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Bosonic

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Bosonic

    Bosonic

    Commun.TheorPhys.(Beijing,China)41(2004)PP.878880 ?InternationalAcademicPublishersV01.41.No.6.June15.2004 BosonicOperatorRealizationofHamiltonianforaSuperc0nductingQuantum InterferenceDevice

    FANHongYil,

    lDepartmentofPhysics

    ,

    ShanghaiJiaoTongUniversity,Shanghai200030,China 2DepartmentofMaterialScienceandEngineering,

    UniversityofScienceandTechnologyofChina,Hefei230026,China (ReceivedJuly19,2003)

    AbstractBasedontheappropriatebosonicphaseoperatordiagonalizedintheentangledstater

    epresentationwe

    constructtheHamiltonianoperatormodelforasuperconductingquantuminterferencedevic

    e.Thecurrentoperatorand

    voltageoperatorequationsarederived.

    PACSnumbers:03.67.a.03.65.w

    Keywords:SQUID,Hamiltonianoperatormodel,currentoperatorandvoltageoperatorequa

    tions

    1Introduction

    Asuperconductingquantuminterferencedevice(SQU

    ID)iscomposedoftwoJosephsonjunctions1and2con

    nectinginparalleltoformaloop.,JTheroleofaSQUID

    istomeasuremagneticflux,withenoughsensitivityto detectonefluxquantum.Magneticfluxthroughthe

    loopinducescurrentsintheloop.Aschanges.the

currentoscillateswithadefiniteperiod西n.Aninterest

    ingquestionthusnaturallyarises:Whatisthedynamic HamiltonianfordescribingtheSQUID?Fromthestruc

    tureofaSQUID.therequiredHamiltonianmustinclude theHamiltonianoftwoJosephsonunctions.Recalling thatinRef.f31wehaveintroducedthebosonicphaseop

    eratorrealizationforJosephsonHamiltonianmodel,and inRef.4wehaveintroducedchargestateandphasestate representationstodevelopthismodel,whichshallbeused asqubitmodelforquantumcomputation.inthisworkwe shallconstructthebosonicSQUIDmodelonthebasis ofRefs.f3andf41.Sincethemacroscopicquantumef_ fectfthequantumpropertiesoftheelectronwavebecome visibleatamacroscopicleve1)ofsuperconductorsarises fromthequantumphase.theJosephsoneffectisastrik. ingphenomenonrelatedtothisquantalphase.Basedon Feynman'shighlight[5thatthesuperconductivitycurrent iscausedbythephasedifference0201'acrossthesu.

    perconductingjunction,andenlightenedbytheideaofthe operationalphaseofNoh,FougeresandMande',,(NFM) inquantumopticstheory[JinRefs.

    f3andf4wehave

    proposedthephasedifierenceoperatoracrossthejunction a.S

    ei

    |abt,.|atbV'V

    c.s西=(ei+e)(1)

    where[n,at=[b,bt=1,(Note[nbt,atb=0,

    sotheycanresideinthesamesquareroot.)Since

    eisunitary.西isHermitian.Intheentangledstate l)representation,.]

    l)=exp[一三lI+r/at--r/*6t+nt6t]loo> =re,r=ll,

    wherel00)istheground

    nihilatedbyaandb,e

    havior,

    becausel)

    statetwo-modeFockspace,

    manifestlyexhibitsitsphase

    e17)=eil)

    satisfiestheeigenvalueequations (nbt)lr/)=I),(atb)ln>=17)

    (2)

    an.

    be

    (4)

    InRefs.[3and[4wehavealsointroducedthenumber. differenceoperatortorepresentthepairchargesbetween

    thetwosuperconductors,

    Q=g?d,?databtb(5)

    Projecting?dontothe(lrepresentationleadstoare. markableproperty

    whichimplies

    (l?d=

    [西,?d=-i

    (6)

    Thisisthefundamentalcommutativerelationwhichwe

    shalluseinconstructingtheHamiltonianoperatorfora

    SQUID.Theworkisarrangedasfollows.InSec.2we

    brieflyreviewtheoperatorJosephsonequationsderived fromthebosonicoperatorHamiltonianforaJosephson junction.InSec.3weconstructthebosonicHamiltonian foraSQUIDandderivethevoltageoperatorequation.In TheprojectsupportedbyNationalNaturalScienceFoundationofChinaunderGrantN0.101

    75057andthePresidentFbundati0n0f

    theChineseAcademyofSciences

    

N0.6BosonicOperatorRealizationofHamiltonianforaSuperconductingQuantumInterfer

    enceDevice879

    Sec.4wededucethecurrentoperatorequationforthe SQIUID.

