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Fluctuation Phenomenon in a Nonlinear Microelectromechanical Oscillator

Permanent Link: http://ufdc.ufl.edu/UFE0024316/00001

Material Information

Title: Fluctuation Phenomenon in a Nonlinear Microelectromechanical Oscillator
Physical Description: 1 online resource (144 p.)
Language: english
Creator: Stambaugh, Corey
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: activation, duffing, equilibrium, escape, far, fluctuations, kramers, mems, nonequilibrium, nonlinear, oscillator, torsional, transition
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: This dissertation details my research on the effects of fluctuations on vibrational systems driven out of equilibrium. Under the right conditions such systems will exhibit multiple stable states. When the fluctuations are large enough, the systems may overcome an activation barrier and switch from one stable state to another. The transitions lead to large changes in the system's vibrational amplitude. These changes can have a large impact on the performance and behavior of the system. The activation barrier plays a major role in determining how the system reacts to noise. We study its dependence on the system parameters, driving amplitude and driving frequency. Previous theoretical work has predicted that the barrier should display specific dependences, on these parameters, that are independent of the system. In this dissertation we test these predictions in our system, along with examining other features that systems possessing multiple stable states and driven out of equilibrium are expected to show. The system used in this study is a micro-electromechanical torsional oscillator. The device is fabricated using a standard commercial process that is reliable and customizable. The device provides a robust system that can be operated at different temperatures, pressures and in different resonant modes. When the damping of the system is small enough and the drive is large enough, the system displays multistable states. The micro-electromechanical torsional oscillators used in these studies are ideal apparatus for studying the transitions that occur between states of a multistable system when fluctuations are present.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Corey Stambaugh.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Chan, Ho Bun.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2011-05-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024316:00001

Permanent Link: http://ufdc.ufl.edu/UFE0024316/00001

Material Information

Title: Fluctuation Phenomenon in a Nonlinear Microelectromechanical Oscillator
Physical Description: 1 online resource (144 p.)
Language: english
Creator: Stambaugh, Corey
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: activation, duffing, equilibrium, escape, far, fluctuations, kramers, mems, nonequilibrium, nonlinear, oscillator, torsional, transition
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: This dissertation details my research on the effects of fluctuations on vibrational systems driven out of equilibrium. Under the right conditions such systems will exhibit multiple stable states. When the fluctuations are large enough, the systems may overcome an activation barrier and switch from one stable state to another. The transitions lead to large changes in the system's vibrational amplitude. These changes can have a large impact on the performance and behavior of the system. The activation barrier plays a major role in determining how the system reacts to noise. We study its dependence on the system parameters, driving amplitude and driving frequency. Previous theoretical work has predicted that the barrier should display specific dependences, on these parameters, that are independent of the system. In this dissertation we test these predictions in our system, along with examining other features that systems possessing multiple stable states and driven out of equilibrium are expected to show. The system used in this study is a micro-electromechanical torsional oscillator. The device is fabricated using a standard commercial process that is reliable and customizable. The device provides a robust system that can be operated at different temperatures, pressures and in different resonant modes. When the damping of the system is small enough and the drive is large enough, the system displays multistable states. The micro-electromechanical torsional oscillators used in these studies are ideal apparatus for studying the transitions that occur between states of a multistable system when fluctuations are present.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Corey Stambaugh.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Chan, Ho Bun.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2011-05-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024316:00001


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IwouldliketorstthankmyadvisorProf.HoBunChan.WhenIrststartedworkinginProf.Chan'slaboratoryhehadonlybeenattheUniversityofFloridaforonesemester.Nonetheless,hewasabletoprovidemewiththedissertationprojectpresentedhere.Overthelastveyears,Prof.ChanhasprovidedmewiththesupportIhaveneededtosucceed.ThisincludestakingmeonasaGraduateResearchAssistant,providingmewithalltheequipmentneededtoperformmyresearch,andprovidingthenancialsupportneededtoattendseveralAPSconferences.Iwouldalsothankhimfortheyearsofmentoringwhichwereandwillbeinvaluableinmyscienticpursuits.IwouldliketothankProf.DykmanofMichiganStateUniversity.Prof.Dykmanprovidedthetheoreticalbasisforthemajorityofmyexperiments.Moreimportantly,heprovidehistimeoverthelastveyearstohelpmeunderstandhiswork.Finally,heprovidedanadditionalvoiceofguidancethatIamverygratefulfor.Ithankallofmycommitteemembers,Prof.Stanton,Prof.Hebard,Prof.Tanner,andProf.Arnold,fortakingthetimetohelpmefulllmyrequirementsforgraduation.Ioweadeepamountofthankstothetechnicalsta.ThemachineshopmembersMarcLink,BillMalphursandEdStorchprovidedmeboththeirtimeandskillsinconstructingtheprecisionpartsneededforseveralexperiments.Alsothecryogenicslab,GregLabbeandJohnGraham,forconstantlyandquicklyprovidinglleddewarsofliquidnitrogenandliquidhelium.Tobothofthesegroupsaspecialthanksfortheextratimegiventohelpmelearnallthelittlethings.Iwishtothankmyocematesovertheyears,YiliangBaoandZsoltMarcet,formakingtheharddaysbearableandmyotherlabmembersKostasNiniosandJheZoufortheirconstanthelp.Finally,athankstomyfamilyforacceptingandsupportingmydecisiontopursuethiscareer,eventhoughitmeansbeingsofaraway. 4

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page ACKNOWLEDGMENTS ................................. 4 LISTOFFIGURES .................................... 8 ABSTRACT ........................................ 10 CHAPTER 1FLUCTUATIONSINANONLINEARSYSTEM ................. 12 1.1Introduction ................................... 12 1.2InterstateSwitchinginMultistateSystems .................. 17 1.2.1KramersRateforaSysteminThermalEquilibrium ......... 19 1.2.2ActivatedEscapeinaNonlinearSystemDrivenOutofEquilibrium 23 1.3UniversalFeaturesinaNonlinearSystem ................... 25 1.3.1MostProbableSwitchingPath ..................... 25 1.3.2CriticalExponent ............................ 26 1.3.3FluctuationsNearaKineticPhaseTransition ............. 27 1.3.4FluctuationRelations .......................... 28 1.4Summary .................................... 28 2THEEXPERIMENTALSYSTEM ......................... 30 2.1FabricationofMEMS .............................. 30 2.2DevicePreparationandElectricalConnections ................ 31 2.3TorsionalOscillator ............................... 33 2.4ExcitationandDetectionSchemes ....................... 37 2.5Summary .................................... 42 3DEVICERESPONSE ................................ 43 3.1LinearDeviceResponse ............................ 43 3.2DungOscillator ................................ 46 3.3ParametricOscillator .............................. 49 3.4Summary .................................... 53 4PATHSOFFLUCTUATIONINDUCEDSWITCHING:MPSP ......... 55 4.1Introduction ................................... 55 4.2QualitativePictureandPreviewoftheResults ................ 56 4.3TheoryoftheSwitchingPathDistribution .................. 61 4.4MicromechanicalTorsionalOscillator ..................... 62 4.4.1DeviceCharacteristics ......................... 62 4.4.2TransformationtoSlowVariablesandParametricResonance .... 63 4.4.3DeterminationofDeviceParameters .................. 64 5

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................. 66 4.5.1MeasuredSwitchingPathDistribution ................ 66 4.5.2GenericFeaturesoftheSwitchingPathDistribution ......... 67 4.5.3LackofTimeReversalSymmetryinaDrivenOscillator ....... 69 4.6Conclusions ................................... 70 5SCALINGOFTHEACTIVATIONBARRIER .................. 73 5.1Introduction ................................... 73 5.2CriticalExponentNearaSpinodalBifurcation ................ 75 5.3CriticalExponentNearaPitchforkBifurcation ............... 82 5.4Summary .................................... 90 6THEKINETICPHASETRANSITION ...................... 92 6.1Introduction ................................... 92 6.2KineticPhaseTransition ............................ 94 6.3SpectralDensitiesofFluctuations ....................... 97 6.4FrequencyMixing ................................ 102 6.5Summary .................................... 109 7FLUCTUATIONTHEOREM ............................ 111 7.1Introduction ................................... 111 7.2DeviceandExperimentalSetup ........................ 112 7.3Theory ...................................... 113 7.4Results ...................................... 115 7.5Summary .................................... 120 8SUMMARY ...................................... 121 APPENDIX APROCEDUREUSEDTOANALYZERESULTS ................. 124 A.1Trajectory .................................... 124 A.1.1Theory .................................. 124 A.1.2Experiment ............................... 127 A.2ActivationBarrierScaling ........................... 128 A.2.1ExtractingSwitchingTimesforDungOscillator .......... 128 A.2.2ExtractingSwitchingTimesforParametricallyDrivenOscillator .. 129 A.2.3DeterminingCriticalExponent ..................... 129 A.3FluctuationSpectrum ............................. 132 A.4WorkFluctuations ............................... 132 BDETERMINATIONOFSYSTEMPARAMETERS ................ 135 B.1LinearOscillator ................................ 135 B.2DungOscillator ................................ 135 6

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.............................. 137 REFERENCES ....................................... 139 BIOGRAPHICALSKETCH ................................ 144 7

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Figure page 1-1Hysteresisinferromagnet .............................. 14 1-2HysteresisinDungoscillator ............................ 15 1-3Basinsofattraction .................................. 18 1-4Cubicpotential .................................... 21 1-5Quarticpotential ................................... 22 2-1BasicstepsinMEMSfabrication .......................... 32 2-2SEMofMEMStorsionaloscillator ......................... 34 2-3Capacitancedependenceond.c.voltage ...................... 36 2-4Cross-sectionalschematicofoscillatorandelectricalsetup ............ 37 2-5Electricalsetupforheterodynemeasurement .................... 39 3-1Responseoflinearoscillator ............................. 44 3-2ResponseofDungoscillator ............................ 47 3-3ResponseofDungoscillatortonoise ....................... 49 3-4Responseofparametricoscillator .......................... 50 3-5Time-translationsymmetryinresponsetoparametricdrive ............ 52 3-6Responseofparametricresonancetonoise ..................... 53 4-1Phaseportraitofatwo-variablesystemwithtwostablestates. .......... 58 4-2Switchingprobabilitydistribution .......................... 60 4-3Harmonicandparametricresonances ........................ 65 4-4MeasuredaveragedvelocityalongtheMPSP .................... 68 4-5Conservationofthestationaryprobabilitycurrent. ................ 69 4-6ComparisonoftheMPSPandthedissipation-reversedpath. ........... 70 5-1Dungoscillator ................................... 74 5-2SwitchinginDungOscillator ........................... 79 5-3ActivationenergyforaDungoscillator ...................... 80 8

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............................. 81 5-5Responseofoscillatortoparametricdrive ..................... 82 5-6Switchinginaparametricallydrivenoscillator ................... 85 5-7Theoccupationinphasespaceatfourdierentdrivingfrequencies ........ 86 5-8Transistionrateinparametricallydrivenoscillator ................. 87 5-9Scalingoftheactivationbarrier ........................... 88 6-1Kineticphasetransition ............................... 94 6-2Dependenceoftheintensityofthesupernarrowspectralpeakonthedrivingfrequency ....................................... 96 6-3Powerspectraldensityofuctuations,viewI .................... 98 6-4Powerspectraldensityofuctuations,viewII ................... 99 6-5Thesupernarrowpeak,viewI ............................ 100 6-6Thesupernarrowpeak,viewII ........................... 101 6-7ResponseandswitchinginDungoscillator .................... 102 6-8DistributionofstatesinDungoscillator ..................... 103 6-9Powerspectraldensityofuctuations ........................ 104 6-10Frequencymixing ................................... 106 6-11Switchingatoptimalnoiselevel ........................... 107 6-12Stochasticresonance ................................. 108 7-1Workvarianceforlinearoscillator .......................... 114 7-2Independenceofworkvarianceondrivingfrequency ................ 116 7-3WorkvarianceandnoiseintensityinDungoscillator .............. 117 7-4Exponentialdependenceofworkvariancenoiseintensity ............. 119 9

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Thisdissertationdetailsmyresearchontheeectsofuctuationsonvibrationalsystemsdrivenoutofequilibrium.Undertherightconditionssuchsystemswillexhibitmultiplestablestates.Whentheuctuationsarelargeenough,thesystemsmayovercomeanactivationbarrierandswitchfromonestablestatetoanother.Thetransitionsleadtolargechangesinthesystem'svibrationalamplitude.Thesechangescanhavealargeimpactontheperformanceandbehaviorofthesystem. Theactivationbarrierplaysamajorroleindetermininghowthesystemreactstonoise.Westudyitsdependenceonthesystemparameters,drivingamplitudeanddrivingfrequency.Previoustheoreticalworkhaspredictedthatthebarriershoulddisplayspecicdependences,ontheseparameters,thatareindependentofthesystem.Inthisdissertationwetestthesepredictionsinoursystem,alongwithexaminingotherfeaturesthatsystemspossessingmultiplestablestatesanddrivenoutofequilibriumareexpectedtoshow. Thesystemusedinthisstudyisamicro-electromechanicaltorsionaloscillator.Thedeviceisfabricatedusingastandardcommercialprocessthatisreliableandcustomizable.Thedeviceprovidesarobustsystemthatcanbeoperatedatdierenttemperatures,pressuresandindierentresonantmodes.Whenthedampingofthesystemissmallenoughandthedriveislargeenough,thesystemdisplaysmultistablestates.Themicro-electromechanicaltorsionaloscillatorsusedinthesestudiesareidealapparatus 10

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1 { 3 ].Thetrendinthesedisciplinesistowardthedevelopmentofsystemswhosesizesareontheatomicscale.However,asdevicesbecomesmaller,therolethatuctuationsplayintheirbehaviorbecomesincreasinglyimportant. Fluctuations,commonlyreferredtoasnoise,playanimportantroleinawidevarietyofsystems.Noisecouldarisefromeitherdissipativeprocessesthatconnecttoathermalreservoirorthediscretenessofparticlesthatmakeuptheowofcharge.Forexample,Johnsonnoiseresultsfromthethermalmotionoftheelectronsacrossaresistor.Inthiscasetheresistoristhedissipativeelement.Noiseresultingfromthermaluctuationsexistsinallsystemsatnitetemperature.Smallsystems,ingeneral,areparticularlysusceptibletotheseuctuationsbecausetheenergyassociatedwiththeuctuationscanoftenbeonthesamescaleastheenergyofthesystem.EvenattemperaturesofacoupleofKelvin,theuctuationsinvoltageintroducedbythethermalmotioncanbelargeenoughtomakeprecisionmeasurementsdicult.Ontheotherhand,noisecanbeadvantageousincertaincases.Forexample,inthephenomenonofstochasticresonance[ 4 ]increasesinthenoisecanactuallyleadtobettersignal-to-noiseratios. Theroleofuctuationsinsystemspossessingtwoorstablestatesisanareaofinterestinavarietyofelds.Forsystemsinthermalequilibriumastablestateisalocalorglobalminimumofthefreeenergy.Averysimpleexamplewouldbeaparticleplacedinadeepwell.Clearly,herethestablestateisthebottomorminimumofthewell.Atnitetemperatures,theparticleuctuatesabouttheminimumofthewellduetothermalmotion.Ifthedisturbancebecomeslargeenough,however,theparticlecanescapeoutof 12

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5 ],includessuchprocessesasproteinandRNAfolding[ 6 ],chemicalreactions,andnucleation[ 7 ]. Quiteoftensystemsdisplayingtwomorestablestatesalsodisplayhysteresis.Inhystereticsystems,theresponsedependsonthehistoryofthesystem.Awellknownexampleofsuchasystemisthemagnetizationofaferromagnet.Here,themagnetizationisdependentontheappliedmagneticeld.WhentheeldHisincreasedpastacriticalvalue,thetotalmagnetizationsaturatesatM.Astheeldisdecreasedthemagnetizationdoesnotdecreaseinaproportionalmanner.Instead,themagnetizationremainsnite,evenwhentheappliedeldbecomeszero.Atanegativecriticalappliedeldthemagnetizationsaturatesat-M.Theresponse,showninFig. 1-1 ,hastwostateswhichcanbeaccessedbycorrectlyadjustingtheappliedeld.ThehystereticdependenceoftheMonHisindicatedbythearrowsinthegure.The\on/o"orMbehaviorseenhereisutilizedinthestorageindustryformagnetictapesandharddrivessincethesystemcan,inacontrollablemanner,bemovedfromonestatetotheother. Inrecentyears,attentionhasturnedtononlinearsystemswhich,whendrivenbyasucientlystrongperiodicmodulation,displayhysteresisandthuspossesstwoormorestablestates.Anexampleofthehysteresistypicallyseeninthistypeofdrivensystem,calledaDungoscillator,isshowninFig. 1-2 .Asinthecaseofthemagnetizationofaferromagnet,acontrolparameterliketheappliedmagneticeldexist.FortheDungoscillatorthisistypicallythedrivingfrequency.Theparametercanbetunedtomovethesystemaroundthehystereticresponse,providingaccesstothedierentstablestates.Withinthehysteresisregion,uctuationsinthesystemcaninduceittoswitchfromonestablestatetoanotherwithouthavingtorstmovealongthehysteresisloop.Theseswitcheswouldbeanalogoustothermaluctuationsinducingthemagnetization,inthepreviousexample,toswitchfromMto-MwhileH=0.Sowhiletheuctuationsmay,onaveragebeweak,thetransitionscanleadtolargechangesintheresponse.These 13

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Hystereticresponseofaferromagnettoanappliedmagneticeld.Dependingonthehistoryoftheappliedmagneticeld,themagnetizationatzeroeldcanbeM. largechangescanproduceunwantedbehaviorinthesystem,suchasalossofstorageinmagneticmemories.However,theycanalsobeusedinanadvantageousmanner.Asalreadymentioned,thisisaprimeexampleofasystemwherestochasticresonancecouldbeutilized,leadingtopossiblesensingapplicationsinthepresenceofnoise. Thetypeofnonlinear,periodicallydrivensystemmentionedisfarfromequilibriumanditsbehaviorisnotaswellunderstood.Whiletheoreticalandsimulationworkdoesexistdealingwiththevarietyoffeaturesthistypeofsystemexhibits[ 8 { 13 13 14 ],onlyrecently,withadvancesinmicro-andnano-scalefabrication,haveseriousexperimentalstudiesbeenpursued.Theexperimentalsystemsincludeatomsinamagneto-opticaltrap[ 15 ],electronsinapenningtrap[ 16 ],micro-andnano-mechanicaloscillators[ 17 { 19 ]andrf-drivenJosephsonjunctions[ 20 ].Thesesystemareusedtoinvestigatefundamentalscienceandapplicationsinawiderangeofelds.Nevertheless,theysharethecommon 14

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HystereticresponseoftheDungoscillator,aperiodicmodulatedsystemwithacubicnonlinearity.Theamplitudeoftheoscillatoryresponsedependsonthehistoryofthedrivingfrequency!. characteristicthatwhenanexternalperiodicmodulationisapplied,theyalldisplayhysteresis.Thehysteresistheyexhibitcanbedescribedbythesameequationsofmotionandaccordingtotheory,theyareexpectedtoexhibitspecicuniversalpropertiesneartheircriticalpoints.Ingeneral,acriticalpointiswhereaphaseboundaryinasystemceasestoexist,forexampleabifurcationisacriticalpointwherethenumberofstateschanges.Thebehaviornearthecriticalpointsisexpectedtobesystemindependent;aremarkablepredictiongiventheapparentdierencesinthesesystems. Inthisdissertationwefocusononeparticulardevice,themicro-electromechanicaltorsionaloscillator.Devicesofsimilardesignshavebeenusedinawiderangeofexperiments,includingthemeasurementoftheCasimirforce[ 21 ]andthemeasurementofthemagnetizationofnovelmaterials[ 22 ].Thetorsionaloscillatorprovidesavery 15

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Fromapracticalstandpoint,understandingthehystereticresponseandtherolethatuctuationsplayintheoscillatorisessentialinordertoexploitthenonlinearoscillatorforsensingapplications.Forinstance,thebifurcationintheJosephsonjunction[ 23 ]isalreadybeingusedtodesignampliersforthereadingofquantumbits.AswillbediscussedinCh. 6 ,astochasticresonancelikeresponsewasseeninourtorsionaloscillator.Possibleapplicationscouldincludetunablenarrowbandltering[ 24 ]orfrequencymixing[ 25 ].AswewillndinCh. 5 ,theprobabilityofswitchingoutofastablestateisexponentiallydependentonanactivationbarrier.Theactivationbarrierhasastrongdependenceonthesystemparameters.Thiscouldbetakenadvantageofindetectionschemesbasedonshiftsinresonantfrequencylikemasssensing[ 26 ].Hereaverysmallchangeinmass,whichwouldshifttheresonantfrequency,wouldproduceanexponentiallysensitivechangeintheprobabilityofswitching.Theworkperformedthroughoutthisdissertationlaysthegroundworkforahostofpossibleuses.Abetterunderstandingofthesetypesofsystemswillalsobenetthesciencecommunityingeneral,sincethebehaviorwestudyisexpectedtobesystemindependent. Inthenextsectionofthischapterthegroundworkforunderstandinguctuationinducedescapeislaidoutinmoredetail.Inthenalsectionofthischapterweintroduce 16

