文献翻译-Number-phase quantization of a mesoscopic RLC circuit

Chin.Phys.BVol.21,No.2(2012)020402Number-phasequantizationofamesoscopicRLCcircuitXuCheng-Lin()DepartmentofMathematics,YunnanNormalUniversity,Kunming650092,China(Received16July2011;revisedmanuscriptreceived29July2011)Withthehelpofthetime-dependentLagrangianforadampedharmonicoscillator,thequantizationofmesoscopicRLCcircuitinthecontextofanumber-phasequantizationschemeisrealizedandthecorrespondingHamiltonianoperatorisobtained.Thentheevolutionofthechargenumberandphasedifierenceacrossthecapacityareobtained.Itisshownthatthenumber-phaseanalysisisusefultotacklethequantizationofsomemesoscopiccircuitsanddynamicalequationsofthecorrespondingoperators.Keywords:mesoscopicRLCcircuit,number-phasequantization,LagrangianfunctionPACS:04.60.Ds,73.23.bDOI:10.1088/1674-1056/21/2/0204021.IntroductionInrecentyears,researchworkonmesoscopiccir-cuitshasattractedgreatinterest.In1973,LouisellflrstquantizedamesoscopicLCcircuit.1Inhisscheme,electricchargeqisquantizedasthecoor-dinateoperatorQandelectriccurrentImultipliedbyLisquantizedasthemomentumoperatorP,andthentheLCcircuitisquantizedasaquantumhar-monicoscillator.TheequationofmotionforanRLCcircuitwithatime-dependentpowersourcehasbeenquantized2andthequantumuctuationsofchargeandcurrentinthevacuumstatecanbeobtainedwhenthecircuithasnopower.Afterthat,muchworkhadbeendoneonthecomplicatedelectriccircuitquantization.310Insteadofthecoordinatemomentumquantumvariable,1references1113adoptedanumber-phasequantizationschemetoconstructtheHamilto-nianoperatorforanLCcircuitandtwoLCcircuitswithmutual-inductance,whereelectricchargetakesdiscretevalues.Inthispaper,wearemotivatedtoadoptthisnewquantizationschemetofurtherstudytheRLCcircuit.Thismotivationalsoarisesfromapreviousnumber-phasequantizationschemeforaJosephsonjunction.14,15AccordingtoFeynman,16theJosephsoncurrentcrossinganinsulatingbarrieriscausedbythephasedifierencebetweentwosupercon-ductors.Vourdas17andFan18consideredthisphasedifierenceasanoperator.Therefore,weexpectthatquantizationofanRLCcircuitcanberecastintotheformalismbasedonanumber-phasecommutativerela-tion.Withthehelpofthetime-dependentLagrangianandusingthenumber-phasequantization,weobtainthedynamicalequationsofchargenumberandphasedifierencecrossingtheinductanceandcapacityoftheRLCcircuit,andthenderivetheirevolutionequationswithtime.2.Number-phasequantizationofamesoscopicRLCcircuitNow,weflrstadoptanumber-phasequantiza-tionschemetoconstructtheHamiltonianoperatorforRLCcircuits.Consideringthediscretenessofelectriccharge,theelectricchargeiswrittenasq=en,wherenisthenumberofelectronsacrosstheinductanceL.TheequationofmotionforanRLCcircuitwithapowersourceisLd2(en)dt2+Rd(en)dt+enC=(t),(1)whereL,R,andCstandfortheinductance,resis-tance,andcapacityrespectively,(t)representsthepowersource.ItshowsthattheRLCcircuitcanberegardedasadampedharmonicoscillator.Forthedampedharmonicoscillator,atime-dependentLa-grangianhasbeenconstructed19,20L=12Le2n2(2C)1e2n2+en(t)expRtL.(2)Correspondingauthor.E-mail:c2012ChinesePhysicalSocietyandIOPPublishingLtd/cpb020402-1Chin.Phys.BVol.21,No.2(2012)020402Therefore,Eq.(1)canbeobtainedbythefollow-ingLagrangeequationtLnLn=0.(3)Whenthecircuithasnopowersource,i.e.(t)=0,fromEq.(2)theLagrangefunctionofthecircuitsys-temisjust21L=12Le2n2(2C)1e2n2expRtL.(4)Inthenumber-phasequantizationscheme,nisre-gardedascanonicalcoordinate.Correspondingly,thecanonicalmomentumconjugatedtonisp=Ln=Le2nexpRtL.(5)Sinceen=L,(6)whereisthemagneticuxcausedbyself-inductance,thensubstitutingEq.(6)intoEq.(5)yieldsp=eexpRtL.