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Controlling and Probing Atoms and Molecules with Ultrafast Laser Pulses

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Title:
Controlling and Probing Atoms and Molecules with Ultrafast Laser Pulses
Creator:
MANESCU, CORNELIU ( Author, Primary )
Copyright Date:
2008

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Subjects / Keywords:
Atoms ( jstor )
Chirp ( jstor )
Electric fields ( jstor )
Electronics ( jstor )
Electrons ( jstor )
Forced migration ( jstor )
Ionization ( jstor )
Lasers ( jstor )
Molecules ( jstor )
Momentum ( jstor )

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University of Florida
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University of Florida
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Copyright Corneliu Manescu. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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4/30/2007
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436097560 ( OCLC )

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CONTROLLINGANDPROBINGATOMSANDMOLECULESWITHULTRAFASTLASERPULSESByCORNELIUMANESCUADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2004

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Copyright2004byCorneliuManescu

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TomyparentsAlexandruandElisabetaManescu

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ACKNOWLEDGMENTSIamfortunatetoworkundertheinspiringguidanceofagreatpersonalityintheeldof`quantumcontrol,'myresearchadvisorandcommitteechair,ProfessorJeffKrause.Iampleasedtoexpressheremygratitudeforhisteachingsandsupport,andforprovidingmewiththeopportunitytoworkinthisexcitingeld.Dr.Krause'sopen-mindedattitudetowardsresearchhasprovidedafreeenvironment,inwhichIcouldpursuescienticsubjectsthatreallyexciteme.ForthatIamdeeplyappreciative.Atthesametime,Iwouldliketothankthemembersofmysupervisorycommittee,ProfessorsDavidReitze,YngveOhrn,PradeepKumarandMichaelFrankfortheirtimeandtheinsightstheyprovided.IalsowanttoextendmythankstoallmembersoftheQuantumTheoryProjectgroupattheUniversityofFlorida,whohavebeenlikeafamilyformeduringthelastveyears.Mythanksalsogototwogreatprofessorswhohaveprovidedmewithbenecialacademicguidanceandencouragementsinthepast,ProfessorVioricaFlorescufromtheQuantumMechanicsCathedraattheUniversityofBucharest,andProfessorTaskoGrozdanovfromthePhysicsInstituteofBelgrade.Likewise,IwanttothankhereprofessorswithwhomIhadthegreatopportunitytocollaborateduringmystudy,ProfessorKennethSchaferfromLouisianaStateUniversity,andProfessorsNielsHenriksenandKlausMllerfromtheTechnicalUniversityofDenmark.Finally,Iwanttoexpressmygreatestthanksanddeepappreciationtomyfamilyfortheircontinuoussupportandencouragementthroughoutmyacademiccareer.Iwouldcertainlynothavemadeitsofarwithoutthem. iv

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TABLEOFCONTENTS page ACKNOWLEDGMENTS ................................ iv LISTOFFIGURES ................................... viii ABSTRACT ....................................... xiii 1INTRODUCTION ................................ 1 1.1ShortHistoricalPerspectiveofAtomism ................. 1 1.2UltrafastLaserPulses ........................... 2 1.3TerahertzHalf-CyclePulses ........................ 3 1.4QuantumControl ............................. 4 1.5ThesisObjectivesandOrganization .................... 5 2EXPERIMENTALANDTHEORETICALASPECTSOFQUANTUMCON-TROL ...................................... 7 2.1Overview ................................. 7 2.2QuantumControlofMolecules ...................... 8 2.2.1TheBrumer-ShapiroMethod ................... 9 2.2.2TheTannor-RiceMethod ..................... 12 2.2.3QuantumControlbyOptimalControlTheoryOCT ...... 14 2.2.4Closed-LoopLearningControl .................. 17 2.3QuantumControlforQuantumStatePreparation ............. 19 2.3.1AdiabaticRapidPassage–ARP .................. 20 2.3.2Stark-ChirpedRapidAdiabaticPassage–SCRAP ......... 24 2.3.3StimulatedRamanAdiabaticPassage–STIRAP ......... 26 2.3.4AdiabaticPassagebyLightInducedPotentials–APLIP ..... 34 2.4QuantumControlforQuantumComputing ................ 38 2.5SomeOtherApplicationsofQuantumControl .............. 41 2.6ClosingWords ............................... 48 3CONTROLOFWAVEPACKETFOCUSINGBYCHIRPEDULTRASHORTLASERPULSES ................................ 49 3.1Introduction–Wavepackets ........................ 49 3.2FocusingControl ............................. 49 3.3WavepacketFocusing ........................... 52 3.3.1GeneralConditionsforWavepacketFocusing .......... 52 v

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3.3.2DynamicsofGaussianWavepacketsinaLinearPotential .... 53 3.4GenericControlofExcitedWavepackets ................. 55 3.5FocusingofWavepacketsCreatedbyChirpedUltrashortPulses ..... 57 3.5.1ExcitedWavepacketDynamicsinaLinearPotential–Theory .. 57 3.5.2NumericalResultsfortheLinearPotential ............ 59 3.5.3ExcitedWavepacketDynamicsinanExponentialPotential ... 63 3.5.4WavepacketFocusinginanExponentialPotential ........ 67 3.5.5NumericalResultsfortheExponentialPotential ......... 68 3.6AnalysisoftheUltrashortPulseLimit .................. 74 3.7DiscussionoftheResultsfortheBreakdownDependenceonChirp.ExplanationfortheMechanismofProducingSelf-FocusingWavepack-etsbyNegative-ChirpPulses. ..................... 80 3.8Conclusions ................................ 85 4HALF-CYCLEPULSES–OVERVIEW ..................... 87 4.1Introduction ................................ 87 4.2TheoreticalandExperimentalBackground ................ 88 4.3StudyGoalsandOverview ........................ 91 5IONIZATIONOFSODIUMRYDBERGSTATESBYHALF-CYCLEPULSES–THEORYANDRESULTS ........................... 93 5.1TheoreticalMethods ............................ 93 5.1.1Field-FreeSodiumStates ..................... 93 5.1.2NaStarkStates .......................... 95 5.1.3NumericalModelofaHCP .................... 96 5.1.4TheImpulseApproximationandtheImpulsiveMomentumRe-trievalIMRMethod ...................... 98 5.1.5LongerHCPPulses ........................ 99 5.1.6IonizationSpectrum ........................ 101 5.1.7SemiclassicalCalculations .................... 102 5.2NumericalResultsandDiscussion .................... 104 5.2.1SemiclassicalInterpretationoftheOscillationsintheDownhillIonizationSpectrum ....................... 104 5.2.2TheInuenceoftheHCPWidthupontheIonization ...... 108 5.2.3WhyAreThereNoOscillationsintheUphillIonizationSpec-trum? .............................. 112 5.3Summary ................................. 116 6CONCLUSIONSANDFUTUREWORK .................... 120 6.1Summary ................................. 120 6.2FutureWork ................................ 123 REFERENCES ..................................... 126 vi

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AATOMICUNITS ................................. 144 BPRODUCTIONOFAHALF-CYCLEPULSE .................. 146 CDISCRETIZATIONOFTHESCHRODINGEREQUATIONONANON-UNIFORMGRID ............................... 147 DTHECHEBYSEVTIME-PROPAGATIONMETHOD ............. 151 ESOMEFORMULADERIVATIONS ....................... 153 BIOGRAPHICALSKETCH .............................. 155 vii

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LISTOFFIGURES Figure page 2Theleftpanelshowstheevolutionoftheadiabaticstateenergiescontinu-ouslineandofthediabaticstatesdashedline.Therightpanelshowstheevolutionofthepopulationsinthetwostates. ............. 21 2Diagramshowingtheevolutionoftheadiabaticstateenergiescontinuouslineandofthediabaticstatesdashedlineforthe5s-5p-5dladderinrubidium. .................................. 22 2Diagramshowingtheevolutionoftheadiabaticstateenergiescontinuouslineforthecaseofagroundstatecoupledbyachirpedpulsetoaman-ifoldofvestates. ............................. 23 2TheupperpanelshowsthepumpandStarkpulsesasafunctionoftime.Thelowerpanelpresentstheevolutionoftheadiabaticstateenergiescontinuouslineforatwo-levelsystemdrivenbythetwopulses. .... 25 2Thisgurepresentsa3-levelsystem.States1and2arecoupledbythepumppulse,andstates2and3arecoupledbytheStokespulse.PandSrepresentthedetuningsofthetwopulseswithrespecttoresonance. . 26 2TheupperpanelshowstheevolutionofthepumpandStokesRabifre-quenciesasfunctionoftime.Themiddlepaneldisplaystheevolutionoftheadiabaticstateenergies.Thebottompanelshowstheevolutionofthepopulationsinthethreestates. ..................... 29 2TheleftsideshowsthedressedpotentialsU1,U2andU3.Thesystemsat-isesthe2-photonresonanceconditionfortheminimumofthepoten-tialsU1andU3.representsthedetuningnegativeherefromreso-nancewiththeintermediarylevel,U2.Therightsideshowstheadia-baticlightinducedpotentials,ULIP+,ULIP0andULIP)]TJ/F23 11.955 Tf 18.345 2.955 Td[(.Theinitialpopu-latedLIPisULIP0correlatestoU1. .................... 35 viii

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2ASnapshotAshowsthelightinducedpotentialswhenthepulsecou-plingU2andU3isturnedon.TherightwellofULIP0whichcorrelatestoU3islifted,andthewavepacketislocalizedintheleftwellwhichcorrelatestoU1.BInsnapshotBtherstpulseisalmostoverandthesecondpulseturnson,couplingU1andU2.ThiscausestheleftwellofULIP0torise,andtherightwelltofall.Thebarrierbetweenthetwowellshasdisappearedandthewavepacketevolvesintotherightwell.CInthissnapshotonlythesecondpulseison,andthewavepacketislocalizedintherightwellofULIP0. .................... 37 3Snapshotsofthewavepacketevolutionforthemaximumfocusingtimecase.xg=0:08a.u.,correspondstogroundvibrationalstateofICN,Pulseparameters:=125:69a.u.,=)]TJ/F15 11.955 Tf 9.298 0 Td[(1:4510)]TJ/F24 7.97 Tf 6.587 0 Td[(4a.u.,Pre-dictedfocusingvalues:tfoc=142:2a.u.,f=1:08,andxfoc=0:02a.u.Fromlefttorightthepropagationtimesare:0,80,142,220,300,380,460,540,620,and700a.u. ......................... 60 3Snapshotsofthewavepacketevolutionforthemaximumfocusingtimecase.Herexg=0:24a.u.,correspondsto=4excitedvibra-tionalstateofICN.Pulseparameters:=172:6a.u.,=)]TJ/F15 11.955 Tf 9.298 0 Td[(7:7110)]TJ/F24 7.97 Tf 6.587 0 Td[(4a.u.Predictedfocusingvalues:tfoc=2169a.u.,f=1:37,andxfoc=4:78a.u.Fromlefttorightthepropagationtimesare:0,600,1150,1700,2169,2600,3000,3400,3800,and4200a.u. ......... 62 3SameasinFig. 3 ,exceptthatthesignofthechirpisreversed.Thatis,=7:7110)]TJ/F24 7.97 Tf 6.587 0 Td[(4a.u. ............................ 62 3Snapshotsofthewavepacketevolutionforthemaximumfocusingcase.Herexg=0:24a.u.,correspondsto=4excitedvibrationalstateofICN.Pulseparameters:=172:6a.u.,=)]TJ/F15 11.955 Tf 9.299 0 Td[(1:6110)]TJ/F24 7.97 Tf 6.587 0 Td[(4a.u.Pre-dictedfocusingvalues:tfoc=867:4a.u.,f=2:56,andxfoc=0:76a.u.Fromlefttorightthepropagationtimesare:0,350,600,867,1100,1500,1900,2300,2700,and3100a.u. ................... 63 3Thisisthemaximumfocusingcase.Herexg=0.24a.u.corresponds=4excitedvibrationalstateofICN.Pulseparameters:=1000a.u.,=-3.010)]TJ/F24 7.97 Tf 6.587 0 Td[(4a.u.Predictedfocusingvalues:tfoc=1580a.u.,f=2:70,andxfoc=2:54a.u.Fromlefttorightthepropagationtimesare:0,600,1100,1579,2100,2500,29003300,3600,and3900a.u. ......... 64 3APanelApresentsawavepacketwhosesidesaremovingrelativelytowardseachother,leadingtothefocusingofthewavepacket.BInpanelBthesidesofthewavepacketaremovingrelativelyaway,andthepacketspreads. ............................. 68 ix

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3Excitedwavepacketscreatedby>0pulseatdifferentmomentsduringtheirpropagation.Eachgraphshowsthewavepacketevery7500a.ufora15000a.u.interval.Fromtoptobottomv0hasthevalues0.08,0.12,0.15. .................................... 70 3Excitedwavepacketscreatedby>0pulseatdifferentmomentsduringtheirpropagation.Eachgraphshowsthewavepacketevery7500a.ufora15000a.u.interval.Fromtoptobottomv0hasthevalues0.05,0.1,0.15. 71 3Excitedwavepacketscreatedby<0pulseatdifferentmomentsduringtheirpropagation.Eachgraphshowsthewavepacketevery7500a.ufora15000a.u.interval.Fromtoptobottomv0hasthevalues0.1,0.18 ... 72 3Excitedwavepacketscreatedby<0pulseatdifferentmomentsduringtheirpropagation.Eachgraphshowsthewavepacketevery7500a.ufora15000a.u.interval.Fromtoptobottomv0hasthevalues0.1,0.15,0.18. 73 3GraphsshowingthepromotedwavepacketEq. 3 asafunctionofthepulsewidthatzerochirp=0a.u..Allgraphshave2=.004a.u.,x20=0.2a.u.Inascendingorderofpeakamplitudesthe2valuesare:3840000,960000,60000,240000a.u. ............................. 76 3AplotofthesamepromotedwavepacketsasinFig 3 withT2inEq. 3–63 neglectedT2=0. ............................ 77 3Graphsshowingthedependenceofthebreakdownwiththeslopeoftheexcitedsurfacepotential.Allgraphsdisplaysmallkinksontheright-bottomside.Weconsiderasbreakdownatagiventhepulsewidthforwhichtheheightoftherstkinkbecomeslargerthan1%ofthepeakheightofthepromotedwavepacket.Inascendingorderofpeakampli-tudesthe2,2valuesare:0:7110)]TJ/F24 7.97 Tf 6.587 0 Td[(5,3840800.,0:20310)]TJ/F24 7.97 Tf 6.586 0 Td[(4,1920800,0:5810)]TJ/F24 7.97 Tf 6.586 0 Td[(4,960800,0:16510)]TJ/F24 7.97 Tf 6.586 0 Td[(3,480800,0:4910)]TJ/F24 7.97 Tf 6.587 0 Td[(3,240800,0:1410)]TJ/F24 7.97 Tf 6.586 0 Td[(2,120800,0:410)]TJ/F24 7.97 Tf 6.586 0 Td[(2,60800. ................ 77 3Graphshowingthedependenceofthebreakdownon .......... 78 3Graphsshowingthedependenceofthebreakdownwiththechirpofthepulse.Allgraphshave2=.004a.u.,x20=0.2a.u.Inascendingorderofpeakamplitudesther,2valuesina.u.are:.82,7640800,.2,3820800,.6,1960800,.2,960800,.9,480800,.4,240800,.5,120800,.7,60800. ................................ 79 3Graphshowingthedependenceofthebreakdownron .......... 79 3Graphsshowingtheabovepromotedwavepacketsinthecasechirpisre-ducedby33%. ............................... 80 x

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3Promotedwavepacketcreatedby=)]TJ/F15 11.955 Tf 9.298 0 Td[(6:4410)]TJ/F24 7.97 Tf 6.586 0 Td[(4a.u.pulseatdif-ferenttimesduringitspropagation.Otherparameters:2=0.42a.u.,x20=0.2a.u.,=1581a.u.Thetimesaregivenbyn750a.u.,wherefromlefttorightnis0,3,4,5,6,7,8,9,10,11,12. ............ 84 3Promotedwavepacketcreatedby=6.4410)]TJ/F24 7.97 Tf 6.587 0 Td[(4a.u.pulseatdifferenttimesduringitspropagation.Otherparameters:2=0.42a.u.,x20=0.2a.u.,=1581a.u.Thetimesaregivenbyn750a.u.,wherefromlefttorightnis0,3,4,5,6,7,8,9,10,11,12. ..................... 84 3Foratime-reversedevolution,theparticlestartsinstate2withamomen-tumdirecteddownhill,itbacksupthehilltoreachtheturningpointx=0,andthenitbacksdowndownhillwithamomentumdirecteduphill. .. 85 5Theoreticalmodelsofhalf-cyclepulses:half-sin2solidline,half-sindashedline,formsuggestedbyJonesdottedline. ............... 97 5Sodiumn=15,k=)]TJ/F15 11.955 Tf 9.299 0 Td[(12top,andk=+12bottomStarkstates. .... 105 5IonizationspectrafortheuphilltopanddownhillbottomstatesTp=200fs.Thek=+12spectrumshowsnooscillations,whereask=-12spectrumpresentsinterference-likeoscillations. ................... 106 5Curvesrepresentingtheclassicalnalenergyasafunctionoftheinitialanglevariable.Continuousatline–initialenergymanifold,continuousline–nalmanifoldforTp=250fs,largedashedline–Tp=100fs,smalldashedline–Tp=50fs. ........................ 107 5TrajectoryIheadsinitiallytowardsnucleusandisslowedbytheeld,whereastrajectoryIIstartsmovingawayfromthenucleusandisacceleratedbytheexternaleld. .............................. 109 5Ionizationcurvesshowingtheeffectofthegobblerandimplicitlyofthenucleusontheoscillationsinthenalenergyspectra. .......... 110 5IonizationcurvesforaHCPwithQ=.129,showingthedependenceofthenalspectraonthedurationofthepulse.aimpulsivelimitTp!0fs,bTp=50fs,cTp=200fs,dTp=800fs. .............. 113 5Electronicdistributionasafunctionoftimeduringtheinteractionwithasin2HCPofQ=0:129a.u.,HCP=100fs.Thepanelsontheleftdisplaytheevolutionforthek=+12uphillstate,andthepanelsontherightshowtheevolutionofthek=)]TJ/F15 11.955 Tf 9.299 0 Td[(12downhillstate. .......... 115 xi

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5Evolutionoftheexpectationvaluesofthesemi-paraboliccoordinatesdur-ingtheinteractionwithasin2HCPofQ=0:129a.u.,HCP=100fs.Thesolidlinesshowtheexpectationvalueofvfortheuphilllledcir-cles,anddownhilllledtrianglesstates.Thedashedlinesshowtheexpectationvalueofufortheuphillopencirclesanddownhillopentrianglesstates. .............................. 116 5IonizationspectraforNan=15,k=)]TJ/F15 11.955 Tf 9.299 0 Td[(12hitbyasin2HCPHCP=100fs,Fpeak=160kV/cmasafunctionofthestaticelectriceld,Fs. .. 117 xii

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyCONTROLLINGANDPROBINGATOMSANDMOLECULESWITHULTRAFASTLASERPULSESByCorneliuManescuMay2004Chair:JeffreyKrauseMajorDepartment:PhysicsRecentadvancesintheultrashortlasertechnologyhaveopenedupaseriesofnewdirectionsofresearchinatomicandmolecularphysics.Thisworkcompoundstwostudiesonthecontrolandprobingofatomicmatterbyultrafastlaserpulses.Onestudyisconcernedwiththecontrolofwavepacketfocusingbychirped,ultrashortlaserpulses,whilethesecondstudyreferstotheionizationofsodiumRydbergstatesviahalf-cyclepulsesHCPs.Theabilitytocontrolwavepacketfocusingisimportantfortheachievementofspecicexcitationpathways,formolecularsynthesis,aswellasforthecontrolofchemicalreactivity.Inthiswork,wepresentageneralmethodforwavepacketcontrolintheweakeldlimit.Forultrashortpulsesandlinearpotentials,wederiveanalyticalformulasthatpredictthefocusingtimeandwidthoftheexcitedwavepacketbasedonthewidthandchirpofthepulse.NumericalsimulationsinvolvingtheICNmoleculeconrmtheseformulas.Furthermore,westudythefocusingofwavepacketsinmodelexponentialpotentials.Analyticalformulasforthecontroloflaserexcitationofawavepacketareobtained. xiii

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Theintricateinterferencemechanismbywhichthepulsechirpcancontrolfocusingisinvestigatedandexplained.FortheionizationofNaRydbergstatesbyHCPs,fullquantum-mechanicalcalcula-tionsdemonstrateanasymmetrybetweentheionizationfromdownhillStarkred-shiftedanduphillStarkblue-shiftedstates.Thetotalionizationratesoftheuphillstatesaremuchreducedcomparedtothosefordownhillstates.Thenal-energyspectrarevealanotherdiscrepancy:thedownhillionizationspectrapresentoscillationsasafunctionofenergy,whiletheuphillspectralackanysimilarpattern.Anintuitivepictureforthefull-quantumresultsisobtainedbyusinga1Dsemiclas-sicalmodel.Theoscillationsobservedinthedownhillionizationspectraareexplainedinthismodelbytheinterferenceoftwoclassicaltrajectoriesleadingtothesamenalstate.Thedependenceoftheionizationprocessonthelengthofthepulseisinvestigated.Fordownhillstates,theresultsobtainedbyfull-quantumcalculationsareexplainedsuccessfullybasedona“twointerferingpaths”theory.Finally,weshowthatthe1Dmodelisnotapplicableinthecaseoftheuphillstatesduetointrinsic2Ddynamicsinthesestates.Multipleionizationtrajectorieswithrandomphaseswashouttheoscillatoryinterferencepattern. xiv

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CHAPTER1INTRODUCTION 1.1 ShortHistoricalPerspectiveofAtomismMorethan2300yearsago,theGreekphilosopherDemocritusbrokewiththeoldquasi-mysticaltheoriesaboutnature,andadvancedtheideathatallmatterismadeupoftinyindivisibleparticles,differinginsizeandshapeandoatingintoaninnitevacuum: Democritusassertedthatspace,ortheVoid,hadanequalrightwithreality,orBeing,tobeconsideredexistent.HeconceivedoftheVoidasavacuum,aninnitespaceinwhichmovedaninnitenumberofatomsthatmadeupBeingi.e.thephysicalworld.Theseatomsareeternalandinvisible;absolutelysmall,sosmallthattheirsizecannotbediminishedhencethenameatomon,or”indivisible”;absolutelyfullandincompressible,astheyarewithoutporesandentirelyllthespacetheyoccupy;andhomogeneous,differingonlyinshape,arrangement,position,andmagnitude. `Democritus'inEncyclopediaBritannicaAlthoughnearlyforgotteninantiquityandtheMiddleAges,atthebeginningofthe19thcentury,theatomisttheorynallyacquireditsrightfulrole,asprincipaltheoryofmatter.AmajorcontributiontothisrevolutionarychangeofviewscamefromtheworkofJohnDalton,“ANewSystemofChemicalPhilosophy,”publishedin1808.Aboutonehundredyearslater,in1912,thephysicistErnestRutherfordwasableto“see”forthersttimetheelusiveatomimaginedbyDemocritus,bybombardingathinfoilofgoldwithchargedalphaparticles.Theresultsofhisexperimentsshowedthattheatomiscomposedofaheavy,small,andpositivelychargednucleus,havinglight,swift,andnegativelychargedelectronsorbitingaroundit.Severalyearslater,thequantummechanicstheorydevelopedbymanygreatphysicistssuchasBohr,Heisenberg,Schrodinger,Diracandothersshowedthattheelectronenergystatesintheatomare 1

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2 quantized,andthatbyjumpingfromonestatetoanothertheatomcanabsorboremitquantaofelectromagneticradiationcalledphotons.Theatomicphysicserawasborn. 1.2 UltrafastLaserPulsesInhis1917article,“ZurQuantentheoriederStrahlung”OntheQuantumTheoryofRadiation[ 1 ],AlbertEinsteinshowedthatintheinteractionbetweenmatterandradiation,complementarytotheprocessesofabsorptionandspontaneousemission,theremustbeathirdprocess:“stimulatedemission.”Besidesestablishingtheexistenceofthisnewtypeofprocess,Einsteinalsoassertedthattheradiationproducedinstimulatedemissionisidenticalinallrelevantaspectstotheincidentradiation.Theremarkableimplicationsofthesendingswerefullyexploitedatthebeginningofthe1960s,inthebuildingoftherstlasersources.Today,afterfourdecadesofadvancesinlasertechnology,ultrafastcomputer-con-trolledlasersallowexperimentaliststointeractwiththeatomicworldatanunprecedentedlevelofswiftnessandaccuracy.Sincetheinceptionofthelaserin1960,thelengthofthelaserpulseshasshrunkmorethansixordersofmagnitude,currentpulseshavingdura-tionsofjustafewfemtoseconds10)]TJ/F24 7.97 Tf 6.586 0 Td[(15seconds.Suchultrafastpulsespermitscientiststostudyprocessesthatoccuroversmalltimescales,muchasastrobelightallowsonetocapturearapidmovementinaseriesoffrozenframes.Ultrashortpulsescanactasan”ultrafastcamera”tostudythemotionofmolecules,theprogressionandintermediatestatesofchemicalreactionsorthedynamicsofchargecarriersinasemiconductor.Pulsesintheattosecond10)]TJ/F24 7.97 Tf 6.586 0 Td[(18secondsrangecanevenprovidereal-timeobservationofthenu-clearmotion,ortheCoulombexplosionofadiatomicmoleculeintheprocessofdoubleionization[ 2 ].Thedevelopmentofultrafastlasershashadamajorimpactontheeldofatomicandmolecularphysics.Inthehighintensityregimeavailablewithultrafastlasers,theradiationeldisinteractingwiththeatomsandmoleculessostronglythatthechangesinducedcannolongerbetreatedbyrstorderperturbationtheory.Examplesofsuch

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3 nonlinearphenomenaaremulti-photonionizationMPIandabovethresholdionizationATI,inwhichmultiplephotonscontributetotheionizationprocess,orhighharmonicgenerationHHGinwhichanatomormoleculeabsorbsseveralphotonstoionize,andsubsequentlydecaystotheinitialstateemittingharmonicsoftheexcitinglaserfrequency.HighharmonicgenerationisactuallyoneofthecandidatemechanismstoproducelaserpulsesintheUVandXraydomains[ 3 ].Otherphenomenaobservableathighintensitiesareatomicstabilizationtoionization,“bondsoftening”and“vibrationaltrapping”inmolecules,stimulatedRamanscattering,hyper-Ramanandhyper-Rayleighscattering,laserinducedcollisionthroughpairexcitationanddoubleionization.Ultrafastlasersbringenormousbenetstoothereldsofactivityaswell.Inmedicalapplications,cornealtissuecanberemovedwithoutanysignicantsignofthermalorstructuraldamagetoadjacenttissue[ 4 ].Functionalneurosurgerymightalsoprotfromthehighlylocalizedandefcientablationprocessachievedwithfemtosecondlasers.Ultrashortpulsesareefcientprobesinthestudyofbio-molecularstructures,suchasgenesincancerresearch,orproteinandaminoacidsinbiochemistry.Inthestudyoftheatmosphere,shortpulsesofmid-infraredwavelengthscanbeusedtorecordabsorptionspectraofatmosphericgasses,whichleadstothepossibilityoflongrangeatmosphericsensing,andenvironmentalandpollutionmonitoring.Fastpulsesalsondapplicationsinmicro-machiningandfabricationofnanostructures,inhighbandwidthdatacommunications,laserfusion,andinmanyotherdomains.Asaresult,theeldofultrafastlasersispresentlyattractingmuchinterest,andnumeroustheoreticalandexperimentalstudiesareunderwaytounlocktheirpotentialinnewapplications. 1.3 TerahertzHalf-CyclePulsesTerahertzhalf-cyclepulsesHCPsareanexoticvarietyofultrashortelectromag-neticpulses,whichresultedfromthelaserresearchinthepastdecade.Unlikenormallaserpulses,HCPsconsistapproximatelyofonlyone-halfofanopticalcycle,thushavinganalmostunipolarcharacter.Asaresult,anHCpulsecanimpartmomentum

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4 toanatomicelectronanywhereinitsorbit,incontrasttoanordinarypulse,whichcaninteractimpulsivelywiththeelectrononlyinthevicinityofthenucleus.Inthisrespect,theinteractionwithHCPsisquitesimilartotheinteractionwithchargedparticleswheretheinteractionismediatedbythetime-varyingeldofthepassingparticle.Duetotheirunrestrictedinteractionwithatomicandmolecularelectrons,HCPsconstituteidealprobesfortheinvestigationofRydbergstates,whicharehighlyexcitedstateswithinterestingcharacteristics.Inparticular,HCPshavebeenusedtoprobethedynamicalpropertiesofRydbergwavepacketsviaexperimentaltechniquessuchastime-delayspectroscopy[ 5 , 6 ]andimpulsivemomentumretrieval[ 7 , 8 ].AbetterunderstandingofthedynamicsofRydbergstatesiscrucialforareasofstudysuchasthequantum-classicalcorrespondenceprinciple,quantumchaosandnon-perturbativeresponseofatomsinexternalelds.HCPsmayalsoprovetobeusefultoolsforquantumcontrolincreatingandshapingexoticwavepackets[ 8 , 9 , 10 , 11 , 12 , 13 , 14 ],controllingTHzemissionfromStarkwavepackets[ 15 ],storingandretrievinginformationinatomicquantumregisters[ 16 , 17 ],andpossiblyperformingselectivechemistryinmolecules. 1.4 QuantumControlWiththeadventoffast,computer-shapedlaserpulses,thedesiretocontrolatomsandmoleculeswasspurred.Atsub-nanometerscalesconceptsofthemacroscopicworldfadeaway,andmatterparticlescanbedescribedaccuratelyonlybyprobabilitywaves.Inthiscontext,thecontrolofaquantumsystemisachievedbymanipulatingtheconstructiveanddestructiveinterferencesofthequantumwavesofmattertoattainadesiredtargetstate.Duetoitsintrinsiccoherence,laserlightistheidealagenttoaffectthecoherenceofanatomicormolecularsystem.Currently,laserpulsescanbesculptedwithgreataccuracyinphase,amplitudeandevenpolarization[ 18 ],andtrainsofpulseswithdenitephaserelationshipscanbeproduced[ 19 ].Whenashapedlaserpulseilluminatesaquantumsystem,throughthecouplingbetweentheEMeldandmatter,itscoherence

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5 istransferredtothequantumwavefunctionofthesystem.MoreexactlythecoherenceofthewavefunctionisreectedinaparticularsuperpositionofeigenstatesofthefreeHamiltonian.Inthisscenariotheprincipleofquantumcontrolisquitestraightforward:thecompositionofthelaserpulseshouldexcitejusttherightinitialsuperpositionstate,which,undertheHamiltonianofthesystem,isensuredtoevolvetotheprescribedtargetstateattheprescribedtargettime.Initialapplicationsofquantumcontrolwerefocusedonthemanipulationofchemicalreactionstooptimizereactionyields[ 20 , 21 , 22 , 23 ],buttechniquesofquantumcontrolwereadoptedrapidlyinnumerousotherareassuchasquantumcomputingandinformation,quantumentanglement,wavepacketcontrolinatoms,quantumstatepreparation,optimalemissionofradiationandnanoscaletechnology.Thesuccessesobtainedtodateintheseareasareextremelypromising,andsuggestthatquantumcontrolwillplayanessentialroleinfuturetechnologicaldevelopments. 1.5 ThesisObjectivesandOrganizationThisdissertationpresentstwotheoreticalstudiesoftheinteractionbetweenatomicandmolecularsystemswithultrafastlaserpulses.Inarststudyweconsiderquantumcontrolofthedynamicsofvibrationalwavepacketsinmolecules.Weinvestigatethefocusingofmolecularwavepacketsbychirpedultrashortlaserpulses,withapplicationstochemicalreactioncontrol.Weseektocharacterizethedependenceofthedynamicsontheparametersofthelaserpulse,andultimatelytoprovideanintuitiveexplanationfortheobservedeffects.ThesecondstudyisconcernedwiththeionizationofsodiumRydbergstatesbyhalf-cyclepulses.Inthisworkweattempttoelucidatethedifferentdimensionsofthisphenomenon,andtoprovideaphysicalexplanationofthecharacteristicsoftheionizationspectrum.

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6 Chapter2providesanoverviewofthetopicofquantumcontrolandbringsinfor-mationaboutprevioustheoreticalandexperimentalstudiesinthiseld.However,thepresentationisnotexhaustiveasthebreadthofthisresearchareaisquitesignicant.Inchapter3wepresentatheoreticalstudyoncontrolofvibrationalwavepacketsinmolecules.Thefocusingofmolecularvibrationalwavepacketshasbeenshownpreviouslytobeoneofthekeyingredientsinthecontrolofchemicalreactions.Chapter4isdedicatedtothesubjectofhalf-cyclepulses,exposinganumberofsig-nicanttheoreticalandexperimentalaspectsrelatedtoHCPionizationandspectroscopy.Inchapter5,welayoutthetheoreticalmethodsusedinourinvestigationonsodiumHCPionization,andtheresultsobtainedbythenumericalsimulationsoftheinteraction.Also,anexplanationofthecharacteristicsoftheionizationspectrumbasedonsemiclassicalcriteriaisprovided.Chapter6concludesthisworkwitharecapitulationofthemainresultsandtheirimplicationsandsignicance.Anumberofideasandprospectsforfutureinvestigationarealsoincluded.

