Diffusion of a massive quantum particle coupled to a quasi-free thermal medium

1 MB·English

by  W. De Roeck

Diffusion of amassive quantumparticle coupled toa quasi-freethermalmedium.
W.DeRoeck1
InstituteforTheoreticalPhysics
K.U.Leuven
B3001Heverlee,Belgium
InstituteforTheoreticalPhysics
ETHZu¨rich
CH-8093Zu¨rich,Switzerland
J.Fro¨hlich
0
1 InstituteforTheoreticalPhysics
0 ETHZu¨rich
2 CH-8093Zu¨rich,Switzerland

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u
A

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2 Abstract: Weconsideraheavyquantumparticlewithaninternaldegreeoffreedommovingonthed-dimensionallatticeZd (e.g.,aheavy
atomwithfinitelymanyinternalstates). Theparticleiscoupledtoathermalmedium(bath)consistingoffreerelativisticbosons(photons
] orGoldstonemodes)throughaninteractionofstrengthλlinearincreationandannihilationoperators. Themassofthequantumparticleis
h assumedtobeoforderλ−2,andweassumethattheinternaldegreeoffreedomiscoupled“effectively”tothethermalmedium.Weprovethat
p themotionofthequantumparticleisdiffusiveind≥4andforλsmallenough.
-
h KEYWORDS:diffusion,weakcouplinglimit,quantumBoltzmannequation,quantumfieldtheory
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a
m
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1 Introduction
3
v
8 1.1 Diffusion
7
1 Diffusion and Brownianmotion arecentralphenomena in the theory of transport processesand nonequilibrium
5 statisticalphysicsingeneral. Onecanthinkofthediffusionofatracerparticleininteractingparticlesystems,the
.
6 diffusionofenergyincoupledoscillatorchains,andmanyotherexamples.
0 Fromaheuristic point of view, diffusionisratherwell-understood inmost of these examples. Itcanoftenbe
9
successfullydescribedbysomeMarkovianapproximation,e.g.theBoltzmannequationorFokker-Planckequation,
0
dependingontheexampleunderstudy. Infact,thishasbeenthestrategyofEinsteininhisgroundbreakingwork
:
v of1905,inwhichhemodeleddiffusionasarandomwalk.
i
X However,uptothisdate,thereisnorigorousderivationofdiffusionfromclassicalHamiltonianmechanicsor
unitaryquantummechanics,exceptforsomespecialchaoticsystems;seeSection1.3.1. Suchaderivationoughtto
r
a allowus,forexample,toprovethatthemotionofatracerparticlethatinteractswithitsenvironmentisdiffusive
atlargetimes. Inotherwords,onewouldliketoproveacentrallimittheoremforthepositionofsuchaparticle.
Inrecentyears,somepromisingstepstowardsthisgoalhavebeentaken. Weprovideabriefreviewofprevious
resultsinSection1.3. Inthepresentpaper,werigorouslyexhibitdiffusionforaquantumparticleweaklycoupled
toathermalreservoir.However,ourmethodisrestrictedtospatialdimensiond 4.
≥
1PostdoctoralFellowFWO-FlandersatK.U.Leuven,Belgium,email:[email protected]
1
1.2 Informaldescription of themodeland mainresults
WeconsideraquantumparticlehoppingonthelatticeZd,andinteractingwithareservoirofbosons(photonsor
phonons) attemperatureβ 1 > 0. Inthe presentsection, we describe the system in awaythat isappropriateat
−
zerotemperature,butisformalwhenβ < . ThetotalHilbertspace,H,ofthecoupledsystemisatensorproduct
oftheparticlespace,H ,withareservoirs∞pace,H . Thus
S R
H :=H H . (1.1)
S R
⊗
TheparticlespaceH isgivenbyl2(Zd) S,wheretheHilbertspaceS describestheinternaldegreesoffreedom
S
⊗
of the particle, e.g., a (pseudo-)spin or dipole moment, and the particle Hamiltonian is given by the sum of the
kineticenergyandtheenergyoftheinternaldegreesoffreedom
H :=H 1+1 H (1.2)
S S,kin S,spin
⊗ ⊗
Thekineticenergyischosentobesmallincomparisonwiththeinteractionenergy,andthisismademanifestinits
definitionbyafactorλ2,whereλisthecouplingstrengthbetweentheparticleandthereservoir(tobeintroduced
below). Henceweset
H =λ2ε(P), (1.3)
S,kin
wherethefunctionεisthedispersionlawoftheparticleandP isthelattice-momentumoperator.Themostnatural
choiceistotakeε(P)tobe(minus)thediscretelatticeLaplacian, ∆. Theenergyofstatesoftheinternaldegreeof
−
freedomistoalargeextentarbitrary
H :=Y, forsomeHermitianmatrixY, (1.4)
S,spin
themainrequirementbeingthatY notbeequaltoamultipleoftheidentity.
