Open quantum systems I: the Hamiltonian approach

346 Pages·2006·3.04 MB·English

by  Alain Joye (auth.)

Lecture Notes in Mathematics 1880
Editors:
J.-M.Morel,Cachan
F.Takens,Groningen
B.Teissier,Paris
· ·
S. Attal A. Joye C.-A. Pillet (Eds.)

Open Quantum
Systems I
The Hamiltonian Approach
ABC
Editors
Stéphane Attal Alain Joye

InstitutCamille Jordan Institut Fourier

Université Claude Bernard Lyon 1 Universitéde Grenoble 1
21 av. Claude Bernard BP 74
69622 Villeurbanne Cedex 38402 Saint-Martin d'Hères Cedex
France France
e-mail:[email protected] e-mail:[email protected]
Claude-Alain Pillet
CPT-CNRS, UMR 6207
Université du Sud Toulon-Var
BP 20132
83957 La Garde Cedex
France
e-mail:[email protected]
LibraryofCongressControlNumber:2006923432
MathematicsSubjectClassification(2000):37A60,37A30,47A05,47D06, 47L30,47L90,
60H10,60J25,81Q10,81S25,82C10, 82C70
ISSNprintedition:0075-8434
ISSNelectronicedition:1617-9692
ISBN-10 3-540-30991-8 SpringerBerlinHeidelbergNewYork
ISBN-13 978-3-540-30991-8 SpringerBerlinHeidelbergNewYork
DOI10.1007/b128449
Thisworkissubjecttocopyright. Allrightsarereserved,whetherthewholeorpartofthematerialis
concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting,
reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication
orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9,
1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violations
areliableforprosecutionundertheGermanCopyrightLaw.
SpringerisapartofSpringerScience+BusinessMedia
springer.com
(cid:1)c Springer-VerlagBerlinHeidelberg2006
PrintedinTheNetherlands
Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnotimply,
evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevantprotectivelaws
andregulationsandthereforefreeforgeneraluse.
Typesetting:bytheauthorsandSPI Publisher Services usingaSpringerLATEXpackage
Coverdesign:design&productionGmbH,Heidelberg
Printedonacid-freepaper SPIN:11602606 V A 41/3100/ SPI 543210
Preface
This is the first in a series of three volumes dedicated to the lecture notes of the
SummerSchool”OpenQuantumSystems”whichtookplaceattheInstitutFourier
in Grenoble from June 16th to July 4th 2003. The contributions presented in these
volumes are revised and expanded versions of the notes provided to the students
duringtheSchool.
Closedvs.OpenSystems
By definition, the time evolution of a closed physical system S is deterministic. It
is usually described by a differential equation x˙ = X(x ) on the manifold M of
t t
possibleconfigurationsofthesystem.Iftheinitialconfigurationx ∈ M isknown
0
thenthesolutionofthecorrespondinginitialvalueproblemyieldstheconfiguration
x atanyfuturetimet.Thisappliestoclassicalaswellastoquantumsystems.Inthe
t
classicalcaseM isthephasespaceofthesystemandx describesthepositionsand
t
velocities of the various components (or degrees of freedom) of S at time t. In the
quantumcase,accordingtotheorthodoxinterpretationofquantummechanics,M is
a Hilbert space and x a unit vector – the wave function – describing the quantum
t
state of the system at time t. In both cases the knowledge of the state x allows
t
to predict the result of any measurement made on S at time t. Of course, what we
mean by the result of a measurement depends on whether the system is classical
or quantum, but we should not be concerned with this distinction here. The only
relevantpointisthatx carriesthemaximalamountofinformationonthesystemS
t
attimetwhichiscompatiblewiththelawsofphysics.
