Theory of nonlinear optical spectroscopy of electron spin coherence in quantum dots

We study in theory the generation and detection of electron spin coherence in nonlinear optical spectroscopy of semiconductor quantum dots doped with single electrons. In third-order differential transmission spectra, the inverse width of the ultra-narrow

Theoryofnonlinearopticalspectroscopyofelectronspincoherenceinquantumdots

Ren-BaoLiu,1S.E.Economou,2L.J.Sham,2andD.G.Steel3

1

arXiv:cond-mat/0610075v1 [cond-mat.mes-hall] 3 Oct 2006

DepartmentofPhysics,TheChineseUniversityofHongKong,Shatin,N.T.,HongKong,China2

DepartmentofPhysics,UniversityofCaliforniaSanDiego,LaJolla,California92093-03193

TheH.M.RandallLaboratoryofPhysics,UniversityofMichigan,AnnArbor,MI48109

(Dated:February6,2008)

Westudyintheorythegenerationanddetectionofelectronspincoherenceinnonlinearopticalspectroscopyofsemiconductorquantumdotsdopedwithsingleelectrons.Inthird-orderdi erentialtransmissionspectra,theinversewidthoftheultra-narrowpeakatdegeneratepumpandprobefrequenciesgivesthespinrelaxationtime(T1),andthatoftheStokeandanti-Stokespinresonances

givesthee ectivespindephasingtimeduetotheinhomogeneousbroadening(T2).Thespinde-phasingtimeexcludingtheinhomogeneousbroadeninge ect(T2)ismeasuredbytheinversewidthofultra-narrowhole-burningresonancesin fth-orderdi erentialtransmissionspectra.

PACSnumbers:76.70.Hb,42.65.An,78.67.Hc

T2isalsomeasuredforQDensembles,2,3,5,12,13givingalowerboundofT2.SpinechoinmicrowaveESRexperimentsisaconventionalapproachtomeasuringthespindecoherencetimeT2excludingtheinhomoge-neous18,19,20which,however,islessfeasi-bleforIII-Vcompoundquantumdotsduetotheul-10 6secandtrafasttimescalesinsuchsystems(T2<～ 9 <T2～10sec).Indeed,theremarkablespinechoex-perimentsincoupledQDsdonebytheMarcusgroupare

performedwithratherlongDCvoltagepulsesinsteadofinstantaneousmicrowavepulses.12Alternatively,picosec-ondopticalpulsesmaybeusedtomanipulateelectronspinsviaRamanprocesses21andrealizethespinecho,which,however,stillneedtoovercomethedi cultyofstabilizingandsynchronizingpicosecondpulsesinmi-crosecondtime-spans.ArecentexperimentbyGreilichetalalsoshowsthattheinhomogeneousbroadeninge ectcanbe lteredoutfromthespincoherencemode-lockedbyaperiodictrainoflaserpulses.5

Inthispaper,wewillstudythefrequency-domainnon-linearopticalspectroscopyasanotherapproachtomea-suringtheelectronspindecoherencetimes.Particularly,

1

thespindephasingrateT2iscorrelatedtothewidthofultra-narrowhole-burningpeaksin fthorderdi erentialtransmission(DT)spectra.Thishole-burningmeasure-mentofthespindephasingtimeisanalogoustotheex-plorationofslowrelaxationofopticalcoherenceinatomicsystemsbythethird-orderhole-burningspectroscopy.22Herethe fthordernonlinearityisneededbecausethecreationofspincoherencebyRamanprocessesinvolvesatleasttwoordersofoptical eldandhole-burningtwomore.Thestate-of-the-artspectroscopyalreadyhastheultra-highresolution(muchbetterthanMHz-resolution)toresolvetheslowspindecoherenceinmicrosecondorevenmillisecondtimescales.Theorganizationofthispaperisasfollow:Afterthisintroductorysection,Sec.IIdescribesthemodelforQDsystemandthemaster-equationapproachtocalculatingthenonlinearopticalsusceptibility.Sec.presentstheresultsanddiscussions.Sec.IVconcludesthispaper.Thesolutionofthemasterequationinfrequencydomain

