Theory of nonlinear optical spectroscopy of electron spin co

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