Theory of nonlinear optical spectroscopy of electron spin co(4)

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)