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CHAPTER2EXPERIMENTALANDTHEORETICALASPECTSOFQUANTUMCONTROL 2.1 Overview“Quantumcontrol”isanemergentparadigm,whichhasthepotentialtoignitethenextglobaltechnologicalrevolution.Theeldofquantumcontrolholdsthekeyforfuturerevolutionarytechnologiessuchasquantumandopticalcomputers[ 24 , 25 ],molecularengineering[ 26 ],andlaser-controlledfusion.Researchinthiseldhasincreasedatanacceleratedrateinthepastyears,andpromisingresultsarealreadystartingtoappear.Theterm`control'isusedingeneraltodenoteanexternalprogrammaticinterven-tionupontheevolutionofasystemwiththegoalofachievingadesiredoutcome.Theattribute`quantum'inquantumcontrolreferstothefactthatcontrolisattemptedonasystemthatobeystherulesofquantummechanics,andforthisreasonthesystemhasaquantum,`nonclassical'behavior.Itisexactlythisquantumbehaviorthatoffersnewpossibilitiesforcontrol,whicharesystematicallypursuedintheeldofquantumcontrol.Whendescendingtothelevelofatomicmatter,onestartstofeelthebizarreeffectsofthe`quantumworld.'Accordingtoquantummechanics,realitycanbedescribedwithstatesvectorsinaninnitedimensionalHilbertspace.Inlayterms,positionandmomentumortrajectoryforthatmatterlosetheirattributesof`obvious'and`basic'quantities,andmatterappearslikeuncreated,danglingundecidedbetweenconcretestates.Thebizarrerulesofquantummechanicsareresponsibleforparadoxessuchasnon-localityandmultipleuniverses,buttheyhavebeenconrmedandirrefutablyvalidatedagainandagain.Asuggestiveillustrationoftheparadoxicalnatureofthelawsofquantummechanicsisthe`poor'Schrodinger'scat,whichcanbebothdeadandaliveatthesametime[ 27 ].A”Schrodingercat”-likeatom,whichislocatedsimultaneouslyintwoseparateplaces, 7

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8 hasactuallybeencreatedexperimentallybyresearchersfromtheNationalInstituteofStandardsandTechnologyNIST[ 28 ].Thisispossiblebecauseinquantummechanicsasystemcanbesimultaneouslyinaninnitesuperpositionof`classical,'`concrete'states,thissuperpositionrepresentingtheactualquantumstateofthesystem.Inthisrespect,aquantumstatecanbeviewedasawaveofmatter,inwhichtheamplitudeofthewaveassociatedwitha`classical'stateisdirectlyrelatedtothecontributionofthatstatetotheoverallsuperposition.Thiswave-likebehaviorallowsforinterferenceeffects,andalsoleadstothequantizationoftheenergylevelsinaboundsystem,muchasairpressurewavesinanenclosedtubehaveonlyadiscretespectrumoffrequencies.Thus,onecanseehowatomsandmolecules,whicharesystemswithanintrinsicallyquantumbehavior,havediscretelevelsofenergyassociatedwiththedynamicsofthenucleivibrations,rotationsandelectronselectronicstates.Inthispictureonecoulduseacoherentelectromagneticexcitationlasertomanipulatethequantumwavesassociatedwiththenucleiandelectronstocreatespeciccongurations.Manipulatingtheconstitutionofmatteratitsmostintimatelevelshasbenetsindiverseapplications.Theseincludecreatingnewmoleculesformedicine,developingnewfuelsandmaterials,makingquantumcomputersandquantuminformationdevices,andbuildingmolecularandatomicmachines.Inthenextparagraphsofthischapterwewillreviewsomeofthemostsignicantdevelopmentsintheeldofquantumcontrol.Thepresentationwillbebrief,andsometheoreticalandexperimentalaspectspertinenttoeachtopicwillbeexposed. 2.2 QuantumControlofMoleculesSincetheinceptionofthelaserinthesixties,chemistslookedatthisnewsourceofcoherentradiationastheultimatetoolformanipulatingchemicalbonds.Theyarguedthatsinceeachchemicalbondhasitscharacteristicresonancefrequency,bytuningthelasertothisfrequency,onecouldexciteorbreakthebond.Thetheoryappearedsound,butearlyexperimentsbasedonthisintuitivestrategywereatotaldisaster.Thereasonforthis

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9 wasthephenomenonofinternalredistributionoftheexcitationenergytoother,coupled,vibrationalmodes,alsoknownasIVRintramolecularvibrational-energyredistribution[ 29 ].Laterstudiesrenedtheapproachandtookintoconsiderationthequantum,wave-likebehaviorofmolecules.Thus,byusingashapedlaserpulsethattakesintoaccountthedynamicalcharacteristicsofthemoleculeandproducesjusttherightquantuminterferences,onecanachieveaparticularcongurationoropenaparticularreactionchannel,whileinhibitingcompetingpathways.Ingeneral,therearetwomajorparadigmsformolecularquantumcontrol,andbothexploitthequantummechanicalinterferencesbetweenthedifferentdynamicalpathsofasystemininteractionwithalasereld.OnemethodwasproposedbyBrumerandShapiro[ 30 ]inlate1980s,andusestwoormorelaserstoexcitemultiplepathwaystowardsanalstate.AnalternativemethodwasproposedbyTannorandRice[ 31 ],alsoinlate1980s,andusesmultiplepulses,withsuitabletime-delays,tosteerthedynamicsintherightdirectionatjusttherighttimes.Inthefollowingpageswewillbrieypresenteachofthesemethods. 2.2.1 TheBrumer-ShapiroMethodTheBrumer-Shapiromethod[ 30 ]forquantumcontrolreliesonusingcoherentlaserradiationtoexcitemultipledynamicpathwaystoadegeneratenalstate.Thedegeneracyinthenalstatecorrelatestothemultiplereactionproductsthatcanresultfromtheprocess.Accordingtoquantummechanics,themultiplereactionpathwayscaninterferetoincreasetheprobabilityoftransitionintoacertainsetofproducts,anddecreasetheprobabilityofformationoftheotherproducts.Inthisscenario,controlisachievedbymanipulatingtherelativephasesandintensitiesofthelasersproducingtheexcitation.Tobetterillustratethiscontrolmechanismweconsiderexcitationfromaninitialsuperpositionstate0=a1j1i+a2j2i;

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10 toadegeneratenalstatef=Xm;qE;m;q)]TJ/F31 11.955 Tf 7.084 4.748 Td[(;bytwoCWlasersproducingthetotaleldtot=1e)]TJ/F26 7.97 Tf 6.587 0 Td[(i!1t+1+2e)]TJ/F26 7.97 Tf 6.587 0 Td[(i!2t+2:Intheabove,jE;m;q)]TJ/F28 11.955 Tf 7.085 -4.338 Td[(irepresentstheoutgoingstatecorrelatingtothenalproductchannelqwheremdenotestherestofthedegeneracyindices,andthetwolasersarechosenresonantwiththetransitionfromthetwolevelsintheinitialsuperposition,!i=E)]TJ/F25 11.955 Tf 11.955 0 Td[(Ei.Fromstandardperturbationtheory,wendthattheprobabilitytoobtainproductqatenergyEisPE;q=2Xmh~1a11+~2a22j^djE;m;q)]TJ/F28 11.955 Tf 7.084 -4.936 Td[(i2;where^disthecomponentofthedipolemomentalongtheelddirection,and~i=iexpii.Thentheratioofproductsinchannels1and2isgivenbyR;2;E=Pm[jA1mj2+jA2mj2+2ReA1mA2m] Pm[jB1mj2+jB2mj2+2ReB1mB2m];whereAim=h~iaiij^djE;m;1)]TJ/F28 11.955 Tf 7.085 -4.338 Td[(iisthetransitionamplitudeintherstpathway,andBim=h~iaiij^djE;m;2)]TJ/F28 11.955 Tf 7.085 -4.338 Td[(iisthetransitionamplitudeinthesecondpathway.NotethatboththedenominatorandthenumeratorinEq. 2 containtermsthatcorrespondtoindependentexcitationfromeachofthetwolevelsintheinitialsuperposition,butalsotheinterferenceterms2ReA1mA2m,2ReB1mB2m.Sincetheinterferencetermsdependonboththeintensitiesandthephasesoftheexcitinglasers,theycanbemanipulatedtoachieveadesireddistributionofnalproducts.Withproperlychosenvaluesfortheintensitiesandphasesonecan,forexample,maximizetheformationofacertaindesiredproduct,whileinhibitingtheformationofanothercompetingproduct.Thisisverysimilartothedouble-slitYoungexperimentwheretheinterferencebetweentwodifferentpathwaysproducesfringesofenhancedandreducedintensity.

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11 TheabovecoherentcontrolschemewasemployedinatheoreticalstudybyChan,Brumer,andShapiro[ 32 , 33 ]toobtainproductformationselectivityinthephotodissocia-tionofCH3asCH3+I2P3=2CH3I!CH3+I2P1=2:AnotherdemonstrationofcoherentcontrolbyBrumerandShapiro[ 34 ]usestwopathwaysdeterminedby1-and3-photontransitionsinIBr.Theirtheoreticalstudyprovesthatbyvaryingtherelativephasebetweenthe1-photonand3-photonpathways,onecanachievesignicantcontroloverthebranchingratioforIBr!I+Br2P3=2vs.IBr!I+Br2P1=2.ChenandElliot[ 35 , 36 ]demonstratedanexperimentalimplementationofthe3-photonschemetocontroltheionizationofHg.Intheirexperimenttherelativephasebetweenthe1-and3-photontransitionsiscontrolledbyvaryingthegasdensityinthepathofthe3-photonresonantlaser.Inaseriesofexperiments,Gordonandcoworkers[ 37 , 38 ]demonstratedexperimen-talcontrolofthevibrationalandrotationalexcitationinbound-boundtransitionsofHClandCO.OthercoherentcontrolexperimentshavebeenperformedbyKleiman,Zhu,Li,andGordon[ 39 ]tocontroltheionizationversusdissociationofH2S,H2+S+H2S!H+HS+:ThesamegroupalsoperformedexperimentstocontrolthedissociationofCH3ICH3+I+CH3I!CH+3+I:ExcellentexperimentalresultswerealsoobtainedbyZhuetal.[ 40 ].TheyusedcoherentcontroltooptimizetheionizationversusdissociationofHIH+I+HI!HI+:

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12 2.2.2 TheTannor-RiceMethodTannorandRicehavedevelopedanintuitiveschemeusingthetimingbetweenapairofpulsestocontrolproductformationinaphotodissociationreaction[ 31 ].Toillustratethemethod,consideratri-atomicmoleculeABCwhichcandissociatethroughtwodifferentreactionchannels:ABC!AB+CorABC!A+BC;toformdifferentproducts.InthiscasethegroundstatepotentialsurfacehasacentralminimumcorrespondingtothestablemoleculeABC,andtwodifferentexitchannelscorrespondingtothetwopossibleproducts.Thegoalofthemethodistomaximizetheevolutionofthesystemthroughoneexitchannel,inhibiting,atthesametime,theotherchannel.Toachieveselectivityinproductformation,themethodusesasequenceoftwoultrashortpulsesseparatedbyapropertimedelay.Thesequenceoftwopulseshasthepurposeofcouplingthegroundstatetoanexcitedstate.Thus,therstlaserpulsepumppulseexcitestheground-statewavepackettotheexcitedsurface,whereitstartstoevolveaccordingtothepotentialofthissurface.Atalatertimethesecondpulsedumppulsecausesstimulatedemissionbacktothegroundstate,andifthetimingiscorrect,theFrank-Condonregionofthesecondinteractioncorrespondsexactlytothedesiredproductchannel.Inthisway,theformationofthedesiredproductisenhanced,whiletheproductionoftheundesiredproductisinhibited.Intheabovescenariothecontrolofproductformationisachievedbysimplyvaryingthetime-delaybetweenthetwopulses.Pulsesareproperlyshapedtoachievequasi-completepopulationtransferbetweenthetwoelectronicstates.Insecondorderperturbationtheoryandthedipoleapproximation,thetransitionamplitudetoachannelq

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13 withnalenergyEisgivenbyAqE;t=)]TJ/F25 11.955 Tf 9.299 0 Td[(i2XmhE;m;q)]TJ/F31 11.955 Tf 9.077 12.399 Td[(Ztdt1e)]TJ/F26 7.97 Tf 6.587 0 Td[(iEt)]TJ/F26 7.97 Tf 6.587 0 Td[(t1E1t1dg;eZt1dt2e)]TJ/F26 7.97 Tf 6.586 0 Td[(i^Het1)]TJ/F26 7.97 Tf 6.587 0 Td[(t2E2t2de;ge)]TJ/F26 7.97 Tf 6.587 0 Td[(iEit2Eii; whereE1andE2arethetwolaserpulses,da;brepresentsthedipolemomentelementbetweenthestatesaandb,and^HeistheHamiltonianoftheexcitedelectronicstate.Intheaboveequation,theterme)]TJ/F26 7.97 Tf 6.586 0 Td[(i^Het1)]TJ/F26 7.97 Tf 6.587 0 Td[(t2representstheevolutionofthewavepacketontheexcitedsurfaceanddependsonthetime-delaybetweenthetwopulses.Thus,byvaryingthetime-delay,onecancontrolthetransitionamplitudetoanalproductchannel.Whenthewavepacketissufcientlylocalized,thedynamicsofthesystemcanbedescribedsatisfactorilybyclassicaltrajectoriesasshownbyHeller[ 41 ].Formorecomplicatedcaseswhenwavepacketdelocalizationcannotbeavoided,classicaltrajectoriescanbeusedasaguidetodetermineaninitialguessofthetime-delayforsubsequentquantumcalculations.AninitialexperimentalvericationoftheTannor-RicemethodwasperformedbyBaumertandcoworkers[ 42 ],whodemonstratedtwo-pulsecontroloverthetwo-channelionizationreactionofNa2:Na+2+e)]TJ/F28 11.955 Tf 10.406 -4.936 Td[(Na2!Na++Na+e)]TJ/F25 11.955 Tf 10.074 -4.936 Td[(:InasubsequentexperimentonNa2[ 43 ],Baumertandcoworkersusedathree-photonprocedure,basedonthesameideas,andobtainedsimilarresults.AnotherexperimentalrealizationoftheTannor-RiceschemewasperformedbythegroupofNobelPrizewinnerAhmedZewail[ 44 ]tocontroltheamountofproductformedinthebimolecularreactionXe+I2!XeI+I:Successfulcontrolwasdemonstratedonceagain.

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14 2.2.3 QuantumControlbyOptimalControlTheoryOCTInOCTquantumcontroltheaimistodesignashapedinfrequency,amplitudeandphasepulsethatguidesthedynamicsofthesystemtowardsadesiredtarget.Sincetheshapedpulsecanbeviewedasasequenceofphase-lockedshorterpulsesofdifferentintensitiesanddurations,theOCTquantumcontrolcanbeviewedasanextensionoftheTannor-Ricescheme.TheneedforOCTarisesinsituationsinwhichtheHamiltonianofthesystemhasacomplexform,andsimpleintuitionisnotsufcienttoobtainthepulsesthatwoulddrivethesystemtoadesiredtargetstate.OCTquantumcontrolwasdevelopedwithcontributionsfromJudsonandRabitz[ 45 ],Kosloffandcoworkers[ 20 ],andJakubetz,ManzandSchreier[ 46 ].Inoptimalcontrol,onedenesaphysicalobjective,suchasthecleavageofaspecicmolecularbond,tobeachievedataspecictimetf,andthenrecaststheobjectiveinamathematicalform.Forexample,ifthewavefunctionassociatedwiththedesiredoutcomeattimetfisji,thentheaimofOCTistomaximizethetransitionprobabilityJ=htfj^Pjtfi;subjecttocertainconstraintsontheformofthepulseguidingthesystem,wherejtfiisthesystem'swavefunctionatmomenttf,and^P=jihjistheprojectoronstate.TherequirementthattheevolutionofthesystemobeystheSchrodingerequationisintroducedasafunctionalconstraint.WiththeadditionalconstraintthatthelaserpulsehasatotalenergyEp=tfRt0jtj2dt,thefunctionalthatmustbemaximizedbecomesJ=J+1Ztft0jtj2dt+iZtft0dth2ji@ @t)]TJ/F15 11.955 Tf 14.99 3.022 Td[(^Hji+c:c::Whenthemaximizationcondition@J=0isimposed,theLagrangemultiplier2satisestheSchrodingerequationi@j2i @t=^Hj2i;

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15 withtheboundaryconditionj2tfi=^P^Hj2tfi:Theoptimumeldresultingfromtheoptimizationconditionisthengivenbyt=Ot1 EpZtft0dtjOTj2)]TJ/F24 7.97 Tf 6.586 0 Td[(1=2;whereOtisdenedasOt=)]TJ/F25 11.955 Tf 9.298 0 Td[(ih2tj@^H @jti:Theabovesetofequationscanbesolvedbyaniterativeprocedure,whichstartswithaguessfortandthendetermines^Htwhichdependson.Using^Ht,theinitialstate,0,ispropagatedtodeterminet,and2tfisobtainedas^Ptf.Then2tfispropagatedbackwardstoobtain2t.Withtand2tdeterminedinthiswayonecancalculateanewestimatefortusingEq. 2 .Theprocedureisthenrepeateduntilconvergenceisobtained.VarioustheoreticalstudieshavebeenemployedtodeterminetheutilityoftheOCTmethodinobtaininganoptimalshapedeldforageneralcontrolproblem.OneimportantresultwasobtainedbyPeirceandcoworkers[ 47 ],whoprovethatbound-statecontrolisalwayssolvable,andaninnitenumberofapproximatesolutionscanbefound.Inasubsequentstudy,ZhaoandRice[ 48 ]extendedtheproofofsolvabilityforthecaseofunbound,continuumnalstates.Anotherstudydemonstratedthatanoptimalsolutioncannotbefoundwhentheinitialstatedoesnotpresentacertaindegreeofphasecoherence[ 49 ].NumerousgroupshavestudiedtheoreticallytheOCTquantumcontroltechnique.Shiandcoworkers[ 50 ]usedOCTtoachievethebondselectiveexcitationanddissocia-tionofaharmoniclinearchainmolecule.ShiandRabitz[ 51 ]usedOCTtodemonstrateselectivevibrationalexcitationofharmonicmoleculesmodeledasMorseoscillators.Kosloffandcoworkers[ 20 ]usedoptimallyshapedpumpanddumppulsestoenhance

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16 selectivityofproductformationononeelectronicsurfaceoptimallycoupledtoanotherelectronicsurface.Grossandcoworkers[ 52 ]usedOCTtocontrolthecurve-crossingbetweentwodissociativediabaticBorn-Oppenheimersurfacesinordertoachievedisso-ciationchannelselectivityenhanceoreliminatedissociationfromonechannelversusanother.Theyaddedafrequencyltertotheoptimizationprocedureinordertorestrictthebandwidthoftheoptimalpulse.ThemechanismisreminiscentoftheTannor-Ricepump-dumpcontrolscheme,butthechannelselectivityismuchimprovedduetothedynamicinterplayofconstructiveanddestructiveinterferences.Ingeneral,theiterativeprocesstosolvetheOCTequationsiscomputationallyexpensiveandoftenproducescomplex,non-intuitivepulses,whicharenon-feasibleexperimentallye.g.theyhaverapidturn-onorturn-off.Toimprovethecomputingtime,Tannorandcoworkers[ 53 , 54 ]usedKrotov'soptimizationtheorytocontrolphotodissociationandmultiphotonionization.Inasimilarvein,ZhuandRabitz[ 55 ]developedafamilyofrapidconvergentiterativemethodstosolvetheOCTproblem.Arobustsolutionwasobtainedbyincludingatnessconstraintfunction[ 56 ].Toforcesolutionswithasmoothturn-onandturn-off,SundermannanddeVivie-Riedle[ 57 ]addaconstraintfunctionwithjustasmallperformancepenalty.AstudybyYanetal.[ 58 ]presentsanOCTformulationspecictotheweakre-sponseregime.Theexcitedstatepopulationscaleslinearlywiththelaserintensityandthesystemdynamicscanbedescribedaccuratelywithinrstorderperturbationtheory.Theoptimizationproblemisconvertedtoanintegraleigenequation.Theeigenvectorscor-respondtooptimalsolutionsfortheeldandtheeigenvaluesrepresenttheachievement.Notably,thetheoryiscastintoadensitymatrixformalism,whichallowsaconsiderationoftheroleoftheenvironmentinthedynamicstemperature,solventetc.Krauseetal.[ 59 ]usedLinearDensityMatrixOCTtocontrolthevibrationaldynamicsofI2.TheyfocusedvibrationalwavepacketsontheBexcitedelectronicstateataspeciedpositionandtime.Theoutgoingcontinuumwavepacketsfocusedbothin

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17 positionandmomentumaretermed“molecularcannons,”whiletheincomingboundandfocusedwavepacketsaretermed“molecularreectrons.”Theresultsobtainedintheirstudyareexplainedinasimpleclassical-trajectorypicture.Accordingtothatpicture,foracannon,thelowerfrequenciestakelongertoreachthetargettheyhavelowerenergyandthereforetheymustbeexcitedrst.Forthisapositivelychirpedpulsemustbeused.Forareectron,thehigherfrequenciestakelongertoreachtheouterpotentialwallandexperienceatterregionsofthepotential.Asaresult,theymustbeexcitedrst,andanegativelychirpedpulseisrequiredinthiscase.Wewillshowinchapter3ofthisthesisthattheabovesimplisticpictureisnotalwaysvalid,andquantuminterferenceeffectsmustingeneralbeconsideredinordertopredicttherightchirpthatachievesfocusing.Inafollowupexperiment,Kohleretal.[ 60 ]provedexperimentallythepredictionsofKrauseetal.[ 59 ]andshowedtherobustnessofthesolution.Subsequently,Krauseetal.[ 61 ]extendedthepreviousworktothestrongresponseregimes,andusedasaninitialguessfortheiterativeOCTproceduretheeldobtainedintheweakresponsecase.Withthisapproachtheyachievedfastconvergence<10iterationstowardstheoptimaleld. 2.2.4 Closed-LoopLearningControlAsshownabove,optimallyshapedlaserpulsesrepresentthemostversatilein-strumenttocontroltheintricateinterferencesintheevolutionofaquantumsystem.However,thedeterminationoftheoptimaleldbyOCTrequiresdetailedknowledgeoftheHamiltonian,whichinmostcasesparticularlyforcomplexsystemsisnotavailable.Furthermore,incasesinwhichtheHamiltonianisknown,calculationoftheoptimaleldmaybeverytime-consuming,andthesolutionmaybetoocomplexandunstabletobeusedinexperiments.ApracticalalternativetotheOCTtechniquewasadvancedbyJudsonandRabitz[ 45 ],whoproposedtheuseoflearningalgorithmstonddirectlyfromexperimenttheoptimallaserpulsethatachievesaparticulargoal.Inthisapproach,thesystemitselfworksasananalogcomputer,solvinginreal-timeitsownSchrodingerequationofwhich

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18 ithasperfectknowledge.Alearningalgorithmworkssequentiallytondthroughtrialanderrortheshapeoftheoptimaleld.Ateachstep,theoutputfromtheexperimentisanalyzedwithrespecttothetargetoutput,anda'smart'decisionismadetoimprovetheshapeofthetriallaserpulse.Eventually,thetrialpulseconvergestotheoptimalshape,whichisrobustwithregardtotheimperfectionsoftheexperiment,andachievesthedesiredtargetwithhighaccuracy.Forthepurposeofacceleratingthelearningprocess,anumberoftheoreticalstudieshaveanalyzedthepropertiesofvarioussearchalgorithms.PhanandRabitz[ 62 , 63 ]investigatedalgorithmsbasedonlinearinput-outputmapping,andfoundthatgeneticalgorithmsGA[ 64 ]andsimulatedannealingSA[ 65 ]arequiteefcientforlearningcontrol.Theynotedthatthesimulatedannealingalgorithmisgenerallyfaster,butthegeneticalgorithmrespondsbettertonoisyexperimentalconditions.Forthisreason,geneticalgorithmsarethemethodofchoiceinmostclose-loopcontrolexperiments.TherstexperimentalimplementationoflearningcontrolforamolecularsystemisduetoWilsonandcoworkers[ 66 ].Theywereabletooptimizetheefciencyandeffectivenessofelectronicexcitationinthe“IR215”dyemolecule.Assionetal.[ 67 ]usedfeedbackloopcontrolwithanevolutionaryalgorithmtocontrolbranchingratiosinthephotodissociationoforganometalliccompoundsFeCO5andCpFeCO2Cl,whereCp=5C5H5.Also,inaquiterecentexperiment,Danieletal.[ 68 ]usedpump-probefeedbackcontroltooptimizeionratiosinthephotodissociationofCpMnCO3.AseriesofexperimentsbyWeinacht,Bucksbaumandassociates[ 69 , 70 ]usedfeedbackwithalearningalgorithmtocontrolRydbergcesiumwavepackets,thepho-todissociationofNa2,andthevibrationaldynamicsinmethanol.OtherexperimentaldemonstrationswereprovidedbyLevis,MenkirandRabitz[ 71 ]whousedclose-loopcontroltooptimizeionyieldsinthephotodissociationofaseriesoforganicmolecules.Inaddition,Weinachtetal.[ 72 ]demonstratedcontrolofRamanscatteringinliquids,and

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19 Bartelsetal.[ 73 ]usedclosed-loopcontroltosuccessfullyoptimizetheefciencyofhighharmonicgenerationforproductionofcoherentX-raypulses. 2.3 QuantumControlforQuantumStatePreparationQuantumstatepreparationisanabsolutedesideratuminmodernatomicandmolecularphysics.Theabilitytoprepareatomsandmoleculesinaspeciccongurationhasapplicationsinmanyelds:controlofchemicalreactions,quantuminformation,statetostatecollisions,atomoptics,andingeneralforthepurposeoflasercontrolinatomicandmolecularprocesses.Thecentralmotifinquantumstatepreparationistobeginwithanatomormoleculeinacertaindiscretequantumcongurationusuallyathermaldistributionofpopulation,andapplyappropriatelaserradiationtodrivethesystemtoadesiredtargetstate.Themostsuccessfultechniquesforpopulationtransfermakeuseofanadiabatictimeevolutionofthesystemalongthedressedpotentialsinducedbythelaserradiation.Thisphenomenonisgenerallyknownasadiabaticpassage.Forincoherent,near-resonantradiation,thepopulationoftheexcitedstateattimetisgivenby[ 74 ]Pet=1=2[1)]TJ/F25 11.955 Tf 11.955 0 Td[(e)]TJ/F26 7.97 Tf 6.587 0 Td[(Ft];whereFt=RtIt0dt0representsthesocalled“uence”ofthepulse,andistheabsorptioncoefcient.Astheuenceofthepulseincreases,thispopulationapproachesmonotonicallythesaturationvalueof0:5,whichisthemaximumefciencyofincoherentpopulationtransfer.Forresonantcoherentradiationactingonatwo-statesystem,thesolutionofthetime-dependentSchrodingerequationshowsanexcitedpopulationgivenby[ 74 ]Pet=1 2[1+cosAt];whereAt=tRt0dt0isthepulseareauptotimet,andt=Et histheRabifrequency.Theexcitedpopulationvariessinusoidallywithtimebetween0and1.Thus

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20 themaximumtheoreticalefciencyofpopulationtransferis100%,butvariousfactorssuchastheDopplerbroadeningofthelevels,variationoftheintensityoftheeldoverthebeamcrosssection,andotherexperimentaluncertaintiesoftenreducethisefciencysignicantly. 2.3.1 AdiabaticRapidPassage–ARPAnalternativetocoherentresonantexcitationisexcitationwithcoherentradiationwhosefrequencyslowlysweepsthroughresonance.Whentheparametersofthelaserpulsearechosenproperly,thisprocedureinducesacontrolledadiabaticpassageofthelowerstatepopulationintotheexcitedstate.TheHamiltoniandescribingthetwo-levelsystemundercoherentexcitationisgivenintherotatingwaveapproximationRWAbyHt=26401 2t1 2tt375;wheretheoff-diagonalelementistheRabifrequency,t=d12Et=h,d12isthedipolemomentofthetransition,andtistheinstantaneousdetuningofthelaserfrequency.TheinstantaneouseigenstatesoftheaboveHamiltonianarecalled“adiabaticstates”andaregivenby+t=1sint+2cost;)]TJ/F15 11.955 Tf 7.085 1.794 Td[(t=1sint)]TJ/F25 11.955 Tf 11.955 0 Td[(2cost; where1,2aretheunperturbeddiabaticatomicstates,andtisthe“mixingangle,”t==2arctan[t=t].Theenergiesoftheadiabaticstatesaregivenby=1 2[tp 2t+2t]:ForasufcientlysmoothpulseandlargeRabifrequency,thecouplingbetweenthetwoadiabaticstatesissmall,andtheevolutionofthesystemisadiabatic.Ifatsometime,t,

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21 Figure2: Theleftpanelshowstheevolutionoftheadiabaticstateenergiescontinu-ouslineandofthediabaticstatesdashedline.Therightpanelshowstheevolutionofthepopulationsinthetwostates. thesystemisinoneofthetwoadiabaticstates,t,itwillremaininthatstateatallulteriortimes.TheadiabaticrapidpassageARPtechniqueusesthepropertyofadiabaticevolutionofthesystemtotransferpopulationbetweentwodiabaticstates.Bymonotonicallyincreasingordecreasingthefrequencyoftheexcitinglaserpulsethroughresonance,themixingangle,tvariessmoothlybetween0and=2.Forexample,inthepositivechirpcasemonotonicincreasingoflaserfrequency,tdecreasesmonotonicallyfrom=2to0,andineffectthe+t=1sint+2costadiabaticstatecoincidesatthestartofthepulsewith1,andattheendofthepulsewith2.Ifconditionsforadiabaticevolutionofthesystemaresatised,theentirepopulationofstate1willbetransferredviaadiabaticstate+tostate2.Agraphshowingtheevolutionofthetwoadiabaticstatesandtheevolutionofthepopulationsinstates1and2isgiveninFig. 2 .AdiabaticpopulationtransferpresentssignicantadvantagesoverRabicyclingexcitationbecausetheprocessismorerobusttovariationsinthelaserfrequencyandintensity,andthetotalinteractiontimeisnolongerastrictparameter.Populationtransferbyadiabaticpassagewasrstdemonstratedinconnectionwithnuclearmagneticresonanceexperiments.Intheopticaldomain,therstsuccessfuladiabaticpassageexperimentwasconductedbyLoy[ 75 ]inNH3.Inthisexperiment,insteadofvarying

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22 Figure2: Diagramshowingtheevolutionoftheadiabaticstateenergiescontinuouslineandofthediabaticstatesdashedlineforthe5s-5p-5dladderinrubid-ium. thelaserfrequency,thecomplementaryapproachofvaryingthetransitionfrequencywithaslowlyvaryingStarkeldwasused.Laterexperimentsdemonstratedadiabaticpassageinthenearinfrared[ 76 ]andthevisibledomain[ 77 ]byDopplershiftingthelaserfrequencythroughaningeniousgeometryofthelaser-beam-molecular-beamsystem.Inthepicosecond-pulseregime,adiabaticpassagewasdemonstratedbychirpingalarge-bandwidthpulsewithapairofparallelgratings[ 78 ].Thephenomenonofadiabaticpassageofpopulationisnotlimitedtotwo-statesystems.Forexample,thismethodhasbeenappliedsuccessfullytoathree-levelladdersystembyNoordamandco-workersinaseriesofexperimentsonthe5s-5p-5dtransitioninrubidium[ 78 , 79 ].Aschematicdiagramofthe3-levelsysteminvolvedisshowninFig. 2 .Weseeinthisgurethateitherred-to-blueorblue-to-redchirpedpulsescanbeusedtoadiabaticallytransferpopulationfrom5sto5d.Thisfactwasalsorecordedbytheactualexperiments,whichhavereportedcompletepopulationtransferinbothcases.However,ifthefrequencychirpisblue-to-red,meaningthatstates2and3getcoupledbeforestates1and2,state2,whichisfarfromtheupperadiabaticstate,isvirtuallyunpopulatedduringtheentireprocess.ThisissimilartothestimulatedRamanadiabatic

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23 Figure2: Diagramshowingtheevolutionoftheadiabaticstateenergiescontinuouslineforthecaseofagroundstatecoupledbyachirpedpulsetoamanifoldofvestates. passageSTIRAPmechanism,inwhichasimilarcounterintuitivepulsesequenceisused,andtheintermediatestateisunpopulatedatalltimes.ARPcanalsobeusedtoachieveselectiveexcitationwhentheinitialstateiscoupledtoamanifoldofcloselyspacedlevels,despitethefactthatthelaserpulsebandwidthislargerthanthespacingbetweenthemanifoldlevels.ThesituationisdepictedinFig. 2 .Fromtheevolutionoftheadiabaticstateswecanseethatthelaserpulseselectivelyexcitesthelowerorhigherstateinthemanifold,dependingonwhetherthechirpisred-to-blueor,respectively,blue-to-red.Onescheme,whichachievedselectiveexcitationinthe3s-3ptransitioninsodium,wasdemonstratedboththeoreticallyandexperimentallybyMelingerandcoworkers[ 80 ].Forared-to-bluechirp,theyobservedthepredominantexcitationofthe3p2P1=2ne-structurelevel,whileforablue-to-redchirpthepredominantexcitationwasinthe3p2P3=2ne-structurelevel.Notably,thebandwidthofthepicosecondlaserpulseusedwaslargerthantheseparationbetweenthetwo3pne-structurelevels.However,itcanbeshownthatwiththismethodonlytheuppermostandlowermoststatesinthemanifoldcanbeexcitedselectively.Amodiedtechnique,whichusesnarrowbandlaserpulsessmallerthanthelevelspacinginthe

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24 manifold,andcanachieveselectivepopulationtransfertoallmanifoldlevels,wasdevelopedbyMalinovskyandKrause[ 81 ].MultistateARPcanalsobeusedtoachieveselectivevibrationalexcitationinmolecules,whichisusefulforcontrollingmolecularreactivityordissociation.Chelkovskyandco-workershaveshowntheoreticallythatselectivevibrationalexcitationcanbeachievedwithultrashortchirped-pulses,whichcancompensatefortheanhar-monicityofthevibrationalpotential,ifthepulseshavesufcientbandwidthtocouplethevibrationallevelsofinterest.AnexperimentalstudybyMaasetal.[ 82 ]usedinfraredsub-picosecondpulsestoachieveselectivevibrationalexcitationinthegroundelectronicstateofNO.Thechirpofthepulsewasblue-to-redinordertofollowtheanharmonicityofthevibrationalladder.Signicantpopulationwastransferredtothe=4vibrationalstate,whichwasthehigheststatewithatransitionfrequencywithinthelaserpulsebandwidth.Theoveralladiabaticevolutioninonedressedstateistheresultofcontrolledin-terferencebetweentheFockstatesdescribingthetwo-levelatominteractingwiththeradiationeld.Theinversionofpopulationcanbeseenasalmostcompleteconstruc-tiveinterferenceforpopulationtransferintotheexcitedstateversusalmostcompletedestructiveinterferenceforpathstransferringpopulationintotheinitialstate. 2.3.2 Stark-ChirpedRapidAdiabaticPassage–SCRAPInthenanosecondregime,laserchirpingbybothactivephasemodulationsuitableinthemicrosecondregimeandspatialdispersionsuitableinthepico-andfemtosecondregimesisquiteinefcient.Atechniqueforadiabaticpopulationtransferthatispar-ticularlysuitableinthenanosecondregimeistheStark-chirpedrapidadiabaticpassageSCRAPtechnique[ 83 ].SCRAPisbasedontheuseoftwoproperlyoffsetpulses:aslightlyoff-resonantpumppulse,whichdrivespopulationintotheexcitedlevel,andaStarkpulsethatcreatestwodiabaticlevelcrossingsonthewingsofthepulse.Forsuccessfulpopulationtransfer

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25 Figure2: TheupperpanelshowsthepumpandStarkpulsesasafunctionoftime.Thelowerpanelpresentstheevolutionoftheadiabaticstateenergiescontinuouslineforatwo-levelsystemdrivenbythetwopulses. tooccur,theevolutionofthesystemmustproceedadiabaticallyatonlyoneoftheStarkcrossings.Thiscanbeachievedbyintroducingashorttimedelaybetweenthetwopulsessothatoneofthediabaticcrossingsoccursatthemaximumofthepumppulse,andtheotherwhenthepumppulseintensityislow.Asaresult,theadiabaticevolution,whichrequiresastrongdrivingeld,isensuredonlyatthemaximumofthepumppulse.Anexampleofsuchasituation,inwhichthepumppulseisoffsetbeforetheStarkpulsealthoughthereverseisalsopossible,isshowninFig. 2 .Inthiscaseweobservethatifthesystemisinitiallyinstate1itwillfollowtheloweradiabaticstate,)]TJ/F23 11.955 Tf 7.085 1.794 Td[(,attherstStarkcrossing,andthediabaticstate2atthesecondcrossing.Eventually,allthepopulationthatwasinitiallyinthelowerstate1endsupintheupperstate2.TheSCRAPtechniquepresentsalltheadvantagesofadiabaticpassagebeingrobusttoexperimentalimprecisionsandotherenvironment-relatedfactors.Inaddition,SCRAPiswellsuitedforexperimentalimplementation[ 84 ]sincethexed-frequencylong-wavelengthpulsedradiationnecessarytoproducetheStarkshiftingisreadilyavailable.