Thereservoirisdescribedbyafreeboson field; creationandannihilation operatorscreating/annihilatingbo-
sonswithmomentumq Rd arewrittenasa ,a ,respectively. Theysatisfythecanonicalcommutationrelations
∈ ∗q q
[a#q ,a#q′]=0, [aq,a∗q′]=δ(q−q′), (1.5)
wherea# standsforeitheraora . Theenergyofareservoirmodeq isgivenbythedispersionlawω(q) 0. To
∗
describethecouplingoftheparticletothereservoir,weintroduceaHermitianmatrixW onS andwewrit≥eX for
thepositionoperatoronl2(Zd).
ThetotalHamiltonianofthesystemistakentobe
Hλ :=HS+ZRddqω(q)a∗qaq+λZ dq(cid:16)eiq·X ⊗W ⊗φ(q)aq +e−iq·X ⊗W ⊗φ(q)a∗q(cid:17) (1.6)
actingonH H . Thefunctionφ(q)isaformfactorandλ Risthecouplingstrength. WewriteH insteadof
S R S
⊗ ∈
H 1,etc.
S
⊗
Weintroducethreeimportantassumptions:
1) Thekineticenergyissmallw.r.t.thecouplingtermintheHamiltonian,ashasalreadybeenindicatedbythe
inclusionofλ2inthedefinitionofH . Physically,thismeansthattheparticleisheavy.
S,kin
2) Werequirealineardispersionlawforthereservoirmodes,ω(q) q ,inordertohavegooddecayestimatesat
≡| |
lowspeed.Thismeansthatthereservoirconsistsofphotons,phononsorGoldstonemodesofaBose-Einstein
condensate.
3) We assume that the amplitude of the wave front of a reservoir excitation (located on the light cone) has
integrable(intime)decay.Thisissatisfiedifthedimensionofspace2isatleast4.
2Sincetheintegrabilityintimeisonlyneededforreservoirexcitations, wecaninprinciple alsotreatmodelsinwhich theparticleis3-
dimensional,butthereservoiriseffectively4-dimensional.
2
Additionalassumptionswillconcernthesmoothnessoftheformfactorφandthe“effectiveness”ofthecoupling
totheheatbath(e.g.,theinteractionbetweenthe internaldegreesof freedomandthereservoir,describedbythe
matrixW,shouldnotvanish.)
The initial state, ρβ, of the reservoir is chosen to be an equilibrium state at temperature β 1 > 0. For math-
R −
ematicaldetailson the construction of infinite reservoirs, see e.g.[10, 4,2]. The initialstate of the whole system,
consistingoftheparticleandthereservoir,isaproductstateρ ρβ,withρ adensitymatrixfortheparticlethat
S⊗ R S
willbespecifiedlater. Thetime-evolveddensitymatrixoftheparticle(’subsystem’)iscalledρ andisobtained
S,t
by “tracing out the reservoir degreesof freedom” after the time-evolution has acted on the initial state during a
timet,i.e.,formally,
ρS,t :=TrHR e−itHλ ρS⊗ρβR eitHλ , (1.7)
whereTrHR isthepartialtraceoverHR. Wewarnhtherea(cid:16)derthatt(cid:17)heaboiveformuladoesnotmakesensemath-
ematically for an infinitely extended reservoir, since the reservoir state ρβ is not a density matrix on H . This
R R
is aconsequence of the factthatthe reservoiris describedfromthe startin the thermodynamic limit and, hence,
the reservoir modes form a continuum. Nevertheless, the LHS of formula (1.7) can be given a meaning in the
thermodynamiclimit.
Thedensitymatrixρ obviouslydependsonthecouplingstrengthλ,butwedonotindicatethisexplicitly. We
S,t
alsodropthesubscriptSandwesimplywriteρ ,insteadofρ ,inwhatfollows.
t S,t
Wewilloftenrepresentρ asaB(S)-valuedfunctiononZd Zd:
t
×
ρ (x ,x ) B(S), x ,x Zd. (1.8)
t L R ∈ L R ∈
Althoughthisisnotnecessaryformanyofourresults,werequiretheinitialstateoftheparticletobeexponentially
localizedneartheoriginofthelattice,i.e.,
kρt(xL,xR)kB(S) ≤Ce−δ′|xL|e−δ′|xR|, forsomeconstantsC,δ′ >0 (1.9)
Ourfirstresultconcernsthediffusionofthepositionoftheparticle.
1.2.1 Diffusion
Wedefinetheprobabilitydensity
µt(x):=TrS ρt(x,x) (1.10)
whereTrS denotesthepartialtraceovertheinternaldegreesoffreedom. Thenumberµt(x)istheprobabilityto
findtheparticleatsitexaftertimet.
