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Correlations in optically-controlled quantum emitters Cristian E. Susa and John H. Reina ∗ † Departamento de F´ısica, Universidad del Valle, A.A. 25360, Cali, Colombia (Dated: January 23, 2012) Weaddresstheproblemofopticallycontrollingandquantifyingthedissipativedynamicsofquan- tum and classical correlations in a set-up of individual quantum emitters under external laser ex- citation. We show that both types of correlations, the former measured by the quantum discord, are present in the system’s evolution even though the emitters may exhibit an early stage disen- tanglement. In the absence of external laser pumping, we demonstrate analytically, for a set of suitableinitial states, thatthereisanentropyboundforwhich quantumdiscord andentanglement of the emitters are always greater than classical correlations, thus disproving an early conjecture thatclassical correlations aregreaterthanquantumcorrelations. Furthermore,weshowthatquan- tumcorrelations canalsobegreaterthanclassical correlations whenthesystemisdrivenbyalaser field. Forscenarios wheretheemitters’quantumcorrelations arebelowtheirclassical counterparts, an optimization of the evolution of the quantum correlations can be carried out by appropriately tailoring the amplitude of the laser field and the emitters’ dipole-dipole interaction. We stress the 2 importance of using the entanglement of formation, rather than the concurrence, as the entangle- 1 mentmeasure,sincethelattercangrowbeyondthetotalcorrelationsandthusgiveincorrectresults 0 on the actual system’s degree of entanglement. 2 n PACSnumbers: 03.65.Ud,03.67.Mn,42.50.Fx,03.65.Ta,34.80.Pa a J I. INTRODUCTION tions using the concept of relative entropy as a distance 0 measure of correlations has been discussed in [14]. An 2 The nature and the optimal dynamical control of the experimental investigationof classicaland quantum cor- ] correlations present in a given interacting quantum sys- relations has been recently reported for a system of two h p tem are of interest to fundamental aspects of quantum polarizedphotonsinthepresenceofaphase-decoherence - physics butalso to the physicalimplementationofquan- non-dissipativechannel [15], and a sudden transitionbe- nt tum computation and algorithms [1–3]. A decade ago, tween classical and quantum correlations dynamics has a twoseparateworksreportedfromLosAlamos[4]andOx- been reported [15], a result in line with a recent theoret- u ford [5] set a formal frameworkthat allowedthe identifi- ical proposal [12]. q cationofquantumcorrelationsdifferenttoentanglement. [ Suchcorrelationsubiquituoslymanifestthemselvesinin- 3 teracting quantum systems, even in the absence of en- v tanglement, and it has been demonstrated that they can 5 play a key role in quantum computing protocols such as 7 DQC1 [6, 7], without the need to invoke quantum en- 2 tanglement. It has become clear that zero entanglement Westartwithabriefdefinitionofcorrelations(Sec. II) 4 does not imply classicality, and the task of classifying followed by the formulation of the dissipative dynamics . 2 the kind of total correlations present in a given system under consideration, in Sec. III. Our main findings are 1 intoquantumandclassicalpartshasledtothe introduc- detailed in Sections IV-VI as follows: We show that the 1 tionofquantifiers(differenttoentanglement)suchasthe entanglement of formation, rather than the concurrence, 1 : quantum discord (QD) [4]. This measure of the ‘quan- isthefreeofambiguitymetricthatshouldbeusedtocor- v tumness’ of correlations is defined in terms of the differ- rectly quantify the entanglement generated in the emit- i X encebetweenthe twopossiblewaystoobtainthemutual ters’ system (Sec. IV). The computed concurrence can information (MI) that arises in the quantum case. reach higher values than the mutual information, which r a More recently, it has been argued that the power of is the highest possible value for the total (classical plus the quantum computer can be attributed to both en- quantum) correlations. In Sec. V we demonstrate ana- tanglement and quantum discord which are distributed lytically,intheabsenceoflaserpumping,thatthereexist following a monogamic relation [8]. Further, other re- anentropyboundforwhichthequantumcorrelationsare portsonthe classificationandquantificationofquantum always greater than classical correlations, a fact which and classical correlations have been put forward [9–14]. disproves an early conjecture posed in [25]. In Section A unified view of quantum, classical and total correla- VIwegivenumericalexamplesforwhichsuchresultalso holds when the emitters are optically-controlled by an externallaserfield. This alsogivesa methodfor station- aryenhancementofthedegreeofcorrelationsbytailoring ∗ [email protected] thelaseramplitudesandthestrengthofthedipole-dipole † [email protected] interaction. 