1、文章编号2097-1842（2023）05-1206-09Decoherence of temporal quantum correlation in electricallycontrollable quantum-dots moleculesXIEJia-ling，YANKai，TANJia，CAOZhao-liang，HAOXiang*（Jiangsu Key Laboratory of Micro and Nano Heat Fluid Flow Technology and Energy Application,School ofPhysical Science and Techno
2、logy,Suzhou University of Science and Technology,Suzhou 215009,China）*Corresponding author，E-mail:Abstract:Thedecoherenceoftemporalquantumcorrelationisexploredinavoltage-controlledquantumdotsmoleculecoupledtoacavity.Thetemporalcorrelationintheoptoelectronichybridsystemisstudiedbasedon Leggett-Garg i
3、nequalities.The inequality violations can be interpreted as the existence of temporalquantumcorrelationduringdynamicalevolution.Thetemporalquantumcorrelationisenhancedbyitselec-trontunnelsstrengthandcavityfrequencydetuning.Itisfoundthatthereisnotemporalquantumcorrela-tionintheregionswherethevaluesof
4、spatialquantumcorrelationarezeroandthemaximalviolationsoccurinconditionswithhighvaluesofquantumcorrelation.Incontrast,thespatialquantumcoherencecanstillex-sitwhenthevalueoftemporalquantumcorrelationiszero.Themethodofopenquantumsystemdynamicisusedtostudytheeffectofreservoirmemoryontemporalquantumcorr
5、elation.Thetemporalquantumcorrel-ationcanbesuppressedduetothespontaneousdecayofthequantumdotsandcavityleakage.Theseresultsarehelpfulforquantuminformationprocessingtechnologyinhybridquantumsystems.Key words:temporal quantum correlation;quantum dots molecule;Leggett-Garg inequalities;hybridquantumsyst
6、ems电控量子点分子的时域量子关联退相干谢佳凌，严凯，谭佳，曹召良，郝翔*（苏州科技大学物理科学与技术学院江苏省微纳热流技术与能源应用重点实验室,江苏苏州 215009）摘要：本文以与光腔耦合的电控量子点分子为研究对象，分析了量子点的时域量子关联退相干特性。基于可测量的 Leg-收稿日期：2022-11-18；修订日期：2022-12-08基金项目：国家自然科学基金资助项目（No.61875145）；“十四五”江苏省重点学科资助项目（No.2021135）；江苏省研究生科研与实践创新计划项目（No.KYCX23_3312）；SupportedbyNationalNaturalScienceFo
7、undationofChina(No.61875145);JiangsuKeyDisciplinesoftheFourteenthFive-YearPlan(No.2021135);PostgraduateResearch&PracticeInnovationProgramofJiangsuProvince(No.KYCX23_3312);第16卷第5期中国光学（中英文）Vol.16No.52023 年 9 月ChineseOpticsSept.2023gett-Garg 不等式，研究光电混合系统的时域量子关联。测量不等式的违背性可以作为动态演化过程中时域量子关联的存在证据。调控电子隧穿强度和
8、光腔频率失谐有利于增强时域量子关联。发现，在空间量子关联值为零的区域内，不存在时域量子关联。当空间量子关联值较高时，量子点动力学演化存在 Leggett-Garg 不等式测量的最大程度违背现象。与之相反，在时域量子关联为零的时间段内，空间量子关联仍然存在。本文采用开放量子系统动力学方法研究环境效应对时域量子关联的影响。量子点的自发衰变和光腔泄漏抑制了时域量子关联。这些结果可用于混合量子系统的量子信息处理技术。关 键 词：时域量子关联；量子点分子；Leggett-Garg 不等式；混合量子系统中图分类号：O431.2文献标志码：Adoi：10.37188/CO.EN-2022-00251Intr
9、oductionQuantum correlations1-2 havebecome neces-saryresourcesinquantuminformationsciencesandtechnologies3-5.Thepresenceofnonclassicalcorrel-ationscandistinguishquantumworldsfromclassic-alworlds6-8.Bellinequalities9playafundamentalroleinthenatureofcorrelationsbetweenspatiallyseparatedsystems.Testoft
10、heviolationsofBellorClauser-Horne-Shimony-Holt(CHSH)inequalitiescandemonstratequantumnonlocality10.Spatialcor-relationscanbequantitativelyevaluatedbyquantumentanglement11-12,quantum discord13-14,quantumcoherence15-17 and other characteristics18-19.Fromthe viewpoint of time sequential measurements,Le
