LimitationsontheUtilityofExactMaster

Equations

G.W.Ford1

,R.F.O’Connell2,∗

SchoolofTheoreticalPhysicsDublinInstituteforAdvancedStudies10Burlington

RoadDublin4,Ireland

Abstract

Thelowtemperaturesolutionoftheexactmasterequationforanoscillatorcoupled

toalinearpassiveheatbathisknowntogiverisetoseriousdivergences.Wenow

showthat,eveninthehightemperatureregime,problemsalsoexist,notablythe

factthatthedensitymatrixisnotnecessarilypositive.

Keywords:

PACS:

∗Correspondingauthor

Emailaddress:oconnell@phys.lsu.edu(R.F.O’Connell).1Permanentaddress:DepartmentofPhysics,UniversityofMichigan,AnnArbor,

MI48109-11202Permanentaddress:DepartmentofPhysicsandAstronomy,LouisianaStateUni-

versity

PreprintsubmittedtoElsevierScience3November2004

1Introduction

Inapreviouspublication[1],wepresentedageneralsolutionoftheexact

masterequationforanoscillatorcoupledtoalinearpassiveheatbath.This

wasachievedbysolvingthegeneralizedquantumLangevinequationforthe

initialvalueproblem[2]whichenabledustoobtainexplicitexpressionsforthe

time-dependentcoefficientsoccuringintheexactmasterequation.Mext,we

illustratedthegeneralsolutionwiththeconstructionofexplicitexpressions

fortheprobabilitydistributionsfortwoexamples:wavepacketspreadingand

”Schrodingercat”stateforafreeparticle.Thisenabledustodiscoverthatthe

lowtemperaturesolutiongaverisetodivergences.Thepurposeofthepresent

paperistoshowthatproblemsalsoexistinthehightemperatureregime.

Thus,wearemotivatedtoextendourpreviousanalysistoobtainexplicit

expressionsfortheWignerfunctionandtheelementsofthedensitymatrix,

andwiththeaidoftheseresultstocriticallyexaminetheequationandits

solution.

Attheoutsetwemustpointoutthattheexactmasterequationhasdrawbacks

thatseriouslylimititsutility.Forthemostpartthesearerelatedtothe

factthattheinitialstateisnecessarilyoneinwhichtheparticleandheat

bathareuncoupled,whilethesubsequentmotionisdescribedbythefully

coupledsystem.Themostseriousoftheseisthattheequation(andtherefore

itssolution)hasanirreparabledivergence,irreparableinthesensethatit

persistsinacut-offmodelwithfinitebathrelaxationtime[1].Webelievethis

divergenceisrelatedtoawellknownphenomenonofquantumfieldtheory:

theHilbertspacespannedbytheeigenfunctionsofthecoupledHamiltonian

isorthogonaltothatoftheuncoupledHamiltonian[3].Itcanbeshownthat

2

thisisthecaseforthesystemofcoupledoscillatorsthatisthemicroscopic

basisforthederivationoftheexactmasterequationanditssolution[4,5].

Theexceptionisthehightemperaturelimit,wherethedivergencedoesnot

appearsince,byconvention,onethereneglectsthezero-pointoscillationsof

thebath.Wethereforeconfineourdiscussiontothatlimit.However,evenin

thatcasetherearefurtherdifficulties.Thefirstoftheseisthattheinitialstate

breaksthetranslationalinvarianceofthesystem.Thisismanifestinaninitial

squeezecenteredattheoriginthat,forafreeparticle,resultsinadriftofthe

meanpositiontowardtheorigin.Forthisreasonintheexamplesweconsider

onlythecaseofafreeparticle,wherethisdifficultyismostapparent.

Aseconddifficultywiththeequationinthehightemperaturelimitisthat

forshorttimesthediagonalelementsofthedensitymatrix,whichhavethe

physicalinterpretationofprobabilities,arenotnecessarilypositive.Thisis

awellknowndifficultywiththeweakcouplingmasterequation,solvedbya

furthertimeaveragetobringittotheso-calledLindbladformthatguarantees

positivity[6].Itwassupposedthattheexactmasterequation,withitstime-

dependentcoefficients,wouldbeimmunebut,asweshallshowexplicitly,in

thehightemperaturelimitthisdifficultyremains.Thesituation,therefore,

isthatthereareconcernsthatseriouslylimittheutilityoftheexactmaster

equation.Theirreparabledivergencecanonlybeavoidedwiththeneglectof

zero-pointoscillationsinthehightemperaturelimit,andeventhereproblems

remainwiththefailureoftranslationalinvarianceandpositivity.

Despitetheseproblems,wearguethatthesolutionoftheexactmasterequa-

tioncanbeuseful.Inparticular,weshowthatwhentheinitialparticletemper-

atureistakentobeequaltothatofthebath,weobtainresultsthatforshort

3

timesareidenticalwiththoseobtainedfromacalculationwhichconsidersen-

tanglementatalltimes[7].Finally,animportantresultisthedemonstration

thatmeasuresofdecoherencebasedontheWignerfunctionareidenticalwith

thosebasedontheoff-diagonalelementsofthedensitymatrix.Wearguethat

theWignerfunction,beingeverywherereal,isthepreferabledescription.