2 II. QUANTIFYING CLASSICAL AND where the conditional entropy S(ρ ) S(ρ ) = QUANTUM CORRELATIONS p S(ρ ), with probability pA|B=≡tr(ΠBAρ|{ΠBjΠ}B), j j AΠB j j AB j | j andthedensitymatrixafterthemeasurementshavebeen Webeginbyconsideringanopenbipartitesystem,say P performed on subsystem B is given by the molecular dimer AB. Its quantum dynamics can be described in terms of the total density matrix ρAB; the ΠBj ρABΠBj pbyarttihaelirnefdourmceadtiodnenasbitoyutmeaatcrhicoefstρhes=ubtsrys(tρems),isig,ijve=n ρA|ΠBj = tr(ΠBj ρABΠBj ). (5) i j AB A,B. Equation (4) gives the amount of information gained about A after a measure has been carried out on B. A. Mutual Information C. Classical correlations In classical information theory, the mutual informa- tion (MI) can be written in terms of the conditional A measure for classical correlations (CCs) was intro- entropy of two random variables, that is, for a system duced in Ref. [5]. This quantity is defined as the maxi- described by two classical random variables, the most mumextractableclassicalinformationfromasubsystem, information that one can gain from the two variables say A, when a set of positive operator valued measures is the MI. Let X and Y be two random variables with (POVMs) [16] has been performed on the other subsys- Shannon entropies H(X)=− xp(X=x)logp(X=x), and tem (B): H(Y)= p logp . The MI is written as: − y (Y=y) (Y=y)P CC(ρ )=max J(ρ ) = (6) AB ΠB AB ΠB P { j} { j } I(X :Y)=H(X)+H(Y) H(X,Y). (1) − max{ΠBj} S(ρA)− pjS(ρjA) , Using Bayes’s rules for the conditional probability (cid:16) Xj (cid:17) p(XY=y) =p(X,Y=y)/p(Y=y), the MI reads whereS(ρj )istheentropyassociatedtothedensityma- | A trixofsubsystemAafterthemeasure. Briefly,ameasure J(X :Y)=H(X) H(X Y), (2) − | ofclassicalcorrelationsshould incorporatethe following: i) CC(ρ ) = 0 if and only if ρ = ρ ρ , and where H(X Y) is the conditional entropy [16]. AB AB A ⊗ B | ii) CC must be non-increasing, and invariant under lo- cal unitary operations. In addition, the set of POVMs that maximize the classicalcorrelationEq. (6) is a com- B. Total correlations plete unidimensional projective measurement ΠB [17]. { j } If 0 , 1 defines the basis states for qubit B, the pro- Themutualinformationhasbeendemonstratedto de- jec{t|oris|cain}be writtenasΠB =1 j j , j =a,b,where scribe the whole content of correlations in a given quan- j ⊗| ih | a = cosθ 0 +eiφsinθ 1 , b = e iφsinθ 0 cosθ 1 , tum system, and at any time t, the total correlations | i | i | i | i − | i− | i and the optimization is carried out over angles θ and φ. present in such a system can be quantified by the quan- tum version of Eq. (1), the so-called quantum mutual information D. Quantum correlations I(ρ )=S(ρ )+S(ρ ) S(ρ ), (3) AB A B AB − 1. Quantum Discord where S(ρ) = tr(ρlog ρ) is the von Neumann entropy. The problem t−hat arise2s in the quantum case is that the Following Sec. IIB, and given that Eqs. (3) and (4) non trivial version of Eq. (2) is not always identical to arenotalwaysequivalent,ameasureofthe quantumness I(ρ ). Thisisbecause,quantummechanically,thecon- of correlations—the so-called quantum discord (QD) is AB ditional entropy implies that the state of a subsystem defined as the minimum of the difference between the (say A) is only known after a set of measures has been Eqs. (3) and (4). The QD then reads carried out on the other subsystem (B). Ollivier and Zurek [4] proposed a generalization of Eq. (2) assum- D(ρAB)=min{ΠBj} I(ρAB)−J(ρAB){ΠBj } = (7) ing a complete unidimensional projective measurement (cid:16) (cid:17) ma(dΠeBo)nΠsBub=sys1t.emThBe,gweintehraplrizoajetciotnorosf{EΠqBj.}(s2a)tiasfllyoiwngs S(ρB)−S(ρAB)+min{ΠBj}Xj pjS(ρA|ΠBj ), j † j another way of writing the quantum mutual information and the measure of total correlations becomes I(ρ )= AB P (whichingeneraldoesnotcoincidewiththatofEq. (3)): D(ρ ) + CC(ρ ), where D(ρ ) is a measure of AB AB AB ‘purelyquantum’correlations,andCC(ρ )isthemax- AB J(ρAB){ΠBj } =S(ρA)−S(ρA|{ΠBj}), (4) imum extractable classical information. Briefly, some 3 properties can be inferred from Eq. (7): i) For pure For mixed states, the EoF is written as [18] states, D = S(ρ ), ii) D = 0 if and only if ρ = B AB ΠBρ ΠB, iii) 0 D S(ρ ), iv) D is an entan- EoF(ρ)= (C(ρ)). (11) j j AB j ≤ ≤ B E glement monotone on pure states. Both QD and CC are Pantisymmetric by definition, i.e., they depend on what Consider, in decreasing order, the eigenvalues λi of the matrix √ρ ρ˜ , where ρ˜ =(σ σ )ρ¯ (σ σ ), subsystem the measures are taken on [9, 10, 14]. With- AB AB AB y⊗ y AB y ⊗ y ρ¯ is the elementwise complex conjugate of ρ, and σ outlossofgenerality,wetakequbitB tobethemeasured AB y is the Pauli matrix. The concurrence C is defined as one. C(ρ )=max 0,λ λ λ λ , (12) AB 1 2 3 4 { − − − } 2. Quantum Entanglement where the λ ’s areas defined aboveor,equivalently (also i in decreasing order), the square root of the eigenvalues Entanglementisaspecialandaveryparticularkindof of the non-Hermitian matrix ρABρ˜AB [18]. correlation[1],radicallydifferenttothequantumdiscord. Itisfundamentalinquantumphenomena,andaprecious resource for performing quantum computing [2, 3] and III. SINGLE QUANTUM EMITTERS quantumcommunicationprotocols[16]. Abipartitepure stateissaidtobeentangled ifatleasttwocofficientsofits We consider dipole-dipole (d-d) interacting quantum Schmidtdecompositiondonotvanish[16],andtheopera- emitters in contact with a surrounding medium that torsρ andρ havethesamenon-vanishingeigenvalues: acts as an environment. This is the case, for exam- A B they are equal to the square of the Schmidt numbers. A ple, of coupled terrylene single molecules embedded in state acting on Hilbert space is separable if it is of a para-terphenyl crystal for which symmetric and anti- AB the form ρ= k c ρA ρB,Hfor some k, where ρA and symmetric(super-andsub-radiant)states,aswellasd-d ρB are states oni=1 iaind⊗ i respectively. If ρ is ai pure interactionstrength,andsingleandcollectivedecayrates