In principle any physical system S that is not closed can be considered as part
ofalargerbutclosedsystem.ItsufficestoconsiderwithS thesetRofallsystems
which interact, in a way or another, with S. The joint system S ∨R is closed and
from the knowledge of its state x at time t we can retrieve all the information on
t
itssubsystemS.InthiscasewesaythatthesystemS isopenandthatRisitsenvi-
ronment.Therearehoweversomepracticalproblemswiththissimplepicture.Since
the joint system S ∨R can be really big (e.g., the entire universe) it may be diffi-
cult,ifnotimpossible,towritedownitsevolutionequation.Thereisnosolutionto
VI Preface
thisproblem.Thepragmaticwaytobypassitistoneglectpartsoftheenvironment
R which, a priori, are supposed to be of negligible effect on the evolution of the
subsystemS.Forexample,whendealingwiththemotionofachargedparticleitis
oftenreasonabletoneglectallbuttheelectromagneticinteractionsandsupposethat
theenvironmentconsistsmerelyintheelectromagneticfield.Moreover,iftheparti-
clemovesinaverysparseenvironmentlikeintergalacticspacethenwecanconsider
that it is the only source in the Maxwell equations which governs the evolution of
R. Assuming that we can write down and solve the evolution equation of the joint
systemS∨Rweneverthelesshitasecondproblem:howtochoosetheinitialconfig-
urationoftheenvironment?IfRhasaverylarge(e.g.,infinite)numberofdegrees
of freedom then it is practically impossible to determine its configuration at some
initial time t = 0. Moreover, the dynamics of the joint system is very likely to be
chaotic,i.e.,todisplaysomesortofinstabilityorsensitivedependenceontheinitial
condition.Theslightesterrorintheinitialconfigurationwillberapidlyamplifiedand
ruinourhopetopredictthestateofthesystematsomelatertime.Thus,insteadof
specifyingasingleinitialconfigurationofRweshouldprovideastatisticalensem-
ble of typical configurations. Accordingly, the best we can hope for is a statistical
information on the state of our open system S at some later time t. The resulting
theoryofopensystemsisintrinsicallyprobabilistic.Itcanbeconsideredasapartof
statisticalmechanicsattheinterfacewiththeergodictheoryofstochasticprocesses
anddynamicalsystems.
TheparadigmofthisstatisticalapproachtoopensystemsisthetheoryofBrown-
ianmotioninitiatedbyEinsteininoneofhiscelebrated1905papers[3](seealso[4]
for further developments). An account on this theory can be found in almost any
textbookonstatisticalmechanics(seeforexample[9]).Brownianmotionhadadeep
impactnotonlyonphysicsbutalsoonmathematics,leadingtothedevelopmentof
thetheoryofstochasticprocesses(seeforexample[12]).
Open systems appeared quite early in the development of quantum mechanics.
Indeed,toexplainthefinitelifetimeoftheexcitedstatesofanatomandtocompute
the width of the corresponding spectral lines it is necessary to take into account
the interaction of the electrons with the electromagnetic field. Einstein’s seminal
paper [5] on atomic radiation theory can be considered as the first attempt to use a
Markov process – or more precisely a master equation – to describe the dynamics
of a quantum open system. The theory of master equations and its application to
radiation theory and quantum statisticalmechanics was subsequently developed by
Pauli[8],WignerandWeisskopf[13],andvanHove[11].Themathematicaltheory
of the quantum Markov semigroups associated with these master equations started
todevelopmorethan30yearslater,aftertheworksofDavies[2]andLindblad[7].
Itfurtherledtothedevelopmentofquantumstochasticprocesses.
Toillustratethephilosophyofthemodernapproachtoopensystemsletuscon-
siderasimple,classical,microscopicmodelofBrownianmotion.Eventhoughthis
model is not realistic from a physical point of view it has the advantage of being
exactly solvable. In fact such models are often used in the physics literature (see
[10,6,1]).
