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Entanglement created by spontaneously generated coherence Zhao-hong Tanga,b, Gao-xiang Lia,∗ and Zbigniew Ficekc aDepartment of Physics, Huazhong Normal University, Wuhan 430079, PR China bSchool of Science, Wuhan Institute of Technology, Wuhan 430073, PR China cTheNationalCentreforMathematicsandPhysics, KACST,P.O.Box6086, Riyadh11442, SaudiArabia Weproposeaschemeabletogenerateondemandasteady-stateentanglementbetweentwonon-degenerate cavitymodes. Theschemereliesontheinteractionofthecavitymodeswithdriventwoorthree-levelatoms whichactasacouplertobuildentanglementbetweenthemodes. Weshowthatinthelimitofastrongdriving, crucialforthegenerationofentanglementbetweenthemodesistoimbalancepopulationsofthedressedstates ofthedrivenatomictransition. Inthecaseofathree-levelV-typeatom,wefindthatastationaryentanglement canbecreatedondemandbytuningtheRabifrequencyofthedrivingfieldtothedifferencebetweentheatomic transitionfrequencies. Theresultingdegeneracyoftheenergylevelstogetherwiththespontaneouslygenerated coherencegeneratesasteady-stateentanglementbetweenthecavitymodes. Itisshownthattheconditionfor themaximalentanglementcoincideswiththecollapseoftheatomicsystemintoapuretrappingstate. Wealso show that the creation of entanglement depends strongly on the mutual polarization of the transition atomic 1 dipolemoments. 1 0 PACSnumbers:42.50.Dv,42.50.Gy 2 n I. INTRODUCTION canbereducedorevencompletelyeliminated,butitcouldbe a J difficulttoeliminateonamacroscopicscalewhereonewould liketocreateentanglementusingmacroscopicatomicensem- 1 Thegenerationofcontinuousvariable(CV)entangledlight has attracted a significant interest due to a potential appli- bles. This raises an important question of how to eliminate ] cation in quantum information science, specifically in quan- the decoherenceor how to maintaina largecoherencein the h presenceofthedecoherence. p tum teleportation[1], quantumtelecloning [2], and quantum - dense coding [3]. Continuous variables offer the possibility Inthispaper,weproposeasystemformedbyathree-level nt tocreateentanglementdeterministicallyanddifferentnonlin- atom located inside a two mode cavity that can generate the a ear processeshave been proposedto generate CV two-mode maximal stationary entanglement between the cavity modes u entangled beams [4–8] including nondegenerate parametric inthepresenceofdecoherence.TheatomismodelledasaV- q down-conversion[9, 10] and nondegenerate four-wave mix- typesystemwherethedipoleallowedtransitionscanbeinde- [ ing processes [11–15]. Recently, the four-wave mixing pro- pendentofeachotherorcanbecorrelatedthroughthesponta- 2 cess hasbeenproposedas a potentialsourceofnarrow-band neouslygeneratedcoherence(SGC)[25]. Theatomisdriven v entangledbeams,animportantresourceforquantummemory by an externallaser field coupled exclusively to only one of 7 storage[16]andlong-distancecommunications[17]. theatomictransitions.Weusethedressed-atomapproachand 6 OfparticularinterestforCVentanglementarecavityQED show that the effective three-level system of dressed states 0 1 systems where entanglement between cavity modes can be comprises a suitable medium for a non-linear coupling be- . created by coupling the modes to an atomic system or non- tweenthecavitymodes. We workinthestrongdrivinglimit 0 linearcrystallocatedinsidethecavity[18–20]. Itwasshown which assumes that the Rabi frequency of the laser field is 1 thatforthegenerationofentanglementbetweencavitymodes, muchlargerthanthetransitiondampingratesandthecoupling 0 1 itisessentialtocreateacoherenceinthecoupling(orentan- strengthsof the cavity modesto the atomic transitions. This : gling)system. Typicalsystems forentanglingthe modesare promptsustoapplythesecularapproximationwhichignores v multi-level atoms or nonlinear crystals where the coherence thecouplingofthepopulationsofthedressedstatestotheco- i X can be established initially by a preparation of the atoms in herences. Itisknownthatnon-secularterms, althoughsmall r alinearsuperpositionoftheirenergystatesorcanbecreated can have a destructive effect on coherence effects [21, 22] a dynamicallybya suitabledrivingofthe atomsthroughfour- ormay evenhaveconstructiveeffectsandlead to interesting wavemixing[11–15]orRaman-typeprocesses[21–24]. novel features [26–28]. However, we are interested in features created by the SGC rather than features created by the The coherenceis subjectedto dissipationdue to the deco- coherenceinduced by the driving field and therefore neglect herence process and over a long time it might be difficult to thenon-secularterms. maintainthecoherencelargeenoughforentanglingthecavity modes.Themainsourceofdecoherenceisspontaneousemis- We consider four scenarios, where the cavity modes cou- sionresultingfromtheinteractionoftheatomswiththeenvi- ple to the same or different atomic transitions that could be ronment. On a microscopicscale, the spontaneousemission correlated or independent of each other. The first scenario represents a situation in which the atomic transitions are in- dependentofeachotherandbothcavitymodescoupletothe sameatomictransitionthat,inaddition,isdrivenbyastrong ∗Electronicaddress:[email protected] andingeneraloff-resonantlaserfield. Physically,thissystem 2 behavesasadriventwo-levelsystemandthedrivingfieldoc- enhancementof entanglementbetween the cavity modes for cursasadressingfieldfortheatoms. Wedemonstratethatthe different coupling configurations of the cavity modes to the necessaryandsufficientconditionsforgenerationofthemax- atomic transitions. We are particularly interested in the role imal entanglement between the