Open Quantum Systems II: The Markovian Approach

258 Pages·2006·2.36 MB·English

by  Luc Rey Bellet (auth.)

Lecture Notes in Mathematics 1881
Editors:
J.-M.Morel,Cachan
F.Takens,Groningen
B.Teissier,Paris
· ·
S. Attal A. Joye C.-A. Pillet (Eds.)

Open Quantum
Systems II
The Markovian 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 3-540-30992-6 SpringerBerlinHeidelbergNewYork
ISBN-13 978-3-540-30992-5 SpringerBerlinHeidelbergNewYork
DOI10.1007/b128451
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Preface
Thisvolumeisthesecondinaseriesofthreevolumesdedicatedtothelecturenotes
of the summer school “Open Quantum Systems” which took place in the Institut
FourierinGrenoble,fromJune16thtoJuly4th2003.Thecontributionspresentedin
thesevolumesarerevisedandexpandedversionsofthenotesprovidedtothestudents
during the school. After the first volume, developing the Hamiltonian approach of
openquantumsystems,thissecondvolumeisdedicatedtotheMarkovianapproach.
Thethirdvolumepresentsbothapproaches,butattherecentresearchlevel.
Openquantumsystems
Aquantum open systemisaquantum systemwhich isinteracting withanother
one.Thisisageneraldefinition,butingeneral,itisunderstoodthatoneofthesys-
tems is rather “small” or “simple” compared to the other one which is supposed to
behuge,tobetheenvironment,agasofparticles,abeamofphotons,aheatbath...
Theaimofquantumopensystemtheoryistostudythebehaviourofthiscoupled
systemandinparticularthedissipationofthesmallsysteminfavourofthelargeone.
Oneexpectsbehavioursofthesmallsystemsuchasconvergencetoanequilibrium
state,thermalization...Themainquestionsonetriestoanswerare:Isthereaunique
invariant state for the small system (or for the coupled system)? Does one always
convergetowardsthisstate(whatevertheinitialstateis)?Whatspeedofconvergence
canweexpect?Whatarethephysicalpropertiesofthisequilibriumstate?
Onecandistinguishtwoschoolsinthewayofstudyingsuchasituation.Thisis
trueinphysicsaswellasinmathematics.Theyrepresentingeneral,differentgroups
ofresearcherswith,uptonow,ratherfewcontactsandcollaborations.Wecallthese
twoapproachestheHamiltonianapproachandtheMarkovianapproach.
In the Hamiltonian approach, one tries to give a full description of the coupled
system. That is, both quantum systems are described, with their state spaces, with
their own Hamiltonians and their interaction is described through an explicit inter-
actionHamiltonian.OnthetensorproductofHilbertspacesweendupwithatotal
Hamiltonian, and the goal is then to study the behaviour of the system under this
dynamics.ThisapproachispresentedindetailsinthevolumeIofthisseries.
VI Preface
In the Markovian approach, one gives up trying to describe the large system.
The idea is that it may be too complicated, or more realistically we do not know
it completely. The study then concentrates on the effective dynamics which is in-
duced on the small system. This dynamics is not a usual reversible Hamiltonian
dynamics, but is described by a particular semigroup acting on the states of the
smallsystem.
BeforeenteringintotheheartoftheMarkovianapproachandallitsdevelopment,
in the next courses, let us have here an informal discussion on what this approach
exactlyis.
TheMarkovianapproach
WeconsiderasimplequantumsystemH whichevolvesasifitwereincontact
withanexteriorquantumsystem.Wedonottrytodescribethisexteriorsystem.Itis
maybetoocomplicated,ormorerealisticallywedonotquiteknowit.Weobserveon
theevolutionofthesystemHthatitisevolvinglikebeingincontactwithsomething
else,likeanopensystem(byoppositionwiththeusualnotionofclosedHamiltonian
systeminquantummechanics).Butwedonotquiteknowwhatiseffectivelyacting
onH.WehavetodealwiththeefffectivedynamicswhichisobservedonH.
