstoquasielastic52scatteringofmodes{offthestudiedmode,whichresultsinabroad-bandfrequencynoiseandintheLorentzianspectrumF0(w);[70,71]thespectrumFF(w)forabroad-bandfrequencynoiseisdiscussedinChap.2.3.7.1Anintermediatenumberofmodes:weakandeffectivelystrongcouplingAmoreinterestingsituationcanariseintheintermediatecaseofalargebutlimitednumberNofmodes{.Weassumethatthemodefrequenciesarewellseparated,jw{w{0j˛G{;G{0,andthefrequencydifferencesdonotresonatewithw0;2w0.BecauseNislarge,themotional-narrowingparametersa{=3jg{jkBT=8w0w2{G{canbesmall.However,a{arenotsmall.ForalargeN,onecanthinkofasituationwherethecumulativeeffectofthecouplingtomanymodesiseffectivelystrong.Foramultimodecoupling,thefactorŸc(t)intheaveragesusceptibilityhcsl(t;0)i(3.22)isgivenbytheproductoftheexpressions(3.22)forŸc(t)calculatedforeachmode{.[59]Fora{˝1Ÿc(t)ˇexpå{g{(t);g{(t)=2ia{G{tsgng{+a2{[12G{texp(2G{t)]:(3.43)ForlargeN,themostsimplerelevantcaseisthecaseofweakcoupling,whereå{a2{˝1.InthiscasethepowerspectrumF0(w)=(2kBT=w0)Imc(w)isclosetoLorentzian.FromEqs.(3.12),(3.22),and(3.43),totheleadingorderinå{a2{,Imc(w)ˇŸG=n2w0[ŸG2+(wŸw0)2]o;Ÿw0=w0+2å{a{G{sgng{;ŸG=G+2å{a2{G{:(3.44)Incontrasttothepreviouswork,[59]wedonotassumeherethatthehalfwidthofthespectrumisclosetoG;evenforsmalla{thedispersive-couplinginducedspectralbroadeningmaybecomecomparabletothedecayrateofthestudiedmodeforG{˛G.ForŸGG˝GoneshouldkeepinImc(w)othercorrectionsµa2{,whichmakethespectrumslightlynon-Lorentzian.[59].53Evenwherea{˝1,thesumå{a2{isnotnecessarilysmall.Wewillnowconsiderthecasewhereå{a2{G2{greatlyexceedsthescaledsquareddecayratesoftheinvolvedmodesG2{(1+2a{)2(typically,thisrequiresthatå{a2{˛1)andå{a2{G2{˛G2.Thisisthecaseofcumula-tivelystrongcoupling,wherethecouplingbecomesstrongbecauseofthelargenumberofmodesinvolved.OnecanseethatthemajorcontributiontotheFouriertransformofŸc(t)inEq.(3.22)[andinEq.(3.43),fora{˝1]comesfromthetimeranget.(å{a2{G2{)1=2.InthisrangeŸc(t)isgivenbyEq.(3.43)withtheexponentexpandedtosecondorderinG{t.ThenthepowerspectrumintheabsenceofdrivingF0(w)=(2kBT=w0)Imc(w)hasaGaussianspectralpeak.FromEqs.(3.12),(3.22),and(3.43),Imc(w)ˇ(p=8w20s2)1=2exp[(wŸw0)2=2s2];s2=4å{a2{G2{:(3.45)AGaussianshapeofthespectruminthecaseofmulti-modedispersivecouplingwasproposedtodescribethespectraofvibrationalmodesincarbonnanotubes.[6]Thisshapewas6]innumericalsimulationsofamodelwhereallG{werethesame.Thenumericalanalysis[6]furthershowedthatthetailsofthespectrumareLorentzian,whichisgenericfornonlinearlycoupledmodes[59]andisseenfromEq.(3.43).3.7.1.1Thedriving-inducedspectrumThespectrumFF(w)isdeterminedbytheFouriertransformoffunctionG2(t;t0;t01)which,asindicatedearlier,isgivenbytheproductofexpressions(3.23)calculatedforeachmode{.Forsmalla{G2(t;t0;t01)ˇexpg{(t1)+g0{(t3)+sgn(t0t01)å{a2{e2G{t21e2G{t11e2G{t3;(3.46)whereg{isgivenbyEq.(3.43),whereasg0{=g{fort0<t01andg0{=g{fort0>t01;therelationbetweent1;2;3andt;t0;t01isexplainedbelowEq.(3.23),seealsoSec.3.7.1.2.54Forsmallå{a2{andsmallfrequency-noiseinducedspectralbroadening,å{a2{G{˝Gandå{a2{G2{˝G2,thedriving-inducedspectrumisgivenbyEq.(3.37)inwhichthefactora2dG3d=[(wwF)2+4G2d]isreplacedbyå{a2{G3{=[(wwF)2+4G2{]andŸw0isgivenbyEq.(3.44).Inthecasewhereå{a2{G{&G,thespectrumFF(w)hasaLorentzianpeaknearŸw0describedbyEq.(3.38)inwhichoneshoulduseEq.(3.44)forŸG;Ÿw0,andc(w),andshouldreplaceinthenumeratora2dGdwithå{a2{G{.Thedriving-inducedtermFF(w)arisesalsointhecaseoftheeffectivelystrongcouplingwhereå{a2{G2{largelyexceedsthescaledsquareddecayratesG2;G2{(1+2a{)2.InthiscaseoneshouldkeepintheexponentinG2inEq.(3.46)onlytermsuptosecondorderint1;t3.FunctionG2thenshouldbeexpandedinaseriesinå{a2{G2{t1t3exp[2G{t2].TheresultofthecalculationisgiveninSec.3.7.1.2.Thegeneralexpressionssimplifyintheimportantcasewherethedecayrateoftheconsideredmodeissmallcomparedtothedecayratesofthe{-modes,G˝G{.InthiscaseFF(w)ˇ(2G)1Imc(w)Imc(wF):(3.47)Equations(3.45)and(3.47)showthat,forfrequencynoisewiththecorrelationtimesmallcomparedtothemodelifetimeG1,theleading-orderterminthedriving-inducedpowerspectrumFF(w)hasthesameshapeasthepeakinthepowerspectrumintheabsenceofdrivingF0(w).ThisbehaviorwasfoundearlierinChap.2forageneralfrequencynoiseprovidedthenoisespectrumismuchbroaderthanthewidthofthespectralpeakofF0(w)andthestandarddeviationofthenoise,inwhichcasethespectrumF0(w)isLorentzian;cf.alsoEq.(3.38).Inthepresentcasethecouplingisstrongandthewidthofthenoisespectrum˘maxG{issmallerthanthestandarddeviations,andasaresultFF(w)isnotproportionaltothesquaredcouplingparameterasinEq.(3.38).OthertermsintheexpressionforFF(w)obtainedinSec.3.7.1.2showthatthedriving-inducedspectrumisnonmonotonicalsonearthedrivingfrequency.Thestructureofthespectrumsensi-tivelydependsonthecouplingandthedecayratesofthe{-modes.Thegeneralexpressions(3.48)and(3.50)simplifyifall{modeshavethesamedecayrate.In55thiscaseitisalsopossibletosimulatethespectranumerically.Wehavecheckedthattheanalyticalresultsareinexcellentagreementwiththesimulations.Itisimportantthatthedispersive-couplinginducedterminthepowerspectrumofthedrivenmodeFF(w)isalwayspositive,whetherthedispersivecouplingismostlytoonemodeortomanymodes.3.7.1.2Driving-inducedspectrumforaneffectivelystrongdispersivecouplingtoalargenumberofmodesCalculationofthedriving-inducedpowerspectruminvolvesatripleintegralovertime,asseenfromEq.(3.13).Itisconvenienttoevaluatetheintegralsovert0;t01separatelyinthreeregions,A1,A2,andA3.RegionA1correspondsto¥<t0t01;¥<t010;theninEq.(3.46)t1=t;t2=t01;t3=t01t0.RegionA2correspondsto¥<t01t0;¥<t00;theninEq.(3.46)t1=t;t2=t0;t3=t0t01.RegionA3correspondsto0<t0t;¥<t010;theninEq.(3.46)t1=tt0;t2=t0;t3=t01.ExpandingtheexponentialsinG2(t;t0;t01)asdescribedinSec.3.7.1andintegratingG2withweightexp[(GidwF)(tt0)+(G+idwF)t01][seeEq.(3.13)],weobtainthecontributionsoftheregionsA1andA2intheformF(A1)F(w)ˇ12Re¥ån=0Kn¶nc(w)¶wn¶nc(wF)¶wnF;Kn>0=4nn!å{1;:::;{na2{1:::a2{nG2{1:::G2{n[2G+2nåi=1G{i]1F(A2)F(w)ˇ12Re¥ån=0Kn¶nc(w)¶wn¶nc(wF)¶wnF:(3.48)InthisequationK0=1=2G.Thesusceptibilityc(w)=(i=2w0)Z¥0dtei(wŸw0)tGts2t2=2(3.49)canbeeasilyexpressedintermsoftheerrorfunction;Ÿw0ands2aregivenbyEqs.(3.44)and(3.45).IntheregionA3itisconvenienttochangefromintegrationovert0tointegrationoverŸt0=tt0.TothespectrumFF,itisconvenienttointegrateG2withtheappropriateweight56overtandthenoverŸt0.OneshouldtakeintoaccountthatŸt0.1=s˝1=G{,buttherangeofthevaluesoftthatcontributetotheintegralisnotlimitedto.1=s.TheresultofintegrationreadsF(A3)F(w)ˇIm[c(w)c(wF)]=[2(wwF)]+12Re¥ån=1K0n¶nc(w)¶wn¶nc(wF)¶wnF:(3.50)Here,thecoefK0naregivenbytheexpression(3.48)forKninwhich2Gisreplacedbyi(wwF).WenotethatF(A3)Fisnotsingularatw=wF,sinceImjc(wF)j2=0andc(w)isasmoothfunctionoffrequency;thecorrespondingtermisimportantprimarilywhereeitherworwFareonthetailofthespectralpeakF0(w).TheseriesoverninEqs.(3.48)and(3.50)generallyconvergesslowlyifthedecayratesofthe{modesG{.G.Forlargen,inEq.(3.50)thederivative¶nc(w)=¶wnshouldbecalculatedwiththedecayrateGreplacedwithG+2åni=1G{iinEq.(3.49).Thesummationoverthemodes{iinthecoefK0nshouldnowbeextendedtoincludethec(w),whichnowitselfdependson{i.Wenotethatc(wF)inEq.(3.50)shouldstillbecalculatedusingEq.(3.49).Theoveralldriving-inducedterminthepowerspectrumFF(w)=F(A1)F(w)+F(A2)F(w)+F(A3)F(w)haspeaksand,generally,morecomplicatedfeaturesnearboththeoscillatoreigenfre-quencyandthedrivingfrequency.3.8PowerspectrumofadrivennonlinearoscillatorAnimportantcontributiontothebroadeningofthespectraofmesoscopicoscillatorscancomefromtheirinternalnonlinearity.[4]Thevibrationfrequencyofanonlinearoscillatordependsonthevibrationamplitude.ThereforethermaloftheamplitudeleadtofrequencyTheanalysisofthespectraiscomplicatedbytheinterplayofthefrequencythatcomefromtheamplitudeandthefrequencyuncertaintythatcomesfromtheoscillatordecay.Neverthelessthelinearsusceptibilitycouldbefoundforanarbitraryrelationbetweenthestandard57deviationofthefrequencyDwandthedecayrateG.