Preface VII
BrownianMotion:ASimpleMicroscopicModel
Inacubiccrystaldenotebyq thedeviationofanatomfromitsequilibriumposition
x
x∈Λ ={−N,...,N}3 ⊂Z3andbyp thecorrespondingmomentum.Suppose
N x
thattheinter-atomicforcesareharmonicandonlyactsbetweennearestneighborsof
thecrystallattice.InappropriateunitstheHamiltonianofthecrystalisthen
(cid:1) (cid:1)
p2 κ
x + xy(q −q )2,
2 4 x y
x∈ΛN x,y∈Z3
where (cid:2)
1if|x−y|=1;
κ =
xy 0otherwise;
andDirichletboundaryconditionsareimposedbysettingq = 0forx ∈ Z3\Λ .
x N
Iftheatomatsitex = 0isreplacedbyaheavyimpurityofmassM (cid:4) 1thenthe
Hamiltonianbecomes
(cid:1) (cid:1)
p2 κ
H ≡ x + xy(q −q )2,
2m 4 x y
x∈ΛN x x,y∈Z3
where (cid:2)
M ifx=0;
m =
x 1otherwise.
We shall consider the heavy impurity at x = 0 as an open system S whose
environmentRismadeofthe(2N+1)3−1remainingatomsofthecrystal.Towrite
downtheequationofmotioninaconvenientformletusintroducesomenotation.We
wseetΛde∗Nno=teΛbyNδ\x{th0e}(cid:3),Kqro=ne(cqkxe)rxd∈eΛl∗Nta,fpun=cti(opnx)axt∈xΛa∗Nn,dQby=|xq|0t,hPeE=ucpl0id.eFaonrnxor∈mZo3f
x.Wealsosetχ= δ .ThemotionofthejointsystemS∨Risgovernedby
|x|=1 x
thefollowinglinearsystem
q˙ =p, p˙ =−Ω2q+Qχ,
0
(1)
MQ˙ =P, P˙ =−ω2Q+(χ,q),
0
where−Ω2 isthediscreteDirichletLaplacianonΛ∗ andω2 =6.Accordingtothe
0 N 0
opensystemphilosophydescribedinthepreviousparagraphweshouldsupplysome
appropriatestatisticalensembleofinitialstatesoftheenvironment.Tomotivatethe
choiceofthisensemblesupposethatintheremotepasttheimpuritywaspinnedat
some fixed position, say Q = P = 0, and that at time t = 0 the resulting system
hasreachedthermalequilibriumatsometemperatureT >0.Thepositionsandmo-
mentainthecrystalwillbedistributedaccordingtotheGibbs-Boltzmanncanonical
ensemblecorrespondingtothepinnedHamiltonianH =H| ,
0 Q=P=0
(cid:4) (cid:5)
1
H = (p,p)+(q,Ω2q) .
0 2 0
VIII Preface
ThisensembleisgivenbytheGaussianmeasure
dµ=Z−1e−βH0(q,p)dqdp,
whereZ isanormalizationfactorandβ =1/k T withk theBoltzmannconstant.
B B
Attimet = 0wereleasetheimpurity.Thesubsequentevolutionofthesystem
isdeterminedbytheCauchyproblemforEqu.(1).Theevolutionoftheenvironment
canbeexpressedbymeansoftheDuhamelformula
(cid:6)
sin(Ω t) t sin(Ω (t−s))
q(t)=cos(Ω t)q(0)+ 0 p(0)+ 0 χQ(s)ds.
0 Ω Ω
0 0 0
InsertingthisrelationintotheequationofmotionforQleadsto
(cid:6)
t
MQ¨ =−ω2Q+ K(t−s)Q(s)ds+ξ(t), (2)
0
0
wheretheintegralkernelK isgivenby
sin(Ω t)
K(t)=(χ, 0 χ), (3)
Ω
0
and (cid:7) (cid:8)
sin(Ω t)
ξ(t)= χ,cos(Ω t)q(0)+ 0 p(0) .