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26 Figure2: Thisgurepresentsa3-levelsystem.States1and2arecoupledbythepumppulse,andstates2and3arecoupledbytheStokespulse.PandSrepresentthedetuningsofthetwopulseswithrespecttoresonance. ThisradiationisusedintherstplacetogeneratethevisibleorUVpumppulsethroughfrequencyconversion. 2.3.3 StimulatedRamanAdiabaticPassage–STIRAPSingle-pulseadiabaticpassagetechniquescannotbeusedtotransferpopulationbetweentwonon-opticallycoupledstates.However,iftwocoherentpulsesareused,alternativeschemesforadiabatictransfercanbedevised.Oneverypopularscheme,whichusestwocoherent,overlappingpulsestoachievepopulationinversionbetweenthetwolowerstatesofa-system,iscalled“StimulatedRamanAdiabaticPassage”STIRAP.Thelowerstatesofthe-systemseeFig. 2 ,j1iandj3i,arenotopticallycoupledandhavelonglifetimes,whereastheupperstate,j2i,isopticallycoupledtobothlowerstatesandcanundergospontaneousemission.InSTIRAP,theorderbetweenthepumpcouplingj1iandj2iandStokescouplingj2iandj3ipulsesiscounterintuitive,withtheStokespulseappliedrst,followedbythepumppulse.Tounderstandwhysuchanorderingisdesiredandbenecialforpopulationinversionbetweenj1iandj3i,wemustexaminetheevolutionofthedressedoradiabaticstatesofthesystem.Intheusualrotatingwave

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27 approximation,the3-statetime-dependentHamiltonianofthesystemisgivenby[ 74 ]Ht=1 22666640Pt0PtPSt0StP)]TJ/F15 11.955 Tf 11.955 0 Td[(S377775;wherePtandStaretheRabifrequenciesofthepumpandStokespulses,respectively,andPandSarethe1-photondetuningsofthetwopulsesfromtheirrespectivetransitions.AnessentialconditionforSTIRAPistwo-photonresonancebetweentheinitialandnalstates,P=S=.Forthissituation,theadiabaticdressedstatestheinstantaneouseigenstatesofthesystemaregivenbyj+ti=sintsin\050tj1i+cos\050tj2i+costsin\050tj3ij0ti=costj1i)]TJ/F15 11.955 Tf 19.261 0 Td[(sintj3ij)]TJ/F15 11.955 Tf 7.085 1.793 Td[(ti=sintcos\050tj1i)]TJ/F15 11.955 Tf 19.261 0 Td[(sin\050tj2i+costcos\050tj3i wherethemixingangles,tand\050t,aredenedmodulo2through:tant=Pt St;tan\050t=p 2Pt+2St :Thecorrespondingeigenvaluesare:0=0;=1 21 2q 2Pt+2St+2:Fromtheformoftheadiabaticstates,weseethatthemostappropriate`conveyor'ofpopulationbetweenstates1and3isj0i.Bothj)]TJ/F28 11.955 Tf 7.084 1.793 Td[(iandj+icontaincontributionsfromtheshort-livedstatej2i,andthereforetheymaylosesomepopulationthroughspontaneousdecay.Statej0i,whichisstableagainstdecay,isanexampleofa`trappedstate'orradiatively`darkstate.'TrappedstatesareessentialforSTIRAP,andtheyarealsoextensivelystudiedinconnectionwithanumberofothercoherentphenomenasuch

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28 aselectromagneticallyinducedtransparency,inducedrefractiveindexchanges,andlasingwithoutinversion.Theexpressionforj0itellsusthatinordertoachievepopulationinversion,wemustmodifyadiabaticallythemixinganglefrom0to=2.ThisvariationoftcanbeachievedwithacounterintuitiveorderingofthepumpandStokespulses,wherethepumppulsefollowstheStokespulse.Inthissituation,fort!onehasPt=0andSt6=0,andthereforelimt!tant=0.Fort!1onehasPt6=0andSt=0,whichimplieslimt!1tant=1.Thus,therequirementthatthemixinganglechangesfrom0atthebeginningoftheinteractionto=2attheendofthepulsesequenceissatised.Asaresult,ifthesystemisinitiallyinj1i,andiftheevolutionisadiabaticp 2Pt+2St>10,whereisthepulsewidth[ 74 ],allpopulationistransferredtoj3iseeFig. 2 .Bergmannandco-workersperformedtherstunequivocalexperimentaldemonstra-tionofSTIRAPin1989[ 85 ].IntheirexperimentabeamofNa2waspassedthroughtwooffset,butpartially-overlappingCWlaserbeams.TherstbeamrepresentstheStokespulse,andthesecondbeamrepresentsthepumppulse,whereastheadjustableoffsetofthetwobeamsgivesthetemporaldelaybetweenthepulses.Theexperimentreportedcompletepopulationtransferbetweentwovibrationallevels=0,J=5and=5,J=5intheelectronicgroundstateX1+gofNa2,viaanintermediatelevel=7,J=6ontherstexcitedelectronicstate,A1)]TJ/F24 7.97 Tf 0 -7.294 Td[(u.Asimilartwo-beamgeometrywasusedbyTheuerandBergmanntoperformSTIRAPbetweenthe2p53s3P0and2p53s3P2metastablestatesinneon,viaanintermediate2p53p3P1state.Remarkably,althoughthetransitiontimewasmorethan20timeslongerthantheintermediatestate'slifetime,theoveralldecayfromthisstatewaslessthan0.5%,conrmingitstrappedcharacter.STIRAPhasalsobeenappliedinthepulsedlaserregimetoachieveselectiveexcitationofhigh-lyingvibrationalstatesinmolecules[ 86 ].Theuseofpulsedlasersintheseexperimentsisdictatedbytwomainfactors:rst,mostmoleculeshavetransition

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29 Figure2: TheupperpanelshowstheevolutionofthepumpandStokesRabifrequen-ciesasfunctionoftime.Themiddlepaneldisplaystheevolutionoftheadia-baticstateenergies.Thebottompanelshowstheevolutionofthepopulationsinthethreestates.

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30 frequenciesbetweenthegroundandrst-excitedelectronicstatesintheUVdomain,whichrequiresfrequencyconversionfromhigh-intensitylaserpulses.Second,electronictransitiondipolemomentsinmoleculesareingeneralmuchsmallerthaninatoms,andadiabaticevolutionofthesystemrequiresmuchhigherintensities,achievableonlywithpulsedlasers.SuccessfulexperimentaldemonstrationofSTIRAPwithpulsedlaserswasperformedinNO[ 87 , 88 ]andSO2[ 89 ].InNOtheadiabaticpopulationtransferwasbetweentheX21=2=0,J=1=2andX21=2=6,J=1=2rovibrationallevelsofthegroundelectronicstate,viatheintermediateA2=0,J=1=2levelintherstexcitedelectronicstate.Forthisconguration,theSTIRAPprocessiscomplicatedbythehypernesplittingofthemainlevelsintoan18-levelhypernestructure.Althoughthehyperneinteractioncaninduceundesiredcouplingsbetweenthedifferentadiabaticstates,thehypernesplittingoftheinitialandnalstatescanstillberesolvedexperimentally.Asaresultnearlycompletepopulationinversionisachieved.Similarly,theSTIRAPprocessinSO2iscomplicatedbytheincreaseddensityoflevels,duetothethreeconstituentnuclei.However,forproperlaserpowersandproperdelaysandsequencingbetweentheStokesandpumppulses,completepopulationtransferisobtained[ 89 ]. ExtensionsofSTIRAPtoLadderSystemsofThreeorMoreLevelsAsshownbyShoreetal.[ 90 ]theSTIRAPtechniquecanbereadilyextendedto3-levelladdersystemsinwhichthenalstatelieshigherthantheintermediatestate.Thisfacthasbeenprovenexperimentallyinexcitationofhigh-lyingstatesofatomsbyseveralgroups[ 91 , 92 ].GeneralizationsoftheSTIRAPtechniquetomulti-levelsystems1;2;:::;Nhavealsobeenproposedandimplemented.Theoreticalstudieshaverevealedthatmulti-statesystemsmustbeapproacheddifferently,dependingonwhethertheyinvolveanevenoroddnumberofstates.Thus,forasystemcontaininganoddnumberofstatesN=2n+1,aSTRAP-likepopulationtransfercantakeplace

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31 fordelayedandcounter-intuitivelyorderedpulses[ 90 , 93 ].Inthiscase,thepopulationtransferismediatedbyamulti-leveldarkstate,whichisacoherentsuperpositionoftheoddstatesinthechain1;3;:::;2n+1.Forillustration,ave-leveldarkstateisformedas[ 90 , 93 ]j0ti=1 t[23t45t1)]TJ/F15 11.955 Tf 11.955 0 Td[(12t45t3+12t34t5];wheretisanormalizationfactor,andtheijsarethevariousRabifrequencies.Fromtheformofthisstate,itcanbeseeneasilythatarstpulseconnectingstates4and5,followedbyasecondpulseconnectingstates1and3,cantransferpopulationadiabaticallyfromstate1tostate5.Inthecaseofmulti-statechains,inwhichtheintermediatestateshavearelativelyshortdecaytime,itisdesirabletoreducetheirpopulation.MalinovskyandTannor[ 94 ]proposedaschemeinwhichthedecayingintermediatestatesarecoupledbymuchstrongerpulsesthanthepulsesfortherstpumpandlastdumptransitions.InthisschemealltheintermediateexcitationpulsesarrivesimultaneouslywiththeStokespulse,andvanishwiththepumppulse.Asaresult,theentireprocessismorerobusttoradiativedecayfromtheintermediarystates,andbettertransferefciencyisachieved.Forasystemwithanevennumberofstates,whentheintermediatetransitionsareresonant,adarkstatedoesnotexist.Therefore,STIRAP-likepopulationtransfercannotoccur,butinsteadthenalstatepopulationexhibitoscillationsasthepulseintensityvaries[ 95 ].Whentheintermediatetransitionsareoff-resonant,buttheinitialandnalstatesstillsatisfytheN)]TJ/F15 11.955 Tf 11.978 0 Td[(1-photonresonance,systemswitheitheranevenoroddnumberofstatesbehavesimilarly.TheSTIRAP-liketransfermayormaynotoccur,dependingonthedetuningsandintensitiesofthelaserpulses[ 95 ]. ApplicationsoftheSTIRAPTechniqueAsanefcientmethodofpopulationtransferinmoleculesandatoms,STIRAPhasfoundnumerousapplicationsinavarietyofdomains.Oneimportantapplicationisin

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32 thecontrolofchemicalreactions,sincethereactioncrosssectiondependsingeneralonthevibrationalstatesofthereactants.Oneexampleofsuchareactioncontrolstudy[ 96 ]investigatedtheprocessNa2+Cl!NaCl+Na:Itwasfoundthatthereactioncross-sectionincreasesbyabout0.75%pervibrationallevelintherange319.Otherstudies[ 97 ]examinedthereactionsNa200;j00+H!NaH0;j0+NaandNa200;j00+e)]TJ/F28 11.955 Tf 10.405 -4.936 Td[(!Na+Na)]TJ/F25 11.955 Tf 9.077 -4.936 Td[(:Inthesecondcase,theexperimentsshowedthatexcitationofNa2tothe00=12levelincreasesthedissociativeattachmentratebymorethanthreeordersofmagnitude.AnotherimportantapplicationofSTIRAPistocreatecoherentsuperpositionsofstates,whichareessentialinquantumcomputing,quantumcryptographyandaseriesofothercoherentphenomena.Asimplewaytocreateasuperpositionstateistointerrupttheadiabaticevolutionofthedarkstatebeforecompletetransferoccurs[ 75 , 98 ].Then,thecompositionofthesuperpositionstatedependsontheratioofthepumpandStokesRabifrequenciesatthetimethetransferwasstopped.ThisfractionalSTIRAPprocesswasdemonstratedexperimentallybyWeitzetal.[ 99 ].Inatoms,thelaserpulsesthatcoherentlytransferpopulationbetweenstatesimpartmomentumtotheatom.Asaresult,STIRAPcanbeusedasanefcienttoolforcoherentmomentumtransferinatomopticstobuildkeyelementssuchasatommirrorsandbeamsplitters.AnexperimentalimplementationofcoherentmomentumtransferbySTIRAP[ 100 ]usestwocircularlypolarizedCWlasersofoppositepolarizationstocouplethe3P2-3D2transitioninmetastableNeatoms.Thetwobeamswereorderedcounterintuitivelysothatthe+beamStokesactsrst,followedbythe)]TJ/F23 11.955 Tf 10.074 -4.338 Td[(beampump.TheNeatomswerepreparedinitiallyintheM=2sublevelofthe3P2metastablelevel.WhenadiabaticconditionsweremetthepopulationwastransferredcompletelybetweentheM=2andM=)]TJ/F15 11.955 Tf 9.299 0 Td[(2sublevelsof3P2.Sincethetwolaserbeamshadoppositepolarizations,a

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33 totalmomentumof4hkwastransferredtotheatoms:2hkduetoabsorptionoftwo)]TJ/F23 11.955 Tf -407.077 -28.246 Td[(photons,and2hkduetorecoilfromstimulatedemissionoftwo+photons.CoherentmomentumtransferbyadiabaticpassageinsimilarchainsofZeemansublevelshasbeendemonstratedinanumberofexperimentsbyPilletatal.[ 101 , 102 ].LawallandPrentiss[ 103 ]havedemonstratedamomentumtransferof4hk,with90%efciency,usingtheM=)]TJ/F15 11.955 Tf 9.298 0 Td[(1!M=1transitionbetweenthesublevelsof23S1coupledtothe23P0levelsofHe.Chuetal.[ 99 , 98 ]havebuilttherstatomicinterferometerbasedonSTIRAP.Forthispurpose,theyuseSTIRAPbetweentwocesiumhypernelevels,6S1=2;F=3;MF=0and6S1=2;F=4;MF=0,viatheexcitedlevel6P1=2,F=3or4,MF=1.Inthisimplementationtheyachieveamultiple-passcoherenttransferofmorethan140photonmomenta,with95%efciencyperexchangedphotonpair.TheSTIRAPtechniquehasalsobeenusedinthemeasurementofveryweakmagneticeldsthroughatechniquecalled“Larmorvelocitylter”[ 100 ].Thetechniqueisbasedonthefactthatamagneticeldmixesthemagneticsublevelsofagivenstateandaffectsthemomentumtransferbetweentheselevels.Magneticeldsoftheorderof1Thavebeenmeasuredbythismethod.STIRAPhasalsofoundapplicationsinlasercooling,tomanipulatetheatomicwavepacketsresultedfromsubrecoillasercooling[ 104 , 105 ].Themomentumdistribu-tionofatomscooledbyvelocityselectivecoherentpopulationtrappinghaspeaksat+hkand)]TJ/F15 11.955 Tf 9.298 0 Td[(hk,bothwithwidthssmallerthanthephotonrecoilmomentumhk.UsingSTIRAP,Esslingeretal.[ 106 ]coherentlytransferredatomscooledbyvelocityselectivetrappingintoasinglemomentumstate,withsubrecoilmomentumspread.CavityquantumelectrodynamicsisanothereldthatbenetsfromtheuseofSTIRAPtechnique.Parkinsetal.[ 107 , 108 ]andParkinsandKimble[ 109 ]haveproposedtheuseofSTIRAPtocreatecoherentsuperpositionsofphoton-numberstatesbystronglycouplinganatomtoacavity.TheideaistomapthegroundstateZeeman

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34 coherencepreviouslypreparedbyanexcitationschemetothecavitymodebytwotime-delayedpulses,oneofwhichhascircularpolarization,andtheotherwithlinearpolarization.Theproposedschemeisreversible,alsoallowingforthemappingofthecavity-modeeldsontotheground-stateZeemancoherence. 2.3.4 AdiabaticPassagebyLightInducedPotentials–APLIPWhenanintenselaserpulseinteractswithamolecule,itspotentialsurfacesaredeformedinresponsetothestrongexternalperturbation.Ifonecanappropriatelycontrolthedynamicsofthisdeformationitispossibletotransferadiabaticallyawavepacketfromaninitialvibrationalstatetoachosentargetstate.ThisideaisusedintheAPLIPtechniquedevelopedbyGarrawayandSuominen[ 110 ]toachieveefcientandrobustpopulationtransferbetweenvibrationallevelsintwocoupledelectronicstates.FollowingSolaandcoworkers[ 111 ],forathree-statesystemconsistingofpotentialsurfacesV1,V2andV3,usingtheBorn-Oppenheimerandrotating-waveapproximations,wehavethefollowingSchrodingerequation:ih@ @t0BBBB@1t2t3t1CCCCA=266664^T+0BBBB@U1x)]TJ/F24 7.97 Tf 10.494 5.699 Td[(1x;t 20)]TJ/F24 7.97 Tf 10.494 5.699 Td[(1x;t 2U2x)]TJ/F24 7.97 Tf 10.494 5.699 Td[(2x;t 20)]TJ/F24 7.97 Tf 10.494 5.699 Td[(1x;t 2U3x1CCCCA3777750BBBB@1t2t3t1CCCCA;where^TisthekineticenergyoperatorandUiarethedressedpotentials,U1x=V1x+h!1+!2,U2x=V2x+h!2andU3x=V3x.Togaininsightonthedynamicsofthesystemitisconvenienttotransformtheaboveequationtoarepresentationinwhichthedressedpotentialmatrixisdiagonal.Using^Rastheunitarytransformationtothisrepresentationgivesx;t=^Rx;tand^R^U^R)]TJ/F24 7.97 Tf 6.587 0 Td[(1=0BBBB@ULIP+000ULIP0000ULIP)]TJ/F31 11.955 Tf 23.326 67.032 Td[(1CCCCA=^ULIP:

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35 Figure2: TheleftsideshowsthedressedpotentialsU1,U2andU3.Thesystemsatisesthe2-photonresonanceconditionfortheminimumofthepotentialsU1andU3.representsthedetuningnegativeherefromresonancewiththeinter-mediarylevel,U2.Therightsideshowstheadiabaticlightinducedpotentials,ULIP+,ULIP0andULIP)]TJ/F23 11.955 Tf 18.345 2.956 Td[(.TheinitialpopulatedLIPisULIP0correlatestoU1. ThenewSchrodingerequationisih@ @tx;t=^R^T^R)]TJ/F24 7.97 Tf 6.586 0 Td[(1+^ULIP)]TJ/F25 11.955 Tf 11.955 0 Td[(i^R@^R)]TJ/F24 7.97 Tf 6.587 0 Td[(1 @t!x;t;whereULIP+,ULIP0andULIP)]TJ/F23 11.955 Tf 21.333 2.956 Td[(arecalled“lightinducedpotentials.”Figure 2 showsadiagramofthedressedandlightinducedpotentials.Thedynamicsare`spatiallyadiabatic'whenthedynamiccouplingintroducedby^R^T^R)]TJ/F24 7.97 Tf 6.587 0 Td[(1issmall,allowingthistermtobeapproximatedas^T.Thedynamicsis`temporallyadiabatic'whenthelastterminEq. 2 ,i^R@^R)]TJ/F18 5.978 Tf 5.756 0 Td[(1 @t,canbeneglected.Whenbothspatialandtemporaladiabaticityisobeyed,theevolutionisfullyadiabatic,andthetransformedSchrodingerequationcanbewrittenasih@ @t0BBBB@+t0t)]TJ/F15 11.955 Tf 7.084 1.793 Td[(t1CCCCA=0BBBB@T+ULIP+t000T+ULIP0t000T+ULIP)]TJ/F15 11.955 Tf 18.345 2.956 Td[(t1CCCCA0BBBB@+t0t)]TJ/F15 11.955 Tf 7.084 1.793 Td[(t1CCCCA:

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36 TheaboveequationshowsthatifthesystemisinitiallyinaparticularLIPstate,itwillfollowthetimeevolutionofthatstateaslongasadiabaticconditionsaremaintained.Animportantpointtorememberaboutthelightinducedpotentialsisthat,inanalogytotheradiation-freecurvecrossingproblem,realcrossingsofthedressedpotentialsbecomeavoidedcrossingsfortheLIPs,whenadiabaticityisobeyed.Therepulsionbetweenthelightinducedpotentialsgrowswiththestrengthofthecoupling,whichinturnisdeterminedbytheintensityoftheeldandthedipolemoment.Thus,forpulsedlaserexcitation,astheeldamplituderisesthegapbetweenLIPsopensup,andastheeldvanishesthegapclosesdown.Thissimplefactcanbeusedinstrong-eldcontrolexperimentstomanipulatethelightinducedpotentialstoachievethedesiredtransitions.TheoreticalstudiesbyGiusti-Suzoretal.[ 112 , 113 ]andAubanelandBandrauk[ 114 ]havedemonstratedtheefciencyofLIPshapingincontrollingthephotodissociationofH+2.Othergroups,Verschuuretal.[ 115 ]andZavriyevetal.[ 116 ],havedemonstratedexperimentallytheLIP-shapingcontroloverthephotodissociationofH2.Intheseandsimilarstudiesanotherphenomenoncalled`bondsoftening'[ 113 ]hasbeenobserved.ThisphenomenoncanbeexplainedusingtheabovepictureofLIPgapdynamics.Thus,ifthemoleculeisinitiallyinitsgroundstate,andinteractswithastronglaserpulse,asthelasereldincreases,thegapbetweenLIPsopensandthelowerLIPcorrelatedtothegroundstateinitiallyispusheddown,whichdecreasestheheightofthepotentialbarriertodissociation.Asaresult,thestrengthofthemolecularbondisreducedandmoderateamountsofexcitationcanphotodissociatethemolecule,providedthegroundwavepacketreachesthegapatjusttherighttime.Inthisscenario,thetimingofthewavepacketiscrucial,andcanbecontrolledbyvaryingthetimedelaybetweentheexcitationandtheLIPpulses.Anoppositeeffectto“bondsoftening”is“bondhardening”or“vibrationaltrapping”wherethemoleculeisobservedtohaveanincreasedlifetime[ 117 ].

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37 Figure2: ASnapshotAshowsthelightinducedpotentialswhenthepulsecouplingU2andU3isturnedon.TherightwellofULIP0whichcorrelatestoU3islifted,andthewavepacketislocalizedintheleftwellwhichcorrelatestoU1.BInsnapshotBtherstpulseisalmostoverandthesecondpulseturnson,couplingU1andU2.ThiscausestheleftwellofULIP0torise,andtherightwelltofall.Thebarrierbetweenthetwowellshasdisappearedandthewavepacketevolvesintotherightwell.CInthissnapshotonlythesecondpulseison,andthewavepacketislocalizedintherightwellofULIP0. ToshowhowlightinducedpotentialshapingcanbeusedforadiabaticpopulationtransferinaschemesimilartoSTIRAP,considerthescenarioinFig. 2 .Thisdiagramcorrespondstoacounterintuitivepulsesequenceandnegativedetuningofthelaserpulses,thoughotherscenariosarepossible[ 111 ].ConsidertheinitiallypopulatedLIPasULIP0,whileU2isthedressedstatewiththelowestenergynegativedetuning.Usingacounterintuitivepulsesequence,thepulsecouplingU2andU3isturnedon.ThiscouplingcorrespondsroughlytotherightwellofULIP0,andthereforeasthepulseintensityincreases,therepulsionbetweenLIPspushestherightwellup.IneffectthewavepacketistrappedintheleftwellofULIP0.Astherstpulsedecreasesinintensityandthesecondpulseincreases,couplingU1andU2,therightwellgoesdownandtheleftwellrisesup.Atsomepointthepotentialbarrierbetweenthewellsdisappearsandthewavepacketevolvesadiabaticallyintotherightwell.Afterbothpulsesareover,thepotentialbarrierisrestoredandthewavepacketremainstrappedintherightwell,whichcorrelatestothenalstateU3.Inthiswayadiabaticpopulation

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38 transfercanbeeffectivelyandrobustlyachievedbyusingpotentialshapingwithstronglaserpulses. 2.4 QuantumControlforQuantumComputingIn1965,GordonMoore,co-founderofIntelCorporation,predictedthatthenumberoftransistorsoncomputerchipswoulddoubleabouteverytwoyears[ 118 ].Moore'slawhasprovedtobequiteaccuratetothisday,whenthelatestPentium4processorsintegratetensofmillionsoftransistorswithfeaturesofabout140nm.Ifthecurrenttrendcontinues,byyear2012thedimensionsofanintegratedtransistorwillapproachthoseofalargemolecule,andquantumeffectswillstarttoaffectitsfunctionality.Althoughquantumeffectsmaynegativelyaffectthefunctionalityofatinyelectroniccomponentdesignedforclassicaloperation,theycouldbeextremelybenecialifharnessedinadeviceoperatingonquantumprinciples.ThersttorealizethatquantumsystemsmightbeusedtoperformcomputationswerePaulBenioff[ 119 ]andRichardFeynman[ 120 ]inearly'80s.Goingonestepfurther,DavidDeutschwasrsttointroducetheconceptofquantumcomputationbasedonmanipulationofquantumsuperpositionstates[ 121 ].HeshowedthatquantumcomputingismorepotentthanclassicalTuringcomputingsinceitcanoperateinparallelonasuperpositionofinputstoproduceasuperpositionofresults[ 122 ].Thisisduetothefactthataquantumsuperpositionisactuallyasinglequantumstate.PowerfulquantumalgorithmsfordiscreteFouriertransform[ 123 , 124 , 125 ],largenumberfactorizationShor[ 126 ],anddatabasesearchGrover[ 127 ]takeadvantagespecicallyoftheabilitytoperforminparallelasuperpositionofcalculations.SinceDeutsch'sseminalwork,anumberofauthorshavecontributedtotheresearchrequiredforthecreationofquantumcomputers.Severalstudieshavebeenconcernedwithidentifyinguniversalquantumgates,onwhichanyquantumcomputationcanbedecomposed[ 128 , 129 , 130 ].Eventually,itwasfoundthatanyquantumcomputationcanbeexpressedintermsoftwo-qubitquantumoperations,likecontrolled-NOTCN,

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39 andsimplequbitrotations[ 129 , 131 ].Basedonthesendings,numeroustheoreticalandexperimentalgroupshavestartedtosearchforimplementationsoftwo-qubitandsingle-qubitgatesinvariousquantumsystems.Implementinguniversalquantumgatesinatomsandmoleculesrequiremanyofthesametechniquesasquantumcontrol.Forexample,wehavealreadyshownabovehowpartialSTIRAPcanbeusedtocreatepredeterminedsuperpositionsofstates,whichareessentialforquantumcomputing[ 75 , 98 ].Inanotherinterestingstudy,STIRAPwasusedtoperformqubitrotationina4-statesystem[ 132 ].NumericalsimulationsshowthatthequbitrotationprocedureinheritstheattributesofrobustnessandefciencycharacteristictoSTIRAP.Iontrapping,lasercoolingandcoherentpopulationtransferarekeyingredientsinanearlymodelforquantumcomputingwithtrappedionsproposedbyCiracandZoller[ 133 ].ExperimentalimplementationsofthismodelwithoneandtwoionshavebeendemonstratedbyMonroeandcoworkers[ 134 ].TheoreticalstudiesbyKielpinskietal.andMonroe[ 135 , 136 ]proposelargescalequantumcomputersbasedoneithertrappedionsoropticallattices.Inordertotransmittheresultsofcomputationsfromoneplacetoanotherinsidethecomputer,theseproposalsusecavityQED[ 137 ]orquantumteleportation[ 138 ].Themainobstaclefortheexperimentalimplementationsofquantumcomputingisthedestructiveeffectoftheenvironmentonthecoherenceofaquantumsystem.Thisphenomenoniscalledquantumdecoherenceanditseffectistodestroytherelativephasesbetweenthestatesinthequantumsuperposition,whichcarrytheinformationofthequantumcomputation.Fortunately,errorcorrectionschemeshavebeendeveloped,someofwhicharebasedonactivelycontrollingtheinteractionwiththeenvironmentbylaserpulses[ 139 ].AnothereffectivemodelforimplementingquantumcomputingisrelatedtotheuseofcavityquantumelectrodynamicsQEDtechniques[ 129 , 140 , 141 ].Inthismodela