[59].Thepowerspectrumofanonlinearoscillatorintheabsenceofdrivingisgenerallyasymmetricandnon-Lorentzian.Findingthedriving-inducedtermsinthepowerspectrumisstillmorecomplicated.Theos-cillatordisplacementisnonlinearinthedrivingamplitudeF,andthedriving-inducedpartofthepowerspectrumF(w)isnotquadraticinF.However,iftheisweak,Eq.(3.1)forF(w)applies.InthecalculationofFF(w)oneshouldtakeintoaccounttermsintheoscillatordisplacementthatarequadraticinF,whichisgenericfornonlinearsystems.[16]Weassumethatthenonlinearpartoftheoscillatorenergyissmallcomparedtothelinearpart.Thenthenonlineartermintheoscillatorenergycanbetakenintheformofgq4=4.[72]TheoscillatorequationofmotionintherotatingwaveapproximationisgivenbyEq.(3.7)withgd=0,u=(G+idwF)u+3ig2w0juj2uiF4w0+f(t):(3.51)Inthissectionwedonotdiscusstheeffectofdispersivecoupling,andthefrequencynoisethatcomesfromthiscouplingisnotincludedintoEq.(3.51).ToFF(w),weconsiderthedynamicsofadrivennonlinearoscillatorwithouttuationsandthentakeintoaccount.ThestationarysolutionustofEq.(3.51)intheabsenceofthenoisef(t)canbefoundbysettingu=0.Forweakdriving,ustisaseriesinF,whichcontainsonlyoddpowersofF.SinceweareinterestedinthetermswhicharelinearorquadraticinF,itissuftokeeponlytheleadingterm,ust=F=4iw0(G+idwF).OnethensubstitutesintoEq.(3.51)u(t)=ust+du(t).Thedeviationdu(t)isdueonlytothenoise,du=(idwF+G)du+3ig2w0jduj2du+2ustjduj2+ustdu2+2justj2du+u2stdu+f(t):(3.52)Timeevolutionofdu(t)dependsonthedrivingintermsofust.Wethistimeevolutioninthetwolimitingcases.583.8.1WeaknonlinearityTheanalysisofthedynamicsinthecaseofsmallnonlinearity-inducedspreadoftheoscillatorfrequencyDwcomparedtothedecayrateG.AsseenfromEq.(3.51),intheabsenceofdrivingthefrequencyshiftisquadraticinthevibrationamplitudeµjuj2,[72],andthereforethefrequencyspreadisdeterminebythestandarddeviationofjuj2duetothethermalnoise.ThisgivesDw=3jgjkBT=8w30.ForDw˝G,itissuftokeeponlythelinearindutermsinEq.(3.52).[73,74]Astraight-forwardcalculationthengivesasimpleexpressionforthethedriving-inducedpowerspectrum,FF(w)ˇ3gkBT8w50(ww0)G(G2+dw2F)[G2+(ww0)2]2:(3.53)Thespectrum(3.53)isproportionaltothederivativeoftheLorentzianspectrumofthehar-monicoscillatorF0(w)µ1=[G2+(ww0)2]overw.Ithasacharacteristicdispersiveshape,beingoftheoppositesignsontheothersidesofw0.ThisistheresultoftheshiftoftheoscillatorvibrationfrequencyµgF2duetothedriving.SuchshiftisthemaineffectofthedrivingforsmallDw=G.3.8.2LargedetuningofthedrivingfrequencyForarbitraryDw=G,theanalysisisifthedetuningofthedrivingfrequencyfromthesmall-amplitudeoscillatorfrequencyjdwFj˛G;Dw.Inthiscase,onecanchangevariablesinEq.(3.52)todŸu(t)=du(t)eidwFt.Theright-handsideoftheresultingequationfordŸu,besidesthenoiseterm,hastermsthatsmoothlydependontimeonthescalejdwFj1andtermsthatoscillateasexp(idwFt);exp(2idwFt).Theseoscillatingtermscanbeconsideredaperturbation.Totheorderoftheperturbationtheory,theequationforthesmoothtermstakestheformdŸu=GdŸu+3igw0justj2dŸu+ 1+9gjustj2w0dwF!3ig2w0jdŸuj2dŸu+Ÿf(t);(3.54)whereŸf(t)=f(t)eidwFt.Wekeepinthisequationthetermsµjustj2µF2.ThesetermscontributetothespectrumFF(w).Thetermsofhigheroderinjustj2havebeendiscarded.59Equation(3.54)hasthesameformastheequationofmotionforthecomplexamplitudeu(t)intheabsenceofdriving,i.e.,Eq.(3.51)withF=0.ThenoiseŸf(t)hasthesamecorrelationfunctionasf(t).ThereforethepowerspectrumofdŸu(t)isthesameasthepowerspectrumofanonlinearoscillatorfoundearlier,[59]withtherenormalizedparameters:theeigenfrequencyisshiftedby3gjustj2=w0andthenonlinearityparameterismultipliedbythefactor1+9gjustj2=w0dwF.Wenotethatthecorrectionµjustj2inthisfactor,whichcomesfromtheperturbationtheoryin1=dwF,issmall.ToFF(w)wehavetoexpandtheresult[59]withtheappropriatelyrenormalizedparame-terstotheorderinjustj2.ThisgivesF2FF(w)=bf¶b[F0(w2bdwF;Dw(1+6b))]gb=0;F0(w;Dw)=kBTw20ÂZ¥0dtexpf[i(ww0)+G]tg[cosh(at)+(G=a)(1+2iasgng)sinh(at)]2:(3.55)Theparametersaandahavethesamestructureandthesamephysicalmeaningastheparametersadandadusedbefore,a=Dw=Ganda=G(1+4iasgng)1=2,whereasb=3gF2=32w30(dwF)3isthescaledintensityofthedrivingThemajorcontributiontoFF(w)asgivenbyEq.(3.55)forlargejdwFj=DwcomesfromthefrequencyshiftofthespectrumwithoutdrivingF0(w)andisdeterminedby2dwF¶wF0(w;Dw).Physically,thisresultsagaincorrespondstotheshiftoftheoscillatoreigenfrequencyassociatedwiththeforcedvibrations,andthespectrumFFagainhasthecharacteristicshapeofadispersivecurve.Tothenextorderin1=dwF,thedrivingbroadensornarrowsthespectrumdependingonthesignofg=dwFbyrenormalizingthenonlinearity-inducedstandarddeviationoftheoscillatorfrequencyDw.3.8.3NumericalsimulationsTheanalyticalresultsonthespectraofthemodulatednonlinearoscillator,Eq.(3.55),arecomparedwiththeresultsofnumericalsimulationsinFig.3.5.ThespectrumFF(w)generallyhasapositive60Figure3.5Thedriving-inducedpartofthepowerspectrumofanonlinearoscillatorforlargede-tuningofthedrivingfrequency,dwF=Dw=40.Thesolidcurvesandthedotsshowtheanalyticalexpressionsandtheresultsofsimulations,respectively.Thevaluesofthenonlinearityparameterandthescaleddrivingstrengthforthecurves1to3are,respectively,aDw=G=0.125,1.25,and5,andb3gF2=32w30dw3F=0.016,0.004,and0.004.TheinsetshowsthechangeofthepowerspectrumintheabsenceofdrivingwithvaryingDw=G.andnegativeparts,inadramaticdistinctionfromthecaseofalinearoscillatordispersivelycoupledtoanotheroscillator.AsDw=Gincreases,theshapeofFF(w)becomesmorecomplicated,inparticular,thepositiveandnegativepartsbecomeasymmetric.Thesimulationswereperformedinthesamewayasforthedispersivelycoupledmodesbyintegratingthestochasticdifferentialequations(3.51).Wevthatthevaluesofthemodu-latingamplitudeFwereintherangewherethedriving-inducedterminthepowerspectrumwasquadraticinF.Asseenfromthisthesimulationsareinexcellentagreementwiththeanalyticalresults.Intheintermediaterange,wherethenonlinearityisnotweakandthedrivingisnottoofardetuned,i.e.,jdwFj˘max(G;Dw),weobtainedthespectrumFF(w)byrunningnumericalsimu-lations.TheseresultsarepresentedinFig.3.6.TheyshowthatthegeneraltrendseeninFig.3.5thatFF(w)changessignsandisasymmetricforanonlinearoscillatorpersistsinthiscaseaswell.61Figure3.6Thedriving-inducedpartofthespectrumofanonlinearoscillatorforsmalldetuningofthedrivingfrequency.Thesolidcurve(red)showstheanalyticalresultsforFF(w)forsmallDw=Gforthesameparametersasthedottedcurve1.Thedotsshowtheresultsofsimulations.Thescaledvaluesofthenonlinearityparameter,thedetuning,andthedrivingstrengthonthecurves1and2are,respectively,aDw=G=0.05,and1.25,dwF=G=0.5and5,andb3gF2=32w30(dwF)3=0.64and0.01.Theinsetshowsthefullspectrumfortheparametersofcurve2(bluedots,simulations);thespectrumwithoutdrivingforthesameDw=Gisshownbythesolidline(analytical)and(green)dotsontopofthisline,whichareobtainedbysimulations.3.9ConclusionsIntermsofexperimentalstudiesofmesoscopicvibrationalsystems,themajorresultofthischapteristhesuggestionofawaytosingleoutandcharacterizethedispersive(nonresonant)couplingbetweenvibrationalmodes.Theproposedmethodallowsrevealingdispersivecouplingevenwherethereisnoaccesstothemodecoupledtothestudiedone.Wehaveshownthatdispersivecouplingleadstoagenerallydouble-peakextrastructureinthepowerspectrumofamodewhenthismodeisdrivenclosetoresonance.Thedispersive-couplinginducedpartofthepowerspectrumisquadraticinthedrivingamplitude.Itvarieswiththedetuningofthedrivingfrequencyfromthemodeeigenfrequency.The"tuneofftoreadoff"approach,whichreliesonchangingthedrivingfrequency,allowsonetostudyseparatelytwoeffects.Oneisthedispersive-couplinginducedbroadeningofthespectralpeakofthelinearresponse,whichisofinterestformesoscopicmodes.