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5 ].Herethebarrierheightneededtobeovercomeforatransitiontooccuristypicallygivenbythefreeenergyofthesystem.Inthisdissertationwefocusonthemoredicultproblemoftransitionswhenthepotentialofthesystemhasaperiodictimedependenceandthebarrierheightcannolongerbecharacterizedbythefreeenergyofthesystem.Inthiscasethesystemisfarfromequilibriumandlacksdetailedbalance[ 8 ].Detailedbalance[ 27 ]isaconditionthatsysteminthermalequilibriumsatisfy.AnindepthexplanationoftheprinciplecanbefoundinRef.[ 27 ]. Inthecaseofasystemwhichpossesstwostablestates,thetwoattractors(A1andA2)areseparatedinphasespacebyaseparatrix,seeFig. 1-3 .Here,attractorreferstoastablestate.Theword\attractor"isusedtomeanapointtowhichthesystemisattractedandtowhichthesystemreturnsafteradisturbance.Theattractorneednotbeapoint.Itcouldalsobeapaththesystemmovesalonginaperiodicfashion.AsshowninFig. 1-3 ,theseparatrix(dashedline)isaboundarythatdenesthebasinsofattraction.Abasinofattractioniscomprisedofasetofpoints(q1;q2)fromwhichasystemevolvesintoanattractor.Forexample,inFig. 1-3 allpointsthatleadtoattractorA1areshaded 17

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Thebasinsofattractionforasystempossessingtwostablestates.Thecoordinatesq1andq2,introducedinSec. 1.2.2 ,aretheinandoutofphasecoordinatesofthesystemintherotatingframe.Thewhiteregionindicatesthebasinofattraction,whereA2istheattractor.IntheabsenceofnoiseasystempreparedwithinthewhiteregionwillrelaxtoA2.Thegrayregionrepresentstheotherbasinofattraction.Thebasinsareseparatedinphasespacebytheseparatrix(dashedline).ThesaddlenodeSliesontheseparatrix.Duringauctuationinducedescapethesystemismostlikelytopassthroughthesaddleduringthetransitiontotheotherstate. ingray.Intheabsenceofuctuations,thesystemrelaxestoeitherattractorA1orA2.OnthislineisasaddlenodeS.Thesaddlenodeisthelocationoftheunstablestate.Aswillbeshownlater,duringauctuationinducedescapethesystemgoesovertheseparatrixatthesaddlenodeasitswitchesintoanotherstate.IfthesystemisinitiallysetwithinthegrayregionofFig. 1-3 ,intheabsenceofuctuations,thesystemwillmovetotheattractorA1inacharacteristicrelaxationtimetr.Thistimeisrelatedtothedissipationinthesystem.Now,inthepresenceofweakuctuationsthetrajectorythesystemstakestotheattractorisnolongerdeterministicbutonaveragewillbethesamepath.Onceneartheattractor,thesystemwilluctuateaboutitforatimemuchgreaterthantr.Thistypeofuctuationwillbereferredtoasanintrastateuctuation,sinceitinvolvesonlyuctuationsassociatedwithmovingwithinthestate.Givenasucientlengthoftime,alargeenoughuctuationwilloccurmovingthesystemuptowardsthesaddle.Ultimately,itwilleitherfallbacktotheinitialattractororcontinueoverthesaddle;whereitwill 18

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Intherestofthissectionweconsidertherateatwhichtransitionsoccur.Weakuctuationsareconsideredsothattr1.Theprobabilityofswitchingbetweenstatesisnotalwaysequalandtheratiooftheindividualratesprovidestherelativeoccupationbetweenthestates.InSec. 1.2.1 aBrownianparticleisconsidered.HerethedistributionisdeterminedbytheBoltzmanformula.InSec. 1.2.2 ,themorecomplexproblemofaperiodicallydriven,nonlinearpotentialisusedtoaddresstheissueoftransitionratesinthecaseofweaknoise. 2p2+U(q);whereU(q)=1 3q3+q:(1{1) Hereqandparethecoordinateandmomentumoftheoscillator.Thevariablerepresentsacontrolparameterofthesystem.ByadjustingthepotentialU(q)canbechanged.InFig. 1-4 thecubicpotential,withaBrownianparticlesittingatanattractor,isplotted.Ifthesystemisconnectedtoathermalbaththerewillbebothdissipationanductuations.Thethermalnoisecausestheparticletomoveabouttheminimumofthewell.InorderfortheparticletoescapefromthestablestateitmustovercomeabarrierUwhich,forsystemsclosetothermalequilibrium,istypicallythefreeenergybarrier.Occasionally,atarate ij=consteU=kBT(1{2) 19

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5 ],theparticlewillswitchoutofthestablestateiandintoanotherstatej.Forthecubicpotential U=U(qs)U(qa)=4 3(c)3=2withqa=1 2andqs=1 2:(1{3) Theparametercrepresentsacriticalpointinthesystem.FromEq. 1{3 itisevidentthatchangingthevalueofthecontrolparameterwillaltertheheightofthebarrierU.Asapproachesctheheightofthebarrierdecreasesandi!jtransitionsaremoreprobable.When=Cthewellnolongerexists.Forthecubictypepotentialthemergingoftheunstableandstablestateoccurs,whiletheotherstatestaysfarway.Byfaraway,wemeanthattheotherstatedoesnotalsomergewiththeunstablestate.Thistypeofbifurcationiscalledaspinodalbifurcation. LookingbackatEq. 1{2 forthecasesof>c,weconsiderthesituationofplacingmanynon-interactingparticlesintothewellofthecubicpotential.Byincreasingthetemperature,theuctuationofeachparticlealsoincreases.Thereforeonaverage,moreparticlescanescapeduringagivenperiodoftime.TheprobabilityofescapefortheparticlesisaPoissonprocess,whereeachswitchisindependentofanotherswitchandswitchesdonotoccursimultaneously.Afamiliarexampleofsuchaprocessisthedecayofradioactiveatoms.Theexampleexaminedherewasforaparticleswitchingoutofastateiandintoanotherstatej.Thisotherstatemaybeanotherstablestatewheretheprobabilityofswitchingbackislow,suchthatijjiandtransitionsfromj!icanbeconsiderednottooccuronthetimescalesofinterest.Severaloftheexperimentsweperformedsatisfythisrequirement,wheretheprobabilityofswitchingfromi!jismuchlargerthanswitchingfromj!i. Anotherpotentialofinterestisthequarticpotential: 4q4q2withqa=1 2andqs=0:(1{4) 20

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ABrownianparticletrappedinacubicpotential.ThecontrolparameterchangesthedepthofthewellandtheheightofthebarrierU.Asapproachescthetwostatesqsandqaalsoapproacheachother.When=c,thetwostatesmergeandaspinodalbifurcationoccurs.Thermaluctuationscaninducetheparticletojumpoutofthestablestatebyovercomingthebarrier(Eq. 1{3 ). Inthispotential(seeFig. 1-5 )therearetwoidenticalstablestatesseparatedbythebarrier.Thebarrierheightsareidenticalforbothstatesaslongas>c,soij=ji.Thisisincontrasttothecaseseenforthecubicpotentialwhereijji.Becausechangingaltersthebarrierheightforbothwellsinthequarticpotentialequivalently,asapproachescthetwoattractorsqa;1andqa;2movetowardstheunstablestate.At=callthreestatesmergetogetherinapitchforkbifurcation. 21

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ABrownianparticletrappedinaquarticpotential.ThecontrolparameterchangesthedepthofthewellandtheheightofthebarrierU.Asapproachescthethreestatesqs,qa;2,andqa;1alsoapproacheachother.When=c,thethreestatesmergeandapitchforkbifurcationoccurs.Thermaluctuationscaninducetheparticletojumpoutofthestablestatebyovercomingthebarrier(Eq. 1{4 ). Thedierenttypesofbifurcationsseeninthesetwomodelswillplayanimportantroleinlaterchapters.Inaddition,thequalitativedescriptionofthestablestatesaspotentialwellsandthefreeenergyasabarriertobeovercomeduringaswitchprovidesanintuitivepictureofuctuationinducedescape.Strictlyspeaking,thisistrueeveninthecasesthatwillbeexaminedinthisdissertation,wherethesystemisfarfromequilibriumandthebarrierheightcannolongerbedescribedbythefreeenergy.Asanote,thestablestatesinthetorsionaloscillatormanifestthemselvesasdistinctmechanicalamplitudes 22

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2(p20+!20q20)+1 4q40qF0cos!t:(1{5) Hereq0andp0arethecoordinateandmomentum,!0istheresonantfrequency,anddescribesthestrengthofthenonlinearity.Weassumethatthereisphenomenologicaldampinginthesystem,whichcouldresultfromavarietyofsources,andthatitismuchlessthanthedrivingandresonantfrequencies;thereforewecantransformtoslowvariablesintherotatingframe.Introducingtheinandoutofphasedimensionlesscoordinatesq1andq2,wecanexpressthevariablesq0andp0as (1{6a)p0(t)=C![q1sin(!t)+q2cos(!t)] (1{6b) whereC=q 4(q21+q221)2p andistheeectiveamplitudesquared.Usingtheconventionalaveragingmethod[ 28 ]termsofexpin!t,wherejnj2,wereignored.NowtheLangevinequation,which 23

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_q1=Kq1+f()withKq1=q1+@q2g (1{8a)_q2=Kq2+f()withKq2=q2+@q1g: Herethedissipationhasbeenaccountedforusingthedimensionlessfrictionparameter==j!!0j,=j!!0jtisthedimensionlesstime,andtheappliednoisefisGaussianwhitenoiseorhf()f(0)i=2D().ThetermDisadimensionlessnoiseparameter,whereforweaknoiseD1.Intheabsenceofuctuations,thevectorKdeterminesthesystemdynamicsandthustheattractorscanbefoundbysettingKq1=0andKq2=0. Thesetypesofnonlinearsystems,whicharedrivenoutofequilibrium,generallylackdetailedbalance.ThereforeKramersequationnolongerdirectlyapplies.Ithasbeenshown[ 16 19 20 29 { 32 ],however,thattherateofescapestillmaintainsanactivationtypebehavior: ij/eRi=D:(1{9) ThetransitionratedependsonthenoiseDwhichcanoriginatefrommanysourcesincludingthermal,electricalandmechanical.Typically,theuctuationshaveasmallamplitude.InorderforthesystemtogoovertheactivationbarrierRialargeuctuationisneeded.ThedeterminationofRiinvolvesndingtheoptimalnoisethatismostlikelytoinduceatransition.AdetailedderivationcanbefoundinRef.[ 8 33 ].Essentiallythough,bychangingtheproblemfromonedependingonthenoisetoonedependingonthetrajectory,theactivationbarriercanbewrittenas 4(_qK)2;K=(Kq1;Kq2);andq=(q1;q2):(1{10) ThischangecanbedoneusingtheFeynmanpathintegralformulation[ 34 ]whereeachrealizationofthenoisehasapathassociatedwithit.Thustheoptimalnoiseshould 24

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Inthissectionanequationfortheactivationbarrierwasfound.Thesolutionisnottrivial,unlikethecaseinSection 1.2.1 .Throughoutthisdissertationtheactivationbarrieranditseectontheprobabilityofatransitionoccuringisexamined.WhiletheproblemiscomplexwendthattheintuitivemodelgiveninSection 1.2.1 canbeappliedtogainaqualitativeunderstandingforthedierentbehaviorsbeingstudied. 1{10 ),anontrivialproblemforsystemslackingdetailedbalance.Thisresultisexpectedtobeindependentofthetypeofsystemandshouldonlyrelyonacoupleofdeterminablesystemparameters. InChapter 4 weshowresultsexperimentallyverifyingtheexistenceoftheMPSP.HerewecalculatetheMPSP,thenperformswitchingexperimentstodetermineiftheactualtrajectorytakenduringaswitchcoincideswiththecalculatedMPSP.Wendourexperimentalresults,usingnoadjustableparameters,tobeinexcellentagreementwiththecalculatedpath.Thisexperimentmarkstherstexperimentalproof,fora 25

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1{10 )needtobedoneinordertodeterminetheactivationbarrier.Thisistypicallyadicultproblem.Instead,scalinglawshavebeenidentiedwhichpredicthowtheactivationbarriershoulddependongeneralsystemparameters.Theactivationbarrierinthesecasescanwrittenas whereisthecriticalexponentandcrepresentsacriticalpointinthesystem.Whilegenerallytheprefactordependsontheparticularsystem,nearbifurcationpointstheexponentisexpectedtobesystemindependent. IntherstpartofCh. 5 thescalingoftheactivationbarriernearaspinodalbifurcationisconsidered.Aspinodalbifurcationoccurswhenastableandunstablestatemergetogether.Thiscaseoccurs,forexample,inthecubicpotential(showninFig. 1-4 )whenapproachesc.Justpriortomergingthemotionintherotatingframe[ 33 ]ischaracterizedasoverdamped.Heretheeectivepotentialbecomesshallowandtheparticlesmotionisslowed.Theoreticalanalysishasestablishedthattheactivationbarrierfoructuationinducedescapeshouldscalenearaspinodalbifurcationpoint[ 33 35 ]withcriticalexponent=3=2.Thisscalingrelationshipissupposedtobesystem-independent.InChapter 5 experimentalresultsareshownfortherstobservationofthisscalinginaperiodicallydrivensystemoutsideofanalogcircuitsimulations. 26

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1-5 ).Likethepreviouscase,thissystemisalsofarfromequilibrium,inthatmultiplestablestatesonlyexistwhenthesystemisstronglydriven.Unlikethepreviousexperimenthowever,allthreestatesmergetogetheratacriticalpointandadierentcriticalexponentisexpected.Theoreticalanalysispredictsthatthecriticalexponent,nearthecriticalpoints,willbe=2.Thisexponentisexpectedregardlessofwhetherthemotionisoverdampedorunderdamped[ 36 ].Additionalsystemdependentscalingsawayfromthetwobifurcationfrequenciesarealsoinvestigated.ThecriticalexponentofthesystemwillbediscussedalongwithexperimentalresultsinChapter 5 37 ]or InEq. 1{12 theprobabilityofndingthesysteminstateiisgivenbywi=ji=(ij+ji).Thisparticularregionisidentiedasthe\kineticphasetransition"[ 37 ].Thetermisusedtodescribeanonequilibriumphasetransition[ 38 ],partiallybecausethephenomenaexhibitsgenericpropertiesthatsharesimilaritieswiththephasetransitionofathermalequilibriumsystem.Therstisthatinathermalequilibriumsystemthesystemwillusuallybeinoneoftwophases,suchaswater,oranother,vapor.Onlyatthephasetransitionwillthetwophasescoexist.Theotherindicationofaphasetransitioninathermalequilibriumsystemistheappearanceoflargeuctuations.Inathermal 27

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39 ]. InChapter 6 theregionwherethetwostatesareequallylikelytobeoccupiedisinvestigated.Herethespectraldensityofuctuations,whichdescribeshowtheuctuationsaredistributedwithfrequency,isanalyzedandalargesupernarrowpeakinthedistributionisidentiedatthe\kineticphasetransition".Additionally,theconstructiverolenoisecanplayinthemixingoffrequencyiscovered.Byaddingasmallsecondarymodulationsignal,atafrequencynearthedrivingfrequency,wendthattheswitchingbetweenstatesbecomessynchronizedforarangeofnoiseintensities.Thesynchronizationleadstoanenhancedsignal-to-noiseratiofornon-zeronoiselevels;aphenomenonassociatedwithstochasticresonance. 7 ,involvesuctuationrelationsforsystemsoutofequilibrium.Theworkdoneonadrivensystemiscalculatedfortwocases:(a)whenaweakdriveisapplied,sothattheresponseislinear,(b)forastrongdrivesothesystemresponseisnonlinear.Noiseisappliedandtheratioofthevarianceoftheworkoverthemeanworkiscalculated.Thelinearcaseisshowntobeconsistentwiththeuctuation-dissipationrelation.Howeverforsystemsdisplayingbistabilitytherelationnolongerholds;yet,universalfeaturesarestillexpected.Weexaminetheratioofthevariancetothemeanworknearthe\kineticphasetransition"andndanexponentialdependenceonthedrivingfrequency,aresultconsistentwiththeoreticalpredictions. 1{9 andprimarilyinvolves 28

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29

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Allexperimentsinthisdissertationwereperformedusingmicro-electromechanicalsystems(MEMS).MEMSarestructures,fabricatedonwafers,withtypicalsizesontheorderoftenstohundredsofmicronsandareusedtoperformtaskssuchassensing,actuating,andsignalacquisition.OvertheyearsMEMShavegrowntoencompassalargecategoryofsmallsystemsincludingthoseinvolvingthermal,magnetic,uidicandopticaldomains[ 40 ].OnecommonfeatureamongMEMSisthattheyactastransducersconvertingenergyfromonedomaintoanother,forexamplethermaltomechanical.Inthecaseofthisdissertation,electricalenergyisconvertedtomechanicalandthenbacktoelectrical.ThesmallscaleofMEMSallowforstudyingrealmsotherwisediculttoinvestigateonthemacroscale;thisincludesstudyingtheeectofsmalluctuationsonsmallsystems.InthischapterthefabricationofMEMSisdiscussed.Thetypeofdeviceusedtocarryouttheexperimentsperformedinthisdissertationispresentedanddierentschemesforactuatingandsensingtheoscillationsofthedeviceareexamined. 41 ]. ThePolyMUMPSprocessbeginswithasiliconwaferwhichhasathin(600nm)siliconnitridelayerdepositedonit(Fig. 2.1 a).Thegroundlayer,composedofpolysilicon,ispatternedusingstandardphotolithographytechniques.Thisinvolves,rst,depositingalayerofpolysiliconusinglowpressurechemicalvapordeposition(Fig. 2.1 b).Next,a 30

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2.1 c).Aphotomaskisthenplacedabovethephotoresistandultravioletlightisshinedthroughthemaskandontothephotoresist(Fig. 2.1 d).Thephotomaskdetermineswhichareasofthephotoresistareexposedtothelight.Dependingonthetypeofphotoresistusedtheexposedareaseitherbecomemoresolubleorlesssoluble.Theappropriatedeveloperisthenusedtowashawaytheundesiredregionsofphotoresist.Nextareactive-ionetch(RIE)isdonetoetchawaytheexposedregionsofthepolysilicon(Fig. 2.1 e).Thepolysiliconcomprisesthebottomconductivestructurallayer.Theremainingphotoresististhenremoved(Fig. 2.1 f)andtherststructurallayeriscomplete.Theprocessisthenrepeated,exceptnowasacriciallayerofphosphosilcateglass(PSG)ispatterned.Again,theprocessisrepeatedforthesecondlayerofpolysilicon,thesecondlayerofPSG,andathirdlayerofpolysilicon.Finally,agoldlmispatternedforelectricconnectionstothedevice.ThenalproductinvolveslayersofpolysiliconandPSG.InthisstatethesacriciallayersofPSGkeepthelayersofpolysilicon,whichmaybeintendedtobeabletomovefreely,inaxedpositionuntiltheproductreachestheconsumer. 42 ].AftertheremovalofthePSG,thesuspendedpolysiliconstructuresarefreetomove.Ifthechipisallowedtodryinairtheselayers,whichhaveagapof2mbetweenthem,maysticktogetherasaresultofthecapillaryeect[ 40 ].Toavoidthisproblemacriticalpointdryerisused.Thedevice,afterthewetetch,issubmergedintomethanolandplacedintothechamberofthecriticalpointdryer. 31

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ThefollowingstepsarerepeatedforeachstructuralandsacriciallayerinthePolyMUMPSprocess.(a)Startwithsilicon(redbox)witha600nmthicklayerofnitride(greenbox).(b)Alayerofpolysilicon(orangebox)isdepositedusingLPCVD.(c)Alayerofphotoresist(bluebox)isspunontop.(d)Shineultravioletlightthroughaphotomask,exposingonlycertainareasofphotoresistandusedevelopertowashawayunexposedregionsofphotoresist.(e)Areactive-ionetchisperformedtoremoveunwantedregionsofpolysilicon(orangebox).(f)Remainingphotoresistcanberemoved,leavingdesiredstructure. 32

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Afterdrying,thedeviceismountedina16pinceramicpackageusingsilverproxy.Usingawirebonder,goldwiresarebondedbetweenthebondingpadsonthedeviceandgoldpadsonthepackage.Thepackageistheninsertedintoareceptaclelocatedatthebottomofaprobe.Theprobeismadeofalongsteeltube(2m)whichhasshieldedwiresrundowntheinsideofit.Atthebottomoftheprobethewiresaresolderedtothereceptacleholdingthepackage.AtthetopoftheprobethewiresaresolderedtoseveralBNCfeedthroughs.Usingthesefeedthroughstheelectricalconnectionscanbemadetothedeviceatthebottomoftheprobe.Avacuumtightsealbetweenthebottomoftheprobeandcapismaintainedbyarubbero-ringoranindiumo-ring[ 43 ].Inthisstatethesamplespacecanbepumpeddowntolessthan106Torr.Usingtheindiumsealtheprobecanalsobesubmergedinliquidnitrogen(77K)orliquidhelium(4K).Amainadvantageformeasuringatlowtemperaturesisthereductionoftemperatureuctuations.Thexedtemperatureremovestemperaturedependentdriftintheresonantfrequencyoftheoscillator.Thevacuumreducesviscousdampingsotheoscillatorisunderdamped. 2-2 ,wasfabricatedusingthePolyMUMPSprocessdescribedinSection 2.1 .Thisparticulardeviceconsistsofthreemainparts:amovabletopplate,twoelectrodespositioneddirectlybeneaththetopplate,andtwotorsionalspringswhichareusedtosuspendthetopplate2mabovetheelectrodes.Themovabletopplatesusedhavedimensionsofeither500m500m3:5mor200m200m3:5m.Themassoftheplateisgivenbym=V,whereVisthevolumeofthemovableplateand,the 33

