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Publié par | technische_universitat_munchen |
Publié le | 01 janvier 2009 |
Nombre de lectures | 13 |
Langue | English |
Poids de l'ouvrage | 2 Mo |
Extrait
TechnischeUniversita¨tMu¨nchen
Max-Planck-Institutfu¨rQuantenoptik
AQuantumInformation
PerspectiveofFermionicQuantum
Many-BodySystems
ChristinaV.Kraus
Vollsta¨ndigerAbdruckdervonderFakulta¨tfu¨rPhysik
derTechnischenUniversita¨tMu¨nchen
zurErlangungdesakademischenGradeseines
DoktorsderNaturwissenschaften(Dr.rer.nat.)
genehmigtenDissertation.
Vorsitzender:Univ.-Prof.Dr.R.Gross
Pru¨ferderDissertation:1.Hon.-Prof.I.Cirac,Ph.D.
2.Univ.-Prof.Dr.H.Friedrich
DieDissertationwurdeam14.09.2009beider
TechnischenUniversita¨tMu¨ncheneingereichtund
durchdieFakulta¨tfu¨rPhysikam02.11.2009angenommen.
Abstract
InthisThesisfermionicquantummany-bodysystemaretheoreticallyinvestigated
fromaquantuminformationperspective.
Quantumcorrelationsinfermionicmany-bodysystems,thoughcentraltomanyof
themostfascinatingeectsofcondensedmatterphysics,arepoorlyunderstoodfrom
atheoreticalperspective.Eventhenotionof”paired”fermionswhichiswidelyused
inthetheoryofsuperconductivityandhasaclearphysicalmeaningthere,isnota
conceptofasystematicandmathematicaltheorysofar.Applyingconceptsandtools
fromentanglementtheory,weclosethisgap,developingapairingtheoryallowing
tounambiguouslycharacterizepairedstates.Wedevelopmethodsforthedetection
andquanticationofpairingaccordingtoourdenitionwhichareapplicableto
currentexperimentalsetups.Pairingisshowntobeaquantumcorrelationdistinct
fromanynotionofentanglementproposedforfermionicsystems,givingfurther
understandingofthestructureofhighlycorrelatedquantumstates.Inaddition,we
showtheresourcecharacterofpairedstatesforprecisionmetrology,provingthat
BCS-statesallowphasemeasurementsattheHeisenberglimit.
Next,thepoweroffermionicsystemsisconsideredinthecontextofquantum
simulations,wherewestudythepossibilitytosimulateHamiltoniantimeevolutions
onacubiclatticeundertheconstraintoftranslationalinvariance.Givenasetof
translationallyinvariantlocalHamiltoniansandshortrangeinteractionswedeter-
minetimeevolutionswhichcanandthosewhichcannotbesimulated.Bosonicand
nite-dimensionalquantumsystems(”’spins”)areincludedinourinvestigations.
Furthermore,wedevelopnewtechniquesfortheclassicalsimulationoffermionic
many-bodysystems.First,weintroduceanewfamilyofstates,thefermionicPro-
jectedEntangledPairStates(fPEPS)onlatticesinarbitraryspatialdimension.
ThesearethenaturalgeneralizationofthePEPSknownforspinsystems,andthey
approximateecientlygroundandthermalstatesofsystemswithshort-rangeinter-
action.WegiveanexplicitmappingbetweenfPEPSandPEPS,allowingtoextend
previoussimulationmethodstofermions.Inaddition,weshowthatfPEPSnat-
urallyariseasexactgroundstatesofcertainfermionicHamiltonians,andgivean
examplethatexhibitscriticalitywhilefulllingthearealaw.
Finally,wederivemethodsforthedeterminationofgroundandthermalstates,
aswellasthetimeevolution,ofinteractingfermionicsystemsusinggeneralized
Hartree-Focktheory(gHFT).Withthecomputationalcomplexityscalingpolyno-
miallywiththenumberofparticles,thismethodcandealwithlargesystems.As
abenchmarkweapplyourmethodstothetranslationallyinvariantHubbardmodel
withattractiveinteractionandndexcellentagreementwithknownresults.