Bydiffusion,wemeanthat,forlarget,
d/2
1 1 x x
µt(x) (detD)−1/2exp D−1 , (1.11)
∼ 2πt {−2 √t · √t }
(cid:18) (cid:19) (cid:18) (cid:19)
wherethediffusiontensorD D isastrictlypositivematrixwithrealentries;actually,iftheparticledispersion
λ
≡
law ε is invariant under lattice rotations, then the tensor D is isotropic and hence a scalar. The magnitude of D
is inferred from the following reasoning: The particle undergoes collisions with the reservoir modes. Let t be
m
the mean time between two collisions, and let v be the mean speed of the particle (the direction of the particle
m
velocityisassumedtoberandom). Thenthemeanfreepathisv t andthecentrallimittheoremsuggeststhat
m m
×
theparticlediffuseswithdiffusionconstant
(v t )2
m m
D × (1.12)
∼ t
m
The meantime t is of ordert λ 2 since the interaction with the reservoircontributes only in second order.
m m −
∼
The mean velocity v is of order v λ2 because of the factor λ2 in the definition of the kinetic energy. Hence
m m
∼
D λ2.
∼
3
Wenowmovetowardsquantifying(1.11). Letusfixatimet. Sinceµ (x)isaprobabilitymeasure,onecanthink
t
ofx asarandomvariablewith
t
Prob(x =x):=µ (x). (1.13)
t t
The claimthatthe randomvariable xt convergesindistribution, ast , toaGaussianrandomvariablewith
√t ր ∞
mean0andvarianceD iscalledaCentralLimitTheorem(CLT).Itisequivalenttopointwise convergenceof the
characteristicfunction,i.e.,
e−√itx·qµt(x) e−12q·Dq, forallq Rd, (1.14)
−t→ ∈
xX∈Zd ↑∞
anditisthisstatementwhichisourmainresult,Theorem3.1. Astrongerversionoftheconvergencein(1.14)(also
includedinTheorem3.1)impliesthattherescaledmomentsofµ converge. Forexample,fori,j =1,...,d,
t
1 1
Tr[ρ X ]= x µ (x) 0 (1.15)
t i i t
t t −t→
x ↑∞
X
1 1
Tr[ρ X X ]= x x µ (x) D , (1.16)
t i j i j t i,j
t t −t→
x ↑∞
X
Infact,thefirstline(vanishingofaveragedrift)isexpectedonlyifoneassumesthatthemodelhasspaceinversion
symmetry,whichisassumedthroughout.
1.2.2 Equipartition
Oursecondresultconcernstheasymptoticexpectationvalueofthekineticenergyoftheparticleandtheinternal
degreesoffreedom. Theequipartitiontheoremsuggeststhattheenergyofalldegreesoffreedomof theparticle,
the translational and internal degrees of freedom, thermalizes at the temperature β 1 of the heat bath. We will
−
establish this property up to a correction that is small in the coupling strength λ. This is acceptable, since the
interactioneffectivelymodifiestheGibbsstateoftheparticle. Weprovethat,forallboundedfunctionsF,
1
ρt(F(HS,kin)) dkF(λ2ε(k))e−βλ2ε(k)+o(λ0) (1.17)
t−ր→∞ Z ZTd | |
1
ρt(F(HS,spin)) F(e)e−βe+o(λ0), as λ 0 (1.18)
t−ր→∞ Z′ e spY | | ց
∈X
where Z,Z arenormalization constants and the sum rangesover alleigenvalues of the Hamiltonian Y.
′ e spY
Wenotethatthefactore βλ2ε(k) canbereplacedby1(asin∈Theorem3.2)sinceweanyhowallowacorrectionterm
− P
thatissmallinλandthefunctionε(k)isbounded. Forthisreason,onecouldsaythat,forverysmallvaluesofλ,
thetranslationaldegreesoffreedomthermalizeatinfinitetemperature(β =0).
1.2.3 Decoherence
Bydecoherencewemeanthatoff-diagonalelementsρ (x,y)ofthedensitymatrixρ inthepositionrepresentation
t t
falloffrapidlyinthedistancebetweenxandy. Ofcourse,thispropertycanonlyholdatlargeenoughtimeswhen
theeffectofthereservoirontheparticlehasdestroyedallinitiallong-distancecoherence,i.e.,afteratimeoforder
λ 2. Thus,thereisadecoherencelength1/γ andadecayrategsuchthat
− dch
kρt(xL,xR)kB(S) ≤Ce−γdch|xL−xR|+C′e−λ2gt, as tր∞ (1.19)
forsomeconstantsC,C . Themagnitudeoftheinversedecoherencerateγ isdeterminedasfollows: Thetime
′ dch
thereservoirneedstodestroycoherenceisoftheorderofthemeanfreetimet ,whilethetimethatisneededfor
m
coherence to be built up over a distance 1/γ is given by (γ v ) 1, where v is the mean velocity of the
dch dch m − m
×
particle. Equatingthesetwotimesyields
γdch (tm vm)−1 (1.20)
∼ ×
andhence,recallingthatt λ 2andv λ2,asarguedinSection1.2.1,wefindthatγ doesnotscalewithλ.