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40 two-stateatomisentangledwiththecavityeld,andquantumoperationsareperformedbyeitherusingtheatomtoprovidenonlinearinteractionsbetweenphotons,orusingphotonstoprovidenonlinearinteractionsbetweenatomsindifferentcavities.AninterestingimplementationforquantumcomputinghasbeenproposedbySundermannanddeVivie-Riedle[ 57 ].Intheirschemeaqubitisformedbyapairofstatesbelongingtosomevibrationalmode.Foreachvibrationalmodeanassociatedqubitisdened.Singlequbitrotationsareachievedusingtechniquesofcoherentpopulationtransferbetweenvibrationallevelse.g.STIRAP,whereas2-qubitgatesareperformedusingshapedpulsesobtainedbyOCT.Arecentproposalforquantumcomputingusesstrongeldalignmenttoimplement1-qubitand2-qubitquantumgates[ 142 ].ThisstudyusesRamantransitionsinalinearlypolarizedeldbetweenrotationalstatestoimplementthedesiredoperations.Thetransitionsareoptimizedwiththehelpofastrongaligningeld.Averyattractivewayofperformingquantumcomputingisbyusingopticallattices[ 143 ].Opticallatticespresentanaturalwaytotrapneutralatomsbasedontheinteractionbetweentheirinduceddipolemomentsandtheelectriceldofthelaserlight.Varyingtheintensityandphaseofthelatticelaserbeamscancontroltheexactpositionsoftheatomsintheopticallattice.Whentwoatomsarebroughtclosetogether,theycanbecomeentangledduetodipole-dipoleinteractions.Theideaistoperformquantumlogicgatesbyusingthecontrolleddipole-dipoleinteractionbetweenatomsatdifferentsitesintheopticallattice[ 144 ].Brennenetal.[ 145 ]proposedanewsystemforimplementingquantumlogicgates:neutralatomstrappedinaveryfar-off-resonanceopticallattice.Pairsofatomsaremadetooccupythesamewellbyvaryingthepolarizationofthetrappinglasers,andthenanear-resonantelectricdipoleisinducedbyanauxiliarylaser.Acontrolled-NOTcanbeimplementedbyconditioningthetargetatomicresonanceonaresolvablelevelshiftinducedbythecontrolatom.Atomsinteractonlyduringlogical

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41 operations,andtherebythedecoherenceofthesystemisreduced.Similarschemeshavealsobeenreported[ 143 , 146 ].QuiterecentlyaschemehasbeenproposedinwhichatomicRydbergstatesareusedasaquantumregisterfordatastorage[ 13 , 147 ].Inthisproposal,optimalcontroltheoryisemployedtodesignshapedlasereldsthatachieveefcientstorageandretrievalofdata.ThelasersworkcoherentlywiththequantumsuperpositionofRydbergsatesaffectingonlythedesiredmembersofthesuperposition.Thereisarapidlyincreasingnumberofexperimentalsystemsinwhichquantumme-chanicaleffectsarebeingusedandinvestigatedforthepurposeofquantumcomputing.Examplesincludemanyopticaldevicesusinglasers,microwavecavities,andentangledphotonpairs,orimplementationsbasedonnuclearmagneticresonancewithmoleculesinliquidorsolidstate,trappedionoratomsystems,Rydbergatoms,quantumdots,superconductingdevicesJosephsonjunctions,SQUIDsandspintronicselectronspinsinsemiconductordevices.Althoughatpresentnoviableexperimentalimplementationofaquantumcomputerexists,theincreasedresearchactivityinthiseldbringshopethatinthenearfutureusefulquantuminformationprocessingtechnologiesmaybedeveloped. 2.5 SomeOtherApplicationsofQuantumControlTheadvancesinlasertechnologyoverthepasttwodecadeshavespurredaplethoraofapplicationsofquantumcontroltovariouseldsinatomicandmolecularphysics.Thenumberofapplicationsissolargethatthispresentationisbynomeansexhaustive,butattemptsonlytobrieyillustratesomeofthemostexcitingexperimentalandtheoreticalachievements.Theinterestedreaderwhowantstondmoreinformationabouttheeldofquantumcontrolmayrefertoseveralbookswrittenonthistopic[ 148 , 149 , 150 ].IntheremainderofthissubchapterIwillsketchsomeofthemostexcitingeldsinwhichquantumcontrolndsspectacularuses.Thereareofcoursemanyothereldsthatarenotincluded,butthisistobeexpectedduetotheoverwhelmingbreadthofresearchin

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42 quantumcontrol.Thepresentationwillbejustabriefbrowsingthroughsomesignicantresultsineacheld,andreferenceswillbegivenforfurtherinformation.Weshowedintheprevioussectionshowthereactivityofamoleculeanditsexcitationcanbecontrolled.Anothermolecularparameterthatcanbecontrolledisorientation.Molecularorientationisamajorfactorinenhancingchemicalreactioncross-section[ 151 , 152 , 153 ],incontrollingsurfaceprocessing[ 154 ]orcatalysis[ 155 ],andfornanoscaledesignbylaserfocalizationofmolecularbeams[ 154 , 156 ].AsuccessfulexperimentonthecontrolofmolecularorientationwasperformedbyDionetal.[ 142 ].Theyusedapairofintenselinearlypolarizedlasereldscombiningafrequency!anditssecondharmonicresonantwithavibrationaltransitiontoorientHCN.Duetopolarizability,bothevenandoddrotationalstatesareexcited,resultinginacontrolledrotationaleffect.Dionandcoworkers[ 157 ]employedanoptimalcontrolschemebasedonageneticalgorithmtodeterminetheoptimalpulseforachievingcontrolledmolecularorientation.Theoptimizationprocedureleadstoasuddenpulseeldwhichcanbeefcientlyproducedusinghalf-cyclepulses.AnotherstudybyHokiandFujimura[ 158 ]usedOCTtoachievecontrolovertheorientationofCO.TheirresultswerecomparedtotheeldscalculatedbyDionetal.,andcommonfeatureswereobserved.AnextensiontothetechniquesformolecularalignmentwasdevelopedbyIvanov,Spannerandcoworkers,whousedastrongeldwithrotatingpolarizationtospinmolecules[ 159 ].Tocreatearotatingeldoffrequency,theyusedtwofemtosecondcounterrotatingcircularlypolarizedelds,withfrequencies!L)]TJ/F15 11.955 Tf 12.708 0 Td[(and!L+.Torelativelyslowlyacceleratetherotationfrequency,,thetwocircularlypolarizedpulseswerechirpedinoppositefrequencydirections.Theyappliedtheentireprocesstoachlorinemolecule,whichwasacceleratedfrom0to6THzinlessthan50ps.AngularmomentumstatesashighasJ=420!wereexcited.Atthispointthecentrifugalforcesinsidethemoleculebrokethemolecularbond,andthemoleculedissociated.

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43 Thesamegroupproposedthebindingofbarenucleiwithstronglaserelds[ 160 ].Theydescribeatechniqueanalogoustothatofastreetperformerbalancingabroomonitsforehead.Althoughthegravityisalwaystippingthebroomfromvertical,s/hecanpreventitfromfallingbyrapidmovementsoftheheadinthedirectionoffalling.Inthebarenucleiscenario,theequilibrationtrickisperformedbyproperlyshapedlaserpulsesthatjostlethenucleiaroundeachotherbeforetheyhaveachancetoseparate.Itwasspeculatedthatbyusinganadditionalstrongandfastpulse,thenucleicouldbesmashedtogethertoachievenuclearfusion.Quantumcontrolhasalsobeenemployedforisotopeseparation.Charronetal.[ 113 ]showthatlinearlypolarizedlaserscontainingabasefrequencyanditssecondharmoniccanbeusedtoseparateisotopesthroughmolecularphotodissociation.Insubsequentstudy,LeibscherandAverbukh[ 161 ]usedOCTtoobtainoptimizedpulsesallowingfortheseparationof79Br2and81Br2.Theirschemeisbasedonthespatialseparationoftheexcitedmolecularwavepacketsduetotheirdifferentmassesandhencedifferentvibrationalperiods.Quantumcontrolofatomsconcernsthecreationandmanipulationofcoherentelec-tronicwavepackets.Krauseandcoworkers[ 162 ]haveshownthattransientnanostructureswithdynamicsonapicosecondtimescalecanbeproducedbycontrollingthequantumdynamicsofRydbergelectrons.Theshapedpulsesthatachievethisobjectivearedeter-minedusingOCTbasedonageneticalgorithm.Fordetectionofthesestructuresultrafastextremeultravioletdiffractionisproposed.StroudandNoel[ 163 ]demonstratedanexperimentinwhichanatomicelectroninterfereswithitself,inmuchthesamewayasabeamoflightintheYoung'sdouble-slitexperiment.Theyusedtwophase-coherentlaserpulsestoexciteasingleelectronintotwowavepacketsinitiallylocatedonoppositesidesoftheorbit.Thetwowavepacketsevolvedandspreaduntiltheycompletelyoverlapped.Athirdpulsewasusedtoprobetheresultingfringepattern.

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44 Weinachtetal.[ 164 ]usedshapedopticalpulsestoengineerRydbergwavepacketsincesium.Theirpulse-shaperwasacomputer-controlledliquid-crystalmodulator,whichallowedcontroloverthephaseandamplitudeofthedifferentspectralcomponentsofthepulse.TheresultingwavepacketwasanalyzedbymonitoringtemporalinterferenceviatheopticalRamseymethod[ 165 ].Areiterationonthesametheme,byWeinachtetal.[ 69 ],demonstratedanautomatedprocessforshapinganatomicwavepacket.Theyusedacomputer-controlledlasertoexciteacoherentstateinatomiccesium.Theshapeofthewavefunctionwasthenmeasuredandtheinformationfedbackintothelasercontrolsystem,whichreprogramsthepulseeld.Theprocesswasrepeateduntiltheshapeofthewavefunctionmatchedthatofatargetwavepacket.Usingavariationofquantumholography[ 166 ]toreconstructthemeasuredwavefunction,theshapedquantumstatewasconvergedwithinonlytwoiterationstothedesiredtarget.Inaninterestingstudy,KrauseandSchafer[ 15 ]presentamethodforcontrollingtheTHzemissionfromStarkwavepacketsinNa.TheyusedageneticalgorithmtondtheoptimalpulsethatproducesthedesiredfrequencyandintensityoftheTHzemission.Thedynamicsofthesystemandtheemittedradiationarefoundtobesensitivetothepulseparameters,asthecontrolspacehasarichandcomplicatedstructure.AnotherinterestingexperimentwasrecentlydemonstratedbyMonroeandcoworkers[ 167 ]whoexcitedaBeionintoaSchrodinger-cat-likestate.Theylaser-cooledatrapped9Be+iontothezero-pointenergy,andthenprepareditinasuperpositionofspatiallyseparatedharmonic-oscillatorstates.Thiswasachievedbyusingasequenceoflaserpulsesthatentangletheexternalmotionalandinternalelectronicdegreesoffreedomoftheion.TheSchrodinger-catstatewasdetectedbyquantuminterferencebetweenthelocalizedwavepackets.Thismesoscopicsystemmayprovideinsightintothequantumworldbyallowingstudiesofquantummeasurementandquantumdecoherence.Thesamegroupreportedthegenerationofnonclassicalmotionalstates,suchasthermal,Fock,andsqueezedstates,inatrapped9Be+ion[ 168 ].

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45 Twostudies,byBialynicki-Birulaetal.[ 169 ]andShapiroetal.[ 170 ],studiedtheformationofbothcircularandnon-circularTrojanstatesbyatomicelectrons.Trojanelectronicstatesarestable,non-spreadingwavepackets,whichareconnedbytheeffectivepotentialwellcreatedbytheCoulombeldandtherotatingeldproducedbyastrongcircularlypolarizedlaser.Trojanwavepacketsareinterestinginthecontextofstudyingquantumchaos,andasatoolfornon-perturbativequantumcontrol[ 171 ].Otherinterestingobjectivesofatomicandmolecularcontrolarerelatedtocooling[ 105 , 172 , 173 , 174 ]andentanglementcontrol[ 175 , 176 ].Winelandandcoworkers[ 172 ]reportedlasercoolingofasingle9Be+ionheldinanrfPauliontraptothegroundstateofvibrationalmotion.Withtheuseofresolved-sidebandstimulatedRamancooling,thezeropointofmotionwasachieved98%ofthetimein1D,and92%ofthetimein3D.Hamannandcoworkers[ 177 ]trappedneutralCsatomsinatwo-dimensionalopticallatticeandcooledthemclosetothezero-pointmotionbyresolved-sidebandRamancooling.Sidebandcoolingoccursviatransitionsbetweenthevibrationalmanifoldsassociatedwithapairofmagneticsublevels,andtherequiredRamancouplingisprovidedbythelatticepotentialitself.QuiterecentlyHopkinsandcoworkers[ 178 ]presentedanactivefeed-backcontrolschemeforcoolingananomechanicalresonatordowntotemperatureswherequantumdynamicaleffectscanbeobserved.TherequiredtemperaturesareontheorderofafewmilliKelvin.ThefeedbackstrategyisbasedoncontinuousobservationoftheresonatorpositionOpticallatticeswerealsousedforstudyingquantumentanglement.OnestudybyDeutschandcoworkers[ 146 ]wasconcernedwithcreatingmultiparticleentanglementofneutralatomstrappedinopticallattices,byusingpairwisecontrolleddipole-dipoleinteractions.Excitationofthedipolescanbemadeconditionalontheatomicstates,allowingfordeterministicgenerationofentanglement.Thestudyalsopresenteddifferent

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46 protocolsforimplementingtwo-qubitquantumlogicgatessuchasthe“controlled-phase”and“swap”gates.Thesamedipole-dipoleinteractionwasusedbyHettichetal.[ 179 ]toentanglemolecules.Theirexperimentemployedcryogeniclaserspectroscopytoresolvetwoindividualuorescentmoleculesseparatedby12nanometersinanorganiccrystal.Thetwomoleculesunderwentastrongcoherentdipole-dipolecouplingthatproducedentangledsub-andsuper-radiantstates.Thisexperimentalschemecanbeusedefcientlytoopticallyresolvemoleculesatthenanometerscale,andtomanipulatethedegreeofentanglementamongthem.Ascenariothatusesmolecularentanglementandnanoscalestructuresforthepurposeofquantuminformationprocessinghasalsobeenproposed[ 180 ].AveryinterestingproposalbySchwabandcoworkers,fromtheLaboratoryforPhysicalSciencesatUniversityofMaryland,attemptedtherstviableschemetoproduceanddetectthepresenceofasuperpositionstateofamechanicaldevice.Theirideawastoentangleanano-mechanicalresonatorandacooper-pairbox,whichhasanintrinsicquantum-coherentbehavior[ 181 ].Bycapacitivelycouplingtheboxtothenanomechanicalresonator,anentangledstatewasformed.Detectionofthisentanglementcanbeaccomplishedbyobservingthedecoherenceandrecoherenceoftheboxquantumstate.Experimentsarenowunderdevelopmenttodemonstratethisdynamics.Theexperimentsareexpectedtobeanexcellenttest-bedformodelsofdecoherenceformacroscopicobjects.Semiconductorphysicsisanothereldthathaswitnessedimportantapplicationsofquantumcontrol.Theobjectivesofquantumcontrolinthisdomainhavebeenmainlyconcernedwiththeactivemanipulationofcarrierdynamicsformodifyingtransportproperties,optimizingTHZemission,orcreatingspecictargetsuperpositions.Notably,thesamequantumcontrolschemesthatwereemployedingas-phasemolecularandatomicdynamicswereemployedsuccessfullyinthesolid-statedomain.

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47 Forexample,aseriesofstudieshavereportedtheuseofmultiplepulseswithappropriatetimingsandrelativephasestoachieveexcitoncontrolinquantumwells[ 182 , 183 ].Othersimilarstudiesthatuseinterferencebetweenwavepacketscreatedbyproperlytimedpulses[ 184 , 185 , 186 , 187 ]havedemonstratedunequivocallythepertinenceoftheTannor-Riceparadigmtocontrolinsemiconductors.Someinterestingapplicationsofquantumcontrol,boththeoreticalandexperimental,havebeenconcernedwiththeTHzemissionfromwavepacketsinquantumwells[ 188 , 189 , 190 , 191 ].OnemethoddevelopedbyPotz[ 192 ]used,inamannersimilartotheBrumer-Shapiromethod,one-andtwo-photontransitionstocontrolthewavepacketemission.DupontandcoworkersreportedanotherimplementationbasedontheBrumer-Shapiroscheme.Theyusedone-andtwo-photontransitionstocontrolphotocurrentsinquantumwells[ 193 ]andinthebulk[ 194 , 195 ].Othertheoreticalandexperimentalstudieshavefocusedoncreatingwavepacketswithpredeterminedstructuresinquantumwellsandbulksemiconductors[ 196 , 197 , 198 ].Theseapplicationsgenerallyusedoptimalcontroltheorybasedongeneticorevolutionaryalgorithms.Anumberofgroupshaveappliedquantumcontroltechniquestoquantumdots.Bonadeoetal.[ 199 ]andPotz[ 200 ]supporttheideathatquantumdotsaremoreamenableforquantumcontrolduetotheirreduceddensityofenergylevels.Actually,thisreduceddensityofenergylevelscanevenallowfortheimplementationofadiabaticpopulationtransfertechniques.TheoreticalstudiesbyHohenesteretal.[ 201 ]andPazyetal.[ 202 ]showedthatSTIRAPcouldbesuccessfullyimplementedindouble-dotstructures.TheimplementationofsuchpowerfulcoherentpopulationtransfertechniquesinquantumdotsmayleadtotherealizationofultrafastTHzoptoelectronicswitchingdevices[ 203 ].

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48 2.6 ClosingWordsTheabovereviewofquantumcontrolapplications,althoughverybrief,bringstoattentiontheimportanceofthisparadigmforfuturescienticandtechnologicaldevelopments.Aslasertechnologyprogressesmanyotherapplicationsofquantumcontrolwillappear.Thedaywhenquantumcontrolwillallowforthecreationofnewmolecules,ofquantummachinesandcomputers,oreventhecreationofbiologicallyactiveagentstocorrectgeneticreactivitymaynotbetoofarinthefuture.Lookingbacktotheextraordinaryachievementsfromthepastfewdecades,these“scientico-fantastic”objectivesmaybecloseraheadthanwerealize.

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CHAPTER3CONTROLOFWAVEPACKETFOCUSINGBYCHIRPEDULTRASHORTLASERPULSES 3.1 Introduction–WavepacketsAnatomicormolecularwavepacketisanintrinsicquantumobjectrepresentingasuperpositionofstationarystatessolutionstothetimeindependentSchrodingerequation.Wavepacketsareparticularlyappealingbecausetheyrelatemorenaturallytoourclassicalintuitionofmotionasmovementalongatrajectory.Thus,whenthewidthpositionuncertaintyofawavepacketissmall,itsmotioncanbefairlywellapproximatedbytheclassicalmotionofaparticleplacedatthecenterofthepacket,withamomentumequaltotheaveragemomentumofthewavepacket.Asemiclassicalmethodforwavepacketpropagation,developedbyHeller[ 204 ],andextendedbyMllerandHenriksen[ 205 ],isbasedonexactlythisquantum-classicalanalogy.Whenanatomormoleculeinteractswithanultrashortlaserpulse,manyenergylevelsareexcitedduetothelargecoherentbandwidthofthepulse.Thissuperpositionofenergystatesisnothingelsebutawavepacket,whichwillcoherentlyevolveintimeasdictatedbytheHamiltonianofthesystem.Inthisscenario,theonlyelementthatcanbecontrolledisthelaserpulse.Byappropriatelyshapingtheexcitationpulse,onecanmanipulatethecompositionoftheexcitedwavepacket,suchthatitevolvesexactlyintothedesiredtargetstate.Withthisapproachitispossibletocontroldifferentaspectsofthemoleculardynamicssuchasvibrationsanddissociation,rotations,electronicexcitation,andevenchemicalreactivity. 3.2 FocusingControlAparticularlyimportantaspectofwavepacketcontrolisthecontroloffocusing,wheretheexcitinglasermustbeengineeredtofocalizethewavepacketataspecied 49

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50 locationandtime[ 58 , 59 , 206 , 207 ].Theabilitytofocalizewavepacketsmayproveusefulifonewantstocontrolchemicalreactivityusingapump-dumpTannor-Ricescheme[ 20 ].Insuchaschemearstlaserpulseexcitesaground-statewavepackettoanexcitedelectronicstate,andafteraproperlytimedevolutionontheexcitedpotential,asecondpulsedumpsthewavepacketbacktothegroundstate.Sincethetransitionamplitudebetweenelectronicstatesdependsontheradialcoordinate,byfocalizingtheexcitedwavepacketatthetimewhenthepumppulseacts,onecanensurethattheentirenucleardistributionfollowsthedesiredpathway.Otherstudiessuggestthatfocusingcontrolcanbeusedtoplaceatomsandelectronsatdesiredspatiallocationswiththecorrectmomenta,suchastocontroltheirchemicalbondingfortheformationofspecialdesignermolecules[ 26 ].Wavepacketfocusingalsohasimportantapplicationsintheeldofnanoscaleprocessingdeposition,etching,doping,etcofmaterials[ 154 , 208 ],inatomoptics[ 209 , 210 ],andpossiblyinthebuildingofsingle-electron“Y-branch”switches[ 211 ].Previousstudiesonfocusingcontrolhavedemonstratedthismethodinbothatoms[ 212 ]andmolecules[ 58 , 59 , 213 ].Theresultsofthesestudieshaveshownthatacontinuumwavepacket,createdwithatransform-limited,Gaussianpulse,spreadsintime.Incontrast,acontinuumwavepacketcreatedwithapositivelychirpedpulsehasaself-focusingtendency,suchthatinarstphaseitspositionvarianceoruncertaintydecreasesintime.Alternatively,ifaboundwavepacketisexcitedintheboundregionofanexcited-statepotentialinamoleculebyanegativelychirpedpulse,thewavepacketfocusesafteronerecoilfromtheturningpointofthepotential[ 59 , 214 , 215 ].Thesepredictionsofwavepacketfocusinghavenowbeenobservedexperimentally[ 60 , 214 , 216 ].Earlyworkonwavepacketfocusingdevelopedasimple,semi-classicalmodelforthefocusingmechanism.Toexplainthevariouscasespresentedabove,thismodellooksatthefocusingprocessintermsoftheindividualfrequenciesthatcomposeanexcited

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51 wavepacket.Forexample,ifthegenerationofanauto-focusingwavepacketisdesired,oneshouldexciterstthelowerfrequencies,whichareslowerhavelowermomentum,followedbythehighfrequencies,whichtravelfasterhavehighermomentum.Inthisscenario,focusingiscausedbythehigherfrequencycomponentscatchingupwiththelowerfrequencycomponents.Thesamemechanismisoperativeinboththecontinuumtermedpreviouslythe“cannon”,andtheboundregionofapotentialthe“paddle-ball”.Incontrast,toobtaina“reectron,”whichisaboundwavepacketthatfocusesafterreectionontheouterwallofthepotential,oneshouldemployanegativelychirpedpulse.Theexplanationisthat,thehigh-frequencycomponentsshouldbeexcitedrst,sincetheyhavetotravelfurthertoreachthepotentialwall,andthelow-frequencycomponentsshouldbeexcitedlatersincetheyhavetotravelashorterdistance.Thesimpleclassicalmodelwasquitesuccessfulinexplainingthesituationsencounteredinthestudiescitedabove.Butisthisreallythecompletepicture?Howaboutinterferenceeffects?Forinstance,whendecomposingawavepacketintoitsfrequencycomponents,onemusttakeintoaccountthequantumphaseofeachcomponent.Asaresult,whenthedifferentfrequencycomponentscatch-upwitheach-other,theyshouldalsobein-phaseinordertointerfereconstructivelyandtoproduceafocusedwavepacket.Sointerferenceeffects,whichareeminentlyquantummechanical,areimportant!Inaddition,foranaccuratepicture,onemustalsoconsidertheinitialmomentumdistributionoftheexcitedfrequencycomponents.Thatmayplayanimportantroleintheracebetweenthedifferentcomponents.Thegoalofthepresentstudyistoanalyzethemechanismofwavepacketfocusingingreaterdetail,bothanalyticallyandnumerically.Theresultsweobtainconrm,insomecases,thesimpleclassicalmodel,butcontradictitinothers.Thereasonforthiscontradictionistheomissionfromtheclassicalmodelofthetwofactorspointedabove:quantuminterference,andtheinitialmomentumdistribution.

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52 3.3 WavepacketFocusing 3.3.1 GeneralConditionsforWavepacketFocusingGivenageneralone-dimensionalsystemdescribedbytheHamiltonian^H=^p2=m+Vx,thetime-derivativeofthepositionuncertaintyisgivenbyseeAppendix E forderivationd dtx2t=2 m[hx^p+^pxi=2)-222(hxih^pi]=mrtwherertrepresentsthecorrelationbetweenthepositionandmomentumobservables.Theaboveequationshowsthatthenecessaryandsufcientconditionforawavepackettofocalizeorcontractisrt<0.Inthecasert>0,awavepacketwillspreadintime.Attimeswhenthecorrelationrtiszero,thewidthiseitherinalocalminimumoralocalmaximum.Next,weapplytheaboveformulastothecaseofalinearpotential,Vx=Vx0)]TJ/F25 11.955 Tf 11.955 0 Td[(x)]TJ/F25 11.955 Tf 11.955 0 Td[(x0:Inthiscase,itcanbeshownseeAppendix E thatthetimederivativeforrtisdrt dt=p20=m;whichimpliesrt=r0+[p20=m]t;wherer0istheinitialcorrelation.UsingthisexpressionandEq. 3 ,weobtainx2t=x20+[=mr0]t+[p20=m2]t2:Weseethatforr00,thewavepacketspreadscontinuously,whereasforr0<0,x2thasaminimumasafunctionoftime,andthepacketwillinitiallyfocusorcontract.Thefactthat,foralinearpotential,drt=dtisalwayspositiveimpliesthatforlargeenoughtimesawavepacketalwaysspreads.Theonlypossibilityforfocusingistohavea

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53 wavepacketthatinitiallyhasanegativeposition-momentumcorrelation,r0<0.Inthissituationthewavepacketwillcontractuntilthetimetfoc,whenthecorrelationrtiszero,i.e.,tfoc=mjr0j=p20;afterwhichitwillstartspreading.ThewidthofthefocalizedpacketattimetfoccanbeobtainedfromEq. 3 asx2tfoc=x20)]TJ/F25 11.955 Tf 11.955 0 Td[(r20=p20: 3.3.2 DynamicsofGaussianWavepacketsinaLinearPotentialTounderstandthedynamics,weseektoderiveanalyticalformulasforthepropaga-tionofaGaussianwavepacketonalinearpotential.Notethatinthismodelmuchoftherelevantphysicsisretained,sincealinearpotentialisoftenareasonableapproximationfortheFranck-Condonregionofdiatomicmolecules,orforwavepacketsmovingathighkineticenergies.WestartwithaGaussianwavepackethavingthefollowingformGx;t=expi hAtx)]TJ/F25 11.955 Tf 11.955 0 Td[(xt2+ptx)]TJ/F25 11.955 Tf 11.955 0 Td[(xt+st;wherextandptaretheexpectationvaluesofpositionandmomentum,respectively,andAtisacomplexparameter.Itisstraightforwardtoshowthattheuncertaintiesinpositionandmomentumaregivenbyx2t=h 4ImAt p2t=hReAt2+ImAt2 ImAt: Foralinearpotential,theintegrationofthetimeSchrodingerequationgives[ 41 ]At=A0 1+2A0=mt

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54 whereA0isAtatt=0.Usingthisrelation,weobtainthefollowingtimeevolutionforthepositionuncertaintyx2t=x20+hReA0 ImA0t m+p20t2 m2:EquationEq. 3 alsoimpliesaxedmomentumuncertaintyp2t=p20:BycomparingEq. 3 andEq. 3 wecanidentifyrtasrt=h=2ReAt=ImAt:WiththisresultandEq. 3 weobtainthatp2t=4jAtj2x2t=h2=4+r2t x2t:NotethatfromEq. 3 ,aGaussianwithrt=0isaminimumuncertaintystate.GivenaGaussianwavepacketwithanegativeinitialposition-momentumcorrelation,r0<0,wederivethetime,tfoc,whenthepacketgetsfocalizedrtfoc=0,andthewidthofthefocalizedwavepacket,x2tfoc.ByinsertingEq. 3 intoEqs. 3 and 3 weobtaintfoc=mx20jr0j h2=4+r20andx2tfoc=x20h2=4 h2=4+r20:Theequationfortfocshowsthattoobtainalargefocusingtimeforagivenx2tfoc,theinitialwidthofthewavepacketmustbelarge.Onceweknowtfocwecanimmediatelyndthefocusingpositionasxfoc=x0+t2foc=m;

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55 whereistheslopeofthepotentialandtheinitialmomentumisp0=0.Wenotethatgivenadesiredfocusingtime,tfoc,andfocusedwidth,x2tfoc,onecaninvertEqs. 3 and 3 toobtaintheinitialconditionsthatproducethesefocusingcharacteristics.Nextwecomputethemaximumfocusingtime,and,implicitly,distance,thatcanbeachievedstartingfromawavepacketwithagiveninitialuncertaintyx0.MaximizingtfocinEq. 3 withrespecttor0foraxedvalueofx0,givestmaxfoc=m hx20forr0=)]TJ/F15 11.955 Tf 9.299 0 Td[(h=2.Thisrelationshowsagainthatalargevalueoftfocrequiresadelocalizedpromotedstate.Suchastatecouldbeobtained,forinstance,byexcitationfromanexcitedvibrationalstate,whichhasalargeuncertainty.ThefocalizedwidthisobtainfromEq. 3 asx2tmaxfoc=x20=2:Thisequationshowsthatformaximumfocusingtimethewidthofthewavepacketisreducedexactlybyafactorofp 2=1:41. 3.4 GenericControlofExcitedWavepacketsInthissectionweexplorethepossibilitiesofcontrollingtheexcitationofawavepacketbyalaserpulse,intheweak-eldlimit.Letusconsiderthelaserexcita-tionofanelectronictransitioninamolecule,fromelectronicstate”toelectronicstate.”ForalasereldEt,withintheelectric-dipoleapproximationandrst-orderperturbationtheory,theexcitedwavepacketassumestheformattimes,t,whenthelaserpulsehasvanished[ 217 ]j2ti=exp)]TJ/F25 11.955 Tf 9.299 0 Td[(i^H2t=hj2i:;wherej2iisthepromotedstate,j2i=i hZ1dt0e)]TJ/F26 7.97 Tf 6.586 0 Td[(i0t0=hEt0expi^H2t0=hji;

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56 and^H2istheHamiltonianfortheexcitedstate.”HerejiistheFranck-Condonwavepacketji=12j1i;wherej1iisthestationaryelectronicstate”ofenergy0,and12istheprojectionoftheelectronictransitiondipolemomentonthepolarizationoftheelectriceld.Thegoalofourcontrolschemeistondthelaserpulse,Et,whichexcitesawavepacketthatevolvestoadesiredtargetstatejfoci,attimet=tfoc.Todothis,weinvertEq. 3 toobtainthepromotedstatethatmustbecreatedbythelaser,j2i=expi^H2tfoc=hjfoci:ConsideringtheFourierrepresentationofthelasereldEt=Z1~E!e)]TJ/F26 7.97 Tf 6.587 0 Td[(i!td!;andusingEq. 3 ,wecanwritej2i=2i hZ~E!EhEjijEidE=2i hZCEjEidE;whereweintroducedthecoefcientsfCEgasCE=~E!EhEji=h 2ihEj2i;where!E=E)]TJ/F25 11.955 Tf 12.018 0 Td[(0=h,andfjEigaretheeigenstatesof^H2.Thus,whentheexpansioncoefcientsoftheexcitedwavepacket,fCEg,areknown,thelasereldproducingthiswavepacketcanbecalculatedbyinvertingEq. 3 .Thewavepacketexcitedbythiseldisguaranteedtoachievethedesiredtarget,jfoci,atthetargettime,tfoc.

