Correlations in emitters coupled to plasmonic waveguides
CCorrelations in emitters coupled to plasmonic waveguides
C E Susa , J H Reina and L L S´anchez-Soto Departamento de F´ısica, Universidad del Valle, A. A. 25360, Cali, Colombia Departamento de ´Optica, Facultad de F´ısica, Universidad Complutense, 28040 Madrid,SpainE-mail: † [email protected] Abstract.
We report on quantum, classical, and total correlations in a set of distant quantumemitters coupled via their interaction with the plasmon modes of a one-dimensional waveguidedriven by an external laser field. The coupling of the emitters with the plasmonic modes andits influence on the collective decay rate suggest that entanglement does not play a significantrole in the qubit dynamics. Rather, discord is the quantity that matters and should be harnessedas a resource.PACS numbers: 03.65.Ud, 03.67.Lx, 42.79.Gn
1. Introduction
Entanglement is a cornerstone of quantum theory [1–4]. It has strong practical implicationsfor futuristic quantum technologies [5–10] and, consequently, it has been investigated in awide variety of physical systems [11–14]. For a thorough description of this phenomenon,a formal framework has been introduced over the past decade to quantify quantum [15] andclassical [16] correlations in a physical system. The former, being of a more general characterthan entanglement, has also been hinted to play a role in quantum information processingthrough the so-called mixed state-based quantum computing [17, 18].Quantum emitters (e.g., single molecules [19, 20] or quantum dots [21–23]) coupledto surface plasmons of conducting nanowires or waveguides have been seen as a promisinghardware for quantum information processing because of the strong coupling that can beachieved between the emitters and the plasmons [21]. The collective spontaneous decayand the plasmon-mediated emitter-emitter coupling have been investigated for a two-qubit(emitter) system close to a nanowire [24, 25], as has the resonance energy transfer mediatedby different plasmonic nanowaveguides [26]. Entanglement of two qubits, mediated bya V-groove plasmonic waveguide, has recently been theoretically studied [27, 28] andquantified via the concurrence [29]. We address this problem within a more generalframework: we calculate the entire spectrum for the entanglement, classical, quantum, andtotal correlations [30–32] for emitters coupled to plasmonic modes.We demonstrate that it is not entanglement but the quantum correlations (measured bythe discord) that provide the relevant robust features that could be harnessed for plasmon-assisted information processing. Our results support the conclusion that the entanglement offormation [29] is a natural metric and that, in the context of entropic measures, quantificationvia the concurrence might be inappropriate. a r X i v : . [ qu a n t - ph ] D ec orrelations in emitters coupled to plasmonic waveguides
2. Quantum emitters, plasmon modes and correlations
We start by briefly delineating the model. We are dealing with two two-level atoms fixedat positions r i ( i = , ω i and separated by the vector r . Wedenote by | i (cid:105) and | i (cid:105) the ground and excited states of the emitter i , with associated transitiondipole moments ˆ µ i ≡ (cid:104) i | D i | i (cid:105) , D i being the corresponding dipole operators.The emitters are embedded in a medium of refractive index n and interact via a dipole-dipole coupling, so that the system Hamiltonian H S can be written as H S = H + H , (2.1)where the free and interaction Hamiltonians are H = ¯ h ω σ ( ) z + ¯ h ω σ ( ) z (2.2) H = ¯ hV (cid:16) σ ( ) x ⊗ σ ( ) x + σ ( ) y ⊗ σ ( ) y (cid:17) . The strength V depends of the configuration of the interacting dipoles and σ ( i ) are the Paulioperators.In addition, an external laser field of frequency ω L drives the emitters, which we representby H L = ¯ h (cid:96) ( ) (cid:16) σ ( ) − e i ω L t + σ ( )+ e − i ω L t (cid:17) + ¯ h (cid:96) ( ) (cid:16) σ ( ) − e i ω L t + σ ( )+ e − i ω L t (cid:17) . (2.3)Here (cid:96) ( i ) is the strength of