m − m dch
∼ ∼
4
1.3 Related resultsanddiscussion
1.3.1 Classicalmechanics
Diffusionhasbeenestablishedforthetwo-dimensionalfinitehorizonbilliardin[6]. Inthatsetup,apointparticle
travelsinaperiodic,planararrayoffixedhard-corescatterers.Thefinite-horizonconditionreferstothefactthatthe
particlecannotmovefurtherthanafixeddistancewithouthittinganobstacle.
Knauf[27]replacedthe hard-corescatterersbyaplanarlatticeofattractiveCoulombic potentials, i.e., thepo-
tential is V(x) = 1 . In that case, the motion of the particle can be mapped to the free motion on a
− j Z2 x j
∈ | − |
manifoldwithstrictlynegativecurvature,andonecanagainprovediffusion.
P
Recently,adifferentapproachwastakenin[24]: theauthorsof[24]considerad=3latticeofconfinedparticles
that interact locally with chaotic maps such that the energy of the particles is preserved but their momenta are
randomized. Neighboringparticlescanexchangeenergyviacollisionsandoneprovesdiffusivebehaviourofthe
energyprofile.
1.3.2 Quantummechanicsforextendedsystems
Theearliestresultforextendedquantumsystemsthatweareawareof,[30],treatsaquantumparticleinteracting
with a time-dependent random potential that has no memory (the time-correlation function is δ(t)). Recently,
thiswasgeneralizedin[25]tothecaseoftime-dependentrandompotentialswherethetime-dependenceisgiven
by a Markov process with a gap (hence, the free time-correlation function of the environment is exponentially
decaying). In[32],wetreatedaquantumparticleinteractingwithindependentheatreservoirsateachlatticesite.
Thismodelalsohasanexponentiallydecayingfreereservoirtime-correlationfunctionandassuch,itisverysimilar
to[25]. Noticealsothat,inspirit,themodelwithindependentheatbathsiscomparabletothemodelof[24],but,
inpractice,itiseasiersincequantummechanicsislinear!
Themostseriousshortcomingoftheseresultsisthefactthattheassumptionofexponentialdecayofthecorrel-
ationfunctionintimeisunrealistic. Inthemodelofthepresentpaper,thespace-timecorrelationfunction, called
ψ(x,t)inwhatfollows,isthecorrelationfunctionoffreely-evolvingexcitationsinthereservoir,createdbyinterac-
tionwiththeparticle. Sincemomentumisconservedlocally,theseexcitationscannotdecayexponentiallyintime
t,uniformlyinx. Forexample,ifthedispersionlawofthereservoirmodesislinear,thenψ(x,t)isasolutionofthe
linearwaveequation. Ind=3,itbehavesqualitativelyas
1
ψ(x,t) δ(ct x), with c thepropagationspeedofthereservoirmodes (1.21)
∼ x | |−| |
| |
Inhigherdimensions,onehasbetterdispersiveestimates,namelysupx|ψ(x,t)|≤td−21 (undercertainconditions),
andthisisthereasonwhy,forthetimebeing,ourapproachisrestrictedtod 4. IntheAndersonmodel,theana-
≥
logueofthecorrelationfunctiondoesnotdecayatall,sincethepotentialsarefixedintime. Indeed,theAnderson
model is different from our particle-reservoir model: diffusion is only expected to occur for small values of the
couplingstrength,whereastheparticlegetstrapped(Andersonlocalization)atlargecoupling.
Finally, we mention a recent and exciting development: in [13], the existence of a delocalized phase in three
dimensionsisprovenforasupersymmetricmodelwhichisinterpretedasatoyversionoftheAndersonmodel.
1.3.3 Quantummechanicsforconfinedsystems
Thetheoryofconfinedquantumsystems,i.e.,multi-levelatoms,incontactwithquasi-freethermalreservoirshas
beenintensivelystudiedinthelastdecade,e.g.by[3,23,12].Inthissetup,oneprovesapproachtoequilibriumfor
themulti-levelatom. Althoughatfirstsight,thisproblemisdifferentfromours(thereisnoanalogueofdiffusion),
thetechniquesarequitesimilarandweweremainlyinspiredbytheseresults. However,animportantdifference
is that, due to its confinement, the multi-level atom experiences a free reservoir correlation function with better
decay properties than that of our model. For example, in [23], the free reservoir correlation function is actually
exponentiallydecaying.