stiate, i.e., ρ =PΨHA Ψ HB, then it is separable if and have been measured [19]. For these, molecular quantum AB AB only if Ψ =| Ψ ih Ψ |. control and entangling quantum logic gates have been AB A B The |entaniglem|entio⊗f|formiation (EoF), arguably the designed [20], in addition to the control of the quantum most prominent measure of entanglement, allows an an- dissipative entanglement dynamics [21, 24]. Moreover, alyticalexpressioninthebipartitecase. TheEoFforthe quantum control experiments and femtosecond single- density matrix ρ is defined as the average of the entan- qubit gates involving single organic molecules at room glement of all the states composing ρ, minimizing over temperature have recently been performed [22]. the whole composition (over pure state decompositions) For the purpose of the analysis of the dynamics of the [18], correlationspresentedhere,weconsiderthattheemitters couple with interaction strength V, and have individ- EoF(ρ)=min piE(ψi), (8) ual transition frequencies νi (molecule i) in interaction with the quantized radiation field. We denote by 0 , i i X | i and 1 , i = 1,2, the ground and excited state of emit- i where E(ψ) = −tr[ρAlog2ρA] = −tr[ρBlog2ρB] is the ter i|, reispectively, and hence the single particle Hamil- entropy of each of the subsystems. The EoF can also tonian Hˆ = hν σ(1) hν σ(2). The two two-level be written in terms of the quantum mutual information emitters a0re se−pa2ra1tedz b−y t2he2vzector r and are char- as EoF(ρAB) = min pi 12I(ρiA : ρiB). Although the acterized by transition dipole moments1µ2ˆi 0i Di 1i , minimization is not trivial in general, the EoF has an withdipoleoperatorsD ,andspontaneouse≡mihssio|nr|ateis P i analytical solution for the two-qubit case [18]. Γ . The emitters are externally controlled by a coherent For pure states, the EoF can be written as i driving laser field which acts on each of the molecules with a coupling amplitude hℓ = µ E , and a fre- EoF(ψ)= (C(ψ)), (9) i − i · i E quency ω . The light-matter interaction Hamiltonian L wherethe concurrenceC(ψ)=|hψ|ψi|,andthe function HtˆuLde=ofhℓth(ie)(cσo−(hi)eerieωnLtt+drσiv+(ii)neg−iaωcLttin),gwohnermeoEleiciuslethieloacmaptelid- 1+√1 C2 at position r . We also allow for the molecular and laser E(C)=h 2− !, (10) detunings ∆i≡ν1−ν2, and ∆+ ≡ ν1+2ν2 −νL. where h(x)= xlog x (1 x)log (1 x) denotes the binary entropy−funct2ion−. i−s mono2toni−cally increasing, IV. DISSIPATIVE DYNAMICS OF QUANTUM E CORRELATIONS and it goes from 0 to 1 when the concurrence C ranges from 0 to 1. This is why C can be used as an entangle- ment quantifier. This said, it should be stressed that it TheemittersinteractionHamiltoniancanbewrittenin is only the EoFthat is an entanglement measure, and C the computational basis of direct product states i j | i⊗| i gets its meaning via its relation to the EoF. (i,j = 0,1) as Hˆ = Hˆ + Hˆ , where Hˆ is set S 0 12 12 4 by the dipole coupled molecules of interaction energy 0.25 Hˆ = hV σ(1) σ(2) + σ(1) σ(2) . Hence, the dis- 1.0 22..550 .0 0.2 0.4 0.6 0.8 1.0 x1 0 -1 sip1a2tive d2ynaxmic⊗s coxrresponydin⊗g tyo the ligth-matter in- 0.20 ) 0.8 1212....5050 teraction [2(cid:0)3, 24] can be described, w(cid:1)ith Hˆ =HˆS +HˆL, sno 0.15 C(Fo 00..46 00101.....05050 by the quantum master equation ita E x 10 -2 le 0.2 ρˆ˙ =−~i Hˆ,ρˆ +L(ρˆ), (13) rroC 0.10 0.00 .0 0. 2 0.4 C0.6 0.8 1.0 0.05 (cid:2) (cid:3) where the dissipative Lindblad super-operator reads [21] Γ 0.000 2 4 6 8 10 12 14 Vt L(ρˆ)= 1 ρˆσ(1)σ(1)+σ(1)σ(1)ρˆ 2σ(1)ρˆσ(1) − 2 + − + − − − + MI EoF Γ (cid:16) (cid:17) QD C 2 ρˆσ(2)σ(2)+σ(2)σ(2)ρˆ 2σ(2)ρˆσ(2) CC − 2 + − + − − − + Γ (cid:16) (cid:17) 12 ρˆσ(1)σ(2)+σ(1)σ(2)ρˆ 2σ(1)ρˆσ(2) FIG. 1. (Color online) Correlations dynamics for a pair − 2 + − + − − − + of interacting emitters initially in the doubly-excited state Γ (cid:16) (cid:17) |11i. MI (solid thick line, purple),QD (solid thin line, blue), 21 ρˆσ(2)σ(1)+σ(2)σ(1)ρˆ 2σ(2)ρˆσ(1) , − 2 + − + − − − + CC (doubly dashed line, green), C (dotted line, pink), EoF (cid:16) (cid:17) (dashed line, red). γ = 0.6884V. Main: Γ = 0.7818V iwnhgearendσl+(oi)we=rin|1giiPha0ui|l,ioapnedraσt−(oir)s=act|i0nigiho1ni|eamreitttehrei.raΓis-, T(rh12es=efλu0n/c8t)io.nIsnaseret:mFounnocttoionnicaalllryeliantciroenasbinetgwfereonmE0otFoa1n.dThCe. i zoompointsoutthedifferencesbetweentheEoF(∈[0,0.025]) and Γ = Γ γ are the individual, and the collec- 12 ∗21 ≡ and C (∈[0,0.1]). tive spontaneous emission rates, respectively. The latter arises from the coupling between the emitters through the vacuumfield. The interactionstrengthandthe inco- rates V Γ which in the main graph corresponds to herent decay rate read ∼ r =λ /8. The correlations last longer times and reach 12 0 V = [µˆ µˆ (µˆ ˆr )(µˆ ˆr )]cosz + (14) higher values the smaller Γ. As expected, the quantum 1 2 1 12 2 12 C − · − · · z mutualinformationisalwaysgreaterthanthe restofthe (cid:16) cosz sinz correlations. Fortheemittersintheinitialdoublyexcited [µˆ µˆ 3(µˆ ˆr )(µˆ ˆr )] + , 1· 2− 1· 12 2· 12 z3 z2 state 11 we find that the entanglementquantifiers EoF | i h sinz i(cid:17) and C exhibit a ‘sudden birth’ of entanglement [21, 27] γ = [µˆ µˆ (µˆ ˆr )(µˆ ˆr )] + (15) C 1· 2− 1· 12 2· 12 z at a time τ 5/V (main graph); this behavior is ‘short- ∼ (cid:16) cosz sinz ened’asthedecayrateincreases(notshown). Incontrast [µˆ µˆ 3(µˆ ˆr )(µˆ ˆr )] , 1· 2− 1· 12 2· 12 z2 − z3 to this, the quantum discord is always