2Generalsolutioninhightemperaturelimit

Theexactmasterequation,intermsoftheWignerfunctionW(q,p,t)forthe

completesystemofoscillatorplusheatbath,maybeexpressedintheform

∂W

∂t=−

1

mp∂W

∂q+mΩ

2(t)q

∂W

∂p

+2Γ(t)∂pW

∂p+~mΓ(t)h(t)

∂2W

∂p2+~Γ(t)f(t)∂

2W

∂q∂p,(1)

andwenotethepresenceoffourtime-dependentcoefficientsforwhichwe

obtainedexplicitexpressions[1].

Thegeneralsolutionoftheexactmasterequationismostsimplyexpressed

intermsoftheWignercharacteristicfunction(theFouriertransformofthe

Wignerfunction).

W(Q,P;t)=∫dq∫dpe−i(qP+pQ)/~W(q,p;t).(2)

Expressedintermsofthisfunction,thegeneralsolutiongiveninEq.(4.15)

of[1]takesthesimpleform,

W(Q,P;t)=exp−〈X

2〉P

2+m

⟨XX+XX

⟩QP+m

2⟨X

2⟩Q

2

2~2×W(mGQ+GP,m

2GQ+mGP;0).(3)

4

InthisexpressionX(t)isthefluctuatingpositionoperator,definedinEq.

(2.16)of[1],whileG(t)istheGreenfunction,definedingeneraltobe

G(t2−t1)=1

i~[x(t1),x(t2)]θ(t2−t1),(4)

inwhichx(t)isthetime-dependentHeisenbergcoordinateoperatorandθis

theHeavisidefunction.

Itwasshownin[1](Sec.V.A.2)thatthequadraticexpectations〈X2〉,⟨XX+XX

⟩

and⟨X

2⟩arelogarithmicallydivergent,whichhastheeffectthatthesolution

(3)vanishesidenticallyfornon-zeropositivetimes.There,too,itwasshown

thatthedivergenceisirreparable,inthesensethatitpersistsinamodelwith

ahighfrequencycut-off.AswasstatedintheIntroduction,itappearsthat

thisisareflectionofthewellestablishedfactthatthestatesoftheuncou-

pledsystem(theinitialstate)areorthogonaltothoseofthecoupledsystem

(statesforlatertimes)[4,5].Thisfact,surprisingfromthepointofviewof

thequantummechanicsoffinitesystems,isafeatureonlyofsystemswithan

infinitenumberofdegreesoffreedom.Inourcaseitistheheatbaththatmust

beinfinite.Inanyevent,theupshotisthat,whilethederivationoftheexact

masterequationisformallycorrect,thereisnousefulresult.Theexception

isthehightemperaturelimit,wherebyconventiononeneglectszero-point

oscillations.Wethereforerestrictourfurtherdiscussiontothislimit.Indeed,

itisthislimit(orbettersaid:approximation)thathasbeenconsideredinall

previousdiscussionsoftheexactmasterequation.

Inconsideringtheresultinthehightemperaturelimit,werestrictourdiscus-

siontotheOhmicmodelofcouplingtotheheatbath,whichisadequatefor

ourpurposes.ThismodelcorrespondstoNewtonianfriction,withretarding

5

forceproportionaltothevelocity.Inadditionweconsideronlythecaseofa

freeparticle,movingintheabsenceofanyexternalpotential.Choosingthe

frictionconstanttobemγ,theGreenfunctionthentakesthesimpleform:

G(t)=1−e−γtmγ

.(5)

Inthehightemperaturelimitthemomentsappearinginthesolutioncanthen

bewrittenintheform:

⟨X

2⟩=2mγkT

t ∫

0

dt′G2(t′)

=kT

mγ2[2γt−(1−e−γt)(3−e−γt)],⟨XX+XX

⟩=2mγkTG

2(t)

=2kT

mγ(1−e−γt)2

,

⟨X

2⟩=2mγkT

t ∫

0

dt′G2(t′)

=kT

m(1−e−2γt

).(6)

WeshouldemphasizethatintheseexpressionsTisthetemperatureofthe

bath.Intheinitialstatethebathandtheparticleareuncoupled,sointhe

generalsolution(3)W(Q,P;0),theinitialWignercharacteristicfunction,may

bechosentohaveanyformconsistentwithasingleparticlenotcoupledto

abath.Intheexamplesweconsidertwopossibilities:theparticleinitiallyat

zerotemperatureandatatemperatureequaltothatofthebath.