this coupling, and is given by ¯ h (cid:96) ( i ) = − ˆ µ i · E i , with E i being theamplitude of the coherent field acting on the i th emitter, and σ ( i )+ = | i (cid:105) (cid:104) i | , and σ ( i ) − = | i (cid:105) (cid:104) i | are the raising and lowering Pauli operators acting on the i th emitter. We next assume that the emitters are coupled with a bath of the radiation field, so that thecorresponding dissipative dynamics is given by the total Hamiltonian H = H S + H L , by meansof the quantum master equation [20]˙ ρ = − i ¯ h [ H , ρ ] − ∑ i , j = Γ i j (cid:16) ρσ ( i )+ σ ( j ) − + σ ( i )+ σ ( j ) − ρ − σ ( i ) − ρσ ( j )+ (cid:17) , (2.4)Without loss of generality, we set Γ ii ≡ Γ i = Γ , and Γ i j = Γ ∗ ji ≡ γ ( i (cid:54) = j ), the individual, andcollective spontaneous emission rates, respectively.The explicit form of the individual emitter decay rate Γ i , as well as the collective decayrate γ associated to the dipole-dipole (qubit-qubit) interaction V depend on the particularphysical setup under consideration. For the case of interacting ‘bare’ quantum emitters, wehave V = √ Γ Γ (cid:26) ˆ µ · ˆ µ − ( ˆ µ · ˆ r )( ˆ µ · ˆ r )] cos zz + [ ˆ µ · ˆ µ − ( ˆ µ · ˆ r )( ˆ µ · ˆ r )] (cid:18) cos zz + sin zz (cid:19)(cid:27) , (2.5) orrelations in emitters coupled to plasmonic waveguides γ = √ Γ Γ (cid:26) [ ˆ µ · ˆ µ − ( ˆ µ · ˆ r )( ˆ µ · ˆ r )] sin zz + [ ˆ µ · ˆ µ − ( ˆ µ · ˆ r )( ˆ µ · ˆ r )] (cid:18) cos zz − sin zz (cid:19)(cid:27) , where z = nk r , k = ω / c , and ω = ( ω + ω ) / ( r , r ) that describes the electromagneticinteraction between the two dipole moments (emitters) µ , of frequency ω = ω ≡ ω asfollows (see [25] for a discussion of the evaluation of these terms) V pl = πε c ¯ h P (cid:90) ∞ ω Im [ µ ∗ G ( ω , r , r ) µ ] ω − ω d ω , Γ pl i j = ω ε c ¯ h Im [ µ ∗ i G ( ω , r i , r j ) µ j ] , (2.6)where the superscript “pl” indicates that the collective parameters are now modified by theinteraction with the plasmonic waveguide. As before, we set Γ pl ii = Γ pl i , and Γ pl i j = ( Γ pl ji ) ∗ = γ pl ,with i , j = , ( r , r , ω ) ≈ G pl ( r , r , ω ) .If we set the plasmon wavelength λ pl = π / k pl , we have [25, 27] V pl = Γ pl ˜ β sin ( πζ ) , γ pl = Γ pl ˜ β cos ( πζ ) , (2.7)where ˜ β = β exp [ − λ pl ζ / ( L )] , with ζ = d / λ pl , d is the distance between the emitters, Γ pl = Γ pl1 = Γ pl2 , L is the propagation length of the propagating mode, and the β factor thatmeasures the fraction of emitted radiation by the propagating mode. It is worth mentioningthat, due to radiative contributions, such a ‘plasmonic approximation’ breaks down foremitters separations shorter than ∼ λ pl / | V pl |≤ Γ pl / | γ pl |≤ Γ pl , which means thatthe interaction that arises due to the coupling to the plasmons is weak and the most importantcontribution due to collective effects comes from the damping γ pl . The definitions involved in the calculation of the correlations are as follows. Thequantum mutual information describes the whole content of correlations in a given quantumsystem [30–32]. It has been shown that quantum correlations (entanglement included) [15]and classical correlations [16], in the sense of entropic measures, add up to give the quantummutual information [31]. Furthermore, this point has been recently emphasized, via the useof the relative entropy, within a unified framework that captures both quantum and classicalcorrelations within the quantum mutual information [32]. For a bipartite system, this can bewritten as: I ( ρ AB ) = S ( ρ A ) + S ( ρ B ) − S ( ρ AB ) , (2.8)where S ( ρ ) = − Tr ( ρ log ρ ) is the von Neumann entropy of density matrix ρ . orrelations in emitters coupled to plasmonic waveguides
