第四章-量子力学密度矩阵

574.112jjp←→ΨjΨ3FˆkkkFFψψ=ˆ2211Ψ+Ψ=ΨCCFˆkF22211)(Ψ+Ψ=kkkCCFWψψ21,ΨΨ21,ΨΨ21,ppFˆkF222121)(ppFWkkkΨ+Ψ=ψψ1ψ1=ψψψψFˆkkkFFψψ=ˆFΨΨΨ=FFˆ{}n∑∑ΨΨ=ΨΨ=nnnFnFnnFˆˆ581ψψρ=ˆ2)ˆˆ(ˆˆρρFTrnFnFn==∑3FΨkFkkkCψρψˆ2=21∑ΨΨ=kkkkpρˆ1=∑kkp2)ˆˆ(ˆˆρρFTrnFnFn==∑3FkFkkkFWΨΨ=ρˆ)(3FˆkkkFFψψ=ˆ{}kψ1ρˆρρˆˆ=+2ρˆ3ρˆ1ˆ=ρTr==)(1)(1ˆ2ρTr401nnρ≤≤nnρn5ψρˆϕϕψϕψψϕρ=ˆρρˆˆ2=6ˆρ1SSρρ−′=4159ρˆ()ρψψˆˆˆFtrFF==21{}n∑=nnnCψρˆ*ˆnmmnCCnmnm===ψψρρρˆmnρψmnψmnmnρmmρψm2∑∑=ΨΨ=kknkmkkkkkmnCCpnmp*ρBloch1p()()pttσψσψ==2BlochBloch11/21/2122×cos0sin122iieeγϕθθψ=+,,θϕγexp()iγcos0sin122ieϕθθψ=+6021/24{}0123σσσσ312123111(1)122ppipppippρσ+−==++−i()21det14pρ=−ρ1p≤3BlochBlochBlochBloch4.2ρˆ1]ˆ()1ˆˆ(),()dtHttdtiρρ=1()()()HSSSFtUtFUt−=11()()()()(0)(0)()()(0)()SSSStttUtUtUtUtρρ−−=ΨΨ=ΨΨ=20iHHH=+0101()()()()()(0)()(0)ISSSSItUttUtUtUt−−Ψ=Ψ=Ψ=Ψ61()()()IIIitHttt∂Ψ=Ψ∂010()()()ISSSFtUtFUt−=0()()1(),()IIIIdFtFtFtHtdtti∂=+∂010()()iiISSHUtHUt−=00100()()ISSHUtHUt−=010ˆˆ()()()()ISStUttUtρρ−=010ˆ()exp(/)SUtiHt−=]ˆ()1ˆˆ(),()iIIIdtHttdtiρρ=001ˆˆ()()()()ISStUttUtρρ−=00ˆ()exp(/)SUtiHt=Hˆn())(1)(tEEittddnmnmnmρρ−=)exp()0()(titnmnmnmωρρ−=nmnmEE−=ωmnω(())FtrtFρ={}()()[(),()]()FtiFtitrHttFtttρ∂∂=+∂∂4.3rq62()qr,Ψ)(ˆrFr1BA+A{}AmAB{}BnBBABAnmnm=AmBnBA+2BA+ABΨ∑=ΨmnBAmnABnmC12=∑mnmnC3∑=ΨΨ=jknmBABAnmjkABABABkjnmCC*ˆρ1AAF1AAFAAAFBAIFF⊗=BIB2AF)ˆˆ(FTrFABABρ=ABBAABIFFΨ⊗Ψ=∑∑⊗=mnBAmnjkBABAjknmCIFkjC*63)ˆˆ(FTrFAAρ=)ˆ(ˆABBATrρρ=2)ˆ(ˆABBATrρρ=31AAρρˆˆ=+21)ˆ(=AATrρ3AρˆAρρ→411()MABABABBBjtrjjρρρ===∑1MBBBjjjI==∑1()NBAABABAAitriiρρρ===∑1NAAAiiiI==∑2,AB{}Ai{}Bj{}ijij=ψ{}ijABρψψ=1Lind-bladLind-blad011ˆ[,]{,}2dHdtiµµµµµρρρρ++=+ΓΓ−ΓΓ∑µΓ+ΓΓµµρµµµµρρΓΓ+ΓΓ++64(0)0ρ()0tρ)(tρ1.(decoherence)(dephasing)2Bloch-Redfield1Bloch-RedfieldnmnmnmnmklklkliRρωρρ+=∑()()()()nmkllmnklmnknklrrmlmnrrkrrRδδ+−−+=Γ+Γ−Γ−Γ∑∑()()()2int,int,00lmilmnklmnkdeHHωτττ∞−−−Γ=∫()()()2int,int,00nkilmnklmnkdeHHωτττ∞−+−Γ=∫()intintexpexpBBiiHtHtHHt=−2RedfieldintsBHHHH=++1sHBHintHxX=⋅xXt=−∞()()()totsBρρρ−∞=−∞⊗−∞2T()[]expBBBHkTρ−∞=−t=−∞X()0BTrXρ−∞=65()()()()†000tottottWtttWttρρ=−−⇒()()()int,tottotdittHtdtρρ=−3()00intintiiHtHtHteHe−=intiHtWe−=0sBHHH=+()()()†00AtUtAUt=()00expiUtHt=−()()()()()()()()††0000tUttUtUtWtψψψψ===()expiUtHt=−3()()()int,tttottotidHdρτρτττ−∞−∞=∫∫⇒()()()()0int,tottottotittyHtydyρρρ−∞−−∞=++∫()()0int,totitHtdρτττ∞=−−−∫()()int0,totitHtdρτττ∞=−−∫⇒()()()()int0,tottottotittHtdρρρτττ∞=−∞+−−∫43()()()()()()intintint0,,,tottottotdiitHttHtHtddtρρρτττ∞=−−∞−−−∫5()()sBtottTrtρρ=(),0BBHρ−∞=X()()()()0BBiiHtHtBBBTrXtTreXeTrXρρρ−−∞=−∞=−∞=6()()()(){}intint201,,sBtotdtTrtHtHtddtρρτττ∞=−−−∫766()()ssiiHtHtssteteρρ−=⇒()()(),ssssiiiiHtHtHtHtssssiteteetHeρρρ−−=−87()()()()(){}intint201,,,ssiiHtHtsssBtotdittHeTrtHtHteddtρρρτττ∞−−=−−−∫()()()()(){}intintint201,ssiiHtHtBtottotdeTrtHtHttHteτρτττρτ∞−=−−−−−−∫()()(){intint201siHtBtotdeTrtHtHtτρττ∞−=−−−∫()()()intinttotHttHtτρτ−−−()()()intinttotHttHtρττ−−−()()()}intintsiHttotHtHtteτρτ+−−()()()()()201ssiiHtHtsBXtXtetxtxteτρττ∞−=−−−−∫()()()()()ssiiHtHtsBXtXtexttxteττρτ−−−−−()()()()()ssiiHtHtsBXtXtexttxteτρττ−−−−−()()()()()ssiiHtHtsBXtXtextxttedττρττ−+−−−()()()()()20100ssiiHtHtsBXXetexxτρττ∞−=−−−−∫()()()()()00ssiiHtHtsBXXxetexττρτ−−−−−()()()()()00ssiiHtHtsBXXxetexτρττ−−−−−()()()()()00ssiiHtHtsBXXxxetedττρττ−+−−−()()()()()20100ssiiHHsBXXetexxτττρττ∞−=−−−∫()()()()()00ssiiHHsBXXxetexττττρτ−−−−67()()()()()00ssiiHHsBXXxetexτττρττ−−−−()()()()()00ssiiHHsBXXxxetedττττρττ−+−−9t[],t−∞Redfield1112,sETT−−∆sH9()()()BXtXtττΦ=−cτ0cτ1112,cTTτ−−()stρτ−()()ssiiHHssteteττρρτ−=−Redfield()()()()()()(){201,00ssssBdittHXXtxxdtρρτρτ∞−=−−∫()()()()()00sBXXxtxττρ−−()()()()()00sBXXxtxτρτ−−()()()()()}00sBXXxxtdττρτ+−10sHnεnmnmωεε=−Bloch-Redfield:ssρρ=()()()nmssditntHHtmdtρρρ−−()()()()()201000ssiiHHBXXntexexmτττρ∞−=−∫()()()()()000ssiiHHBXXnexetxmτττρ−−()()()()()000ssiiHHBXXnxtexemτττρ−−68()()()()()000ssiiHHBXXnxexetmdτττρτ−+()()()()()20,10lriinmnmnmnllrrmBlrtitXXtexexετετρωρτρ∞−+=−∑∫()()(),0nkiinkkllmBklXXexetxετεττρ−−∑()()(),0lmiinkkllmBklXXxtexeετεττρ−−∑()()(),0kriinrrkkmBkrXXxexetdετεττρτ−+∑()()20,10lrinmnmnmlrrmnlBlridXXexxωτρωρττρ∞−+=−∑∫()()20,10nkinklmklBkldXXexxωτττρ∞−+∑∫()()20,10lminklmklBkldXXexxωτττρ∞−+∑∫()()20,10rkinrrkkmBkrdXXexxωτττρ∞−−∑∫()()20,10lrinmnmnmnklrrmklBklridXXexxωτρωρδττρ∞−+=−∑∑∫()()20,10nkilmnkklBkldXXexxωτττρ∞−+∑∫()()20,10lmilmnkklBkldXXexxωτττρ∞−+∑∫()()20,10rkilmnrrkklBlkrdXXexxωτδττρ∞−−∑∑∫()()int,int,20,10lrinmnmnmnklrrmklklrideHHωτρωρδττρ∞−+=−∑∑∫()()int,int,20,10nkilmnkklkldeHHωτττρ∞−+∑∫()()int,int,20,10lmilmnkklkldeHHωτττρ∞−+∑∫69()()int,int,20,10rkilmnrrkkllkrdeHHωτδττρ∞−−∑∑∫()()()2int,int,00lmilmnklmnkdeHHωτττ∞−−−Γ=∫()()()2int,int,00nkilmnklmnkdeHHωτττ∞−+−Γ=∫()intintexpexpBBiiHtHtHHt=−()()()(),,,,nmnmnmnklrrmkllmnkkllmnkkllmnrrkklklrklklklriρωρδρρρδρ−−+++=−Γ+Γ+Γ−Γ∑∑∑∑∑∑()()()(),lmnklmnknklrrmlmnrrkklklrrδδρ+−−+=Γ+Γ−Γ−Γ∑∑∑nmklklklRρ=∑nmnmnmnmklklkliRρωρρ+=∑11RedfieldRRedfieldRedfieldR()()()()nmkllmnklmnknklrrmlmnrrkrrRδδ+−−+=Γ+Γ−Γ−Γ∑∑