greater than en- h i(cid:17) tanglement(measuredbyEoFandC),andonlyvanishes where z = nk r , n denotes the medium refraction in- 0 12 aroundatime 3/V: onlyaroundasmalltimeneighbour- dex, k0 = ωc0, ω0 = ω1+2ω2, and C = 3√8Γπ1Γ2. hood centered at this value do the classical correlations (CC) become slightly greater than quantum correlations (QD). For the remaining parts of the evolution, QD > A. Identical emitters without laser field CC.Weshallreturntothispointlater,whereweaddress thequestionofwhetheritisatallfeasibletocharacterize Webeginwiththesimplestcasescenario,thatofiden- a dynamical order relation between classical and quan- tical emitters or qubits, ∆ = 0, no external laser field: tum correlations. We remark that even in the absence ℓ =0, and ∆ =(ν +ν )−/2. of entanglement (t < τ) there are both, quantum and i + 1 2 We consider the emitters dipolar moments to be par- classical correlations present in the dynamics, and that allel amongst them and perpendicular to the interqubit the QD is the only quantum correlation present in such position vector r . In particular, if r λ , the a time frame. 12 12 0 ≪ maximum (minimum) interaction strength and decay rate are obtained for parallel(perpendicular) dipole mo- ments as hV = 3h√Γ1Γ2 [µˆ µˆ 3(µˆ ˆr )(µˆ ˆr )], B. Entanglement of Formation vs Concurrence 8πz3 1· 2− 1· 12 2· 12 γ = √Γ Γ µˆ µˆ . However, our calculations make di- 1 2 1 2 · rect use of Eqs. (14) and (15), since we investigate the The concurrence has been widely used as an entan- dependence of correlations on the interqubit separation. glement metric due to the fact that, according to Eq. Theeffectofdissipationonthecorrelationsdynamicsis (11), the function EoF(C) is monotonically increasing initially showninFig. 1forthe decayrateγ =0.6884V, and ranges from 0 to 1 as C goes from 0 to 1. This said, and Γ = Γ Γ. Here, weakly coupled qubits are con- the difference in value between these quantifiers can be 1 2 ≡ sideredforinteractionstrengthsoftheorderofthedecay significantandthisisreflectedintheinterpretationofthe 5 0.25 in the system at any given time, concurrence cannot be a good quantifier of entanglement in this context. This 0.20 is an important point to bear in mind when performing s n quantum information protocols, where entanglement is o0.15 ati required as a physical resource. As already pointed out, el it is only the EoF that is an entanglement measure, and r0.10 r the significance of C is actually revealed through its re- o C lation to the EoF. Hereafter, we shall consider the EoF 0.05 as the quantifier of the emitters’ entanglement. 0.00 0 2 4 6 8 10 12 14Γt V. HIERARCHY FOR THE EMITTERS CLASSICAL AND QUANTUM CORRELATIONS FIG. 2. (Color online) Same parameters and notations as in Fig. 1 (main), but in the presence of an external continuous laser field ofamplitudel=0.4Γ in resonancewith thequbits Regardingtheconjecturethatclassicalcorrelationsare transition energies: ∆+ =0 (ωL=ω1 =ω2 ≡ω). The initial always greater than quantum correlations, and the re- condition is taken to be thedoubly-excitedstate |11i. cent experimental evidence that the opposite situation can also arise for a specific scenario in bipartite entan- gled photons subjected to a decoherent source (thus dis- correlation dynamics. The EoF changes at a slower rate proving the conjecture) [15], we ask whether it is at all than C at values close to 0; for instance, in the interval feasible,andunderwhatcircumstances,tofindandorder C [0,0.1], the EoF is much smaller than C, as can be hierarchy for the emitters’ classical and quantum corre- see∈n in the zoom of the inset of Fig. 1. The whole range lations. for the metric is plotted in the inset of Fig. 1, where The following analysis of dynamical behavior for the the dashed line is a guide to the eye and indicates the correlations is given in terms of the inherent physical equality EoF = C, to better appreciate the differences propertiesoftheemitters,suchasthecollectivedamping between EoF and C (solid curve). (decay) rate γ, and the d-d interaction V. We demon- strate, in a simple manner, that the dimer system can As for the physical interpretation of the correlations be naturally dominated by quantum correlations at all dynamics, we relate the above discussion with that pre- times, that is, that QD or EoF are greater than classical sented in Fig. 1. Here, the dotted (pink) curve denot- correlations during the whole time evolution. For this, ing C gives the impression that entanglement is almost we solve the emitters’ master equation (13) for identi- the only quantum correlation present in the system for cal qubits (ν =ν ), with emitters only coupled through timesaroundorgreaterthantwicethesuddenbirthtime 1 2 their dipolar interaction (there is no laser excitation). (t > 2τ), since for this, C and QD are almost, or actu- The calculation gives, for the non-trivial density matrix ally, superimposed. The solid (red) curve for the EoF elements, the following result: clearly shows that most quantum correlations, however, in the system are far from entanglement. As a matter 1 ρ = e (γ+Γ)t 2e(γ+Γ)t 2f(1 e2γt)cosϕ of fact, entanglement, as measured by EoF, contributes 00,00 − 2 − − very little to the full quantum correlations dynamics. h e2γt 1 This point is made even more evident when we con- − − sider, for example, the influence of an external continu- 1 i ous wave laser field of amplitude l, and frequency ωL in ρ01,01 = 4e−Γt e−γt 1+e2γt+2f(1−e2γt)cosϕ resonance with the qubits transition energies: ωL = ω (2 h4α)c(cid:0)osΘ+4fsinϕsinΘ (cid:1) (below we shall return to the discussion of the laser ef- − − fects), as shown in Fig. 2. The physical parameters and 1 i ρ = e Γt e γt 1+e2γt+2f(1 e2γt)cosϕ the notations are the same as those shown in the main 10,10 4 − − − graphinFig. 