5
1.3.4 Scalinglimits
Up to now, most of the rigorous results on diffusion starting from deterministic dynamics are formulated in a
scalinglimit.Thismeansthatonedoesnotfixonedynamicalsystemandstudyitsbehaviourinthelong-timelimit,
but, rather, one comparesa family of dynamical systems at differenttimes. The precise definition of the scaling
limitdiffersfrommodeltomodel,but,ingeneral,onescalestime,spaceandthecouplingstrength(andpossibly
also the initial state) such that the Markovian approximation to the dynamics becomes exact. In our model the
naturalscaling limit is the so-called weakcoupling limit: one introducesthe macroscopic time τ := λ2t and one
takesthelimitλ 0,t whilekeepingτ fixed. Inthatlimit,thedynamicsoftheparticlebecomesMarkovian
ց ր∞
in τ (asif the heatbath had no memory) and it is describedbya Lindblad evolution. The long-time behavior of
thisLindbladevolutionisdiffusive. ThisisexplainedindetailinSection4. Onemaysaythat,inthisscalinglimit,
the heuristic reasoning employed in the previous sections to deduce the λ-dependence of the diffusion constant
and the decoherence length becomesexact. The same scaling is known verywell in the theoryof confined open
quantumsystemsasitgivesrisetothePaulimasterequation. Thiswasfirstmadeprecisein[9].
IfwehadsetupthemodelwithakineticenergyofO(1)(insteadofO(λ2)),thenoneshouldalsorescalespace
byintroducingthemacroscopicspace-coordinateχ:=λ2x. Thereasonforthisadditionalrescalingisthat,between
twocollisions,aparticlewithmassoforder1movesduringatimeoforderλ 2,andhenceittravelsadistanceof
−
orderλ 2. Theresultingscalinglimit
−
x λ−2x, t λ−2t, λ 0 (1.22)
→ → ց
isoftencalledthekineticlimit. InthekineticlimitthedynamicsoftheparticleisdescribedbyalinearBoltzmann
equation(LBE)inthevariables(χ,τ). TheconvergenceoftheparticledynamicstotheLBEhasbeenprovenin[14]
for a quantum particle coupled to a heatbath, and in [17] for a quantum particle coupled to a random potential
(Anderson model). The long-time, large-distance limit of the Boltzmann equation is the heat equation, which
suggeststhatoneshouldbeabletoderivetheheatequationdirectlyinthelimitingregimecorrespondingto
x λ−(2+κ)x, t λ−(2+2κ)t, λ 0, forsome κ>0. (1.23)
→ → ց
Thiswasaccomplishedin[16,15]fortheAndersonmodel. Ananalogousresultwasobtainedin[28]foraclassical
particlemovinginarandomforcefield.
1.3.5 Limitationstoourresult
Twostrikingfeaturesofourmodelarethelargemass,oforderλ 2,andtheinternaldegreesoffreedomdescribed
−
bytheHamiltonianH = Y. Physicallyspeaking,thesechoicesareofcoursenotnecessaryfordiffusion,they
S,spin
justmakeourtaskofprovingiteasier. Letusexplainwhythisisso. Firstofall,oncethemassischosentobeof
orderλ 2,theinternaldegreesoffreedomarenecessarytomakethemodeldiffusiveinsecondorderperturbation
−
theory. Withouttheinternaldegreesoffreedom,itwouldbeballistic. ThisisexplainedinSection4.2;inparticular,
it can be deduced immediately fromconservation of momentum and energyfor the processesin Figure 3. Note
alsothatthedependenceonλischosensuchthatthekinetic termH = λ2ε(P)iscomparabletotheparticle-
S,kin
reservoir interaction in second order of perturbation theory (both are of order λ2). The large mass ensures that
the position of the particle remains well-defined for a time of order λ 2, which permits us to sum up Feynman
−
diagramsinrealspace.
Further, we note thatour resultrequiresananalyticity assumption on the formfactor φ, see Assumption 2.3.
Thisassumption ensuresthatthefreereservoircorrelationfunction ψ(x,t)isexponentiallydecayingfor smallx,
eventhoughithasslowdecayonthelightcone,asexplainedinSection1.3.2.
1.4 Outline of the paper
ThemodelisintroducedinSection2andtheresultsarestatedinSection3.InSection4,wedescribetheMarkovian
approximationtoour model. Thisapproximationprovidesmost of the intuition and itisa keyingredientof the
proofs. Section5 describesthe mainideasof the proof, which iscontained inthe remainingSections6-9and the
fourappendicesA-D.