modes is to create the com- ofthemutualpolarizationoftheatomicdipolemomentsand plete population inversion between the dressed states of the the conditionsfor the generation of a large stationary entan- couplingatomicsystem. Apopulationdifferencebetweenthe glementbetweenthemodes. Thephysicaloriginofentangle- dressed states occursfor an off-resonantdrivingfield. Since mentbetweenthecavitymodesisexplainedintermsofpop- for a strong driving field there is no coherence between the ulationtrappinginalinearsuperpositionoftheatomiclevels. dressedstates, one couldconcludethatthe entanglementoc- Finally,wesummarizeourresultsinSec.IV. curswithoutcoherencein this case. However,for a detuned driving field, a coherence actually occurs between the two bare states of the system. In other words, in the bare atom picture,theentanglementiscreatedwithcoherence. Wefind II. GENERALFORMALISM thatthemaximalentanglementcannotbecreatedinthissce- nariosinceitisnotpossibletocreatealargepopulationdiffer- We considera three-levelatom locatedinside a two-mode encebetweenthedressedstatesandatthesametimemaintain cavity. TheatomismodelledasaV-typesystemwithground a strongcouplingbetweenthe cavitymodesmediatedbythe state 3 , and two excited states 1 and 2 separated in fre- atom. | i | i | i quencyby ∆ = ω ω , where ω and ω are atomic 0 13 23 13 23 In the second scenario, we include the coupling between transitionfrequenciesb−etweenstates 1 3 and 2 3 , theatomictransitionsthroughtheSGC,acloseanalogonthe respectively. We shall assume that ω| i↔> |ωi so|thiat↔∆| iis 13 23 0 schemesofquantum-stateengineeringbydissipation[29–35]. positive. Thischoice,of course,involvesno lossofgeneral- We findthatinthiscase, the dissipationisusedto createthe ity. Theatomactsasacoupling(orentangling)mediumthat required coherence in the atomic system. The maximal sta- couples two non-degeneratecavity modes of frequenciesω 1 tionaryentanglementcan be createdondemandevenforthe andω throughthe interactionofthe modeswith the atomic 2 resonantdrivingfieldbytuningtheRabifrequencyofthefield dipoletransitions 1 3 and 2 3 . In addition, the to the difference between the atomic transition frequencies. transition 2 3| iis↔dri|veinbya| sitr↔ong|laiserfieldofangu- As a result, the atomic system evolves into a pure trapping lar frequen|ciy↔ω |aind the amplitude determined by the Rabi L statewhichisanasymmetricsuperpositionofthedegenerate frequency2Ω,asillustratedinFig.1. Thedipolemomentsof energystates. Theparticularpurestateintowhichtheatomic thetwoallowedatomictransitionscanbeorthogonalornon- systemevolvesdependsupontheratioofthedampingratesof orthogonaltoeachother. Thelattercasecanleadtoquantum theatomictransitionsandthedetuningofthelaserfrequency interference effects induced by the SGC. The cavity modes fromtheatomictransitionfrequency. Thetrappingeffectre- cansimultaneouslycoupletooneoftheatomictransitionsor sultsinthecompletepopulationinversionbetweenthedressed to different transitions. One can also arrange a situation in statesofthesystem. Inotherwords,themaximalsteadystate which each of the cavity modes could couple to both of the entanglementisgeneratedwhenthepopulationoftheatomic atomictransitions.Inthiscase,thecouplingandtheresulting systemistrappedinapuresuperpositionstate. entanglementbetweenthemodescandependonwhetherthe Inthethirdscenario,weassumethatthe cavitymodesare transitiondipolemomentsareparalleloranti-paralleltoeach coupled to different atomic transitions. The new feature of other. thisscenarioisthatnowthegenerationofentanglementisin- dependentofthepopulationofthedressedstates. Theneces- !"#"$ sary condition for entanglementis the creation of coherence % betweentheatomictransitions,thecoherencethatcanbecre- 0 δ2 atedbytheSGC. !"%"$ % Finally,inthefourthscenario,weconsiderthemostgeneral ω ! configuration in which each of the cavity modes is coupled 13 δ ω 1 tobothatomictransitions. Weshowthatthisscenariocanbe ω 2 treatedasacombinationofthesecondandthirdscenarios,and 23 ω ! findthatthegenerationofentanglementdependsnowonthe ω 1 mutualpolarizationof the atomic dipole moments. Depend- ing on whether the transition dipole momentsare parallelor !"&"$ anti-parallel,theentanglementcanbeenhanced(reduced)by theconstructive(destructive)interferencebetweentheatomic FIG.1: Schematicdiagramoftheatomiclevelsandoneofpossible transitionamplitudes. couplingconfigurationsofthelaserandthecavityfields.Alaserfield Thepaperisorganizedasfollows.WebegininSec.IIwith offrequencyωL drivesthe|3i → |2itransitionwithdetuning∆L a description of the proposed schemes for the generation of andtwonon-degeneratecavitymodesoffrequenciesω1andω2cou- entanglement between two nondegeneratecavity modes and pletothedriventransitionwithdetuningsδ1 andδ2 fromthelaser derivethemasterequationforthereduceddensityoperatorof frequency. the cavity modes. In Sec. III, we study the generation and 3 Foranopencavityinwhichtheatomandthecavitymodes are the atomic transition operators between energy states i | i arecoupledtotheoutsidevacuummodes,thedynamicsofthe and j , (i,j =1,2,3)oftheatom. | i drivenatom plus the cavity modes is convenientlydescribed Sincethetransition 2 3 isdrivenbyastrong,nearly | i ↔ | i bythedensityoperatorρ, whichinaframerotatingwiththe resonant laser field, it is convenient to work in the dressed- laser frequency frequency ω satisfies the following master statepicture[36,37]. Weintroducedressedstates,whichare L equation(~=1) theeigenstatesoftheHamiltonian(3): d ˜1 = 1 , ρ= i[Hc+Ha+V,ρ]+Lcρ+Laρ, (1) | i | i dt − ˜2 = sinφ 2 cosφ 3 , | i | i− | i where ˜3 = cosφ 2 +sinφ 3 , (7) | i | i | i Hc =−δ1a†1a1+δ2a†2a2 (2) where isthefreeHamiltonianofthecavitymodes, cos2φ= 1 + ∆L, (8) 2 2Ω 0 H =(∆ +∆ )A +∆ A Ω(A +A ) (3) a L 0 11 L 22 23 32 − andΩ = ∆2 +4Ω2 istheRabifrequencyofthedetuned 0 L istheHamiltonianofthedrivenatom, field.Inthepdressed-statebasis,theoperatorsAij arereplaced bydressed-stateoperatorsR = ˜i ˜j , andthedensityop- V =(g a +g a )A +(g a +g a )A +H.c. (4) ij 1 1 2 2 23 3 1 4 2 13 erator of the system can be transf(cid:12)or(cid:11)m(cid:10)ed(cid:12) to the dressed-atom (cid:12) (cid:12) picturebytheunitarytransformation is the interaction Hamiltonian of the cavity modes with the atomictransitions, ρ˜=exp iH˜ t ρexp iH˜ t , (9) 0 0 2 (cid:16) (cid:17) (cid:16)− (cid:17) L ρ= κ 2a ρa† a†a ρ ρa†a (5) c Xj=1 j(cid:16) j j − j j − j j(cid:17) where H˜ =(∆ +∆ )R +Ω R δ a†a +δ a†a , (10) and 0 L 0 11 0 z − 1 1 1 2 2 2 andR =(R R )/2isthepopulationinversionoperator Laρ = γ1[A31,ρA13]+γ2[A32,ρA23] betweeznthed2r2es−sed3s3tates ˜2 and ˜3 . + η([A ,ρA ]+[A ,ρA ])+H.c. (6) | i | i 31 23 32 13 Applying the unitary transformation (9), we find that the commutator part of the master equation for ρ˜ contains ex- are operators representing the damping of the cavity-field plicitly time dependentterms that oscillate at frequenciesδ modesbycavitydecaywithratesκ andκ ,andoftheatomic 1 1 2 and δ , and the atomic dissipative part contains terms oscil- transitionsbyspontaneousemissionwithratesγ andγ . The 2 1 2 latingwithΩ and2Ω . InthelimitoflargeRabifrequency parametersg (i=1,2,3,4)arecouplingstrengthsofthecav- 0 0 i Ω g ,γ ,theoscillatingtermsinthedissipativepartmake itymodestotheatomictransitions.Weassumethatingeneral 0 ≫ i i contributions of order γ /Ω , where i = 1,2. These terms the modes couple with strengths g and g to the transition i 0 1 2 can be neglected in the secular approximation. The errors 2 3 andandalsocanbesimultaneouslycoupledtothe |1i ↔ |3itransitionwithstrengthsg andg ,respectively. of the secular approximationare of order γi/Ω0 and gi/Ω0. | i↔| i 3 4 Thus, it is reasonable to neglect these terms on time scales Thecoefficientη =p√γ1γ2 isameasureoftheamountof t γ−1 when Ω g ,γ . This approximation permits coherence,theso-calledSGC,inducedbydissipationbetween ≫ i 0 ≫ i i importantmathematicalsimplifications,and”exact”solutions the 1 3 and 2 3 atomictransitions.Thesourceof | i↔| i | i↔| i forthesteady-statedensitymatrixelementsmaybeobtained thiscoherencehasanobviousinterpretation.Namely,sponta- thatcouldprovideimmediateinsightintothephysicsinvolved neouslyemittedphotonononeoftheatomictransitiondrives intheproblem. the other transition. The degree of the coherence, measured Thus,themaserequationinthedressed-atombasisandun- bythecoefficientη,dependsexplicitlyonthemutualpolariza- derthesecularapproximationsimplifiesto tion of the transition dipolemomentswith p = cosθ, where θ istheanglebetweenthetwodipolemoments. Thus,p = 0 d when the transition dipole moments are orthogonal to each ρ˜= i V˜,ρ˜ +L ρ˜+L ρ˜, (11) d c other and p attains its maximal value of p = 1 when the dt − h i ± dipolemomentsareparalleloranti-paralleltoeachother.Ob- where viously, the SGC vanisheswhen p = 0 and attains maximal valuewhenp= 1. V˜ = d sin(2φ)R +sin2φR eiΩ0t cos2φR e−iΩ0t Theparameter±∆ =ω ω isthedetuningofthelaser 1 z 23 − 32 frequencyωL fromLtheato2m3i−ctraLnsitionfrequencyω23,δ1 = + (cid:8)d2 (cid:2)sinφR13ei[∆0+12(Ω0+∆L)]t (cid:3) ω ω andδ =ω ω aredetuningsofthecavitymodes (cid:16) ω1La−ndω12from2thela2s−erfrLequency,respectively;Aij =|iihj| −cosφR12ei[∆0−21(Ω0−∆L)]t(cid:17)o+H.c. (12) 4 is the interaction of the dressed atom with the cavity modes The master equation (15) is of a form characteristic for a with systemcomposedoftwofieldmodescoupledtoamulti-mode squeezed vacuum [38]. For this reason, to quantify entan- d1 = g1a1eiδ1t+g2a2e−iδ2t, glement between the modes, we shall use the Duan’s crite- d = g a eiδ1t+g a e−iδ2t, (13) rion [39], which relates entanglement to squeezing between 2 3 1 4 2 the modes. If the cavity modes were initially in a vacuum and state,whichisanexampleofaGaussianstate,thestateofthe modes governedby Eq. (15) will remain a two-mode Gaus- Ldρ˜=γ1 sin2φ[R31,ρ˜R13]+cos2φ[R21,ρR12]+H.c. sianstateforalltimest. Thequantumstatisticspropertiesof +γ (cid:0)sin2(2φ)([R ,ρ˜R ]+H.c.) (cid:1) atwo-modeGaussianstate areconvenientlystudiedin terms 2 z z ofquadratureoperatorsofthetwocavitymodes +γ sin4φ[R ,ρ˜R ]+cos4φ[R ,ρ˜R ]+H.c. 2 32 23 23 32 ++ηη0(cid:0)csions22φφ(([[RR31,,ρ˜ρ˜RR23]]++[[RR32,,ρ˜ρ˜RR13]]++HH..cc..)) (1(cid:1)4) Xl = √12(cid:16)a†leiθl +ale−iθl(cid:17), 0 21 22 22 12 i Y = a†eiθl a e−iθl , l=1,2, (16) is an operatorrepresenting the damping of the dressed-atom l √2(cid:16) l − l (cid:17) system. whereθ isthephaseanglesofthemodes.Ifweintroducetwo Obviously,thecavitydampingtermremainsunchangedun- l operators derthedressed-atomtransformation,buttheatomicdynamics are now determined in terms of the dressed-atom operators. 1 1 Here, we are interested in the case of the two cavity modes u=aX X , v =aY + Y , (17) 1 2 1 2 − a a beingnon-degeneratedi.e., ω = ω , forwhichthe timede- 1 2 pendenceofV˜ isquitecomplic6ated. Thisrendersthemaster wherea is a state-dependentrealnumber,then, accordingto equationdifficulttosolveexactly,exceptinaspecialcaseof theDuan’scriterion,atwo-modeGaussianstate isentangled aweakcouplingofthecavitymodestotheatomictransitions, ifandonlyifthesumofthevariancesΣ= (∆uˆ)2 + (∆vˆ)2 gi Ω0. Inthiscase,wecantreattheinteractionasaweak satisfiestheinequality h i h i ≪ perturbationtothestrongatom-laserinteractionandfind,af- tertracingovertheatomicvariables,thattheeffectivemaster 1 Σ=2na2+2m/a2 4c<a2+ , (18) equationforthereduceddensityoperatorofthecavitymodes, − a2 ρ =Tr ρ˜,isoftheform c A with a2 = (2m 1)/(2n 1), n = a†a +1/2, m = − − h 1 1i dρ =i 2 δ B¯ a†a ,ρ i 2 A¯ a a†,ρ ha†2a2i+1/p2,andc = |ha1a2i|. Sincetheright-handsideof dt c Xj=1(cid:0) 12− j(cid:1)h j j ci− Xj=1 jh j j ci Eq.