4A measure of classical correlations was introduced in Ref. [16] as the maximumextractable classical information from a subsystem, say A , when a set of positive operatorvalued measures [7] has been performed on the other subsystem (say B ):CC ( ρ AB ) = sup { Π Bj } (cid:34) S ( ρ A ) − ∑ j p j S ( ρ jA ) (cid:35) , (2.9)where S ( ρ jA ) is the entropy associated to the density matrix of subsystem A after the measure.Such correlations must be non-increasing, and invariant under local unitary operations, andCC ( ρ AB ) = ρ AB = ρ A ⊗ ρ B .If {| (cid:105) , | (cid:105)} define the basis states for the qubit B , the projectors can be written as Π Bj = ⊗ | j (cid:105) (cid:104) j | , j = a , b , where | a (cid:105) = cos θ | (cid:105) + e i φ sin θ | (cid:105) , | b (cid:105) = e − i φ sin θ | (cid:105) − cos θ | (cid:105) ,and the optimization is carried out over angles θ and φ . The measure CC is antisymmetric bydefinition, and, without loss of generality, we take the qubit B to be the one measured.Following the definition for CC ( ρ AB ) , a simple way to define the total quantumcorrelations in a bipartite system is D ( ρ AB ) = I ( ρ AB ) − CC ( ρ AB ) . In terms of the von Neumannentropies, the quantum correlations, which coincide with the definition for the quantumdiscord given in Ref. [15], read D ( ρ AB ) = S ( ρ B ) − S ( ρ AB ) + inf { Π Bj } ∑ j p j S ( ρ A | Π Bj ) . (2.10)For pure states, D = S ( ρ B ) , and D = ( ρ ) = h (cid:32) + (cid:112) − C ( ρ ) (cid:33) , (2.11)where h ( x ) = − x log x − ( − x ) log ( − x ) denotes the binary entropy function [29].Additionally, consider, in decreasing order, the eigenvalues λ i of the matrix √ ρ AB ˜ ρ AB ,where ˜ ρ AB = ( σ y ⊗ σ y ) ¯ ρ AB ( σ y ⊗ σ y ) and ¯ ρ AB is the elementwise complex conjugate of ρ . Theconcurrence C can be defined as C ( ρ AB ) = max { , λ − λ − λ − λ } , (2.12)where the λ i ’s are as introduced above or, equivalently (also in decreasing order), the squareroot of the eigenvalues of the non-Hermitian matrix ρ AB ˜ ρ AB [29].
3. Dynamics of correlations
We calculate the exact dynamics of correlations by solving the master equation (2.4),and taking into account the definitions introduced in section 2.2. The notation for thecorrelations reported in the graphs throughout this paper is as follows: total correlationor mutual information (purple, thick-solid line), classical correlation (green, doubly-dashedline), quantum discord (blue, thin-solid line), entanglement of formation (red, dashed line),and concurrence (pink, dotted line).To compare the physical scenarios described in Sec. 2, we first present correlations whenthe emitters interact solely with the vacuum electromagnetic field, i.e., in the absence ofcoupling to plasmons. In terms of the collective effects, such a dynamics depends on theseparation between emitters and also on the orientation of their dipole moments.A typical situation where the dipole-dipole interaction is much greater than theincoherent part V (cid:29) Γ (cid:29) γ [figure 1(a)] can be observed, for example, in a system of diluted orrelations in emitters coupled to plasmonic waveguides Figure 1. (Color online) Correlations for a two-emitter system in the absence of couplingto plasmons. (a) V = Γ and γ = . Γ . (b) Dipoles parallel to each other and perpendicularto their separation vector; r = λ , λ ≡ π / k . The considered initial state is separable: ρ ( ) = | (cid:105)(cid:104) | . In all graphs presented in this work: Total correlation or mutual information(purple, thick-solid line), classical correlation (green, double-dashed line), quantum discord(blue, thin-solid line), entanglement of formation (red, dashed line), and concurrence (pink,dotted line). molecules of Terrilene in a dispersive crystal [19], for which a rich entanglement dynamicshas been predicted [20]. Figure 1(a) shows that correlations reach high values and oscillaterapidly because of the strong dipole-dipole interaction energy V ; however, the correlationsdecay within a short time and the system becomes uncorrelated quite rapidly.A different scenario, depicted in figure 1(b), considers a pair of dipoles parallel toeach other and perpendicular to their separation vector, which allows the calculation of thecollective parameters V and γ directly from Eqs. (2.5). Although the correlations initiallyshow smaller values than before, they persist for a longer time.An interesting issue that arises from figure 1 is the