11、ggettandGargputforwardasimilarinequalitywhoseviolationcanbequalitativelyusedtoestim-atetemporalcorrelations20.Leggett-Garginequalit-ies(LGIs)mustbeverifiedbysequentialmeasure-mentsactingonasinglesystematdifferenttimes,whichisdistinctfromBellinequalitiesthatconcernmultiplepartiesspatiallyseparatedfro
12、meachother.TheviolationsofLGIsareinconsistentwithtwoas-sumptionsofmacroscopicrealismandnoninvasivemeasurability21.Themacroscopicrealismofphys-icalstatesimpliesthatmeasurementsonasystemsimply reveal the values which exist at previoustimes.Noninvasivemeasurabilityensuresthatsuchmeasurementscanbeperfor
13、medwithoutdisturbingthedynamicalevolutionofthestates.In recent years,many experimental tests ofLGIshavebeencarriedoutinsetupsincludingRy-dbergatoms22,photonicsystems23,superconduct-ingcircuits24andaquantumcomputer25.Withtherapiddevelopmentofquantumengineeringtechno-logies,feasible quantum systems ha
14、ve extendedfrom qubits to multilevel quantum systems26-28.Amongthese fantastic quantum systems,electric-allycontrollablequantumdotsmoleculeshavebeenextensivelyapplied in quantum optics and con-densedmatterphysicsbecauseoftheirhighnonlin-earopticalsusceptibility,largeelectricaldipolemo-ments of band
15、transition and great flexibility indesigning devices29-30.A unity of two or morequantumdotswithcloselyspacedcouplingscanbereferred to as a quantum dots molecule which ismodeledasonekindofartificialmultilevelatom31.Electricvoltagescanbeusedtogenerateelectrontunnelsbetweentwoneighboringquantumdots32.M
16、eanwhile,leveltransitionscanbeinducedbydir-ectcouplingsbetweenacavityandaquantumdotsmolecule33.Howdoelectrontunnels,cavitycoup-lingsandenvironmentalnoisesaffectquantumcor-relationsinsuchsystems?Thisquestionmotivatesus to theoretically evaluate temporal and spatialquantumcorrelationsinhybridmultileve
17、lsystems.l1Thestructureofthispaperisorganizedasfol-lows.InSec.2,wepresentwitnessesabouttempor-alandspatialquantumcorrelations.Thegeneralex-pressionofLGIbasedontwo-timecorrelationfunc-tionsisgivenbytheopensystemapproach.Tem-poralcorrelationsarequalitativelyestimatedbyvi-olationsofLGIandquantumcoheren
18、cebasedon-normisusedtomeasurespatialcorrelations.InSec.3,we describe the model and Hamiltonian of anelectricallycontrollablequantumdotsmoleculeinacavity.Quantumcorrelationswithrespecttolevel第5期XIEJia-ling,et al.:Decoherenceoftemporalquantumcorrelationinelectrically.1207transitionsareestablishedbyext
19、ernalvoltagesandcavity couplings.In Sec.4,we provide our mainanalyticalandnumericalresultsaboutthedynamicsoftemporalandspatialcorrelations.Theexplicitex-pressionsoftheLGIviolationandquantumcoher-enceareanalyticallyobtainedbythequantumdy-namicmethod.Thequantummasterequationforaquantumdotsmoleculeisex
20、ploitedtonumericallystudyenvironmentaleffectsontemporalcorrelation.Wewilldiscussourconclusionsinthelastsection.2Quantum correlation in a generalevolutiont0 t1 t2(t)LGIAsatemporalanalogversionofBellinequalit-ies,onesimpleformofLGI21canbeobtainedbyperforming three measurements of time such that.Accord
