Entanglement created by spontaneously generated coherence

0.3 MB·English

by  Zhao-hong Tang

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 fea-
tures 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 tran-
sition. 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=rtsC˜ojf=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