6
Acknowledgements
W.DeRoeckthanksJ.BricmontandH.Spohnforhelpfuldiscussionsandsuggestions,andforpointingoutseveral
references. He has also greatly benefited from collaboration with J. Clarkand C. Maes. In particular, the results
describedinSection4andAppendixCwereessentiallyobtainedin[7].Afterthefirstversionwassubmitted,some
inaccuracieswerepointedoutbyL.Erdo¨s,A.Knowles,H.-T.YauandJ.Yin. Mostimportantly,theremarksofan
anonymous refereeallowed for serious improvements in the presentation of the proof. Finally, W.D.R. acknow-
ledgesthefinancialsupportoftheFlemishresearchfund‘FWO-Vlaanderen’.
2 The model
AfterfixingconventionsinSection2.1,weintroducethemodel. Section2.2describestheparticle,whileSection2.3
dealswiththereservoir. InSection2.4,wecoupletheparticletothereservoir,andwedefinethereducedparticle
dynamics . Section2.5introducesthefiberdecomposition.
t
Z
2.1 Conventions and notation
GivenaHilbertspaceE,weusethestandardnotation
Bp(E):= S B(E),Tr (S∗S)p/2 < , 1 p , (2.1)
∈ ∞ ≤ ≤∞
n h i o
withB (E) B(E)theboundedoperatorsonE,and
∞ ≡
1/p
S p := Tr (S∗S)p/2 , S := S . (2.2)
k k k k k k∞
(cid:16) h i(cid:17)
For bounded operators acting on B (E), i.e. elements of B(B (E)), we use in general the calligraphic font:
p p
, , ,.... AnoperatorX B(E)determinesanoperatorad(X) B(B (E))by
p
V W T ∈ ∈
ad(X)S :=[X,S]=XS SX, S B (E). (2.3)
p
− ∈
ThenormofoperatorsinB(B (E))isdefinedby
p
(S)
p
:= sup kW k . (2.4)
kWk S∈Bp(E) kSkp
WewillmainlyworkwithHilbert-Schmidtoperators(p = 2)and,unlessmentionedotherwise,thenotation
kWk
willrefertothiscase.
Forvectorsυ Cd,weletReυ,Imυ denotethevectors(Reυ ,...,Reυ )and(Imυ ,...,Im υ ),respectively.
1 d 1 d
Thescalarproduc∈tonCdiswrittenasυ υ andthenormas υ :=√υ υ.
′
ThescalarproductonageneralHilbe·rtspaceE iswritten|a|s , ,o·r,occasionally,as , E. Allscalarproducts
h· ·i h· ·i
aredefinedtobelinearinthesecondargumentandanti-linearinthefirstone. Weusethephysicists’notation
ϕ ϕ′ fortherank-1operatorinB(E)actingas ϕ′′ ϕ′,ϕ′′ ϕ (2.5)
| ih | 7→h i
WewriteΓ (E)forthesymmetric(bosonic)FockspaceovertheHilbertspaceE andwereferto[10]fordefin-
s
itions and discussion. If ω is a self-adjoint operator on E, then its (self-adjoint) second quantization, dΓ (ω), is
s
definedby
n
dΓ (ω)Sym(ϕ ... ϕ ):= Sym(ϕ ... ωϕ ... ϕ ), (2.6)
s 1 n 1 i n
⊗ ⊗ ⊗ ⊗ ⊗ ⊗
i=1
X
whereSymprojectsonthesymmetricsubspaceof nE andϕ ,...,ϕ E.
1 n
⊗ ∈
WeuseC,C todenoteconstantswhoseprecisevaluecanchangefromequationtoequation.
′
7
2.2 The particle
We choose a finite-dimensional Hilbert space S, which can be thought of as the state space of some internal
degreesoffreedomoftheparticle,suchasspinoradipolemoment. ThetotalHilbertspaceoftheparticleisgiven
byH :=l2(Zd,S)=l2(Zd) S (thesubscriptSrefersto’system’,asiscustomaryinsystem-reservoirmodels).
S
Wedefinethepositionope⊗rators,X ,onH by
j S
(X ϕ)(x)=x ϕ(x), x Zd, ϕ l2(Zd,S), j =1,...,d (2.7)
j j
∈ ∈
In what follows, we will almost always drop the component index j and write X (X ) to denote the vector-
j
valuedpositionoperator. WewilloftenconsiderthespaceH initsdualrepresenta≡tion,i.e.asL2(Td,S),where
S
Tdisthed-dimensionaltorus(momentumspace),whichisidentifiedwithL2([ π,π]d,S). Weformallydefinethe
‘momentum’operatorP asmultiplicationbyk Td,i.e., −
∈
Pϕ(k)=kϕ(k), k [ π,π]d,ϕ L2(Td,S) (2.8)
∈ − ∈
AlthoughP iswell-definedasaboundedoperator,itdoesnotcorrespondtoacontinuous functiononTd,andit
isnottruethat[X ,P ]= i. Throughoutthepaper,wewillonlyuseoperatorsF(P)whereF isafunctiononTd
j j
thatisextendedperiodica−llytoRd. Wechoosesuchaperiodicfunction,ε,ofP todeterminethedispersionlawof
theparticle. Thekineticenergyofourparticleisgivenbyλ2ε(P),whereλisasmallparameter,i.e.,the’mass’of
theparticleisoforderλ 2
−
The energyof the internal degreesof freedomis given by a self-adjointoperator Y (S), actingon H as
S
∈ B
(Yϕ)(k)=Y(ϕ(k)). TheHamiltonianoftheparticleis
H :=λ2ε(P) 1+1 Y (2.9)
S
⊗ ⊗
AsinSection1,wewillmostlywriteε(P)insteadofε(P) 1andY insteadof1 Y.