discrepancy between the entanglementof formation and the concurrence as quantifiers of entanglement. Figure 1(b) shows thatconcurrence reaches much higher values (and decays much slower) than the EoF; moresurprising is the fact that concurrence reaches, for almost any time, higher values than the totalcorrelations, a result that contrasts with the definition of the mutual information as a quantitythat accounts for all the correlations (classical and quantum), entanglement included [31, 32].Thus, within this framework, concurrence can indicate results that are well above the EoF, andhence does not allow a direct comparison with other entropic measures such as the quantumdiscord; in contrast, the latter can be compared, on the same grounds, to the EoF [33]. This isexplicitly shown in figure 1(b).The influence of the plasmonic waveguide is illustrated in figure 2, and can be directly orrelations in emitters coupled to plasmonic waveguides Figure 2. (Color online) Correlations for the emitters interacting with a plasmonic waveguidewith the experimentally feasible set of parameters β = . L = µ m, obtained for anoperational wavelength λ =
640 nm; this gives ˜ β (cid:39) .
82, and a plasmonic wavelength λ pl (cid:39)
542 nm. (a) Separation between emitters d = λ pl , V pl =
0, (b) ζ ≡ d / λ pl = / γ pl = (cid:96) = . Γ pl . The inset corresponds to case (b), but in the absence of laser pumping. Thenotations for the graphs are as given in figure 1. compared with the results in figure 1. According to the relations derived from Eqs. (2.6) [25,27], one of the parameters V pl or γ pl can, in principle, be maximized by ‘switching off’ theother one. For the interqubit distance d = λ pl [Fig 2(b)], the plasmonic modes allow for aneffective enhancement of correlations, as can be seen from a direct comparison between theinset of figure 2(b) and figure 1(b). This said, note that the correlations decay more rapidlyin the plasmon-assisted case because here the emitters distance is such that the collective rateis switched off, and the nonlocal effects are purely due to the weak dipole-dipole interaction V pl . The correlations are enhanced as ˜ β tends to 1. For an emitters separation d = λ pl , therealistic value ˜ β = .
82 can be obtained for L = µ m, and this is shown in Fig. 2(a): for thisinterqubit distance, V pl is switched off and the correlations reach larger values than beforeand are maintained for a much longer time. We also point out that for the parameter windowconsidered in figure 2, classical correlations do not play a major role.Interestingly, the quantum discord, which is in its essence different to entanglement, doesplay a role (and is more robust than the EoF) during the qubit dissipative evolution. We remarkthat is not only the existence of quantum correlations, but the way they relate to each other,that matters, since the latter can lead to operational interpretations [33]. In fact, figure 2(a)shows that for a time τ = / Γ pl , entanglement, as calculated by the EoF, has almost vanishedand the quantum discord quickly approximates to the value given by the total correlations. Incontrast to this, Ref. [27] uses the concurrence to show that a large amount of entanglement orrelations in emitters coupled to plasmonic waveguides t = τ (figure 2(a)) corresponds to the mixed state ρ ( τ ) = . | (cid:105) (cid:104) | + . | Ψ − (cid:105) (cid:104) Ψ − | ,where | Ψ − (cid:105) = ( | (cid:105) − | (cid:105) ) / √ | Ψ − (cid:105) , and so it provides a small degree of entanglement.However, the mixture of this with the product state | (cid:105) ⊗ | (cid:105) , produces a quantum correlatedstate with a type of correlation—the so-called quantum discord, which is different, in its veryessence, to entanglement [15, 32].Although a larger value of ˜ β can enhance entanglement as well as the other correlations(not shown), we point out that care must be taken when choosing such a parameter; inparticular, the value ˜ β = . d = λ pl . Indeed, if we include the ‘best’achievable β ( (cid:39) . ) factor, in the expression for ˜ β → .