(18)isapositivenumber,wemayintroduceaparameter +Xj=21(cid:16)B˜j +κj(cid:17)(cid:16)2ajρca†j −a†jajρc−ρca†jaj(cid:17) Υ=Σ−a2− a12, (19) and then the condition for entanglement between the cavity 2 + A˜ 2a†ρ a ρ a a† a a†ρ modesisthattheparameterΥmustbenegative. Xj=1 j(cid:16) j c j − c j j − j j c(cid:17) FromEqs.(18)and(19)itisobviousthatinordertocalcu- latetheparameterΥ,itisnecessarytohaveavailablethecav- 2 ity field correlation functions n,m and c. These correlation + C a†a† ρ +D ρ a† a† j j j′ c j c j′ j functions are readily found using the master equation (15), j6=Xj′=1n fromwhichwecanderiveequationsofmotionfortherequired (C +D )a† ρ a†+H.c. , (15) correlationfunctionsandfindthattheysatisfyasetofcoupled − j j j′ c j o differentialequations iwmhaegrienδa1r2y=par(tδs2o−fcδo1m)/p2l,exA˜jc,oBe˜fjficainedntAs¯jA,Bj¯,jBajr,ertehsepercetailvaenlyd. ddtha†jaji=− Γj +Γ∗j ha†jaji ThecoefficientsA˜j andB˜j haveobviousinterpretationasab- +χ (cid:0)a†a† +(cid:1)χ∗ a a +2A˜ , sorptionandgainrates,whereasA¯jandB¯jareradiativeshifts jh 1 2i jh 1 2i j ofthecavitymodefrequencies.Correspondingly,thecomplex d a a = (Γ +Γ ) a a +χ a†a coefficientsC andD determinetermsrepresentingdesired dth 1 2i − 1 2 h 1 2i 2h 1 1i j j correlations between the cavity modes. The expressions for +χ a†a +(C +C ), (20) thecoefficientsdependstronglyonthecouplingconfiguration 1h 2 2i 1 2 of the cavity modes to the atomic transitions and also on a whereΓ =κ +iδ (A B )andχ =C D . The j j 12 j j j j j − − − particularchoiceofotherparameters. Theexplicitanalytical set ofthe differentialequations(20) can be easily solvedfor formsofthecoefficientsfordifferentcouplingconfigurations arbitrary initial conditions. Since we are interested in a sta- ofthecavitymodestotheatomswillbegiveninSec.III. tionary entanglementbetween the cavity modes, we analyze 5 the stability condition and find that the system is stable and Westartbyintroducingtheexplicitformofthecoefficients reachesitssteady-stateast when ofthemasterequation(15),whichread →∞ Re(cid:20)Γ1+Γ2−q(Γ1−Γ∗2)2+4χ1χ∗2(cid:21)>0. (21) A1 =g12(cid:20)−14F1(δ1)sin2φ+ ff∗1∗((−δδ1))ρs33η2 cos4φ 12 − 1 − 0 f (δ )ρs η ρs Theabovestabilityconditionmaybesimplifiedsubstantially + 1 1 22− 0 12 sin4φ , forparticularchoicesofthedetuningsandtheRabifrequency f12(δ1)−η02 (cid:21) suchasδ ,δ γ andΩ γ . 1 f (δ )ρs 1 2 ≫ i 0 ≫ i B =g2 F (δ )sin2φ+ 1 1 33 sin4φ 1 1(cid:20)−4 2 1 f (δ ) η2 12 1 − 0 f∗( δ )ρs η ρs III. ENTANGLEMENTBETWEENCAVITYMODES + 1 − 1 22− 0 21 cos4φ , f∗ ( δ ) η2 (cid:21) 12 − 1 − 0 ItisclearfromEq.(15)thatthedynamicsandentanglement C = 1g g sin2φ F (δ )+ f1(δ2)ρs33 ofthecavitymodesareasensitivefunctionofthepropertiesof 1 4 1 2 (cid:20) 2 2 f (δ ) η2 12 2 − 0 thedrivenatomicsystem. Inordertostudythisdependence, f∗( δ )ρs η ρs weshallexaminefourscenariosofthecouplingconfiguration + 1 − 2 22− 0 21 , f∗ ( δ ) η2 (cid:21) ofthecavitymodestotheatomictransitions,twoscenariosin 12 − 2 − 0 1 f∗( δ )ρs whichbothmodescoupletothesamedrivenatomictransition D = g g sin2φ F (δ )+ 1 − 2 33 and the other two in which the cavity modes are coupled to 1 4 1 2 (cid:20) 1 2 f1∗2(−δ2)−η02 differenttransitions. Aparticularattentionwillbepaidtothe f (δ )ρs η ρs + 1 2 22− 0 12 , (22) roleofaspecificdrivingoftheatomsandtheSGCinentan- f (δ ) η2 (cid:21) glingthecavitymodes. 12 2 − 0 where A. Thecaseofbothmodescoupledtothedriventransition F (δ )=[M (δ ) M (δ )]ρs [M (δ ) M (δ )]ρs 1 j 32 j − 22 j 22− 33 j − 23 j 33 +[M (δ ) M (δ )]ρs , Inthissection,weexaminetheentanglementpropertiesof 34 j − 24 j 12 F (δ )=[M (δ ) M (δ )]ρs [M (δ ) M (δ )]ρs thecavitymodeswhenbothmodesarecoupledtoonlyoneof 2 j 32 j − 22 j 22− 33 j − 23 j 33 the atomic transitions, the laser driven transition |2i ↔ |3i, +[M35(δj)−M25(δj)]ρs21, (23) asillustratedinFig.1. Inotherwords,allthefieldscoupleto onlyoneoftheatomictransition. Thisisachievedbyputting and thecouplingstrengthsg andg intheHamiltonian(4)equal 3 4 to zero. We shallbe particularlyinterested in the generation f12( δj)=f1( δj)f2( δj), j =1,2, (24) ± ± ± ofentanglementbetweenthecavitymodeswhenthecoupling systemisreducedtoasimpletwo-levelsystemandtheroleof with thespontaneousemissionincouplingofthetwo-levelsystem 1 to the auxiliary level 1 . Therefore, we consider separately f ( δ )=γ +γ cos2φ+i ∆ + (∆ +Ω ) δ , | i 1 ± j 1 2 (cid:18) 0 2 L 0 ± j(cid:19) twocasesoforthogonal(p = 0)andnon-orthogonal(p = 0) 6 dipole moments of the atomic transitions. When the dipole 1 f ( δ )=γ 1+ sin22φ +i(Ω δ ). (25) moments are orthogonal to each other, p = 0, and then the 2 ± j 2(cid:18) 2 (cid:19) 0± j atomic transition 1 3 decouples from the driven transition. In this ca|sei, t↔he|syistem reduces to that of a driven Here, ρs , ρs , ρs are the steady-state values of the atomic 22 33 12 two-levelatom. Ontheotherhand,whenthedipolemoments density matrix elements under the condition of ignoring the arenonorthogonal,p=0,andthenthespontaneousemission effect of the weak coupling between the cavity modes and 6 onthe 1 3 caninfluenceonthetwo-leveldynamicsof the atom, and M (δ ) are elements of the inverse matrix mn j | i ↔ | i thedriven 2 3 transition. ofU(δ ): j | i↔| i 2γ +iδ 0 0 η η 1 j 0 0 2γ cos2φ 2γ sin4φ+iδ 2γ cos4φ η cos2φ η cos2φ  1 2 j 2 0 0  − − − − U(δj)= 2γ1sin2φ 2γ2sin4φ 2γ2cos4φ+iδj 2η0sin2φ 2η0sin2φ . (26) − η − η 0 −b+iδ − 0   0 0 j   η η 0 0 b∗+iδ   0 0 j  