⊗ ⊗
OurfirstassumptionensuresthattheHamiltonianH =Y +λ2ε(P)hasgoodregularityproperties
S
Assumption2.1(Analyticityoftheparticledynamics). Thefunctionε,definedoriginallyonTd,extendstoananalytic
functioninaneighborhoodofthecomplexmultistripofwidthδ >0. Thatis,whenviewedasaperiodicfunctiononRd,εis
ε
analytic(andbounded)inaneighborhoodof(R+i[ δ ,δ ])d. Moreover,εissymmetricwithrespecttospaceinversion,i.e.,
ε ε
−
ε(k)=ε( k). (2.10)
−
Furthermore, weassumethereisno υ Rd such thatthefunctionk υ ε(k)vanishesidenticallyandthatεdoesnot
haveasmallerperiodicitythanthatofT∈d,i.e.,weassumethat 7→ ·∇
ε(k)=ε(z+k) forall k Td z (2πZ)d. (2.11)
∈ ⇔ ∈
Themostnaturalchoiceforεisε(k)= d 2(1 cos(k )),whichcorrespondsto ε(P)beingthelatticeLapla-
i=1 − i −
cian. AsalreadyindicatedinSection1.2.1,thesymmetryassumption(2.10)isnecessarytoexcludeanasymptotic
P
driftoftheparticle.
Bya simple Paley-Wiener argument, Assumption 2.1implies thatone has exponential propagationestimates
fortheevolutiongeneratedbytheoperatorε(P). Indeed,fromtherelation
(eiν·Xe−itε(P)e−iν·X) = e−itε(P+ν) eqε(|Imν|)|t|, for Imν <δε (2.12)
≤ | |
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13) (cid:13) (cid:13) (cid:13)
withqε(γ):=sup|Imp|(cid:13)≤γ|Imε(p)|,oneobtai(cid:13)ns (cid:13) (cid:13)
(e−itε(P))(xL,xR) e−γ|xL−xR|eqε(γ)|t|, for γ δε (2.13)
S ≤ ≤
(cid:13) (cid:13)
wherewewriteS(xL,xR)fo(cid:13)(cid:13)raB(S)-valued’(cid:13)(cid:13)matrixelement’ofS B(HS).
∈
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2.3 The reservoirs
2.3.1 Thereservoirspace
We introduce a one-particle reservoir space h = L2(Rd) and a positive one-particle Hamiltonian ω 0. The
coordinateq Rd shouldbethoughtofasamomentumcoordinate,andω actsbymultiplicationwitha≥function
∈
ω(q),
(ωϕ)(q)=ω(q)ϕ(q) (2.14)
In other words, ω is the dispersion law of the reservoir particles. The full reservoir Hilbert space, H , is the
R
symmetricFockspace(seeSection2.1or[10])overtheone-particlespaceh,
H :=Γ (h) (2.15)
R s
ThereservoirHamiltonian,H ,actingonH ,isthenthesecondquanitzationofω
R R
HR :=dΓs(ω)= dqω(q)a∗qaq. (2.16)
ZRd
withthecreation/annihilationoperatorsa ,a tobeintroducedbelow.
∗q q
2.3.2 Thesystem-reservoircoupling
The coupling between system and reservoir is assumed to be translation invariant. We choose a ‘form factor’
φ L2(Rd)andaself-adjointoperatorW =W B(S)with W 1,andwedefinetheinteractionHamiltonian
∗
∈ ∈ k k≤
H by
SR
HSR := dq eiq·X ⊗W ⊗φ(q)aq +e−iq·X ⊗W ⊗φ(q)a∗q on HS⊗HR, (2.17)
Z (cid:16) (cid:17)
wherea ,a arethecreation/annihilationoperatorsonhsatisfyingthecanonicalcommutationrelations(CCR)
q ∗q
[aq,a∗q′]=δ(q−q′), [a#q ,a#q′]=0 (2.18)
witha#standingforeitheraora . Wealsointroducethesmearedcreation/annihilationoperators
∗
a∗(ϕ):=ZRddqϕ(q)a∗q, a(ϕ):=ZRddqϕ(q)aq, ϕ∈L2(Rd). (2.19)
InwhatfollowswewillspecifyourassumptionsonH ,butwealreadymentionthatweneed[W,Y] = 0for
SR
6
theinternaldegreesoffreedomtobecoupledeffectivelytothefield.