6

3Example:Gaussianwavepacket

3.1Initialparticletemperaturezero

WeconsiderfirstaninitialstatecorrespondingtoaGaussianwavepacket,

withwavefunctionoftheform

φ(x,0)=1

(2πσ2)1/4exp−(x−x0)

2

4σ2.(7)

Thisisastationarywavepacket,centeredatx0withmeansquarewidthσ2.It

isaminimumuncertaintywavepacket,sincethemeansquaremomentumhas

theminimumvalue~2/4σ

2.ThecorrespondingWignercharacteristicfunction

is

W(Q,P;0)=

∞∫

−∞dxe−iPx/~φ(x−

Q

2,0)φ∗(x+

Q

2,0)

=exp−Q

2

8σ2−σ

2P

2

2~2−ix0P

~.(8)

Thisinitialstateiswhatistermedapurestate.Theconditionforapurestate

isusuallystatedintermsofthedensitymatrix:Trρ2=1.Intermsofthe

Wignercharacteristicfunction,thisconditionbecomes

1

2π~

∞∫

−∞dQ

∞∫

−∞dP

∣∣∣W(Q,P)

∣∣∣2

=1.(9)

Itiseasytoshowthatastateisapurestateifandonlyifitcanbeassociated

toawavefunction.Suchapurestatenecessarilycorrespondstoaparticleat

zerotemperature;afinite-temperaturestateisamixedstate.

7

Puttingtheexpression(8)fortheinitialWignercharacteristicfunctioninthe

generalsolution(3)weseethat

W(Q,P;t)=exp−A

(0)11P

2+2A

(0)12PQ+A

(0)22Q

2

2~2−ix0(m

2GQ+mGP)

~,(10)

wherewehaveintroduced

A(0)11=

⟨X

2⟩+σ

2m

2G

2+~

2G

2

4σ2

=kT

mγ2[2γt−(1−e−γt)(3−e−γt)]+σ2e−2γt

+~

2(1−e−γt)2

4m2γ2σ2,

A(0)12=m

⟨XX+XX

⟩

2+σ

2m

3GG+

~2mGG

4σ2

=kT

γ(1−e−γt)2

−mγσ2e−2γt

+~

2(e−γt−e−2γt

)

4mγσ2,

A(0)22=m

2⟨X

2⟩+σ

2m

4G

2+~

2m

2G

2

4σ2

=mkT(1−e−2γt)+m

2γ

2σ

2e−2γt

+~

2e−2γt

4σ2.(11)

Herewehaveusedthesuperscript“(0)”toindicatethattheparticleisinitially

atzerotemperaturebeforesuddenlybeingcoupledtothebathattemperature

T.

WenowformtheWignerfunctionwiththeinverseFouriertransform

W(q,p;t)=1

(2π~)2

∫dQ

∫dPe

i(qP+pQ)/~W(Q,P;t).(12)

Withtheexpression(10)forW(Q,P;t),theintegrationisastandardGaussian

integral[1],withtheresult

W(q,p;t)=W(0)

(q−mGx0,p−m2Gx0;t),(13)

8

where

W(0)

(q,p;t)=1

2π√A

(0)11A

(0)22−A

(0)212

exp−A

(0)22q

2−2A

(0)12qp+A

(0)11p

2

2(A(0)11A

(0)22−A

(0)212)

(14)

istheWignerfunctioncorrespondingtoasingleminimumuncertaintywave

packetinitiallyattheorigin.

TheWignerdistribution(13)correspondstoaGaussiandistributioninphase

space,centeredatthepoint(〈x(t)〉,〈p(t)〉),where

〈x(t)〉=mGx0=x0e−γt,

〈p(t)〉=m2Gx0=−mγx0e−γt.(15)

ThewidthofthedistributionischaracterizedbythecoefficientsA(0)jk,which

havetheinterpretation

A(0)11=∆x

2≡⟨x

2(t)⟩−〈x(t)〉

2,

A(0)22=∆p

2≡⟨p

2(t)⟩−〈p(t)〉

2,

A(0)12=

1

2〈x(t)p(t)+p(t)x(t)〉−〈x(t)〉〈p(t)〉.(16)

Thebreakingoftranslationalinvarianceisevident,sincethecenterofthe

distributiondriftstotheorigin,whichisaspecialpoint.Wegetmoreinsight

intothisphenomenonbyformingthelimitastgoestozerothroughpositive

values,

W(q,p;0+

)=1

π~exp−

(q−x0)2

2σ2−2σ

2(p+mγq)

2

~2.(17)

Hereweseethattheparticlehasreceivedanimpulse−mγqproportionalto

thedisplacementanddirectedtowardtheorigin.Thisisexactlytheactionthat

producesasqueezedstate[8,9].Indeed,asonecanreadilyverify,theWigner

9

function(17)correspondstoapurestatewithassociatedwavefunction

φ(x,0+

)=1

(2πσ2)1/4exp−(x−x0)

2

4σ2−imγ

2~x

2.(18)

Thus,thestateimmediatelyafterthecouplingtothebathisstillazero

temperaturepurestate.However,asaresultofthesqueezetowardtheorigin,it

isnolongeraminimumuncertaintystate.Itstillcorrespondstoawavepacket

centeredatx0withmeansquarewidthσ2,butthemeansquaremomentum

uncertaintyisnow∆p2

=~2/4σ

2+m

2γ

2σ

2,greaterthantheminimumvalue.