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(a)Scanningelectronmicroscopeimageofamicro-electromechanicaltorsionaloscillator.Thelargesquarepaddleissuspendedabovetwoelectrodesbytwotorsionalsprings.Thethreesmallersquarepadsallowattachmentofgoldwiresforelectricalconnectionstothetopplateandthetwoelectrodes.Onthetopplate,thesmalldotsareetchholestoallowthewetetchanttoremovethePSGeectively.(b)Amagniedviewofoneofthetorsionalsprings.Inthisdevicethespringisconnectedtothesecondlayerofpolysilicon. densityofpolysilicon,is2330gm3.ThemomentofinertiafortheplateisI=1 12mb2,wherebistheeectivemomentarm.Springsofdierentsizeswereused;atypicalsizewas40m4m3:5m.Thespringsareanchoredtothesubstrateandprovideanelectricalconnectiontothetopplate.Thetwoelectrodesbeneaththemovableplatehaveanareaapproximatelyhalfthatofthetopplate.Asmallgap,directlybeneaththeaxisofrotation,isolatestheelectrodesfromeachother.Electricalleadsrunfromeachelectrodeandfromtheanchorofonethespringstothethreegoldbondingpads.Asmentionedintheprevioussection,goldwiresconnectthesepadstothepackage. db;(2{1) 34

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2{1 d1sotheparallelplateapproximationisreasonable. TheenergyEstoredinacapacitorwhenad.c.voltageVdcisappliedacrossitisgivenby 2CV2dc(2{2) andcanberelatedtoanappliedelectrostatictorqueTdcthrough Solvingforthetorquegives Byputtingad.c.voltageonthebottomelectrode,anattractivetorquecanbeappliedtothemovabletopplate.Thisattractiveelectrostatictorqueisbalancedbytherestoringtorqueofthetorsionalsprings: wherekisthetorsionalspringconstant.InFig. 2-3 aplotofthecapacitanceversustheappliedd.c.voltageisshownfora500m500m3:5mdevice.Inthisgurethed.c.voltageisappliedtooneelectrodewhilethecapacitanceismeasured,usinganAndeen-Hagerlingcapacitancebridge,betweenthetopplateandtheotherelectrode.Astheplateisattractedtowardtheelectrodewithad.c.voltageittiltsawayfromtheplatewherethecapacitanceisbeingmeasured.Thisleadstoadecreasingcapacitanceforanincreasingvoltage.InFig. 2-3 itisshownthataboveacertainvoltagethecapacitancedropssuddenlyandstaysataxedvalueforhighervoltagelevels.Ifthevoltageisdecreasedbackdownthecapacitancenolongerreturnstoitsinitialvalue.Thisresult,knownasthepull-ineect[ 44 ],occurswhentheplatetiltssuchthatbd=3.This 35

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Thecapacitancebetweenthemovabletopplateandabottomelectrodeisshownasafunctionoftheappliedd.c.voltagetotheotherbottomelectrode.AtVd=1:45Vtheplatesuccumbstothepull-ineect.Forhigherd.c.voltagesthecapacitanceshowsasmallincreaseasthemiddlesectionoftheplateispulledslightlydown.Asthevoltageisreduced( coincideswiththeelectrostatictorqueexceedingtherestoringcapabilitiesofthetorsionalspring,causingtheplatetosnapdown.Afterthevoltageisremovedthemovableplateisoftenleftstuckinthetiltedposition,possiblyasaresultofstaticchargebetweenthetwoplates.Usingasmallglasscapillaryandamicromanipulatorthedevicecanbemanuallyreleasedandreturnedtoitstartingposition.Asecondmethodtofreeastuckdeviceinvolvesapplyingalargea.c.voltagetooneoftheelectrodeorthetopplateforashortperiodoftime,typicallylessthan2s.Whilenotalwayssuccessful,thismethodcanbeusedevenwhenthedeviceisintheprobe.Thishelpseliminatetheneedtodismantletheentireexperimentwheneverthedevicegetsstuck. 36

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Cross-sectionalschematicofoscillator(nottoscale)andelectricalsetup.Ad.c.biasisappliedbetweenbothelectrodesandthetopplate.Thedrivesignalisappliedtooneoftheelectrodes,whiletheotherisusedtomeasuretheresponsecapacitively. InFig. 2-4 across-sectionalschematicoftheoscillatorwithelectricalconnectionsandmeasurementcircuitryisshown.AdrivingvoltageVd=Vdc+Vaccos!dtisappliedtooneelectrodesettingupaperiodicelectrostatictorque,seeCh. 3 .AnSRS345functiongeneratorwasusedtocreatethea.c.voltagewhilealownoised.c.source,custombuiltbytheUniversityofFlorida'sPhysicsElectronicShop,wasusedtoproduceasteadyd.c.voltage.Toensurethattheappliedtorque(seeEq. 3{2 )islinearlyrelatedtothe 37

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ThedetectionelectrodeisconnectedtoaAmptek250chargesensitivepre-amplier,whichislocatedontheoutsideoftheprobe.Astheplateoscillatesthecapacitancechanges.Thevirtualgroundfromthepre-ampliferxesthevoltageacrossthetopplateandthedetectionelectrode.SinceQ=CVthechargemustvaryintimewiththecapacitance.Thepre-amplierdetectsthevaryingchargeandoutputsanampliedtimevaryingvoltage.Theoutputisfedintoalock-inamplier(SRS830)whichmeasurestheamplitudeBandphaseoftheresponseatthereferencefrequency.ThemeasuredresponsecanberelatedtotheinandoutphasequadraturesX=Bcos()andY=Bsin().ThemeasuredsignalsXandYareproportionaltothecoordinatesq1andq2introducedinSec. 1.2.2 .Thelock-inamplieressentiallyltersoutsignalsatfrequenciesawayfromthereferencefrequencyandampliesthoseatthereferencefrequency.Thisisolatestheresponsesignalfromtheelectronicnoise.Themajordrawbackofthissetupisthatcapacitivecouplingbetweenthendetectionandexcitationelectrodescanleadtoosetsinthemeasuredsignalwhichdonotrepresentmechanicalmotion.Heretheappliedexcitationvoltageleaksthroughtothedetectionelectrodeandgetsaddedintothedetectedmechanicalresponse.Typicallythisisoflittleconsequencesincethemeasuredmechanicalresponseismuchlargerthantheleakedsignal,howeverforparticularexperimentsitisnecessarythattheschemebealteredtominimizethisbackgroundsignal. Theeasiestwaytoimprovethesignaltonoiseratioistoreducethestraycapacitancebyamplifyingthesignalclosertothedevice.Thiscanbeachievedbyconnectingahighelectronmobilitytransistor(aFHx35XGaAsHEMTwasused)tothedeviceat 38

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Cross-sectionalschematicofoscillator(nottoscale)andelectricalsetup.Ad.c.biasisappliedbetweenbothelectrodesandthetopplate.Thedrivesignalisappliedtooneoftheelectrodesalongwithahighfrequencycarriersignal;theotherelectrodealsohasahighfrequencycarriersignalshiftedby180appliedtoit.Theresponseismeasuredatthetopplatethroughatransistor. thepackagelevel.ThiswasincorporatedintothedetectionschemefortheexperimentsperformedinCh. 4 and 7 7 requiresmeasurementofthemechanicaloscillationwithahighsignal-to-noiseratioandminimalbackground.Toachievethisaheterodynemeasurementtechniquewasused.Theapproachissimilartothepreviousmethod,exceptthatthedetectedsignalismodulatedintoahighercarrierfrequency,seeFig. 2-5 .Thismethodinvolvesaddingacarriersignal!ctoeachofthebottomelectrodes.Thefrequencyofthecarriersignalischosensuchthat!c1lockin;1!dwherelockin;1isthetimeconstantofthelock-inamplier.Theappliedcarriersignalsare180outofphaseandtheiramplitudesareapproximatelyequal.Withouttheexcitationsignal!d, 39

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44 ].TheoutputvoltageonthetopplateisrelatedtotheappliedcarriervoltageVcacby whereC1andC2arethecapacitancesbetweeneachelectrodeandthetopplate.Forsimplicity,weassumethatbothcarriersignalshavethesameamplitudeVcac.Theoutputvoltageismonitoredusingalock-inamplierwhichisreferencedatthefrequencyofthecarrierwave.IftheplateisnotmovingthemeasuredvoltageVoutisxedbecausethelock-inmeasurestheamplitudeoftheperiodicvoltageVout.Ifad.c.voltageisappliedtheresultingtorquetiltstheplateandthesignalonthelock-inchangesbyanamountproportionaltothechangeincapacitance. Now,iftheexcitationsignalisaddedin,thedevicewilloscillateandthecapacitancewillbemodulatedatthetheexcitationfrequency.Thetotaloutputsignalfromthedeviceiscomprisedofsignalsatseveralfrequencies:!c,!c!d,and!d.The!c!dcomponentsresultfromthemixingofthecarriersignalandthemechanicalresponseattheexcitationfrequency: sin!ctsin!dt=[cos(!c!d)t+cos(!c+!d)t]=2:(2{7) The!dcomponentistheexcitationsignalthatleakedthroughthedeviceandmanifestsitselfasastraycapacitance.Therstlock-inamplierisreferencedatthecarrierfrequencywiththetimeconstantlockin;1.Thetimeconstantischosensuchthatthe!c1lockin;1!dsothatfrequenciesoutsideof!c!darelteredout.Theoutputsignalfromtherstlock-inisthemechanicalresponseat!d.Thisoutputisfedintoasecondlock-in,referencedattheexcitationfrequency,whichmeasurestheamplitudeandphaseofthemechanicaloscillationofthedevice.Bymixingthemechanicalresponseoftheoscillatorintothecarrierfrequency,thebackgroundprobleminthepreviousdetectionschemeiseliminated.Thisisbecausethestraycapacitance,whichoccurredat!d,isnolongerpresentsinceitwaslteredoutattherstlock-in.Thesamegoesfortheicker 40

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2{6 suchthatVout=0forthestaticoscillator,thesensitivityofthelock-incanbeincreasedtoavoidtheproblemofdigitizationintherecordedsignal. 1.2.2 thedimensionlessparameterDwasdenedtodescribethenoiseintensityneededtoinduceatransitionovertheactivationbarrier.Typicallytheparameterdependsonthetemperatureoftheenvironment.Aswillbeshown,foroursystemthermaluctuationsarenotlargeenoughtoinduceatransition.ThereforeanampliedJohnsonnoisesourcewasused.A50resistorisusedastheprimarynoisesource.TherootmeansquarenoisevoltageorJohnsonnoiseis vn=p whereRistheresistanceofthedissipativeelementandfisthefrequencybandwidthacrosswhichthenoiseisapplied.Atroomtemperature(300K)overa1Hzbandwidththe50resistorproducesanoisevoltageof0:9nVHz1=2.Thisnoisevoltage,ingeneral,istoosmalltoinduceatransitiononthetimescaleoftheexperiments.ThereforetwoSRS560lownoiseamplierswereusedinseriestoamplifytheprimarynoisesourceintoarangeof10to1000VHz1=2.Theamplierwasalsousedtolterthenoise.Thebandwidthofthelterednoisewasselectedtobemuchlargerthanthewidthoftheresonancepeaksothataboutthefrequencyrangeofinterestthenoisecanbeconsideredwhite.Insomeexperimentsanadditionaldigitallterwasusedtolterthenoise;againwithinthefrequencyrangeofinterest,thenoiseremainswhite.Thelteringwasperformedtoremovefrequencycomponentsofthenoisethatcouldinteractwithothermodesoftheoscillator.ThiswasveriedbybothmeasuringthefrequencydistributionusinganetworkanalyzerandperformingfastFouriertransformsontimetracesofthenoisesignal. WhennoiseisinjectedintothedrivingvoltageVd=Vdc+Vaccos!dt+Vnoise.TheappliedampliedandlterednoiseisVnoise.FromtheTaylorexpansioninEq. 3{5 the 41

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Toensurethattheintrinsicnoiseduetodampingdoesnotexceedtheappliednoise,acomparisonofthenoisetorqueforbothsituationscanbedone.Assumingaminimumappliednoisevoltageof10VHz1=2,theassociatednoisetorqueforadevicewitha500m500mareaandVdc=0:2Visnoise;171017NmHz1=2.Theresultcanbecomparedtothemechanicalnoisetorqueassociatedwiththedissipation[ 45 ] whereIisthemomentofinertiaandisthedampingcoecient(seeCh. 3 ).InorderforthemechanicalnoiseinthissituationtocomparewiththeminimalappliednoisethetemperaturewouldhavetobeatleastT103K.However,allexperimentsarecarriedoutatatemperatureofeither4Kor77K,soanexternalnoisesourcewithamuchlargereectivetemperatureisusedtoinducetransitions. 42

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Inthischaptertheresponseofthetorsionaloscillatortoaperiodicexcitationisexamined.Thedrivingtorqueandexcitationfrequencyareshowntohaveadramaticeectonthesystemresponse.Whenthedrivingtorqueexceedsacriticalvaluetheoscillatorresponsebecomeshysteretic.Withinthehysteresismultiplestablestatescoexists.Eachstablestaterepresentsanactualmechanicaloscillationamplitudeanditsphase.Inthepresenceofasucientlyhighnoiseintensity,thesystemcanbeinducedtoswitchbetweenthestablestates.Examplesofthisbehavior,knownasinterstateswitching,willbegiven. +2_+!20=Taccos!dt;(3{1) whereistheangularrotationofthetopplate,isthedampingcoecient,!0=p istheamplitudeofthedrivingtorque.Also,Aistheoverlappingareaacrosswhichthevoltageisapplied,d=2mistheinitialplateseparation,andbistheeectivemomentarm.Theappliedtorque,asmentionedinSec. 2.4 ,actuallyhasatotaldrivingvoltageofVd=Vdc+Vaccos!dt.IngeneralthetorquedependsonthesquareofthevoltageVdwhere 43

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Theresponseoftheoscillatorforsmalldrivesislinearandwell-ttedbythesquarerootofaLorentzian.Theresponsewasmeasuredonadevicewithatopplatemeasuring500m500m3:5mandeachtorsionalspringmeasuring40m4m2m. IntheexperimentscoveredinthisdissertationVdcVac,thereforeV2accos2!dtprovidesanegligibled.c.componentandisignoredinthisanalysis.InthelinearresponseregimeV2dcleadstoashiftintheresonantfrequencyand2VdcVaccos!dtistheperiodicdrive. Inthisdissertationtheideaofcontrolparametersaectingthesystemisdiscussed;anexampleoftwocontrolparametersareTacand!d.Ofparticularinterestishowtheresponseofthesystemdependsonthetwoparameters.FollowingLandau[ 28 ]asolutionissoughtfortheamplitudeBandthephasesuchthat=Bei(wd+)t.Byassuming!0asolutionfortheamplitudeandphasecanbewritten: !d!0:(3{4) 44

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3{4 providesaclearindicationofhowtheamplitudeoftheoscillatorresponsedependsonspeciccontrolparameters.Theoverallmagnitudeoftheresponseincreaseslinearlyasthetorqueisincreased.Asthedrivingfrequencyissweptneartheresonantfrequencytheresponsebecomeslargeandisamaximumwhere!d=!0.Thedampingcoecientkeepsthedenominatorfromgoingtozero.Thedampingcoecientisone-halfthewidthofthetheresponseB2.Typicalvaluesforourexperimentsrangefrom0:1Hzto1Hz. Thedampingcoecientistheinverseoftherelaxationtimeofthesystem.Denedastr=1=,itisthecharacteristictimescaleoverwhichthesystemreturnstotheattractorafteradisturbance.So,ifthedrivingfrequencyischangedsuddenly,acorrespondingchangeinamplitudewilloccuroveratimescaleequivalenttotherelaxationtime.Asaresult,experimentaldicultiesarisewhentherelaxationtimeisgreaterthanacoupleofseconds.Hereanymeasurementbeingmade,whereparametersarechanged,musttakeintoaccounttherelaxationtime.InChapters 4 7 theeectofnoiseonthesystemisanalyzed.Heretherelaxationtimeplaysamajorpartinboththeanalysisandexperimentalprocedure.Thisisbecauseeverytimeauctuationagitatesthesystemitismovedawayfromtheattractor.Whenthenoiseisremoveditcantakeuptoseveraltimestherelaxationtimeforthesystemtosettlebackdown. Thedampinginthesystemresultsfromtwomainsources:squeezelmdamping[ 46 ]andthermoelasticdamping[ 47 ].Theeasiestwaytochangetherstsourceisto 45

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3{1 .ThesquaredresponseiswellttedbyaLorentzian.InFig. 3-1 theresponseisttedusingEq. 3{4 ,whichisthesquarerootoftheLorentzianlineshape.AsthedrivingtorqueisincreasedEq. 3{1 nolongercompletelydescribesthemotionoftheoscillatorbecauseofthestrong,nonlineardistancedependencesintheelectrostatictorque.ItisnecessarytoperformaTaylorexpansionoftheelectrostatictorqueaboutb d.TheTaylorexpansionis (1b d)2=boAV2d d+3(b d)24(b d)3+:::):(3{5) AgainthetermscontainingV2accos2!dtareignored.InsertingtheresultintotherighthandsideofEq. 3{1 gives +2_+[!20]+2+3=T0cos(!t);(3{6) where=3b30A Id5V2dc1,=b20A Id3V2dc1,andT0=boAVdcVac 3{6 ,therstordertermfromtheTaylorexpansionofthetorqueleadstoashiftintheoscillator'snaturalfrequency,whilethehigherordertermsleadtononlinearcontributionstotherestoringtorque.Thesenonlineareectsarisingfromandcanbecharacterizedbyaconstant[ 28 ]=3=8!052=12!30.Thiscoecientrelatesthesquaredamplitude 46

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Asthedrivestrengthisincreasedtheresponseoftheoscillatorbecomesasymmetricasthepeakshiftstolowerfrequencies.Atacriticalamplitudeadiscontinuityappearsintheresponse,forlargerdrivestheresponseishysteretic.Thearrowsindicatethedirectioninwhichthefrequencywasswept.Thebifurcationfrequenciesaref1=!b1 oftheresponsetothesecondordertermintheapproximationoftheappliedfrequency[ 28 ].ForourdevicegeometryisdominatedbythetermandthusintheabsenceofuctuationsthedevicecanberegardedasaDungoscillator.Also,includingtheshiftfromthed.c.voltagetheresonantfrequencyisredenedas!200=!20. FollowingLandau[ 28 ],theamplitudedependenceoftheDungoscillatoronthesystemparametersis 47

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3{7 ,issimpliedtoEq. 3{4 .Asthedriveincreases,however,thesetermscannotbeignoredandtheresonantlineshapebecomesasymmetricandthepeakbeginstoshifttolowerfrequenciesasexpectedforanegative.Thenegative,knownasasoftspringnonlinearity[ 48 ],isassociatedwithanexternalnonlinearitywhichresultsfromthehighordertermsintheappliedelectrostatictorque.Intrinsicnonlinearity,whichleadstopositive,tiltsthepeaktotheright.Thistypeofnonlinearity,knownasahardspringnonlinearity,canresultfromthegeometryofthespringandtheboundaryconditionsofitsconnectiontotheanchorandthetopplate. Whenthepeakoftheresponsereachesafrequencyof!=!00p 3-2 andiswelldescribedbyEq. 3{6 .Thetwooscillationamplitudes,whichexistbetweenthelowerbifurcationfrequency!b1 3-3 ashowsanexampleofthesystemswitchingfromoneoscillationamplitudetotheotherandback 48

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(a)Histogramofoscillationamplitudeshowingtherelativeoccupationofthetwoamplitudestates.(b)Theoscillatorswitchingbetweenthetwodistinctoscillationamplitudesstatesofthesysteminthepresenceofnoise. foraparticulardrivingfrequencyanddrivingamplitude.Bytakingalongertimetraceoftheoscillationresponseandbinningtheamplitudeoftheresponseinahistogram,asshowninFig. 3-3 b,therelativeoccupationbetweenthetwostatescanbeinferred.Inthiscase,thesystemfavorstheupperoscillationamplitude.Thisindicatesthattheactivationbarrierneededtobeovercomebythenoiseintensity,whengoingfromthelowerstatetotheupperstate,issmallerthanthebarrierseparatingtransitionsfromtheupperstatetothelowerstate. 49 ],wherethechildmodulateshismomentofinertiaattwice 49

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Parametricresonanceinatorsionaloscillator.Theamplitudeoftheoscillationisplottedversusthedrivingfrequency.Theresponseisathalfthedrivingfrequency.Above!b1onlyonestablestate,withazeroamplitude,exists.Between!b2and!b1thezeroamplitudestatebecomesunstable,however,amonotonicallydecreasingamplitudeappearswhichpossessestwoidenticalstatesthatareoutofphasebyrad.Finallyat!b2thezero-amplitudestatebecomesstableagainandthreestatescoexist. theswingfrequency,resultingintheswingreachingalargerheight.Inthissectionwediscusshowthesameideaisachievedusingatorsionaloscillator,theonlydierencebeingtheeectivespringconstantismodulatedinsteadofthemomentofinertia. FollowingLandau[ 28 ],theequationofmotionforaresponseat!d=2whenadriveof!d2!00isappliedcanbewrittenas +2_+!200+ke backdownbythetimetheswingreachesthepeakheightofitsforwardmotion.Astheswingreturnstotheminimumthechildrepeatstheupanddownmotion.Bycontinuallyraisingandlower,thechildeectivelyismodulatingthemomentofinertiaoftheswingattwicetheswingingfrequency.Thisactstoparametricallyexciteoscillations. 50