21、ingtotwo-timecorrelationfunc-tionsforadichotomicobservable,wedefinetheas,K3=C(t0,t1)+C(t1,t2)C(t0,t2)1.（1）C(ti,tj)=(ti)(tj)(t)H|,=1,2,d=M+Mm=1|t0M+=|M=,|m=1|K31VLGI=max(K31,0)VLGImaxHerethetwo-timecorrelationfunctioniswrit-tenaswheredenotesthetimeevolutionoftheobservableintheHeisen-bergpicture.Foram
22、ultilevelsystem,theHamiltoni-an has a complete set of energy levels with.Thedichotomicobservableischosentobewiththecorrespondingvalues.Ifthesystemofinterestisinaspe-cificenergystateataninitialtime,wecandefine and26.In thiscondition,theperformanceofthedichotomicoper-atordetermineswhetherthesystemisst
23、illinthestate with a measurement value of orwhether it has undergone a transition from tootherorthogonalstateswiththeotheroutcome.Theviolationofisasignatureofthequantumnatureofthesystemduringtimeevolution.Wecandefine the violation of the inequality ascharacterizingtheoccurrenceoftem-poralcorrelation
24、s.Thequantityofrepres-entsthemaximumdegreetowhichasystemcanvi-olatetheLeggett-Garginequality.titjtjti(tj)=tjti(ti)tjtiti0=tj0(tj)=U(ti,tj)(ti)U+(ti,tj)U(ti,tj)=T expiwtjtiH(t)dtTt(t)=L(t)LHC(ti,tj)C(ti,tj)=m,n=(mn)p(tmi)q(tnj tmi)p(tmi)mtiq(tnj tmi)ntjtiToobtaintheexplicitexpressionoftimecor-relatio
25、nfunctions,weneedtoknowthetimeevolu-tionprocessofthesystem.Ingeneral,adynamicalmapgoverningthetimeevolutionofasystemfromto isgivenby,i.e.,.Inthefollowing,wetakeintoaccountaquantumdynamic-al semigroup which satisfies.Themapforaclosedsystemcanbeexpressedbythe unitary operatorwhereistheunitaryoperatora
26、nd istheorderoftime.Foranopensys-temcoupledtoitssurroundings,thetimeevolutionof the system can be described by the quantumLindblad equation where is aLindbladiansuperoperator.ThegeneralformofthesuperoperatorconsistsofoneunitarypartgeneratedbytheHamiltonianandtheotherpartgeneratedby a dissipator.The
27、two-time correlation functioncanbedefinedintermsofthejointprobabil-ities,.Here is the probability of obtaining the measure-mentvalueatatime andisthecon-ditionalprobabilityofgettingthevalue atthelat-tertime,giventhatthepreviousvaluemismeas-uredat.Incombinationwithadynamicalmap,themeasurementprobabili
28、tiesarewrittenasp(tmi)=TrMmti0(0)q(tnjtmi)=TrMntjtiMm(ti)Mmp(tmi).（2）Jointmeasurability18playsanimportantroleindeterminingtwo-timecorrelationfunctions.l1l1|jl1Spatial quantum correlation can be treatedquantitativelybydefiningthe-normasacoher-encemeasurement17.The-normofamatrixisre-latedtotheabsolute
29、valueoftheelementsofthematrix.WhereadefiniteHilbertspacehasarefer-enceorthonormalbasisanda-normofcoher-1208中国光学（中英文）第16卷=jkjk|jk|Q()=j,k|jk|LGIl1ence of a state,the measure ofquantumcoherence for a density matrix is de-scribedas.Thevalueofspatialcor-relationcanbegivenbythesummationovertheab-soluteva
30、luesofalltheoff-diagonalelementsofthedensity matrix.According to temporal and spatialcorrelations,wewillinvestigatethequantumnatureof a system in the two approaches.The quantumnatureisrevealedifthestatesofthesystemoccurintheformofacoherentsuperpositionofHamiltonianeigenstates.Theviolationisconsidere