I.INTRODUCTION

Electronspincoherenceinsemiconductorquantumdots(QDs)isaquantume ecttobeexploitedinemerg-ingtechnologiessuchasspin-basedelectronics(spintron-ics)andquantumcomputation.1Theelectronspinde-coherenceisakeyissueforpracticalapplicationoftheelectronspinfreedomandisalsooffundamentalinter-estinmesoscopicphysicsandinquantumphysics.TheelectronspindecoherenceinQDs,however,isyetpoorlycharacterized.Byconvention,thespindecoherenceisclassi edintothelongitudinalandthetransverseparts,whichcorrespondtothespinpopulation ipandtheZee-manenergy uctuationprocessesandareusuallychar-acterizedbytherelaxationtimeT1andthedephasingtimeT2,respectively.Mostcurrentexperimentsarecar-riedoutonensemblesofspins,composedofeithermanysimilarQDsormanyrepetitionsof(approximately)identicalmeasurementsonasingleTheensemblemeasurementsaresubjectedtotheinho-mogeneousbroadeningoftheZeemanenergywhichre-sultsfromthe uctuationoftheQDsize,shapeandcom-poundcomposition(andinturntheelectrong-factor)andfromtherandomdistributionofthelocalOver-hauser eld(duetothehyper neinteractionwithnuclearspinsinthermalstates).Theinhomogeneousbroadening

leadstoane ectivedephasingtimeT2.Thethreetimescalescharacterizingtheelectronspindecoherencecandi erbyordersofmagnitudeusuallyintheorder

T1 T2 T2.Forexample,inatypicalGaAsQDatalowtemperature(<～4Kelvin)andunderamoderatetostrongmagnetic eld(0.1～10Tesla),therelaxationtimeT1canbeintheorderofthedephasingtimeT2isuptoseveralmicroseconds,5,12,14

andthee ectivedephasingtimeT2canbeasshortasa

3,5,12,13

fewnanoseconds.

TheissueishowtomeasurethecharacteristictimesofelectronspindecoherenceinQDs.TherehavebeenmanyexperimentsbothinopticsandinwhichestablishthespinrelaxationtimeT1inQDsofdi erentmaterials.Thee ectivedephasingtime

We study in theory the generation and detection of electron spin coherence in nonlinear optical spectroscopy of semiconductor quantum dots doped with single electrons. In third-order differential transmission spectra, the inverse width of the ultra-narrow

ispresentedinAppendixA.

g( )=e ( T2)

2

/2

√/(

II.MODELANDTHEORY

ThesystemtobestudiedisasemiconductorQDdopedwithasingleelectron.ThegeometryoftheQDun-deranexternalmagnetic eldandopticalexcitationisshowninFig.1(a)and(c).TheQDisassumedofashapewithsmallthicknessinthegrowthdirectionandrelativelylargeradiusinthelateraldirections,asintheusualcasesof uctuationQDsandself-assembledQDs.3,4,9,10,11Toenablethegenerationandmanipula-tionoftheelectronspincoherencethroughRamanpro-cesses,amagnetic eldisappliedalongalateraldirec-tion(x-axis).Thepropagationdirectionsofthepumpandprobelaserbeamsareclosetothegrowthdirection(z-axis).Thetwoelectronspinstates|± aresplitbythemagnetic eldwithZeemanenergyω0.Thestrongcon nementalongthez-axisinducesalargesplittingbe-tweentheheavyholeandthelightholestates,thustherelevantexcitonstatesarethegroundtrionstates|τ and|τ¯ whichconsistoftwoelectrons(includingthedopedoneandonecreatedbyopticalexcitation)inthesingletspinstateandoneheavyholeinthespinstate|+3/2 and| 3/2 (quantizedalongthez-axiswithnearlyzeroZeemansplitting),respectively.Similarly,wecanalsoneglecttheexcitationofhigherlyingtrions,bi-excitonandmulti-excitonstatessincetheenergyofaddinganexcitonineachcaseiswellseparatedfromenergyofthelowesttrionstates.Theselectionrulesfortheopticaltransitionsaredeterminedbythe(approximate)conser-vationoftheangularmomentumalongthegrowthdirec-tionsothatacircularlypolarizedlightwithpolarizationσ+orσ connectsthetwoelectronspinstatestothetrionstate|τ orτ¯ ,respectively[seeFig.1(b)].TherelaxationprocessesinthesystemareparameterizedbytheexcitonrecombinationrateΓ1,theexcitondephas- 1