1. Theresultthatmarkstheimportanceof h (cid:0) (cid:1) +(2 4α)cosΘ 4fsinϕsinΘ usingEoFasthemetricofentanglementisthat,whilein − − Fig. 2theEoF(redcurve,justbelowthediscordwhichis ρ =ρ = i 01,10 ∗10,01 plottedinblue)showsabehaviorthatcanbeunderstood 1 following the lines of reasoningin Fig. 1 (but now in the e−Γt e−γt 1 e2γt+2f(1+e2γt)cosϕ 4 − presence of a laser field; note the differences), the con- h (cid:0) (cid:1) currence (dotted pink curve) exhibits a ‘birth’ of entan- 2i 2fsinϕcosΘ+(1 2α)sinΘ , (16) − − glement (following the qualitative behavior of the EoF), (cid:0) (cid:1)i justaroundatimethatmarksthebeginningofthe dom- where f = (1 α)α, Θ = 2Vt, ϕ is a relative phase, − inance ofclassicaloverquantumcorrelations,thatgrows and α defines the set of initial states αϕ √α01 + p | i ≡ | i well beyond the mutual information: since the latter is, eiϕ√1 α10 ,0 α 1,achoicemotivatedfromstates − | i ≤ ≤ by definition, the measure of total correlations present thatcanbe‘naturally’preparedinthequantumemitters, 6  Returningtoourcase,quantumcorrelationsarealways greater than classical correlations. This can be seen in    Fig. 3, where we have plotted the entropies condition   that make D(ρ ) CC(ρ ). With the parametriza-  AB ≥ AB  tion x = exp[ (Γ+γ)t], in Fig. 3 we show that the   quantumdiscor−dis greaterthanclassicalcorrelationsfor    thewholedynamics. Theequalityisonlyobtainedatthe    iinnittihael tliamtteer(,puthreesstyasttee)mantdenadtsthtoe ‘afinsaepl’a(rta→ble∞an)dtimune-;  correlated state. In a similar manner, we can also show    x    that the entanglement of formation EoF(ρ ) is always AB greater than the classical correlations. FIG. 3. (Color online) Hierarchy of correlations: D(ρAB) ≥ The same hierarchy of correlations can be proven by CC(ρAB) for all t. x = exp[−(Γ +γ)t]. The top (solid) direct calculation. For the density matrix (17), the con- curve is the quantity 2minBj{S(ρA|{ΠBj})}, and the bottom ditional entropy, after the measurement, admits an ana- (dashed) curvecorresponds to theentropy S(ρAB). α=1/2, lytical expression and therefore both CC and D can be ϕ=0; S(ρA)=S(ρB). analytically computed: the conditional entropy has two possible solutions: ρ e.g., separable states α = 0,1, and symmetric (α = 1/2, S(1)(ρ )= ρ log 00,00 and ϕ = 0) and anti-symmetric (α = 1/2, and ϕ = π) A|{ΠBj} − 00,00 2(cid:18)ρ00,00+ρ10,10(cid:19) entangledstates. Forthe classofαϕ initialstates,whose ρ10,10 ρ log , time-evolution has been derived in Eq. (16), the density − 10,10 2 ρ +ρ matrix adopts the structure (cid:18) 00,00 10,10(cid:19) 1 1 ρ00,00 0 0 0 S(2)(ρA|{ΠBj})=−2(1−ξ)log2 2(1−ξ) 0 ρ ρ 0 (cid:18) (cid:19) ρ(t)= 00 ρ∗00110,,1001 ρ10010,,1100 00, (17) −21(1+ξ)log2 12(1+ξ) , (20)   (cid:18) (cid:19)   whereξ = (1 2ρ )2+4ρ 2. Then,DandCC and thus the correlations can be analytically calculated. 10,10 01,10 − | | are determined by the minimum of the two expressions In particular, we are interested in finding a dynamical p in (20). order hierarchy amongst classical and quantum correla- Forα= 1,wefindthatS(2)(ρ ) S(1)(ρ ) tions. Equations (6) and (7) allow to show that there is 2 A|{ΠBj} ≤ A|{ΠBj} an entropy bound: is always satisfied and thus S(2)(ρ ) is the entropy A ΠB |{ j} used for computing the correlations. D(ρ ) CC(ρ ), (18) AB AB The concurrence for the density matrix (17) takes the ≥ simpleformC(ρ)=2max 0, ρ . Theentanglement 01,10 which reads { | |} of formation is then directly computed from Eq. (11). InFigure4weshowtheevolutionofthesethreecorre- S(ρ )+2min S(ρ ) S(ρ ) S(ρ ) 0. B Bj{ A|{ΠBj} }− A − AB ≥ lationsasafunctionofthenormalizedtimeτ =(Γ+γ)t. (19) The exponential decay is due to the particular choice Whenthesystembeginsinthemaximallyentangledsym- of maximally entangled initial states α = 1/2 for (a) metric Bell state α = 1, ϕ = 0, the density matrix be- the symmetric state ϕ = 0 (main), and (b) the anti- 2 comes independent of the dipolar interactionV, and γ is symmetric state ϕ=π (inset). Although the qualitative justanadditionaldecayratethatmakesthesystemdecay behavioroftheinsetissimilartothatofthemaingraph, with the new rate Γ+γ; furthermore, S(ρA)=S(ρB). wenoticethatsuchstateevolutionhasamuchslowerde- The independence of ρˆ with respect to V can be cayrateofthe correlations;this happensatthenew rate formally understood by recalling that the considered Γ γ. This result is in agreement with what would be Bell states are part of a bigger family of states—ρM, ex−pected for sub-radiant states and explains the higher which have maximally mixed marginals, i.e., trA(ρM) = value of the correlations at all times, when compared to mtrBat(rρixM;)he=nc21eIS2×(ρ2,A)w=ithSI(2ρ×B2) b=ei1n.gStuhcehidsteantteitsyca2n×b2e theTihneitiaablosvuepearn-raalydsiaisntisϕe=xt0ensdtaedte.to all initial states writtenasρ = 1 I+ 3 h σ σ [10,28],whereσ 0 α 1, see Figs. 5 and 6. We also find, for iden- M 4 i=1 i i⊗ i i ≤ ≤ arethe Paulimatrices andthe coefficients h satisfy tical qubits, that quantum correlations are stronger and i (cid:0) P (cid:1) ∈R constraints suchthat ρ is a well defined density opera- morerobusttodecaythanclassicalcorrelations. Thedy- M tor. Itcanbe shownthatfor the whole family of density namical behavior is richer than the exponential decay of matricesρ , the Markoviantime evolutiongivenby Eq. Fig. 4 because different initial conditions for the den- M (13) is completely independent of the interaction V. sitymatrixallowfordifferentrolesforthe corresponding 7                  τ      α   Γ      τ       FCisICGth(.