31、daqual-itativetestandquantumcoherencebasedon-normisreferredtoasaquantitativeestimation.3Modeland Hamiltonian of a hy-bridsystem(In,Ga)As/GaAsRapiddevelopmentinlaserandsemiconductortechnologieshasmadeitpossibletofabricatehy-brid systems applied to quantum informationtasks33-36.Among these hybrid syst
32、ems,quantumdots molecules coupled to cavities have attractedmuchattentioninrecentyears.Itisknownthatleveltransitions between different electronic states inquantumdotsmoleculescanbeexcitedbyexternalvoltages and laser fields.We consider a quantumdots molecule composed of two self-assembledquantumdotsw
33、ithalateralquan-110GaAs|0|2g|1Te21tuminteraction.Theinterdotcouplingiscontrolledbyanelectricalvoltage.Thissystemiscoupledtoamicroscopiccavity.Aschemeforaquantumdotsmolecule coupled to the cavity is illustrated inFigure1(a)(coloronline).Somepresenttechnolo-giescanbeexploitedtoproduceahomogeneousen-se
34、mbleofmoleculesconsistingoftwodotsalignedalongthedirectionofthesubstrates37.Aschemeforadoublecoupledquantumdotssys-temisillustratedinFigure1(b)(coloronline).Thegroundstatedenotesalevelwheretwoquantumdotsarenotexcited.Thestaterepresentsalevelinwhichanelectronisdirectlyexcitedbyalasertotheconductionba
35、ndintheleftdot.Theparameteristhefrequencydetuningbetweenthelasersdriv-ingandleveltransition.denotesthestrengthofelectrical dipole interactions between a moleculeand cavity fields.The application of electricalvoltagesinducesthetransferofanelectronfromtheleftdottotheconductionbandoftherightdot.Theindi
36、rectlyexcitedstatestandsfortheleveltrans-itionbyinterdottunnels.isthestrengthofthetunnelcouplingwhichcanbevariedbytheexternalvoltage.Theleveltransitionisconsideredtobezeroduetothenegligibleenergydifferencebetweenthedirectandindirectexciton.By means of the electric dipole and rotatingwaveapproximatio
37、n,theHamiltonianofthehybridsystemintheinteractionpicturecanbeexpressedas(a)(b)VQDM120gTe21Fig.1(a)Aquantumdotsmoleculeiscoupledtoacavity.ThesymbolVdenotesanelectricalvoltagewhichisusedtocon-troltheelectrontunnel.(b)Schematicdiagramofbandstructureandlevelconfigurationofaquantumdotsmolecule.Thesystemo
38、fthequantumdotsmoleculeconsistsoftwodotswithlateralcouplings.Theelectronandholearerepres-entedbytheredcircleandblackcircle,respectively第5期XIEJia-ling,et al.:Decoherenceoftemporalquantumcorrelationinelectrically.1209H=|22|+(21)|11|+(g a|20|+Te|12|+H.c.),（3）H.c.a|(0)=|1|0c=|10|0cH|1|0c,|2|0c,|0|1cEj(j
39、=1,2,3)E3(221)E2+(221g2T2e)E+g2(21)=0|j=cj1|10+cj2|20+cj3|01cj1=TeE+21cj2cj3=gEcj2k|cjk|2=1U(ti,tj)=U(tjti)=kexpiEk(tjti)|k|=0 21=0whereistheHermitianconjugatepartandrepresents the annihilation operator for the cavitymode.Inordertoobtainadynamicmapofthesys-tem,wecanassumethattheinitialstateofthetota
40、lsystemwhereisthevacu-umstateforthecavitymode.Thetotalsystemwillevolve in the Hilbert space spanned by.Theeigenvaluesfor the Hamiltonian can be obtained by the threereal roots of the equation.Thecorres-ponding eigenstate is given by where three coefficients satisfy,and.Fortheclosedquantumsystem,theu