Bysuchadynamics,wemeanthatwelookattheevolutionofthestatesofthe
system H. That is, for an initial density matrix ρ at time 0 on H, we consider the
0
stateρ attimetonH.Themainassumptionhereisthatthisevolution
t
ρ =P (ρ )
t t 0
isgivenbyasemigroup.Thisistosaythatthestateρ attimetdeterminesthefuture
t
statesρt+h,withoutneedingtoknowthewholepast(ρs)s≤t.
EachofthemappingP isageneralstatetransformρ (cid:1)→ρ .Suchamapshould
t 0 t
be in particular trace-preserving and positivity-preserving. Actually these assump-
tionsarenotquiteenoughandthepositivity-preservingpropertyshouldbeslightly
extendedtoanaturalnotionofcompletelypositivemap(seeR.Rebolledo’scourse).
Weendupwithasemigroup(P ) ofcompletelypositivemaps.Undersomecon-
t t≥0
tinuityconditions,thefamousLindbladtheorem(seeR.Rebolledo’scourse),shows
thattheinfinitesimalgeneratorofsuchasemigroupisoftheform
(cid:2) (cid:3)
(cid:1)
1 1
L(ρ)=i[H,ρ]+ L ρL∗− L∗L ρ− ρL∗L
i i 2 i i 2 i i
i
forsomeself-adjointboundedoperatorH onHandsomeboundedoperatorsL on
i
H.Theevolutionequationforthestatesofthesystemcanbesummarizedinto
d
ρ =L(ρ ).
dt t t
This is theso-called quantum master equation inphysics. Itisactually the starting
point in many physical articles on open quantum systems: a specific system to be
studiedisdescribedbyitsmasterequationwithagivenexplicitLinbladgeneratorL.
Preface VII
ThespecificformofthegeneratorLhastounderstoodasfollows.Itissimilarto
thedecompositionofaFellerprocessgenerator(seeL.Rey-Bellet’sfirstcourse)into
afirstorderdifferentialpartplusasecondorderdifferentialpart.Indeed,thefirstterm
i[H, ·]
istypicalofaderivationonanoperatoralgebra.IfLwerereducedtothattermonly,
thenP =etLiseasilyseentoactasfollows:
t
P (X)=eitHXe−itH.
t
Thatis,thissemigroupextendsintoagroupofautomorphismsanddescribesausual
Hamiltonian evolution. In particular it describes a closed quantum system, there is
noexteriorsysteminteractingwithit.
Thesecondtypeoftermshavetobeunderstoodasfollows.IfL=L∗then
1 1
LXL∗− L∗LX − XL∗L=[L,[L,X]].
2 2
It is a double commutator, it is a typical second order differential operator on the
operatoralgebra.Itcarriesthediffusivepartofthedissipationofthesmallsystemin
favoroftheexterior,likeaLaplacianterminaFellerprocessgenerator.
When L does not satisfy L = L∗ we are left with a more complicated term
whichismoredifficulttointerpretinclassicalterms.Ithastobecomparedwiththe
jumpingmeasureterminageneralFellerprocessgenerator.
Now,thatthesemigroupandthegeneratoraregiven,thequantumnoises(seeS.
Attal’s course) enter into the game in order to provide a dilation of the semigroup
(F.Fagnola’scourse).Thatis,onecanconstructanappropriateHilbertspaceF on
which quantum noises dai(t) live, and one can solve a differential equation on the
j
spaceH⊗F whichisoftheformofaSchro¨dingerequationperturbedbyquantum
noisesterms: (cid:1)
dU =LU dt+ KiU dai(t). (1)
t t j t j
i,j
This equation is an evolution equation, whose solutions are unitary operators on
H⊗F,soitdescribesaclosedsystem(ininteractionpictureactually).Furthermore
itdilatesthesemigroup(P ) inthesensethat,thereexistsa(pure)stateΩ onF
t t≥0
suchthatifρisanystateonHthen
<Ω, U (ρ⊗I)U∗Ω>=P (ρ).