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57 3.5 FocusingofWavepacketsCreatedbyChirpedUltrashortPulses 3.5.1 ExcitedWavepacketDynamicsinaLinearPotential–TheoryHereweconsidertheexcitationofawavepackettoalinearpotentialsurface,viaalinearlychirpedlaserpulse.TheformofthepulseeldisassumedtobeEt=E0exp[)]TJ/F25 11.955 Tf 9.298 0 Td[(t2=2)]TJ/F25 11.955 Tf 11.955 0 Td[(i!0t)]TJ/F25 11.955 Tf 11.955 0 Td[(it2=2];wheretheinstantaneousfrequencyis!t=!0+t,andisthelinearchirp.WealsopresumethattheinitialstateisaGaussian,i.e.,x=Nexp[)]TJ/F15 11.955 Tf 9.299 0 Td[(x)]TJ/F25 11.955 Tf 12.131 0 Td[(x02=4x2g],andthatthedurationofthepulsecanbeconsideredshortwithrespecttothecharacteristictimeofthenuclearmotion.Thisshort-pulseapproximationallowsustoneglecttheactionofthekineticenergyoperator,whichgivestheevolutioninposition,andconsideronlytheactionofthepotentialenergyoperator,whichaccountsforthemomentumspaceevolution.UsingEq. 3 togetherwiththeconditionforresonantexcitation,h!0=Vx0)]TJ/F25 11.955 Tf 11.955 0 Td[(0,thepromotedpacketisgivenby[ 205 ]2x=i h~EVx)]TJ/F25 11.955 Tf 11.956 0 Td[(0 hx=expi hA0x)]TJ/F25 11.955 Tf 11.955 0 Td[(x02+s0; whereReA0=r0 2x20=2 2h =22+2ImA0=h 4x20=h 4x2g+2 2h1=2 =22+2: Theaboveequationsclearlycontradicttheclassicalmodeldiscussedinthesecondsectionofthischapter.Namely,anegativechirp<0excitesaself-focusingGaussianwavepackethasnegativecorrelation,r0<0.AnotherfactimpliedbyEq. 3 isthatthepromotedstateissqueezedcomparedtotheinitialvibrationalgroundstatexg>x0.Asaresultitsfocusingtime,tfoc,isrelativelyshort.Notealsothatfor

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58 =0i.e.,aconstantpotentialthepromotedstategeneratedbyanultrashortpulseisidenticaltothegroundstate,andfocusingisnotpossibleintheshort-pulselimit.ThetimetfocatwhichthewavepacketfocusescanbedeterminedusingEq. 3 andEq. 3 astfoc=2m02x2g hj=2j a2+=22;whereweusethenotations0=xg=h;anda=02+1=22:ThechirpthatrendersamaximumfocusingtimetmaxfocisfoundbycancelingthederivativeofEq. 3 withrespecttotime,toobtain=max=)]TJ/F15 11.955 Tf 9.298 0 Td[(2a:Forthisvalueofthechirp,tmaxfoc=mx2g=h 1+b;whereb=h2=x2g22:Theassociateduncertaintyinpositionisgivenbyx2tmaxfoc=1+2b 1+bx2g=2:Forageneralvalueofthechirp=kmax,usingEqs. 3 , 3 and 3 weobtainthefocusingtimeanduncertaintyastfoc=2k 1+k2tmaxfoc;andxt2foc=k+2b k2+b+bx2tmaxfoc;where,asexpected,tfoc
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59 NotethatEqs. 3 and 3 arequitesimilartotheresultsinEqs. 3 and 3 .Inthiscase,however,theyrepresenttheconnectionbetweentfocandxtmaxfocandthesetofparametersformedbythegroundstatewidthxg,thepulsewidth,andthepotentialslope.Intheseformulasthepulsechirpisnotaparameter,butinstead,thechirpissetimplicitlybytheconditionforamaximumtfoc=)]TJ/F15 11.955 Tf 9.299 0 Td[(2a.Inadifferentscenario,ratherthanmaximizingthefocusingtimewecanchoosetomaximizetheeffectofthefocusing.Todothis,wedeneafocusingparameter,f,astheratiobetweenthevarianceofthepromotedstate,x20,andthevarianceatthefocusingtime,x2tfoc.UsingEq. 3 ,fisgivenbyf=x20 x2tfoc=1+r0=h2:Ascanbeseeninthisequation,tomaximizef,wemustmaximizejr0ji.e.,theabsolutevalueoftheinitialcorrelationbetweenpositionandmomentumwithrespecttothechirp.RecallingEq. 3 ,andsettingthechirpas=)]TJ/F15 11.955 Tf 9.299 0 Td[(2kawehaveseeAppendix E 2r0 h=ReA0 ImA0=k k2+b+b;whereaandbaredenedinEqs. 3 and 3 above.Themaximumofthisratioisobtainedfork=p b=+b,whichimpliesthatfmax=1+[4b+b2])]TJ/F24 7.97 Tf 6.586 0 Td[(1Weremarkthatintheaboveformulas,bothtmaxfocandfmaxdependontheparameterb,whichimpliesthatallcaseswiththesamevalueof22areequivalent. 3.5.2 NumericalResultsfortheLinearPotentialInthissection,wepresenttheresultsofseveralnumericalsimulationsdesignedtotestthevalidityoftheanalyticalexpressionsderivedinsection 3.5.1 .Thenumericalre-sultsareobtainedviaadirectintegrationofthetime-dependentSchrodingerequation.Asamodelsystem,weconsiderthephotodissociationoftheICNmolecule.Inthismolecule,

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60 Figure3: Snapshotsofthewavepacketevolutionforthemaximumfocusingtimecase.xg=0:08a.u.,correspondstogroundvibrationalstateofICN,Pulseparameters:=125:69a.u.,=)]TJ/F15 11.955 Tf 9.299 0 Td[(1:4510)]TJ/F24 7.97 Tf 6.587 0 Td[(4a.u.,Predictedfocusingvalues:tfoc=142:2a.u.,f=1:08,andxfoc=0:02a.u.Fromlefttorightthepropagationtimesare:0,80,142,220,300,380,460,540,620,and700a.u. dissociationoccursalongtheI–Ccoordinate,andtheCNbondcanbeconsideredrigid.AlinearizationofoneofthepurelyrepulsiveexcitedelectronicstatesofICN[ 218 ]intheFranck-Condonregionleadsto=0:08a.u.,whereisdenedinEq. 3 .Thelinearapproximationisreasonablefortherst20fs[ 218 ]i.e.,about1000a.u..Atlongertimes,thelinearpotentialissimplyamodelpotential.ForthegroundvibrationalstateofICN,xg=0:08a.u.Thereducedmass,m,ofICNis21.6amu.Figure 3 showssnapshotsofthewavepacketdynamicsforthemaximumfocusingtimecaseforthegroundvibrationalstateofICN.Thepulsewidth,,is126a.u.,andthechirp,,hastheoptimalvalueof)]TJ/F15 11.955 Tf 9.299 0 Td[(1:4510)]TJ/F24 7.97 Tf 6.586 0 Td[(4a.u.Withtheseparameters,Eq. 3 predictsthattmaxfoc=142a.u.,andthelocationatwhichthefocusingoccursisxfoc=0:02.Ascanbeseeninthegure,thesepredictionsagreeperfectlywiththenumericalresults.Thefocusing,f,forthiscaseismodest,withf=1:08.WhiletheresultsinFig. 3 conrmthevalidityoftheanalyticalformulas,theyaresomewhatdisappointinginthesensethatthemaximumfocusingtimeoccursjust

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61 aftertheendofthepulse,andthefocusingdistanceisveryclosetotheFranck-Condonregion.ThereasonforthisdifcultycanbeseenbyexaminingEqs. 3 and 3 .Themaximumfocusingtime,foraxedslopeofthepotentialandmass,dependsonxgand.Themaximumpossibletmaxfocisobtainedas!1,inwhichcasetmaxfoc/x2g.Consequently,whenx2gissmall,tmaxfocissmall,andxfocmustalsobesmall.Withashortpulse,tmaxfocisfurtherreducedduetothesmallpulsewidth.Toobtainmoreimpressivefocusingresults,weconsiderxg=0:24a.u.,whichcorrespondstothevarianceassociatedwithanexcitedvibrationalstateofICNwith=4[ 205 ].Figure 3 showssnapshotsofthewavepacketpropagationfortmaxfoc,withpulseparametersaslistedinthecaption.Inthiscase,thefocusingtimeis2169a.u.,whichiswellaftertheendoftheexcitationpulse,andthefocusingpositionis4.78a.u.,whichiswellbeyondtheFranck-Condonregion.Thefocusing,f,is1.37.Onceagain,theagreementwiththeanalyticalformulasisperfect.WhilethefocusinginFigure 3 appearstobefairlylow,ifthesignofthechirpisreversed,asinFig. 3 ,thewavepacketdynamicsarecompletelydifferent.Thewavepacketspreadscontinuously,andthewidthatthetargettimeismuchgreaterthaninthenegativelychirpedcase.ThecomparisonofFigs. 3 and 3 showsthatthenegativechirpdoesindeedcounteractthenaturaltendencyofawavepackettospread.Figure 3 showssnapshotsofthedynamicsforthesamesituationasinFig. 3 ,exceptthatthechirphasbeenchosenaccordingtoEq. 3 ,suchthatthefocusingismaximized.Thefocusing,fis2.56andthewavepacketisfocusedattfoc=867:4a.u.,intheregioninwhichthelinearapproximationtotheexcited-statepotentialofICNisreasonable.Figure 3 showssnapshotsofthedynamicsforthemaximumfocusingcaseEq. 3–40 ,wherethepulsewidth,,is1000a.u.muchlongerpulsethaninFig 3 .Thefocusingtimeis1579a.u.,which,asexpected,isconsiderablyshorterthanthemaximumfocusingtime.However,thefocusing,fis2.70,whichissignicantlylargerthanthe

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62 Figure3: Snapshotsofthewavepacketevolutionforthemaximumfocusingtimecase.Herexg=0:24a.u.,correspondsto=4excitedvibrationalstateofICN.Pulseparameters:=172:6a.u.,=)]TJ/F15 11.955 Tf 9.298 0 Td[(7:7110)]TJ/F24 7.97 Tf 6.587 0 Td[(4a.u.Predictedfocusingvalues:tfoc=2169a.u.,f=1:37,andxfoc=4:78a.u.Fromlefttorightthepropagationtimesare:0,600,1150,1700,2169,2600,3000,3400,3800,and4200a.u. Figure3: SameasinFig. 3 ,exceptthatthesignofthechirpisreversed.Thatis,=7:7110)]TJ/F24 7.97 Tf 6.586 0 Td[(4a.u.

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63 Figure3: Snapshotsofthewavepacketevolutionforthemaximumfocusingcase.Herexg=0:24a.u.,correspondsto=4excitedvibrationalstateofICN.Pulseparameters:=172:6a.u.,=)]TJ/F15 11.955 Tf 9.298 0 Td[(1:6110)]TJ/F24 7.97 Tf 6.587 0 Td[(4a.u.Predictedfocusingvalues:tfoc=867:4a.u.,f=2:56,andxfoc=0:76a.u.Fromlefttorightthepropagationtimesare:0,350,600,867,1100,1500,1900,2300,2700,and3100a.u. previouscase.Onceagain,theagreementbetweentheanalyticalpredictionsandthenumericalresultsisexcellent. 3.5.3 ExcitedWavepacketDynamicsinanExponentialPotentialInthissectionwestudythedynamicsoftheexcitedwavepacketinthecaseofapotentialsurfacewithexponentialformVx=v0expf)]TJ/F25 11.955 Tf 15.276 0 Td[(bx)]TJ/F25 11.955 Tf 11.955 0 Td[(0g:Weconsideragainexcitationbyanultrashort,linearlychirpedlaserpulseasinEq. 3 .Intheshort-pulselimit,wealsoconsideralinearapproximationoftheexcitedpotentialintheFranck-Condonregion,withaslope,,givenbythederivativeofthepotentialatthecenteroftheregion.Withtheseapproximationsweareabletocalculatethepromoted

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64 Figure3: Thisisthemaximumfocusingcase.Herexg=0.24a.u.corresponds=4excitedvibrationalstateofICN.Pulseparameters:=1000a.u.,=-3.010)]TJ/F24 7.97 Tf 6.586 0 Td[(4a.u.Predictedfocusingvalues:tfoc=1580a.u.,f=2:70,andxfoc=2:54a.u.Fromlefttorightthepropagationtimesare:0,600,1100,1579,2100,2500,29003300,3600,and3900a.u. wavepacketas[ 205 ]2x=i h~EVx)]TJ/F25 11.955 Tf 11.956 0 Td[(0 hx=expi hA0x)]TJ/F25 11.955 Tf 11.955 0 Td[(x02+s0; whereReA0=r0 2x20=2 2h =22+2ImA0=h 4x20=h 4x2g+2 2h1=2 =22+2: Weareinterestedtostudythefocusingoftheabovewavepacketintheexcitedexponen-tialpotentialgivenbyEq. 3 .Asweshowedinsection 3.3.2 thefreeevolutionoftheexcitedwavepacketisdeterminedbytheinitialformofthepromotedstate.Inordertoassessthedifferentdynamicalscenarios,wemustparameterizethepromotedwavepacketinasuitablemanner.Todothis,wendasetofparametersthatuniquelydeterminesthe

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65 focusingparametersofthewavepacket,tfocandxtfoc,inalinearpotential.UsingEq. 3 ,wecanrewriteEqs. 3 and 3 astfoc=)]TJ/F25 11.955 Tf 10.494 8.088 Td[(m 2ReA0 ReA02+ImA02andx2tfoc=h 4ImA0 ReA02+ImA02:Theseformulasclearlyshowthatthefocusingvalues,tfocandxtfoc,dependdirectlyonthephaseandamplitudeofthequadraticcoefcient,A0,oftheinitialwavepacketEq. 3 .Hence,weparameterizethepromotedwavepacketbasedonthisvariables.Furthermore,tocontroltheexcitationprocess,wedeterminethepulseparameters,and,thatstartingfromagivenstateproduceapromotedwavepacketwithaspeciedcomplexA0.Consideragroundstatewavepacketx=Nexp)]TJ/F15 11.955 Tf 9.298 0 Td[(x)]TJ/F25 11.955 Tf 11.956 0 Td[(x02=4x2g;andapromotedGaussianpacketcharacterizedbyagivenA02x=expi hA0x)]TJ/F25 11.955 Tf 11.955 0 Td[(x02+s0:Theobjectiveistondthepulseparameters,and,thatleadtotheexcitationoftheabovepromotedpacket.Withthenotationsag=h=x2g,a0=jA0j,=argA0orA0=a0expi,u=2=h,and0=1=2,Eq. 3 becomesa0cos=u 2+02=xa0sin=ag+u0 2+02=y: Fromtherstequationabovewendthat2+02=u=x,whichinsertedintothesecondequationleadsto0=y)]TJ/F25 11.955 Tf 12.49 0 Td[(ag=x=r,wherer=y)]TJ/F25 11.955 Tf 12.49 0 Td[(ag=x.Replacing

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66 0asrintheequationReA02+ImA02=x2+y2=a20,weobtainasecondorderequation:+r2a2g)]TJ/F25 11.955 Tf 11.955 0 Td[(a202+2ua0r+u2=0;or+r2)]TJ/F25 11.955 Tf 11.955 0 Td[(k2a2g2+2uagr+u2=0;wherek=a0=ag.Inthecaseofapositivechirp>0,rwillbepositiveandthesolutionis=u agp k2+r2)]TJ/F15 11.955 Tf 11.955 0 Td[(1)]TJ/F25 11.955 Tf 11.955 0 Td[(r;0=r:Foranegativechirp<0,rwillbenegativeandwehave=)]TJ/F25 11.955 Tf 68.879 8.088 Td[(u agp k2+r2)]TJ/F15 11.955 Tf 11.955 0 Td[(1+r;0=r:IntermsofthephaseofthequadraticcoefcientA0rcanbewrittenasr=ksin)]TJ/F15 11.955 Tf 11.955 0 Td[(1 kcos:ReplacingthisresultinEqs. 3 and 3 ,weobtain=s agk2)]TJ/F25 11.955 Tf 11.955 0 Td[(ksin+1 uksin)]TJ/F15 11.955 Tf 11.955 0 Td[(1and=s ukcos agk2)]TJ/F25 11.955 Tf 11.955 0 Td[(ksin+1:Theaboverelationsprovidethedesiredrelationshipsbetweenthepromotedwavepacketandthelaserpulseparameters.BasedontheserelationswecancontroltheexcitationprocesstoobtainanydesiredpromotedGaussianwavepacket,inthelimitofultrashortpulses.WiththeseresultsweproceedtoperformnumericalsimulationsfortheexcitationofawavepackettoapotentialsurfacewithexponentialformseeEq. 3 .Butrst,wediscussafewaspectsrelatedtowavepacketfocusingonanexponentialpotential.

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67 3.5.4 WavepacketFocusinginanExponentialPotentialTherearetwoimportantfactorsthatcontributetothedynamicsofaGaussianwavepacketx=expi hA0x)]TJ/F25 11.955 Tf 11.955 0 Td[(x02+px)]TJ/F25 11.955 Tf 11.955 0 Td[(x0+s0onanexponentialpotential.Onefactorisrelatedtotheamplitudeofthequadraticcoefcient,A0.Thus,itcanbereasonedthataGaussianwavepacketforwhichReA0<0hasaself-focusingtendency,whereasapacketwithReA0>0hasaspreadingtendency.ThisfactisimpliedbytheanalyticalformulasforthelinearpotentialEq. 3–30 ,butcanalsoberationalizedonanintuitivebasis.Toseethis,letuswritethetermexpfiReA0x)]TJ/F25 11.955 Tf 11.955 0 Td[(x02ginEq. 3 aseiReA0=eiPxx)]TJ/F26 7.97 Tf 6.587 0 Td[(x0;wherePx=ReA0x)]TJ/F25 11.955 Tf 12.844 0 Td[(x0.Intheaboveequation,Pxappearsasapositiondependentmomentum,which,forReA0<0,ispositivewhenxx0.Thus,fornegativeReA0,thewavepacketcomponentsatpositionsxx0.Asaresult,thexx0componentstherightsideofthewavepacket,andthepacketwillfocalizeseeFig. 3 A.Incontrast,forapositiveReA0,Pxisnegativewhenxx0.Therefore,thewavepacketcomponentsatpositionsxx0.Thisimpliesthattheleftandrightsidesofthewavepacketwillmoveapart,andthewavepacketwillspreadseeFig. 3 B.Theotherimportantfactorthatinuencesthedynamicsofthewavepacketcomesfromthenon-uniformityoftheexponentialpotential.Normally,foradecreasingpotential,thepartsofthewavepacketwithxx0.Inalinearpotentialthisdifferenceofpotentialismaintainedatalltimes,andallpositionsexperiencethesameacceleration.Butforadecreasing

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68 Figure3: APanelApresentsawavepacketwhosesidesaremovingrelativelyto-wardseachother,leadingtothefocusingofthewavepacket.BInpanelBthesidesofthewavepacketaremovingrelativelyaway,andthepacketspreads. exponentialpotential,theslopeofthepotentialdecreasesasthepacketevolvesintime,andthereforethepotentialenergydifferencebetweenthetwosidesxx0decreasesintime.Fromenergyconservationconsiderations,thelossinrelativepotentialenergyistransformedintoadifferenceinkineticenergybetweentheleftandrightsides.Hence,intime,ifonlytheeffectofthepotentialisconsidered,therelativespeedbetweentheleftsideandtherightsideofthewavepacketincreases,andtheleftsidestartscatchingupwiththerightside.Thistranslatesintoafocusingofthewavepacket.Basedontheabovefacts,wecansaythatadecreasingexponentialpotentialtendstofocalizeawavepacket.However,theeffectoffocusingisreducedbecause,ingeneral,thedifferenceinpotentialenergybetweenthetwosidesofthewavepacketissmall. 3.5.5 NumericalResultsfortheExponentialPotentialNext,wepresentnumericalsimulationsfortheexponentialpotential.Inallsim-ulations,thepotentialhastheforminEq. 3 ,where0=)]TJ/F15 11.955 Tf 9.299 0 Td[(2:14a.u.andb=8:0.Also,withrespecttotheparameterizationofthepromotedGaussianwavepacket,wevaryindependentlyonlythephaseoftheA0quadraticcoefcient,.ThevariationoftheamplitudeofA0isconditionedbytherequirementtohaveaminimumwidthpulse,whichimpliesfromEq. 3 k=a0 ag=1+jcosj sin:

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69 InFig. 3 weshowapromotedwavepacketcreatedwithapositive>0chirp.Thephaseofthequadraticcoefcient,A0,is==4andag=6:66a.u.Fromthegure,wecanseethatthefocusingofthewavepacketbecomesmorepronouncedastheslopeoftheexponentialpotentialincreases,whichsupportstheobservationsintheprecedingparagraph.Wecanalsoseethatthepositivelychirpedpulsecreatesawavepacketthatinitiallyspreads.Intime,thespreadingiscounteractedbythepotentialasperourconsiderations,whichfocusesthewavepacket.IngureFig. 3 weshowapromotedwavepacketcreatedwithapositivechirp,where=0:42andag=6:66a.u.Hereweobserveatransitionfromanon-focusingpromotedwavepackettoafocusingpromotedwavepacketastheslopeofthepotentialincreases.Forlowerslopes,thepotentialdoesnothavesufcientstrengthtocanceltheinitialspreadingtendency,whichisduetothepositivechirp.Inthiscasetheinitialspreadingishamperedforawhilebythepotential,butnottotallycancelled,andnofocusingoccurs.Forhigherpotentialslopes,thefocusingeffectofthepotentialcanovercomethespreadingeffectofthepositivechirp,andfocusingoccurs.FigureFig. 3 presentssnapshotsofawavepacketcreatedbyanegativechirp,wheretheinitialphaseis=0:58.Inthiscaseweseethatthewavepackethasatearlytimesafocusingtendency,duetothenegativeReA0,asexplainedinthebeginningofthissection.Theexponentialpotentialonlycompoundsthefocusingeffect,whichresultsinashortfocusingtime,andaccentuatedfocusing.InFig. 3 wepresentanothercaseofawavepacketexcitedviaanegativelychirpedpulse,where=3=4.Inthissituationthefocusingisevenfaster,duetothemorenegativevalueofReA0=a0cos.Apparently,thefocusingofthewavepackethappensatlongertimeswhentheslopeofthepotentialincreases,whereonewouldexpectittohappenfaster.Infact,thefocusingonlyhappensatalargerdistancebutsomewhatshortertimesbecauseofthehighervelocitiesinducedbythehigherpotentialslope.

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70 Graphsfor=0:25,ag=6:66a.u. Figure3: Excitedwavepacketscreatedby>0pulseatdifferentmomentsduringtheirpropagation.Eachgraphshowsthewavepacketevery7500a.ufora15000a.u.interval.Fromtoptobottomv0hasthevalues0.08,0.12,0.15.

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71 Graphsfor=0:42,ag=6:66a.u. Figure3: Excitedwavepacketscreatedby>0pulseatdifferentmomentsduringtheirpropagation.Eachgraphshowsthewavepacketevery7500a.ufora15000a.u.interval.Fromtoptobottomv0hasthevalues0.05,0.1,0.15.

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72 Graphsfor=0:58,ag=6:66a.u. Figure3: Excitedwavepacketscreatedby<0pulseatdifferentmomentsduringtheirpropagation.Eachgraphshowsthewavepacketevery7500a.ufora15000a.u.interval.Fromtoptobottomv0hasthevalues0.1,0.18

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73 Graphsfor=0:75,ag=6:66a.u. Figure3: Excitedwavepacketscreatedby<0pulseatdifferentmomentsduringtheirpropagation.Eachgraphshowsthewavepacketevery7500a.ufora15000a.u.interval.Fromtoptobottomv0hasthevalues0.1,0.15,0.18.

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74 3.6 AnalysisoftheUltrashortPulseLimitAllthecalculationspresentedintheprevioussectionhaveassumedanultrashortlaserpulse,whichactsonatime-scaleshorterthanthetime-scaleforthenuclearmotion.Thisassumptionallowedustoneglectthewavepacketevolutionincoordinatespace,duringtheactionofthepulse.Inthissectionwedeterminetheconditionsthatguaranteethevalidityoftheshort-pulseapproximation.Tondthelimitsofapplicabilityforthe!0approximation,wecomputethefullformofthepromotedwavepacketinEq. 3 ,andcompareittothereducedformobtainedintheshort-pulseapproximationEqs. 3 and 3 .TocomputethetermeiH2tjiinEq. 3 itisconvenienttostartfromtherepresentationoftheground-statewavepacketEq. 3 inmomentumspace~p=ag)]TJ/F24 7.97 Tf 6.587 0 Td[(1=2expf1 hp2=agg;whereag=h=x2g.Usingtheaboveformula,wecancomputethemomentumspacerepresentationofj2i=ei^H2tjias2p;t=ag)]TJ/F24 7.97 Tf 6.586 0 Td[(1=2exp)]TJ/F15 11.955 Tf 10.494 8.088 Td[(p)]TJ/F25 11.955 Tf 11.955 0 Td[(pt2 4p2g+i ht2 2mp)]TJ/F25 11.955 Tf 19.083 8.087 Td[(t 2mp2)]TJ/F25 11.955 Tf 13.151 8.088 Td[(2t3 6m)]TJ/F25 11.955 Tf 11.956 0 Td[(V0t;wherep2g=h2=x2gandV0isthevalueoftheexcitedpotentialatthecenteroftheground-statewavepacket.ThisequationcanbeFourier-transformedtoobtain,uptoanormalizationconstant,2x;texpit 01 12x2t+xt 3)]TJ/F25 11.955 Tf 13.151 8.088 Td[(x 2)]TJ/F25 11.955 Tf 11.955 0 Td[(x+x 4+itV0)]TJ/F15 11.955 Tf 15.085 8.088 Td[(1 0x)]TJ/F25 11.955 Tf 11.956 0 Td[(xt2 4x20;wherext=t2 2m;=1 2mx40;0=1+t2 mx202:Weprefertheaboveformfortheback-propagatedgroundwavepacket,j2i,becauseitsmagnitudeandphaseareevident.Forthefollowingdiscussionwerewritetheabove

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75 formulaas:2xexpit 0[T3+T2)]TJ/F25 11.955 Tf 11.955 0 Td[(x+T1]+itVx0)]TJ/F15 11.955 Tf 15.086 8.088 Td[(1 0x)]TJ/F25 11.955 Tf 11.955 0 Td[(T02 4x20;whereT3=1=12x2t,T2=xt=3)]TJ/F25 11.955 Tf 12.217 0 Td[(x=2,T1=x=4,andT0=xt.Notethat,inthe!0approximation,T0,T1,T2,T3areall0,and0is1.Toillustratetherelativemagnitudesofthedifferenttermscontributingtotheaboveexpression,weprovideanumericalexampleforthecase2=0:0032a.u.,x20=0:2a.u.,Vx0=0a.u.,andm=58313:85a.u.Forthissetofvalues,2takestheform2xexpit:4210)]TJ/F24 7.97 Tf 6.587 0 Td[(17t4+:48510)]TJ/F24 7.97 Tf 6.587 0 Td[(6t2:01885)]TJ/F25 11.955 Tf 11.955 0 Td[(:000107x)]TJ/F25 11.955 Tf 11.955 0 Td[(x:0565+:53510)]TJ/F24 7.97 Tf 6.587 0 Td[(4x)]TJ/F15 11.955 Tf 9.298 0 Td[(1:25x)]TJ/F25 11.955 Tf 11.955 0 Td[(:48510)]TJ/F24 7.97 Tf 6.586 0 Td[(6t22:Thisexampleshowsthat,forthewavepacketphase,theT3termhasaverysmallmagnitude,andtheessentialdeviationfromthe!0approximationisgivenbythetermT2,whichcanbecomeimportantfor1000a.u.TheT0term,whichcentersthewavepacketattheinstantaneouspositionxt,canalsobecomeimportantfor1000a.u.However,ourcalculationsshowthatthemaintermwhichleadstothedeparturefromthe!0limitisT2inthewavepacketphase.Belowweprovidegraphsshowingresultsthatsupportthisanalysis.IngureFig. 3 wepresentthedependenceofthepromotedwavepacketEq. 3–22 onthewidthoftheexcitinglaserpulse,,forzerochirp=0.Asthewidthoftheexcitationpulseincreases,theamplitudeofthepromotedwavepacketdecreases,andthewavepacketismoredelocalized.Wealsonoticethatforlonger-stheoscillationsontherightsideofthepacketbecomemoresignicant,increasingthedeviationfromaGaussianform.Figure 3 showsthesamepromotedwavepacketsasinFig. 3 withtheT2terminEq. 3 neglectedT2=0.Thesideoscillationshavedisappeared,andthe

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76 Figure3: GraphsshowingthepromotedwavepacketEq. 3 asafunctionofthepulsewidthatzerochirp=0a.u..Allgraphshave2=.004a.u.,x20=0.2a.u.Inascendingorderofpeakamplitudesthe2valuesare:3840000,960000,60000,240000a.u. wavepacketshaveaclearGaussianform.ThisresultsuggeststhattermT3ismainlyresponsiblefordeviationsfromtheGaussianform,whichwasobtainedinthe!0limit.NextwestudytheinuenceofthepotentialslopeintheFranck-CondonregiononthefullformofthepromotedwavepacketEq. 3 .InFig. 3 weshowthelimiting-sforwhichthebreakdownoftheshort-pulselimitoccursasafunctionofthepotentialslope.Foragiven,weconsiderasbreakdown,thepulsewidthforwhichtheheightoftherstsidekinkofthepromotedwavepacketisapproximately1%ofitspeakheight.Thegureshowsthatastheslopeofthepotentialincreases,thebreakdownofthe!0limithappensatevenshorterpulsewidths.Toshowthismoreclearly,inFig. 3 wepresentagraphofthesquareofthebreakdownpulsewidthasafunctionofthesquaredslope2.Anotherissuestudiedisthedependenceofthebreakdownofthe!0approxi-mationwiththechirpofthepulse,.Inthiscaseweconsiderthebreakdownlimitasthe

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77 Figure3: AplotofthesamepromotedwavepacketsasinFig 3 withT2inEq. 3–63 neglectedT2=0. Figure3: Graphsshowingthedependenceofthebreakdownwiththeslopeoftheexcitedsurfacepotential.Allgraphsdisplaysmallkinksontheright-bottomside.Weconsiderasbreakdownatagiventhepulsewidthforwhichtheheightoftherstkinkbecomeslargerthan1%ofthepeakheightofthepromotedwavepacket.Inascendingorderofpeakamplitudesthe2,2valuesare:0:7110)]TJ/F24 7.97 Tf 6.587 0 Td[(5,3840800.,0:20310)]TJ/F24 7.97 Tf 6.587 0 Td[(4,1920800,0:5810)]TJ/F24 7.97 Tf 6.586 0 Td[(4,960800,0:16510)]TJ/F24 7.97 Tf 6.586 0 Td[(3,480800,0:4910)]TJ/F24 7.97 Tf 6.586 0 Td[(3,240800,0:1410)]TJ/F24 7.97 Tf 6.587 0 Td[(2,120800,0:410)]TJ/F24 7.97 Tf 6.587 0 Td[(2,60800.

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78 Figure3:Graphshowingthedependenceofthebreakdownon casewheretheshapeofthewavepacketlosesitscentralsymmetry.Herethechirpsareexpressedintermsofareferencechirp0=2=m,whichdependsonthewidthofthepulseandtheslopeofthepotential.TheideaistocomparetheaveragephaseshiftintroducedbythetermT2inEq. 3 ,whichis23=m,withtheaveragephaseshiftduetothechirp,2=2.Thetwoshiftsareequalwhen=2=m.Hence,weintroducearelativechirprasr= 2 2m:Figure 3 showsnumericalresultsforthedependenceofthebreakdownonrfordifferentvaluesof.Asobservedhere,evenlongpulsescanbetreatedintheshort-pulseapproximationifthechirpofthepulseisadjustedappropriately.Fig. 3 showsthedependenceofthebreakdownvalueofronthesquareofthepulsewidth.Itisinterestingtonotethefactthatafterarstrapidincrease,rseemstodecreasetowards0forlongerpulses.InFig. 3 weshowresultsforthecasewhererisreducedby33%belowthebreakdownvalue.Theirregularityintheshapeofthepromotedwavepacketdevelopsrapidly.Theeffectismorevisibleforlargerpulsewidths.