Theresultofthesqueezeisthatthedistribution(13)driftstowardtheorigin.

Lessobviousperhaps,thesamephenomenonappearsinashrinkingwidthof

theprobabilitydistributionatshorttimes.Atlongertimesthewidthexpands

duetothermalspreading

3.2Initialparticletemperatureequaltothatofthebath

Aswehavenoted,theminimumuncertaintywavepacket(7)consideredabove

correspondstoaparticleattemperaturezero.Aperhapsmorerealisticinitial

statewouldbethatforaparticleinitiallyattemperatureT,equaltothatof

thebath.In[1]weshowedthattheinitialstateattemperatureTisobtained

fromthatattemperaturezerobythereplacement

W(Q,P;0)→exp−mkTQ

2

2~2W(Q,P;0).(19)

Thisfinitetemperaturestateisamixedstate,nolongersatisfyingthecondi-

tion(9)forapurestate.Itcannotbeassociatedtoawavefunction.Making

thisreplacementwiththezerotemperatureform(8)andrepeatingthesteps

10

leadingtotheresult(13),weobtain

W(q,p;t)=W(T)

(q−mGx0,p−m2Gx0;t),(20)

wherenowtheWignerfunctioncorrespondingtoasingleGaussianwave

packetinitiallyattheorigintakestheform

W(T)

(q,p;t)=1

2π√A

(T)11A

(T)22−A

(T)212

exp−A

(T)22q

2−2A

(T)12qp+A

(T)11p

2

2(A(T)11A

(T)22−A

(T)212)

,(21)

inwhich

A(T)11=A

(0)11+mkTG

2

=2kT

mγ2(γt−1+e−γt)+σ2e−2γt

+~

2(1−e−γt)2

4m2γ2σ2,

A(T)12=A

(0)12+m

2kTGG

=kT

γ(1−e−γt)−mγσ2

e−2γt+~

2e−γt(1−e−γt)

4mγσ2,

A(T)22=A

(0)22+m

3kTG

2

=mkT+m2γ

2σ

2e−2γt

+~

2e−2γt

4σ2.(22)

Notethatthemotionofthecenterofthedistributionisthesameasthat

giveninEq.(15)forthecaseofinitialparticletemperaturezero.Theonly

changeisanincreaseinthewidthoftheGaussiandistributionduetothermal

spreading,thewidthsbeinggivenbytheexpressions(16)withA(T)jkinplace

ofA(0)jk.Weagaingetmoreinsightintothisbyformingthelimitastgoesto

zerothroughpositivevalues,

W(q,p;0+

)=1

π~√

1+4σ2

λ2T

exp−(q−x0)

2

2σ2−(p+mγq)

2

2(~2

4σ2+mkT).(23)

HereweseethatthatthereisthesameinitialsqueezeasinEq.(17),butthe

squeezeactsonastatewithmeansquaremomentumuncertaintyincreased

11

bythemeansquarethermalmomentum,mkT.Theresultisthatimmediately

afterthesqueezethemeansquaremomentumuncertaintyis∆p2

=~2/4σ

2+

mkT+m2γ

2σ

2.However,theinitial∆x

2=σ

2andisunaffected.bythesqueeze.

4Example:PairofGaussianwavepackets

4.1Initialparticletemperaturezero

Thewavefunctionforaninitial”Schrodingercat”stateconsistingoftwo

separatedminimumuncertaintyGaussianwavepacketshastheform

φ(x,0)=1

(8πσ2)1/4(1+e−d2/8σ2)1/2(exp−

(x−d2)

2

4σ2+exp−(x+

d2)

2

4σ2).(24)

TheinitialWignercharacteristicfunctionisthengivenby

W(Q,P;0)=1

1+e−d2/8σ2exp−Q

2

8σ2−σ

2P

2

2~2(cosPd

2~+e−d2/8σ2

coshQd

4σ2).(25)

Therefore,fromthegeneralsolution(3)wecanwrite,

W(Q,P;t)=1

1+e−d2/8σ2exp−A

(0)11P

2+2A

(0)12PQ+A

(0)22Q

2

2~2

×(cos(m

2GQ+mGP)d

2~+e−d2/8σ2

cosh(mGQ+GP)d

4σ2). (26)

HeretheA(0)ijareagaingivenby(11).