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3{8 isgivenby 2(!2!00)s Fordrivingamplitudesbelowacriticalamplitude,givenbytheconditionthatthedeterminantofEq. 3{9 iszero,thesystemshowsnoresonantresponse.WhenthedeterminantofEq. 3{9 isreal,twobifurcationfrequenciesappearatfrequencies!b2and!b1givenby AsshowninFig. 3-4 ,whentheoscillatorisdrivenatfrequencies!d=2>!b1onlyasinglestablestatewithzeroamplitudeexist.Werefertothisstateasthezeroamplitudestate.At!b1apitch-forkbifurcationoccursandtwostablestatesemergeandthezerostatebecomesunstable.Thestablestatesareperiod-twovibrationsthatareareoutofphasebyradandhavethesameamplitudewhichincreasesasthefrequencyisdecreased.Againthecurvingoftheresponsetowardlowerfrequenciesisaresultofthenegativecoecientofnonlinearity.Atthelowerbifurcationfrequency!b2asecondpitchforkbifurcationoccursastheunstablezerostatebecomesstableagainandtwounstablestatescorrespondingtoperiod-twovibrationsemerge.Thehighamplitudestate,possessingthestableperiod-twovibrations,stillexistshere.Thetwounstableperiod-twovibrationsseparatethestableperiod-twovibrationsfromthestablezeroamplitudestate. MuchliketheDungoscillatorcase,theresponseishysteretic.Asthefrequencyisincreasedpast!b2thesystemjumpsuptothehighamplitudestate.Whenthefrequencyissweptbackdownpast!b2itwillstayintheupperamplitudestate.Thecaseofthestableperiod-twovibrationsisdierentfromtheDungoscillator.Whenthesystem 51

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Thesystemisdrivenat!dwhiletheresponseisat!d=2.Shiftingtheresponsebyleadstoanequivalentandvalidsolutiontotheequationofmotion.Thesetwosolutionshaveanidenticalamplitudebutrepresenttwouniquestates. crosses!b1fromthehighfrequencysidethetwocoexistingstatesareoccupiedwithequalprobability;thisresultsfromatime-translationsymmetryintheresponse,whereshiftingtheresponsebyradgivesanothervalidsolution.Ultimately,thesystemdiusesoutoftheunstablestateandintoonethestablestatesduetothebackgroundthermalnoise.InFig. 3-5 aplotofthedrivingfrequencyisshownwiththe!d 3-6 ashowsahistogramresultingfromthesystemswitchingbetweentheperiod-2vibrationsatafrequency!b2
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(a)Whennoiseisapplied,thesystemisinducedtotransitionbetweenthetwostates.Fortheperiod-2vibrationstheprobabilityofswitchingis50/50.Ahistogramshowingthisoccupationisplotted.(b)Theamplitudeforthistypeoftransitiononlychangeswhenthesystemisswitchingfromonestatetotheother. Whenaswitchoccursthesystemmovesthoughthezeroamplitudeunstablestate.Forfrequenciesbelow!b2thezerostatealsobecomesstable.SimilartothecaseoftheDungoscillatortheamplitudecanswitchbetweentwodistinctvalues.Inthisregionthetwostableperiod-2statesstillexist. 53

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54

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Theresultsandderivationsinthischaptercanbefoundinthearticles,PathsofFluctuationInducedSwitching,H.B.Chan,M.I.Dykman,andC.Stambaugh,PhysicalReviewLetters100,130602(2008)\Copyright2008bytheAmericanPhysicalSociety"andSwitching-pathdistributioninmultidimensionalsystems,H.B.Chan,M.I.Dykman,andC.Stambaugh,PhysicalReviewE79,051109(2008)\Copyright2008bytheAmericanPhysicalSociety". 10 37 50 { 58 ].Despiteitscentralrolefortheunderstandingofuctuation-inducedswitchingandswitchingrates,theideaoftheMPSPhasnotbeentestedexperimentallyinmultivariablesystems.Thequestionofhowthepathsfollowedinswitchingaredistributedinphasespacehasnotbeenaskedeither. Perhapsclosesttoaddressingtheaboveissueswastheexperimentondropouteventsinasemiconductorlaserwithopticalfeedback[ 59 ].Inthisexperimenttheswitchingpathdistributioninspaceandtimewasmeasuredandcalculated.However,thesystemwascharacterizedbyonlyonedynamicvariable,andthusallpathslieononelineinphasespace.Inanothereort,electroniccircuitsimulations[ 60 ]comparethedistributionofuctuationalpathstoandrelaxationalpathsfromacertainpointwithinonebasinof 55

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59 60 ]donotapplytotheswitchingpathdistributioninmultivariablesystems. InthisChapterweintroducetheconceptofswitchingpathdistributioninphasespaceandthequantitythatdescribesthisdistribution,calculatethisquantity,andreportthedirectobservationofthetubeofswitchingpaths.AmoredetailaccountofthecalculationcanbefoundinRef.[ 61 ].Theexperimentalandtheoreticalresultsontheshapeandpositionofthepathdistributionareinexcellentagreement,withnoadjustableparameters.Theresultsopenthepossibilityofecientcontroloftheswitchingprobabilitybasedonthemeasurednarrowpathdistribution. InSec. 4.2 weprovidethequalitativepictureofswitchingandgiveapreviewofthecentraltheoreticalandexperimentalresults.Sec. 4.3 presentsatheoryoftheswitchingprobabilitydistributioninthebasinsofattractiontotheinitiallyoccupiedandinitiallyemptystablestatesaswellassomesimpleresultsforsystemswithdetailedbalance.InSec. 4.4 thesystemusedintheexperiment,amicromechanicaltorsionaloscillator,isdescribedandquantitativelycharacterized.Section 4.5 presentstheresultsoftheexperimentalstudiesoftheswitchingpathdistributionforthemicromechanicaloscillator,withthecoexistingstablestatesbeingthestatesofparametricallyexcitednonlinearvibrations.Genericfeaturesofthedistributionarediscussed,andthelackoftime-reversalsymmetryinswitchingofsystemsfarfromthermalequilibriumisdemonstratedforthersttime.Section 4.6 containsconcludingremarks. 4-1 .Forlowuctuationintensity,thephysicalpictureofswitchingisasfollows.ThesystempreparedinitiallyinthebasinofattractionofstateA1,forexample,willapproachqA1overthecharacteristicrelaxationtimetrandwillthenuctuateaboutqA1.Weassumetheuctuationintensitytobe 56

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Eventhoughuctuationsaresmallonaverage,occasionallythereoccurlargeuctuations,includingthoseleadingtoswitchingbetweenthestates.Theswitchingrate12fromstateA1toA2ismuchlessthanthereciprocalrelaxationtimet1r,thatis,thesystemuctuatesaboutA1foralongtime,onthescaleoftr,beforeatransitiontoA2occurs.InthetransitionthesystemmostlikelymovesrstfromthevicinityofqA1tothevicinityofqS.Itstrajectoryisexpectedtobeclosetotheoneforwhichtheprobabilityoftheappropriatelargerareuctuationismaximal.ThecorrespondingtrajectoryisillustratedinFig. 4-1 .FromthevicinityofqSthesystemmovestostateA2closetothedeterministicuctuation-freetrajectory.ThesetwotrajectoriescomprisetheMPSP. Forbrevity,wecallthesectionsoftheMPSPfromqA1toqSandfromqStoqA2theuphillanddownhilltrajectories,respectively.ThetermswouldliterallyapplytoaBrownianparticleinapotentialwell,withA1;2correspondingtotheminimaofthepotentialandStothebarriertop(seeSec. 1.2 ). WecharacterizetheswitchingpathdistributionbytheprobabilitydensityforthesystemtopassthroughapointqonitswayfromA1toA2, Here,theintegrandisthethree-timeconditionalprobabilitydensityforthesystemtobeatpointsqfandqattimestfandt,respectively,giventhatitwasatq0attimet0.Thepointq0lieswithindistancelDofqA1andisotherwisearbitrary.Integrationwithrespecttoqfgoesovertherange2ofsmalluctuationsaboutqA2;thetypicallinearsizeofthisrangeislD. 57

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Phaseportraitofatwo-variablesystemwithtwostablestatesA1andA2.ThesaddlepointSliesontheseparatrixthatseparatesthecorrespondingbasinsofattraction.Thethinsolidlinesshowthedownhilldeterministictrajectoriesfromthesaddletotheattractors.Aportionoftheseparatrixnearthesaddlepointisshownasthedashedline.ThethicksolidlineshowsthemostprobabletrajectorythatthesystemfollowsinauctuationfromA1tothesaddle.TheMPSPfromA1toA2iscomprisedbythisuphilltrajectoryandthedownhilltrajectoryfromStoA2.Theplotreferstothesystemstudiedexperimentally,seeSec. 4.5 Wecallp12(q;t)theswitchingprobabilitydistribution.Ofutmostinterestistostudythisdistributioninthetimerange 112;121tft;tt0~tr:(4{2) Here,~tristheSuzukitime[ 62 ].Itdiersfromtrbyalogarithmicfactorlog[q=lD].Thisfactorarisesbecauseofthemotionslowingdownnearthesaddlepoint.Thetime~trismuchsmallerthanthereciprocalswitchingrates,andthesmallertheuctuationintensitythestrongerthedierence,becausethedependenceofijontheuctuation 58

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Fortt0tr,bytimetthesystemhasalready\forgotten"theinitialpositionq0.Thereforethedistribution(qf;tf;q;tjq0;t0),andthusp12,areindependentofq0;t0.Ontheotherhand,ifthesystemisonitswayfromA1toA2andattimetisinastateqfarfromtheattractors,itwillmostlikelyreachthevicinityofA2overtime.~trandwillthenuctuateaboutqA2.ThiswillhappenwellbeforethetimetfatwhichthesystemisobservednearA2,andthereforep12isindependentoftf. Itisclearfromtheaboveargumentsthat,inthetimerangegivenbyEq. 4{2 ,thedistributionp12(q;t)forqfarfromtheattractorsisformedbyswitchingtrajectoriesemanatingfromthevicinityofA1.Itgivestheprobabilitydensityforthesetrajectoriestopassthroughagivenpointqattimet.Inotherterms,thedistributionp12(q;t)isformedbytheprobabilitycurrentfromA1toA2andisdeterminedbythecurrentdensity. 2?bQ?;(4{3) wherekand?arecoordinatesalongandtransversetotheMPSP,andv(k)isthevelocityalongtheMPSP.Thematrixelementsofmatrix^Q=^Q(k)are/l2D,andZ=[(2)N1=det^Q]1=2.ItfollowsfromEq.( 4{3 )thattheoverallprobabilityuxalongtheMPSPisequaltotheswitchingrate,Zd?p12(q;t)v(k)=12: 4-2 .Theywereobtainedusingamicro-electromechanicaltorsionaloscillatordescribedinSec. 4.4 andCh. 2 .Thepathdistributiondisplaysasharpridge.Wedemonstratethatthecross-sectionoftheridgehasGaussianshape.Asseen 59

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(a)Switchingprobabilitydistributioninaparametricallydrivenmicro-electromechanicaloscillator.Theprobabilitydistributionp12(X;Y)ismeasuredforswitchingoutofstateA1intostateA2.(b)Thepeaklocationsofthedistributionareplottedasblacksquaresandthetheoreticalmostprobableswitchingpathisindicatedbythegreenline.AlltrajectoriesoriginatefromwithinthecyancircleinthevicinityofA1andlaterarriveatthegreencirclearoundA2.Theportionofthedistributionoutsidethemaroonlinesisomitted. fromFig. 4-2 ,themaximumoftheridgeliesontheMPSPwhichwascalculatedforthestudiedsystem.InApp. A.1 thecodenecessarytogeneratetheMPSPisgiven. Equation( 4{3 )iswrittenforagenerallynonequilibriumsystem,butthesystemisassumedtobestationary.Intheneglectofuctuationsitsmotionisdescribedbyequationswithtime-independentcoecients.Inthiscasep12(q;t)isindependentoftimet. 60

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TheModelofaFluctuatingSystem 4{3 forasystemdescribedbytheLangevinequationofmotion _q=K(q)+f(t);hfn(t)fm(t0)i=2Dnm(tt0): Here,thevectorKdeterminesthedynamicsintheabsenceofnoise;K=0atthestablestatepositionsqA1,qA2andatthesaddlepointqS.WeassumethatqSliesonasmoothhypersurfacethatseparatesthebasinsofattractionofstatesA1andA2(seeFig. 4-1 ).Thefunctionf(t)inEq. 4{4 iswhiteGaussiannoise;theresultscanbealsoextendedtocolorednoise.ThenoiseintensityDisassumedsmall.ThedependenceoftheswitchingratesnmonDisgivenbytheactivationlaw,lognm/D1[ 10 37 50 { 54 ].Thisisalsothecasefornoise-drivencontinuoussystems,cf.Ref.[ 63 { 65 ]andpaperscitedtherein.Thereexistsextensiveliteratureonnumericalcalculationsoftheswitchingrateandswitchingpaths,cf.Ref.[ 66 { 70 ]andpaperscitedtherein. Inthemodel( 4{4 ),thecharacteristicrelaxationtimetrandthecharacteristicdiusionlengthlDare wherekaretheeigenvaluesofthematrix@Km=@qncalculatedatqA1;qA2andqS. Forawhite-noisedrivensystem(seeEq. 4{4 ),thethree-timeprobabilitydistribution(qf;tf;q;tjq0;t0)inEq.( 4{1 )canbewrittenasaproductoftwo-timetransitionprobabilitydensities, whichsimpliesfurtheranalysis.TheanalysisisdoneseparatelyforthecasewheretheobservationpointqlieswithintheattractionbasinsoftheinitiallyemptyattractorA2andtheinitiallyoccupiedattractorA1. 61

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71 ].Incontrasttosystemswithoutdetailedbalance[ 72 73 ],whereforsmoothU(q)theMPSPnearthesaddlepointisdescribedbyananalyticfunctionofcoordinatesand^kisperpendiculartotheseparatinghypersurface. 4.4.1DeviceCharacteristics 3.3 ).Themovabletopplate(200m200m3.5m)issuspendedbytwotorsionalrods(4m2m36mandk=3:96108Nm).Therearetwoxedelectrodesonthesubstrate,oneoneachsideofthetorsionalrod.The2mgapunderneaththemovableplateiscreatedbyetchingawayasacricialsiliconoxidelayer. AsdescribedinChapter 3 torsionaloscillationsofthemovabletopplateareexcitedbyapplyingadrivingvoltageVd=Vdc+Vaccos(!dt)+Vnoise(t)tooneofthelowerelectrodeswhilethetopplateremainselectricallygrounded.Thedrivingfrequency!d=2!00+"isclosetotwicethenaturalfrequency!00.ThedcvoltageVdc(1V)ismuchlargerthantheamplitudeVac(141mV)ofsinusoidalmodulationandtherandomnoise 62

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+2_+!200+kecos(!dt)+2+3=N(t):(4{7) Torsionaloscillationsofthetopplatearedetectedcapacitivelybytheotherelectrode.Thiselectrodeisconnectedtoad.c.voltagesourcethroughalargeresistor.Ahighelectronmobilitytransistorisplacedincloseproximitytothedevicetomeasuretheoscillatingchargeonthedetectionelectrodeinducedbymotionofthetopplate.Theoutputofthetransistorisconnectedtoalock-inamplierreferencedathalfthedrivingfrequency!d.Forthechosentimeconstantof300s,themeasurementuncertaintyis80rad,about0.6%ofthefullscaleinFig. 4-2 andmuchsmallerthanthewidthofthepathdistributions.Theoscillationamplitudesin-phase(X)andout-of-phase(Y)withthereferencefrequencywererecordedevery2ms.Allmeasurementswereperformedat77Kand<106Torr. dt=!2dke Thequadraturesq1andq2aredirectlyproportionaltothesignalcomponentsXandYmeasuredwiththelock-inamplier,withtheproportionalityconstantEdeterminedbythemeasuringapparatus, 63

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4{8 )intoEq.( 4{7 )andneglectingfastoscillatingterms,wecanwritetheequationsofmotionforq=(q1;q2)intheform( 4{4 ).ThefunctionKindimensionlesstimeisgivenby Here=ke=2!d,=!d(2!00!d)=ke,andg=q4=4(1+)q21=2+(1)q22=2,where^"isthepermutationtensor.Equation_q=KgivesthedownhillsectionoftheMPSPoftheoscillator.TheuphillsectionoftheMPSPcanbecalculatedbysolvingtheHamiltonianequationsofmotionthatfollowfromtheHamilton-Jacobiequationforanauxiliaryparticle[ 61 ]. 28 ].Oscillationsareinducedathalfthemodulationfrequencyinarangecloseto!00.Betweenthetwobifurcationfrequencies!b1(139318:11rad/s)and!b2(139384:74rad/s)thereexiststwostablestatesofoscillationsatfrequency!d=2.Theydierinphasebybuthaveidenticalamplitude.BothstatesarestablesolutionsofEq.( 4{7 ).Theirbasinsofattractionintherotatingframeareseparatedbyaseparatrixthatgoesthroughtheunstablestationarystate,whichinthelaboratoryframehaszerovibrationamplitudeatfrequency!d=2.ThephaseportraitintherotatingframeisillustratedinFig. 4-1 .Thedrivingfrequencyischosentobe278639.16rad/sformeasurementoftheswitchingpathdistribution. Wenotethatparametricresonanceinnano-andmicro-electromechanicalsystemshasattractedconsiderableattention[ 19 74 { 78 ].Sincehereweareinterestedinthestudiesoftheprincipalfeaturesofnoise-inducedswitching,wechosethesimplestnontrivialregimewherethesystemhasonlytwostablestates,whichoccursfor!b1
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Harmonicandparametricresonancesofthemicromechanicaltorsionaloscillator.Forresonantdriving(),theoscillationamplitudeisplottedasafunctionoftheoscillationfrequency.Thethinlineisattotheharmonicoscillatorresponse.Itgivesdeviceparametersand!00.Forparametricresonance(),thedrivingfrequencyistwicetheoscillationfrequency.Thet(thickline)yieldsnonlinearandtheeectiveparametricmodulationamplitudeke. frameisunderdamped,whichisadvantageousforstudyingagenericfeatureofuctuationsinsystemsfarfromthermalequilibrium,thebreakingoftimereversalsymmetry. CalculationoftheMPSPrequiresanumberofdeviceparametersincluding,!00,theparametricmodulationamplitudeke,andthenonlinearconstantnonlinear=3=8!00[ 79 ].Theseparametersareobtainedfromthelinearandnonlinearresponsesofthedevice.Whenthedeviceisresonantlydrivenwithsmallamplitudeatfrequencycloseto!00,itrespondsasaharmonicoscillator.Fromtheresonancelineshape(Fig. 4-3 ),and!00aredeterminedtobe6.99rad/sand139352.118rad/srespectively.Theremainingtwoparametersareextractedfromtheparametricresonanceoftheoscillatorfor!dcloseto2!00.Specically,theparametricmodulationamplitudekeisdeterminedfromthebifurcationfrequencies!b1;2=2!00!p,where!p=(k2e(4!00)2)1=2=2!00.Thisgiveske=1:94107s2.Thenonlinearparameternonlinear(1.08106s1)isobtainedfromtheproportionalityconstantbetweenthesquareoftheparametricoscillationamplitude 65

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4-3 .SeeAppendix B foramoredetailedanalysis. Usingthesemeasureddeviceparameters,thedimensionlessconstantscontainedinKinEq.( 4{10 )andinEq.( 4{9 )canbedirectlycalculatedtobeE=176:349,=4.968,and=-0.9367.ThetheoreticaloptimalescapepathinFig. 4-2 iscalculatedwiththeaboveparametervalues.Noadjustableparametersareused. 4.5.1MeasuredSwitchingPathDistribution 4-2 a)andsubsequentlyarrivesatstateA2(withintherightcyancircle2).Figure 4-2 showstheswitchingprobabilitydistributionderivedfrommorethan6500transitions.Whileineachtransitionthesystemfollowsadierenttrajectory,thetrajectoriesclearlyliewithinanarrowtube. ThemaximumofthedistributiongivestheMPSP.InFig. 4-2 b,thelocationofthismaximumisplottedontopoftheMPSPobtainedfromtheory.Theoscillatorisunderdampednotonlyinthelaboratoryframe,butalsointherotatingframe.ThereforeboththeuphillanddownhillsectionsoftheMPSParespirals.Ontheuphillsection,theMPSPemergesclockwisefromA1andspiralstowardthesaddlepointattheorigin.Uponexitfromthesaddlepoint,itmakesanangleand,onthedownhillsection,continuestospiralclockwisetowardA2. TheagreementbetweenthemeasuredpeakintheprobabilitydistributionandtheMPSPobtainedfromtheoryisexcellent.Therearenoadjustableparameterssincealldeviceparametersareaccuratelydeterminedfromtheharmonicandparametric 66

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ClosetothestablestatesthepeaksofthedistributionatsuccessiveturnsoftheMPSPoverlap,preventingtheaccuratedeterminationoftheMPSP.TheplotinFigs. 4-2 aand 4-2 bhasexcludedtheportionsoftrajectoriespriortoescapingfromtheinitialstateA1anduponarrivingatthenalstateA2,whichareboundbythetwomaroonlines.SuchcutoalsoeliminatesthelargepeaksofthedistributioncenteredatA1andA2,whicharisebecausetheoscillatorspendsmostofitstimeuctuatingaboutA1andA2.Thesepeaksarenotrelevanttoswitchingdynamics. Figure 4-4 comparesthemeasuredandpredictedvelocityalongtheMPSP.Here,again,thegoodagreementisdemonstratedwithnoadjustableparameters.Asexpected,themeasuredvelocitydecreasesnearthesaddlepoint,k=0.However,itdoesnotbecomeequaltozero,inagreementwiththeargumentthatthetotalprobabilitycurrentremainsconstant.Motionnearthesaddlepointisdominatedbydiusion. 4-5 ashowsthedistributioncross-sectionalongthebluelinetransversetotheMPSPinFig. 4-2 b.Itiswell-ttedbyaGaussian.Gaussiandistributionswithdierentheightandareaareobservedalsoinothercross-sectionsexceptclosetothesaddlepoint.Figure 4-5 bplotstheareaundertheGaussiandistributionversusthereciprocalmeasuredvelocityontheMPSP,fordierentcross-sections.ThelineardependenceagreeswithEq. 4{3 andindicatesthattheprobabilitycurrentfromtheinitiallyoccupiedattractortotheemptyoneisconstant.Thiscurrentgivestheswitchingrate12[ 5 ]. WendthattheprobabilitycurrentconcentrateswithinanarrowtubedeepintothebasinsofattractionofA1andA2.InthebasinofattractiontoA2butnottooclosetoA2,muchoftheprobabilitydistributioncarriestheswitchingcurrent.However,the 67