whereb=γ +γ sin2φ+i[∆ (Ω ∆ )/2]. TheremainingcoefficientsA ,B ,C andD areobtained 1 2 0 0 L 2 2 2 2 − − 6 from Eq. (22) by exchanging δ with δ and g with g . Wewouldliketopointoutthatthemagnitudeoftheentan- 1 2 1 2 − We should pointouthere thatin the derivationof the coeffi- glementisnotlargeandtherearenoparametervaluesatwhich cients (22), we have assumed that the states ˜1 and ˜3 are the entanglement could reach the optimal value Υ = 1. | i | i − separatedin energyby∆ +(Ω +∆ )/2, while the states Moreover, the maximal entanglement occurs at large detun- 0 0 L ˜1 and ˜2 are separated in energy by ∆ (Ω ∆ )/2. ings,∆ 40γ ,atwhichthedrivingfieldisweaklycou- 0 0 L L 1 | i | i − − ≈ ± Thus,ingeneral,thedressedstatesarenon-degenerate.How- pled to the atoms. We shall demonstrate in the second sce- ever, by varying the Rabi frequency Ω or the splitting ∆ , nario,thatthemagnitudecanbeenhancedtoitsoptimalvalue 0 0 onemayturn thestates ˜1 and ˜2 intodegeneracy,whereas Υ = 1bycouplingthe two-levelsystem tothe thirdlevel. thestates ˜1 and ˜3 wil|lialways| riemainfarfromresonance. Tosum−marize,webrieflydiscusstheparameterscharacteriz- | i | i Thiswould happenwhen∆ = (Ω ∆ )/2. As we shall ingthesystemandtherangesoftheseparametersexperimen- 0 0 L − demonstrateinthispaper,thedegeneracyconditionisanop- tallyaccessible. Theparametersareexpressedinunitsofthe timalconditionforentanglementbetweenthecavitymodes. spontaneousemissionrateγ. Inthecaseofalkaliatoms,γ is Having defined the coefficients of the master equation for of the order of 10 MHz. Driving lasers used in experiments the case of both cavity modes coupled to the driven atomic are usually tunable, providing for arbitrary detuning ∆ , so L transition,wenowturnourattentiontothepossibilityofgen- thattherange∆ 100γiseasilyaccessible. Thelasersare L ≤ eratingastationaryentanglementbetweenthemodes. Indo- sufficientlypowerfultogenerateRabifrequenciesupto100γ. ing that we shall consider separately two cases, p = 0 and p=0. 6 2. Thecaseofp6=0 1. Thecaseofp=0 We now turn to illustrate the role of the SGC on entan- glementcreationbetweenthe cavitymodes. We assume that Letusfirstdeterminehowmuchentanglementcanbegen- the driven transition to which the cavity modes are coupled, erated when the atom behaves as a two-level system. The is coupled by spontaneous emission to the auxiliary level masterequation(15)canbeappliedtothissimplifiedcaseby 1 . This coupling can occur for the case of non-orthogonal puttingp = 0. Figure 2 showsthe entanglementmeasure Υ |(pi= 0) dipole moments of the atomic transitions, and then asafunctionof∆L forη0 =0,fixeddetuningsδ1,δ2andthe the6spontaneousemission on the 1 3 can influenceon RabifrequencyΩ0. Thefigureshowsthatunderresonantex- thetwo-leveldynamicsofthedriv|eni ↔2 | i3 transition. citation,thecavitymodesareseparableandbecomeentangled Since the spontaneous emission o|nith↔e a|toimic transitions for an off-resonantexcitation. The entanglementexhibitsan occurs at different frequencies and with different rates, the interesting behavior, in that it has two maxima which occur createdentanglementbetweenthe cavitymodesmaydepend for certain nonzero values of ∆L, and then rapidly declines stronglyonthesplitting ∆0. As weshallsee, thecrucialfor thereafter. A small differenceδ12 = 0.61 between the de- entanglement between the cavity modes is the relation be- − tuningsδ1andδ2isintroducedtocanceltheeffectoftheStark tween Ω0 and ∆0. Figure 3 illustrates the variation of Υ shiftsA¯j andB¯j. Asweseefromthefigure,theStarkshifts with gradually increasing ∆0 for the case of resonant driv- haveadistractiveeffectonentanglement. ing,∆ =0. Weseethatthecavitymodesbecomeentangled L onlyforp = 0andforacertainvalueof∆ = Ω /2,theen- 0 0 6 tanglementbecomesoptimal. Intermsoftheenergiesofthe dressed states, the condition of ∆ = Ω /2 corresponds to 0 0 0.0 thesituationwherethedressedstates ˜1 becomesdegenerate -0.1 withthedressedstate ˜2 [40,41].The|cionditionofp=0cor- | i 6 -0.2 respondstothepresenceofdirectcouplingbetweenthestates -0.3 ˜1 and ˜2 . Notethatthiscouplingisinducedbythedissipa- | i | i -0.4 tiveprocessofspontaneousemission. Sincethisisaresonant -0.5 coupling,itcreatesastrongcoherencebetweenthestates ˜1 and ˜2 . Underthiscircumstance,themodesbecomestrong|lyi -0.6 | i entangledandthedegreeofentanglementismaximalincom- -0.7 parisonwithFig.2.Theamountofthegeneratedentanglement -0.8 dependsalso on the ratio of the spontaneousemission rates, -0.9 γ /γ ,andthemaximalentanglementofΥ 1isachieved 2 1 -1.0 ≈− -80 -60 -40 -20 0 20 40 60 80 at∆0 =Ω0/2andp 1forγ2 γ1. Inotherwords,alarge ≈ ≪ L entanglementoccurswhenthemostofthepopulationresides inthe driventransitionratherthaninthe undriventransition. FIG.2: ThedegreeofentanglementΥplottedasafunctionof∆L Wemaysummarizethatbyusingcarefullydesigneddriving, forthecasecorrespondingtoatwo-levelsystem,g3 = g4 = 0and such that∆0 = Ω0/2 and carefullychosenatoms, such that p = 0, with γ2 = 0.02,Ω = 50,δ1 ≈ δ2 = 50,κ1 = κ2 = γ2 γ1,alargeentanglementcanbeproducedbetweenthe 0.63,g1 = g2 = 10anddifferentδ12: δ12 = 0(solidline),δ12 = cav≪itymodesviadissipationcreatedcoherenceintheatoms. −0.61(dashedline).Allparametersarenormalizedtoγ1. We nowproceedto explainthe physicaloriginof the pro- 7 the population is unequally distributed between the dressed states. Thus,theonlyonefactordeterminesthemagnitudeof 0.1 entanglementbetween the cavity mode, the populationmust 0.0 beinvertedbetweenthedressedstatesofthesystem. Forthe -0.1 case of p = 0, this can be achievedif the laser frequencyis -0.2 detunedfromtheatomictransitionfrequencyω . Itisinter- 23 -0.3 