OCTNNETSContents
i1Introduction1
2PairinginFermionicSystems7
2.1Fermionicstates..............................9
2.1.1Basicnotation...........................9
2.1.2Quantumcorrelationsoffermionicstates............10
2.1.3FermionicGaussianstates....................13
2.1.4Numberconservingfermionicstates...............15
2.2Pairingtheory...............................16
2.2.1Motivationandstatementofthedenition...........16
2.2.2Relationofpairingandentanglement..............18
2.2.3Methodsfordetectingpairing..................20
2.2.4Pairingmeasures.........................23
2.3PairingforGaussianstates........................23
2.3.1PairingwitnessesforGaussianstates..............23
2.3.2Completesolutionofthepairingproblemforfermionic
Gaussianstates..........................24
2.3.3AngularmomentumalgebraforGaussianstates........25
2.3.4ApairingmeasureforGaussianstates.............26
2.4Pairingofnumberconservingstates...................27
2.4.1PairingofallBCSstatesandgeometryofpairedstates....27
2.4.2Eigenvaluesofthetwo-particlereduceddensitymatrix....33
2.4.3Pairingmeasurefornumberconservingstates.........33
2.5Interferometry...............................35
2.5.1Ramseyinterferometrywithfermions..............36
2.5.2Interferometryinvolvingapair-interactionHamiltonian....42
2.6Applicationtoexperimentsandconclusion...............46
3QuantumSimulationsinTranslationallyInvariantSystems49
3.1Statementoftheproblem........................50
3.2QuadraticHamiltonians.........................51
3.3Simulationsinfermionicsystems.....................54
3.3.1Simulationsinone-dimensionalfermionicsystems.......55
3.3.2Simulationsind-dimensionalfermionicsystems........58
3.4Simulationsinbosonicsystems.....................59
iiOCTNNETS3.4.1Simulationsinone-dimensionalbosonicsystems........61
3.4.2Simulationsind-dimensionalbosonicsystems.........61
3.5Simulationsinspinsystems.......................62
3.6Summaryoftheresultsandconclusion.................65
4FermionicProjectedEntangledPairStates(fPEPS)67
4.1MPSandPEPSforspinsystems....................68
4.2ConstructionoffPEPS..........................71
4.3RelationbetweenfPEPSandPEPS...................72
4.4FermionicGaussianstatesandparentHamiltonians..........78
4.5ExampleofacriticalfPEPS.......................81
4.6Conclusion.................................89
5InteractingFermionicSystemsinGeneralizedHartree-FockThe-
19yro5.1ToolboxofgeneralizedHartree-FockTheory..............93
5.2Realtimeevolution............................94
5.3Groundstates...............................95
5.3.1Minimizationoftheenergy....................96
5.3.2Imaginarytimeevolution.....................97
5.4Thermalstates..............................98
5.5Application:The2d-Hubbard-Model..................100
5.6Conclusion.................................103
6ConclusionsandOutlook
AAstandardformforpurefermionicGaussianstates
BProofofThm.2.23
501
901
111
CInterferometrywithpairedstates117
C.1Quasi-bosoniclimit............................121
C.2Interferometryfarfromthebosoniclimit................122
DDerivationoftheevolutionequation127
Chapter1
Introduction
1Quantummechanicalcorrelationsthathavenoclassicalanalogueareoneofthe
mostcompellingphysicaldiscoveriesofthe20thcentury.Theexistenceofsuch
correlations,calledentanglement(”Verschra¨nkung”),wasrevealedbythefamous
Einstein-Podolsky-Rosen(EPR)gedankenexperimentinthe1930s[1].Considered
bysomeasa”spookyactionatadistance”atthattime,thenotionofentanglement
hastransformedintoawell-establishedconcepttoday.Astartingpointforthisde-
velopmentweretheinequalitiesformulatedbyBellin1964[2].ViolationofaBell
inequalityimpliestheexistenceofquantummechanicalcorrelationsthatcannotbe
simulatedbyanyclassicaltheory.Nearlytwentyyearslater,Aspectmanagedto
performtherstconvincingexperimentprovingtheviolationofaBellinequality
[3,4].Aspect’sstrikingexperimentresultedintheadventofquantuminformation
theoryintheearly1990s,wherethequantumcorrelationsplaytheroleofare-
sourcefortechnologicalapplications.Sincethattimequantuminformationscience
hasdevelopedintoavibrantresearcharea,rangingfromfoundationalquestions
oftheinterpretationonquantummechanicstowardsthesearchfortechnological
applicationofentanglement[5].Usingknowledgefromvariouseldsofphysics,
mathematicsandcomputerscience,theunderstandingandthecontrolofquantum
mechanicalsystemsisattheheartofquantuminformationtheory.
Thecoreofquantuminformationscienceistheuseofquantummechanicalpar-
ticlesasthecarrierofinformation.Anyquantummechanicaltwo-levelsystemcan
encodeoneunitbitofquantuminformation,andsuchasystemiscalledqubitin
analogytothebitofclassicalinformationtheory.Theimmensesuccessofquan-
tuminformationtheoryisduetotheinterplayoftheoryandexperimentwith
thosequbits:Protocolsforquantumcryptography[6],quantumdensecoding[7]
orquantumteleportation[8]couldallbedemonstratedinpioneeringexperiments
[9,10,11,12,13,14,15,16].Furthermore,ithasbeenpredictedthatcertain
computationaltasks,suchasfactoringnumbersorsimulatingquantummechanical
systems,canbecarriedoutexponentiallyfasterusingaquantumcomputerbased
onqubitsthanbyanyknownalgorithmrunningonaclassicalcomputer.However,
theexperimentalrealizationofalarge-scalequantumcomputercapableofaccom-
plishingthosetasksis,despitemajorexperimentalprogress,stillanunsolvedtask.
Nevertheless,thecompellingprogressintheeldofquantuminformationsciences
2HCATPRE.1nItordutcinojusti