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MeasuredaveragedvelocityalongtheMPSP()andthevelocitypredictedbytheory(line).Thevelocitydecreasestozeroatthesaddlepoint(k=0). overallquasistationaryprobabilitydistributiondeepinsidethebasinofattractionofA1islargelyassociatedwithuctuationsaboutA1thatdonotleadtoswitching.Thepartofthedistributionresponsiblefortheswitchingcurrentisanexponentiallysmallfractionofthetotaldistribution.Neverthelessourformulationmakesitpossibletosingleoutanddirectlyobservethisfraction. TheslowingdownnearthesaddlepointshowninFig. 4-4 leadstostrongbroadeningandincreaseinheightoftheswitchingprobabilitydistributionseeninFig. 4-2 a.Becausemotionnearthesaddleisdiusive,switchingpathsloosesynchronization.Inotherwords,thedistributionoftimesspentbythesystemnearthesaddlepointiscomparativelybroad.Thisiswhyitisadvantageoustostudythedistributionofswitchingpathsinthespaceofdynamicalvariablesratherthanintime. 68

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(a)Thecross-sectionalongthebluelineinFig. 4-2 transversetotheMPSP.ThesolidlineisaGaussiant.(b)VelocityontheMPSPvs.inverseareaundercross-sectionsoftheswitchingprobabilitydistribution.Thesolidlineisalineartforcedthroughtheorigin. 71 80 ],i.e.,toreplacingwithinEq.( 4{7 ).Inoverdampedequilibriumsystemswithdetailedbalance,thesetwopathscoincideinspace(butareoppositeindirection). Ourparametricoscillatorisdrivenfarfromthermalequilibrium.ThereforetheuphillsectionoftheMPSPdoesnotsimplyrelatetothedeterministictrajectorywithreversedsignofdissipation.ThissectionoftheMPSP,i.e.,themostprobableuctuationalpathfromA1tothesaddlepointattheoriginisplottedasthethicksolidlineinFig. 4-6 .Uponsignreversalofthedissipation,theattractorbecomesarepeller,asinthecaseofsystemsinthermalequilibriumdescribedearlier.However,incontrasttoequilibrium 69

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ComparisonoftheMPSPandthedissipation-reversedpath.ThesectionofthemostprobableswitchingpathfromA1toSisshownasathickblueline.Uponchangingthesignofdissipation,theattractorisshiftedtoanewlocationA01andbecomesarepeller.Theuctuation-freepathwithreverseddissipationfromA01toSisshownasthethinorangeline. systems,itisalsoshiftedawayfromitsoriginallocation(fromA1toA01inFig. 4-6 ).Thedissipation-reversedpathisshownasthethinsolidlineinFig. 4-6 .Inaddition,Fig. 4-1 allowsonetocomparetheuphillsectionoftheMPSPwiththedeterministicdownhillpathfromStoA1.OurdatashowthattheuphillsectionoftheMPSP,whichisformedbyuctuations,thedissipation-reversedpath,andthedownhillnoise-freepathfromthesaddletothestablestatearealldistinct.Thetimeirreversibilityoftheswitchingpathsisdirectlyrelatedtothelackofdetailedbalanceofourdrivenoscillator,distinguishingitfrombistablesystemsinthermalequilibrium. 70

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Theswitchingprobabilitydistributionwasshowntheoreticallytohaveashapeofanarrowridgeinphasespace.Farfromthestationarystates,thecross-sectionoftheridgeisGaussian.Themaximumoftheridgeliesonthemostprobableswitchingpath(MPSP). Experimentalstudiesoftheswitchingpathdistributionweredoneusingahigh-Qmicromechanicaltorsionaloscillator.Allparametersoftheoscillator,includingthenonlinearityconstant,weredirectlymeasured.Theoscillatorwasdrivenintoparametricresonance,whereithadtwocoexistingvibrationalstatesthatdierinphase.Thepathsfollowedinswitchingbetweenthesestateswereaccumulatedandtheirdistributioninthespaceofthetwodynamicalvariables(theoscillationquadratures)wasobtained.ItwasfoundthatthedistributionhasindeedtheshapeofaGaussianridge. Thereisexcellentagreementbetweentheexperimentalandtheoreticalresults,withnoadjustableparameters.ThemeasuredmaximumoftheswitchingpathdistributionliesontopofthetheoreticallycalculatedMPSP.ThemeasuredvelocityofmotionalongtheMPSPasafunctionofthepositionontheMPSPalsoquantitativelyagreeswiththetheory.Animportantpropertyofthepathdistributionisthetotalcurrentconservation:theproductofthevelocityofmotionalongtheMPSPandthecross-sectionareaofthepathdistributionremainsconstant.Thisconservationofprobabilitycurrentwasdemonstratedexperimentally.Inaddition,weobserved,forthersttime,thatthelackofdetailedbalanceleadstothedierencebetweentheuphillsectionoftheMPSPandthenoise-freepathwithreversedsignofdissipation. 71

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72

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TheresultsinthischaptercanbefoundinNoiseactivatedswitchinginadriven,nonlinearmicromechanicaloscillator,C.StambaughandH.B.Chan,PhysicalReviewB,73,172302(2006)\Copyright2006bytheAmericanPhysicalSociety"andActivationBarrierScalingandCrossoverforNoise-InducedSwitchinginMicromechanicalParametricOscillators,H.B.ChanandC.Stambaugh,PhysicalReviewLetters,99,060601(2007)\Copyright2007bytheAmericanPhysicalSociety". 33 ].Forinstance,inaDungoscillatorresonantlydrivenintobistability,spinodalbifurcationsoccurattheboundariesofthebistableregion.Onestablestatemergeswiththeunstablestatewhiletheotherstablestateremainsfarawayinphasespace(seeFig. 1-4 ).Theorypredicts[ 33 35 ]thattheactivationbarrierscaleswithcriticalexponent3/2nearspinodalbifurcationsindrivensystems.Ontheotherhand,adierentcriticalexponentof2isexpectedatapitchforkbifurcation[ 36 ]insystemswhereallthreestatesmerge(seeFig. 1-5 ).Suchbifurcationcommonlytakesplaceinparametricallydrivensystemswhereperioddoublingoccurs.Forinstance,uctuation-inducedphaseslipswereobservedinparametricallydrivenelectronsinPenningtraps[ 16 ]betweentwocoexistingattractorsandtransitionsbetweenthreeattractorswerestudiedinmodulatedmagneto-optical 73

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FrequencyresponseofsampleAscaledbytheexcitationvoltageamplitudesof47V()and450V().Theredlinerepresentsattothedataatsmallerexcitationusingtheresponseofadampedharmonicoscillator.Forthelargeexcitation,twodynamicalstatescoexistfrom3278:76Hzto3280:8Hz.Thebluelinetsthedatatoadampedoscillatorwithcubicnon-linearity[ 28 ],yielding=1:21014rad2s2. traps[ 15 ].Toourknowledge,theactivationbarriershavenotbeenmeasuredoverawideenoughparameterrangeintheseparametricallydrivensystemstodemonstratetheuniversalscalingatdrivingfrequenciesnearthetwocriticalpointsandthecrossovertosystem-specicdependenceatlargefrequencydetuning. InSec. 5.2 wereportourinvestigationofnoise-activatedswitchinginsystemsfarfromequilibrium.Awell-characterizedsystem,anunderdampedmicromechanicaltorsionaloscillatorperiodicallydrivenintononlinearoscillations.Heretheoscillatorhastwostabledynamicalstateswithdierentoscillationamplitudewithinacertainrangeofdrivingfrequencies.Weinducetheoscillatortoescapefromonestateintotheotherbyinjectingnoiseinthedrivingforce.Bymeasuringtherateofrandomtransitionsasafunction 74

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12 33 81 82 ].Hereacriticalexponentof3/2isexpected. InSec. 5.3 ,wereportmeasurementsoftheactivationbarrierfoructuation-inducedswitchinginparametrically-drivenmicromechanicaltorsionaloscillators,asystemthatisfarfromthermalequilibrium.Thespringconstantofourdeviceismodulatedelectrostaticallyneartwicethenaturalfrequency.Undersucientlystrongparametricmodulation,twopitchforkbifurcationpointsexist.Atthesupercriticalbifurcation,thereemergetwostableoscillationstatesthatdierinphaseby.Atthesubcriticalbifurcation,anadditional,stablestatewithzerooscillationamplitudeappears.Noiseinducestransitionsbetweenthecoexistingattractors.Bymeasuringtherateofrandomtransitionsasafunctionofnoiseintensity,wededucetheactivationbarrierforswitchingoutofeachattractorasafunctionoffrequencydetuning.Nearbothbifurcationpoints,theactivationbarriersarefoundtodependonfrequencydetuningwithcriticalexponentof2,consistentwiththepredicteduniversalscalinginparametricallydrivensystems[ 36 ].Awayfromtheimmediatevicinityofthebifurcationpoint,universalscalingrelationshipsfortheactivationbarriernolongerhold.Wendthatinourparametricoscillator,thedependenceoftheactivationbarrieronfrequencydetuningchangesfromquadraticto3=2thpower. DeviceandExperimentalSetup 75

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2 InFig. 2-4 across-sectionalschematicoftheoscillatorisshownwithitselectricalconnectionsandmeasurementcircuitry.Theapplicationofaperiodicvoltagewithd.c.biasVdc1tooneoftheelectrodesleadstoanelectrostaticattractionbetweenthegroundedtopplateandtheelectrode.Torsionaloscillationsofthetopplateareexcitedbytheperiodiccomponentoftheelectrostatictorque.Thedetectionelectrodeisconnectedtoad.c.voltageVdc2througharesistorR.Astheplateoscillates,thecapacitancebetweentheplateandthedetectionelectrodechanges.Thedetectionelectrodeisconnectedtoachargesensitivepreamplierfollowedbyalock-inamplierthatmeasuresthesignalattheexcitationfrequency.Measurementsforbothsampleswereperformedatpressureoflessthan2107Torr.SamplesAandBweremeasuredatliquidnitrogenandheliumtemperaturesrespectively.Themaineectofdecreasingthetemperaturefromliquidnitrogentoliquidheliumisthereductionofthedampingconstant,yieldingqualityfactorsQofabout4;000forsampleAand16;000forsampleB. TheexcitationvoltageVconsistsofthreecomponents: Vd=Vdc+Vacsin(!t)+Vnoise(t):(5{1) ThethreetermsontherightsideofEq. 5{1 representthedcvoltage,periodicacvoltagewithangularfrequency!andrandomnoisevoltagerespectively.AsdescribedinCh. 3 ,VdcischosentobemuchlargerthanVacandVnoise.VnoiseisGaussianwithabandwidthof100Hzaboutthenaturalfrequencyoftheoscillator.Thestrongdistancedependenceoftheelectrostaticattractionbetweenthetopplateandtheelectrodeleadstononlinearcontributionstotherestoringtorque.ATaylorexpansionoftheelectrostatictorqueleads 76

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3{6 withtheadditionofanoisesourceN(t):+2_+[!20]+2+3=T0cos(!t)+N(t): 5-1 bthefrequencyresponseofsampleAattwodierentoscillationamplitudes.Bothresponseshavebeenscaledbytheirrespectiveexcitationvoltages.Forthesmallerexcitation,theresonancepeakisttedwellbytheredlinethatcorrespondstotheresponseofadampedharmonicoscillator.Astheperiodicexcitationisincreased,thecubicterminEq. 3{6 leadstononlinearbehaviorintheoscillations.Theresonancecurvebecomesasymmetricwiththepeakshiftingtolowerfrequencies,consistentwithanegativevalueof.Atahighenoughexcitation,hysteresisoccursinthefrequencyresponse,asshownbythesquaresinFig. 5-1 b.Withinacertainrangeofdrivingfrequencies,therearetwostabledynamicstateswithdierentoscillationamplitudeandphase.Dependingonthehistoryoftheoscillator,thesystemresidesineitherthehigh-amplitudestateorlow-amplitudestate.Intheabsenceofuctuations,theoscillatorremainsinoneofthestablestatesindenitely. AsdescribedinSec. 1.2.2 ,whensucientnoiseisappliedintheexcitation,theoscillatorisinducedtoescapefromonedynamicstateintotheother.Sincethisdriven,bistablesystemisfarfromthermalequilibriumandcannotbecharacterizedbyfreeenergy,calculationoftheescaperateisanon-trivialproblem.Theoreticalanalysis[ 12 33 ]suggeststhattherateofescapeataparticulardrivingfrequencydependsexponentiallyontheratioofanactivationbarrierEatothenoiseintensityIN: =0eEa=IN:(5{2) ThenoiseintensityINwasintroducedinSec. 2.4 andinthischapterhasarbitraryunits.LikewiseEa/Ri,wheretheunitsofEacanbeassumedtobethesameastheunitsof 77

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Closetothebifurcationfrequencywherethehigh-amplitudestatedisappearstheactivationbarrierisexpectedtodisplaysystem-independentscaling: wherethefrequencydetuningisthedierencebetweenthedrivingfrequencyandthebifurcationfrequency.Thefrequencyparameterisusedhereinlieuofthegenericsystemparameter.Theactivationbarrierispredicted[ 12 33 ]toincreasewithfrequencydetuningwithcriticalexponent=3=2.Thisscalingrelationisgenericandisexpectedtooccurinanumberofnon-equilibriumsystems.Wedescribebelowourcomprehensiveexperimentalinvestigationofactivatedswitchingfromthehigh-amplitudetothelow-amplitudestatefortwomicromechanicaloscillatorswithdierentresonantfrequenciesanddampingcoecients.Thecriticalexponentsmeasuredforbothsampleswereingoodagreementwiththeory. 5-2 ashowstypicalswitchingeventsatanexcitationfrequencyof3278:81HzforsampleAwheretheoscillatorresidesinthehigh-amplitudestateforvariousdurationsbeforeescapingtothelow-amplitudestate.Duetotherandomnatureofthetransitions,alargenumberofswitchingeventsmustberecordedtodeterminethetransitionrateaccurately.DuringthetimeintervalbetweenswitchingeventsinFig. 5-2 a,theoscillatorisresettothehigh-amplitudestateusingthefollowingprocedure.First,thenoiseisturnedoandthedrivingfrequencyisincreasedbeyondtherangeoffrequencieswherebistability 78

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(a)Inthepresenceofnoiseintheexcitation,theoscillatorswitchesfromthehigh-amplitudestatetothelow-amplitudestateatdierenttimeintervals.Thesystemisresettotheupperamplitudestatebetweenswitchingevents.Thedetuningfrequencyis=0:05Hz.(b)Histogramoftheresidencetimeintheupperstatebeforeswitchingoccurs,atadetuningfrequencyof0:05HzforsampleA.Thelineisanexponentialtusingthedecayrateequation. occurs(>3280:8HzasshowninFig. 5-1 b).Thedrivingfrequencyisthendecreasedslowlytowardsthetargetfrequencysothattheoscillatorremainsinthehigh-amplitudestate.Oncethetargetfrequencyisreached,thenoiseisturnedbackonandthetimefortheoscillatortoescapefromthehigh-amplitudestateisrecorded.Thisprocessisthenrepeatedmultipletimestoaccumulatethestatisticsforswitching.Suchaprocedureisnecessarybecausetheenergybarrierfortransitionsfromthelow-amplitudestatebacktothehigh-amplitudestateismuchlargerthanthebarrierfortransitionsintheoppositedirection.Thus,noiseinducedtransitionsfromthelow-amplitudestatetothehigh-amplitudestatewillfailtooccurinthedurationoftheexperimentandtheoscillatormustberesettothehigh-amplitudestateusingthestepsdescribedabove.Figure 5-2 bshowsahistogramoftheresidencetimeinthehigh-amplitudestatebeforeatransitionoccurs.TheexponentialdependenceontheresidencetimeindicatesthatthetransitionsarerandomandfollowPoissonstatisticsasexpected. 79

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Logarithmofthetransitionratefromthehigh-amplitudestateasafunctionofinversenoiseintensityatadetuningfrequencyof0:25HzforsampleA.Theslopeofthelineartyieldstheactivationbarrier. Todeterminetheactivationbarrierforaparticulardetuningfrequency,werecordalargenumberoftransitionsformultiplenoiseintensitiesIN.Theaverageresidencetimeateachnoiseintensityisextractedfromtheexponentialttothecorrespondinghistograms.Figure 5-3 plotsthelogarithmoftheaveragetransitionrateasafunctionofinversenoiseintensity.Thetransitionratevariesexponentiallywithinversenoiseintensity,demonstratingthatescapefromthehigh-amplitudestateisactivatedinnature.AccordingtoEq. 5{2 ,theslopeinFig. 5-3 yieldstheactivationbarrierforescapingfromthehigh-amplitudestateattheparticulardetuningfrequency. Werepeattheaboveproceduretodeterminetheactivationbarrierforotherdetuningfrequencies(isthedierencebetweenthedrivingfrequencyandthebifurcationfrequencyatwhichthehigh-amplitudestatedisappears)andshowtheresultsinFig. 5-4 forsampleAandsampleB.AllthedetuningfrequencieschosenforsampleAaresmaller 80

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B DependenceoftheactivationbarrierondetuningfrequencyforsampleAandsampleB.Thesolidlinesarepowerlawts,yieldingcriticalexponentsof1:380:15and1:40:15respectively. thanitsresonancepeakwidthwhilethemaximumdetuningfrequencyforsampleBisabout4timesitsresonancepeakwidth.Fittingtheactivationenergieswithapowerlawdependenceonthedetuningfrequencyyieldscriticalexponentsof1:380:15and1:40:15forsamplesAandBrespectively.Despitethedierentresonantfrequenciesandafactorof4dierenceindamping,thecriticalexponentsobtainedforbothsamplesareingoodagreementwiththeoreticalpredictions[ 12 33 81 82 ].Suchscalingbehaviornearaspinodalpointisexpectedtobeuniversalinallsystemsfarfromthermalequilibrium.Apartfromperiodicallydrivenmicromechanicaloscillators[ 83 ],acriticalexponentof3=2wasrecentlyobservedinrf-drivenJosephsonjunctions[ 84 ].Othernon-equilibriumsystemssuchasnanomagnetsdrivenbypolarizedcurrent[ 85 ]anddoublebarrierresonanttunnelingstructures[ 86 ]arealsoexpectedtoobeythesamescalingrelationship. RecentlyAldridgeandCleland[ 19 ]measurednoise-inducedswitchingbetweendynamicalstatesinananomechanicalbeam.Theyfoundaquadraticdependenceoftheactivationbarrieronthedistancetothecriticalpointwherethetwostablestatesofforcedvibrationsandtheunstableperiodicstateallmergetogether.Suchquadraticdependence 81

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ResponseofsampleAat!d=2versusthefrequencyofparametricdriving!d.Thesolidanddashedlinesrepresentthestableattractorsandtheunstableoscillationstatesrespectively. arises[ 82 ]whentheparametersarechangedalongthelinewherethepopulationsofthetwostablestatesareequaltoeachother.Thisrequireschangingsimultaneouslyboththeamplitudeandthefrequencyofthedrivingeld.Incontrast,inourexperiment,weapproachabifurcationpointwhereastablelarge-amplitudestateandtheunstablestatemergetogether,whilethestablesmall-amplitudestateisfaraway.Wevaryonlyoneparameter,thedetuningfrequency,whilemaintainingtheperiodicdrivingamplitudeconstant.Wefoundthattheactivationbarrierforescapeisreducedtozerowithcriticalexponentof3=2. DeviceandExperimentalSetup 2 .Twoelectrodesare 82

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3{8 :+2_+!200+ke WhentheamplitudeofthespringmodulationexceedsathresholdvaluekT=4!00I,perioddoublingoccursintheoscillatorresponse.Oscillationsareinducedathalfthemodulationfrequencyinarangecloseto!00.AsshowninFig. 5-5 ,therearethreerangesoffrequencieswithdierentnumberofstableattractors,separatedbyasupercriticalbifurcationpoint!b1=2!00+!pandasubcriticalbifurcationpoint!b2=2!00!p,where!p=p 5-5 ).Atfrequenciesbelow!b2(41150rads1),thezero-amplitudestatebecomesstable,resultinginthecoexistenceofthreestableattractors.ThesestablestatesareseparatedinphasespacebytwounstablestatesindicatedbythedashedlineinFig. 5-5 Thepresenceofnoiseallowstheoscillatortooccasionallyovercometheactivationbarrierandswitchbetweenthedierentattractors.Sincetheparametricallydriven 83