[6,55,49,7,56]Theotheristhedecayofthefiinvisible"modethatisdispersivelycoupledtothestudiedmode.Thedouble-peakstructureofthedriving-inducedpowerspectrumsensitivelydependsbothonthe62strengthofthedispersivecouplingandtheparametersoftheinvisiblemode.Anotherimportantfeatureofthedriving-inducedspectrumisthequalitativedifferencebe-tweentheeffectsofnonlineardispersivecouplingtoothermodesandtheinternalnonlinearityofthestudiedmode.Bothnonlinearitiesareknowntobroaden,inasomewhatsimilarway,[59]thelinearresponsespectruminthepresenceofthermalHowever,inthecaseofinternalnonlinearity,thedriving-inducedpartofthepowerspectrumchangessignasafunctionoffre-quency,i.e.,ithaspeaksoftheoppositesignsandissimilar(andisclose,inacertainparameterrange)tothederivativeofthepowerspectrumwithoutdriving.Wehaveextendedtheresultstothecaseofdispersivecouplingtoseveralmodes.Thecon-tributionsofdifferentmodestothefrequencyofthestudiedmode,andthereforetotherandomaccumulationofitsphase,areadditiveandmutuallyindependent.Thentheaveragingoverthephaseaccumulationcanbedoneindependentlyforeachmode.Theextensiontothecaseofafewmodesisthereforestraightforward.Newfeaturesemergeiftherearemany,butnottoomanymodes.Thecumulativeeffectofweakdispersivecouplingtomanymodesmayleadtoaneffectivelystrongcoupling.Asaresult,thespectrumwithoutdrivingbecomesclosetoGaussianinthecentralpart,aswassuggestedinRef.[6].Thedriving-inducedpartofthepowerspectrumdis-playsacharacteristicstructure,whichsensitivelydependsbothontheparametersofthedispersivecouplingandthedissipationparametersoftheinvolvedmodes.Intermsofthetheory,thechapterdescribesapath-integralmethodthatenablesinanexplicitformthespectrumofadrivenoscillatorinthepresenceofnon-Gaussianofitsfrequency,whichresultfromdispersivecouplingtoothermodes.Theresultsapplyforanarbitraryratiobetweentherelevantparametersofthesystem.Theseparametersarethemagnitude(standarddeviation)ofthefrequencyDw,theirreciprocalcorrelationtime,whichisgivenbythedecayrateofthedispersively-coupledmodethatcausesthethedecayrateofthedrivenmodeitself,andthedetuningofthedrivingfrequency.Itisthepresenceofseveralparametersthatmakesitcomplicatedtoidentifythebroadeningmechanismsfromthelinearresponsespectra.Theresultsofthechaptershowthequalitative63differencebetweentheeffectsoftheseparametersonthepowerspectrumwhentheoscillatorisdriven.ThisenablestheirGenerally,inmesoscopicvibrationalsystems,andinparticularinnanomechanicalsystems,theinternal(Dufanddispersivenonlinearitiescanbeofthesameorderofmagnitude.Ifthestudiedmodehasamuchhigherfrequencythanthemodetowhichitisdispersivelycoupled,itscanbecomparativelyweakermakingtheeffectofthedispersivecouplingstronger.Alsoifthereareseveralmodesdispersivelycoupledtothemodeofinterest,theircumulativeeffectcanbestrongerthantheeffectoftheinternalnonlinearity.Thismakesitevenmoreimportanttobeabletodistinguishtheeffects,whichtheproposedapproachallows.Theresultsimmediatelyextendtotheparameterrangewherethedrivenmodehashighfre-quencyandisinthequantumregime,¯hw0>kBT.Thisisbecause,aslongasthemodeitselfislinear,itsdisplacementisasuperpositionofthedisplacementwithoutdriving,whichisaffectedbyquantumandtheclassicaldriving-induceddisplacement.Theeffectofdispersivecouplingtoaclassicalmode(¯hwd˝kBT)onthedriving-induceddisplacementisindependentof¯hw0=kBT.Dispersivecouplingofaquantummodetoaclassicalmodeisofparticularinterestforoptomechanics,wherethehigh-frequencyopticalcavitymodecanbedispersivelycoupledtoalow-frequencymechanicalmode.[50,58]Drivingthecavitymodeleadsinthiscasetoacharac-teristicradiationdescribedinthischapter.64CHAPTER4FLUCTUATIONSPECTRAOFDRIVENOVERDAMPEDNONLINEARSYSTEMS4.1IntroductionFluctuationspectraandspectraofresponsetoperiodicdrivingaremajortoolsofcharacterizingphysicalsystems.Thespectraareconventionallyusedtosystemfrequenciesandrelaxationratesandtocharacterizeinthesystem.Forexample,opticalabsorptionspectragivethetransitionfrequenciesofatomicsystemsandthelifetimesoftheexcitedstates,andthespec-trumofspontaneousradiationisawell-knownexampleofthe(power)spectrum[75].Inmacroscopicsystemsthespectraareoftencomplicatedbytheeffectsofinhomogeneousbroad-ening.Recentprogressinnanosciencehasmadeitpossibletostudythespectraofindividualdynamicalsystems.Awell-knownexampleisprovidedbyopticallytrappedBrownianparticlesandbiomolecules[76,77],wherethepowerspectraareamajortoolforcharacterizingthemotioninthetrap[78,79].Spectraofvarioustypesofindividuallyaccessiblemesoscopicsystemsarestudiednowadaysinoptics[80,81],nanomechanicsandcircuitquantumelectrodynamics,cf.[4],biophysics,cf.[82,83],andmanyotherareas;thetechniquebasedonspectralmeasurementshasfoundvariousapplications,photonicforcemicroscopybeingarecentexample,seeRef.[84].Afamiliareffectofweakperiodicdrivingisforcedvibrationsofthesystem.Whenensemble-averaged,theyarealsoperiodicandoccuratthedrivingfrequencywF.Theyleadtoad-shapepeakatfrequencywFinthesystempowerspectrum.However,thedrivingalsothepowerspectrumawayfromwF.Atextbookexampleisinelasticlightscatteringandresonancecence.Inthebothcases,thesystemdrivenbyaperiodicelectromagneticemitsradiationatfrequenciesthatdifferfromthedrivingfrequency[10].Thisradiationisoneofthemajorsourcesofinformationaboutthesysteminopticalexperiments.Inthischapterwestudythespectraofperiodicallydrivennonlinearsystems.Weshowthat,65inthepresenceofnoise,alongwiththed-shapepeakatthedrivingfrequencywF,thesespectradisplayacharacteristicstructure.Weareinterestedintheregimeofrelativelyweakdriving,wherethedriving-inducedchangeofthepowerspectrumisquadraticintheamplitudeofthedriving,asininelasticlightscattering.Inviewoftheinterestinthepowerspectraofsystemsopticallytrappedinweconsidersystemswhereinertialeffectsplaynorole.Intheabsenceofdrivingthepowerspectraofsuchsystemsusuallyhaveapeakatzerofrequency.Inparticularitisthispeakthatisusedtocharacterizethedynamicsofopticallytrappedparticles.Foralinearsystem,likeaBrownianparticleinaharmonictrap,thed-shapepeakatwFistheonlyeffectofthedrivingonthepowerspectrum.ThisisbecausemotionofsuchasystemisalinearsuperpositionofforcedvibrationsatwFandintheabsenceofdriving.Theamplitudeandphaseoftheforcedvibrationsdependontheparametersofthesystemanddeterminethestandardlinearsusceptibility[85].Innonlinearsystemsforcedvibrationsbecomerandom,becausetheparametersofthesystemareThepowerspectrumofsuchrandomvibrationsisnolongerjustad-shapepeak(althoughthed-shapepeakisnecessarilypresent).Thedriving-inducedspectralfeaturesawayfromwFresultfrommixingofandforcedvibrationsinanonlinearsystem.4.1.1QualitativepictureTheideaofthedriving-inducedchangeofthepowerspectrumcanbegainedbylookingataBrownianparticleinapotential,atypicalsituationforopticaltrapping.Themotionoftheparticle,afterproperrescalingoftimeandparticlecoordinateq,isdescribedbytheLangevinequation[86]q=U0(q)+f(t);U0(q)dU=dq;(4.1)whereU(q)isthescaledpotentialandf(t)isthermalnoise.IfpotentialU(q)isparabolicandthesystemisadditionallydrivenbyaforceFcoswFt,forcedvibrationsaredescribedbythetextbook66expressionhq(t)i=12Fc(wF)exp(iwFt)+c.c.;c(w)=[U00(qeq)iw]1;(4.2)whereqeqistheequilibriumposition[theminimumofU(q)]andc(w)isthesusceptibility.ForanonlinearsystemthepotentialU(q)isnonparabolic.BecauseofthermalthelocalcurvatureofthepotentialU00(q)isIntuitively,onecanthinkoftheeffectofthermalonforcedvibrationsasifU00(qeq)inEq.