TheinverseFouriertransform(12)againinvolvesonlystandardGaussian

integrals.WethuseasilyseethattheWignerfunctionforawave-packetpair

is

12

W(q,p;t)=1

2(1+e−d2/8σ2)W

(0)(q−

mGd

2,p−

m2Gd

2)

+W(0)

(q+mGd

2,p+

m2Gd

2)

+2e−A(0)(t)W

(0)(q,p)cosΦ

(0)(q,p:t),(27)

whereW(0)

istheWignerfunction(14)forasingleminimumuncertaintywave

packetattheoriginandwehaveintroduced

A(0)

(t)=(A

(0)11−~2G2

4σ2)(A(0)22−~2m2G2

4σ2)−(A(0)12−~2mGG

4σ2)2

A(0)11A

(0)22−A

(0)212

d2

8σ2

Φ(0)

(q,p;t)=(GA

(0)22−mGA

(0)12)q+(mGA

(0)11−GA

(0)12)p

A(0)11A

(0)22−A

(0)212

~d4σ2.(28)

Thefirsttwotermsinbracketsin(27)correspondtoapairofGaussianwave

packetsoftheform(13),centeredinitiallyatx0=±d/2andpropagating

independently.Wecancallthesethedirectterms.Thethirdtermisanin-

terferenceterm.Thedirecttermsareamaximumattheircenters,whichare

driftingtowardtheoriginbutfordÀσwillforshorttimesbewellawayfrom

theorigin.Theinterferencetermisamaximumattheoriginandinitiallyits

peakheightisexactlytwicethatofeitherofthedirectterms.(However,as

onecanreadilyverify,theareaundertheinterferencetermisindependentof

timeandafactore−d2/8σ2smallerthantheareaunderthedirectterms.)The

factore−A(0)(t),equaltotheratioofthepeakheightoftheinterferencetermto

twicethepeakheightoftheeitherofthedirectterms,istakenasameasure

oftheinterferenceinphasespace.InFig.1weshowA(0)

asafunctionofγt

andthereweseethatforγt¿1,A(0)

islinearint.Usingtheexpressions(11)

forthequantitiesA(0)11and(5)forGwecanreadilyshowthatforshorttimes

A(0)

(t)∼=d2

λ2th

γt,(29)

13

whereλthisthemeandeBrogliewavelength,

λth=~ √mkT

.(30)

ForaseparationlargecomparedwiththethermaldeBrogliewavelength,this

correspondstoadecayofthepeakheightoftheinterferencetermonatime

scaleshortcomparedwithγ−1,

τd=λ

2th

d2γ−1.(31)

Thisisthefrequentlyappearing”decoherencetime”[7].

Itisofinteresttoformtheprobabilitydistribution,whichingeneralisgiven

by

P(x;t)=

∞∫

−∞dpW(q,p;t)

=1

2π~

∞∫

−∞dPe

ixP/~W(0,P;t).(32)

Usingtheexpression(26)fortheWignercharacteristicfunction,thiscanbe

written

P(x;t)=1

2(1+e−d2/8σ2)[P

(0)(x−mG

d

2)+P

(0)(x+mG

d

2)

+2a(t)exp−m

2G

2d

2

8A(0)11

P(0)

(x)cos~Gdx

4σ2A(0)11

),(33)

where

P(0)

(x)=1

√2πA

(0)11

exp−x

2

2A(0)11

(34)

istheprobabilitydistributionforasingleminimumuncertaintywavepacket

initiallycenteredatheoriginandwehaveintroducedtheattenuationcoeffi-

14

cient,

a(t)=exp−〈X2〉d

2

8σ2A(0)11

.(35)

Theattenuationcoefficientisameasureoftheinterferenceasitwouldbe

observedintheprobabilitydistribution.Weshouldemphasizethattheproba-

bilitydistributioncanbedirectlymeasured.Thisistobecontrastedwiththe

Wignerdistributioninphasespace,whichisnotdirectlyobservable.

Theintegratedprobabilityintheinterferencetermisequaltoe−d2/8σ2,inde-

pendentoftime.FordÀσthisisnegligiblysmall.Butitistheamplitude

oftheinterferencefringesthatisobservedandmeasuredbytheattenuation

coefficient(35).

Forshorttimes,γt¿1,usingtheexpressions(6)for〈X2〉and(11)forA

(0)11,

weseethattheattenuationcoefficienttakestheform

a(t)∼=exp−t3

3τd[t2+(2mσ2/~)2].(36)

Thisinitiallyfallsveryslowlyfromunity,butafteratime2mσ2/~(thetime

forthemeansquarewidth∆x2

todouble)willdecaywithacharacteristictime

3τd,whereτdisthecharacteristictime(31)forthedecayoftheinterference

termintheWignerfunction.

Withnocouplingtothebaththeinterferencepatternintheprobabilitydis-

tributionwouldnotappearuntilthedirecttermsbegintooverlapduetowave

packetspreading.Takingγ→0intheexpression(11)forthemeansquare

width,weseethatthiswouldbeatimeofordertmix=2mσd/~¿τd.What

thismeansisthatneithertherapiddisappearanceoftheinterferenceterm

15

intheWignerdistribution,northecorrespondingrapiddecayoftheattenu-

ationcoefficient,cancouldbedirectlyobserved.Allthatcanbeseenisthe

non-appearanceoftheinterferencetermintheprobabilitydistribution.