ρeA4s.Bo)li(dtChoceluodrrovueo,bnlltyihnede)aEsDhoeFydn(aρcmuArBivc)es.iosTfthcheoerinrdeiatlaisahtlieosdntsac.tuerDvise(,gρiAavnBend)   by the super-radiant state αϕ=0 = 1/2 (main), and the sub-  radiant state αϕ=π =1/2 (inset).         Γ          α          Γ t  Γ t      FIG.6. (Color online) Quantumdiscord dynamicsasa func-  tion of the initial state preparations (a) αϕ=0, and (b) αϕ=π  α states(main). α=1/2givethesuper-radiantandsub-radiant states, respectively. The inset notations are as in Fig. 5, for  Γ t α=0. γ =0.91Γ, and V =2.03Γ.   FIG.5. (Color online) Quantumdiscord dynamicsas afunc- moleculesinadispersiveenvironmentforwhichtheircol- tion of the initial αϕ=0 state preparations. The inset shows lective rate is much less than both the spontaneous indi- thedynamicsof correlations D(ρ ) (solid blue), EoF(ρ ) vidualemissionratesandthedipolarinteraction[19,22]; AB AB (dashed, red), and CC(ρAB) (doubly dashed, green), for the or in a pair of emitters that interact with a plasmonic (separablestate)α=0-plane. Thereisnocollectivedamping: channel where the collective stregths V and γ can effec- γ =0, and V =2Γ. tivelybe switchedonandoffasthedistancebetweenthe emitters is varied, a mechanism that produces an effec- tive phase difference between V and γ [29]. collectiveparameters. Forα=1/2,thelasttermsofma- The main graph in Fig. 5 plots the quantum discord 6 trix elements ρ01,01 and ρ10,10, and the imaginary part dynamicsasafunctionofαfortwodifferentsetofinitial of ρ01,10, are non-zero and hence the dynamics exhibits state preparations; the same qualitative behavior is ob- oscillationsdependingonthevalueofthed-dinteraction, servedforCCandEoF(notshown). InFig. 5weseethat which is weak for the case of the diluted molecules first oscillations due to the d-d interaction begin to appear considered;accordingly,weconsiderinthefirstplacethe and to coherently affect the dynamics of the quantum scenario of no collective decay due to γ (decay contri- discordfor α values closer to 0 or 1; this is so because in butions are only those giving by Γ), as shown in Fig. 5 these cases the amplitude of the trigonometric functions for the initial state dynamics αϕ=0. We also consider, in the matrix elements Eq. (16) are closer to their max- in Fig. 6, the situation where γ = 0 plays an important ima. The inset in Fig. 5 shows this for the discord, the 6 role, for the initial state preparations (a) αϕ=0, and (b) EoF and CCs for the plane α=0 (separable state 10 ). | i αϕ=π. The correlation dynamics for these two cases is For the maximally entangledstate (α=1/2)all the cor- shown in Figs. 5 and 6. relationshavea monotonic exponentialdecay,indicating Whenthecollectivedecayrateisnoteffectivelypresent anindependencefromtheinteractionV. Forotherinitial in the dynamics, as, for example, in the case of diluted states set by α, this decay exhibits oscillations which in- 8 creasetheir amplitude as α tends to the separablestates 01 or 10 . The oscillations eventually vanish and the | i | i correlations decay to 0 for all α. Although the main graphs of Fig. 5 only plot the discord, our calculations show that the correlations hierarchy CC < EoF < QD  holds for all time t and for all α (see the inset for the  case α=0).   The case γ = 0 (Fig. 6) plays an important role in   the system’s ev6 olution. Here, two emitters interact via   /λ0  a dipole force throughout all modes of the electromag-  netic field in the vacuum state: the collective (incoher- Γ   ent) decay γ (15) has to be taken into account, and we  introduce the full expressions given for γ and V, and their relation to the Einstein coefficients, dipole orienta- tion, and the dispersive medium, as given by Eqs. (14) and (15). When compared to Fig. 5, the effect of the collective decay rate translates into a much longer time  for the survival of correlations. In fact, the inset, which   is common to Figs. 6(a) and (b), shows that quantum    correlations decay very slowly and thus last for a much  longer time than that obtained in the absence of collec-   /λ0  tive decay in Fig. 5. This said, in Fig. 6 there is a  Γ  clear distinction between the initial (a) α , and (b) ϕ=0  α states. In the latter, the correlations last for a far  ϕ=π longer time before their final decay; in particular, the case of the anti-symmetric α = 1/2 state exhibits a ϕ=π veryslowdecayofthediscord,ascomparedtothecaseof the symmetric α = 1/2 state (note the difference in ϕ=0 thetimescaleofthetwographs). Thisisalsoreflectedin  the inset of Fig. 6(b), for the case α=0. The difference   in physicalbehaviorbetween graphs(a) and(b) owesits    origin to the nature of the super-radiance phenomenon,  which is manifest here in graph (b) for α = 1/2. In any   / λ0  case, in both graphs, we have quantum correlations al-  Γ  ways greater than classical correlations.   