    2BriefReviewofBosonicOperatorModelof

    JosephsonJunction

    BasedonFeynman'sexplanationaboutCooperpair ,

    i-e-"aboundpairactingasaBoseparticle".bycombining Eqs.(1)and(5)wehaveproposedthebosonicHamilto

    nianoperatorofaJosephsonjunction,

    Hj=QEjcos,(8)

    whereEjistheJosephsoncouplingconstant.

    Project

    ingHjontothe(Irepresentationandusingtheproperty givesriseto

    fr

    q-02

    Ejc.s,(9)

    whichis

    ,j1usttheeffectiveHamiltonianproposedbyVourdasI10Using [COS西,Nd=isin西

    andtheHeisenbergequation,wehavederivedthetunnel

    currentequationsinoperatorform ,

    i.e.,

    (g)=g[?djHA=gEjsin西.(.)

    ThesecondoperatorJosephsonequation(thevoltage

    equation)thatcontrolsthephaseis

    0西

    :

    1

    Hj]=qz?d:一去q,where

    :Q/c=(qNa)/C(12)

    istheinducedvoltageassociatedwiththechargingofthe

    unc

    .

    tion-Further,byintroducingthecommoneigenvector

    fn,7')0fNdand(at6)(0bt),[11

    ^rdln,r)=jXff()

    ×f=7'ei)=nf,7'),

    n,r)=dl=re)en

    ei,I,l,7'):l凡一l,r)

    Formallywealsohave

    西I,7')=

    eIn,r):In+l,r)

    (13)

    (14)

    (15)

:

    ir)_(17)

    ItthenfollowsfromEqs.

    (11)and(17)that

    ((q?d)=qEJ<n,rJin西

    and

    (n,rlHj=

    :g((几十l,r_(凡一l,rI),(18)

    1,2

    qn)'(,rZL/

    1

    

    Ej((n+1,7'I+(凡一l,7'f),(19)

    whichexhibitstheloweringandascendingofCooperpair

    numberinthetunnelingprocess. 3FullBosonicOperatorHamiltonianfor aSQUIDandVoltageOperatorEquation FordescribingaSQUID,wemusttaketheectof

    magneticfieldintoaccount.Consideringthepresenceof

    avectorpotentialAthatproducesamagneticfieldand

    anelectricfieldaccordingtotheMaxwellequations.

    B:V×A

    fromwhichweknowthe

    tionintheSQUIDis

    :

    ld/ct/dr,

    .

    E:

    additionalvo1

    :

d

    wheretheintegralsaretakenalongpathlforthefirst

    Josephsonjunctionlocatingandpath2inspace.

    Follo,v-

    ingFaraday'slawV=A~/cAtwehave drA-\2drA=(22)

    whereisthemagneticfluxpassingthroughtheloop0f

    SQUID.Wecanlet

    drA=110+)

    ,drA=-

    2

    1

    -

    (~),(23)

    wherekisanintegralconstant. ForaSQUIDwethink

    thatthefullHamiltonianshouldbe

    2

    =+H0,H=?

    2

    =?qNdi

    =1

    where

    Qt=qYdt,

    Inthisaase,using

    ageequationis

    i:1

    (12c.s)

    (24)

?dnntj6i,i=1,2.(25)

    Eq.(22)weseethattheoperatorvolt

    d

    西t1=一一

    qd/drwhere :

    =.

    IntegratingEq.

    (28)wehave ()=一罢/ddr. (26)

    (27)

    (28)

    c=)

    m

    A

    

    a

    S

    S

    0,?

    ar

    edgr2

    ,/r

    

    r

    =

    ,/

    r

    n

    ,l, f

    6

    

    0

    ,,I,

    ,J6

    tS 0.1 ,,/ r

    n

    e

    r

    e

    h

    w

    

    o. p‰陴 :

    叽? d

    

    b .

    m

880FANHongYiV0l41

    4CurrentOperatorEquationforaSQUID Using

    eiA?.eiA?d=0ei

    .eiANdbteiANd=bteiA(29)

    andtakingH0inEq.(25)asthefreeHamiltonian(thecontributionfromthemagneticfield)wet

    ransformourdiscussion

    intotheinteractionpicture.Using and

    weobtain

    Nd,COS]=isin,Na,sin]=iCOS

    ei?dcoseiANd

    eiHoA/^eiHoAt/h

    c.sn西一2COS西+in西+

    COS西COSAsin西sinA=cos(~+A),

    2

    [-Ejcos()],

    wherethesuperscriptImeanstheinteractionpicture.Letq

    currentoperatorequationsintheinteractionpictureare

    =tI1N'=Ejsin(西

    z=EJsin(+)

    (30)

    (32)

    2e,usingEqs.(22)^(24)wecanderivethatthe

    qV~At):Ejshhin(西一/\e(+k)ch

    When西l=42=西,addingEqs.(33)and(34)togethergivesthecurrentoperatorofaSQUID

    =

    Ejsin(~一篆)cos():Easin(~)cos()/l?J

    (33)

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