4.2Initialparticletemperatureequaltothatofthebath

ThecorrespondingresultsfortheparticleinitiallyatthetemperatureTof

thebathareobtainedusingtheprescription(19).Theresultis

W(q,p;t)=1

2(1+e−d2/8σ2)

W

(T)(q−

mGd

2,p−

m2Gd

2)

+W(T)

(q+mGd

2,p+

m2Gd

2)

+2e−A(T)(t)W

(T)(q,p)cosΦ

(T)(q,p:t)

,(37)

where

A(T)

(t)=(A

(T)11−~2G2

4σ2)(A(T)22−~2m2G2

4σ2)−(A(T)12−~2mGG

4σ2)2

A(T)11A

(T)22−A

(T)212

d2

8σ2

Φ(T)

(q,p;t)=(GA

(T)22−mGA

(T)12)q+(mGA

(T)11−GA

(T)12)p

A(T)11A

(T)22−A

(T)212

~d4σ2.(38)

ThisexpressionfortheWignerfunctionhasthesameformasthatfoundabove

forthecasewheretheinitialparticletemperatureiszero.Theonlydifference

isthatA(0)jk→A

(T)jk.Theareaundertheinterferencetermisstillindependent

oftimeandsmallerthantheareaunderthedirecttermsbythesamefactor

e−d2/8σ2.Themaindifferenceisthatthepeakheightoftheinterferenceterm

ismuchreduced.Theratioofthepeakheightoftheinterferencetermtotwice

thatofthedirecttermsisgivenbythefactore−A(T)(t)but,asweseeinFig.

16

1,A(T)

nolongervanishesatt=0.Instead,wenowfindforγt¿1,

A(T)

(t)∼=d2

2λ2th+8σ2.(39)

ForseparationlargecomparedwithboththethermaldeBrogliewavelength

andtheslitwidth,thiswillbealargenumberandthepeakheightofthe

interferencetermwillbecorrespondinglysmall.Thus,weseethatthereis

notimescaleforthedisappearanceoftheinterferenceterm,itisalready

negligiblysmallatt=0.Wemightsaythatthisisjustthepoint.What

hasoccurredisthatanincoherentsuperpositionofpurestateshaswiped

outtheinterferenceterm.Inthepresentcasethisistheresultoftheinitial

preparationofthestate.Inthepreviouscaseofaninitialpurestate,this

occursasaresultofthe”warmingup”oftheparticleinatimeτd.Ineither

casetheinterferencetermisgonelongbeforeitseffectcouldbeobservedin

theprobabilitydistribution.

Theprobabilitydistributionisofthesameform(33),butwithA(T)11(t)replac-

ingA(0)11(t).Theattenuationcoefficientisnowgivenby

a(t)=exp−(〈X

2〉+mkTG

2)d

2

8σ2A(T)11

(40)

Forγt¿1thisbecomes

a(t)∼=exp−kTd

2

8mσ4t2(41)

Theseresultsforshorttimesareidenticalwiththoseobtainedfromanexact

calculationassumingentanglementoftheparticlewiththebathatalltimes

[7]).Sincethecouplingtothebath,asmeasuredbythedecayrateγ,does

notappeartheycanalsobeobtainedfromacalculationwithoutdissipation

17

usingonlythefamiliarformulasofelementaryquantummechanics[10].Thus

weseethat,whiletheexactmasterequationgivesherecorrectresults,itdoes

soonlyfortimessoshortthatcouplingtothebathcanbeneglected.

5Densitymatrix

ThedensitymatrixisaHermitianoperator,ρ,definedsuchthatforanywave

functionψcorrespondingtoastateoftheparticleintheHilbertspace〈ψ,ρψ〉

istheprobabilitythatthesystemisinthestate.Fromthenotionofprobability,

weseeimmediatelythethedensitymatrixmustbeapositivedefiniteoperator.

Considertheeigenfunctionsφaandthecorrespondingeigenvaluespaofthe

densitymatrix,

ρφa=paφa.(42)

Clearly,theeigenvaluepaistheprobabilitythatthesystemisinstateφaand

isthereforenecessarilypositive.Weassumethedensitymatrixisnormalized,

sothat

∑

a

pa=1.(43)

Thisisasumofpositivetermsequaltounity.Itfollowsimmediatelythat

Trρ2=

∑

a

p2a≤1.(44)

Moreover,theequalityholdsifandonlyifthereisexactlyoneeigenstate

withprobabilityone,allothershavingprobabilityzero.Thisisexactlythe

condition(9)forapurestate.

18

Incoordinatespacetheelementsofthedensitymatrixaregivenintermsof

theWignercharacteristicfunctionbytheformula:

〈x|ρ|x′〉=∞∫

−∞dpe

i(x−x′)p/~W(

x+x′

2,p)

1

2π~

∞∫

−∞dPe

i(x+x′)P/2~W(x′−x,P).(45)

Thenormalizationcondition(43)becomes

Trρ=

∞∫

−∞dx〈x|ρ|x〉=W(0,0)=1.(46)

Thatis,theWignercharacteristicfunctioncorrespondingtoanormalized

densitymatrixmusthavethevalueunityattheorigin.