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79 Figure3: Graphsshowingthedependenceofthebreakdownwiththechirpofthepulse.Allgraphshave2=.004a.u.,x20=0.2a.u.Inascendingor-derofpeakamplitudesther,2valuesina.u.are:.82,7640800,.2,3820800,.6,1960800,.2,960800,.9,480800,.4,240800,.5,120800,.7,60800. Figure3:Graphshowingthedependenceofthebreakdownron

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80 Figure3: Graphsshowingtheabovepromotedwavepacketsinthecasechirpisre-ducedby33%. 3.7 DiscussionoftheResultsfortheBreakdownDependenceonChirp.Explanationfor theMechanismofProducingSelf-FocusingWavepacketsbyNegative-ChirpPulses. Forboththelinearpotentialcaseandtheexponentialpotentialcase,wefoundthatnegativelychirpedpulsesexcitewavepacketswithaninitialself-focusingtendency.Wealsosawintheprevioussectionthatachirpedpulse,althoughlong,canpreservetheresultsobtainedinthe!0limit.Inthissectionwepresentanexplanationforbothobservations.WestartfromequationEq. 3 forthepromotedwavepacket.Next,wedenethenotionof`instantaneousj2ti-contribution'tothepromotedstateinEq. 3 .Forthisweconsideradivisionoftheinterval[)]TJ/F25 11.955 Tf 9.298 0 Td[(T,T]inwhichthepulse,Et,hassignicantamplitude.Lett0=)]TJ/F25 11.955 Tf 9.299 0 Td[(T
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81 andwehaveEt=XkPkt:Usingtheaboveequationwecanwritej2i=iZ1dt0Et0j2t0i=XkjIki;wherejIki=iR1dt0Pkt0j2t0i.Whenthespacingofthetimedivision,t,isverysmallwhere`small'iscase-dependent,wedenethe`instantaneousj2ti-contribution'atmomenttktothepromotedstatej2iasjIki=iZ1dt0Pkt0j2t0i:Notethattheaveragemomentumofaninstantaneouscontribution,Ikx,atmomenttk,isthesameastheaveragemomentumof2x;tk,sincethepulsePkisextremelyshort.Hence,theaveragepositionandmomentumofanIkareforalinearpotentialofslope,andx0=0,p0=0xtk=t2k 2mandptk=)]TJ/F25 11.955 Tf 9.299 0 Td[(tk;wherethenegativeresultforptkisduetothetimereversedpropagationofjiinEq. 3 .Next,weshowhowanextrafactorofe)]TJ/F26 7.97 Tf 6.586 0 Td[(itintheintegrandinEq. 3 modiestheenergyoftheinstantaneouscontributionvectorjIki.ConsideradecompositionofjiinEq. 3 inthenon-degeneratecontinuumbasisof^H2:ji=XEEjEi:ThenwehavejIki=XEuEjEi;whereuE=iEZ1dt0Pkt0eiEt0;

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82 andtheenergyofjIkiisE=DIkj^H2jIkE=XEEjuEj2:Iftheinitialintegrandacquiresanextrafactorofe)]TJ/F26 7.97 Tf 6.586 0 Td[(it,thenewexpansioncoefcientsforjIkiareu0E=iEZ1dt0Pkt0eiE)]TJ/F26 7.97 Tf 6.586 0 Td[(t0=uE)]TJ/F25 11.955 Tf 11.955 0 Td[(:AsaresultthenewenergyisE0=DIkj^H2jIkE=XEEjuE)]TJ/F25 11.955 Tf 11.955 0 Td[(j2=XEE+juEj2=E+;whereweusedthecompletenessofthebasisPEjuEj2=1.Hence,e)]TJ/F26 7.97 Tf 6.587 0 Td[(itmodiestheenergyoftheinstantaneouscontributionjIkiby.SincethekineticenergyofjIkiisnotmodiedbyanadditionalintegrandfactore)]TJ/F26 7.97 Tf 6.586 0 Td[(itisindependentofx,itspotentialenergymustchange.Butachangeinpotentialenergyisequivalenttoachangeinaverageposition,foralinearpotential.For>0thepotentialenergyofjIkiincreases,and,assumingadecreasingpotentialdV=dx<0,theinstantaneouscontributionIkx=hxjIkiisshiftedtowardslowerxhigherpotential.For<0thepotentialenergydecreases,andIkxisshiftedtowardshigherxlowerpotential.Goingbacktotheformulafor2x;tEq. 3 ,andmakingtheapproximationsT3=0,T1=0and0=1,weobtain2x;texpit[T2)]TJ/F25 11.955 Tf 11.955 0 Td[(x]+itVx0)]TJ/F15 11.955 Tf 13.151 8.088 Td[(x)]TJ/F25 11.955 Tf 11.955 0 Td[(T02 4x20:Nextweconsideratimedivisionoftheintervalofinterest,withaspacingsuchthatduringeachsubpulse,Pk,T2andT0canbeconsideredconstant.Intheseconditions,thetermeitT2producesashiftoftheaveragepositionofaninstantaneouscontributionIkbyx=T2tk=downhillT2>0and>0.Fortwosymmetricwithrespecttot=0moments,tkand)]TJ/F25 11.955 Tf 9.298 0 Td[(tk,theinstantaneouscontributionsIkxandI)]TJ/F26 7.97 Tf 6.586 0 Td[(kxhaveequalandoppositemomenta,andareshiftedbythesameamount)]TJ/F25 11.955 Tf 9.298 0 Td[(T2tk=because

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83 T2)]TJ/F25 11.955 Tf 9.299 0 Td[(tk=T2tk.Note:Inthepreviousresultweconsider=0,otherwisethechirpwillalsointroduceshiftsoftheinstantaneouscontributions.Becausethesetwoinstantaneouspacketshavethesamefrequencyaveragemomenta,theywillinterfereandproducepositiondependentoscillations.ThisexplainsthefeaturesinthegraphsinFig. 3 ,whereis0.Whenalinearpositivechirp,>0,ispresent,withtheapproximationt2=2tk=2t=ktduringeachsubpulsePk,theinstantaneouscontributionIkxwillbedisplacedduetoanextrafactore)]TJ/F26 7.97 Tf 6.587 0 Td[(it2=2e)]TJ/F26 7.97 Tf 6.587 0 Td[(ikt.Thedisplacementisgivenbyx=)]TJ/F25 11.955 Tf 9.299 0 Td[(k=theshiftinpotentialenergy/theslopeofthepotential.Forexample,aninstantaneouscontributionIux,atnegativetimetu<0,willbedisplaceddownhillbecauseu=tu=2<0.Incontrast,aninstantaneouscontributionIvx,atpositivetimetv>0,willbedisplaceduphillbecausev=tv=2>0.Consequently,ifthequadraticchirpisstrongenough,thecontributionswithsimilarfrequenciesdonotoverlap,asinthezero-chirpcase,andnointerferenceoccurs.ThisexplainsthegraphsinFig. 3 ,whereevenwavepacketscreatedwithlongerpulsesarequitewelltrimmedbytheeffectofthechirp,andhaveareduceddeviationfromthe!0limit.Figures 3 and 3 presentnumericalsimulationsthatdemonstratetheagree-mentwiththe!0limit,eveninthecaseofapulselengthofapproximately40fsor1581a.u.!Aspredictedintheshort-pulselimit,awavepacketexcitedwithanegative-chirppulsewillfocalizeFig 3 ,whereasawavepacketpromotedviaapositive-chirppulsespreadsFig. 3 .Wecannowexplainthisbehaviorindetail.Sincetheinstantaneousj2tkisinEq. 3 areobtainedthroughtime-reversedpropagationofthegroundstatej2ti=ei^H2tji,fort<0theiraveragespeedisdownhill,andfort>0theiraveragespeedisuphillsincex=0istheturningpoint,anditisreachedatt=0,seeFig. 3 .Alsonotethattheaveragemomentumofaninstantaneouscontribution,IkxEq. 3 ,atmomenttk,isthesameastheaveragemomentumofj2tki,sincethepulsePkis

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84 Figure3: Promotedwavepacketcreatedby=)]TJ/F15 11.955 Tf 9.298 0 Td[(6:4410)]TJ/F24 7.97 Tf 6.587 0 Td[(4a.u.pulseatdifferenttimesduringitspropagation.Otherparameters:2=0.42a.u.,x20=0.2a.u.,=1581a.u.Thetimesaregivenbyn750a.u.,wherefromlefttorightnis0,3,4,5,6,7,8,9,10,11,12. Figure3: Promotedwavepacketcreatedby=6.4410)]TJ/F24 7.97 Tf 6.586 0 Td[(4a.u.pulseatdifferenttimesduringitspropagation.Otherparameters:2=0.42a.u.,x20=0.2a.u.,=1581a.u.Thetimesaregivenbyn750a.u.,wherefromlefttorightnis0,3,4,5,6,7,8,9,10,11,12.

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85 Figure3: Foratime-reversedevolution,theparticlestartsinstate2withamomentumdirecteddownhill,itbacksupthehilltoreachtheturningpointx=0,andthenitbacksdowndownhillwithamomentumdirecteduphill. extremelyshort.Hence,forapositivechirp,thenegativetimet<0instantaneouscontributionshaveamomentumdirecteddownhillandaredisplaceddownhilltowardsx>0,whilethepositivetimet>0instantaneouscontributionshaveamomentumdirecteduphillandaredisplaceduphilltowardsx<0.SincethepromotedwavepacketisthesumofalltheinstantaneouscontributionsEq. 3 ,onecanseehowapositivelychirpedpulseproducesapromotedwavepacketthattendstospreadintimebecauseitssidesmoveawayfromeachotherseeFig. 3 B.Foranegativechirp,thet<0instantaneouscontributionsaredisplaceduphilloppositethanfrompositivechirpandhaveanaveragemomentumdownhill,andthet>0instantaneouscontributionsaredisplaceddownhillandhaveanaveragemomentumuphill.Thismeansthatthetwosidesofthepromotedwavepacketaremovingtowardseachother,andthepromotedwavepackethasaninitialself-focusingtendencyseeFig. 3 A. 3.8 ConclusionsInthischapterwepresentedatheoreticalstudyofthecontrolofwavepacketfocusingbyultrashortpulses.Theabilitytocontrolwavepacketfocusingisimportantfor

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86 theachievementofspecicexcitationpathways,aswellasforthecontrolofchemicalreactivity.Intheshort-pulselimitandalinearpotential,wehavederivedimportantanalyticalresultsthatpredictthefocusingtime,andthewidthofthefocalizedexcitedwavepacket,basedontheparametersofthepulse.Toverifythevalidityoftheseresultsweperformednumericalsimulationsforarealisticmolecularsystem,ICN.Theresultsshowexcellentagreementbetweenthesimulationandtheanalyticalequations.Theshort-pulselimitwasobservedtoholdforallcasesstudiedhere.Wehavealsoderivedgeneralequationsforthecontrolofwavepacketfocusing,whicharevalidbeyondtheultrashortpulselimit.However,inthegeneralcase,theoptimumlaserpulsemayhaveacomplicatedformdependingontheconstitutionofthetargetintermsoftheeigenstatesoftheexcitedpotential.Furthermore,westudiedthefocusingofwavepacketsinexponentialpotentials.Importantfactorsthatcontributetothefocusingdynamicshavebeenidentied.Ana-lyticalformulasforthecontroloflaserexcitationofawavepackethavebeenderived.Oursimulationsfortheexponentialpotentialareintotalagreementwiththeanalyticalformulasandwiththetheoreticalconsiderationspresented.Thebreakdownlimitfortheultrashortpulseapproximationhasalsobeeninvesti-gated.Wendthatthebreakdownlimitdependsonthelengthofthepulse,thepotentialslope,aswellasthepulsechirp.Thechirpofthepulsewasfoundtobeanimportantfactorthat,forappropriatevalues,mayassurethevalidityofthe!0limitevenforlongpulses.Attheendofthechapterwestudytheintricateinterferencemechanismbywhichthepulsechirpcancontrolthefocusingofawavepacket,aswellasenforcethevalidityoftheultrashortpulselimitevenforlongpulses.Ourtheoreticalconsiderationsareinfullagreementwiththeresultsobtainedinnumericalsimulations.

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CHAPTER4HALF-CYCLEPULSES–OVERVIEW 4.1 IntroductionOverthepasttwodecadesthephysicsofultrafastphenomenahaswitnessedrapidgrowth,andhasbecomeanincreasinglyimportanteldofscienticresearch.Majoradvancesintheultrashortlaser-pulsetechnologyhaveled,inthersthalfofthe1990s,tothedevelopmentofnewsources,thatemitanearlyunipolarelectromagneticpulsetheelectriceldmainlyliesinonehalfoftheopticalcycle,genericallytermeda“halfcyclepulse”HCP[ 219 , 220 ].Forafreelypropagatingelectromagneticeldthetermhalf-cyclepulseisnotstrictlycorrectsince,accordingtotheMaxwell'sequations,thetimeintegraloftheeldoverthepulsedurationmustbezero[ 221 ].Experimentalmeasurementsofhalf-cyclepulses[ 222 , 223 ]showthattheyconsistofashortsteeplobe,ofapproximately0:5ps,followedbyalongandshallowtailofoppositepolaritythatcanlastupto70ps.However,theeffectofthetailisnotsignicantincasesinwhichtheinteractionbetweenaHCPandanatomoccursonatime-scaleoftheorderoftheclassicalorbitalperiod[ 221 ].Insuchcases,onlythesteepinitiallobeofthepulseimpactsthedynamics,andtheHCPcanbeaccuratelyapproximatedasaunipolar,ultrafastpulse,withadurationHCPapproximatelyequaltothedurationoftheshort,steeplobe.Conventionalacelectromagneticpulsescannottransfermomentumwheninteractingwithafreeelectron,sincethetotalmomentumtransferredoveranentireperiodaveragestozero.ForanatomicelectroninteractingwithanEMpulse,themomentumtransfercanonlyoccuriftheatomiccoreparticipatesintheinteraction.Accordingly,intraditionalspectroscopicmethodssuchasphotoionization,theelectronionizationoccursonlywhentheelectronicdistributionhasahighprobabilitydensitynearthenucleus.Thisrestriction 87

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88 makesitdifculttoinvestigatethepropertiesofatomicRydbergstates,inwhichtheelectronicwavefunctionisconcentratedatlargeradialdistances.Incontrast,duetotheirunipolarnature,HCPscanchangethemomentumofanatomicelectron,andthereforetheycanexciteorionizetheelectronanywhereinitsorbit[ 5 ].Owetothischaracteristic,HCPpulseshaveimportantapplicationsinthephysicsofRydbergatoms,wheretheyareusedextensivelytoobserveandcontrolthedynamicsofelectronicwavepackets.Powerfulexperimentaltechniques,suchastime-delayspectroscopy[ 5 , 6 ]andimpulsivemomentumretrieval[ 7 , 8 ],havebeendevelopedbasedonthespecialpropertiesoftheHCPs.AbetterunderstandingofthedynamicsofRydbergstatesiscrucialforareasofstudysuchasquantum-classicalcorrespondenceprinciple,quantumchaos,andthenon-perturbativeresponseofatomsinintenseexternalelds.Asidefromtheirutilityinprobingatomicproperties,HCPpulsesalsoprovetobeusefulasatoolforquantumcontrolincreatingandshapingexoticwavepackets[ 8 , 9 , 10 , 11 , 12 ],controllingtheTHzemissionfromStarkwavepackets[ 15 ],andpossiblyperformingselectivechemistryonlargemolecularsystems.RecentworkhasproposedusingRydbergwavepackets,controlledbyengineeredHCPlaserpulses,asaphysicalrealizationofqubitregisters[ 16 , 13 , 17 ],insupportofquantumcomputing. 4.2 TheoreticalandExperimentalBackgroundAnumberofexperimentalandtheoreticalstudiesinthelastfewyearshavefocusedontheexcitationandionizationofRydberg-StarkatomicstateswithHCPelectromag-neticpulses.Theregimesinvestigatedarecharacterizedbyadurationofthepulse,Tp,shorterthan,oroftheorderof,theKeplerorbitalperiod,Tn.IonizationbyHCPsisofparticularinterestsinceitrepresentsanintermediateregimebetweentheionizationbyelectromagneticradiationphotoionization,andionizationbycollisionwithhighenergyparticles,wheremomentumtransferismediatedbytheshortunidirectionalelectriceldofthepassingprojectile.DuetothepeculiarcharacteristicsoftheHCPs,newfeatures,

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89 notpresentinthecaseofordinaryelectromagneticpulses,havebeenrevealed.Examplesarethebroadionizationthresholdsandtheunconventional1=n2or1=nintheimpulsiveregimelimitscalinglawofthethresholdeld[ 224 , 225 ].EarlystudiesconcerningtheinteractionofHCPsandextremeStarkstatesinhydrogen,haverevealedamanifestasymmetrybetweenthetotalionizedfractionsoftheuphillStark-blueshiftedanddownhillStark-redshiftedstates.Thisasymmetrywasexplainedinrelationtothelocalizationpropertiesofthesestates[ 225 , 226 ].AnumberofstudiesontheinteractionofHCPswithextremeparabolicstateshaveinvestigatedthedependenceoftheexcitationandionizationprocessesonthepropertiesofthepulse.Inthisdirection,onestudyinvestigatingbound-boundtransitionscausedbyHCPsfoundevidenceofsocalled`Starkbeats'[ 227 ],whichareoscillationsinthetransitionprobabilityasafunctionoftheHCPintensity.Twolaterstudies[ 228 , 229 ],employingquantumandsemiclassicalcalculations,extendedtheseresultsbyshowingthattheionizationratesofthehydrogendownhillstatesoscillateasafunctionofboththeHCPintensity,andthedurationofthepulse.Uphillstateionizationprobabilitiesshowednodiscernibleoscillationsasafunctionofthepulseintensity.AnotherinterestingstudyconcernedtheionizationofellipticRydbergstatesbyhalf-cyclepulses[ 230 ].EllipticstatesareRydbergcoherentstatesinhydrogen-likeatoms,whichcanbepreparedbylaserexcitationincrossedelectricandmagneticelds[ 231 ].Theyarethemostgeneralstationarystatesofahydrogen-likeatom,includingasspecialcasestheStarkandcircularstates.Thestudyinvestigated,inparticular,thedependenceoftheionizationprobabilitiesontheeccentricityandorientationoftheellipsewithrespecttothepolarizationoftheeld.Theresultshaveshownthattheionizationisstronglydependentonthecharacteristicsoftheellipticstate,whereasthedependenceontheshapeoftheHCPisweak.Severaltheoreticalstudies[ 16 , 13 , 17 ]concernedthepossibilityofusingRydbergwavepacketsasquantumregisters.Thesestudiesshowthatinformationcanbestored

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90 intherelativephasesofthestatesinaquantumsuperpositionbyusingultrashortHCPs.Furthermore,theretrievalofinformationorquantumphasecanbeperformedbyusingsubsequentHCPpulses,oralternativelybyusingoptimallyshapedusingOCTTHzpulses.Ontheexperimentalside,HCPshavebeenusedinanumberofRydbergspec-troscopyexperiments.Anearlystudy[ 11 ]reportsatime-delayspectroscopyexperimentonhighly-excitedRydbergwavepackets.Ahalf-cyclepulseinitiallyexcitesawave-packetcomprisingvery-high-lyingRydbergstates.ThepropertiesofthiswavepacketarethenprobedbyusingasecondHCP,appliedafteravariabletimedelay.Itisfoundthatthesurvivalprobability,aftertheionizationwiththesecondpulse,exhibitspronouncedoscillations,whichareassociatedwiththequasiperiodicevolutionofthewavepacket.AnotherexampleofRydbergwavepackettime-delayspectroscopyisprovidedbyRamanetal.[ 5 ].InthisexperimenttheionizationofaRydbergpacketinCsisobservedasthewavepacketorbitstheatom.Inanotherexperiment,sub-picosecondhalf-cyclepulsesandasingle-shotimagingdetectorwereusedtomonitortheevolutionofanelectronicwavepacketincalcium[ 7 ].Thetime-dependentmomentum-spaceprobabilityofthewavepacketisobtainedbyanoveltechniquecalled“ImpulsiveMomentumRetrieval”IMR.Inarelatedexperiment[ 232 ]RydbergwavepacketsincalciumwereexcitedbyanHCPinthepresenceofastaticelectriceld.ThedynamicsoftheStarkwavepacketswasobservedbytheIMRtechnique.Thisallowsfortheobservationofinterestingdynamicprocessessuchasthefullprecessionoftheorbitalangularmomentum,theappearanceofalargeamplitudeoscillationinthedipolemoment,andaperiodicup-downasymmetryinthemomentumdistribution.Anexperimentalimplementationofakickedatom[ 233 ]hasalsobeendemonstratedwithhalf-cyclepulses.ThekickedatomwasrealizedbyexposingpotassiumRydbergwavepackets,withn388toasequenceofupto50HCPs.ThedurationoftheHCPs

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91 waschosentobeshorterthantheKeplerorbitalperiodofthepacket.TheRydbergatomsurvivalprobabilitywasseentohaveabroadmaximumforpulserepetitionfrequenciesclosetotheorbitalperiodoftheelectron.Finally,wementionarecentexperiment,whichndsthatvery-highly-excitedRydbergatomsn100arequiterobustagainstionizationbyHCPs[ 221 ].Instead,theatomicelectronisdisplacedtoevenhigher-lyingRydbergstates.Forthelow-lyingRydbergstates,wherethelengthoftheHCPiscomparabletotheorbitalperiod,theHCPcanbeaccuratelytreatedasafast,unipolarpulse. 4.3 StudyGoalsandOverviewInthepresentstudyweinvestigateindetailtheionizationofextremeStarkstatesofsodiumviahalf-cyclepulses.ForregimesinwhichHCP>n,ionizationissup-pressed[ 147 ],whereasforveryshortpulsesHCP<
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92 theionizationspectraforthedownhillstateskickeduphillDkUaresimilartothoseforUkD.Concentratingonthetwocasesmentionedabove,DkDandUkD,capturesmostoftherelevantphysicsoftheproblem.Inthisstudyweshowthattheenergy-resolvedionizationspectraofthedownhillstatesexhibitanoscillatorypattern.Thisphenomenonappearstoberelatedtotheelddependentoscillationsfoundinearlierstudies[ 228 , 229 ].Incontrast,theionizationenergyspectraoftheuphillstateslackanydiscernibleoscillations.Weperformfullyquantum-mechanicalcalculationstostudythedependenceoftheoscillationsinthedownhillspectrumonthescaleddurationHCP=noftheHCP.Theroleplayedbytheatomiccoreintheionizationprocessisinvestigatedbyemployingamaskfunction,oruxgobbler,thatabsorbstheelectronicuxmovingtowardsthenucleus.Ourresultsshowthattheinteractionwiththeatomiccoreisessentialincreatingtheoscillationsinthedownhillspectrum.Apictureofthequantum-classicalparallelismisobtainedwiththehelpofa1Dclassical-trajectorymodel[ 234 , 235 ].Theclassical-trajectoryinterpretationoftheionizationismoreintuitive,allowingustointerprettheelusivequantum-mechanicalresults.Wendgoodagreementbetweenthefullquantumandthe1Dcalculationsforthecaseofionizationfromthedownhillstates.However,thequantumand1Dmodelsdonotagreeinthecaseofionizationfromtheuphillstates.Thisisanindicationthatthe1Danalysisisinadequatehere,mainlybecausetheelectronisnotconnedtoonesideoftheatomcoreasimpliedbythe1Dmodel.Finally,weexaminethespatialevolutionofthedownhillanduphillstatesduringandfollowingthelaserpulse.Theresultsobtainedprovidesupportfortheapplicabilityoftheone-dimensionalsemiclassicalmodelandtheinterfering-trajectorymethod[ 229 , 235 , 236 ]forthedownhillstatedynamics,andillustratehowthismodelfailstoexplaintheuphillionizationdynamics.

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CHAPTER5IONIZATIONOFSODIUMRYDBERGSTATESBYHALF-CYCLEPULSES–THEORYANDRESULTS 5.1 TheoreticalMethods 5.1.1 Field-FreeSodiumStatesThealkali-metalatomsareinmanyrespectssimilartohydrogen,sincetheyhaveonlyonevalenceelectronorbitinginthenon-Coulombicpotentialcreatedbythenitesizedcore.Comparedtohydrogen,thepresenceofthecoresimplycausesadownwardshiftoftheenergylevels,whichislargerforlevelswithloworbitalmomentum.Theenergylevelsoftheeld-freealkali-metalsaregivenbyaformulasimilartohydrogen:Enl=)]TJ/F15 11.955 Tf 9.298 0 Td[(1 2n)]TJ/F25 11.955 Tf 11.955 0 Td[(l2whereliscalledquantumdefect.Inthecaseofsodium,thequantumdefectissignicantonlyforstateswithloworbitalquantumnumbers=1:35,p=0:85.Forhigherlvaluesl>1,thecentrifugalbarrierpreventsthewavefunctionfrompenetratingthecore,andconsequentlythequantumdefectispracticallyzero.Theeld-freespectrumofNashowsthatthestatesofsandpcharacter,withinsomemanifoldn,aresignicantlylowerthantheircounterpartsinhydrogen,whiletherestofthestatesarenotaffected,andformaquasi-degeneratemanifold.Whenanexternalelectriceldisapplied,thequasi-degeneracyofthestateswithl>1islifted,andaStarkmanifold,similartothatformedbythecorrespondingstatesinhydrogen,isobtained.TondtheeigenstatesofNainanexternalelectriceld,westartbysolvingthestationarySchrodingerequationoftheeld-freeatom: 93

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94 ^H0r=)]TJ 10.494 8.088 Td[(r2 2+Vrr=Er;whereatomicunitswithe=me=h=1seeAppendix A areused.FortheNacorepotential,Vr,weuseanon-localform[ 237 ]:Vr=XlVlrjlihlj+Vpolr)]TJ/F15 11.955 Tf 13.151 8.087 Td[(1 r;whereVlrisanl-dependentshortrangecontributionrepresentingtheeffectofthenuclearshieldingbythecoreelectrons,andVpolisapolarizationtermaccountingforthepolarizationofthecoreuptoquadrupolecontributions.NotethattheHamiltonianpreservesazimuthalsymmetry,whichmeansthez-componentoftheangularmomentumisaconstantofmotionandmisagoodquantumnumber.Duetothenite-sizecore,thewavefunctionisnolongerseparableinparaboliccoordinates,asitisinhydrogen,andtheparabolicquantumnumbern1isnolongeragoodquantumnumber.InordertosolvetheSchrodingerequationnumerically,weconsideranexpansionofthewavefunctionintermsofsphericalharmonics:r;;=lmaxXl=0lrYml;;wherelmaxrepresentstheexpectedmaximumvaluefortheorbitalmomentum.InsertinginEq. 5 wehaveXl)]TJ/F15 11.955 Tf 10.494 8.088 Td[(1 2@2 @r+ll+1 r2+VrlrYml=EXllrYml:Theaboveequationshowsthatwecansolvefortheradialfunctionsindividually,foreachvalueofl.TheresultingsetofequationsforlarediscretizedseeAppendix B onanon-uniformgrid[ 238 ].Thismethodrendersanefcientdiscretizationoftheproblem,witharesultinggridthathasahigherdensityclosetothenucleuswherehigherkineticenergiesareexpected,andissparseatlongerradialdistances.Byusingasecond-orderapproximationinthediscretizationprocedure,weobtainasymmetrictridiagonalmatrix

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95 foreachvalueofl.Thismatrixissubsequentlydiagonalizedtoobtaintheeld-freeenergiesandeigenstates.ThediagonalizationprocedurescalesasONr,whereNristhenumberofpointsinthegrid.Theresultsobtainedareconvergedto0.01%errorbyoptimizingthevariousparametersofthegrid.Comparisonwithpreviouscalculations[ 239 ]showexcellentagreement. 5.1.2 NaStarkStatesTheapplicationofanexternalelectriceldproducesamixingoftheNaunperturbedstates,andliftsthezero-elddegeneracies.WecanndtheneweigenstatesofthesystembydiagonalizingthematrixofthefullHamiltonian,^H=^H0+Fs^zwhereFsisthestaticeldpolarizedalongthe^zaxis,inarestrictedbasisofeld-freestates.Forinstance,tocomputetheStarkmanifoldn=15,atypicalrestrictedbasisofapproximately90states,includingmanifoldsn=12throughn=18,isused.Consideringthezaxisinthedirectionoftheappliedeld,theStarkpotentialisVr=Fsz=Fsrcos:ThematrixelementsoftheStarkpotentialintheunperturbedbasisaregivenbyhVi=hn0l0jrjnlihl0jcosjli;wherethermatrixelementcanbeeasilycomputedthroughaquadratureoverthegridpoints,andthecostermamountstohl+1jcosjli=l+1 p l+1l+1:Asaresult,thediagonalizationofthefullHamiltonian^H0+Fs^rcosreducestondingtheeigenvaluesEkandeigenvectorsakll=0;lmaxofareal,symmetric,bandedmatrixof

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96 dimensionNrl.Theeigenstatesofthesystemarethencomputedas:jki=Xlaklrjli:ConcerningthelabelingoftheStarkstates,forhydrogentheeigenstatesarelabeledbyapairofindicesn;k,wherenistheprincipalquantumnumber,andkisequaltothedifferencen1)]TJ/F25 11.955 Tf 12.698 0 Td[(n2betweenthetwoparabolicquantumnumbers.Foracertainmanifoldn,theStarkindexktakesthevalues)]TJ/F15 11.955 Tf 9.299 0 Td[(n)]TJ/F15 11.955 Tf 12.564 0 Td[(1;)]TJ/F15 11.955 Tf 9.299 0 Td[(n)]TJ/F15 11.955 Tf 12.564 0 Td[(1+2;:::;n)]TJ/F15 11.955 Tf 12.564 0 Td[(1.InthecaseofNa,thesandpstateshavelargequantumdefectsandaredisplacedfromtherestofthemanifold,andonlythestateswithl2areeffectivelycoupledbytheelectriceld,formingaStarkmanifold.Theconventionisthattheindexkisusedtolabelonlythesestates,andtherefore,foracertainmanifoldn,ktakesthevalues)]TJ/F15 11.955 Tf 9.299 0 Td[(n)]TJ/F15 11.955 Tf 12.229 0 Td[(3;)]TJ/F15 11.955 Tf 9.299 0 Td[(n)]TJ/F15 11.955 Tf 12.229 0 Td[(3+2;:::;n)]TJ/F15 11.955 Tf 12.229 0 Td[(3.Forinstance,inthecaseofn=15Namanifold,wehavetheisolatedsplitoff15sand15pstates,andthehigherangularmomentumstatesformaStarkmanifoldlabeledbypairsn=15;k,wherektakesthevalues)]TJ/F15 11.955 Tf 9.299 0 Td[(12;)]TJ/F15 11.955 Tf 9.298 0 Td[(10;:::;+12. 5.1.3 NumericalModelofaHCPIonizationbyhalf-cyclepulsesHCPsisaspectroscopictechniquedevelopedrecentlyforstudyingRydbergstatedynamics[ 219 , 220 ].AnHCPisessentiallyaunipolarelectro-magneticpulsewithalarge,intenselobeinonehalfoftheopticalcycle.Thepulseconsistsofasteeplobeofshortduration,followedbyalongtailofoppositepolarityandmorereducedamplitude.Inlaboratoryexperiments,theHCPisalocallyfocusedpulse,andtheelectronusuallyleavesthefocalregionbeforeexperiencingthetailofthepulse.MoredetailsabouttheproductionandexperimentaluseofHCPsaregiveninAppendix C .

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97 Figure5: Theoreticalmodelsofhalf-cyclepulses:half-sin2solidline,half-sindashedline,formsuggestedbyJonesdottedline. Mostofthecalculationsinourstudyassumeasquaredhalf-sinusoidformfortheHCPasgivenintheequationbelow:FHCPt=8><>:^zFpeaksin2 HCPt0tpulse0pulse;;whereHCPisthefull-widthathalfmaximumFWHMoftheelectriceld,andFpeakisthemaximumamplitudeoftheHCP.However,inordertochecktheinuenceofthepulseshapeontheionizationresults,wealsoperformedcalculationsusingahalf-sinshapemoregradualslope,andashapesuggestedbyJones[ 226 ]evenmoregradualslopefortheHCPseealsoFig. 5 .Inthisstudy,mostofthesimulationsusedinitialSodiumStarkstateswithn=15staticeldFs'400V/cm,althoughseveralcomputationswithn=16andn=20stateswereperformedaswell.ThesestateswerecomputedbydiagonalizingtheStarkHamiltonianinazero-eldeigenstatebasis,aspresentedabove.TheevolutionoftheinitialatomicstateduringtheHCPpulsewascomputedwithaChebysevtime-propagationscheme[ 240 ]Appendix D .

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98 5.1.4 TheImpulseApproximationandtheImpulsiveMomentumRetrievalIMRMethodWhentheinteractionbetweentheHCPpulseandanatomicelectrontakesplaceonatime-scalemuchshorterthanthecharacteristictimefortheCoulombinteractionwiththenucleus,thetotalmomentumtransferredtotheelectronis:Q=)]TJ/F31 11.955 Tf 11.291 16.273 Td[(ZHCP0dtFHCPt:Thisregimeisusuallyreferredastheimpulseapproximationorimpulsivelimitregime.Intheimpulsivelimitconditions,theeffectoftheHCPpulseissimplytoboosttheelectronbyanetmomentumQ:fr;t=eiQrir;t;whereiandfaretheinitialandnalstates,respectively,oftheelectron.OnewaytocomputethenalstateistondthesimilaritytransformationthatdiagonalizestheeiQzoperatorinther)]TJ/F25 11.955 Tf 12.289 0 Td[(lbasis.WeemployinsteadaChebysevpolynomialexpansionoftheexponential,whichyieldstheresultviamultiplicationsofthe^zmatrixwithstatevectorsseeAppendix D .UsingaFourierexpansionoftheinitialwavefunctionintermsofmomentumeigenfunctions,itcanbeshownthatthetotalenergytransferduringthisinteractionis:E=hp2if 2)]TJ 13.15 8.087 Td[(hp2ii 2=hpziiQ+Q2 2;wherehxiyrepresentstheaverageof^xinstatejyi.FortheHCPtoionizetheelectron,thetransferredenergymustbegreaterthanthebindingenergy,jEbj;orintermsofmomentum,theionizationoftheatomicelectronoccursonlyifithasaninitialz-componentofthemomentumgreaterthanp0z=jEbj)]TJ/F25 11.955 Tf 19.128 8.088 Td[(Q2 2=Q:

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99 Theaboverelationcanbeusedinionizationexperimentstoextractinformationabouttheinitialmomentumdistributionoftheelectronicstate.Italsorepresentsthebasisforthe“impulsivemomentumretrievaltechnique”IMR[ 7 , 8 ]. 5.1.5 LongerHCPPulsesWhenthewidthoftheHCPpulseiscomparabletothecharacteristictimeoftheinteractionwiththecoreorTn-theorbitalperiod,theelectronexperiencestheinuencesofboththehalf-cyclepulseandthecore.ThesolutionofthetimedependentSchrodingerequationihdt dt=^H=p2 2+Vr+[Fs+FHCPt]ztcanbeexpressedasaperturbativeseriesjti=1Xn=0)]TJ/F25 11.955 Tf 9.298 0 Td[(in n!Zt0dt1Zt0dt2:::Zt0dtn^T[^Ht1^Ht2:::^Htn]j0i;where^Trepresentsthetime-orderingoperator.Tosolveforji,wesplittheentiretimeintervalintoanumber,N,ofequalandshortsubintervalsti;ti+1,suchthatduringeachsubintervalarstorderapproximationcanbeused.Therelationbetweenthewavefunctionsattwosuccessivetimesisjti+1i=e)]TJ/F26 7.97 Tf 6.587 0 Td[(i^Hti+t=2tjtiii=0;:::;N)]TJ/F15 11.955 Tf 11.955 0 Td[(1;wheret=t=Nisthelengthofonesubinterval.Ahintabouttheproperlengthofthesubintervalsisprovidedbytherequirementthatthefollowingrelationissatisedtoagooddegreeofaccuracy:rmaxZt0FHCPtdt=rmaxXiFHCPtiti;wheretiarethedivisionpoints,ti=ti+1)]TJ/F25 11.955 Tf 12.004 0 Td[(ti,andrmaxisthemaximumradialdistanceofthegrid.