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33 ](seeEq. 5{2 ).Ingeneral,theactivationbarrierEaisdeterminedbythedeviceparameterssuchasthedampingconstant,nonlinearitycoecientsandthedrivingfrequency.Nearthebifurcationpoints,thesystemdynamicsischaracterizedbyanoverdampedsoftmodeandEadecreasestozeroaccordingtoj!d2!bj,wherethecriticalexponentisuniversalanddependsonlyonthetypeofbifurcation.Inaparametricoscillator,thesupercriticalandsubcriticalbifurcationsinvolvethemergingoftwostableoscillationstatesandanunstablezero-amplitudestate(at!b1)andthemergingoftwounstablestatesandazero-amplitudestablestate(at!b2)respectively.Whenthreestatesmergetogetherinsuchpitchforkbifurcations,thecriticalexponentispredictedtobe2[ 36 ].Awayfromthebifurcationpoints,thescalingrelationshipnolongerholdsanddierentexponentswereobtaineddependingonthenonlinearityanddampingofthesystem. Inordertoinvestigatethetransitionsbetweenstablestatesinourparametricoscillator,weinjectnoisewithabandwidthof600rads1centeredat!00.Figures2aand2cshowrespectivelytheoscillationamplitudeandphaseatadrivingfrequencyintherangeoftwocoexistingattractors.Transitionscanbeidentiedwhenthephaseslipsby.Thetwooscillationstateshavethesameamplitude.ThesetwoattractorscanalsobeclearlyidentiedintheoccupationhistogramsinFigs. 5-7 aand 5-7 b.Figures 5-6 band 5-6 dshowswitchingeventsatadrivingfrequencywiththreeattractors,wherethezero-amplitudestatehasalsobecomestable.IncontrasttoFig. 5-6 a,theoscillatorswitchesbetweentwodistinctamplitudes.Athighamplitude,thephasetakesoneitheroneoftwovaluesthatdierby.Whentheoscillatorisinthezero-amplitudestate,therearelargeuctuationsofthephaseasafunctionoftime.ThecoexistenceofthreeattractorsinphasespaceisalsoillustratedinFigs. 5-7 cand 5-7 dfortwootherdrivingfrequencies. 84

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Oscillationamplitude(a)andphase(c)for!d=41163.0rads1.For!b2
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Theoccupationinphasespaceatfourdierent!d's.XandYdenotethetwoquadraturesofoscillation.Thegreyscalerepresentsthenumberoftimesthattheoscillatorismeasuredtoliewithinacertainlocationinphasespace.(a)!d=41171.6rads1.Apairofoscillationstatesemergesnear!b1.(b)!d=41153.0rads1.As!ddecreases,thetwostatesmovefurtherapartinphasespace.(c)!d=41139.8rads1.When!d
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(a)Histogramoftheresidencetimeinoneoftheoscillationstates(!d=41163.0rads1)beforeswitchingoccurs.Thesolidlineisanexponentialt.(b)Logarithmofthetransitionrateasafunctionofinversenoiseintensity. respectively.AtthehighfrequencyendofFig. 5-9 a,onlythezero-amplitudestateisstable.As!disdecreased,twostableoscillationstates(separatedbyanunstablestate)emergeat!b1.Withincreasingfrequencydetuning!1=!b1!d,thepairofoscillationstatesmovefurtherapartinphasespace(Figs.3aand3b)andEa1increases.At!b2,thezero-amplitudestatebecomesstable.Theappearanceofthestablezero-amplitudestateisaccompaniedbythecreationoftwounstablestatesseparatingitinphasespacefromthetwostableoscillationstates.Ea2initiallyincreaseswithfrequencydetuning!2=!b2!dinafashionsimilartoEa1.Closeto!b2,Ea1exceedsEa2andtheoccupationoftheoscillationstatesishigherthanthezero-amplitudestate(Fig. 5-7 c).As!ddecreases,Ea2continuestoincreasemonotonicallywhileEa1remainsapproximatelyconstant.Asaresult,Ea1andEa2crosseachotherat41140rads1,beyondwhichtheoccupationofthezero-amplitudestatebecomeshigherthantheoscillationstates.Thedependenceoftheoccupationonfrequencydetuningwasalsoobservedinparametricallydrivenatomsinmagneto-opticaltraps[ 15 ]. 87

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(a)DependenceoftheactivationbarriersEa1()andEa2()onthefrequencyofparametricmodulationforsampleC.(b)Ea2vslog!2forsampleC.Thelinesarepowerlawtstodierentrangesof!2.(c)and(d)logEa1vslog!1forsamplesCandDrespectively. Ingeneral,theactivationbarriersEa1andEa2dependonvariousparametersofthedevice.Nonetheless,atfrequenciesclosetothebifurcationpoints,theoreticalanalysisindicatesthattheactivationbarriersexhibituniversalscaling,withEa1;a2/j!b1;2!dj.Forpitchforkbifurcationsinaparametricoscillatorthatinvolvemergingofthreestates,ispredictedtobe2[ 36 ].Figures5band5cshowthedependenceofEa1;a2onfrequencydetuning!1;2onlogarithmicscales.Atsmalldetuning,bothactivationbarriersshowpowerlawdependenceondetuning.Thecriticalexponents 88

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16 ]andatomsinmagneto-opticaltraps[ 15 87 ].Awayfromthevicinityofthebifurcationpoint,however,thevariationoftheactivationbarrierwithfrequencydetuningisdevice-specic.Figures5band5cshowcrossoversfromthequadraticdependencetodierentpowerlawdependencewithexponents1.430.02and1.530.02forEa1andEa2respectively.ThesevaluesaredistinctfromtheexponentsobtainedinparametricallydrivenelectronsinPenningtraps[ 16 ]becausethenonlinearityanddampingaredierentforthetwosystems. Recenttheoreticalpredictionsindicatethatthesymmetryintheoccupationofthetwooscillationstatesinaparametricallydrivenoscillatorwillbeliftedwhenanadditionalsmalldriveclosetofrequency!d=2isapplied[ 79 ].Anumberofphenomena,includingstrongdependenceofthestatepopulationsontheamplitudeofthesmalldriveanductuation-enhancedfrequencymixing,areexpectedtooccur.Furtherexperimentsarewarrantedtotestsuchpredictionsandrevealotheructuationphenomenainparametricallydrivenoscillators. Parametricpumpingiswidelyusedtoimprovethesensitivityofmicromechanicaldetectorsbymechanicallyamplifyingasignal[ 74 ]orbyreducingtheresonancelinewidthinviscousenvironments[ 88 ].Thesharpjumpinoscillationamplitudeatthebifurcationpointsisutilizedforaccuratedeterminationofdeviceparameters[ 19 84 89 ].Asthedissipationincreasesinaresonantly-drivenDungoscillator,thenaturalresonancelinewidthbecomesverybroadandthehysteresisregionshrinksforcomparableoscillationamplitude.Parametricallydrivenoscillators,ontheotherhand,maintainthesharpjump 89

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90 ].Therefore,parametricoscillatorsareparticularusefulforsensinginliquidorgaseousenvironments.Apartfromtherelevancetootherparametricallydrivennonequilibriumsystems[ 15 16 87 ],thecomprehensivestudyofthedependenceofthetransitionrateonfrequencyreportedheremayproveusefulinsensingapplications[ 91 ]withparametricallydrivenmicromechanicaldevices. Inconclusion,wedemonstratedtheactivatedbehaviorofnoiseinducedswitchinginanonequilibriumsystem,anonlinear,underdamped,micromechanicaltorsionaloscillatorsmodulatedbyastrongresonanteld.Themeasuredcriticalexponentfortheactivationbarriernearthebifurcationpointagreeswellwiththepredictedvalueof3=2,consistentwiththesystem-independentscalingoftheactivationbarrierinthevicinityofthebifurcationpoint.Suchscalingrelationshipalsoappliestoothersystemsthatarefarfromequilibriumnearthespinodalpoint,includingrf-drivenJosephsonjunctions[ 20 31 ],nanomagnetsdrivenbypolarizedcurrent[ 85 ]anddoublebarrierresonanttunnelingstructures[ 86 ].Incontrast,weseethatwhenthesystemisparametricallydrivenand 90

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91

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TheresultsinthischaptercanbefoundinarticlesSupernarrowspectralpeaksneara\kineticphasetransition"inadriven,nonlinearmicromechanicaloscillator,C.StambaughandH.B.Chan,PhysicalReviewLetters,97,110602(2006)\Copyright2006bytheAmericanPhysicalSociety"andFluctuation-enhancedfrequencymixinginanonlinearmicromechanicaloscillator,H.B.ChanandC.Stambaugh,PhysicalReviewB,73,224301(2006)\Copyright2006bytheAmericanPhysicalSociety". 5 itwasshownthatwhenthenoiseisweak,theescaperateioutofstatei(i=1or2)dependsexponentiallyontheratioofanactivationbarrierRiandthenoiseintensityD: i/eRi=D:(6{1) Forclarity,wereturnheretothedimensionlessformsofthenoiseandactivationbarrierintroducedinCh. 1 .AlsoRiwasshowntodependononethecontrolparametersofthesystem:thedrivingfrequency.Theactivationbarrierisalsoknowntodependontheamplitude,aswellastheshapeofthepowerspectrumofthenoise.Theratioofthepopulationsofthetwodynamicalstatesisgivenby: Asaresultoftheexponentialdependenceofthepopulationratioonthedierenceintheactivationbarriers,thesystemwillbefoundineitherstate1orstate2withoverwhelminglylargeprobabilityovermostoftheparameterspace[ 33 92 ].ThiswasseeninCh. 5 whentheactivationbarrierwasanalyzednearthebifurcationfrequency.Theoccupationsofthetwostatesarecomparableonlyoveraverynarrowrangeofparameters. 92

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33 ]. Similartothermodynamicsystems,uctuationsincreasesignicantlywhenthesesystemsfarfromequilibriumundergo\kineticphasetransition's".Arangeofgeneric,system-independentphenomena,includingtheappearanceofasupernarrowpeakinboththesusceptibilityandthespectraldensityofuctuations[ 33 37 ],isexpectedtotakeplace.However,thesephenomenahavesofaronlybeenobservedinanalogcircuitsimulations[ 81 ].Additionallynearthe\kineticphasetransition"noiseisexpectedtoplayaconstructiveroleinenhancingthemixingofoscillationsatdierentfrequencies,animportantuseofanonlinearsystem.Well-knownexamplesarehigherharmonicgenerationandfour-wavemixinginnonlinearoptics.Frequencymixingisalsoofgreatimportanceinhighsensitivitydetectionandsignalprocessing. InSec. 6.2 thesystemisintroducedanddataispresentedthatshowsthataregionexistswheretheoccupationofthetwostatesisapproximatelythesame.InSec. 6.3 wemeasurethespectraldensitiesofuctuationsofanunderdamped,nonlinearmicromechanicaltorsionaloscillatornearthe\kineticphasetransition".Themostprominentfeatureintheuctuationspectrumisanarrowpeakcenteredatthefrequencyoftheperiodicexcitation.Wedemonstratethatthisnarrowpeakisassociatedwithnoise-inducedtransitionsbetweenthetwoattractors.Thewidthofthepeakvarieslinearlywiththetransitionrateandismorethanafactorof10smallerthanthenaturallinewidthoftheresonanceinourexperiment.Awayfromthe\kineticphasetransition",theintensityofthepeakdecreasesexponentially.Apartfromthenarrowpeak,wealsoobservesmaller,muchbroaderpeaksinthespectrumthatareassociatedwithuctuations 93

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(a)Timetraceofthenormalizedoscillationamplitudeneara\kineticphasetransition"demonstratingtransitionsbetweenthetwostates.(b)Histogramoftheoscillationamplitudeoftheoscillatorneara\kineticphasetransition"showingapproximatelyequaloccupationofthetwostates. withineachattractor.Thesebroadpeaksarepresentforalldrivingfrequencieswithinthehysteresisloopandtheirdependenceonthenoiseintensityisdistinctlydierentfromthenarrowpeaksatthe\kineticphasetransition". InSec. 6.4 wedemonstratethatwhenasecond,weakperiodictrialexcitationwithfrequency!s!disappliedinadditiontothestrongperiodicdrive,uctuation-inducedinterstatetransitionsbecomesynchronouswiththebeatingbetweenthetwodrivingfrequenciesatcertainnoiseintensities,resultinginenhancedresponseatboththefrequencyoftheweakexcitationandthedown-convertedfrequency.Responsesatbothfrequenciesinitiallyincreasewithnoiseandsubsequentlydecreaseasthenoiseintensitybecomesverystrong.Theoccurrenceofsuch\highfrequencystochasticresonance"isinagreementwiththeoreticalpredictions[ 33 81 ]. 94

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3 avoltageVisappliedtooneoftheelectrodesgeneratinganelectrostatictorquethatexcitesthetorsionaloscillations.Theotherelectrodeisusedtocapacitivelydetecttheoscillations.Allmeasurementsweretakenatatemperatureof77Kandpressure<107Torr.ThedrivingvoltageVisasumofabiasingd.c.voltage,aperiodica.c.voltagewithfrequencyfd=!d 3{6 withtheadditionofthenoisetermN(t). Intheabsenceofnoiseandatsmalldrivingtorque,theresponseoftheoscillatorcorrespondstothatofadampedharmonicoscillatorwitharesonantfrequencyof3286HzandQ8000.Whenthedrivingtorqueisincreasedbeyondacriticalvalue,thefrequencyresponsebecomeshystereticduetothecubicnonlinearity.Withinarangeofdrivingfrequencies,twostabledynamicalstatescoexist.Intheabsenceofuctuations,notransitionsoccurbetweenthetwostates.Whensucientnoiseisinjectedintothedrivingvoltage,theoscillatorswitchesbetweenthetwoattractors.Overmostofthehysteresisloop,theactivationbarriersR1;2forescapefromthetwostatesaresignicantlydierent.ThehollowandsolidtrianglesinFig. 6-2 bshowstheoccupationofthehigh-amplitudeandlow-amplitudestatesrespectively.Onthelowfrequencysideofthehysteresisloop,theoccupationofthelowamplitudestateisconsiderablyhigherthanthehighamplitudestateandtheprobabilityofndingtheoscillatorinthelowamplitudestateisclosetounity.Asthedrivingfrequencydecreases,theactivationbarrierR1forswitchingoutofthehighamplitudestatesdecreases.Atthebifurcationfrequency(f1=3283:3Hz),R1goestozeroandthelow-amplitudestatebecomestheonlyattractor.InChapter 5 ,weshowedthatR1dependsonthedetuningfrequency(fdf1)inthevicinityofthebifurcationpointwithacriticalexponentof3/2inagreementwiththeoreticalpredictions[ 33 ].Onthehighfrequencysideofthehysteresisloop,asimilarargumentappliesexceptthatthehigh-amplitudestateisthestableattractor. 95

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(a)Normalizedfrequencyresponseoftheoscillator()ttedtoadampedoscillatorwithcubicnonlinearity(solidline).(b)Occupationofthetwostatesversusfrequency.Atthe\kineticphasetransition"(fp=3284:8Hz),theoccupationofthehighamplitudestate(M)andthelowamplitudestate(H)arecomparable.(c)Dependenceoftheintensityofthesupernarrowspectralpeakonthedrivingfrequencyfd.Theintensityfallsoexponentiallyasthedrivingfrequencyismovedawayfromfp. Whiletheoscillatorispredominantlyinoneoftheattractorsovermostofthehysteresisregion,thereexistsasmallrangeoffrequencieswheretheoccupationofthetwoattractorsisofthesameorderofmagnitude.Figure 6-1 ashowstheoscillationamplitudeasafunctionoftimeatadrivingfrequencyof3284:8Hzandclearlyillustratesthesystemswitchingbetweenthetwostates.Therelativeoccupationofthetwostatesatthisdrivingfrequencyisdeducedbycalculatingtheareaunderthetwopeaksinthehistogramoftheoscillationamplitude(Fig. 6-1 b). 96

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33 ].Animportantfeatureassociatedwithphasetransitionsisthelargeuctuationsarisingfromtransitionsbetweenthetwostates.Weperformasystematicstudyoftheuctuationsofthenonlinearoscillatoratthe\kineticphasetransition"byexaminingthepowerspectraldensitiesoftheoscillatorresponse. whereX(t)andY(t)aretheslowlyvaryingamplitudesofoscillationsinandoutofphasewiththedrivingtorqueatfrequencyfd.Thespectraldensityofuctuationsisgivenby[ 93 ] whereNisanormalizationconstant. Intheabsenceofinjectednoise,oscillationsoccuronlyattheperiodicdrivingfrequencyfdandthemeasuredspectrumconsistsofadeltafunctioncenteredatfd.Whennoiseisaddedtotheexcitation,thespectraldensitiesofuctuationsbecomedramaticallydierent.Figure 6-3 showsthespectraldensityofuctuationsat3dierentperiodic 97

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Powerspectraldensityofuctuationsforthreedierentdrivefrequenciesfd:(a)3284Hz,(b)3284:7Hz,(c)3285:1Hz.Noticethattheverticalscaleof(b)ismorethan20timeslargerthan(a)and(c). drivingfrequencies.Tofocusonuctuationsabouttheensembleaverageresponse,thedeltafunctionpeakatfdobtainedwithnoinjectednoiseisremovedfromthespectrum.Inotherwords,thedatapointatfdisomittedforeachpanelinFig. 6-3 .Figure 6-3 bshowsthespectraldensityofuctuationsatthedrivingfrequencyfdfpwheretheoccupationsofthetwostatesareequal.Themostprominentfeatureisaverysharppeakcenteredatthedrivingfrequency.Thewidthofthispeakisafactorof10smallerthanthenaturalwidthoftheresonancepeak.Thissharppeakispredictedtoarise[ 33 ]duetouctuation-inducedtransitionbetweenthetwodynamicalstates.Figures 6-3 aand 6-3 cshowsthespectraldensityattwootherdrivingfrequenciesthatarecomparativelyfarawayfromfp.Astheperiodicdrivingfrequencyischangedsothattheoscillatormoves 98

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Powerspectraldensityofuctuationsforthreedierentdrivefrequenciesfd:(a)3284Hz,(b)3284:7Hz,(c)3285:1Hz.Noticethattheverticalscaleof(b)ismorethan20timeslargerthan(a)and(c).TheaxesinthesegureshavebeenrescaledtorevealthesmallerandbroaderpeaksintheuctuationspectrumsofFig. 6-3 (a),(b)and(c)respectively. awayfromthephasetransitionpoint,thesharppeakshiftsaccordinglytoremaincenteredatthedrivingfrequency.Theareaunderthepeak,however,dropssignicantly.Figure 6-2 cplotstheareaunderthenarrowpeakasafunctionofperiodicdrivingfrequency,clearlydemonstratingthattheintensityofthesupernarrowpeakattainsmaximumatthe\kineticphasetransition"anddecreasesexponentiallyastheoccupationofoneofthestatesexceedstheotherandtransitionsbetweenthestatesbecomelessfrequent. Figure 6-5 ashowsthebehaviorofthesupernarrowpeakwithdierentnoiseintensitieswhentheperiodicdrivingfrequencyisheldconstanttomaintaintheoscillatoratthe\kineticphasetransition".Thenoiseintensitydiersbyafactorof4between 99

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Thesupernarrowpeakfortwointensitiesofinjectednoiseatthe\kineticphasetransition".Theinjectednoiseintensityforthehollowcirclesis4timesgreaterthanthesolidcircles.ThedottedandsolidlinesareLorentziantstothedata.Leftinset:Dependenceoftheareaofsupernarrowpeakoninjectednoiseintensity.Rightinset:Thepeakwidthvs.transitionrate. solidandhollowcircles.BothsetsofdataarettedwellbyLorentzians[ 37 81 ].Asthenoiseincreases,thepeakwidthincreasesandthepeakheightdecreases.Wefoundthattheareaunderthepeaksremainsaboutconstant,changingbylessthan10%whenthenoiseintensitychangesbymorethanafactoroffour(leftinsetofFig. 6-5 ).TherightinsetofFig. 6-5 plotsthepeakwidthasafunctionofthesumofthetransitionratesioutofeachstate,wherethetransitionratesaredetermineddirectlyfromtheresidencetimebetweentransitions.Thelineardependenceofthepeakwidthonthetransitionrateandtheexponentialdecreaseintheareaofthepeakawayfromthe\kineticphase 100

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Thebroad,smallpeakattwoinjectednoiseintensitiesthatdierbyafactorof2,atfd=3284Hz.Inset:Lineardependenceofthepeakareaoninjectednoiseintensity. transition"clearlyidentifythesupernarrowpeakwithnoise-inducedtransitionsbetweentheattractors. Inadditiontothesupernarrowpeak,thereareotherpeaksintheuctuationspectrumthatareweakerandmuchbroaderthanthesupernarrowpeakatthe\kineticphasetransition".Thesepeaks,unlikethesupernarrowpeak,arepresentforallexcitationfrequencieswithinthehysteresisloop,asshowninFigs. 6-3 aand 6-3 cwhentheoscillatorisfarfromthe\kineticphasetransition"andinFig. 6-3 bwhentheoscillatorisatthe\kineticphasetransition".Thesesmallerpeaksrepresentcharacteristicfrequenciesofuctuationsabouteachdynamicalstatewhentheuctuationsarenotstrongenoughtoinduceatransitionovertheactivationbarrier.Incontrasttothesupernarrowpeak,the 101

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(a)OscillationamplitudeAvs.drivingfrequencyatVdof460V()and51V(#).Thesolidlinerepresentsattothedataatthesmallerexcitationusingtheresponseofadampedharmonicoscillator.Forthelargeexcitation,bistabilityoccursbetween20539.25rads1and20559.97rads1.Thedashedlinetsthedatatoadampedoscillatorwithcubicnonlinearity[ 28 ].(b)Atdrivingfrequencyof20555.75rads1,theoscillationamplitudeswitchesbetweentwodistinctvalueswhennoiseisinjectedintothesystem. shapeofthesesmallpeaksdonotchangeconsiderablyasthenoiseintensityincreases(Fig. 6-6 )andtheirareavariesproportionallywithnoiseintensity(insetofFig. 6-6 ). 6-7 a).Thepresenceofnoiseenablesthesystemtooccasionallyovercometheactivationbarrierandswitchbetweenthesetwostates.Figure 6-7 bshowsthattheoscillationamplitudejumpsbetweentwovaluesasafunctionoftimewhennoiseisinjectedintotheexcitationvoltage.Theoccupationsofthetwostatesdependonthedrivingfrequency.AsshowninFig. 6-8 a,theoscillatorresides 102

