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1、AppendixASemianalyticalDerivationoftheAtomicPolarizabilityThepurposeofthissectionistoderivethelinearpolarizabilityofatwo-levelquan-tumsysteminthedipoleapproximation.Thequantumsystemmightbeanatom,amolecule,oraquantumdot.Forsimplicity,wewilldenotethesystemasatom.Once
2、theatomicpolarizabilityisknown,theinteractionbetweenatomandradiationfieldcanbetreatedinmanyapplicationsclassically.Agenerallyvalidanalyticalexpres-sionforthepolarizabilitycannotbederived.Instead,onehastodistinguishbetweenseveralapproximateexpressionswhichdependonthe
3、relativespectralpropertiesofatomandfield.Thetwomostimportantregimesareoff-resonanceandnear-resonanceexcitation.Intheformercase,theatomresidesmostlyinitsgroundstatewhereasinthelattercase,saturationoftheexcitedlevelbecomessignificant.Accordingtoquantummechanics,thebehav
4、iorofasystemofNparticlesisde-scribedbythewavefunctionΨ(r,t)=Ψ(r1,...,rN,t),(A.1)whereridenotesthespatialcoordinateofparticleiandtrepresentsthetimevariable.Tomakethenotationsimpler,theentiresetofparticlecoordinatesisrepresentedbythesinglecoordinaterwhichalsoincludes
5、spin.However,itshouldbekeptinmindthatoperationsonrareoperationsonallparticlecoordinatesr1,..,r2.ThewavefunctionΨisasolutionoftheSchr¨odingerequationdHˆΨ(r,t)=i¯hΨ(r,t).(A.2)dtHˆdenotestheHamiltonoperator,alsocalledHamiltonian.Itsformdependsontheconsideredsystem.12A
6、PPENDIXA.ATOMICPOLARIZABILITYForanisolatedatomwithnoexternalperturbationtheHamiltonianistimeinde-pendentandithasthegeneralformXh¯h2iHˆ=−∇2+V(r,r).(A.3)oiij2mii,jThesumrunsoverallparticlesinvolvedinthesystem.Theindexin∇ispecifiesoper-ationonthecoordinateri.V(ri,rj)is
7、thepotentialinteractionenergyoftheithandjthparticle.Ingeneral,Vhascontributionsfromallfourfundamentalinteractionssofarknown,namelystrong,electromagnetic,weak,andgravitationalinteraction.Forthebehaviourofelectronsonlytheelectromagneticcontributionisofimportance,andw
8、ithintheelectromagneticinteractiontheelectrostaticpotentialisofdominance.Sincethemassesofnucleiaremuchgreaterthanthemassofanelectron,thenucleimov