ingrateΓ2,thespinrelaxationrateT1,andthespin

1

dephasingrateT2.Theinhomogeneousbroadeningleadstoarandomcomponent totheZeemansplitting:ωc=

FIG.dot,(b)op-tical

We study in theory the generation and detection of electron spin coherence in nonlinear optical spectroscopy of semiconductor quantum dots doped with single electrons. In third-order differential transmission spectra, the inverse width of the ultra-narrow

matrixcanbeexpandedas

ρα,β( )=2πEj···EkEm···El

j,...,k;m,...,l

×ρα,β

(j···km¯···¯l)

δ j···km¯···¯l,

(2)

where j···km¯···¯l≡ j+···+ k ( m+···+ l).The

¯···¯l)

derivationofthedensitymatrixcomponentρ(j···kmuptothe fthorderislengthybutstraightforward.The nalresultisaveragedwiththeinhomogeneousbroadeningdistributiong( ).

III.

RESULTSANDDISCUSSIONS

Thelinearopticalsusceptibilityisgivenby

g( )(j)

ρ=

We study in theory the generation and detection of electron spin coherence in nonlinear optical spectroscopy of semiconductor quantum dots doped with single electrons. In third-order differential transmission spectra, the inverse width of the ultra-narrow

Whentheinhomogeneousbroadeningisincluded,theultra-narrowresonancewillbesmearedintoabroad-enedpeakalongthedirection 3¯1= 43¯21withwidth

～1/T2.Butintheperpendiculardirection(de nedby 3¯1= 43¯21),thepeakwidthremainsunchanged.Sowhen 43¯21is xedaroundω0and 3¯1isscanned,orviceversa,theDTspectrumwillpresentasharppeakwhose

1

widthmeasurestheinversespindephasingtimeT2.Thispeakhasthecharacterofhole-burning:The rstfrequencydi erenceactsjustasaselectionofQDswithZeemanenergyωc= 43¯2¯1fromtheinhomogeneouslybroadenedensemble.Thehole-burningresonanceresult-ingfromtheexcitationpathwayinEq.(8),however,emergestogetherwiththeresonanceassociatedwiththe

1

spinpopulation( 4¯asgiveninEq.(4a).To2+i/T1)

avoidthecomplicationofmixingtwotypesofresonancestructures,wewouldrathermakeuseanothermechanismforspincoherencegeneration,namely,thespontaneousemissionthatconnectsthetrionstatetothetwospinstatesthroughthevacuum eld[relatedtothe rsttermintherighthandsideofEq.(1d)].3,29,30,31

Thegenerationofspincoherenceinthe fthorderopti-calresponseinvolvingthespontaneousemissioncantakeaquantumpathwaylike

4231(4321)ρτ, →ρτ,τρτ, →ρ+, → ρ ,+,ρ , →

(9)

wherethelaststepisthespontaneousemission.ThisopticalprocessisillustratedbytheFeynmandiagraminFig.4(b).Thespincoherencegeneratedbythesponta-neousemissionandthatbyopticalexcitationcanhave

(3¯1)(43¯2¯1)

oppositespinindices[ρ+, →ρ ,+],whichisimpossi-bleinquantumpathwayswithoutthespontaneousemis-sion[ascanbeseenfromFig.4(a)].Thusthedoubleresonancebecomes

E(3)

E

(3¯1)E

(43¯1)E

¯¯

Γ1

(43¯2¯1)