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(a)Occupationsofthehigh-amplitudestate(M)andthelow-amplitudestate(H)asafunctionofdrivingfrequencyatVd=460V.(b)OccupationsasafunctionofVdatxeddrivingfrequencyof20555.44rads1. predominantlyinthehigh-orlow-amplitudestateatthehighandlowfrequencysidesofthehysteresislooprespectively.Theoccupationsofthetwostatesarecomparableonlyatasmallrangeoffrequencies.AswasshowninSec. 6.3 ,noise-inducedinterstatetransitionsatthe\kineticphasetransition"leadtoanarrowpeakintheuctuationspectrumthatiscenteredatthedrivingfrequency(Fig. 6-9 a).Thewidthofthispeakisinverselyproportionaltotheresidencetimeofthestatesanditsheightdropsexponentiallyastheexcitationfrequencyistunedawayfromthevalueatwhichtheoccupationsofthetwostatesarecomparable.Iftheexcitationamplitudeisvariedwiththedrivingfrequencykeptconstant,theoccupationsofthetwostatesexhibitasimilarbehavior:theoccupationofthelow-amplitudestateishighatsmalldrivingamplitudeandviceversaforlargedrivingamplitude.Theoccupationsofthetwostatesbecomecomparableatsomeintermediateamplitude(pointAinFig. 6-8 b). Weinvestigatetheresponseoftheoscillatorwhenaweakperiodicdrive(120V)atfrequency!scloseto!disappliedontopofthestrongperiodicdrive.Sincethenonlinearityoftheoscillatormixestheprimaryandsecondarydrivingfrequencies,theweakdriveinducesoscillationsnotonlyatitsownfrequency!s,butalsoatotherfrequencies.Theresponsetotheweakdriveisthestrongestatthefrequencies!sand 103

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(a)Atthedrivingfrequency,wheretheoccupationsofthetwostatesareequal,asharppeakdevelopsinthepowerspectraldensityofuctuations.(b)Theadditionofasecondary,weakexcitationat!s1=!d+0:05rads1leadstospectralpeaksat!s1and2!d!s1.(c)Thefrequencyofthesecondaryexcitationischangedto!s2=!d+0:20rads1. 2!d!s.Wefoundthatatthe\kineticphasetransition"wheretheoccupationsofthetwooscillationstatesareequal,thepresenceofanoptimalamountofnoiseenhancesthemixing,resultinginampliedresponsesatboth!sand2!d!s,inagreementwithpredictionsfromtheoreticalanalysis[ 81 ].Figures 6-9 band 6-9 cshowthespectralresponseoftheoscillatorwhenweakperiodicdrivesattwodierentfrequencies(!s1=!d+0:05rads1and!s2=!d+0:20rads1respectively)wereappliedinthepresenceofnoise.Spectralpeaksatboththeweakdrivingfrequencies!s1;2andthedown-convertedfrequencies2!d!s1;2canbeclearlyidentied.Thepeaksareaboutafactorof6strongercomparedtothecasewhennoiseisnotappliedtotheexcitation.Inbothcases,thefrequencydierence!s1;2!dismuchsmallerthantherelaxationrateoftheoscillator(!R=2Q1:3s1).AsshowninFig. 6-9 c,bothspectralpeaksdiminishwhenthe 104

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Next,weexaminethedependenceofthespectralpeaksonthenoiseintensity.Figure 6-10 (a)showsthatthespectralpeaksat!srstincreasewithnoiseintensity,attainingamaximumatanoisepowerof0:012mV2=Hzandsubsequentlydecrease.Figure 6-10 bdemonstratesasimilarbehaviorforthespectralpeakatthedown-convertedfrequency2!d!swhereanoptimalamountofnoiseenhancesthefrequencymixing.Weemphasizethatthereisnoperiodicexcitationat2!d!sandthespectralresponseinFig. 6-10 barisesfromthenonlinearmixingoftheprimaryandsecondaryfrequencies.Figures 6-10 cand 6-10 dplotthesignal-to-noiseratio,denedasthespectralresponsesat!sand2!d!sdividedbythespectralresponsesintheabsenceoftheweakperiodicexcitation(Fig. 6-9 a)atthecorrespondingfrequencies.Atboth!sand2!d!s,thesignal-to-noiseratiosachieverelativemaximaatsomeintermediatenoiseintensity. Theenhancementofthespectralresponsesat!sand2!d!swaspredictedtooccurwhennoise-inducedinterstatetransitionsbecomesynchronouswiththebeatingbetweentheprimaryandsecondarydrivingfrequencies[ 81 ].Foranintuitiveunderstanding,weconsiderthebeatingbetweentheprimaryandsecondarydrivesthatleadstomodulationoftheamplitudeofthestrongeldatfrequencyj!s!dj.Sincetherelaxationrateoftheoscillator(!R=2Q1:3s1)ismuchlargerthanj!s!dj,theoscillationamplitudeattheprimaryfrequencyvariesadiabaticallywiththedrivingamplitudeandisthereforemodulatedatthebeatingfrequency(Fig. 6-11 a).Intheabsenceofnoise,theoscillatorremainsinthelow-amplitudestateprovidedthatthemodulationenvelopeissmall.Whensucientnoiseisintroducedintothesystem,theoscillatorswitchesbetweenthetwooscillationstates(Fig. 6-11 b).Slowamplitudemodulationsoftheprimaryexcitationchangetheprobabilityofuctuationaltransitionsandhencetheoccupationsofthetwostates. 105

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(a)Theheightofthespectralpeakat!sasafunctionofthenoiseintensityfor!s1=!d+0:05rads1(B)and!s2=!d+0:20rads1(J).(b)Thespectralpeakatthedown-convertedfrequencyof2!d!s1;2.(c)Thesignal-to-noiseratiosat!s1;2asafunctionofthenoiseintensity.(d)Thesignal-to-noiseratiosat2!d!s1;2. AsillustratedinFig. 6-7 (e),theoccupationprobabilityofthelow-amplitudestate,forexample,decreasesastheexcitationamplitudeincreases(pointB)andviceversa(pointC).Theoccupationprobabilitieshencevaryperiodicallyasafunctionoftime.Attheoptimalnoiseintensity,interstatetransitionsoccurapproximatelyonceeveryhalf-cycleofthebeating(Fig. 6-11 b).Therefore,theamplitudemodulationoftheresponsebecomessignicantlylargercomparedtoFig. 6-11 a,resultinginspectralcomponentsatboththesecondaryfrequency!sandthedown-convertedfrequency2!d!sthatareenhancedcomparedtothecasewhennoiseisabsent.Uponfurtherincreaseofthenoiseintensity, 106

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(a)Oscillationamplitudeasafunctionoftimeintheabsenceofnoise.(b)Underoptimalnoise,theoscillationamplitudeswitchesbetweentwostablevaluesasafunctionoftimeandinsyncwiththebeatingfrequency(squarewaverepresentedbythesolidline)(c)Withlargeintensityofnoise,interstateswitchinglosessynchronizationwiththebeatingfrequency. thetransitionratebetweenthetwostatesexceedsthebeatingfrequencyandtheswitchingeventsarenolongersynchronouswiththemodulationindrivingamplitude(Fig. 6-11 c).Asaresult,thespectralresponsesat!sand2!d!sdecrease. Thenoiseenhancementofspectralresponsesat!sand2!d!soccursonlyinthevicinityofthe\kineticphasetransition"wheretheoccupationsofthetwostatesarecomparable.Figure 6-12 showsthespectralresponsesat!sandat2!d!sasafunctionof!d,wherethefrequencydierencebetweentheprimaryandsecondarydrives!s!dismaintainedconstant.Thespectralresponsesatbothfrequenciesattainmaximumwhentheoccupationsofthetwostatesarenearlyequal.Asthedrivingfrequencyismovedawayfromthe\kineticphasetransition",theoccupationofoneofthestatesbecomemuchlargerthantheotherandinterstatetransitionsbecomesignicantlylessfrequent.Asaresult,theenhancementofspectralresponsesbecomesweaker. Such\highfrequencystochasticresonance"waspredictedtooccurwhenaweak,periodicdriveisappliedtosystemswherebistabilityarisesasaresultofaprimary,strongperiodicmodulation[ 81 ].Forordinarysystemswithbistablepotential,stochasticresonance[ 1 4 ]takesplacewhentherateofnoise-inducedtransitionsbecomescomparable 107

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Dependenceofthespectralpowerat!s1(B)and2!d!s1(J)onthedrivingfrequency!d.Thefrequencydierencebetweentheprimaryandsecondarydrives!s1!dismaintainedconstant.Thearrowindicatesthefrequencyatwhichtheoccupationsofthetwostatesareequal.ThisfrequencyisslightlyshiftedfromFig. 6-7 becausethedevicehasbeenwarmeduptoroomtemperatureandre-cooleddown. tothefrequencyoftheperiodicdrive.Ouroscillator,incontrast,ismonostableunderweakperiodicdrivinganddevelopsbistabilityonlywhenitisdrivenstrongly.Boththeprimaryandsecondaryperiodicdrivesareappliedatfrequenciesmuchhigherthattherateofnoiseinducedtransitions.Anotherimportantdierencefromconventionalstochasticresonanceisthatnoise-enhancedresponsealsotakesplaceat2!d!satwhichnoperiodicexcitationisapplied.Suchecientdown-conversionofexcitationfrequencyoccursduetosynchronizationoftheswitchesintheoscillationamplitudewiththebeatingfrequency. 108

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81 ]shouldbeobservableinouroscillatoraftermodifyingthedetectioncircuittoallowmeasurementoflowfrequencysignals. 36 ].Thebroadeningofthesepeakswithincreasingnoiseintensityleadstothewellknownphenomenonofstochasticresonance[ 1 ].Eventhoughthesupernarrowspectralpeakobservedinourexperimentalsooriginatesfromtransitionsbetweencoexistingstates,thereareimportantdierencesbetweenournonlinearoscillatorandsystemsinthermalequilibrium.First,noise-inducedswitchinginouroscillatoroccursbetweentwooscillationstateswithdierentamplitude.Ouroscillatorisbistableonlyunderstrongperiodicdrive.Itisfarfromequilibrium[ 1 11 94 ]andisnotcharacterizedbyfreeenergy.Second,atthekineticphasetransition,thesharpspectralpeakiscenteredatthedrivingfrequency.Thelackoftimereversalsymmetryresultsinacharacteristicasymmetryinthebroadpeaksoftheuctuationspectrumaboutthedrivingfrequency(seeFig. 6-11 b).Incontrast,forbistablesystemsinthermalequilibrium,theuctuationspectrumiscenteredatzerofrequencyandnosuchasymmetricfeaturesarepresent.Criticalkineticphenomenasuchastheemergenceofthenarrowpeakatthedrivingfrequency[ 33 81 95 ]andnoiseenhancedfrequencymixingreportedhereoccurasaresultoftheinterplaybetweennoiseandnonlinearity.Otherbistablesystemsfarfromequilibrium,includingrf-driven 109

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20 ],nanomagnetsdrivenbypolarizedcurrent[ 85 ]anddoublebarrierresonanttunnelingstructures[ 86 ],areexpectedtoexhibitsimilarbehaviorintheparameterrangewherethetwostateshavecomparableoccupations.Whiletheoreticalanalysis[ 33 ]consideredaDungoscillatorwithcubicnonlinearity,ourmeasurementsindicatethatthesupernarrowpeakisrobustevenwhenhigherordernonlinearitiesarepresent.Thestudyofsuchcriticalkineticphenomenacouldopennewopportunitiesintunablenarrowbandlteringanddetectionusingmicromechanicalandnanomechanicaloscillators,orfrequencymixingapplications. 110

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2.4 Johnsonnoisewasintroducedandtherelationshipbetweentheresistance,adissipativeelement,andvoltageuctuationswaspresented.Thisresultwasadirectconsequenceoftheuctuation-dissipationtheorem;atheoremwhichingeneraldescribesawiderangeofsuchrelationships.Morebroadlyuctuationrelationsseektoidentifyanddescribethegeneralfeaturesofallsystems[ 2 96 ]thatexperienceuctuations.Whiletheuctuation-dissipationrelationsexistforsystemsinthermalequilibrium,theyfailtoaccountfortheeectofuctuationsonsystemsdrivenfarfromequilibrium. Atopicofinterestwithinthebroadercontextofnonlinearsystemsistheroleuctuationsplayinvibratingsystemswhichexhibitbistability.AswasshowninCh. 3 ,whenthedampingofasystemissmall,weakforcescanleadtolargevibrationalamplitudes.Inthisregionwherethenonlinearitycanbeespeciallystrong,itispossiblefortwoormorestablestatestocoexist.Switchingbetweenthestatesmayoccuriftheuctuationsinthesystemarelargeenough[ 33 ].Thisleadstodramaticeectsontheamplitudeofthevibrationseventhoughonaveragetheuctuationsmaybesmall.Thisswitchingbehaviorhasbeenpreviouslystudiedinseveralsystemsincludingmagneto-opticaltraps[ 15 ],electronsinapenningtrap[ 16 ],micro-andnano-mechanicaloscillators[ 17 { 19 ]andJosephsonjunctions[ 20 ].Recenttheoreticalwork[ 97 ]hasidentiedspecicfeaturesintheworkvarianceofthesesystemsthatshouldoccurnearcriticalfeaturesrelatingtothebistability. Inthischapterweinvestigateworkuctuationsinanonlinearoscillatordrivenoutofequilibrium.Webeginbyoperatingthesysteminalinearregimetodemonstratethatheretheuctuation-dissipationrelationholds.Weshowthatthevarianceisproportionaltotheaverageworkandtheproportionalitycoecientbetweenthemisindependent 111

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33 ],wheretherelativeoccupationofthetwostatesisthesame.Moreonthe\kineticphasetransition"canbefoundinCh. 6 .Ourresultsaretherstexperimentalevidenceofthisuniversalfeature. 18 95 98 ].Thesystemisoperatedinvacuumatapressure<106Torrandatatemperatureof77K. ByapplyingadrivingvoltageV=VDC+VACt+Vnoisetooneoftheelectrodesatimevaryingelectrostatictorqueissetupwhichexcitesmechanicaloscillationsinthemovabletopplate.Thedeviceresponseis +2_+(2f20)+3=T0cos2fdt+N(t)(7{1) wheref0istheresonantfrequency,isthenonlinearcoecientandT0isthedrivingtorque.Thedampinginthesystemis 99 ].ThenoisetorqueisN(t)andwaspreviouslydenedinSec. 2.4 ;abouttheregionofinterestthenoisecanbeconsideredwhite.ThenoisetorqueisrelatedtothenoiseintensitywhereIN/N(t)2.AsinCh. 5 ,INwillhasarbitraryunits.ForsmalldrivestheresponseislinearanditssquaredamplitudecanbettedbyaLorentzianlineshape.Whenasucientlystrongdriveisappliedthecubictermbeginstoeecttheresponse;itbecomesasymmetricasthepeakshiftstoalowerfrequency[ 28 ].Aboveacritical 112

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whereistheresponseofthesystemandweassumeattimet=0thesystemisinthesteadystate.Thevarianceoftheworkdistributioncanbewrittenas2=h(WhWi)2i.ForbothstrongandweakdrivesthedistributionoftheworkisGaussian: 2exp(WhWi)2=22:(7{3) Furthermore,itcanbeshownthatW/2/.Thevariancecanalsoberelatedtotheaverageworkandnoiseintensity: TheparameterCisconstantwhenthesystemisinthelinearresponseregime[ 100 ].Whenthesystemdisplaysbistability,Cisfoundtonolongerbeconstant[ 97 ]andtheuctuation-dissipationrelationcannolongerbeapplied. Inourexperimentalsetuptheresponseismeasuredusingaheterodynemethodinordertominimizenoiseandcross-talkintheresponsesignal.Thisisachievedbyapplying180outofphase,highfrequency(100kHz)sinewavestothetwoelectrodes.Onecarriersignalisaddedtothedrivingsignalononeoftheelectrodes,whiletheotherisaddedtotheotherelectrodeinadditiontoad.c.voltage.Theoveralleectofthehighfrequencysignalsonthesystemisashiftintheresonantfrequency.Theresponseismeasuredatthetopplateusingalock-inamplier,lockedtothecarriersignalfrequency.Thesignalisdemodulatedandsenttoasecondlock-inamplierwhichmeasuresatthedriving 113

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(a)SEMofMEMSdevice.(b)Theratio2=hWiisplottedversustheintegrationtimeoftheworkforfd=3362:35.Forlongerthan1=theratioisconstant.(c)Theworkdistribution(#)atfd=3362:23forintegrationtimesof=0:08s(),0:28s(#),0:80s(M),1:70s(),10:0s(O).Thedistributionistted(solidline)toEq. 7{3 .Theaverageworkanditsvariancecanbeextractedfromthet.(d)Theratio2=hWi(#)atfd=3362:23and=10sisplottedversusthenoiseintensityandistted(solidline)toEq. 7{4 .TheslopegivesthecoecientC. 114

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whereXandYaretheinandoutofphasesignalsmeasuredbythelock-inandBisaconversionfactorfromvoltagetoradians.Datacollectionisdonebysettingthedesiredsystemparametersandthenrecordingtheresponsemeasuredbythelock-in.Datawascollectedatarateof500Hzwhenthesystemwasdrivenweaklyand16Hzwhenitwasdrivenstrongly.Aslowerratewasusedinthestronglydrivencasesincetherelevantfeature(interstatetransitions)occurredonalongertimescale.Hundredsofthousandsofdatapointswerecollectedforeachdesiredsystemsetting.WhenisinsertedintoEq. 7{2 ,theworkisfoundtobeproportionaltothesumofY+_X=2f.Thesecondtermprovidesanegligiblecontributiontotheresult.FollowingthemethodprescribedinRef.[ 101 ],thesetofworksisdenedasW=fW1;W2;:::;Wkgwhere ThesizeofthesetWisthendeterminedbyandthenumberofdatapointscollected.TheaverageworkanditsvariancecanbefoundbyttingthedistributiontoEq. 7{3 orequivalentlybytakingthemathematicalmeanandvariance.. 102 ].Thesystemissetatafrequencyclosetotheresonantfrequencyofthesystemandusingtheproceduredescribedinthepreviousparagraphdataiscollected.TodeterminetheproperintegrationtimethesetWiscalculatedforseveral's.AsseeninFig. 7.3 b,forintegrationtimesapproximatelythreetimeslongerthantherelaxationtime, 115

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ThecoecientCisplottedversusthedrivingfrequency.Plottedabovethisistheresonantresponse(#)oftheoscillatorttedtoaLorentzian(solidline).Cisindependentofthedrivingfrequencydespitethestrongdependenceoftheresponseonthedrivingfrequency. ThedistributionoftheworkisplottedinFig. 7.3 cforseveralvaluesof.Whenissmallnegativeworkproductioncanbeseen,butastheintegrationtimeincreasesthecurvenarrowsuntilnegativeworknolongeroccurs.ThelineshapesareGaussianandttedbyEq. 7{3 .ThesetWiscollectedatseveraldierentnoiseintensitieswhilekeepingthedrivingfrequencyxed.Thedependenceof2=hWionthenoiseintensityisshowninFig. 4-2 d.ThelineartisanagreementwithEq. 7{4 andtheslopeprovidesthecoecientC.Thisprocessisrepeatedforseveralotherdrivingfrequenciesaroundtheresonantfrequency.Thecoecient,C,isplottedinFig. 7.3 asafunctionofdrivingfrequency.Abovethisgureisaplotoftheresonantresponseoftheoscillator.Thecoecient,C,isshowntobeindependentofthedrivingfrequency,whiletheoscillationresponseshowsastrongdependenceonthedrivingfrequency.Theresultisinagreement 116

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(a)Lineardependenceof2=hWionthenoiseintensityforintrastateuctuations.ThisresultagreeswithEq. 7{4 .(b)ForinterstateuctuationsEq. 7{4 nolongerholds,insteadaclearexponentialdependencecanbeseen.Thisdependenceleadstothelargevarianceseenforinterstateuctuations. withtheuctuation-dissipationrelationforalinearsysteminequilibrium[ 97 100 ].Wecomparethisresulttothecasewhenthesystemisdrivennonlinearlyandoutofequilibrium. Underasucientlystrongdrivetheoscillatordisplaysahystericresponse,asshowninFig. 7.4 ,wheretwostablevibrationalstatescoexist.Inthisregiontwotypesofbehaviorcanbeexamined:intrastateandinterstateuctuations.Intrastateuctuationsconsistofuctuationsaboutanattractorthatdonotleadtoatransition.Interstateuctuationsarecomprisedoftransitionsbetweenthetwostablesanductuationsabouttheattractors.Therateatwhichthesystemmovesbetweenthetwostatesisgivenbyi;j/expEa=IN[ 5 33 ],whereEaistheactivationbarrierwhichmustbeovercomefortransitionsi!jtotakeplace.Fromthis,thestationarypopulationscanbewrittenas wheretr=21+12.Asshowninpreviouswork[ 18 ],theprobabilityofthesystemoccupyingeitherofthetwostatesisstronglydependentonthedrivingfrequencyof 117