Nextconsider

Trρ2=

∞∫

−∞dx

∞∫

−∞dx′〈x|ρ|x′〉〈x′|ρ|x〉

=1

2π~

∞∫

−∞dQ

∞∫

−∞dP

∣∣∣W(Q,P)

∣∣∣2,(47)

wherewehaveusedthatfactthatW(−Q,−P)=W(Q,P)∗.Thus,thecon-

dition(44)takestheform

1

2π~

∞∫

−∞dQ

∞∫

−∞dP

∣∣∣W(Q,P)

∣∣∣2≤1,(48)

withtheequalityholdingifandonlyifthedensitymatrixisthatofapure

state.Thisisthecondition(9)forapurestate.

19

5.1Gaussianwavepacket

Considernowthecaseofaninitialstatecorrespondingtoasingleminimum

uncertaintywavepacket,forwhichtheWignercharacteristicfunctionattime

tisgiveninEq.(10).Theevaluationoftheformula(45)forthematrixelement

ofthedensityoperatorinvolvesasingleGaussianintegral.Theresultcanbe

written

〈x|ρ(t)|x′〉=expim

2G

~x0(x−x′)R(x−mGx0,x′−mGx0;t),(49)

wherewehaveintroduced

R(x,x′;t)=1

√2πA

(0)11

exp−4~2(A

(0)11A

(0)22−A

(0)212)(x−x′)2

+(x+x′)2

8A(0)11

+i~A

(0)12(x

2−x′2)

2~2A(0)11

,(50)

whichisthematrixelementcorrespondingtoasinglewavepacketinitially

centeredattheorigin.Usingtheexpressions(5)fortheGreenfunctionand

(11)forA(0)jk(t),weseethatviewedinthexx′planetheabsolutevalueofthe

densitymatrix,|〈x|ρ(t)|x′〉|,consistsofasingleGaussianpeak.Thispeakis

initiallycenteredatx=x′=x0andcircularlysymmetricwithwidthσ.In

thecourseoftimethecenterofthepeakdriftstotheoriginwhilestretching

inthedirectionx=x′.

Forshorttimes,thedensitymatrixisnotnecessarilypositive.Thesimplest

waytoseethisistoformTrρ2,usingtheaboveexpressionforthedensity

matrixelementorevaluatingdirectlytheexpression(47).Ineithercasewe

find

Trρ2=

~

2√A

(0)11A

(0)22−A

(0)212

.(51)

20

InFig.2weshowaplotoverashortinitialtimeintervalofthisresult,

evaluatedusingtheexpressions(11)fortheA(0)jkwithparametersλth=4σ,

kT=5~γ.Thereweseeclearlythatthecondition(44)isviolatedforshort

times.Itfollowsthatforsuchtimestheremustbenegativeeigenvalues;the

densitymatrixisnotpositive-definite.Wegainsomeinsightintothisphe-

nomenonifweusetheexpressions(11)toexpand

Trρ2∼=1+(1−4

σ2

λ2th

)γt+···.(52)

FromthisweseethatTrρ2willincreaseaboveunitywheneverλth>2σ.

Thefailureofpositivityoccursforasharplypeakedwavepacket.

Toseeexplicitlythisfailureofpositivity,itwillbesufficienttotakex0=0.We

thenformtheexpectationofthedensitymatrixwithawavefunctionψ(x)∝

dφ(x,0+

)/dx,whereφ(x,0+

)isthestateimmediatelyafterthecouplingto

thebath(giveninEq.(18)withx0=0).Explicitly,

ψ(x)=x

σ(2πσ2)1/4exp−(1

4σ2+imγ

2~)x

2.(53)

Withthis,

〈ψ,ρ(t)ψ〉=∞∫

−∞dx

∞∫

−∞dx′ψ∗(x)〈x|ρ(t)|x′〉ψ(x′)

=−2[1−4~2(A

(0)11A

(0)22−A

(0)212)]

[(1+A11

σ2)(1+4σ2(A

(0)11A

(0)22−A

(0)212)

A11)+4σ2(A

(0)12+mγA

(0)11)2

~2A11]3/2.(54)

Thisexpectationwillbenegativewhenever4(A(0)11A

(0)22−A

(0)212)/~

2<1,thatis

exactlyforthosetimeswhenTrρ2givenby(51)isgreaterthanone.

Wehaveseenthatgeneralsolutionoftheexactmasterequationresultsina

densitymatrixthatcanhavenegativeeigenvalues.Howcanitbethatanexact

21

resultcanleadtosuchanunphysicalconsequence?Theanswer,ofcourse,is

thatwebeenabletogiveanon-trivialmeaningtothesolutiononlyinthe

hightemperaturelimit,whichinvolvesaseriousapproximation(neglectof

zero-pointoscillations).Itisthisapproximateresultthatfailsthepositivity

test.