By computing the correlation dynamics for two emit- ters of identical dipole moments (parallel to each other), FIG. 7. (Color online) Correlations D, EoF, and CC, as and perpendicular to the interqubit distance r , basic 12 a function of time and the interqubit distance, the latter features can be identified. The relevant values of the in- normalised by the emitters characteristic wavelength λ0 = teraction energy and collective decay rate are given by 2π/k0 = 2πc/ω0. The initial state preparation corresponds Eqs. (14) and (15): in Fig. 6 we consider γ = 0.91Γ to α=0. and V = 2.03Γ, which corresponds to a distance r = 12 0.108λ . Even though we now have less oscillations in 0 the discord, a higher degree of correlations than those correlations dynamics: the degree of correlations is in- of Fig. 5 is obtained due to the presence of collective creasedasthecollectivedampingisaugmented. Figure7 damping. Note that for initial configurations of the type showsthe behaviorofthethreerelevantcorrelationsasa α 1orα 0,quantumdiscordandentanglementhold functionofthenormalizeddistancer /λ ,intheregime → → 12 0 a greater value than classical correlationsand, as can be where V goes from 0.2Γ to 2.6Γ, and 0.1 γ 0.9, or, seen from Figs. 5 and 6, the discord is the correlation equivalently, for 0.1 r12 0.4. In Fig.≤7 Γwe≤consider that most benefits due to this collective effect. The en- theinitialconditionα≤=λ00,≤followingtheresultsshownin tropy condition Eq. (19) is always satisfied, and hence Figs. 5and6,assuchastatemakesthestrongestcasefor quantumcorrelationscontinue to be alwaysgreaterthan the influence of the collective parameters. As expected, classical correlations: CC <EoF <QD, for all α. a signature in the correlations dynamics shows up after Accordingtotheaboveinspection,acalculationabout theinteractionstrengthbecomeseitheroftheorderofor thewaythecollectiveparametersinfluencethedynamical greaterthanthedecayrateΓ,whichoccursfor r12 0.2. λ0 ≤ behavior of correlations is in order. From Eqs. (16) it is As in the previous case, the most robust correlation is clearthatavariationofV producesachangeintheoscil- the quantum discord and the classical correlation is the lationsofthedensitymatrixelementsandthereforeinthe leastassistedby the collectiveeffects throughoutthe en- 9                                                         Γ         Γ      FIG. 8. (Color online) Dynamics of correlations for emitters FIG. 9. (Color online) Emitters correlations (notation as in under resonant laser field excitation: νL = ν0 = ν; ℓi = ℓ. Fig. 8) under resonant laser excitation ν = ν. Parameters NotationsforD,EoF,andCC areasinpreviousfigures. (a) L are as in Fig. 8. Initial state: antisymmetric sub-radiant ℓ=1.5Γ, γ =0.91Γ, V =2.03Γ; (b) ℓ=10.0Γ, γ =0.91Γ, V = 2.03Γ; (c) ℓ = 10.0 Γ, γ = 0.97Γ, V = 10.45Γ. The αϕ=π =1/2. initial state is the super-radiant αϕ=0 =1/2. VI. EMITTERS CORRELATIONS UNDER COHERENT LASER EXCITATION We now turn to the external all-optical control of the quantumemitterscorrelationdynamics. Weconsiderthe tire dynamics. Hence, the degree of correlations can be case of a continuous laser field ℓicosωLt locally applied dynamically enhanced when the collective (incoherent) to each of the emitters, where ℓi denotes the amplitude damping is of the order of the (weak) interaction energy of the laser field acting on qubit i, and ωL is the laser and comparable to the spontaneous decay rates of the frequency. Without loss ofgenerality,we considerqubits single quantum emitters. of the same transition frequency ω, under the resonant condition ω =ω =ω. L 0 Thesolutionofthemasterequationdescribingthe dy- Lindblad [25] conjectured that for a general density namical system is fully numerical because the structure matrixρ,classicalcorrelationsaregreaterthanhalfofthe ofthedensitymatrix,whentakingintoaccountthelaser mutual information and as such, they are alwaysgreater field, does not admit a general analytical solution. than quantum correlations. He showed that for an inco- Figure 8 shows the results for initial state preparation herent mixture state, CC(ρ)=I(ρ), and that for a pure in the super-radiant 1 (01 + 10 ) state. For external √2 | i | i state, CC(ρ) = 1I(ρ), it seemed natural to conjecture laserfieldamplitudebelowtheinteractionenergyV (Fig. 2 that CC(ρ) 1I(ρ) [25]. Simple numerical examples, 8(a), and γ, and V as in Fig. 6), the emitters dynamics ≥ 2 for several different configurations of the constants h in exhibits a change inthe correlationshierarchyand, after i ρ ,confirmthatforthecasesofincoherentmixturesand a brief transient period, quantum correlations go below M pure states, this result holds. There are states for which the classical ones. This change in physical behavior, for the conjecture does not hold, however: it is straightfor- weaklaserexcitation,occursinalmostthewholedynam- wardtoshowthatforthesetofBellstates h =1(pure ics. Eventhough suchlaserexcitationcanoverdampthe i | | correlated states), I(ρ ) = 2, hence CC(ρ ) = 1. evolutionofthequantumcorrelationswellbelowtheclas- Bell Bell On the other hand, for h =h = 0, and h 1 (inco- sical correlations, such evolution can be controlled and 1 2 3 | |≤ herent mixture), CC(ρ ) = I(ρ ). For example, the degree of the quantum correlationscan be optimised incoh incoh for h = 0.6, CC(ρ ) = I(ρ ) = 0.278. Finally, byappropriatelytailoringboththeamplitudeofthefield 3 incoh incoh thechoiceh =h =0.8,h = 0.6givesI(ρ )=1.078, andthedipolarinteractionbetweenthequbits;thelatter 1 2 3 M − and quantum correlations D(ρ ) = 0.547> CC(ρ ) = being obtained by