0 Ω
0
Sinceq(0),p(0)arejointlyGaussianrandomvariables,ξ(t)isaGaussianstochastic
process.Itisasimpleexercisetocomputeitsmeanandcovariance
1 cos(Ω (t−s))
E(ξ(t))=0, E(ξ(t)ξ(s))=C(t−s)= (χ, 0 χ). (4)
β Ω2
0
Wenoteinparticularthatthisprocessisstationary.Thetermξ(t)inEqu.(2)isthe
noisegeneratedbythefluctuationsoftheenvironment.Itvanishesiftheenvironment
isinitiallyatrest.TheintegralinEqu.(2)istheforceexertedbytheenvironmenton
theimpurityinreactiontoitsmotion.Notethatthisdissipativetermisindependent
ofthestateoftheenvironment.Thedissipativeandthefluctuatingforcesarerelated
bythesocalledfluctuation-dissipationtheorem
K(t)=−β∂ C(t). (5)
t
The solution zt = (Q(t),P(t)) of the random integro-differential equation (2)
defines a family of stochastic processes indexed by the initial condition z0. These
processesprovideastatisticaldescriptionofthemotionofouropensystem.Anin-
variantmeasureρfortheprocessztisameasureonR3×R3suchthat
(cid:6) (cid:6)
f(zt)dρ(z0)= f(z)dρ(z),
Preface IX
holds for all reasonable functions f and all t ∈ R. Such a measure describes a
steadystateofthesystem.Ifonecanshowthatforanyinitialdistributionρ which
0
isabsolutelycontinuouswithrespecttoLebesguemeasureonehas
(cid:6) (cid:6)
lim f(zt)dρ (z0)= f(z)dρ(z), (6)
0
t→∞
then the steady state ρ provides a good statistical description of the dynamics on
largetimescales.Oneofthemainprobleminthetheoryofopensystemsistoshow
thatsuchanaturalsteadystateexistsandtostudyitsproperties.
TheHamiltonianApproach
Remark that in our example, such a steady state fails to exist since the motion of
thejointsystemisclearlyquasi-periodic.However,inarealsituationthenumberof
atomsinthecrystalisverylarge,oftheorderofAvogadro’snumberN (cid:6)6·1023.
A
Inthiscasetherecurrencetimeofthesystembecomessolargethatitmakessenseto
takethelimitN → ∞.Inthislimit−Ω2 becomesthediscreteDirichletLaplacian
0
ontheinfinitelatticeZ3\{0}.Thisisawelldefined,bounded,negativeoperatoron
theHilbertspace(cid:11)2(Z3).Thus,Equ.(2),(3),(4)and(5)stillmakesenseinthislimit.
Inthesequelweonlyconsidertheresultinginfinitesystem.
Wedistinguishtwomainapproachestothestudyofopensystems.Thefirstone,
theHamiltonianapproach,dealsdirectlywiththedynamicsofthejointsystemS∨R.
Webrieflydiscussthesecondone,theMarkovianapproach,inthenextparagraph.
IntheHamiltonianapproachwerewritetheequationofmotion(1)as
Z˙ =−iΩ˜Z,
whereΩ˜2 =m−1/2Ω2m−1/2withm=I+(M−1)δ (δ , ·)theoperatorofmulti-
0 0
plicationbym and−Ω2isthediscreteLaplacianonZ3.ThecomplexvariableZis
x
givenbyZ =Ω˜1/2m1/2q˜+iΩ˜−1/2m−1/2p˜andq˜=(qx)x∈Z3,p˜=(px)x∈Z3.Itfol-
lowsfromelementaryspectralanalysisthatforM >1theoperator√Ω˜ isself-adjoint
withpurelyabsolutelycontinuousspectrumσ(Ω˜)=σ (Ω˜)=[0, 2ω ]on(cid:11)2(Z3).
ac 0
AsimpleargumentinvolvingthescatteringtheoryforthepairΩ2⊕ω2/M,Ω˜2shows
0 0
thatthesystemS hasauniquesteadystateρsuchthat(6)holdsforallρ whichare
0
absolutelycontinuouswithrespecttoLebesguemeasure.Moreover,ρisthemarginal
onS oftheinfinitedimensionalGaussianmeasureZ−1e−βHdpdqdPdQwhichde-
scribesthethermalequilibriumstateofthejointsystemattemperatureT =1/k β.
B
ThisiseasilycomputedtobetheGaussianmeasure
ρ(dP,dQ)=N−1e−β(P2/2M+ω2Q2/2)dPdQ,
whereN isanormalizationfactorand
1
ω2 = .