ρ ,+

¯1)¯(432

～

Γ1/( 43¯2¯1+i2Γ1)

We study in theory the generation and detection of electron spin coherence in nonlinear optical spectroscopy of semiconductor quantum dots doped with single electrons. In third-order differential transmission spectra, the inverse width of the ultra-narrow

asafunctionoftheprobefrequencywithpumpfrequencies xedtobesuchthat 1=9µeV, 5=2µeV,and 1¯3= 5¯2=ω0=20µeV.Inboth gures,thespindephasingtimeT2=20,50,100and200nsforthesolidcurvesfromtoptobottom,andthedottedlineiscalculatedwiththespontaneouslygeneratedspincoherencearti ciallyswitchedo (forT2=100ns).TheparametersarethesameasinFig.5.

o -diagonalcoherencedemonstratethemselvesinthird-orderdi erentialtransmissionspectraasultra-narrowresonances.TheinhomogeneousbroadeningsmearsoutthesharpStokeandanti-Stokepeaksrelatedtotheo -diagonalspincoherence.Thusthespinrelaxationtime

T1andthee ectivedephasingtimeT2aremeasuredbythethird-orderspectra.Inthe fth-orderopticalresponse,thegenerationofthespincoherencebybothsecond-andfourth-orderopticalprocessesleadstodou-bleresonancestructuresintwo-dimensionalDTspec-tra,whicharesmearedbytheinhomogeneousbroadeningalongonedirectioninthefrequencyspacebutpresentsultra-narrowhole-burningresonancesalongtheperpen-diculardirection.SothespindephasingtimeT2ismea-suredastheinversewidthoftheholeburningpeak.Thespontaneousemission-generatedspincoherence3,29isusefultoproduceholeburningresonanceswellsepa-ratedfromthespin-populationresonancesinthe fth-orderspectra.Thefrequenciesoftheoptical eldcanbecon guredproperlytoenablethedetectionofthesig-nalintheDTsetupinsteadofthemulti-wavemixingones.Inpractice,thepumpandprobefrequenciesmaybegeneratedfromasinglecontinuous-wavelasersourceby,e.g.,acousto-opticalmodulation.25Sincetheultra-narrowholeburningpeaksareratherinsensitivetotheglobalshiftofthelaserfrequenciesandvariationofthehole-burningfrequency,non-stabilizedlasersourcesmaybeusedtoresolvetheslowspindecoherence.25

Acknowledgments

ThisworkwaspartiallysupportedbytheHongKongRGCDirectGrant2060284andbyARO/NSA-LPS.

withsingleelectrons,whichisstudiedinthispaperuptothe fthordernonlinearitywithaΛ-typethree-levelmodel.Theelectronspincoherenceisgeneratedbytheoptical eldthroughRamanprocessesaswellasbyspon-taneousemissionofthetrion.Thespinpopulationand

APPENDIXA:SOLUTIONOFTHEMASTER

EQUATION

ThemasterequationinEq.(1)canbesolvedinthefrequencydomainbyFouriertransformationtobe

ρτ,±( )=ρτ,τ(ω)=

+E ( ω)ρτ,±( ) E( +ω)ρ

τ,±( )

±

E( ω)ρ±,±(ω) E( ω)ρ ,±(ω)+E( ω)ρτ,τ(ω)

2π,

2π

,(A1a)(A1b)

ρ±,±(ω)=p±2πδ(ω) (ω+iΓ1+ip±/T1)

E ( ω)ρτ, ( ) E( +ω)ρ τ, ( )

+(iΓ1 ip±/T1)ρ+, (ω)=

iΓ1ρτ,τ(ω)

ω ωc+i/T2

d

E ( ω)ρτ,±( ) E( +ω)ρ τ,±( )

2π2π,

(A1c)

We study in theory the generation and detection of electron spin coherence in nonlinear optical spectroscopy of semiconductor quantum dots doped with single electrons. In third-order differential transmission spectra, the inverse width of the ultra-narrow

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