estingthattheentanglementiscreatedwithoutanycoherence -0.4 betweenthedressedstates. Thereisnocoherencebetweenthe -0.5 dressedstatessincetheRabifrequencyΩ ismuchlargerthan 0 -0.6 allrelaxationrates, Ω γ ,κ . However,we shouldpoint 0 i i -0.7 ≫ outthatinthecaseofanoff-resonatdriving,thereisacoher- -0.8 encebetweenthebareatomicstates. Thus,onecanarguethat -0.9 thepredictedentanglementactuallyoccursduetoanon-zero -1.0 45 46 47 48 49 50 51 52 53 54 55 coherencebetweenthebareatomicstates. To calculate the populationinversion between the dressed 0 states, we introduce density matrix elements with respect to FIG.3: Thedegreeofentanglement Υplottedasafunctionof∆0 the three atomic dressed states in the absence of the cavity for∆L = 0, γ2 = 0.02, Ω = 50, δ1 ≈ δ2 = 50, δ12 = −0.61, modes,denoting ˜1ρ˜˜2 byρ12,etc. Theequationsofmotion κ1 =κ2 =0.63,g1 =g2 =10,andvariousvaluesofp: p=0.98 are h | | i (solid line), p = 0.7 (dashed line), p = 0.4 (dashed-dotted line), p=0(dottedline).Allparametersarenormalizedtoγ1. ρ˙11 = 2γ1ρ11 η0(ρ12+ρ21), − − ρ˙ =2γ cos2φρ +2γ cos4φρ sin4φρ 22 1 11 2 33 22 − cess responsible for entanglement of the cavity modes pre- +η0cos2φ(ρ12+ρ21),(cid:0) (cid:1) dictedintheabovetwoscenarios.Asweshallsee,thephysics ρ˙ =2γ sin2φρ 2γ cos4φρ sin4φρ 33 1 11 2 33 22 oftheprocesscanbequantitativelyexplainedonthelevelof − − +2η sin2φ(ρ +ρ )(cid:0), (cid:1) thestationarypopulationoftheatomicsystem. Inthefirstin- 0 12 21 stance, a simple analytical expressioncan be derivedfor the 1 ρ˙ = γ +γ sin2φ+i ∆ (Ω ∆ ) ρ master equation as follows. When the frequency difference 12 −(cid:26) 1 2 (cid:20) 0− 2 0− L (cid:21)(cid:27) 12 δ andtheRabifrequencyΩ aremuchlargerthanthedamp- 0 η (ρ +ρ ). (29) ing rates of the atomic transitions, δ1 δ2 = δ γi and − 0 11 22 ≈ ≫ Ωgi0bl≫e,i.γei.,At˜hje=reaB˜ljpa=rtsCõjf=theD˜pjaram0,eatenrdst(h2e2)imbaegcoinmareynpeagrltis- Iintdisuceevdidebnytsfproomntathneeoaubsoveemeisqsuioatnioonssctihllaattetshewciothhefrreenqcueenρc1y2 btoecshoomwetAh¯ajt≈theB¯mjaasntedrCe¯qju=atio−nD¯(≈1j5.)Imtiasythbeenapstprraoigxhimtfaotrewdabryd ∆ing0−tha(tΩth0e−c∆ohLe)r/en2c.eTahtitsaifnasctmhaaxsimthaelovbavluioeuwshpehnys∆icalm(Ωean- 0 0 − − ∆ )/2 = 0. For ∆ = 0, the coherence maximizes at L L ddtρc =−i(cid:0)δ12+2A¯(cid:1)ha†1a1+a†2a2,ρci ∆effi0c=ienΩt D0¯/2eqaunadlssitmou1l,tacnoenosuesqluyetnhtelyftahcetovrasliune2aφt winhtihcehcthoe- iD¯ a†a† +a a ,ρ +L ρ , (27) entanglement,showninFig.3,attainsthemaximalvalue. − h 1 2 1 2 ci c c In the steady-state, the dressed state populationdifference canbeworkedoutexplicitlyforbothp = 0 andp = 0. For where 6 the case of p = 0, the steady state population difference is g2Ω 1+cos22φ givenbytheexpression A¯ = 0 (ρs ρs ), 4((cid:0)Ω2 δ2) (cid:1) 22− 33 0− cos4φ sin4φ D¯ = g2Ω0sin22φ(ρs ρs ), (28) ρs22−ρs33 = cos4φ+−sin4φ, (30) 2(Ω2 δ2) 22− 33 0− which clearly shows that the populationsamong the dressed and,forsimplicity,wehaveassumedequalcouplingconstants states are imbalanced only for a nonzero detuning ∆ = L g =g =g. 0(φ = π/4). In this case the parameter D¯ responsible fo6 r 1 2 6 This shows that the atomic variables contribute to the co- the nonlinear coupling between the modes is different from herentevolutionof the cavity modesand the onlyrelaxation zero. Itiseasytocheckthatthemaximalentanglementseen inthesystemisthedampingofthecavitymodes.Achoiceof inFig.2isattainedatthedetuningscorrespondingtothemax- δ = 2A¯simplifiesfurtherthemasterequationandleaves imalvalueofD¯. Thus,wehaveasimplephysicalinterpreta- 12 − only the parametric amplifying term in its commutator part. tionoftheentanglementcreationbyadetunedlaserfield. Thistermisresponsibleforcorrelationsandsoforentangle- Westressthatinthecaseofthedetuneddriving(∆ = 0) L 6 mentbetweenthemodes. Themagnitudeofentanglementat- and in the limit p = 0, i.e. in the two-level situation, the tains maximal value when D¯ maximizes. It is evident from populationisunequallydistributedbetweenthedressedstates, Eq. (28) that the parameter D¯ is different from zero only if butitisnotpossibletoproduceatomsinapuredressedstate 8 in which ρs ρs = 1 and at the same moment having the coeffic|ie2n2t−D¯ d3i3ff|erentfrom zero. However, for the case !"#"$ of thee-levelatomswith p = 1, it ispossible to have ρs % δ2 ρs = 1, in which case the population is trapped in|on2e2o−f 0 33| the dressed states. The condition of the population trapping !"%"$ % isuniquetotheSGCandcanbeachievedevenforaresonant ω ! 2 ω driving,∆L =0. 13 δ1 Wenowproceedtoevaluatethepopulationinversionwhen ω 23 ω p = 1. A carefulanalysisof the steady-statesolutionshows ! ω thatinthecaseofthelevelcrossingat∆0 =Ω0/2andinthe 1 limitp=1,thepopulationisnottrappedinoneofthedressed statesbutratherinoneoflinearsuperpositions !"&"$ s = α˜2 +β ˜1 , FIG.4:Schematicdiagramofthecouplingconfigurationofthecav- | i | i | i a = β ˜2 α˜1 , (31) itymodesandthedrivenlaserfield.Thecavitymodeoffrequencyω1 | i | i− | i is coupled to the laser driven transition with detuning δ1 from the where laserfrequency,whilethecavitymodeoffrequencyω2iscoupledto theundriventransitionwithdetuningδ2fromthelaserfrequency. 1 1 γ sin2φ 2 γ 2 α= 2 , β = 1 .(32) (cid:18)γ +γ sin2φ(cid:19) (cid:18)γ +γ sin2φ(cid:19) 1 2 1 2 coefficientsofthemasterequation(15)areoftheform Itiseasytocheckthatatthelevelcrossingconditionandin thelimitp=1,thepopulationistrappedintheantisymmetric sratatitoeb|aeit,wie.ee.nρtsahae=dam1pirirnegspraetcetsivγe1oafntdheγ2d.etTuhniisnrges∆uLltiamndpltihees A1 =g12(cid:20)−41F1(δ1)sin2φ+ f2∗ρ(s3−3δc1o)s4−φη02 thattheSGCisessentialfortheatomicsystemtobecapable ρs sin4φ η ρs sin4φ ofachievinga purestate. In otherwords, the trappingeffect + 22 0 12 , f∗(δ ) η2 − f (δ ) η2(cid:21) isa directmanifestationofthe presenceofthe SGCthatcan 2 1 − 0 12 1 − 0 beemployedtomaintainthecompleteinversionbetweenthe B =g2 1F (δ )sin2φ+ ρs33sin4φ dressed states even in the case of zero detuningbetween the 1 1(cid:20)−4 2 1 f (δ ) η2 2 1 − 0 laserandtheatomictransitionfrequencies. Ifweincorporate f∗( δ )ρs η ρs thesolutionρs = 1intoEq.