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95 ]thereexistsafrequencybetweenthetwobifurcationswhereR1R2andthesystemoccupiesbothstateswithequallyprobability.This\kineticphasetransition"regionissimilartothephasetransitionbetweenwaterandvaporwherebothstatescancoexist.Therelativeoccupationofthetwostatesherehasanexponentialdependenceonthedrivingfrequency,asshowninFig. 7.4 b. Thetotalworkvariancethatresultsfrominterstateswitchinganductuationsaboutthestableattractorsiscalculatedinamannersimilartotheprocedureusedinthelinercase.Thesystemisplacedatafrequencywithinthebistableregionandnoiseisapplied.Thenoiseintensityischosensothatatthe\kineticphasetransition"thesystemswitches,onaverage,every10s.Theresponseisrecordedandthevarianceandaverageworkareextracted.Inthelinearcase,Fig. 7.3 d,andforintrastateuctuations,Fig. 7.4 a,theprocesscanberepeatedforseveralnoiselevels.Equation 7{4 canbeusedtothancalculateCateachdrivingfrequency.However,forinterstateuctuationsthisrelationnolongerholds.Instead2=hWihasanexponentialdependenceonIN,asseeninFig. 7.4 b,so2=hWimustbeexaminedatasinglenoiseleveltodetermineitfrequencydependence.Theintegrationtimeusedis=240s,whichfarexceedsthataveragetimethesystemspendsineachstatenearthekineticphasetransition.ThisresultiscomparedtotheexpectedbehaviorpredictedbyDykman[ 97 ]. Inthiscasetheaveragetotalworkistheweightedaverageoftheworkresultingfromintrastateuctuationsand2jistheassociatedvarianceforeachstate.Sincetheintegrationtimeusedforinterstateuctuationsexceedsthetimespentinagivenstate,asmallerisusedforintrastateuctuations.UsingthelineardependenceofhWioncorrespondingvaluescanbeextrapolatedforcomparisontointerstateuctuations.Datathatcoincideswiththesystemtransitioningfromonestatetotheotherisdisregarded. 118

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(a)Hystereticfrequencyresponseofthedevice.Betweenthebifurcationfrequenciestwopossibleoscillationamplitudesexist.(b)Therelativeoccupationofthetwovibrationstates.Thetwostatesarethelargeamplitude(H)andthesmallamplitude(M)ofoscillation.Atfd=3360:35Hztherelativeoccupationisapproximatelythesame,thispointisthe\kineticphasetransition"ofthesystem.(c)Theratio2=hWiINisplotted(#)asafunctionofthedrivingfrequency.Theratio2=hWiINwhichresultsfromEq. 7{8 isplottedtoo().Thetworesultsareingoodagreementwhoeachother. 119

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97 ], where Equation 7{8 iscomposedofthreeparts:thesquareddierenceofthemeanworksfromintrastateuctuations,thefactorMwhichis1/2atthe\kineticphasetransition"andtr.Thefactortrisresponsiblefortheexponentialdependenceofthevariance,forinterstateswitching,onnoiseintensity.InFig. 7.4 c,Eq. 7{8 dividedbythesumoftheweightedworkisplottedandcomparedtotheexperimentalworkvariance.ThisresultshowsthattheratioisstronglydependentonfrequencyunlikethelinearcaseshowninFig 7.3 .Theagreementisgoodandtheexponentialdependenceisevident. 103 ].Anadditionaldependenceoftheworkvarianceisexpectednearthebifurcationfrequency[ 97 ].Thediscoveryofuniversalfeaturesisimportantasdevicesizescontinuetodecreasesincemoresystemswilldisplaysuchbehavioranductuationswillhavealargerroleintheirresponse. 120

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Inthisdoctoraldissertationtheeectofuctuationsonasmallsystem,drivenoutofequilibrium,wasinvestigated.Severalgeneric,systemindependentphenomenonwereexamined.Themajorityofourworkmarkedtherstexperimentalobservationsoftheseuniversalfeatures,thustheresultsshouldbeofinteresttoawiderangeofthescienticcommunity. InChapter 1 weintroducedsystemspossessingtwoormorestablesstates.WefoundthatKramersequation,whichdescribestherateatwhichasystemtransitionsoutofastablestateforasystemnearthermalequilibrium,hasanexponentialdependenceonthesizeofanenergybarrier.Thebarriertypicallycanbedescribedbythefreeenergyofthesystem.However,forperiodicallymodulatedsystems,drivenfarfromequilibrium,thebarrierheightisnolongerbegivenbythefreeenergy.Insteadavariationalproblem,whichseekstondtheoptimalpathforaswitch,mustbesolved.Thisnon-trivialactivationbarrierwasthemainfocusofthedissertation. InChapter 2 MEMSwereintroduced.ThedeviceusedinthisdissertationwasatorsionaloscillatorfabricatedusingthePolyMUMPSprocessoeredbyMEMSCAPR.Thetorsionaloscillatorwasshowntobeabletomakesensitivecapacitivemeasurements.ByutilizingtheelectricalsetupsdescribedinChapter 2 thedevicecanbeexcitedintomechanicaloscillationsandtheresponsecanbedetected.Thetypeofnoiseusedtoinducetransitionswasalsopresented. Byapplyingana.c.voltagetothesystemitwasshowninChapter 3 thattheoscillatorresonates.Whenthedrivestrengthisaboveacriticalvalue,theresponsebecomesnonlinear,andthesystemdisplayshysteresis.Inthecasewhenthedrivingfrequencyisneartheresonantfrequency,theresponseisthatofaDungOscillator.Herethetwostatesarerepresentedbytwooscillationamplitudes:onesmallandonelarge.Whennoiseisinjectedintothesystemtheoscillatorcanbeinducedtoswitchfromone 121

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SwitchingintheparametricallydrivenoscillatorwasexaminedcloserinChapter 4 .Theoptimalpathtakenwhenthesystemswitchesfromonestablestateintotheotherwasintroduced.Byanalyzingthousandsoftheswitchingevents,wewereabletoshowthat,withnoadjustableparameters,oursystemagreeswiththepredictedMPSP.Inaddition,thesystemwasshowntolackdetailedbalancesincethetime-reverseddownhillpathdoesnotcoincidewiththeuctuationinduceduphillpath. InChapter 5 welookedathowtheactivationbarriernearabifurcationscalesasafunctionofthecontrolparametersofthesystem.Dependingonthetypeofbifurcation,adierentcriticalexponentwasexpected.FortheresonantlydrivenDungoscillatoraspinodalbifurcationoccurswhereonestableandoneunstablestatemergetogether.Whileinthecaseofaparametricallydrivensystem,apitchforkbifurcationoccurs,wheretwostablestatesmergewiththeunstablestate.Theoreticalanalysispredictsthatthescalingoftheactivationbarrierinthesetwocasesshouldbeindependentofthetypeofsystembutshoulddependonthetypeofbifurcation.Forthesaddle-nodebifurcationwefoundexponentsof1:380:15and1:40:15,whileforthepitchforkbifurcationcasetheexponentswere2:00:1and2:000:03.Inbothcasesourresultsagreedwiththetheoreticalpredictionsof3=2and2respectively. InChapter 6 theactivationbarrierwasexaminedinaregionwheretheprobabilityofswitchingbetweenthetwostatesinaDungoscillatorwasapproximatelythesame. 122

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Finally,inChapter 7 ,wetookacloserlookatuctuationrelations.Werstanalyzedtheworkvariancewhenthesystemwasweaklydriven,thencomparedourresultstopreviouswork.Wefoundagreementwiththeuctuation-dissipationrelation.Thesystemwasthendrivenstronglyandtheregionnearthe\kineticphasetransition"wasanalyzed.Resultsfortheworkvariancewereshowntohaveanexponentialfrequencydependencewhichpeaksatthe\kineticphasetransition",aswaspredictedinrecenttheoreticalwork. Thisworkrepresentstherststepincharacterizingtheeectnoisehasonsystemspossessingmultiplestablestates.Futureworkcouldinvolveexaminingseveralfeatures.Thisincludelookingatfrequencymixinginthecaseoftheparametricdriveatthepointwherethreestablestatescoexistwiththesameprobability.VerifyingthecriticalexponentsdependenceforawiderrangeroffrequencyintheDungoscillator;herethecriticalexponentof=3=2isexpectedtochangeforfrequenciesawayfromthebifurcation.Also,lookingatthedependenceofthecriticalexponentsonothercontrolparameterslikethedrivingamplitude.Detailedquantitativecalculationsoftheactivationbarrierandotherresultsneedtobedone.Thisisanimportantstepindeterminingwaystoexploitthestudiedbehaviorforapplications. Inconclusion,thisworkdemonstratesthatthedriventorsionaloscillatorisindeedarobustsystem.ThevarietyofusesbothinthisdissertationandintheotherworkbeingdoneintheChanlabclearlywarrantfurtherinvestigationofitsproperties.Thisdissertationprovidesastrongfoundationonwhichtocontinueresearchinboththeuseofthetorsionaloscillatorandinthestudyofuctuationinducedescape. 123

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4 theMPSPwaspresented.Thepaththatrepresentstheoptimaltrajectorythesystemtakesduringanoiseinducedtransitioniscomposedoftwoparts:theuctuationinduceduphillpathandthedeterministicdownhillpath.InFig. 4-2 boththetheoreticalMPSPandtheexperimentalMPSPwereplotted.Inthissectiontheproceduresusedtodeterminethesepathsarepresented. x, of the original system w0=139352.118; Gam=6.991; wF=2 z=F/2/(wF2)/Gam; mubif= mu=((wF2)/2w0)/Gam/z; kappa=1.03910^6; gamma=8w0kappa/3; C1=1/ Eq 3 ( ; q1 and q2 correspond to Q and P g := 1/4( q2 ^2 + q1 ^2)^2 + 1/2(1 ) ^2 + mu ) ^2 ,q2 ]:=1/zq1+q2(q1^2+q2^2+1mu); K2[q1 ,q2 ]:=1/zq2q1(q1^2+q2^21mu); derivatives : Kij = Ki qj ,q2 ]:=1/z+2q1q2; K12[q1 ,q2 ]:=3q2^2+q1^2+1mu; K21[q1 ,q2 ]:=q2^23q1^2+1+mu; K22[q1 ,q2 ]:=1/z2q1q2; and saddles ( the formulas Solutions provide both stable and unstable

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( = NSolve [f [ Q P ] == 0, K2 [ Q P ] == 0g, P ( and saddles ( q2a=zq1a(mubif1); second attractor is at ( ) ( saddle is at (0, 0) ( eigenvalue at qa1 2 and qs q1a,q2a],K12[q1a,q2a]g,fK21[q1a,q2a],K22[q1a,q2a]gg]; real part of two eigenvalues or Re r2= B at the attractor : this matrix gives the equation for the matrix A ^f1g, therefore it is a 4 x4 matrix It allows one to find the matrix A ^f1g relates p to q at the attractor ; everything is real complex conjugation K11[q1a,q2a]+K22[q1a,q2a],0,K12[q1a,q2a]g,fK21[q1a,q2a], 0,K11[q1a,q2a]+K22[q1a,q2a],K12[q1a,q2a]g,f0,K21[q1a,q2a], K21[q1a,q2a],2K22[q1a,q2a]gg; b= Ainv=ffb[[1]],b[[2]]g,fb[[3]],b[[4]]gg; A= path equations ( from the saddle ( of the matrix K at the saddle are zeta ^f1gn [1 ^2] ( determines what side of saddle to start descent dq1=R; dq2=R is the time needed to go from the saddle ; Set tm to be negative to see seperatix tm=31; EQ 6,7 for no noise force ==dq1,q2[t0]n[ Qdet[t ]:=Flatten1[ Pdet[t ]:=Flatten1[ in two ways Qd= [" fromsaddle3 dat ", Qd ] ( provides velocity along path t +6? is used at offest play with it

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]:=Flatten1[ Pdet1[t ]:=Flatten1[ vtd= path equations ( from the attractor is a logical variable Initially g =0 and you try so many directions of extreme paths Then you choose two directions which are closest to the saddle set g =1, and put arr [ n ] and arr [ n +1] with the appropriate n the direction of the optimal path from phiI to phiF into K pieces Then find the extreme path for each phi phiI=?0; phiF=2 ]; phiI=?arr[1]; phiF=arr[2]; ]; is the time needed to get to the saddle ; it must be kept such that the trajectory goes past the saddle otherwise the trajectory may be not the right one Care must be taken about stability of the solution tm=24; = phiI / K + phiF / K ); arr[k]=phi; direction of initial ascent dq2=R k = ,k]; Results to file kdq1dq2 dat ; q10=q1a+dq1; q20=q2a+dq2; p10=A[[1]][[1]]dq1+A[[1]][[2]]dq2; p20=A[[2]][[1]]dq1+A[[2]][[2]]dq2; s0=p10^2+p20^2; opt= q2[t]]p2[t]K21[q1[t],q2[t]],q2'[t]?2p2[t]+K2[q1[t], q2[t]],p2'[t]==p1[t]K12[q1[t],q2[t]]p2[t]K22[q1[t], q2[t]],s'[t]?p1[t]^2+p2[t]^2,q1[t0]?q10,q2[t0]?q20,p1[t0] ?p10,p2[t0]?p20,s[t0]?s0g,fq1,p1,q2,p2,sg,ft,t0,tmg,

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]:=Flatten1[ Popt[t ]:=Flatten1[ tosaddle= s = ,Flatten1[ Phi = ,phi]; k++; ]; Qd1= [" fromsaddle dat ", Qd ] 4-2 .Usingthiscompletearraythefollowinganalysiscanbedone: 127

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Thevelocityeldisdenedasthedierencebetween(Xi+1;Yi+1)and(Xi1;Yi1).Eachvelocityisassociatewithapoint(Xi;Yi).Agridiscreatedwhichcontainsseveralhundredpixels.Ineachpixelthecomponentsofthevelocities,whose(Xi;Yi)fallwithinapixelaresummed.Afterthisisrepeatedforeachpixel,theresultisavelocityeldwhichcanbeassociatedwiththedistribution. 5 involvesndingtheactivationbarrieratseveralfrequenciesnearabifurcation.Afterextractingthetransitiontimeforeachevent,themethodologyusedfortheDungOscillatorandparametricallydrivenoscillatorarethesame.Asaresultofthemethodforcollectingdatahowever,theextractionofindividualswitchesdiersbetweenthetwosystems.Belowwerstdescribehowacollectionofswitchingtimesataparticulardrivingfrequencyandnoiselevelarecollectedforeachsystem.Thendetailsaregivenforgoingfromthiscollectionoftimestothenalresultofacriticalexponent.Thisprocessisthesameregardlessofthesystemused. 5 aprocesswasdescribedwherethesystemwaspreparedinthelargevibrationalstateataparticularfrequency.Noisewasaddedtothethesystemanddata(oscillationamplitudeasfunctionoftime)wascollected.Whentheamplitudedropped 128

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A.2.1 .Thesecondcaseinvolvesswitchingbetweenthetwoperiod-2states.Sincetherateofswitchingbetweenthesetwostatesisthesameresettingthesystemwasunnecessary.Insteadlargeleswerecreatedcontainingseveralswitches.Afterwardthedatawasprocessedtoextracttwodatasetscontainingtheswitchingtimesbetweenthetwostates. Fitting with Poisson variables for low counts % By minimizing chi % Finds A and B for Y = Aexp ( / B ) also finds % error on B inital guess % chi ( counts ( x (1). ( ./ x (2))) (1). ( ./ x (2))) ; myfun=inline(funstr, x time counts ); options and minimization MaxFunEvals ,50000, MaxIter ,30000);

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find error on B by locate deviation from B that leads to change % in chi = 1 x0=[1]; funstr= ( counts ( x (1). ( ./ tau )) (1). ( ./ tau )) ; myfun=inline(funstr, x time counts tau ); dt=j/20; [aresult,nchi]=fminsearch(@(x)myfun(x,time,counts,tau+dt)... ,x0,options); diffchi=nchichi; [aresult,nchi]=fminsearch(@(x)myfun(x,time,counts,tau+dt)... ,x0,options); diffchi=nchichi; else end break else end Given noise ( sq of noise force ), time and error % of time fit 1/ time =1/ time o ( R / D ) returns % R and error on R y= er=timeErr./time; error for log of 1/ time % The chisqln gives information on goodness of fit

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Given frequency detuning activation energy and its % error fit R = A ( dfr )^ zeta returns zeta error x= y= er= [B2,stdB2,A2,stdA2]=chisqln(x,y,er); Take into account error in bifurcation frequency er= [B2,stdB2,A2,stdA2]=chisqln(x,y,er); B2 is critical exponent and stdB2 is the erro perform chi sq linear fit provides goodness of fit [B,stdB,A,stdA]=LinearWeightedFit(x,y,er); chi w=1./er.^2; chi= rchi=chi/(n2); stdB=stdB stdA=stdA Prob=1chi2cdf(chi,n2) Check for goodness = Fit line using weights for y % Assumes x has neglible error and different uncertainties in y s

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weight denominator y = A + B A=( Uncertainty in A and B stdB= Get fft of spectral density of response of % device result shift so f =0 is driving frequency A=t(:,3)+jt(:,4); N= Get fft ( A )j^2/ N define sampling rate define frequency range remove zero frequency ( d c ) component s=[s(1:(N1)/2);s((N1)/2+2: 7 asetofworksiscalculated.FromthissetthemeanandvarianceoftheworkcanbefoundbyttingthatsettoaGaussianfunction.Belowiscodetoextractthesetofworksandcodetotforthemeanandvariance. Integrate work for data file over time % Repeat for multiple files % rate depends on experiment force % tau set to 1 ( arb units ) period=2 scale is used since W is for single cycle % this figure number of cycles per cycle of % acquistion rate sumW=[];

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Load file assume max of res in Y signal only % need Y component Y=t(:,2); points=timerate; cycless= sX=500; Xn=0; Iterate through entire file step by step stc=it; enc=it+sX1; [x,y]= b=x+y; This is work per cycle time scale sumW(( if b=b+(cyclessenc); Yn=Y(b); W= sumW(( tot,sigma tot]=kptwork(t1,t2,time,sumW1,sumW2) Follow Dykman PRE 2008 calculate work and variance for % interstate switching use to compare against experiment % this program returns value for one frequency W2= sigma 1=var(sumW1); sigma 2=var(sumW2); v12=1./t1; rate transfer into state 1 rate transfer into state 1 tr=v12+v21; %%%%% eq 9 tr); expect w1 small for low freq eq 5 tr); expect w1 small for high freq %%%% eq .10 tr=M./(v trtime).(W1W2).^2; sigma tot=w1.sigma 1+w2.sigma 2+sigma tr; eq .11

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tot=w1.W1+w2.W2; A ((( x )/ C )^2) gauss1 ); A=fit3.a1; B=fit3.b1; C=fit3.c1;

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TherstthreeparametersarefoundbyttingtheresonantcurvetoEq. 3{4 .ThefollowingcodeisusedinMatlabtoperformthistask: f0=freq( Frequency at maximum of response Starting guess lower bound upper bound % Define Function see Landau 1976 pg 78 x (1)./(2 (3) (( x (2).^2+( freq (3)).^2))) x freq ); Options need adjusting TolX ,1e24, TolFun ,1e21,... MaxFunEvals ,1e5, MaxIter ,1e5); Perform fit The fitting results are in array x = Damping Resonant Frequency 135

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82 ]. See Dykman Physica 104 A p 490 (1980) % wbf1 is bifurcation frequency for switching from % upper to lower amplitude opposite for wbf2 % kappa1 is nonlinear coefficient calculated from wbf1 % ACratio is applied ac voltage of bistable respone divided % by applied ac voltage for resonance % omega lambda force come from fit to resonance flag flag bfr=wbf1; flag bfr=wbf2; eta=Lambda./OmB; flag bet=2/27(1+9eta.^2+(13eta.^2).^(3/2)); kappa1=bet4.Omega.^2.OmB.^3/Force^2; flag bet=2/27(1+9eta.^2(13eta.^2).^(3/2)); kappa2=bet4.Omega.^2.OmB.^3/Force^2; end 28 ]. Plot using parameters from lorensq and getkappa % Plot is done for freq as function of response Load data from file response= freq=t(:,2); Set range to of x b=(0.00:0.01:stop/1e5)1e5; Convert kappa to beta assume alpha is negligable m=1; set mass to 1 % Following is from Landua 1976 p 88 epsilon1=kappab.^2+ epsilon2=kappab.^2 f1=omega+epsilon1;

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bb=[b,b]; Plot data plus result from fitted values ko ,f1,b, k ,f2,b, k ); XLim([ Set viewing limits Frequency ) Amplitude ) lambda,force,ACratio,linpara,para); ACratio is applied ac voltage of parametric respone divided % by applied ac voltage for resonance % omega lambda force come from fit to resonance % linpara and para are data files m=1; set mass to 1 % gamma is driving frequency range ep is detuning from resonant frequency Load data from portion of paramtric response between % super sub bifurcation frequencies x=t(:,2)omega0; y= [B,stdB2,A,stdA2]=LinearFit(2x,y, From Landua 1976 p 91 DeT=Akappa; aa=DeT^2+lambda^2; ke= bp=ep/2/kappa+DeT/kappa; bm=ep/2/kappaDeT/kappa; b= Locate super sub bifurcation frequencies % based on above fit UpperC=ep( wU=omega0+UpperC(1); wL=omega0+LowerC(1) Load full parametric response and plot t(:,2)=(t(:,2)omega0); b .: ); r .: ); k );

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' k ); k ); 104 ]usedintheabovecode: denominator fit y = A + B A=( Uncertainty in y uncertainty in fit parameters A and B stdA=stdy

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CoreyStambaughwasbornandraisedinAkron,Ohio.Inthefallof1999hebeganattendingTheOhioStateUniversity.DuringhissecondyearhejoinedtheexperimentalhighenergyresearchgroupofProf.WinerandProf.Hughes.Enjoyingtheworkthere,hedecidedtogotograduateschool.Inthespringof2003hegraduatedwithabachelorsdegreeinphysicsandsubsequentlybeganhisgraduateworkinphysicsattheUniversityofFlorida.Uponmeetinghisfutureadvisor,hedecidedtoshifthisinterestawayfromhighenergyphysics.InMayof2004hejoinedProf.Chan'sgroup.AtthetimehewasProf.Chan'srstandonlygraduatestudent.Afterseveralyearsoftoilingawayinthebasementofthephysicsbuilding,hecompletedhisgraduateworkandinMayof2009hegraduatedfromtheUniversityofFloridawithadoctoraldegreeinphysics. 144