(4.2)forthesusceptibilitywerereplacedbyacurvature,seeFig.4.1.IfthedrivingfrequencywFlargelyexceedsthereciprocalcorrelationtimeofthet1c,thewouldleadtotheonsetofastructureinthepowerspectrumnearfrequencywFwithtypicalwidtht1c.Thequantityt1calsogivesthetypicalwidthofthepeakinthepowerspectrumatzerofrequencyintheabsenceofdriving[foralinearsystem,t1c=U00(qeq)].Figure4.1Sketchofapotentialofanonlinearsystemnearthepotentialminimum.BecauseoftheinterplayofnonlinearityandthecurvatureofthepotentialThesetionsareshownasthesmearingofthesolidline,whichrepresentsthepotentialintheabsenceofAnothereffectoftheinterplayofdriving,nonlinearity,andcanbeunderstoodbynoticingthattheperiodicforcecausesaperiodicchangeinthesystemcoordinate.Foranonlinearsystem,roughlyspeaking,thisleadstoaperiodicmodulationofthelocalcurvature,andthusoft1c.Sincet1cdeterminestheshapeofthezero-frequencypeakinthepowerspectrum,suchmodulationcausesachangeofthispeakproportionaltoF2,tothelowestorderinF.67Evenfromtheabovesimplisticdescriptionitisclearthatthedriving-inducedchangeofthespectrumissensitivetotheparametersofthesystemandthenoiseandtothenonlinearitymech-anisms.Explicitexamplesgivenbelowdemonstratethissensitivityandsuggestthattheeffectswediscusscanbeusedforcharacterizingasystembeyondtheconventionallinearanalysis.AfterformulatinghowthepowerspectrumcanbeevaluatedinSection4.2,wedemonstratetheeffectsoftheinterplayofdrivingandforthreeverydifferenttypesofnonlinearsystems:anoverdampedBrownianparticle(Section4.3),asystemthatswitchesatrandombetweencoexistingstablestates(Section4.4),andathresholddetector(Section4.5).Allthesesystemsareofbroadinterest,andallofthemdisplayadriving-inducedchangeofthepowerspectrum.4.2GeneralformulationWeconsidersystemsdrivenbyaperiodicforceFcos(wFt)andassumethatareinducedbyastationarynoise,likeinthecaseofanopticallytrappedBrownianparticle,forexample.Afteratransienttimesuchsystemreachesastationarystate.Thestationaryprobabilitydistributionofthesystemwithrespecttoitsdynamicalvariableq,rst(q;t),isperiodicintimetwiththedrivingperiodtF=2p=wF.Thetwo-timecorrelationfunctionhq(t1)q(t2)i[himpliesensembleaveraging]isafunctionoft1t2andaperiodicfunctionoft2withperiodtF.ThepowerspectrumusuallymeasuredinexperimentisoftheformF(w)=2ReZ¥0dteiwthhq(t+t0)q(t0)ii;hhq(t+t0)q(t0)ii=1tFZtF0dt0hq(t+t0)q(t0)i:(4.3)ThecorrelationfunctioninEq.(4.3)canbeexpressedintermsofrst(q;t)andthetransitionprobabilitydensityr(q1;t1jq2;t2)thatthesystemthatwasatpositionq2attimet2isatq1attimet1t2,hq(t1)q(t2)i=Zdq1dq2q1q2r(q1;t1jq2;t2)rst(q2;t2):(4.4)68Forweakdriving,functionrst(q;t2)canbeexpandedinaseriesinFexp(iwFt2)withtime-independentcoefwhereasr(q1;t1jq2;t2)canbeexpandedinFexp(iwFt2)withcoefcientsthatdependont1t2.Thereforethepowerspectrum(4.3)doesnothavetermslinearinF.TothesecondorderinFforw0wehaveF(w)=F0(w)+p2F2jc(wF)j2d(wwF)+F2FF(w):(4.5)ThetermF0(w)describesthepowerspectrumofthesystemintheabsenceofdriving.Thetermµd(wwF)describestheconventionallinearresponse,cf.Eq.(4.2).However,theexpressionforthesusceptibilityc(w)innonlinearsystemsisfarmorecomplicatedthanEq.(4.2);generally,thesusceptibilityisdeterminedbythelinearinFterminrst(q;t).Intheopticallanguage,thetermµd(wwF)in(4.5)correspondstoelasticscatteringoftheFcoswFtbythesystem.OfprimaryinteresttousisthetermFF(w).Thistermisoftendisregardedintheanalysisofthepowerspectraofdrivensystems,whilethemajoremphasisisplacedonthed-functioninEq.(4.5).FunctionFF(w)describestheinterplayofanddrivinginanonlinearsystembeyondthetriviallinearresponse.Intheconsideredlowest-orderapproximationinthedrivingamplitude,FFdoesnotcontainad-peakat2wF.However,itmaycontainad-peakatw=0,whichcorrespondstothestaticdriving-inducedshiftoftheaveragepositionofthesystem.Inwhatfollowswedonotconsiderthispeak,asthestaticequilibriumpositioncanbemeasuredindependently.FunctionFFcanbefoundfromEq.(4.4)bycalculatingthetransitionprobabilitydensityandthestationaryprobabilitydistribution.ThiscanbedoneforMarkovsystemsnumericallyandalso,inthecaseofweaknoise,analytically,seeSecs.4.3and4.4.Alternatively,functionFFcanberelatedtooflinearandnonlinearresponseofthesystemandexpressedintermsofthelinearandnonlinearsusceptibility,seeSec.4.6.Weemphasizethatthenonlinearresponsehastobetakenintoaccountwhenareconsideredeventhoughwearenotinterestedinthebehaviorofthepowerspectrumnear2wForhigherovertonesorsubharmonicsofwF.694.3PowerspectrumofadrivenBrownianparticleAsimpleexampleofasystemwhereFF(w)displaysanontrivialbehaviorisaperiodicallydrivenoverdampedBrownianparticleinanonlinearpotentialU(q),seeEq.(4.1).Thismodelimmediatelyrelatestomanyexperimentsonopticallytrappedparticlesandmolecules.Wewillassumethatthermalnoisef(t)iswhiteandGaussianandthatitisnotstrong,sothatitsuftokeepthelowest-ordernontrivialtermsinthepotential,U(q)=12kq2+13bq3+14gq4+:::;hf(t)f(t0)i=2Dd(tt0);(4.6)whereDµkBTisthenoiseintensity.Intheabsenceofdrivingthestationaryprobabilitydistribu-tionisoftheBoltzmannform,r(0)stµexp[U(q)=D].ForsmallDandweakdrivingforceequationofmotionq=U0(q)+f(t)+FcoswFtcanbesolveddirectlybyperturbationtheoryinthenoisef(t)andinF,asindicatedinSec.4.6.Herewedevelopadifferentmethod,whichisparticularlyconvenientifonewantstogotohighordersoftheperturbationtheoryinDandF.4.3.1MethodofMomentsSystemsinwhichareinducedbywhitenoisecanbestudiedusingtheFokker-Planckequation¶tr=¶qU0(q)+FcoswFtr+D¶2qr:(4.7)Thisequationcanbesolvednumerically.Aconvenientanalyticalapproachisbasedonthemethodofmoments,whichareasMn(w;t0)=Z¥0dteiwtZdqqnZdq0r(q;t+t0jq0;t0)q0rst(q0;t0):(4.8)FromEq.(4.5),thepowerspectrumisF(w)=(2=tF)ReRtF0dt0M1(w;t0).ThemomentsMnsatisfyasetofsimplelinearalgebraicequationsiwMn(w)+n‹F[Mn(w)]=Dn(n1)Mn2(w)+12FheiwFt0nMn1(w+wF)+eiwFt0nMn1(wwF)i+Qn+1(t0):(4.9)70Here,weskippedtheargumentt0inMnandintroducedfunction‹F[Mn]kMn+bMn+1+gMn+2:.FunctionsQn(t)=Zdqqnrst(q;t)(4.10)intheright-handsideofEq.(4.9)canthemselvesbefoundfromasetoflinearequationssimilarto(4.9).TheyfollowfromEq.(4.7),ifonesetsr=rst(q;t)andtakesintoaccountthatrst(q;t)isperiodicint.TothelowestorderinFitsuftokeepinQn(t)onlytermsthatareindependentoftoroscillateasexp(iwFt);respectively,inEq.(4.10)Qn(t)ˇQ(0)n+hQ(1)nexp(iwFt)+c.c.i,and‹F[Q(0)n]=D(n1)Q(0)n2+FReQ(1)n1;iwFQ(1)n+n‹F[Q(1)n]=Dn(n1)Q(1)n2+12nFQ(0)n1:(4.11)ThesystemofcoupledlinearequationsforthemomentsMnandQncanbequicklysolvedwithconventionalsoftwaretoahighorderinthenoiseintensityD.NontrivialresultsemergealreadyifwekeeptermsµDF2:thesearethetermsthatcontributetothepowerspectrumFF(w)tothelowestorderinD.TothemitsuftoconsidertermsMnwithn3andQnwithn4.ThisgivesFF(w)ˇ2D(k2+w2F)(k2+w2)2(2b2(4k2+w2F)(k2+w2+w2F)[k2+(wwF)2][k2+(w+wF)2]3gkg:(4.12)Thisexpressionreferstojwj>0;functionFF(w)containsalsoad-peakatw=0,whichcomesfromthedriving-inducedshiftoftheaveragestaticvalueofthecoordinate.Thesolutionoftheequationsforthemomentsintheconsideredapproximationgivesacorrec-tionµD2tothepowerspectrumintheabsenceofdrivingF0(w).TothelowestorderinDthisfunctiondisplaysaLorentzianpeakatw=0,F0(w)=2D=(k2+w2).ThispeakisusedintheanalysisofopticaltrapsforBrownianparticles[78,79];withaccounttakenofthetermµD2thezero-frequencypeakofF0(w)becomesnon-Lorentzian.714.3.2PowerspectrumforcomparativelylargedrivingfrequencyTheinterpretationofEq.(4.12)isinthecasewherethedrivingfrequencyexceedsthedecayrate,wF˛k.Inthiscase,periodicdrivingleadstotwowell-resolvedfeaturesinthespectrumFF.Oneislocatedatw=0andhastheformFF(w)ˇ(2D=w2F)(2b23gk)(k2+w2)2(w˝wF):(4.13)Thisequationcanbeeasilyobtaineddirectlybysolvingtheequationofmotionq=U0(q)+FcoswFt+f(t)byperturbationtheoryinwhichq(t)isseparatedintoapartoscillatingathighfrequencywF(anditsovertones)andaslowlyvaryingpart.TothelowestorderinFandD,thefastoscillatingpartrenormalizesthedecayrateoftheslowlyvaryingpartofq(t),withk!k(F=wF)2hk1b2(3=2)g)i.UsingthiscorrectionintheexpressionforthepowerspectrumofalinearsystemF(0)0(w)=2D=(k2+w2),oneimmediatelyobtainsEq.(4.13)totheleadingorderink=wF.Interestingly,Eq.(4.13)describesapeakoradipdependingonthesignof2b23gk.Thatis,thesignofFFisdeterminedbythecompetitionofthecubicandquarticnonlinearityofthepotentialU(q).Thisshowshighsensitivityofthespectrumtothesystemparameters.Thetypicalwidthofthepeak/dipofFFnearw=0isk;theshapeofthepeak/dipisnon-Lorentzian.TheotherspectralfeatureislocatedatwFandnearthemaximumhastheformofaLorentzianpeak,FF(w)ˇ(Db2=w4F)[k2+(wwF)2]1.Theheightofthispeakissmallerbyafactork2=w2F˝1thantheheightofthefeaturenearw=0.WenotethattheheightofthepeakatwFisproportionaltothesquaredparameterofthecubicnonlinearityofthepotentialb,butisindependentofthequartic-nonlinearityparameterg,totheleadingorderinthenoiseintensityD.InFig.4.2wecomparetheanalyticalexpression(4.12)withtheresultsofnumericalsimula-tions.Thesimulationsweredonebyintegratingthestochasticdifferentialequationq=U0(q)+f(t)+FcoswFtusingtheHeunscheme(cf.