t t t
This is to say that the effective dynamics (P ) we started with on H, which we
t t≥0
didnotknowwhatexactexteriorsystemwasthecauseof,isobtainedasfollows:the
smallsystemHisactuallycoupledtoanothersystemF andtheyinteractaccording
to the evolution equation (1). That is, F acts like a source of (quantum) noises on
H. The effective dynamics on H is then obtained when averaging over the noises
throughacertainstateΩ.
VIII Preface
ThisisexactlythesamesituationastheoneofMarkovprocesseswithrespectto
stochasticdifferentialequations(L.Rey-Bellet’sfirstcourse).AMarkovsemigroup
isgivenonsomefunctionalgebra.Thisisacompletelydeterministicdynamicswhich
describes an irreversible evolution. The typical generator, in the diffusive case say,
containstwotypesofterms.
Firstorderdifferentialtermswhichcarrytheordinarypartofthedynamics.Ifthe
generatorcontainsonlysuchtermsthedynamicsiscarriedbyanordinarydifferential
equationandextendstoareversibledynamics.
Second order differential operator terms which carry the dissipative part of the
dynamics.Thesetermsrepresentthenegativepartofthegenerator,thelossofenergy
infavorofsomeexterior.
But in such a description of a dissipative system, the environment is not de-
scribed. The semigroup only focuses on the effective dynamics induced on some
systembyanenvironment.Withthehelpofstochasticdifferentialequationsonecan
give a model of the action of the environment. It is possible to solve an adequat
stochastic differential equation, involving Brownian motions, such that the result-
ing stochastic process be a Markov process with same semigroup as the one given
atthebegining.Suchaconstructionisnowadays naturalandoneoftenuseitwith-
out thinking what this really means. To the state space where the function algebra
acts,wehavetoaddaprobabilityspacewhichcarriesthenoises(theBrownianmo-
tion).Wehaveenlargedtheinitialspace,thenoisedoesnotcomenaturallywiththe
function algebra. The resolution of the stochastic differential equation gives rise to
a solution living in this extended space (it is a stochastic process, a function of the
Brownian motions). It is only when avering over the noise (taking the expectation)
thatonerecoverstheactionofthesemigrouponthefunctionalgebra.
Wehavedescribedexactlythesamesituationasforquantumsystems,asabove.
Organizationofthevolume
Theaimofthisvolumeistopresentthisquantumtheoryindetails,togetherwith
itsclassicalcounterpart.
ThevolumeactuallystartswithafirstcoursebyL.Rey-Belletwhichpresentsthe
classical theory of Markov processes, stochastic differential equations and ergodic
theoryofMarkovprocesses.
ThesecondcoursebyL.Rey-Belletappliesthesetechniquestoafamilyofclas-
sicalopensystems.Theassociatedstochasticdifferentialequationisderivedfroman
Hamiltoniandescriptionofthemodel.
ThecoursebyS.Attalpresentsanintroductiontothequantumtheoryofnoises
andtheirconnectionswithclassicalones.Itconstructsthequantumstochasticinte-
grals and proves the quantum Ito formula, which are the cornerstones of quantum
Langevinequations.
R.Rebolledo’scoursepresentsthetheoryofcompletelypositivemaps,theirrep-
resentationtheoremsandthesemigrouptheoryattachedtothem.Thisendsupwith
thecelebratedLindblad’stheoremandthenotionofquantummasterequations.
Preface IX
Finally,F.Fagnola’scoursedevelopsthetheoryofquantumLangevinequations
(existence, unitarity) and shows how quantum master equations can be dilated by
suchequations.
Lyon,Grenoble,Toulon Ste´phaneAttal
September2005 AlainJoye
Claude-AlainPillet