(7)Thuswenaturallyobtaintherelationbetweenthecanonicalmomentumpandthetotalmagneticux.Now,bytheFaradaytheorem,thetotalvoltagecausedbyinductanceisU=ddt,(8)whichisequaltothesumofthevoltageacrosstheca-pacitorandtheresistance.Thatis,thetotalvoltageisU=UC+UR,(9)whereUR=Ren,thevoltageoftheresistance.ThevoltageUC,fromthequantummechanicalwavefunc-tionspoint,isrelatedtothephasedifierencebetweentwoplatesofthecapacitorwithinatimeintervaldt,i.e.,assumingthewavefunctionsontheflrstplate,thesecondplateofthecapacitorisi=iexpiEit=iexp(iit)=iexp(ii),i=1,2.(10)Notingthatdi=Eidt,d=d2d1=E2E1dt.(11)Thentheenergygapbetweentwoplatesinthecapac-itorisE1E2=eUC=(d1d2)/dt=d/dt.(12)SowecanobtainfromEqs.(8)(12)U=dedt+Redndt.(13)Afterintegration,wecanobtainfromEqs.(8)and(13)=eRen.(14)SubstitutingEq.(14)intoEq.(7),thecanonicalmo-mentumcanberewrittenasp=Re2nexpRtL.(15)ThisequationindicatesthatRe2nexp(Rt/L)isquantizedasthecanonicalmomentum,whilethenumbernisquantizedasthecanonicalcoordinate.Thereforethecanonicalquantizationconditionisn,Re2n=iexpRtL,orn,=iexpRtL.(16)Thentheuncertainrelationisjustnexp(Rt/L)/2,whichmeansquantumuctuation.WhenR=0,wecanobtainthesameresultasthatinRef.11,whichisalsoconsistentwiththeresultofRef.12theoretically.Nowbasedontheoperatorsn,andwiththehelpofthetime-dependentLagrangian,theHamilto-nianoperatorofthiscircuitsystemcanbewrittenash=npLH=12Le2p2expRtL+12Ce2n2expRtL=12Le2Re2n2+12Ce2n2expRtL,(17)whichisconsistentwiththatinRef.22.Wein-troducethefollowingtime-dependentannihilatorandcreatoroperatorsa=e2L021iRL0n+ie2L0exp(Rt/2L),(18)a+=e2L021+iRL0nie2L0exp(Rt/2L),(19)020402-2Chin.Phys.BVol.21,No.2(2012)020402where0=1/LCistheresonantfrequencyofanLCcircuitintheabsenceoftheresistance.ThentheHamiltonianofthecircuitsystemcanberewritteninthesimpleformH=(a+a+12)0.(20)Sowiththehelpofthetime-dependentLagrangianforadampedharmonicoscillatorandusingthenumber-phasequantization,theRLCcircuitcanbequantizedasaquantumharmonicoscillatortoo.3.EvolutionsofchargenumberandphasedifierenceInquantummechanicstheHeisenbergequationiswrittenasdA(t)dt=1iA(t),H.(21)SubstitutingEq.(17)intoEq.(21),weobtainthefol-lowingequationsdndt=Re2nLe2,(22)ddt=RL12LRe2+e2Cn,(23)whichrepresentthecurrentequationandFaradaythe-orem,respectively.FromEqs.(22)and(23),wecanobtaind2ndt2=1LCR22L2n,(24)d2dt2=1LCR22L2.(25)ThenwederivetheevolutionofchargenumberbyEq.(24)n=n(t=0)exp(it),(26)where=1/(LC)R2/(2L2)istheresonantfre-quencyoftheRLCcircuit.Thisequationindicatesthebehaviouroftheelectromagneticallydampedos-cillationoftheRLCcircuit.Ontheotherhand,theevolutionofphasedifierenceoncapacitycanbede-rivedfromEq.(25)=(t=0)exp(it).(27)WhenR=0,weobtainthesameresultasthatofRef.11,whichisalsoconsistentwiththeresultofRef.12theoretically.Therefore,quantizationoftheRLCmesoscopiccircuitscanalsobeexplainedinthecontextofnumber-phasequantizationmethod,inwhichthecanonicalobservablesaretheelectronnum-berandthephasedifierenceacrossthecapacitor.4.ConclusionInsummary,withthehelpofthetime-dependentLagrangian,wehavequantizedtheRLCcircuitbymeansofthenewnumber-phasequantizationscheme,whichisacomplementaryviewofcoordinatemomentumquantizationscheme.Thenwehavede-rivedtheevolutionofthechargenumberandphasedifierence,separately.Itisshownthatnumber-phaseanalysisisusefulfordealingwiththequantizationofsomemesoscopiccircuitsanddynamicalequationsofthecorrespondingoperators.AcknowledgementsTheauthoracknowledgesthehelpfuldiscussionswithDr.WangShuai.References1LouisellWH1973QuantumStatisticalPropertiesofRa-diation(NewYork:Wiley)2FanH,JiaoZKandZhangQR1995Phys.Lett.A2051213FanHYandLiangXT2000Chin.Phys.Lett.171744JiYH,LeiMSandOuyangCY2002Chin.Phys.111635WangJS,FengJandZhanMS2001