[69]).Panel(a)showsthatthecubicnonlinearityofthepotentialleadstoapeakatw=0andacomparativelysmallpeakatwF.Thespectrumbecomesmoreinterestinginthegenericcasewherebothcubicandquartictermsinthepotential72Figure4.2ScaleddrivinginducedtermsinthepowerspectrumofanoverdampedBrownianparti-clemovinginthequarticpotentialU(q)givenbyEq.(4.6),ŸFF(w)=102k2FF(w)=2D.Panels(a),(b),and(c)refertothescaledcubicnonlinearityb2D=k3=0:002andquarticnonlinearitygD=k2=0.0006,0.00147,and0.002,respectively.TheblackdotsandredsolidcurvescorrespondtothenumericalsimulationsandEq.(4.12).ThescaleddrivingfrequencyiswF=k=5andthedrivingstrengthiskF2=w2FD=20.Forthisdrivingstrengthandthenoiseintensity,thesimula-tionresultsinpanels(b)and(c)deviatefromthetheoreticalcurve.Thedeviationdecreasesforweakerdriving.Thisisseenfromthesimulationdatainpanel(b)thatrefertokF2=w2FD=5(bluetriangles)and1.25(greensquares).Thecorrespondingspectraarescaledupbyfactors4and16,respectively..arepresentandb2iscomparabletogk.Here,asseenfrompanel(b),asaresultofthecompetitionbetweentheseterms,FF(w)canhaveadipatw=0andtwopeaks,onenearwFandtheotherwiththepositiondeterminedbyb2=gkandwF=k.Wherethequarticnonlinearitydominates,gk˛b2,seepanel(c),itishardtodetectthepeakatwFforsmallnoiseintensity.Ouranalyti-calcalculationsandnumericalsimulationsshowthat,forlargernoiseintensity,thispeakbecomesmorepronounced.Adeviationofsimulationsandtheasymptoticexpression(4.12)inpanel(b)forsmallwisaconsequenceofthenearcompensationofthecontributionstoFF(w)fromthecubic73andquarticnonlineartermsinU(q)tothelowestorderinF2andD.Thetermsofhigher-orderinDandF2becomethensubstantial.Panel(b)illustrateshowthedifferenceisreducedifF2isreduced.WecheckedthatbyreducingalsothenoiseintensityweobtainaquantitativeagreementofsimulationswithEq.(4.12).Insomecasesthepotentialofanoverdampedsystemhasinversionsymmetry,andthenb=0inEq.(4.6).InsuchcasesspectralfeaturesofFFatthedrivingfrequencyareµ(gD)2.TheycanbefoundbysolvingtheequationsforthemomentsMnwithn5andQnwithn6orbysolvingtheequationsofmotionbyperturbationtheorytothesecondordering,seeSec.4.6.4.4Powerspectrumofadriventwo-statesystemWenowconsidertheeffectofdrivingonatwo-statesystem.Varioustypesofsuchsystemsarestudiedinphysics,fromspin-1/2systemstotwo-levelsystemsindisorderedsolidstoclassicalBrownianparticlesmostlylocalizedattheminimaofdouble-wellpotentials.WewillassumethatthesystemdynamicsarecharacterizedbytheratesWijofinterstatei!jswitching,wherei;j=1;2.Inthecaseofquantumsystems,thismeansthatthedecoherenceratelargelyexceedsWij;inotherwords,thetypicaldurationofaninterstatetransitionissmallcomparedto1=Wij.Forclassicalsystems,thisdescriptionmeansthatsmallaboutthestablestatesaredisregarded.4.4.1Themodel:modulatedswitchingratesAmajoreffectofperiodicdrivingismodulationoftheswitchingrates.Itcanbequitestrongal-readyforcomparativelyweakdriving.Indeed,iftheratesaredeterminedbytheinterstatetunnel-ing,sincethechangesthetunnelingbarrier,itseffectcanbeexponentiallystrong.Similarly,itmaybeexponentiallystrongintheclassicallimitiftheswitchingisduetothermallyactivatedoverbarriertransitions,becausethedrivingchangesthebarriersheights.Nevertheless,forweaksinusoidaldrivingFcoswFtthemodulatedratesW(F)ij(t)canstillbeexpandedinthedrivingam-74plitude,W(F)ijˇWijaijFcoswFt;i;j=1;2:(4.14)Thisequationiswrittenintheadiabaticlimit,wherethedrivingfrequencywFissmallcomparedtothereciprocalcharacteristicdynamicaltimes,liketheimaginarytimeofmotionunderthebarrierinthecaseoftunneling[87]ortheperiodsandrelaxationtimesofvibrationsaboutthepotentialminimainthecaseofactivatedtransitions.TheratesWijarealsoassumedtobesmallcomparedtothereciprocaldynamicaltimes.ThedrivingfrequencywFisoftheorderofWij.ParametersaijinEq.(4.14)describetheresponseoftheswitchingratestothedriving.Theycontainfactors˘Wij.Indeed,foractivatedprocessesWijµexp(DUi=kBT),whereDUiistheheightofthepotentialbarrierforswitchingfromthestatei.IfFcoswFtistheforcethatdrivesthesystem,thenaijˇWijdi=kBT,wherediisthepositionoftheithpotentialwellcountedofffromthepositionofthebarriertop[88].ThetermsµF2,whichhavebeendisregardedinEq.(4.14),areµWij(di=kBT)2inthiscase;apartofthesetermsthatareµcos2wFtdonotcontributetoFF(w)tothesecondorderinF,whereasthecontributionofthetime-independenttermsµF2comestorenormalizationoftheparametersWijinF0(w),seebelow.Forincoherentinterstatequantumtunneling,aijµWij,too.Wewillusequantumnotationsjii(i=1;2)forthestatesofthesystem.Onecanassociatethesestateswiththestatesofaspin-1/2particlebysettingj1ij"iandj2ij#i.Thesystemdynamicsismostconvenientlydescribedbythedynamicalvariableqasq=j1ih1jj2ih2jsz;(4.15)whereszisthePaulimatrix.Foraparticleinadouble-wellpotential,qisthecoordinatethattakesondiscretevalues1and1atthepotentialminima1and2,respectively.Thepowerspectraofdriventwo-statesystemshavebeenattractingmuchinterestinthecontextofstochasticresonance,see[89,90,91,92]forreviews.Bynowithasbeengenerallyacceptedthat,forweakdriving,thepowerspectrumofasystemhasad-peakatthedrivingfrequencywithareaµF2,whichisdescribedbythestandardlinearresponsetheory[93].Thispeakisofcentral75interestforsignalprocessing.However,asweshowinthisSection,alongwiththispeak,thespectrumhasacharacteristicextrastructure,whichisalsoµF2,totheleadingorderinF.4.4.2KineticequationanditsgeneralsolutionItisconvenienttowritetheanalogofEq.(4.4)forthecorrelationfunctionofthediscretevariableqashq(t1)q(t2)i=åi;jhijsz‹r(t1jt2)sz‹rst(t2)jji(4.16)Here,‹r(t1jt2)isthetransitiondensitymatrix,‹r(t1jt2)åjiirij(t1jt2)hjj,and‹rståjii(rst)iihijisthestationarydensitymatrix.Byconstruction(inparticular,becauseofthedecoherenceinthequantumcase)thestationarydensitymatrixisdiagonal.Itsmatrixelements(rst)iigivethepopulationsofthecorrespondingstatesandperiodicallydependontime,‹rst(t+2p=wF)=‹rst(t).Thetransitionmatrixelementsrij(t1jt2)givetheprobabilitytobeinstateiattimet1giventhatthesystemwasinstatejattimet2.Atequaltimeswehave‹r(t2jt2)=‹I,where‹Iistheunitmatrix.Equation(4.16)doesnothavetheformofatraceoverthestatesjii;ratheritexpressesthecorrelatorintermsofthejointprobabilitydensitytobeinstatejjiattimet2andinstatejiiattimet1,withsummationoveri;j[94].Inthequantumformulation,theapplicabilityofthisexpressionisaconsequenceofthedecoherenceandMarkoviankinetics.Matrixelementsrij(tjt0)satisfyasimplebalanceequation,whichinthepresenceofdrivingreads¶tr1j(tjt0)=W(F)12(t)r1j+W(F)21(t)r2j;r1j+r2j=1;(4.17)wherej=1;2.Equationforthematrixelementsof‹rst(t)hasthesameform,exceptthatsubscriptjhastobesetequaltothesubscript.FromEqs.(4.16)and(4.17)weobtainageneralexpressionforthecorrelatorofinterest,hq(t1)q(t2)i=expZt1t2dtW(F)+(t)+hsz(t2)istZt1t2dtnW(F)(t)expZt1tdt0W(F)+(t0);W(F)(t)=W(F)21(t)W(F)12(t):(4.18)76Here,hsz(t)isthq(t)istTr[sz‹rst(t)]isthetime-dependentdifferenceofthestatepopulationsinthestationarystate.Generally,hsz(t)istisnonzeroevenintheabsenceofdrivingunlesstheswitchingratesareequal,W12=W21.Inthepresenceofdrivingthereemergesaperiodicterminhsz(t)ist,whichdescribesthelinearresponse,forweakdriving.Disregardingtermsoscillatingasexp(2iwFt),tothesecondorderinFweobtainfromthebalanceequation(4.17)writtenfor(rst)iihsz(t)istˇWW++F2hc1(wF)eiwFt+c.c.i+a+F22W+Rec1(wF);c1(w)=2(a12W21a21W12)=[W+(W+iw)]:(4.19)Hereweintroducednotationsa=a21a12;W=W21W12:(4.20)Functionc1(w)givesthelinearsusceptibility.Inthecaseofthermallyactivatedtransitions,Eq.