(28),wefindthattheresulting + 1 − 1 22− 0 21 cos4φ , aa f∗ ( δ ) η2 (cid:21) coefficientD¯ takestheform 12 − 1 − 0 f∗( δ )ρs η ρs C =g g sinφcos2φ F (δ )+ 1 − 2 12− 0 11 , D¯ = g2Ω0 γ1sin22φ , (33) 1 1 4 (cid:20) 3 2 f1∗2(−δ2)−η02 (cid:21) 2(Ω2 δ2)γ +γ sin2φ η ρs 0− 1 2 D =g g sinφcos2φ F (δ ) 0 33 , (34) 1 1 4 (cid:20) 4 2 − f∗ ( δ ) η2(cid:21) fromwhichonecaneasilyshowthatthecoefficientD¯ isgreat- 12 − 2 − 0 estwhenφ = π/4(∆ = 0)andγ γ . Thisprediction clearlyexplainsournuLmericalresult2sp≪rese1ntedinFig.3. withF1(δ1)andF2(δ1)giveninEq.(25), To clarify the issue of the mechanism responsible for F (δ )=[M (δ ) M (δ )]ρs creation of the stationary entanglement between the cavity 3 2 32 2 − 22 2 12 modes, we may refer to the equationsof motion for the cor- +[M35(δ2)−M25(δ2)]ρs11, relationfunctions(20). Itisstraightforwardto showthatthe F (δ )=[M (δ ) M (δ )]ρs olifmtihteocfoδrr≫elatγioinanfdunΩct0io≫nsγisi,ththeecoanvliytyddaammppininggm. Techhuasn,itshme 4 2 +[M3351(δ22)−−M2251(δ22)]ρs2122, (35) SGCfacilitiescorrelationsbetweenthecavitymodesthatthen and decaywiththecavitydampingtoastationaryentangledstate. f ( δ )ρs η ρs A =g2 h (δ )+ 2 − 2 11− 0 21 sin2φ , 2 4(cid:20) 1 2 f ( δ ) η2 (cid:21) B. Thecaseofthemodescoupledtodifferentatomic 12 − 2 − 0 transitions B =g2 h (δ )+ f2(−δ2)ρs33sin2φ , 2 4(cid:20) 2 2 f ( δ ) η2 (cid:21) 12 − 2 − 0 Wenowproceedtoevaluateentanglementbetweenthecav- η ρs itymodeswhenoneofthecavitymodes,a1,iscoupledtothe C2 =g1g4sinφcos2φ(cid:20)h3(δ1)− f ( 0δ 3)3 η2(cid:21), driven 2 3 transitionandtheothermodea iscoupled 12 − 1 − 0 | i ↔ | i 2 f ( δ )ρs to the undriventransition 1 3 , as illustrated in Fig. 4. D =g g sinφcos2φ h (δ )+ 2 − 1 12 , (36) In this case, the coupling|stire↔ngth|sig2 = g3 = 0, then the 2 1 4 (cid:20) 4 1 f12(−δ1)−η02(cid:21) 9 with where 1 sin4φ cos4φ h1(δ2)=[M42(−δ2)ρs21+M44(−δ2)ρs11]cos2φ, A¯ = 4g2(cid:20)(cid:18)Ω0+δ + Ω0 δ(cid:19)(ρs22−ρs33) hhh234(((δδδ211)))===[MMM444331(((−−−δδδ112)))ρρρs3s3s2331−−+MMM444224(((−−−δδδ112)))ρρρs2s2s2222−−]coMMs24454φ((,−−δδ11))ρρ(s1s13127,.) D¯ =+ ΩΩs0i0ng−22φsδin(ρφs1c1o−s2ρφs3ρ3)s+,−cosδ2φ(ρs22−ρs11)(cid:21), (39) (Ω δ)δ 12 0 − We may further simplify the master equation by choosing Figure5showstheresultsfortheentanglementmeasureΥ δ = 2A¯,whichleavesonlythenon-linearterminitscom- asafunctionof∆ forvariousvaluesofp.Sinceinthecaseof 12 − 0 mutator part. Note that comparing to the case A, there is a p=0,thecreationofentanglementbetweenthecavitymodes qualitative differencein the dependenceof the coefficientD¯ wasassociatedwithanon-zerodetuning,∆L 6=0,theroleof onthedensitymatrixelements. ThemagnitudeofD¯ depends SGCisillustratedmostclearlyifoneassumesaresonantlaser nowonthecoherencebetweenthestates ˜1 and ˜2 butnoton field. Consequently,wechoosetolimitourillustrationofthe | i | i thepopulationdifference. Thecoherenceisinducedbyspon- creationofentanglementtoasituationinwhich∆ =0. L taneousemissionandcanbedifferentfromzeroonlyifp=0. 6 This means that the SGC is crucial for creation of entangle- mentbetweenthecavitymodeswhenthemodesarecoupled to differentatomic transitions. As it is seen fromFig. 5, the entanglementmaximizesat∆ =Ω /2andp = 1. Itiseasy 0.1 0 0 to show from Eqs. (29) and (31) that for ∆ = Ω /2 and 0.0 0 0 p=1,inthesteadystatethepopulationistrappedintheanti- -0.1 -0.2 symmetricstate a . Thus,similartothecaseA,thecondition | i -0.3 forthe maximalentanglementcoincideswith the collapseof -0.4 theatomicsystemintothepuretrappingstate.Inthiscase,the -0.5 coherenceρs = αβ andthentheparameterD¯ reducesto 12 − -0.6 -0.7 D¯ = Ω0g2sin22φ √γ1γ2 . (40) -0.8 − 4(Ω0−δ)δ γ1+γ2sin2φ -0.9 -1.0 Itis easily verified thatthe coefficientD¯ attainsits maximal 45 46 47 48 49 50 51 52 53 54 55 valueforφ=π/4andγ =2γ . Thus,thesimpleformulain 2 1 0 Eq.(40)predictsaccuratelytheparametervaluesofthemaxi- malentanglementinFig.5. FIG.5: Thedegree ofentanglement Υasafunctionof∆0 forthe Inconcludingthissection, we wouldlike to pointoutthat caseofthecavitymodescoupledtodifferentatomictransitions,g2 = g3 = 0andg1 = g4 = 10,with∆L = 0,γ2 = 2,Ω = 50,δ1 ≈ the qualitative features of entanglement between the cavity δ2 =50,δ12 =−0.38,κ1 =κ2 =0.67,anddifferentp: p=0.98 modesdependonwhetherthedipolemomentsof theatomic (solid line), p = 0.7 (dashed line), p = 0.4 (dashed-dotted line), transitionsare parallel (p = 1) or anti-parallel(p = 1) to − p=0(dottedline).Allparametersarenormalizedtoγ1. each other. We have alreadyseen thatin the case of parallel dipolemomentsand∆ =Ω /2,thepopulationistrappedin 0 0 theantisymmetricstate irrespectiveofthelaserdetuning∆ L As before, for the case IIIA2, the entanglement occurs andtheratiobetweentheatomicspontaneousemissionrates. for p = 0 and the optimal entanglement can be obtained at However,fortheanti-paralleldipolemoments,thesituationis 6 ∆0 = Ω0/2. However, in contrast to the case A, the entan- different. It is not difficult to show from Eqs. (29) and (31) tghleemenentatnmglaexmimeniztemsaaxtiΥmiz≈es−w1hfeonrtγh2e t=ran2sγi1ti.onItrmateeasnosfththaet tohfatthfeosrtpate=sa−re1and∆0 =Ω0/2,thesteadystatepopulations dressed transition resonant with the undressed transition are equal. ρaa = α2 β2 2, ρss =4α2β2, ρ33 =0, (41) − In order to understand this behavior of entanglement, we (cid:0) (cid:1) whereαandβ aregiveninEq.