9, the relation between the plasmonicwavelength and the propagation length should read λ pl / L (cid:39) . { λ pl , L } satisfies this constraint. This simple estimation leadsus to conclude that, if β ∼ .
94, the largest correct realistic value that should be used for theconsidered emitters separation is ˜ β (cid:39) . V pl and γ pl , become even more interesting when they arepumped with a continuous laser field of amplitude (cid:96) i and frequency ω i (targetting the emitter i ). If we consider the laser excitation to be in resonance with the emitters transition frequency ω i = ω , stationary correlations can be obtained by making the amplitudes (cid:96) = (cid:96) ≡ (cid:96) , asshown in figure 2(b) for a distance ζ = / γ pl =
0. In the same spirit, a higher stationarybehaviour of correlations is obtained for ζ =
1, by introducing a relative phase between thelaser amplitudes: (cid:96) = − (cid:96) = (cid:96) , as shown in figure 3(a). The correlations are much largerthan those obtained in figure 2(b) because the laser excitation assists the very slow decay ofthe ‘naturally created’ antisymmetric state | Ψ − (cid:105) = ( | (cid:105) − | (cid:105) ) / √
2. This figure also showsthat the concurrence reaches higher values than the total correlations, which might be seen asinappropriate when compared to the use of entropic metrics.We have shown that rather than entanglement, quantum discord is the most robustcorrelation during the dissipative dynamics, and, depending on the separation betweenemitters, the qubit-plasmon modes coupling can enhance the degree of correlations via thecollective parameter γ pl . For the sake of completeness, in figure 3(b) we consider, for a fixedlaser amplitude, the role of the initial state preparation on the correlations dynamics. Here,we plot the quantum discord as a function of time, for initial states √ α | (cid:105) + √ − α | (cid:105) , α ∈ R . Although it is clear that, at t =
0, the discord takes the value 1 for α = /
2, wehave plotted it only for a range up to D = .
3, in order to appreciate comparatively thecorrelations behaviour for all α . The trend in correlations is clearly influenced by α but,overall, the states converge to a common stationary value. The quantum discord, classicalcorrelations, and EoF are plotted for the particular case α = orrelations in emitters coupled to plasmonic waveguides Figure 3.
Optical control of correlations dynamics. (a) Total spectrum of correlations for laseramplitude (cid:96) = − (cid:96) = (cid:96) = . Γ pl , and initial state | (cid:105) . (b) Quantum discord dynamics for (cid:96) = − (cid:96) = (cid:96) = . Γ pl , and initial state configurations √ α | (cid:105) + √ − α | (cid:105) . (c) Comparisonbetween QD, CC, and EoF for α =
1. In all graphs, ζ = V pl = β = .
94, and L = µ m. Bearing in mind that the quantum correlation, as measured by the discord, gives theamount of information that is not accessible to local measurement, our results show that mostof the information stored in the emitters is purely quantum (compare the thin-solid blue withthe doubly-dashed green lines in all the figures). Also, we note that, in most cases, this typeof information does not arise from entanglement, which is kept to a minimum, as can be seenin Fig.s 1(b), 2(a), and 3(c). This said, a possible optical control of the amount and class ofinduced quantum correlations can be carried out with external laser pumping, as depicted infigure 3.