(4.19)forc1coincideswiththeclassicalresult[88].ThetermW=W+givesthedifferenceofthestatepopulationsintheabsenceofdriving,whereasthetermµF2givesthetimeindependentpartofthedriving-inducedcorrectiontothisdifference.4.4.3Thedriving-inducedpartofthepowerspectrumEquation(4.18)allowsonetocalculatetheperiod-averagedcorrelatorhhq(t1)q(t2)iiintheexplicitformandtoobtainthepowerspectrum.Asbefore,wewillnotconsiderthed-peakinF(w)forw=0.Thespectrumisanevenfunctionofw,andwewillconsideritforw>0:F0(w)=8W12W21W2+W+W2++w2;FF(w)=F(r)F(w)+F(c)F(w);F(r)F(w)=a+åm;n=fF(mw;nwF);fF(w;wF)=[W+i(wwF)]1a+W12W21w2FW2++iW2wFW+c1(wF):(4.21)ThetermF0isthefamiliarpowerspectrumofatwo-statesystemintheabsenceofdriving[88].Ithasapeakatw=0withhalfwidthW+equaltothesumoftheswitchingrates.ThetermF(c)F77describesthedriving-inducedofthepeakcenteredatw=0,F(c)F(w)=(a2+=2w2F)F0(w)jc1(wF)j2W+=(W2++w2):(4.22)OfmajorinteresttousisthepartF(r)F(w)ofthedriving-inducedterminthepowerspectrum(4.21).Forw>0,itshowsaresonantpeak(oradip,dependingontheparameters)atthedrivingfrequencywF.Incontrasttothed-peakofthelinearresponse,thepeakhasahalfwidth˘W+=W12+W21.Itiswellseparatedfromthepeakatw=0forwF˛W+andgenerallyisofanon-Lorentzianshape.Westressthat,totheorderofmagnitude,thepeakhasthesameoverallareaasthed-peakofthelinearresponse(inthecaseofadip,theabsolutevalueoftheareashouldbeconsidered).Anotherimportantfeatureofthepeak/dipseenfromEq.(4.21)isthatitisproportionaltotheparametera+=a12+a21.Thisparameterdescribesthechangeofthesumoftheswitchingratesduetothedriving.Foractivatedswitchingbetweenpotentialminimaconsideredintheclassicalstochasticreso-nancetheory,a+=(kBT)1(W12d1+W21d2).Forasymmetricpotentiala+=0,sinceW12=W21andd1=d2.ThenF(r)F=0,inagreementwith[95]whereasymmetricpotentialwasconsidered.Ontheotherhand,forstrongdrivingitwasfound[96]thatthepowerspectrumforanasymmetricpotentialdisplayspeaksclosetooddmultiplesofthedrivingfrequencyanddipsclosetoevenmultiplesofdrivingfrequency.Inourweak-drivinganalysiswedonotconsiderpeaks/dipsneartheovertonesofwF;however,asseenfromEq.(4.21),thesignofF(r)F(w)nearwFcanbepositiveornegative,dependingontheparameters.Examplesofthedriving-inducedspectraFF(w)areshowninFig.4.3.Onecanclearlyseethepeaksordipsbothatw=0andatthedrivingfrequencywF.InagreementwithEqs.(4.21)and(4.22),thesignsofthefeaturesofFFaredeterminedbytheinterrelationbetweentheparametersofthetwo-statessystem.Forillustrationpurposewechosethevaluesoftheratiooftheresponseparametersa21=a12toliebetweenplusandminustheratiooftheswitchingratesintheabsenceofdriving,W21=W12.AsseenfromFig.4.3,thespectraareverysensitivetotheratioa21=a12.WehaveseenthissensitivityalsofordifferentvaluesofW21=W12.78Unexpectedly,aspectrumFF(w)emergesevenwherethelinearsusceptibilityisequaltozero,whichhappensfora12W21=a21W12.ThisisseenfromEq.(4.21)andalsofromFig.4.3.Theredlinewitha21=a12=7=3referstothiscase,andtheareaofd-peakinthespectrumiszero.Asseenfromthenumericalsimulationsareinexcellentagreementwiththeanalyticalexpressions.Figure4.3Thedrivinginducedtermsinthepowerspectrumofthetwo-statesystemfortheratiooftheswitchingratesW21=W12=7=3.ThescaleddrivingfrequencyandamplitudearewF=W+=5andFa12=W12=1:Onthethicksolid(red),dot-dashed(black),long-dashed(blue),short-dashed(green),andthinsolid(purple)linestheratioa21=a12is7/3,7/6,0,7=6,and7=3.TheverticallineatwFshowsthepositionofthed-peakatwF.Theareasofthed-peaksfordifferenta21=a12aregivenbytheheightsoftheverticalsegments.Theheightsarecountedofffromthelinestothesymbolsofthesamecolor,i.e.,tothecircle,triangle,andopenandfullsquare,intheorderofdecreasinga21=a12;thereisnosymbolfora21=a12=7=3asthereisnod-peakinthiscase.Theinsetshowsthefullspectrumwith(red)andwithout(black)drivingfora21=a12=7=3.Thecurvesandthedotsshowtheanalyticaltheoryandthesimulations,respectively.Thestructureofthespectrumnearw=0willbeifonetakesintoaccounttermsµF2intheexpressionsfortheswitchingrates(4.14).Intheconsideredleading-orderapproximationinFthesetermshavetobeaveragedoverthedrivingperiodandarethusindependentoftime.ThecorrectionduetothesetermscanbeimmediatelyfoundfromEq.(4.21)forF0(w)byexpandingF0totheorderinthecorrespondingincrementsofWij;thiscorrectionisofanon-Lorentzianform.794.5ThresholddetectorAninsightintothedynamicalnatureofthedriving-inducedchangeofthepowerspectrumcanbegainedfromtheanalysisofthespectrumofathresholddetector.Suchdetectorsarebroadlyusedinscienceandengineering,andtheiranalogsplayanimportantroleinbiosystems.Wewillemploythesimplestmodelwheretheoutputofathresholddetectorisq=1ifthesignalattheinputisbelowathresholdvalueh,whereasq=1otherwise,andwillconsiderthecasewheretheinputsignalisasumoftheperiodicsignalFcoswFtandnoisex(t),q(t)=2Q[F(t)+x(t)h]1;(4.23)whereQ(x)istheHeavisidestepfunction.Toavoidsingularitiesrelatedtonon-differentiabilityoftheQ-function,wewillmodeltheoutputbyq(t)=tanhLF(t)+x(t)h;L˛1;(4.24)andintheexpressionswillgotothelimitL!¥.Muchworkontheinterplayofnoiseanddrivinginthresholddetectorshasbeendoneinthecontextofstochasticresonance,cf.[97,98,99].Inthesepapersofprimaryinterestwasthesignaltonoiseratio;theissuesweareconsideringhere,i.e.,theoccurrenceoftheeffectivefiinelasticscattering"andasaresultofinterplayofnonlinearityandnoise,havenotbeenaddressed,tothebestofourknowledge.Intheabsenceofnoise,thepowerspectrumofq(t)isaseriesofd-peaksatwFanditsovertones(includingw=0),providedthedrivingamplitudeF>h,whereasforF<hwehaveq=1andthepowerspectrumisjustad-peakatw=0.Ontheotherhand,ifF=0andx(t)iswhitenoise,inthelimitL!¥inEq.(4.24)thecorrelatorhq(t)q(t0)i=0fort6=t0,sincethevaluesofq(t)atdifferentinstantsoftimeareuncorrelatedandhqi!0,whereashq2i!1.Thesingularbehaviorofthecorrelatorhq(t)q(t0)iinthecaseofwhitenoisepersistsalsointhepresenceofdriving.Thisisaconsequenceoftheabsenceofdynamics,i.e.,anymemoryeffectsinthevariableq(t)(4.23),andthesingulardistributionofwhitenoise,wheretheintensityhx2(t)idiverges.80Dynamicscanbebroughtintothesystembythenoisecolor.Suchnoisecanbethoughtofascomingfromadynamicalsystemwithretardedresponse,whichisdrivenbywhitenoise.Wewillbeinterestedinthecorrelatorhq(t)q(t0)iandthepowerspectrumF(w)forweakdriving,wherethedrivingamplitudeisF˝h(subthresholddriving),andforasimplecolorednoise,theOrnstein-Uhlenbecknoise.ThisisGaussiannoisewithcorrelatorhx(t)x(t0)i=(D=k)exp(kjtt0j):(4.25)Parameterkcharacterizesthedecayrateofnoisecorrelations.Becausethethresholddetectorhasnodynamicsonitsown,thevalueofthevariableq(t)isdeterminedbytheinstantaneousvalueofthenoisex(t).Wecanwriteq(t)Ÿqt;x(t),whereŸq(t;x)isgivenbyEqs.(4.23),(4.24)withx(t)replacedwithx.Thenthegeneralexpressionforthecorrelatorofq(t),Eq.(4.4),canberewrittenashq(t1)q(t2)i=Zdx1dx2Ÿq(t1;x1)Ÿq(t2;x2)r(x)(x1;t1jx2;t2)r(x)st(x2;t2):(4.26)Here,thesuperscriptxindicatesthatthecorrespondingtransitionprobabilitydensityandthesta-tionarydistributionrefertotherandomprocessx(t).Theformofthetransitionprobabilityfortheprocess(4.25)iswell-known[100],r(x)(x1;t1jx2;t2)=sk2pD(1e2kjt1t2j)exp(k(x1x2ekjt1t2j)22D(1e2kjt1t2j)):(4.27)Thestationarydistributionr(x)st(x1;t1)isgivenbythesameexpressionwitht2!¥.SubstitutingtheseexpressionsintoEq.(4.26)andexpandingŸq(t;x)inF(t),afteraveragingoverthedrivingperiodweobtaintosecondorderinF(t)fort1>t2hhq(t1)q(t2)ii=C+4R¥hdx1R¥hdx2hr(x)(x1;t1jx2;t2)r(x)st(x1)ir(x)st(x2)+2F2coswF(t1t2)r(x)(h;t1jh;t2)r(x)st(h)2F2R¥hdx2r(x)st(x2)ddhhr(x)(h;t1jx2;t2)r(x)st(h)i:(4.28)Here,Cisaconstantindependentoftime;itleadstoadpeakatw=0inthepowerspectrumandwillnotbeconsideredinwhatfollows.Theremainingtermsaretime-dependent.Theydecay81withincreasingjt1t2j,exceptforthetermthatoscillatesasexp[iwF(t1t2)]anddescribesthestandardlinearresponsetoperiodicdriving.AsseenfromEq.(4.28),thistermhastheform2F2coswF(t1t2)hr(x)st(h)i212F2jc(wF)j2coswF(t1t2);c(w)=2r(x)st(h)(2k=pD)1=2exp[kh2=2D);(4.29)wherec(w)isthestandardlinearsusceptibility[85]ofthethresholddetector.Interestingly,thissusceptibilityisindependentoffrequency.Thisisbecausethedetectorhasnodynamics,itsre-sponsetothedrivingisinstantaneous.Analternativederivationoftheexpressionforthesus-ceptibility,whichprovidesausefulinsightintotheresponseofthethresholddetector,isgiveninSec.4.6.ItalsoshowshowtodealwiththesingularitiesinEq.(4.28)fort1!t2,whichemergeafterthetransitionL!¥inEqs.(4.24)and(4.26).ThepowerspectrumF(w)isobtainedfromEq.