(32). Itisevidentthatingen- considerthecoefficientsofthemasterequationinthelimitof eralthepopulationisredistributedbetweenthesymmetricand δ ≫ γi and Ω0 ≫ γi and find that in this limit, the master antisymmetric states and only in the case of γ1 = γ2sin2φ equation(15)reducestothefollowingform thepopulationistrappedinone,thesymmetricsuperposition state. A consequence of this population redistribution is the d reductionoftheentanglementbetweenthecavitymodes.This ρ = i δ +2A¯ a†a +a†a ,ρ dt c − (cid:0) 12 (cid:1)h 1 1 2 2 ci is shown in Fig. 6, wherewe plotthe entanglementmeasure +i D¯a†1a†2+D¯∗a1a2,ρc +Lcρc, (38) tfaonrepou=s em−i1ssaionnd draiftefesr.enFtorratγios =betw2γee,ntthheemataogmniictusdpeoonf- h i 2 1 6 10 the entanglement is reduced and attains the maximal value sultingin anenhancedorreducedeffectivemagnitudeofthe of Υ = 1 forγ = 2γ . This is an anotherdemonstration nonlinearprocess. Forp = 1theconfigurationsinterferede- 2 1 thatthem−aximalentanglementbetweenthemodesisachieved structively such that for γ = γ the effective coefficient D¯ 1 2 onlywhentwocorrelatedatomictransitionsdecayratesobey vanish. Ontheotherhand,forp = 1theconfigurationsin- − γ =2γ . terfereconstructivelywhichresultsinanenhancedamplitude 2 1 ofthenonlinearprocess.However,theresultingmagnitudeof the effective coefficient depends strongly on the ratio γ /γ 2 1 suchthatD¯ islargeforγ /γ 1,butbecomesverysmall, 2 1 0.0 ≪ proportionalto γ /γ inthe oppositelimitofγ /γ 1. 1 2 2 1 -0.1 In other words,pthe three-level system can strongly enta≫ngle -0.2 thecavitymodesonlyifthespontaneousemissionrateonthe -0.3 undriventransitionismuchlargerthanthatofthedriventran- -0.4 sition. -0.5 Wefinishthissectionwithashortdiscussionofapossibility -0.6 tocreateentanglementbetweenthecavitymodesbytheSGC inthree-levelatomsintheLambdaorcascadeconfigurations. -0.7 Aswehaveshown,thecrucialforthemaximalentanglement -0.8 is to trap the population in a pure superposition state of the -0.9 atoms. However, it is well known that the SGC has a con- -1.0 46 48 50 52 54 structive effect on trapping of the population in a pure state 0 onlyintheV-typeatoms[25]. IntheLambdaorcascadetype atoms,theSGChasadestructiveratherthanconstructiveef- FIG.6: Thedegreeofentanglement Υplottedasafunctionof∆0 fectonthetrappingphenomenon[42,43]. forthecaseofanti-paralleltransitiondipolemoments,p=−1,with The crucialfor the entanglementis three-levelatoms with ∆L = 0, Ω = 50, δ1 ≈ δ2 = 50, κ1 = κ2 = 0.72, and differ- parallel or nearly parallel dipole moments between the two ent γ2/γ1: γ2/γ1 = 0.5 (solid line), γ2/γ1 = 1.0 (dashed line), atomictransitions.ItisdifficultinpracticetofindV-typesys- γ2/γ1 = 2.0(dashed-dotted line), γ2/γ1 = 3.0(dottedline). All tems with parallel or anti-parallel dipole moments. One of parametersarenormalizedtoγ1. thepossibilityistousesodiumdimers,whichcanbemodeled asafive-levelmoleculeinwhichtransitionswithparalleland anti-paralleldipolemomentscanbeselected[44,45]. Anal- ternativesolutionistoengineeratomicsystemswithparallel C. Otherpossiblecouplingsofthemodestotheatomic dipolemoments. Forexample,ZhouandSwain[46]showed transitions thattransitionswithparalleldipolemomentscanbeachieved inathree-levelatomcoupledtoacavityfieldwithpre-selected Finally,webrieflycommentontheotherpossiblecoupling polarizationinthebadcavitylimit. Agarwal[47]hasdemon- configurationsof the cavity modes to the atomic transitions. stratedthatananisotropyinthevacuumcanleadtoquantum Thetwocasesdiscussedabovepredictalargeentanglementat interference among the decay channelsof close lying states. practicallythesameconditions,withonlydifferentconditions Anotherpossibilityisto alignthedipolemomentsbya slow imposed on the damping rates of the atomic transitions. An motionof the atoms throughthe medium [48], or to apply a anotherpossibleconfigurationistocouplethecavitymodeω 1 dcfieldtocoupletheupperlevelsofathree-levelV-typeatom to the undriventransition 1 3 and the mode ω to the 2 withperpendiculardipolemoments[49]. | i ↔ | i driventransition 2 3 . OnecanseefromFig.4,thatthis | i↔| i configurationisobtainedfromthecaseBsimplybyreplacing δ by δ. Thus, a large entanglementcould be generated in − IV. CONCLUSIONS thisconfigurationforthesameconditionasinthecaseB. Themostgeneralconfigurationofthecouplingconstantsis We have proposeda scheme for generation on demand of the case correspondingto all of the cavity modessimultane- a steady-stateentanglementbetweentwo opticalmodescou- ouslycoupledto bothatomic transitions. Itis easily verified pledtoaV-typethree-levelatom. Wehavedemonstratedthat that this general case can be treated as a sum of two cases theconditionforgenerationofthemaximalentanglementbe- B with opposite detuning δ. By combining the two cases tweenthemodesistocreatethecompletepopulationinversion together, we find that the magnitude of the effective coeffi- cientD¯ dependsstronglyonthesignoftheparameterp. For betweenthedressedstatesofthecouplingatomicsystem. In p= 1,theeffectivecoefficientD¯ takesthefollowingform thecaseofatwo-levelatomcomposingtheentanglingatomic ± system, we have shown that the sufficient condition for en- D¯p=±1 = g22(ΩΩ02sin2δ22)φ√γγ1(cid:0)+√γγ1s∓in√2φγ2(cid:1). (42) tfaenregnlecmebeenttwbeeetwnederensstheedsmtaotdesesofisthtoedcrrievaetneaatopmopicultaratinosnitidoinf-. 0− 1 2 However,wehavefoundthatthemaximalentanglementcan- We see that depending on the sign of p these two coupling notbecreatedinthissystemsinceitisnotpossibletocreate configurationcaninterfereconstructivelyordestructivelyre- thecompletepopulationinversionbetweenthedressedstates