41、nitaryoperatorisrelated to the time interval.In a simple case of,wecanobtaintheeigenvaluesandcorrespondingeigenstatesofthehybridsystemwithE1=0,E2,3=J|1=gJ|10TeJ|01|2,3=12(TeJ|10+|20gJ|01),（4）J=g2+T2ewheretheparameter.Theunitarygen-eratorforaninterval canbewrittenasU()=U11U12U13U12U22U23U13U23U33.（5）
42、U11=g2+T2ecosJJ2U12=iTesinJJU13=gTeJ2.(1cosJ)U22=cosJU23=igsinJJU33=T2e+g2cosJJ2For formula 5,the elements are obtained by,and.Therefore,thedynamicevolu-tionofthesystemisdeterminedbytheaboveunit-arygenerator.4Dynamicsoftemporalandspatialcorrelation=M+MM+=|11|m=Mm=IC(ti,tj)M+C(ti,tj)Toevaluatethetemp
43、oralcorrelation,wechosethedichotomicoperatorwhere.Becausethemeasurementssatisfythecom-pleteness,wecanworkoutasimplifiedexpressionofonlyintermsofmeasure-ment.Thegeneralexpressionofcanbesim-plifiedas,C(ti,tj)=12TrM+tjtiti0(0)TrtjtiM+ti0(0)Trtjtiti0(0)M+2TrM+tjtiM+ti0(0)+2TrM+tjtiti0(0)M+.（6）|(0)=|1|0c
44、TheclosedsystemevolvesinaccordancewiththeunitarygeneratorofEq.(5).Foraninitialstate,the reduced density matrix of thequantumdotsmoleculeafter6measurementscanbegivenby,tjtiti0(0)=TrcU(tjti)U(ti)(0)U+(ti)U+(tjti)tjtiM+ti0(0)=TrcU(tjti)M+U(ti)(0)U+(ti)U+(tjti)tjtiti0(0)M+=TrcU(tjti)U(ti)(0)U+(ti)M+U+(t
45、jti),（7）Trcwheredenotesthepartialtracingoverthede-greesoffreedomofthecavity.t2t1=t1t0=t0=0=0,21=0Assumingthatand,thetwo-timecorrelationfunctionsforthesimplestcase,canbeobtainedanalyticallyasC(0,)=2U2111,C(0,2)=2U2111,C(,2)=12U211+2U4114U211|U12|22U11j=2,3Ujj|U1j|2,（8）U11=g2+T2ecos2JJ2LGILGITeLGIwher
46、e the coefficient.Basedontheabovecorrelationfunctions,thedynamicbe-havior of the violation of can be shown inFigure2.Itisseenthattheviolationofcanbeenhanced by the tunnel coupling.The fluctu-ationsoftheviolationoccurwithtime.Atagiventime,theoptimalvalueofanelectrontunnelcangiverisetothemaximalviolat
47、ion.Theeffectof1210中国光学（中英文）第16卷LGILGI|1thefrequencydetuningontheviolationisdemonstratedinFigure3.Itisshownthatthemax-imalviolationofincreaseswithavariationinthe detuning.In a sense,the tunnel coupling andcavityfrequencydetuningcontributetothegenera-tion of the coherent superposition of the exciteds
48、tateandotherlevelstates.VLGI0.50.40.30.20.10012345Te00.20.40.60.81.0LGITeg=1 =021=0VLGIFig.2Thedynamicsoftheviolationareplottedasafunctionofthetunnelstrengthwhen,and.Thenonzerovaluesofdemon-stratethepresenceoftemporalquantumcorrelation0.500.480.460.440.420.400.380.360.340.320.3042024VLGImaxLGIg=1Te=
49、0.5g21=0Fig.3Themaximalviolationvarieswithcavityfre-quency detuning.The parameters,andarechosenQ()=TrcU()|(0)(0)|U+()The reduced density matrix for the quantumdots molecule is given by.The quantum coherence as ameasurementofspatialcorrelationisobtainedasQ()=2J3|TesinJ(g2+T2ecosJ)|.（9）LGILGIFigure4(c
50、oloronline)illustratesthefluctuat-ingbehavioroftemporalandspatialquantumcor-relationsforaquantumdotsmolecule.Thereisare-lationshipbetweentemporalandspatialcorrelations.Inthetimedomainwherethemaximalviola-tionshappen,thevaluesofquantumcoherencere-mainhigh.Whennospatialcorrelationisdetected,thereis al
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