(4.28)byaFouriertransform.TheFindepen-dentterminEq.(4.28)givesthepowerspectrumF0(w)intheabsenceofdriving.Ithasapeakatw=0.ThetermµcoswF(t1t2)givesad-peakandalsoapeakF2F(r)F(w)atfrequencywF.ThelastterminEq.(4.28)givesadriving-inducedfeatureinthepowerspectrumatzerofrequencyF2F(c)(w).Theshapeofthespectraisdeterminedbythedimensionlessparameterthatcharacterizestheratioofthethresholdtothenoiseamplitudeh(k=D)1=2.Forweaknoise,whereh(k=D)1=2˛1,thepeaknearwFhastheformF(r)F(w)ˇ1Dp2pRe kh24D+iwwFk!1=2ekh2=2D:(4.30)HereweassumedthatwF=kissuflarge,sothatthefeaturesofFFcenteredatwFandw=0arewellseparated;Eq.(4.30)appliesforjwwFj˝wF.Thespectrum(4.30)hasacharacteristicnon-Lorentzianformwithtypicalwidthk2h2=4D.However,itsareaissmall.Intheoppositelimitoflowthreshold,h(k=D)1=2˝1,totheleadingorderF(r)F(w)ˇ12ppDReGiwwF2k=G12+iwwF2k(4.31)82nearwF.Thisspectrumfallsoffslowlyawayfromthemaximum,asjwwFj1=2forjwwFj˛k.Equation(4.31)doesnotcontainthethresholdh.Thesmall-hcorrectionto(4.31)forw=wFis(1ln2)kh2=pD2.Itispositive.FromthecomparisonofEqs.(4.30)and(4.31),oneseesthattheheightofthepeakatwFincreaseswiththeincreasingh(k=D)1=2,butthenstartsdecreasing.InFig.4.4weshowanalyticalresultsforthepowerspectraobtainedfromEq.(4.28)forseveralparametervaluesandcomparethemwiththeresultsofsimulations.Immediatelyseenfromthisisthatthedrivingtheoverallspectrummostnearw=0andnearwFforlargewF=k.ThereemergesapeakatwF.Asseenfromtheinsetinpanel(b),thewidthofthispeakincreaseswithdecreasingnoiseintensity,thatis,withincreasingh(k=D)1=2.Thisisacounterintuitiveconsequenceoftheunusualinterplayofnoiseanddrivinginathresholddetector.Theheightofthepeakdisplaysanonmonotonicdependenceonh(k=D)1=2.Figure4.4Powerspectrumofthethresholddetector.(a):Thefullpowerspectrum;thescaledfrequencyandtheintensityofthedrivingarewF=2pk=100andF2k=D=0:0025.Thescaledthresholdish(k=D)1=2=0:5.Inset:thespectrumnearthedrivingfrequency.Thedeltapeakhasbeensubtracted.Thecurvesandblackdotsrefertothetheoryandsimulations,respectively.(b):Thelow-frequencypartofthedriving-inducedterminthepowerspectrumforwF=k=50asgivenbyEq.(4.28).Thesolid(black),long-dashed(red),short-dashed(blue)anddot-dashed(green)curvescorrespondtothescaledvalueofthethresholdh(k=D)1=2=0:1;0:8;1:2,and2.Inset:thespectrumnearthedrivingfrequency,wF=k=50.Thelow-frequencyspectrumFF(w)ˇF(c)F(w)alsodisplaysapronouncedfeaturenearw=0.OnecanshowfromtheanalysisofthelastterminEq.(4.28)that,forsmallh(k=D)1=2,thisfeatureisadip,withF(c)F(0)=1=Dforh(k=D)1=2!0.Theshapeofthedipisnon-Lorentzian,with83typicalwidthk.Ash(k=D)1=2increases,thedepthofthedipdecreases.Ultimatelytheshapeofthespectrumchangescompletely.Forlargeh(k=D)1=2thespectrumF(c)Fbecomesbroadandshallow.Totheleadingorderin[h(k=D)1=2]1,itcanbewrittenas(2=pD)(D=kh2)1=2exp(kh2=2D)ŸF(c)F(2Dw=k2h2);wherethedimensionlessfunctionŸF(c)F(x)iszeroforx=0,hasaminimumatxˇ1:7,whereitisˇ0:6,andthenapproacheszerowithincreasingxasx1=2.4.6FormulationintermsofsusceptiblitiesThechangeofthepowerspectruminducedbythedrivingcanbeanalyzedintermsofthetuatinglinearandnonlinearsusceptibilityofthesystem,seeSec.(2.2.1).Fornonlinearsystems,thetermF(2)Fisgenerallynonzero,andshouldbekeptinthepowerspectrumincontrasttolinearsystems.Aconvenientwaytocalculatethesusceptibilitiesc1;2isbasedonsolvingdynam-icalequationsofmotionofthesystem.Forexample,foranoverdampedBrownianparticlede-scribedbytheLangevinequationq=U0(q)+f(t)+FcoswFtwithnonlinearpotential(4.6),onecanproceedbyrewritingthisequationintheintegralform,q(t)=Zt¥dt0ek(tt0)expˆZtt0dt00hbq(t00)+gq2(t00)i˙FcoswFt0+f(t0):(4.32)ForsmallfandF,onecanthenexpandtheq-dependentexponentialintheright-handsideandusesuccessiveapproximationsinFandf.Thesusceptibilityc1isgivenbylinearinFterms,whereasc2isgivenbythetermsquadraticinF.Theadvantageousfeatureofthismethodisthatitisnotlimitedtowhitenoise.However,themethodbecomesimpracticalifthenoiseintensityisnotweak,andevenforweaknoiseitbecomescumbersomeifonegoestohigh-ordertermsinthenoiseintensity.84WehavecheckedthatthecalculationbasedonEq.(4.32)givesthesameresultforthedriving-inducedpartofthepowerspectrumFF(w)asthemethodofmoments.Wehavealsofoundthat,inthesecondorderinthenoiseintensityD,thetermgq4=4inU(q)leadstotheonsetofapeakinFF(w)atwF.4.6.1FluctuatingsusceptibilityofathresholddetectorFluctuatinglinearsusceptibilityhasaparticularlysimpleformforathresholddetector.Bylin-earizinginF(t)expression(4.23)fortheoutputofthedetector,weobtainfromtheofthesusceptibility(2.1)c(t;t0)=2d(tt00)dx(t)h;(4.33)wherehisthethresholdandx(t)isthenoise.Zeroind(tt00)causality:thedetectoroutputq(t)isdeterminedbythevalueofthedrivingjustbeforetheobservationtime;theveryd-functionindicatesthattheeffectofthedrivingisnotaccumulatedovertime,theresponseisinstantaneous(butcausal).Thestandardlinearsusceptibilityc(w)isgivenbyexpressionc(w)=Z¥0dteiwthc(t;0)i:FromEq.(4.33),c(w)=2rxst(h).whererxst(h)isthestationaryprobabilitydensityofthenoisex(t),cf.Eq.(4.29).Itappliesforanarbitrarynoisex(t),notjustfortheexponentiallycorrelatednoiseconsideredinSec.4.5.Similarly,thenonlinearsusceptibilityofthedetectorisc2(t;t0;t00)=d(tt00)d(tt000)¶hdx(t)h:(4.34)SubstitutingEqs.(4.33)and(4.34)intothegeneralexpressionsforthepowerspectrumintermsofsusceptibilities,Eqs.(2.3)and(2.4),weobtainthepowerspectruminthesameformaswhatfollowsfromEq.(4.28).854.7ConclusionsTheresultsofthischapterdemonstratethattheinterplayofdrivingandleadstotheonsetofspectralfeaturesinthepowerspectraofdynamicalsystems.Suchfeaturesareanalogsofinelasticlightscatteringandinoptics,whereanelectromagneticcanexciteradiationatafrequencyshiftedfromitsfrequencyandalsoatthecharacteristicsystemfrequency.Ourresultsshowthat,inclassicalsystemsandinincoherentquantumsystems,thespectralfeaturesemergeasaresultofthemodulationoftheresponsetothedriving.Suchmodulationiscommontononlinearsystems.Sincenonlinearityandnoisearealwayspresentinrealsystems,theoccurrenceofthedriving-inducedspectralfeaturesinthepowerspectrashouldbealsogeneric.However,thesefeaturesareforparticularsystems,whichallowsusingthemforsystemcharacterization.Wehavestudiedthreetypesofsystems,allofwhichareattractinginterestinmeso-scopicphysicsandinseveralotherareasofscience.TheoneisanoverdampedBrownianparticleinanon-parabolicpotentialwell.Thismodeldescribes,inparticular,smallparticlesandmoleculesopticallytrappedinaliquid.Wethat,whentheparticleisperiodi-callydriven,thenonparabolicityofthepotentialleadstoanextraspectralpeakoradipatzerofrequency.Forcomparativelyweaknoise,thesignofthedriving-inducedterminthespectrumatsmallwisdeterminedbythecompetitionofthecubicandquarticnonlinearityofthepotential.Theoverallshapeofthelow-frequencyspectrumstronglydependsontheformofthepotentialaswell.Inaddition,alongwithad-peakatthedrivingfrequency,thedriving-inducedspectrumdisplaysapeakatthisfrequencywithawidthoftheorderoftherelaxationrateofthesystem.Wehavealsostudiedatwo-statesystemthatatrandomswitchesbetweenthestates.Weas-sumedthatthedrivingmodulatestheratesofinterstateswitching.Thedriving-inducedspectrumhasarichform.Dependingontheinterrelationbetweentheswitchingrateswithoutdrivingandthedriving-inducedcorrectionstotherates,itcanhavepeaksordipsbothatw=0andatthedriv-86ingfrequency.Thetypicalwidthofthepeaks/dipsisgivenbythesumoftheinterstateswitchingrateswithoutdriving.Interestingly,thesespectralfeaturescanemergeevenwherethed-peakatthedrivingfrequencyhasverysmall(orzero)intensity.Thethirdsystemwestudiedisathresholddetector.Herethedynamicalnatureofthedriving-inducedspectralchangeisparticularlypronounced,asthischangedoesnotoccurifthenoiseinthedetectoriswhite,exceptforthed-peakatthedrivingfrequency.Ontheotherhand,forcolorednoisedrivingdoeschangethepowerspectrumnontrivially.Asinothersystems,weadriving-inducedspectralfeaturenearzerofrequency.Itcanbeapeakoradipdependingontheratioofthethresholdtotheappropriatelyscalednoiseintensity.Also,theheightofthepeakatthedrivingfrequencydisplaysanon-monotonicdependenceonthisratio,asdoesthewidthofthepeak,too,i.e.,noisecanbothincreaseordecreasethewidth.Inallstudiedsystemsinertialeffectsplayednorole:thepeaksofthepowerspectraarelo-catedatzerofrequencyintheabsenceofdriving.Thereforedriving-inducedspectralfeaturesnearthedrivingfrequencyandzerofrequencycorrespondtoinelasticscatteringandre-spectively.However,incontrasttotheconventionaldrivingcaninduceadipinthespectrumatzerofrequency,aswehaveseeninallstudiedsystems(thetotalpowerspectrumre-mainspositive,ofcourse).Theoccurrenceofthediplooksasifthedrivingweredecreasingthenoiseinthesystem,althoughinfactthediphasdynamicalnature.Thepowerspectraofweaklydampednonlinearsystemsshouldalsodisplayextrafeaturesinthepresenceofweakperiodicdriving.Theeffectshouldbemostpronouncedwherethedrivingisresonant.Alongwiththefeaturesnearthedrivingfrequencyandnearw=0,thereshouldarisefeaturesneartheeigenfrequenciesofslowlydecayingvibrationsaboutthestablestates.Severalfeaturesofthepowerspectrahavebeenstudiedfornonlinearoscillatorsintheregimeofstrongdriving,seerecentpapers[101,102]andreferencestherein.Interestingly,theresultsdonotim-mediatelyextendtotheweak-drivingregime,andthefeaturesoftheinterplayofnonlinearityanddrivingwheretheyareofcomparablestrengthremaintobeexplored.However,itisclearfromthepresentedresultsthatthedriving-inducedchangeofthespectraisageneraleffectthatprovidesa87sensitivetoolforcharacterizingsystemsandtheirparameters.88CHAPTER5CONCLUSIONSFluctuationsandnonlinearitiesarekeyfeaturesofmesoscopicvibrationalsystems.Alongwiththereasonablywellunderstoodthermalofthevibrationamplitude,ofinterestareoftheoscillatoreigenfrequency.Theylieattherootofclassicalandquantumcoherenceofmesoscopicsystems.Frequencyleadtobroadeningoftheoscillatorspectra,butthisbroadeningishardtoseparateandidentify,becauseitismixedwithotherspectralbroadeningmechanismssuchasdissipation.Inthisthesis,westudiedhowtorevealandcharacterizefrequencyinmesoscopicvibrationalsystems.Weshowedthattheinterplayofanear-resonantdrivingandfrequencynoiseleadstofeaturesintheoscillatorpowerspectrum.Thesefeaturesallowonetodistinguishfrequencyofdifferentbandwidthsandsources,including1/f-typenoise,broadbandnoise,ornonlinearity-inducedfrequencynoise.Besidestheimmediaterelevancetothedecoher-enceofmesoscopicoscillators,theresultsbearonthegeneralproblemofresonanceandlightscatteringbyoscillators.Theandperhapsmostgenericsysteminwhichtheeffectoffrequencycanbeinvestigatedisaharmonicoscillator.Ofinterestisthegeneralcasewherethespectrumandstatisticsofthecanbearbitrary.Weshowedthat,whentheoscillatorisdrivenbyanear-resonantforce,inthepresenceoffrequencythedriving-inducedpartoftheoscillatorpowerspectrumcontainsnotonlyadpeakatthedrivingfrequency,butalsosomeextrastructureawayfromthedrivingfrequency.Thisextrastructureisaresultoftheinterplayofthedrivingandthefrequencynoise,anditsshapeandstrengthdependsensitivelyonthecharacteristicsofthefrequencynoise.Inthecasewherethefrequencynoisecorrelationtimeismuchlongerthantheoscillatorrelaxationtime,theextrastructurelookslikeafipedestal"atthebottomofthedpeak.Thewidthofthepedestalisdirectlydeterminedbythebandwidthofthefrequencynoise.89Intheoppositecasewherethefrequencynoisecorrelationtimeismuchshorterthantheoscillatorrelaxationtime,thecombinedeffectofdrivingandfrequencynoiseistoinduceabroadpeakneartheoscillatoreigenfrequency.Itsshapeisthesameastheoscillatorpowerspectrumintheabsenceofdriving,anditsintensityisdirectlyproportionaltothefrequencynoiseintensity.Ourtheorywasappliedtoacarbonnanotuberesonator.Bycomparingtheexperimentalobser-vationswiththetheory,wefoundthatahalfoftheobservedspectralwidthcamefromabroadbandfrequencynoise.Alsoforthetimea1/f-typefrequencynoisewasfoundanditsspectrumwasanalyzedinthissystem.Westudiedthespectraleffectsofdispersivemodecouplingindrivenmesoscopicsystems.Wefoundthat,ifthedrivingfrequencyistunedawayfromtheresonantfrequency,thereemergesacharacteristicdouble-peakstructureinthepowerspectrum.Itresultsfromtheinterplayofthedispersive-coupling-inducedfrequencynoiseandthedriving.Thepeaksenablecharacterizationofnotonlythecouplingstrength,butalsothedecayrateofthemodecoupledtothedrivenmode.Thiscanbedoneevenwherethemodeisfihidden"andisnotaccessibletoadirectmeasurement.Wedevelopedapath-integraltechniquetoaverageoverthecoupling-inducedfrequencynoise.Wealsostudiedthepowerspectrumofadrivenoscillatorwithintrinsicnonlinearity.Becausetheoscillatoramplitudeexperiencesthermalandthefrequencydependsontheampli-tudeduetothenonlinearity,thefrequencyisalsoWefoundthatthedriving-inducedchangesofthepowerspectrumarequalitativelydifferentforthecasesofdispersive-couplingin-ducedfrequencyandfrequencyduetotheintrinsicoscillatornonlinearity.Thisisinspitethefactthat,intheabsenceofthedriving,thenonlinearity-relatedchangesofthespectraarenoteasytodistinguishbetweenthetwocases.Ourtheoryonharmonicoscillatorswithfrequencynoiseappliestotheregimewheretheoscil-latorsbecomequantum(i.e.kBT˝¯hw0)andthefrequencynoiseremainsclassical.Inthecaseofdispersivelycoupledoscillators,thetheoryapplieswhentheoscillatorunderstudyisquantumwhilethefihiddenfloscillatorremainsclassical.Wealsostudiedtheinterplayofdrivingandinoverdampednonlinearsystems,90whereinertiaplaysnorole.Weshowedthatthisinterplayalsoleadstocharacteristicfeaturesinthepowerspectrum.Unlikevibrationalsystems,thesefeaturesoccuratzerofrequencyandthedrivingfrequency,andtheycanrepresentadiporapeakinthespectrumdependingontheparametersoftheandthemechanismsofnonlinearities.Inthecourseofthisworkwedevelopednewandfairlysophisticatedmathematicalmethodsincludingapath-integraltechniquetoaverageovernon-Gaussianamethodofmo-mentstocomputethecorrelationfunctionsinnonlinearsystems,andseveralasymptoticmethodstoanalyzethespectraleffectsof5.1OutlookStudiesonfrequencynoiseofmesoscopicvibrationalsystemsarecurrentlyattractingmuchinter-ests.HereImentiontwoimmediatedirectionstoextendourtheory.Animportantextensionofourtheoryondrivenoscillatorsisatheoryofthespectraintheregimeofnonlinearresponse,i.e.comparativelystrongdrive.Thisisofparticularrelevancetothesuperconductingcavityresonators.Itwasfoundthattheintensityofthefrequencynoiseinsuchsystemsisin-verselyproportionaltothedrivingamplitude.Thisnonlinearresponseissometimesattributedtothecouplingbetweentheresonatorandabathoftwoleveldistributedinthedielectric.However,thereisnofulltheorythatdescribesthiseffectnorisitclearhowtoseparateitfromothernonlineareffects.Itwouldbenaturaltoextendourformalismtosuchcases.Anotherdirectionistostudynotjustthespectrumofthefrequencynoise,butalsothestatisticsofthenoise.Thisrequirescalculatinghigherordercorrelatorsormomentsoftheoscillatordis-placement.Itisaninterestingquestionhowthestatisticsofthefrequencynoisewouldmanifestitselfinthedrivinginducedvibrations.91BIBLIOGRAPHY92BIBLIOGRAPHY[1]K.L.EkinciandM.L.Roukes.Nanoelectromechanicalsystems.Rev.Sci.Instrum.,76(6):061101,June2005.[2]MarkusAspelmeyer,TobiasJ.Kippenberg,andFlorianMarquardt.Cavityoptomechanics.Rev.Mod.Phys.,86(4):1391Œ1452,December2014.[3]M.H.DevoretandR.J.Schoelkopf.Superconductingcircuitsforquantuminformation:Anoutlook.Science,339(6124):1169Œ1174,March2013.[4]M.I.Dykman,editor.FluctuatingNonlinearOscillators:fromNanomechanicstoQuantumSuperconductingCircuits.OUP,Oxford,Oxford,2012.[5]M.Sansa,E.Sage,E.C.Bullard,M.Gely,T.Alava,E.Colinet,A.K.Naik,G.L.Vil-lanueva,L.Duraffourg,M.L.Roukes,G.Jourdan,andS.Hentz.Frequencyinsiliconnanoresonators.ArXive-prints,June2015.[6]ArthurW.Barnard,VeraSazonova,ArendM.vanderZande,andPaulL.McEuen.Fluctu-ationbroadeningincarbonnanotuberesonators.PNAS,109(47):19093,2012.[7]T.F.Miao,S.Yeom,P.Wang,B.Standley,andM.Bockrath.Graphenenanoelectrome-chanicalsystemsasstochastic-frequencyoscillators.NanoLett.,14(6):2982Œ2987,June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