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100 Withtheabovedivision,thetotaltimepropagatorisdecomposedaccordingto^Ut;0=N)]TJ/F24 7.97 Tf 6.587 0 Td[(1Yi=0^Uti+1;ti;wheret0=0,tN=t.Hence,theinteractionwiththetimedependentHCPeldistreatedasaseriesofNultrafastkicksoftheelectron,eachtransferringmomentumFHCPti+t=2t.Onceasuitabledivisionoftheentiretimeintervalisfound,theChebysevtime-propagationschemeAppendix D isemployediseachsubinterval.Asaresult,thewavevectorattheendofthei-thsubintervalisgivenbyaseriesexpansionjti+1i=XnantjXni;wherejX0i=jtiiisthewavevectoratthebeginningofthei-thsubintervalandjXni=n^HnormjX0i;wherenisthecomplexChebysevpolynomialofn-thorder.Intheaboveequation^HnormisthescaledHamiltonian^Hnorm=^Hti+t=2)]TJ/F25 11.955 Tf 11.956 0 Td[(Hmed H;whereHmedandHareapproximatevaluesfortheaverageandthehalf-widthoftheHamiltonianspectrum.Thecoefcientsanaregivenbyant=e)]TJ/F26 7.97 Tf 6.586 0 Td[(iHmedt)]TJ/F25 11.955 Tf 11.956 0 Td[(n0JntH;whereJnisthen-thorderBesselfunction.TheresultofthepropagationovertheentireHCPpulseisgivenbythewavefunctioncalculatedattheendofthelastsubintervalofthedivision,jtNi.

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101 ThemainadvantagesoftheChebysevmethod[ 241 ]arestability,space-timecomputationalefciency,andeaseofimplementation.AlltheseremarkablepropertiesaremainlyduetothestablerecurrencerelationsatisedbythecomplexChebysevpolynomialsnx=2xn)]TJ/F24 7.97 Tf 6.586 0 Td[(1x+n)]TJ/F24 7.97 Tf 6.587 0 Td[(2x:ThemaincomputationaloperationintheChebysevschemeistheproductoftheHamilto-nianmatrixwithastatevector.ThesparsenessoftheHamiltonianmatrix,asensuredbythenon-uniformgridrepresentation,makesthisoperationscalelinearlywiththedimen-sionofthegridONrl.ThetotalpropagationprocedurescalesasONNrl. 5.1.6 IonizationSpectrumAfterthetime-propagatedwavefunctionisperformed,thephotoelectronspectrumiscomputedusingan“energywindow”method[ 242 ].Basically,themethodconsistsofprojectingthenalstateontotheeigenstatesofthestationaryHamiltonian^H0usingasocalledwindowfunction^fE=2n ^H0)]TJ/F25 11.955 Tf 11.955 0 Td[(E2n+2n:Forn=1thisfunctionhasaLorentzianprole,while,asnincreases,itapproachesarectangularprole.TheapplicationofthewindowoperatorofargumentEtothenalelectronicwavefunction,selectsonlythosecontributionsfromenergieswithinarangefromE,anddiscardsallothercomponents.TheprobabilitythatthenalelectronhasanenergyEi,withresolution2,isgivenbyPEi=hfj2n H0)]TJ/F25 11.955 Tf 11.955 0 Td[(Ei2n+2njfi:Ingeneral,itwouldbemoreaccurateandmorecumbersometoprojectontheeigenstatesoftheStarkHamiltonian,ratherthantheeld-freeHamiltonian.However,theerrorentailedinusingtheeld-freestatesisroughlyFsz,whichusingavalueforthestaticeldof400V/cm,andanaveragezof400a.u.,isoftheorderof10)]TJ/F24 7.97 Tf 6.587 0 Td[(4a.u.

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102 Thisrepresentsonlyabout1%oftheenergywidthoftheelectronafterthekick,whichisoftheorderof10)]TJ/F24 7.97 Tf 6.587 0 Td[(2a.u.Forn=2,applicationofthewindowoperatorcanbeevaluatedbysolvingtwosuccessivelinearequationsforthevectorjiiH0)]TJ/F25 11.955 Tf 11.955 0 Td[(Ei+p iH0)]TJ/F25 11.955 Tf 11.955 0 Td[(Ei)]TJ 11.955 10.54 Td[(p ijii=jfi:PEiisthengivenbyPEi=4hijii.Thetotalionizationprobabilityiseasilyseentobetheareaunderthegraphintheenergy-resolvedspectrum,PiPEi.InEq. 5 ,theprocessofsolvingforjiiisgreatlysimpliedbythesparseformoftheHamiltonianmatrix.ThecomplexityofthisoperationisONrl 5.1.7 SemiclassicalCalculationsToprovideanintuitivepictureoftheresultsobtainedbythefull-quantumcalcula-tions,wehaveperformedaseriesofclassicaltrajectorycalculationswithintheframeworkofaone-dimensionalmodeloftheNaextremeparabolicstatesdevelopedpreviously[ 234 , 235 ].Thefull2-DclassicalHamiltonianofthesystemincylindricalcoordinatesisgivenbyH2D=1 2p2+p2z+l2z 2)]TJ/F15 11.955 Tf 38.57 8.088 Td[(1 2+z21=2+Vd)]TJ/F15 11.955 Tf 5.48 -9.684 Td[(2+z21=2+z[Fs+FHCPt];wherelzisaconstantofmotion,FsandFHCPtarethestaticandHCPelds,respectivelybothparallelto^z,andVdistheshort-range,non-CoulombiccontributionoftheNacore.AlthoughVdmakesthestaticeldHamiltoniannon-separableinparaboliccoordinates,forstateswithsmallquantumdefectsll2,thistermcanbeneglectedduetothelimitedpenetrationtothecorebythesestates.TransformingEq. 5 tosemi-paraboliccoordinatesu=r+z1=2;v=r)]TJ/F25 11.955 Tf 11.955 0 Td[(z1=2;

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103 andextendingthephasespaceasiscustomaryfortime-dependentproblemsbyintroducingthemomentumconjugatetotime,pt=)]TJ/F25 11.955 Tf 9.298 0 Td[(Ht,theHamiltonianofthesystembecomesH2D=1 2p2u+p2v)]TJ/F15 11.955 Tf 11.955 0 Td[(4 u2+v2+1 2u2)]TJ/F25 11.955 Tf 11.955 0 Td[(v2[Fs+FHCPt]+pt=0:IntheabsenceoftheHCP,theequationsofmotionforuandvareseparable.However,thepresenceofthetime-dependenteldcouplesthetwodegreesoffreedomthroughthetimevaryingenergy,)]TJ/F25 11.955 Tf 9.299 0 Td[(pt.Nevertheless,forinitialconditionsinwhichv=0andpv=0,thecouplingtermsremainzeroatalltimes,andthemotioninvcanbeneglected.Similarly,anapproximateseparationoftheequationsofmotionforuandvcanbeperformedwhenthepvandvcoordinatesremainsmallthroughoutthedurationoftheHCPpulse[ 234 ].Thedegreeofseparabilityischaracterizedbyaquantitydenedas:p2u 2+u2pt)]TJ/F15 11.955 Tf 10.353 0 Td[(2+1 2u4[Fs+FHCPt]=)]TJ/F31 11.955 Tf 11.291 16.857 Td[(p2v 2+v2pt)]TJ/F15 11.955 Tf 13.151 8.088 Td[(1 2v4[Fs+FHCPt]=)]TJ/F25 11.955 Tf 9.298 0 Td[(;Thus,whenthemotioninthevcoordinateisnegligiblesmallvaluesof,thesystemcanbedescribedapproximatelybya1-DHamiltoniangivenasH=1 2p2u+u2pt)]TJ/F15 11.955 Tf 11.955 0 Td[(2+1 2u4[Fs+FHCPt]=0;orincylindricalcoordinatesasH=p2z 2)]TJ/F15 11.955 Tf 13.21 8.088 Td[(1 z+z[Fs+FHCPt]+pt=0:NotethatintheworkofDelosetal.[ 236 ]aneffectivenuclearcharge,Zeff=Z)]TJ/F25 11.955 Tf 12.183 0 Td[(=2,wasused.Inthiswork,weassume0andtakeZeff=1.TheaboveHamiltonianleadstoasystemofcoupleddifferentialequations,whichisintegratedusingafthorderRunge-Kuttaalgorithm[ 243 ].Theresultsoftheintegrationareusedtoobtainthenalenergymanifoldofthesystem.Thismanifoldhelpsininvestigatingthepropertiesoftheenergy-resolvedionizationspectrum.

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104 Animportantpointtorememberaboutthe1Dmodelisthatthemotionoftheelectronisconnedtoonesideofthenucleus.Theelectronstartsononesideofthecore,z>0orz<0,andneverleavesthehalf-spaceinwhichitbegins.Thatis,whentheelectronscattersfromthecore,itmustbackscatter.Asaresult,inthe1Dmodeluphillstatesmustionizeuphill,anddownhillstatesmustionizedownhill,regardlessofthedirectioninwhichtheHCPisapplied. 5.2 NumericalResultsandDiscussionThepresenceoflocalizedstatesisknowntobeageneralfeatureinsystemswithhighlydegeneratemanifolds,undertheinuenceofexternalinteractions[ 244 , 245 ].Forsodiuminanexternalelectriceld,theStarkeffectmixesthedegeneratel-statesinannmanifold,liftingtheinitialdegeneracy.Consideringthez-axisparallelwiththeexternaleld,theextremeblue-andred-shiftedstatesintheStarkmanifoldarecharacterizedbyapronouncedlocalizationalongthe+zand-zaxis,respectivelyFig. 5–2 .AsaresultthedownhillStarkred-shiftedstatespossessapermanentelectricdipoleparalleltotheexternaleld,whiletheuphillStarkblue-shiftedstateshaveadipolemomentantiparalleltoit.Theterminology“uphill”and“downhill”derivesfromthelinearpotential+Ezcreatedbythestaticeld,whichappearsasa“potentialhill”risingmonotonicallyinthe+zdirection. 5.2.1 SemiclassicalInterpretationoftheOscillationsintheDownhillIonizationSpectrumOneeffectofthelocalizationpropertiesoftheextremeStarkstatesisthemanifestasymmetryobservedbetweenthetotalionizationratesoftheuphillanddownhillstates[ 225 , 226 ].Thisasymmetrycanbeattributedtothefactthattheuphillstatesmustpasstheatomiccoretoreachthesaddle-pointregionofthecombinedCoulomb-Starkpotential,whereasthedownhillstatesareatrelativelyclosedistancestothisregion.Asaresult,thedownhillstatesareeasilykickedoverthepotentialbarriertoionizationbytheHCP,whereastheuphillstatesneedamuchstrongereldtoovercometheoppositionofthecoreandtopassoverthepotentialbarrier.

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105 Figure5:Sodiumn=15,k=)]TJ/F15 11.955 Tf 9.299 0 Td[(12top,andk=+12bottomStarkstates.

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106 Inadditiontotheasymmetryinthetotalionizationrates,thenal-energyspectrafortheionizationfromdownhillFig. 5 aanduphillFig. 5 bstatesexhibitanotherimportantdifference:thedownhillspectrumshowsoscillationinthenalenergyprobabilities,whereasnooscillationsarepresentintheuphillspectrumFig.4. Figure5: IonizationspectrafortheuphilltopanddownhillbottomstatesTp=200fs.Thek=+12spectrumshowsnooscillations,whereask=-12spectrumpresentsinterference-likeoscillations. Toelucidatethisdiscrepancyweemployaunidimensionalsemiclassicalmodeldevelopedpreviously[ 234 , 236 , 235 ].Thismodelexplainsthedownhillspectrumoscillationsintermsoftwointerferingsemiclassicaltrajectories.Tobetterillustratethepredictionsofthe1Dmodel,weperformclassicalcalculationsthatshowtheevolutionoftheinitialenergymanifoldEnduringtheHCPpulse.Asinpreviouscalculations[ 235 ]westartwithanumberofinitialtrajectoriesuniformlydistributedintheanglevariable

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107 n=e)]TJ/F15 11.955 Tf 12.494 0 Td[(sine,whereerepresentsthesocalledeccentricanomalyoftheelectronorbit.Theinitialvaluesforzandpzarethendeterminedbyz=n2)]TJ/F15 11.955 Tf 12.605 0 Td[(coseandpz=nsine=z.Withthesevalues,weintegratetheclassicalequationsofmotionEq. 5 ,tondthenalenergyoftheelectron.TheresultsareshowedinFig. 5 .Sincetheenergyisaperiodicfunctionintheanglevariable,itisclearthatforanyenergy Figure5: Curvesrepresentingtheclassicalnalenergyasafunctionoftheinitialanglevariable.Continuousatline–initialenergymanifold,continuousline–nalmanifoldforTp=250fs,largedashedline–Tp=100fs,smalldashedline–Tp=50fs. inthenalmanifoldexceptfortheextremalvaluesthereareanevennumberinourcasetwooftrajectoriesthathavethisenergy.WeseeinFig. 5 thatahorizontallinecorrespondingtosomenalenergycrossesthenalmanifoldattwopoints.Theabscissaeofthesepointsaretheinitialanglevariablesofthetwotrajectories.InaclassicaldescriptionpresentedinFig. 5 theelectronintrajectoryImovesinitiallytowardsthecore,beingsloweddownbytheHCPeld.Afterscatteringoffthenucleus,theelectronregainsthelostenergyand,ifsufcientenergyistransferred,itionizes.IntrajectoryII,theelectronmovesinitiallyawayfromthenucleus,andis

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108 continuouslyacceleratedbytheHCPeld.Again,ifsufcientenergyistransferred,theelectroneventuallyescapesoverthepotentialbarrierandionizes.Althoughtheelectroninthersttrajectoryinitiallylosesenergy,itcompensatesforthislossbyscatteringoffthecore,duetothehigherelectronicvelocitiesreachedinthevicinityofthenucleusE=)]TJ/F31 11.955 Tf 11.291 9.63 Td[(R~F~vdt.Eventually,itmayionizewiththesamenalenergyastheelectronintrajectoryII.Inasemiclassicaltheory,thetwotrajectoriesarerepresentedbytwosemiclassicalwavefunctions,whichcaninterferetoproducetheobservedoscillatorypatternintheionizationspectrum.Notethatintheabovemodelanimportantroleisplayedbythecore,whichscattersthersttrajectorysothatitlaterinterfereswithtrajectoryII.Tostudytheinuenceofthecoreinthefull-quantumcalculations,weuseasmoothabsorbingpotentialor`gobbler'.Theeffectofthegobbleristoremovetheelectronicuxmovingtowardsthenucleus.TheresultsofthesetestsarepresentedinFig. 5 .Notethatasthegobblerbecomesmoreeffectiveincancelingthecomponentofthewavefunctionreachingthenucleushighergobblerradius,theinterferencepatterndisappears.Thisresultshowsthattheclose-rangeinteractionwiththecoreisessentialinproducingtheobservedinterferencebehavior,andthereforeitsupportsthesemiclassicalviewof`twointerferingpaths.' 5.2.2 TheInuenceoftheHCPWidthupontheIonizationAnotherionizationfeatureofinterestisthedependenceoftheionizationspectrumontheHCPpulselengthTp.WhenthepulselengthismuchsmallerthantheKeplerperiodTp<
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109 Figure5: TrajectoryIheadsinitiallytowardsnucleusandisslowedbytheeld,whereastrajectoryIIstartsmovingawayfromthenucleusandisacceler-atedbytheexternaleld.

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110 Figure5: Ionizationcurvesshowingtheeffectofthegobblerandimplicitlyofthenucleusontheoscillationsinthenalenergyspectra.

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111 where0istheinitialwavefunction.UsingaFourierexpansionoftheinitialwavefunctioninmomentumspace,itcanbeeasilyshownthatthetotalenergytransferredduringinteractionisE=hp2if 2)]TJ 13.151 8.088 Td[(hp2ii 2=hpziiQ+Q2 2;wherehxiysigniestheaverageofoperator^xinstatejyi.Tostudyionizationcharacteristicsintheimpulsivelimitregime,wehaveperformedseveralsimulationsforionizationintheimpulsivelimit.Forinstance,weshowinFig. 5–7 athenalwavefunctioneiQz0~r,where0isthen=15,k=)]TJ/F15 11.955 Tf 9.298 0 Td[(12Nastate,andQ=0:129a.u.Notethatintheimpulsivelimitthespectraloscillationsdisappear.Semiclassically,thiscanberelatedtothefactthatthetransitionamplitudescorrespondingtothetwointerferingtrajectoriesareproportionalto@If @i)]TJ/F24 7.97 Tf 6.586 0 Td[(1or@Ef @i)]TJ/F24 7.97 Tf 6.587 0 Td[(1[ 236 ],wherehIfjI0ti=)]TJ/F15 11.955 Tf 9.298 0 Td[(2i@If @i)]TJ/F24 7.97 Tf 6.587 0 Td[(1=2expiZtft0)]TJ/F25 11.955 Tf 9.299 0 Td[(tdI dt)]TJ/F25 11.955 Tf 11.955 0 Td[(Htdt:Inthelimitofultrashortpulses,onecanseeinFig. 5 thatthequantity@Ef @i)]TJ/F24 7.97 Tf 6.587 0 Td[(1approacheszero1slopefortrajectoriesontherightbranch0>min,wheremincorrespondstothetrajectorywithminimumnalenergyofthenalmanifold.Asaresult,oneofthetwotrajectorieshasamuchsmallertransitionamplitudethantheother,andtheinterferenceisgreatlyreduced.Intuitively,thelackofoscillationsinthespectracalculatedintheimpulsivelimitspectrumcanberelatedtothefactthattheelectrondoesnothavetimetointeractwiththecorewhiletheHCPpulseison.Therefore,thetwo-trajectoryinterferencemechanismdescribedaboveisnotoperative.AstheHCPlengthincreases,theoscillationsintheionizationspectrumhavemorecontrast,andthefrequencyoftheoscillationsincreasesslightlytowardshigherenergies.ThiscanagainberelatedtothesemiclassicalresultsinFig. 5 ,whereweobservethatforlongerpulsesthenalmanifoldtendstobemoresymmetricabouttheminimumvalue.Ineffect,theslopefortrajectoriesontherightbranch>minof

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112 themanifolddecreases,andbecomescomparabletotheslopefortrajectoriesontheleftbranchTn=2Fig. 5 d.Thisisexpected,sinceTn=2istheaverageelectron-coreinteractiontime.Thefrequencyoftheoscillationsisalsoseentodecrease,whichcanbeexplainedbyadecreaseinthevariationwithenergyofthephasedifferencebetweenthetwointerferingtrajectories,=E. 5.2.3 WhyAreThereNoOscillationsintheUphillIonizationSpectrum?Onequestionthatwetriedtoanswerisconcernedwiththeabsenceofoscillationsintheionizationspectrafortheuphillstates.Ifthesemiclassicalmodelisappliedtothedynamicsoftheuphillstates,thesameargumentofperiodicityofthenal-energymanifold,evokedforthedownhillstates,wouldpredictoscillationsintheuphillionizationspectrum.However,thequantumcalculationsshownotraceofsuchoscillations.Tounderstandthedifferenceinthedynamicsforthetwotypesofstates,weperformquantum-mechanicalsimulationsoftheinteractionwithanHCP.TheresultsofthesesimulationsarepresentedinFig. 5 .ImportantdifferencesarerevealedattheexaminationoftheelectronicwavefunctionsfortheNa,n=15,m=0downhill

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113 Figure5: IonizationcurvesforaHCPwithQ=.129,showingthedependenceofthenalspectraonthedurationofthepulse.aimpulsivelimitTp!0fs,bTp=50fs,cTp=200fs,dTp=800fs.

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114 k=)]TJ/F15 11.955 Tf 9.298 0 Td[(12anduphillk=+12statesastheypropagateundertheinuenceofthehalf-cyclepulse.Fortheuphillstate,alargepartofthewavefunctionispulledbytheHCPeldbeyondtheatomiccore,intothe)]TJ/F25 11.955 Tf 9.299 0 Td[(zregion.Theshort-rangeinteractionwiththeatomiccorestronglyscatterstheelectronicwavefunction.Incontrast,theevolutionofthedownhillstateiscompletelycontainedinthez<0half-space.TheHCPeldsmoothlypullsthiswavefunctiontowardsionization.Theresultingmotionhasaquasi-1Dcharacter.Arevealingpictureofthetwotypesofmotionisobtainedbyanalyzingthewavepacketevolutioninsemi-paraboliccoordinates,uandvconformEq. 5 .TheevolutionoftheexpectationvaluesoftheuandvcoordinatesisshowninFig. 5 .Thesevaluesareobtainedfromthetime-dependentwavefunctionspresentedabove.Noticethatthemotionofthedownhillstateinthevcoordinateisnegligible,whiletheuphillstateexhibitsasignicantspreadinthiscoordinate.Becausethe1Dsemi-classicalmodelrequiresquasi-1Ddynamicsatalltimes,itisclearwhythismodelcannotdescribeappropriatelythe2Devolutionoftheuphillstate.Asaresultthetwo-trajectorytheoryisnotvalidfortheuphillcase,andinterferenceeffectsarenotobserved.Inthesimpleclassicaltrajectorypicture,thenal-energymanifoldfortheuphillionizationdependsontwoparameters,andthereforeitsintersectionwithaconstantenergyplaneisrepresentedbyacontourinthisplane.Thismeanstherewillbeamultitudeoftrajectoriesthatreachtothesamenalenergy,andtherandomphasesofthesetrajectoriesconspiretoaverageouttheinterferencepattern.Furthersupportfortheaboveconsiderationscomesfromcalculationsofthedependenceofthedownhillionizationspectrumontheintensityofthestaticeld.TheresultsarepresentedinFig. 5 .Thestaticeldcanbeseenasanalignmentinstrumentthatinducesaninitiallocalizationoftheelectronicwavefunctionalongthezaxis.Forlowelds,the15dstate,duetoitssmallquantumdefect,isnotyetmixedwiththehigher-lstatesintheStarkmanifold.Asaresult,thewouldbek=)]TJ/F15 11.955 Tf 9.298 0 Td[(12stateordownhillstate,

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115 Figure5: Electronicdistributionasafunctionoftimeduringtheinteractionwithasin2HCPofQ=0:129a.u.,HCP=100fs.Thepanelsontheleftdisplaytheevolutionforthek=+12uphillstate,andthepanelsontherightshowtheevolutionofthek=)]TJ/F15 11.955 Tf 9.299 0 Td[(12downhillstate.

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116 Figure5: Evolutionoftheexpectationvaluesofthesemi-paraboliccoordinatesduringtheinteractionwithasin2HCPofQ=0:129a.u.,HCP=100fs.Thesolidlinesshowtheexpectationvalueofvfortheuphilllledcircles,anddown-hilllledtrianglesstates.Thedashedlinesshowtheexpectationvalueofufortheuphillopencirclesanddownhillopentrianglesstates. hasnearlypuredl=2character,andisnotwelllocalizedalongthezaxis.Inthissituation,theevolutionofthisstateduringtheinteractionwiththeHCPpresentsnon-negligiblemotioninthesemi-parabolicvcoordinate.Thus,thequasi-1Dapproximationbreaksdown,andtheinterferencepatterniswashedout.WhenthemagnitudeofthestaticeldincreasesFs>100V/cm,the15dstatejoinstheStarkmanifoldandbecomesthek=)]TJ/F15 11.955 Tf 9.298 0 Td[(12Starkstate.Theconnementofthisstatealongthezaxisbecomesmorepronounced,andthe1DapproximationisvalidEq. 5 .Theneteffectistherestorationoftheinterferencepatternintheionizationspectra. 5.3 SummaryInthischapterwestudytheionizationofNaRydbergstatesbyhalf-cyclepulses.Anumberofefcienttheoreticalandnumericaltechniquesaredescribed.Thefullquantum-mechanicalcalculationsperformedprovideevidenceforseveralimportantcharacteristics

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117 Figure5: IonizationspectraforNan=15,k=)]TJ/F15 11.955 Tf 9.298 0 Td[(12hitbyasin2HCPHCP=100fs,Fpeak=160kV/cmasafunctionofthestaticelectriceld,Fs.

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118 oftheionizationprocess:thedependenceoftheionizationspectraonthewidthoftheHCP,theinuenceoftheelectronicwavefunctionlocalizationontheionizationprocess,andalsotheroleplayedbytheatomiccoreinproducinginterferenceeffectsintheionizationofdownhillstates.Onefeaturerevealedbythecalculationsistheasymmetryintheionizationprocessofthesodiumdownhillanduphillstates.Thisasymmetryismanifestedinthediscrep-ancybetweenthetotalionizationratesofthetwostates,andalsointhecharacteristicsoftheenergy-resolvedionizationspectra.Thus,thedownhillionizationspectradisplayoscillationsasafunctionofenergy,whiletheuphillspectralackanysimilarpattern.Toprovideabridgebetweenthenon-intuitivequantumdescriptionandtheclassicaldescription,weemployedaunidimensionalsemiclassicalmodel.Thismodelprovidesanintuitiveinterpretationoftheresultsintermsofclassical-trajectories.Theoscillationsobservedinthedownhillionizationareexplained,inthismodel,astheinterferenceoftwoclassicaltrajectoriesleadingtothesamenalstate.Inrevealingthemechanism,the1Dmodelemphasizestheeffectoftheatomiccoreintheinterferenceprocess.ToisolatetheroleplayedbythecoreduringtheinteractionwiththeHCP,weusedauxgobbler,whichremovestheelectronicuxnearthenucleus.Theobservedeffectisadisappearanceoftheoscillatorypatternasthegobblerradiusincreases.Thisfactshowsthattheshort-rangeinteractionwiththecoreisessentialinproducingtheoscillationsintheionizationspectrum.Forthecaseofdownhillstates,weinvestigatedthedependenceofthespectraloscil-lationsonthelengthofthepulse,andsuccessfullyrelatedtheresultstothepredictionsofthe“twointerferingpaths”theory.Weshowthatthegoodagreementbetweenourresultsandthe1Dmodelisduetothequasi-one-dimensionalcharacterofthedynamicsinthedownhillstates.

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119 Finally,weshowedthattheunidimensionalmodelisinadequatetorepresentthedynamicsofuphillstates.Thisexplainswhytheoscillatoryinterferencepatternisabsentintheionizationspectrumofthesestates.

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CHAPTER6CONCLUSIONSANDFUTUREWORK 6.1 SummaryInconclusion,thisworkpresentstwostudiesonthecontrolandspectroscopyofatomicmatterbyultrafastlaserpulses.Onestudyisconcernedwiththecontrolofwavepacketfocusingbychirped,ultrashortlaserpulses,whileasecondstudyinvolvestheionizationofsodiumRydbergstatesbyhalf-cyclepulses.Efcienttheoreticalmethodsaredevelopedinbothcases,andthepredictionsofthesemethodsaretestedinnumericalsimulations.Wheneverpossible,classicaltrajectoryargumentscomplementthequantum-mechanicaldescriptiontoprovideamoreintuitivepictureofthedynamics.Inreview,inchapter2ofthisworkwepresentacompendiumofprevioustheoreticalandexperimentalworkintheeldofquantumcontrol.Quantumcontrolisanemergentparadigmthathasdemonstratedgreatpotentialforthedevelopmentofnewtechniquesandtechnologiessuchasmolecularsynthesis,quantumcomputingandquantuminforma-tion,nano-fabricationofmoleculardevices,andcontrolofchemicalreactivity.Chapter2givesashortoverviewofthetwomainparadigmsforquantumcontrol:theBrumer-Shapiromethod,andtheTannor-Ricemethod.Italsodiscussesthefeedback-loopcontroltechnique,whichisextremelyusefulwhentheHamiltonianofthesystemisunknownorcomplex.Inthistechniquealearningorevolutionaryalgorithmcontrolsthelaser-shapingdevice.Thealgorithmproceedsinaseriesofsteps,andateachsteptheexperimentaloutputisanalyzed,andthelaserparametersarechangedtoconvergetoadesiredtargetstate.Thetopicofadiabaticpopulationtransferinatomsandmoleculesispresentedindetail.TechniquessuchasadiabaticrapidpassageARP,stimulatedRamanadiabaticpassageSTIRAP,Stark-chirpedrapidadiabaticpassageSCRAP,adiabaticpassage 120

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121 bylightinducedpotentialsAPLIPandRamanchirpedadiabaticpassageRCAParereviewed.Attheendofthechapterwegiveashortreviewonthetopicofquantumcomputing,andalsopresentanumberofinterestingapplicationsofquantumcontrollikeforexamplea`molecularcentrifuge.'Inchapter3wepresentatheoreticalstudyofthecontrolofwavepacketfocusingbyultrashortpulses.Theabilitytocontrolwavepacketfocusingisimportantfortheachievementofspecicexcitationpathways,aswellasforthecontrolofchemicalreactivity.Intheshort-pulselimitandforlinearpotentials,wederiveimportantanalyticalresultsthatmaketheconnectionbetweenthefocusingtimeandthewidthofthefocusedwavepacket,ononehand,andthechirpanddurationofthelaserpulse,ontheotherhand.Thevalidityoftheanalyticalformulasisveriedinanumberofnumericalsimulationspertainingtoarealisticmolecularsystem,ICN.Theresultsshowexcellentagreement.Generalequationsforfocusingcontrol,whicharevalidbeyondtheultrashortpulselimit.arealsoderived.Furthermore,westudythefocusingofwavepacketsinexponentialpotentials.Importantfactorsthatcontributetothefocusingdynamics,suchasthenonuniformityofthepotentialandthechirpofthelaserpulse,areidentiedanddiscussed.Analyticalformulasforthecontroloflaserexcitationofawavepacketareobtained.Sincemostofourresultsareobtainedintheultrashort-pulselimit,weinvestigatethebreakdownofthislimitasafunctionofvariousparameters.Wendthatthebreakdownlimitdependsonthelengthofthepulse,theslopeofthepotential,andthechirpofthepulse.Thechirpwasfoundtobeanimportantfactorthat,forappropriatevalues,mayensurethevalidityofthe!0limitevenforquitelongpulses.Attheendofchapter3,westudytheintricateinterferencemechanismbywhichthepulsechirpcancontrolthefocusingofawavepacket,aswellasenforcethe!0forlongpulses.Ourtheoreticalconsiderationsareinfullagreementwithresultsobtainedinnumericalsimulations.

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122 Inchapter4weprovideageneraloverviewofhalf-cyclepulsesHCPs.Ahalf-cyclepulsehastheuniquepropertythatitselectricoccursinonehalfofanopticalcycle.ThisconferstheHCPanapproximatelyunipolarcharacter,whichincreasesitsprobabilityofinteractionwithaRydbergelectronfarfromtheatomiccore.Incontrast,ordinarylaserpulsescannottransfermomentumtoanelectronfarfromthecore,becausetheircycle-averagedelectricaleldiszero.DuetotheirincreasedinteractionwithRydbergelectrons,HCPsaresensitivetoolsforcreatingandprobingRydbergwavepackets.Chapter5presentstheoreticalandnumericalresultsfortheionizationofsodiumRydbergstatesbyhalf-cyclepulses.Anumberofefcienttheoreticalandnumericaltechniquesarepresented.Fullquantum-mechanicalcalculationsrevealseveralimportantcharacteristicsoftheionizationprocess;thedependenceoftheionizationspectraonthewidthoftheHCP,theinuenceoftheelectronicwavefunctionlocalizationpropertiesontheionizationprocess,andtheroleplayedbytheatomiccoreinproducinginterferenceeffectsintheionizationofdownhillstates.Arstfeaturerevealedbythecalculationsistheasymmetryintheionizationprocessofthesodiumdownhillanduphillstates.Thetotalionizationratesoftheuphillstatesarefoundtobemuchreducedcomparedtotheionizationratesfordownhillstates.Thenal-energyspectrarevealanotherdiscrepancy:thedownhillionizationspectradisplayoscillationsasafunctionofenergy,whiletheuphillspectradonot.Toprovideanintuitivepicturefortheresultsobtainedbythefull-quantumcalcu-lationsweadoptaunidimensionalsemiclassicalmodel.Theoscillationsobservedinthedownhillionizationspectraareexplained,inthismodel,bytheinterferenceoftwoclassicaltrajectoriesleadingtothesamenalstate.Toisolatetheroleplayedbythecoreduringtheinteractionwiththehalf-cyclepulse,weemploya“gobbler”potential,whichabsorbstheelectronicuxtowardsnucleus.Whentheradiusofgobblerisincreased,theoscillatorypatterndisappears.