(δ ,Ω−2δ )
0 0
X Preface
TheMarkovianApproach
AremarkablefeatureofEqu.(2)isthememoryeffectinducedbythekernelK.Asa
resulttheprocessztisnon-Markovian,i.e.,fors>0,zt+sdoesnotonlydependon
zt and{ξ(u)|u ∈ [t,t+s]}butalsoonthefullhistory{zu|u ∈ [0,t]}.Theonly
waytoavoidthiseffectistohaveKproportionaltothederivativeofadeltafunction.
By Relation (5) thismeans that ξ should be a whitenoise. This iscertainly not the
case with our choice of initial conditions. However, as we shall see, it is possible
toobtainaMarkovprocessinsomeparticularscalinglimits.Thisisnotauniquely
defined procedure: different scaling limits correspond to different physical regimes
andleadtodistinctMarkovprocesses.
Asasimpleillustrationletusconsidertheparticularscalinglimit
Q (t)≡M1/4Q(M1/2t), M →∞.
M
ofourmodel.ForfiniteM theequationofmotionforQ reads
M
(cid:6)
t
Q¨ (t)=−ω2Q (t)+ K (t−s)Q (s)ds+ξ (t),
M 0 M M M M
0
where
K (t)≡M1/2K(M1/2t),
M
andthescaledprocessξ (t)≡M1/4ξ(M1/2t)hascovariance
M
C (t)≡M1/2C(M1/2t).
M
(cid:9)
OnecanshowthatC(t)isinL1(R)andthatσ = C(t)dt > 0.Itfollowsthat,in
distributionalsense,
lim C (t)=σδ(t), lim K (t)=0.
M M
M→∞ M→∞
WeconcludethatthelimitingequationforQis
Q¨(t)=−ω2Q(t)+σ1/2η(t),
0
whereη iswhitenoise,i.e.,E(η(t)η(s)) = δ(t−s).Thesolution(Q(t),Q˙(t))isa
MarkovprocessonR3×R3withgenerator
σ
L=− ∆2 −P ·∇ +ω2Q·∇ .
2 P Q 0 P
Itisasimpleexercisetoshowthattheuniqueinvariantmeasureofthisprocessisthe
Lebesgue measure. Moreover, one can show that for any initial condition (Q ,P )
0 0
andanyfunctionf ∈L1(R3×R3,dQdP)onehas
(cid:6)
lim E(f(Q(t),Q˙(t)))= f(Q,P)dQdP,
t→∞
ascaledversionofreturntoequilibrium.
Preface XI
Itisworthpointingoutthatinmanyinstancesofclassicalorquantumopensys-
tems the dynamics of the joint system S ∨R is too complicated to be controlled
analytically or even numerically. Thus, the Hamiltonian approach is inefficient and
the Markovian approximation becomes the only available option. The study of the
Markovian dynamics of open systems is the main subject of the second volume in
thisseries.Thethirdvolumeisdevotedtoapplicationsofthetechniquesintroduced
inthefirsttwovolumes. Itaimsatleading thereader tothefrontof thecurrent re-
searchonopenquantumsystems.
OrganizationofthisVolume
ThisfirstvolumeisdevotedtotheHamiltonianapproach.Itspurposeistodevelop
the mathematical framework necessary to define and study the dynamics and ther-
modynamicsofquantumsystemswithinfinitelymanydegreesoffreedom.
ThefirsttwolecturesbyA.Joyeprovideaminimalbackgroundinoperatorthe-
oryandstatisticalmechanics.ThethirdlecturebyS.Attalisanintroductiontothe
theoryofoperatoralgebraswhichisthenaturalframeworkforquantummechanics
ofmanydegreesoffreedom.Quantumdynamicalsystemsandtheirergodictheory
are the main object of the fourth lecture by C.-A. Pillet. The fifth lecture by M.
Merklidealswiththemostcommoninstancesofenvironmentsinquantumphysics,
theidealBoseandFermigases.FinallythelastlecturebyV.Jaksˇic´ introducesone
ofthemaintoolinthestudyofquantumdynamicalsystems:spectralanalysis.
Lyon,GrenobleandToulon, Ste´phaneAttal
September2005 AlainJoye
Claude-AlainPillet