varying the interqubit distance. The M M 0.531. results plotted in Fig. 9(a), for the initial state prepara- 10 tion in the sub-radiant 1 (01 10 ) state are in clear  √2 | i−| i    contrastwith those of Fig. 8(a), and, althoughquantum correlations are no longer greater than classical corre-    lations, they become of the same order throughout the   entiredynamics. Theearlystagedisentanglement(ESD)   experienced by the emitters in Fig. 9(a) is delayed for   about ten times that of Fig. 8(a). It is worth pointing        outthateventhoughthereisnoentanglementafterESD  occurs, from this point on there are non-zero classical       and quantum correlations present in the evolution. The Γ  order hierarchy for the anti-symmetric state is such that QD > CC at all times. This is not always the case for FIG. 10. (Color online) Correlations dynamics under the ef- the symmetric state, and the hierarchy depends on the fectofanon-resonanceexternalfieldωL=ω0,withamplitude ℓ=10Γ; V =10.45Γ. (a) Emitters with collective damping interplay between V and ℓ. ‘switchedoff’;(b)Emitterswithidenticalandparalleldipoles, InFigs. 8(b)and9(b)wehavekepttheinteractionen- with γ =0.97Γ. Initial state: α=0. ergyconstant(fixedcollective parametersV, andγ) and increased the value for the laser amplitude well above V (ℓ = 10Γ). Now the discord evolves as the most ro- quantum discord more robust than classical correlations bust correlation. For the case of the symmetric state, emerges. This kind of quantum information or correla- however, not always QD > CC (initially, CC > QD), tions is of an entirely different nature to entanglement, whereas for the anti-symmetric state, we always have which decreases to zero much earlier than QD and CC. QD > CC. The dynamics becomes more complex with We conclude that also for quantum emitters under laser theinclusionofthelaserfieldandadependencewiththe pumping,quantumcorrelationscanbecomegreaterthan d-d interaction now appears, in contrast to the case of classicalcorrelationsand that this feature is more‘natu- no laser excitation. This is seen in Figs. 8(b) and (c): ral’ in antisymmetric sub-radiantstates, thus disproving forafixedlaseramplitude,thevariationinthed-dinter- Linblad’s conjecture [25]. action produces a qualitative change which favours the degree ofdiscord,above classicalcorrelationsand entan- glement,asV increases. Theearlystagedisentanglement VII. SUMMARY doesn’t show major changes. Now we turn to Figs. 9(b) and (c): quantum correlations are always higher than We have given a thorough analysis of the dissipative classical correlations and the effect of an increment in dynamics of total, quantum, and classical correlations V noticeably prolongues the lifetime and the degree of for a set of interacting quantum emitters characterized correlations. For example, ESD in graph 9(c) is around by their coupling energy and their collective and indi- 50 times that of 8(c) (notice the difference in the corre- vidual decay rates, and analytically demonstrated the sponding time scales). conditions under which quantum correlations are always In short, the same laser field can reverse the behavior greater than classical correlations, and the role played observed in Figs. 8(a) and 9(a) by appropriately chang- by entanglement during such dissipative quantum evo- ing the interqubit distance and by shining the laser with lution. We have shown that it is the entanglement of higher intensity. This control procedure replicates the formationthe metrictobeusedasthe emmiters’quanti- earlierresult ofquantum correlationsbeing above classi- fier of entanglement and that the use of concurrence for calcorrelations. Itisalsoclearthatatalltimes,quantum thispurposecangivemisleadingresultsabouttheactual correlationsarefargreaterthanentanglement. Thepres- degree of generated entanglement. We have also shown ence of the external field perturbs the asymptotic decay that the dynamics of correlations followed by the emit- ofentanglementand inturn produces anearlystage dis- ters can be controlled and engineered in the presence of entanglementphenomenon[21,27]whichisverifiedinall acoherentlaserfield. We havequantifiedandillustrated the graphs of Figs. 8 and 9. theinterplaybetweenentanglementandquantumdiscord Forcompleteness,inFig. 10wecomparethegraphsfor stressingtheimportanceofthepersistentquantumcorre- aseparablestateasthe initialcondition,andfor(a)‘ab- lations in several different scenarios where entanglement sent’γ and(b)‘active’γ. The correlationsdynamicshas vanishes. beenplottedforV =10.45Γ,withlaseramplitudeℓ V, In cases where classical correlations are greater than ∼ following the parametersof graph8(c) which give a high quantumcorrelations,a coherentcontrolmechanismcan degreeofdiscordforsymmetricstates. Unlikeinthepre- be implemented so that the latter are enhanced and be- viouscases,wenowobtainoscillationsinthediscordand comegreaterthantheformer. Thisresultiscorroborated entanglement. Stationary correlations, which are null in for drivenemitters inthe case ofstronglaser-dipolecou- the absence of collective damping (see Fig. 10(a)), can pling: by preparing the initial symmetric state, once en- now be obtained due to the presence of damping γ in tanglementvanishesthereisarapidstabilisationofcorre- Fig. 10(b). Interestingly, as the correlations stabilise, a lationsandthediscordbecomesgreaterthantheclassical

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