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123 Theseobservationsindicatethattheshort-rangeinteractionwiththecoreisessentialinproducingtheoscillationsintheionizationspectrum.Thedependenceoftheionizationprocessonthelengthofthepulseisalsoin-vestigated.Alloftheresultsobtainedbyfullquantummechanicalcalculationscanbeexplainedsuccessfullybasedona“twointerferingpaths”theory.Weshowthatthegoodagreementbetweenourresultsandthe1Dsemiclassicalmodelisduetothequasi-one-dimensionalcharacterofthedynamicsinthedownhillstates.Finallyweshowthattheunidimensionalmodelisnotapplicableinthecaseoftheuphillstatesduetointrinsic2Ddynamicsinthesestates.Multipletrajectories,whichevolvethroughaseconddimension,arenowcontributingtotheionization.Theirrandomphasesconspiretowashouttheoscillatoryinterferencepatternintheionizationoftheuphillstates. 6.2 FutureWorkThetheoreticaltechniquesusedinthisworkcanbeusedtostudyanumberofinterestingissuesrelatedtotheinteractionbetweenultrashortlaserpulsesandatomicmatter.Forinstance,wavepacketfocusingcouldbeappliedtothestudyofrealisticatomicandmolecularsystems.Wecanimagine,forexample,amechanismformolecularsynthesis,inwhichwavepacketcontrolisusedtobringtheconstituentatomsandelectronstotherightpositionsthatmolecularbondscanform.Atalargerscalethesamemechanismcouldbeusedtobuildnano-machines,orbiologicallyactivemoleculesthatcouldhelpinrectifyinggeneticmaladiesordestroyingtumors.Wavepacketcontrolcanalsobeusedtocontrolchemicalreactivitybyemploying,forexample,apump-dumpTannor-Ricescheme[ 113 ].Insuchaschemearstlaserpulseexcitesagroundwavepackettoanexcitedelectronicstate,andafteraproperlytimedevolutionontheexcitedpotential,asecondpulsedumpsthewavepacketbacktothegroundstate.Sincethetransitionamplitudebetweenelectronicstatesdependsontheradialcoordinate,focusingtheexcitedwavepacketatthetimewhenthepumppulse

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124 actsensuresthattheentirenucleardistributionfollowsthedesiredpathway.Guidingamolecularwavepacketalongaspeciallydesignedpathcanhaveastronginuenceontheformationandbreakingofamolecularbond.Similarwavepackettechniquescanbeemployedforisotopeseparation[ 113 ]asshowninchapter2.Anotherscenarioforwavepacketcontrolwouldbeinatomopticstostudycoldcolli-sionsoruniquequantumeffectsinducedbyspatialconnement.Controlledwavepacketscanbeusedtoachievethedepositionofnanoscalestripsofatomsusingweakeldstand-ingwaves[ 113 ].Techniquesfornanoscaleetchingbased,forexample,onthefocusingofI2onsiliconfollowedbythephotodesorptionofSi-I,havealsobeenproposed[ 154 ].Schemesthatusehalf-cyclepulsestoprobethedynamicsofatomicandmolecularwavepacketsarealsopossiblefutureprospects.ForexampletwoHCPscanbeusedinapump-probestudy,inwhichtherstpulseexcitessomeparticularsuperpositionofstates,andthesecondextractsthroughionizationtheinformationaboutthecreatedwavepacket.AnotherdirectionofstudyisthepossibilityofcoherentlycontrollingtheatomicexcitationbyHCPs,toobtainRydbergwavepacketswithinterestingcoherentproperties.Examplesofsuch“exotic”targetsare:classical-limitatomicstateswavepacketsthatdisplay3Dlocalizationsimilartotheclassicalpictureofanelectronorbitingtheatomiccore,Trojanwavepacketsnon-dispersivewavepacketsthatorbitatlargedistancesfromthenucleus,andevenSchrodingercatstatessuperpositionofatomicstateswithdifferentspatiallocalizations.Thestudyofthesestatescanprovidevaluableinsightsforareassuchasthequantum-classicallimit,quantumchaos,andquantumcontrol.Alongthesamelines,thepossibilityofcoherentlycontrollingTHzemissionusingspecicallydesignedHCPshasbeendemonstratedrecently[ 15 ],andfurtherstudyinthisareamayalsobeconducted.Lastbutnotleast,weareconsideringresearchinanewareaconcernedwiththepossibilityofusingatomicwavepacketscontrolledbyspeciallysculptedultrashortpulsestoperformquantumcomputations.RecentworkbyBucksbaumetal.[ 16 ]has

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125 showedthataquantumregistercanbeimplementedphysicallyasaRydbergwavepacket,inwhichtheinformationisstoredintherelativephasesofthedifferentcomponentstates.TheroleoftheHCPistocoherentlymanipulatethesephasestostorethedesiredinformation.Asubsequentpaper[ 17 ]showsthatasecond,properlyshapedHCPcanbeusedtoselectivelyretrievetheinformationstoredasquantumphaseinsuchaquantumregister.Whatremainstobeaccomplishedforquantumcomputationtobefeasible,istondawaytoimplementauniversalquantum-logicalgatewithintheconstraintsofthissystem.

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142 [217] N.E.Henriksen,“Theoreticalconceptsinmolecularphotodissociationdynamics,”Adv.Chem.Phys.,vol.91,p.433,1995. [218] S.O.WilliamsandD.G.Imre,“Determinationofrealtimedynamicsinmoleculesbyfemtosecondlaserexcitation,”J.Phys.Chem.,vol.92,pp.6648,1988. [219] R.R.Jones,D.You,andP.H.Bucksbaum,“IonizationofRydbergatomsbysubpicosecondhalf-cycleelectromagneticpulses,”Phys.Rev.Lett.,vol.70,pp.1236,1993. [220] G.M.LankhuijzenandL.D.Noordam,“FrequencyandtimeresolvedstudyofthedynamicsofrubidiumRydberg,”Phys.Rev.A,vol.52,pp.2016,1995. [221] C.Wesdorp,F.Robicheaux,andL.D.Noordam,“DisplacingRydbergelectrons:Themono-cyclenatureofhalf-cyclepulses,”Phys.Rev.Lett.,vol.87,p.083001,2001. [222] D.You,R.R.Jones,D.R.Dykaar,andP.H.Bucksbaum,“Generationofhigh-powerhalf-cycle500femtosecondelectromagneticpulses,”Opt.Lett.,vol.18,p.290,1993. [223] N.E.Tiekling,T.J.Bensky,andR.R.Jones,“EffectsofimperfectunipolarityontheionizationofRydbergatomsbysubpicosecondhalfcyclepulses,”Phys.Rev.A,vol.51,p.3370,1995. [224] C.O.Reinhold,M.Melles,HaiShao,andJ.Burgdorfer,“IonizationofRydbergatomsbyhalf-cyclepulses,”J.Phys.B,vol.26,pp.L659–L664,1993. [225] R.R.Jones,N.E.Tielking,D.You,C.Raman,andP.H.Bucksbaum,“IonizationoforientedRydbergstatesbysub-picosecond,halfcycleelectromagneticpulses,”Phys.Rev.A,vol.51,p.R2687,1995. [226] A.Bugacov,B.Piraux,M.Pont,andR.Shakeshaft,“AsymmetryinionizationoforientedRydbergstatesofhydrogenbyahalfcyclepulse,”Phys.Rev.A,vol.51,pp.4877,1995. [227] C.O.Reinhold,H.Shao,andJ.Burgdorfer,“EvolutionofRydbergangularmomentumwavepacketsinhalf-cyclepulses,”J.Phys.B:At.Mol.Opt.Phys.,vol.27,p.L469,1994. [228] C.O.Reinhold,J.Burgdorfer,R.R.Jones,C.Raman,andP.H.Bucksbaum,“Scaled-timedynamicsofionizationofRydbergStarkstatesbyhalf-cyclepulses,”J.Phys.B,vol.28,pp.L457–L464,1995. [229] C.O.ReinholdandJ.Burgdorfer,“InterferenceoscillationsinionizationofextremeStarkstatesbyhalf-cyclepulses,”Phys.Rev.A,vol.51,p.R3410,1995. [230] T.F.JiangandC.D.Lin,“IonizationoforientedellipticRydbergstatesbyhalf-cyclepulses,”Phys.Rev.A,vol.53,p.2172,1997.

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143 [231] C.O.Reinhold,M.Melles,andJ.Burgdorfer,“IonizationofRydbergatomsbyshortelectromagneticpulses,”Phys.Rev.Lett.,vol.70,p.4026,1993. [232] M.B.Campbell,T.J.Bensky,andR.R.Jones,“CompleteviewofStarkwave-packetevolution,”Phys.Rev.A,vol.59,pp.R4117,1999. [233] M.T.FreyandF.B.Dunning,“Realizationofthekickedatom,”Phys.Rev.A,vol.59,p.1434,1999. [234] C.D.SchwietersandJ.B.Delos,“Semiclassicaltreatmentofahalf-cyclepulseactingonaone-dimensionalRydbergatom,”Phys.Rev.A,vol.51,p.1023,1995. [235] M.MallalieuandS.I.Chu,“SemiclassicalstudyoftheionizationspectraofRydbergatomsdrivenbyintensehalf-cyclepulses,”Chem.Phys.Lett.,vol.258,pp.37,1996. [236] C.D.SchwietersandJ.B.Delos,“Half-cyclepulseactingonaone-dimensionalRydbergatom:Semiclassicaltransitionamplitudesinactionandanglevariables,”Phys.Rev.A,vol.51,p.1030,1995. [237] J.N.Bardsley,“Pseudopotentialsinatomicandmolecularphysics,”inCaseStudiesinAtomicPhysicsvol.IV,E.W.McDanielandM.R.C.McDowell,Eds.,pp.302.NorthHolland,Amsterdam,1975. [238] J.L.KrauseandK.J.Schafer,“ControlofTHzemissionfromStarkwavepackets,”J.Phys.Chem.,vol.103,pp.10118,1999. [239] M.L.Zimmerman,M.G.Littman,M.M.Kash,andD.Kleppner,“StarkstructureofRydbergstatesofalkali-metalatoms,”Phys.Rev.A,vol.20,pp.2251,1979. [240] H.Tal-EzerandR.Kosloff,“AnaccurateandefcientschemeforpropagatingthetimedependentSchrodingerequation,”J.Chem.Phys.,vol.81,pp.3967,1984. [241] R.Kosloff,“Time-dependentquantum-mechanicalmethodsformoleculardynamics,”J.Phys.Chem.,vol.92,p.2087,1988. [242] K.J.Schafer,“Theenergyanalysisoftime-dependentnumericalwavefunctions,”Comput.Phys.Comm.,vol.63,pp.427,1991. [243] S.E.Koonin,ComputationalPhysics,BenjaminCummings,MenloParkandCA,1986. [244] A.R.P.Rau,“Localizationandothercommonfeaturesofdegenerateperturba-tions,”Phys.Rev.Lett.,vol.63,p.244,1989. [245] A.R.P.Rau,“Dissectingtheadiabaticinvariantofatomicdiamagnetism,”J.Phys.B,vol.27,p.2719,1989.

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APPENDIXAATOMICUNITSAtomicunitsa.u.wererstintroducedbyHartreein1928.Intheseunitsthechargeoftheelectron,theelectronmass,andthe“rationalizedPlanckconstant”hareconsideredtohaveunitaryvalue:e=me=h=1:AAtomicunitsofsomephysicalquantities 1a.u.ofcharge=1:610)]TJ/F24 7.97 Tf 6.587 0 Td[(19C;1a.u.ofmass=9:110)]TJ/F24 7.97 Tf 6.587 0 Td[(31kg;1a.u.oflength=5:2910)]TJ/F24 7.97 Tf 6.586 0 Td[(11m;1a.u.ofenergy=27:21eV;1a.u.oftime=2:4188910)]TJ/F24 7.97 Tf 6.586 0 Td[(17s;1a.u.ofelectriceldstrength=5:142109V/cm;1a.u.ofmagneticeldstrength=2:35105T.Afewusefulconversions 1a.u.ofenergyequalstwicetheabsolutevalueofthegroundstateenergyinhydrogen=27.2eVor219474:6cm)]TJ/F24 7.97 Tf 6.587 0 Td[(1.1eVisequalto8068:92cm)]TJ/F24 7.97 Tf 6.586 0 Td[(1.1psisequalto41341.27a.u.oftime.6Tisequalto15:3210)]TJ/F24 7.97 Tf 6.586 0 Td[(5a.u.ofmagneticeldstrength.400V/cmisequalto7:77310)]TJ/F24 7.97 Tf 6.587 0 Td[(8a.u.ofelectriceldstrength.Classicalscalingofatomicproperties FollowingisalistwithpropertiesofRydbergatomsthatclassicallyscalewiththeprincipalquantumnumbern: 144

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145 bindingenergy/)]TJ/F24 7.97 Tf 6.586 0 Td[(1 n2;energylevelspacing/1 n3;theradiativelifetime/n3;theorbitalradius/n2;Keplerorbitalperiod/n3;theclassicalmomentum/1 n;Starkcrossingeld/1 3n5;thresholdofeldionization/1 9n4;thediamagneticenergy/n4;Zeemancrossingeld/1 n7=2;

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APPENDIXBPRODUCTIONOFAHALF-CYCLEPULSEAnHCPisproducedusingathinGaAssemiconductorwaferof'0:5mmthicknessand'3cm2surfacearea[ 222 ].Anelectriceldofabout10kV/cmisappliedparalleltothewafersurface.Thisbiasedsemiconductoractsasaphotoconductingswitch.Thebiaseldisshortedacrossthesemiconductorsurfacebyilluminatingthesystemwithan100fspulse,producedbyaTi:saphirelaseroperatingnear780nmabovethebandgapinGaAs'880nm.AsaresulttheGaAswaferisdrivenintoconductionandtheelectronsacceleratedbythebiaseldstartradiatingenergy.Asignicantratioofthisenergyupto'80%isemittedasaspatiallycoherent,almostunipolarelectricpulse,ofduration'0:5ps.TheHCPispolarizedinthedirectionofthebiaseldanditspeakamplitudeisdeterminedbythebiaseldintensity.TheHCPpulsehasabandwidthoftheorderof1THz,whichcanbemeasuredbyusinganelectriceldautocorrelator.TheshapeoftheHCPcanbefoundusingcrosscorrelationtechniques[ 222 ].TheHCPpulseproducedbytheabovemechanismhasashortunipolarcomponent,followedbyalongweaktailofoppositepolarity.Inrecentexperimentsthelongtailofthepulsehasbeencutoffbyasecondphotoswitch.Duringitsspatialpropagation,theHCPchangesshapeduetothediffractionofitslowfrequencycomponents.TheunipolarcharacterofanHCPisrestrictedtoasmallspatiallocationwhichdependsonthefocalizationprocedure.TheinteractionoftheatomswiththemainunipolarlobeoftheHCPisensuredbyanarrowextractionslit. 146

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APPENDIXCDISCRETIZATIONOFTHESCHRODINGEREQUATIONONANON-UNIFORMGRIDWeuseanon-uniformradialgridtodiscretizethesodiumeldfreeHamiltonian.Sinceweexpecttheradialwavefunctionstobeoscillatoryneartheoriginhigherkineticenergyandrelativelysmoothatlargerradialdistances,weusearadialgridthattakesintoaccountthesecharacteristics.Thegridthatweusedisgivenbyri=ri)]TJ/F24 7.97 Tf 6.586 0 Td[(1+min+)]TJ/F25 11.955 Tf 11.955 0 Td[(e)]TJ/F26 7.97 Tf 6.587 0 Td[(ri)]TJ/F18 5.978 Tf 5.756 0 Td[(1max)]TJ/F15 11.955 Tf 11.955 0 Td[(min;Cwiththeconventionthatr0=)]TJ/F15 11.955 Tf 9.298 0 Td[(min;r1=min:CIntheaboverelationwenoticethatminandmaxaretheminimum,and,respectively,maximumspacingsofthegrid,andisanaccelerationparameterthatdeterminestherateatwhichthegridspacingchangesfromminneartheorigintomaxattheouteredge.ThetypicalgridparametersusedinourstudywereN=11001500,min=0:01,max=5:and=510)]TJ/F24 7.97 Tf 6.587 0 Td[(4,whichensuredanaccuraterepresentationforstateswithn=15throughn=20.WestartthediscretizationoftheSchrodingerequationwiththefollowingassump-tions:rmax=0-thegridcontainsthewavefunctionlimr!0r=0 C limr!0r@ @r=0: 147

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148 TheactionSisdenedasS=Zt2t1L;0;CwhereLisL==:CThestationarityoftheactionSimpliesd dt@L @_)]TJ/F25 11.955 Tf 15.946 8.088 Td[(@L @=0:CInsphericalcoordinates,thewavefunctioncanbewrittenasasumoverradialfunctionslrtimessphericalharmonicsYlm=XllrYlm;:CTheexpressionforLcanbewrittenseparatelyforeachvalueofl.Thusthekineticandpotentialenergiesforsomelare=1 2Zrmax0drr2j@ @rj C =Zrmax0drr2Vlr+ll+1 2r2: C Usingtheseintegralexpressionspermitsustodealefcientlywiththeboundarycondi-tionsatr!0.TodiscretizeLonanonlineargrid,wedeneanintegrationrule.Functionsareevaluatedateachgridpointrj.Derivativesareevaluatedatthemidpointbetweengridpoints.Distanceincrementsarecalculatedasthedistancebetweenhalf-points.Thuswehavefr!fjrj@f @r2r2dr!fj+1)]TJ/F25 11.955 Tf 11.955 0 Td[(fj rj+1)]TJ/F25 11.955 Tf 11.955 0 Td[(rj2rj+1)]TJ/F25 11.955 Tf 11.956 0 Td[(rjrj+1+rj 22 C rj!rj+1+rj 2)]TJ/F25 11.955 Tf 13.15 8.088 Td[(rj+rj)]TJ/F24 7.97 Tf 6.587 0 Td[(1 2=rj+1)]TJ/F25 11.955 Tf 11.955 0 Td[(rj)]TJ/F24 7.97 Tf 6.586 0 Td[(1 2:

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149 Thisthreepointintegrationruleensuresthatthediscretizedequationiscorrecttothesecond-orderinthegridspacing.Forasinglel,LiswrittenasL=NPj=1ji_jr2jhrj+1)]TJ/F25 11.955 Tf 11.955 0 Td[(rj)]TJ/F24 7.97 Tf 6.586 0 Td[(1 2i)]TJ/F26 7.97 Tf 14.858 11.358 Td[(NPj=1j~Vjjr2jhrj+1)]TJ/F25 11.955 Tf 11.955 0 Td[(rj)]TJ/F24 7.97 Tf 6.587 0 Td[(1 2i)]TJ/F24 7.97 Tf 10.494 4.708 Td[(1 2NPj=1j+1)]TJ/F25 11.955 Tf 11.955 0 Td[(jj+1)]TJ/F25 11.955 Tf 11.955 0 Td[(j rj+1)]TJ/F25 11.955 Tf 11.955 0 Td[(rjhrj+1+rj 2i2;Cwhere~Vjistheeffectivepotentialincludingthecentrifugalbarrier.Bydeninganewradialfunctiongj=rjp rjjCwegetL=NPj=1gji_gj)]TJ/F26 7.97 Tf 15.523 11.357 Td[(NPj=1gj~Vj_gj)]TJ -104.796 -30.995 Td[()]TJ/F24 7.97 Tf 10.494 4.707 Td[(1 2NPj=1gjgj+1 rj+1)]TJ/F25 11.955 Tf 11.955 0 Td[(rjr 4 rj+2)]TJ/F25 11.955 Tf 11.955 0 Td[(rjrj+1)]TJ/F25 11.955 Tf 11.955 0 Td[(rj)]TJ/F24 7.97 Tf 6.586 0 Td[(11 rj+1rjrj+1+rj 22+gj)]TJ/F24 7.97 Tf 6.586 0 Td[(1 rj)]TJ/F25 11.955 Tf 11.955 0 Td[(rj)]TJ/F24 7.97 Tf 6.587 0 Td[(1r 4 rj+1)]TJ/F25 11.955 Tf 11.955 0 Td[(rj)]TJ/F24 7.97 Tf 6.587 0 Td[(1rj)]TJ/F25 11.955 Tf 11.955 0 Td[(rj)]TJ/F24 7.97 Tf 6.587 0 Td[(21 rjrj)]TJ/F24 7.97 Tf 6.587 0 Td[(1rj+rj)]TJ/F24 7.97 Tf 6.587 0 Td[(1 22)]TJ/F15 11.955 Tf 24.948 7.049 Td[(2gj rj+1)]TJ/F25 11.955 Tf 11.955 0 Td[(rj1 rj)]TJ/F25 11.955 Tf 11.955 0 Td[(rj)]TJ/F24 7.97 Tf 6.587 0 Td[(1rj+rj)]TJ/F24 7.97 Tf 6.587 0 Td[(1 2rj2+1 rj+1)]TJ/F25 11.955 Tf 11.955 0 Td[(rjrj+1+rj 2rj2:CWritingtheEuler-Lagrangeequationsintermsofthenewfunctionsgjweobtainthetimeevolutionequationforgji_gj=~Vjgj)]TJ/F15 11.955 Tf 13.151 8.087 Td[(1 2[Cjgj+1)]TJ/F15 11.955 Tf 11.955 0 Td[(2Djgj+Cj)]TJ/F24 7.97 Tf 6.586 0 Td[(1gj)]TJ/F24 7.97 Tf 6.587 0 Td[(1];CwhereCj=2 rj+1)]TJ/F25 11.955 Tf 11.955 0 Td[(rjr 1 rj+2)]TJ/F25 11.955 Tf 11.955 0 Td[(rjrj+1)]TJ/F25 11.955 Tf 11.956 0 Td[(rj)]TJ/F24 7.97 Tf 6.586 0 Td[(11 rj+1rjhrj+1+rj 2i2;Dj=1 rj+1)]TJ/F25 11.955 Tf 11.955 0 Td[(rj)]TJ/F24 7.97 Tf 6.586 0 Td[(11 rj)]TJ/F25 11.955 Tf 11.955 0 Td[(rj)]TJ/F24 7.97 Tf 6.587 0 Td[(1hrj+rj)]TJ/F24 7.97 Tf 6.587 0 Td[(1 2rji2+1 rj+1)]TJ/F25 11.955 Tf 11.955 0 Td[(rjhrj+rj)]TJ/F24 7.97 Tf 6.586 0 Td[(1 2rji2;~Vj=Vj+ll+1 2r2j:C

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150 SincethewavefunctionisnormalizedXjjr2jjjj2=1;CwegetXjjgjj2=1:CForagivennonlineargrid,thecoefcientsCjandDjcanbecomputed,andtheeigenval-uesEnlandeigenfunctionsgljoftherealsymmetrictridiagonalmatrixinEq. C canbefoundbyusingstandardalgorithms.

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APPENDIXDTHECHEBYSEVTIME-PROPAGATIONMETHODTheChebysevmethodisaglobaltime-propagationmethod,whichmeansitprovidesanalgorithmforcomputingthetime-evolutionoperator^Ut.Inthecaseofatime-independentHamiltonian,thetime-propagatorisgivenby^Ut=e)]TJ/F26 7.97 Tf 6.586 0 Td[(i^Ht.ThesimplestschemetocomputethisoperatorwouldbetoexpandtheexponentialoperatorinaTaylorseries:exp)]TJ/F25 11.955 Tf 9.299 0 Td[(i^Ht=1)]TJ/F25 11.955 Tf 11.955 0 Td[(i^Ht+:::;Dbutunfortunatelynumericalschemesbasedonthisexpansionarenotstable.Itcanbeshownthatstabilitycanbeachievedbyusingasymmetricmodicationoftheexpansion.Nevertheless,forlongdurationtimeserrorswillaccumulateinthephase,affectingtheaccuracyoftheresult.IntheChebysevtime-propagationschemethetime-evolutionoperatorisexpandedinaseriesofcomplexChebysevpolynomialsn^X.ThesepolynomialsareacomplexversionoftheChebysevpolynomialsTkbeingdenedask!=Tk)]TJ/F25 11.955 Tf 9.298 0 Td[(i!:DNotethattheyareorthogonalwithrespecttothefollowinginnerproduct=)]TJ/F25 11.955 Tf 9.299 0 Td[(iZi)]TJ/F26 7.97 Tf 6.587 0 Td[(if!g! p )-222(j!j2d!:DBecausethedenitiondomainofthispolynomialsis[)]TJ/F25 11.955 Tf 9.299 0 Td[(i;i],theHamiltonianmustbenormalizedto[)]TJ/F15 11.955 Tf 9.299 0 Td[(1;1].WedenethenormalizedHamiltonianasHnorm=H)]TJ/F25 11.955 Tf 11.955 0 Td[(Hmed H;D 151

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152 whereHmedandHaretheaverage,respectivelythehalf-widthoftheHamiltonianspectrum.WithaguessfortheminimumenergyVmin,themaximumvalueintheHamiltonianspectrumcanbecomputedbyapplyingH)]TJ/F25 11.955 Tf 12.414 0 Td[(Vminrepeatedlytoarandominitialvector.Thisprocedurewillrapidlyremoveallthecontributionsinthevector,excepttheonecorrespondingtothemaximumeigenvalue.Oncetheextremumspectralvaluesareknown,HmedandHarecomputedeasily.Withtheabovenotationtheexpansionfor^Utisgivenby^Ut=e)]TJ/F26 7.97 Tf 6.587 0 Td[(iHmedte)]TJ/F26 7.97 Tf 6.586 0 Td[(i^HnormtH'e)]TJ/F26 7.97 Tf 6.586 0 Td[(iHmedtNXn=0antHn)]TJ/F25 11.955 Tf 9.298 0 Td[(i^Hnorm;DwherethecoefcientsanareantH=Zi)]TJ/F26 7.97 Tf 6.586 0 Td[(ieitHxnx )]TJ/F25 11.955 Tf 11.955 0 Td[(x21=2dx=)]TJ/F25 11.955 Tf 11.955 0 Td[(n0JntH:DTocomputethetermn)]TJ/F25 11.955 Tf 9.299 0 Td[(i^Hnormintheaboveexpansion,wecanmakeuseoftherecursionrelationsatisedbythecomplexChebysevpolynomials:n+1)]TJ/F25 11.955 Tf 9.299 0 Td[(i^Hnorm=)]TJ/F15 11.955 Tf 9.299 0 Td[(2i^Hnormn)]TJ/F25 11.955 Tf 9.298 0 Td[(i^Hnorm+n)]TJ/F24 7.97 Tf 6.587 0 Td[(1)]TJ/F25 11.955 Tf 9.299 0 Td[(i^Hnorm;Dwhere0=^Iand1)]TJ/F25 11.955 Tf 9.299 0 Td[(i^Hnorm=)]TJ/F25 11.955 Tf 9.298 0 Td[(i^Hnorm.Inordertosavestorageonlytheresultsforthenandn)]TJ/F15 11.955 Tf 12.518 0 Td[(1termsintheexpansionmustbesaved.ThenumberoftermsintheexpansionneededtoachieveconvergenceisdeterminedbythequantitytH.BecausetheBesselfunctionsJnxhaveanexponentialdecayassoonasnbecomeslargerthantheargumentx,thenumberoftermsrequiredisusuallyslightlylargerthantH.TheChebysevpolynomialexpansionisoptimalsincetheerrorinapproximationisminimalcomparedtootherpolynomialexpansions.

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APPENDIXESOMEFORMULADERIVATIONSDerivationofEq. 3 Thesquareoftheuncertaintyisgivenbyx2=h^x2i)-222(h^xi2:EThen,thetime-derivativeoftheuncertaintyisdx2 dt=d dth^x2i)]TJ/F15 11.955 Tf 19.261 0 Td[(2h^xid dth^xi=D)]TJ/F25 11.955 Tf 9.298 0 Td[(ih^x2;^HiE)]TJ/F15 11.955 Tf 11.955 0 Td[(2h^xiD)]TJ/F25 11.955 Tf 9.298 0 Td[(ih^x;^HiE=D)]TJ/F25 11.955 Tf 9.298 0 Td[(i^xh^x;^Hi)]TJ/F25 11.955 Tf 11.955 0 Td[(ih^x;^Hi^xE)]TJ/F15 11.955 Tf 11.955 0 Td[(2h^xiD)]TJ/F25 11.955 Tf 9.298 0 Td[(ih^x;^HiE=2 mh^x^p+^p^xi 2)-222(h^xih^pi; E where^HistheHamiltonianofthesystem,andweusedthefactthat)]TJ/F25 11.955 Tf 9.299 0 Td[(iDh^x;^HiE=h^pi m:EDerivationofEq. 3 Similartotheabovederivationwehavedrt dt=d dth^x^p+^p^xi 2)]TJ/F25 11.955 Tf 15.264 8.088 Td[(d dtfh^xih^pig=D)]TJ/F25 11.955 Tf 9.298 0 Td[(ih^x^p+^p^x;^HiE 2)]TJ/F31 11.955 Tf 11.955 13.271 Td[(D)]TJ/F25 11.955 Tf 9.298 0 Td[(ih^x;^HiEh^pi)-222(h^xiD)]TJ/F25 11.955 Tf 9.298 0 Td[(ih^p;^HiE=D2^p2 m+2h^xiE 2)-222(h^pih^pi=m)]TJ/F25 11.955 Tf 11.955 0 Td[(h^xi=1 mh^p2i)-222(h^pi2=p2 m; E 153

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154 whereweusedthefactthat)]TJ/F15 11.955 Tf 9.299 0 Td[(ih^p;^Hi=)]TJ/F15 11.955 Tf 9.298 0 Td[(i[^p;V^x]=i[^p;^x]=.DerivationofEq. 3 UsingEqs. 3 andthevaluesa,0andbdenedinEqs. 3 and 3 ,wehaveReA0=2 2h NImA0=2 2h1 202+1=2 N; E whereN==22+2.Fromtheaboveequations,theratioReA0=ImA0becomesReA0 ImA0= N 202+1 2:EUsingthefactthatinEq. 3 wehave=)]TJ/F15 11.955 Tf 9.299 0 Td[(2ka,andthata=02+bandb=1=022,weobtainN==22+2==22+404k2+b2=404"1 20222+k2+b2#=404b2+k2+b2: E InsertingthisresultinEq. E wendReA0 ImA0=)]TJ/F15 11.955 Tf 9.298 0 Td[(2k02+b 202[b2+k2+b2]+1 2=)]TJ/F25 11.955 Tf 9.299 0 Td[(k+b b2+k2+b2+b=)]TJ/F25 11.955 Tf 9.298 0 Td[(k k2+b+b: E

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BIOGRAPHICALSKETCHCorneliuManescuwasbornonOctober21st,1968,inTurnuSeverin,arelativelysmalltowninthesouthwestofRomania.ThethirdchildofMr.andMs.Al.Manescu,hegrewupinanenvironmentwhereeducationisvalued.AftergraduatingfromoneofthebesthighschoolsinBucharest,theMathematics-PhysicsHigh-SchoolNo.3,hepassedtheadmissionexamatthePhysicsFacultyoftheUniversityofBucharestinthe9thposition.Atgraduation,in1995,hereceiveda“LicentiateinPhysics”diplomainthespecialty“theoreticalphysics.”HecontinuedfurtherintheMasterofScienceprogramatthesamefaculty,andin1996hewasbestowedthe“MasterofScienceinPhysics”diplomafromtheUniversityofBucharest.InFall1998,CorneliuManescustartedtopursueaPh.D.degreeinphysicsattheUniversityofFloridainU.S.Familiarwithcomputersandprogramming,inthesummerof1999,healsostartedtakingclassesincomputerscienceattheComputerandInformationScienceandEngineeringCISEdepartmentofthesameuniversity.InSpring2000,hewasadmittedintheMasterofScienceprograminCISE.InparallelwithhisstudiesfortheMSincomputerscience,CorneliuManescuworkedtowardshisPh.D.inphysics,anddefendedhisoralexaminationinphysicsinFall2001.SincehisarrivalattheUniversityofFlorida,CorneliuhasworkedasateachingassistantTAinthePhysicsdepartment,teachingtheelectricity,magnetismandopticslaboratory.IntheSpringof2003,CorneliucompletedtheMasterofScienceprogramincomputerscience,withanMSdiploma.IntheSummerofthesameyearhewasacceptedintothePh.D.programincomputerengineering,attheUniversityofFlorida.Lately,CorneliuhasalsobeenaTAintheCISEdepartment,forthe“AlgorithmsandDataStructures”and“ComputerArchitecture”courses. 155

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156 Inphysics,hiscurrentresearchinterestsincludequantumcontrolofatomicandmoleculardynamics,andlaser-atominteractions.Incomputerscience,Corneliuisinter-estedintheeldofmobilemultimedia,withemphasisonvideoandsoundprocessingandcompression.