2.3.3 Thermalstates
Next, we put some tools in place to describe the positive temperature state of the reservoir. We introduce the
densityoperator
Tβ =(eβω 1)−1 on h=L2(Rd). (2.20)
−
LetCbethe algebraconsistingofpolynomialsinthecreationandannihilationoperatorsa(ϕ),a (ϕ)withϕ,ϕ
∗ ∗ ′ ′
∈
h. Wedefineρβ asaquasi-freestatedefinedonC. Itisfullyspecifiedbythefollowingproperties:
R
1) Gauge-invariance
ρβR[a∗(ϕ)]=ρβR[a(ϕ)]=0 (2.21)
2) Thechoiceofthetwo-particlecorrelationfunction
ρβ [a (ϕ)a(ϕ)] ρβ [a (ϕ)a (ϕ)] ϕ T ϕ 0
R ∗ ′ R ∗ ∗ ′ = h ′| β i (2.22)
(cid:18) ρβR[a(ϕ)a(ϕ′)] ρβR[a(ϕ)a∗(ϕ′)] (cid:19) (cid:18) 0 hϕ|(1+Tβ)ϕ′i (cid:19)
9
3) The state ρβ is quasifree. This means that the higher correlation functions are related to the two-particle
R
correlationfunctionviaWick’stheorem
ρβ a#(ϕ )...a#(ϕ ) = ρβ a#(ϕ )a#(ϕ ) (2.23)
R 1 2n R i j
(cid:2) (cid:3) πX∈Pn(i,Yj)∈π (cid:2) (cid:3)
wherea# standsforeithera ora,and isthesetofpairingsπ,partitionsof 1,...,2n intonpairs(r,s).
∗ n
P { }
Byconvention,wefixtheorderwithinthepairssuchthatr<s.
ThereasonthatitsufficestospecifythestateonChasbeenexplainedinmanyplaces,seee.g.[4,20,10]
2.3.4 Assumptionsonthereservoir
Next,westateourmainassumptionrestrictingthetypeofreservoirandthedimensionalityofspace.
Assumption2.2(Relativisticreservoirandd 4). Weassumethatthe
≥
dimensionofspace d 4 (2.24)
≥
Further,weassumethedispersionlawofthereservoirparticlestobelinear;
ω(q):= q (2.25)
| |
Forsimplicity,wewillassumethattheformfactorφisrotationallysymmetricandwewrite
φ(q) φ(q ), q Rd (2.26)
≡ | | ∈
Wedefinethe”effectivesquaredformfactor”as
1 φ(ω )2 ω 0
ψˆ(ω):= ω (d−1) 1−e−βω | | | | ≥ (2.27)
| | ( 1 φ(ω )2 ω <0
e−βω−1| | | |
whereweareabusingthenotation bylettingω denote avariableinR. Previously, ω wasthe energyoperatoron
theone-particleHilbertspaceandassuch,itcouldassumeonlypositivevalues. Indeed,atpositivetemperature,
thefunctionψˆ(ω)playsasimilarroleas φ(ω )2 atzero-temperature: Itdescribestheintensityofthecouplingto
| | | |
thereservoirmodesoffrequencyω. Modeswithω < 0appearonlyatpositivetemperatureandtheycorrespond
physicallyto”holes”. Onechecksthat ψˆ(ω) = eβω,whichisEinstein’semission-absorptionlaw(i.e.detailedbal-
ψˆ( ω)
−
ance). Thisparticle-holepointofviewcanbeincorporatedintotheformalismbytheAraki-Woodsrepresentation,
seee.g.[4,20,10].
Thenextassumptionrestrictsthe“effectivesquaredformfactor”ψˆ.
Assumption2.3(Analyticformfactor). Lettheformfactorberotation-symmetricφ(q) φ(q ),asin(2.26),andletψˆbe
≡ | |
definedasin(2.27). Weassumethatψˆ(0) = 0andthatthefunctionω ψˆ(ω)hasananalyticextensiontoaneighborhood
ofthestripR+i[δ ,δ ],forsomeδ >0,suchthat →
R R R
sup dω ψˆ(ω) < . (2.28)
| | ∞
−δR≤χ≤δRRZ+iχ
WenotethatAssumption2.3issatisfied(ind 4)ifonechooses:
≥
1
φ(q ):= ϑ(q ) (2.29)
| | q | |
| |
with ϑ a function on R with ϑ( ω) = ϑ(ω)and analyticpinthe strip of width δ , and such that(2.28) holds with
R
−
ϑ(ω)2substitutedfor ψˆ(ω).
| | | |
10