Theory of nonlinear optical spectroscopy of electron spin co(5)

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)