4. Discussion
A comparison of the two physical scenarios considered here —emitters dissipative dynamicsin the i) absence and ii) presence of coupling to a plasmonic bus— leads us to the following.In the former case, the correlations exhibit a longer lifetime in figure 1(b) because of thepresence of the incoherent interaction γ for the distance r = λ ; this means that thelifetime of the symmetric correlated state | Ψ + (cid:105) = ( | (cid:105) + | (cid:105) ) / √ H S ) is more robust ( γ < γ pl = V pl . The plasmon-mediated coupled emittersbecome more interesting because for large distances ( d ∼ λ pl ), the incoherent decay γ pl cantake values close to Γ pl , holding the correlations for longer times. A particular case is shown orrelations in emitters coupled to plasmonic waveguides d = λ pl , γ pl = . Γ pl , hence V pl is ‘switched off’. The correlations are presenteven for times of the order of Γ pl t ∼
10 thanks to the slow decay Γ pl − γ pl of the antisymmetricstate | Φ − (cid:105) . For this time scale, entanglement (as measured by the EoF) is almost zero, andthe correlation that prevails is the quantum discord, which in turn tends to the same value asthe total correlations.Aside from the physics that arises in the qubit-plasmon setup here considered, wehighlight the discrepancy between the EoF and the concurrence as quantifiers of the emittersentanglement. This is due to the entropic origin of the EoF compared to that of concurrence(the difference between the two is explicitly stated in equation 2.11). Furthermore, in mostof the plots of this work, the concurrence is larger than the mutual information (purplelines) in several different time frames. Since the quantum mutual information captures allthe possible correlations (entanglement included), this result seems incorrect. Again, theexplanation for this lies in the fact that the concurrence does not have an entropic originlike the other correlation measurements calculated here. Strictly speaking, the concurrenceis an entanglement monotone but is not an ‘actual’ entanglement metric in the sense that itobtains its meaning from its relation to the entanglement of formation and not the opposite[35]. With the emergence of quantum correlations beyond the entanglement, it is crucial tocoherently quantify the latter in order to give a consistent physical interpretation of quantumphenomena. The understanding of whether a qubit system is entangled or discord-correlatedis thus relevant for deciding the way in which the emitters could be operated as a physicaldevice for performing information processing tasks.The relation between the correlations and the physical properties of the hybrid systemanalyzed here is clearly showed in Figs. 1-3: the lifetime of the correlations is enhanced thanksto the illumination with the coherent laser field; also, a large β -factor (due to the plasmonchannel) increases the degree of correlations, especially that of the quantum discord. It isevident from figure 3 that an adequate manipulation of the light-matter interaction strengthallows control of the dynamics followed by the quantum discord and entanglement, andtherefore of their metric value. This degree of quantum control is possible due to the interplaybetween the laser illumination intensity and the coherent emitter-emitter interaction [20, 36].
5. Concluding remarks
We have computed entanglement, classical, quantum, and total correlations for a hybridsystem composed of largely separated emitters coupled to the plasmonic modes of a one-dimensional waveguide, which is externally driven by a laser field. We have illustrated, bydirect calculation, that classical correlations are the least assisted by the plasmon bus and thatquantum discord is the dominant correlation that prevails throughout the whole dissipativedynamics and that this is enhanced by the presence of the plasmonic collective excitationsand by the external laser pumping. This tendency in the correlations has been analyticallydemonstrated for bare emitters coupled to the vacuum electromagnetic field in Ref. [36].We have emphasized the entropic origin of the entanglement of formation, the quantumdiscord, and that of the quantum mutual information as quantifiers of correlations; we alsocomputed the concurrence for direct comparison of our results with those reported in [27,28].We found that the latter can reach values well above the EoF and the total correlations. Thisshows that care must be taken when confronting or using specific quantifiers of entanglementfor describing a physical process, especially because the interpretation of the EoF as the costof creation of an entangled state (with no regularization), leads to an upper bound on the actualdegree of entanglement in a quantum system [37]. orrelations in emitters coupled to plasmonic waveguides V pl or γ pl goes to zero, an appropriate choice of initial conditionsand laser tailoring, enables the enhancement or suppression of the existing correlations. Thisresult is beyond the scope of this work and will be addressed elsewhere [38].Recent experiments have demonstrated the control of light-matter interaction by meansof plasmonic resonators [39] and silver nanowires [40] to enable the enhancement of thePurcell effect in quantum emitters, and the strength of the coupling to the plasmonic modes ofsuch nanostructures. This offers the possibility of testing the results here reported with state-of-the-art technology. Furthermore, another experimental demonstration of communicationbetween two distant single emitters (organic molecules) via single photons has been recentlyreported [41]. Thus, the setup proposed in this work has also the potential for demonstratingan additional degree of quantum control in which sensitive quantum information encodedin single photons can be transmitted and processed between coupled (but largely separated)single emitters. Acknowledgments
C.E.S. is grateful for a Colciencias fellowship. We acknowledge partial financial supportfrom Universidad del Valle (Grant CI 7859), the Spanish DGI (Grant FIS2011-26786), andthe UCM-BSCH program (Grant GR-920992).
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