5.2Wavepacketpair

ForthewavepacketpairtheWignercharacteristicfunctionattimetisgiven

by(26).Withthistheformula(45)forthematrixelementofthedensity

operatorbecomesastandardGaussianintegral.Theresultis

〈x|ρ(t)|x′〉=1

2(1+e−d2/8σ2)

×[expiLd

2(x−x′)R(x−mG

d

2,x′−mGd

2)

exp−iLd

2(x−x′)R(x+

mGd

2,x′+mGd

2)

+e−A(0)

expiMd

2(x+x′)R(x−K

d

2,x′+K

d

2)

+e−A(0)

exp−iMd

2(x+x′)R(x+K

d

2,x′−Kd

2)],(55)

whereR(x,x′;t)isisthematrixelement(50)correspondingtoasinglewave

packetinitiallycenteredattheoriginandtoshortentheexpressionswehave

introduced

K=~

2(mGA

(0)11−GA

(0)12)

4σ2(A(0)11A

(0)22−A

(0)212)

,L=m

2G

~.,M=

~(mGA(0)12−GA

(0)22)

4σ2(A(0)11A

(0)22−A

(0)212)

.(56)

Viewedinthexx′plane,thedensitymatrixelement(55)willshowsfourpeaks

exactlyoftheform(49)ofsinglewavepackets;twodiagonalpeakscentered

atx=x′=±mGd/2andtwooff-diagonalpeakscenteredatx=−x′=

22

±Kd/2.Theoff-diagonalpeaksaremultipliedbythesamefactore−A(0)(t)

thatmultipliestheinterferencetermintheWignerfunction(37).Theresult

isthatundertheconditiondÀλth,whilethepeaksslowlybroadenanddrift

towardtheorigin,theoffdiagonalpeaksrapidlyshrinkinamplitude.Thus,the

twodescriptionsofthedisappearanceoftheinterferenceterm,thatinterms

oftheWignerfunctionandthatintermsofthedensitymatrixelement,are

entirelyequivalent.Inthisconnectionnotethatthediagonaldensitymatrix

elementistheprobabilitydistribution(33).Thedescriptionintermsofthe

Wignerfunctionhastheadvantageofbeingreal,sosomewhatsimpler.Neither

theWignerdistributioninphasespacenorthedensitymatrixelementinxx′

spaceisdirectlyobservable,soeitherdescriptionistheoretical.

6Conclusion

Wehaveexhibitedanumberofsignificantfailuresoftheexactmasterequa-

tion:Duetotheirreparabledivergencethesolutionstrictlydoesnotexist

exceptinthehightemperaturelimit.Eveninthatapproximationthesolution

breakstranslationalinvariance.Finally,thesolutioncorrespondstoaden-

sitymatrixthatcanhavenegativeexpectedvalues.Butwemustremember

thatthederivationoftheequationaswellasitssolutionareformallycorrect.

Rather,thefailurescanallbetracedbacktotheassumptionofanunen-

tangeledinitialstate.Thepictureofaparticlesuddenlyclampedtoabath

correspondstonophysicallyrealizableoperation.Evenwhentheinitialpar-

ticletemperatureisadjustedtothatofthebathitreceivesaviolentsqueeze

towardtheorigin.Fromthisfollowsthe”unphysical”consequences:correctly

describedbytheexactmasterequation.Unfortunately,thefinalassessment

23

isthattheexactmasterequationisofverylimitedutility.

Infact,inanexactdescriptiontheparticlemustbecoupledtothebath(en-

tangled)atalltimes.Thinkoftheexampleoftheblackbodyradiationfield,

whichiscoupledtotheparticlethroughitschargethatcaninnowaybe

switchedonoroff.Inphysicalapplicationsthecouplingtotheradiationfield

issufficientlyweakthatadescriptionintermsofthefamiliarweakcoupling

masterequation(ofLindbladform)isappropriate,buttheeffectsofrenor-

malizationandzero-pointoscillationsareneverthelesspresentatalltimes.

24

References

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[2]G.W.FordandM.Kac,J.Stat.Phys.45,803(1987).

[3]L.vanHove,Physica18,145(1952).

[4]H.ArakiandS.Yamagami,Publ.Res.Inst.Math.Sci.18,283(1982).

[5]G.V.EfimovandW.vonWaldenfels,AnnalsofPhysics233,182(1994).

[6]G.Lindblad,Commun.Math.Phys.48,119(1976).

[7]G.W.FordJ.T.LewisandR.F.O’Connell,Phys.Rev.A64,032101(2001).

[8]G.A.Garrrett,A.G.Rojo,A.K.Sood,J.F.WhittakerandR.Merlin,Science

275,1638(1997).

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25

Fig.1.ThefunctionsA(0)(t)andA(T)(t)multipliedbyλ2th/d2plottedvs.γt.The

parameterschosenareλth=4σandkT=5~γ.

Fig.2.Trρ2foraninitialminimumuncertaintywavepacket.Theparameters

chosenareλth=4σ,kT=5~γ,thesameasforFig.1.Notethatatshorttimes

Trρ2>1.

26