1. INTRODUCTIONA global state of a composite system is entangled when it can-not be written as a product of the states of the individualsystems [1]. This is the basic quantum-mechanical property,with no classical counterpart, for quantum information tech-nologies [2,3]. Two coupled qubits, or two-level systems withground (jgi) and excited (jei) states, is the smallest andsimplest composite system that can display entanglement. Itis, therefore, themost suitablemodel to investigate its creationand processing as well as how environmental noise anddecoherence brought by spontaneous decay and the externalexcitation affects it [4], which is a key point for quantumapplications.
Two qubits can form four independent maximallyentangled states, the so-called Bell states: jϕ�i ¼ ðjggi �jeeiÞ= ffiffiffi
2p
and jψ�i ¼ ðjegi � jgeiÞ= ffiffiffi2
p. The last two are af-
fected by a possible coupling between the qubits and are alsoknown in the atomic literature as the symmetric and antisym-metric collective states [5]. The formation and degradation ofsuch states when subjected to spontaneous emission has beenthe object of much recent research [6–8], focusing on the pre-servation of entanglement into decoherence-free subspacesand taking advantage of the collective damping or effectivecoupling created between the qubits by interaction with com-mon reservoirs [9–23].
The idea of environmentally induced entanglement has alsobeen applied to the case where the two-qubit interaction ismediated by a cavity mode (harmonic oscillator) which is ex-cited by white noise (a thermal reservoir) [24], borrowing theidea from [25] where steady-state entanglement is enhancedbetween two harmonic modes by mediation of a two-level sys-tem excited by white noise. In both cases, entanglement isvery small (<0:4%). A two-level system has also been pro-posed as a mediator (or coupler) to build entanglementbetween qubits [26,27].
Another possibility, close to theone addressed in thepresenttext, is to consider two qubits already coupled, whose entan-glement builds in the steady state despite dissipation and de-coherence from two independent environments [28–37].Entanglement, being essentially a property that requires great
purity of the entangled state realized, is very sensitive to suchdecoherence. It is therefore important to look for optimization.Having a high degree of entanglement in the steady statemeans that it is robust and independent of the initial state; itremains stored forever in the system like a quantum batteryof entanglement [36].
The coupling between the qubits can have different physi-cal origins depending on the realization [3], e.g., Rydbergatoms couple through dipole-dipole interaction, weaker in thecase of cold atoms [38], and so do excitons in single quantumdots or molecules [39–41]; superconducting qubits couplethrough mutual inductance [42]. Moreover, in all these imple-mentations, the coupling can appear effectively through thevirtual mediation of a coupler (a cavity or a wire mode) in thedispersive limit, in which case it is given by geff ≈ G2
cav=Δcav,where Gcav is the coupling of the qubits to the coupler andΔcav the energy detuning to the qubits (considered much lar-ger than the coupling) [43–45]. This scheme requires, for in-stance, placing the qubits into a cavity where the cavitymode acts as the coupler. One can take advantage of theQED techniques while obtaining an effective coupling essen-tially insensitive to the cavity decay and thermal fluctuations.The effective coupling between two Rydberg atoms throughvirtual photon exchange, while crossing a nonresonant cavity,was achieved in 2001 [46]. The final entangled state could becontrolled by adjusting the atom-cavity detuning. A similar ef-fective coupling was obtained between two superconductingqubits on opposite sides of a chip using microwave photonsconfined in a transmission line cavity [47]. The cavity was alsoused to perform multiplexed control and measurement ofboth qubit states. Effective coupling between two distantquantum dots embedded in a microcavity has also been re-cently achieved [48,49].
Taking for granted that the two qubits are coupled, we cen-ter our attention on the situation where the qubits are also incontact with two independent excitation sources. Xu and Li[28] found that with two equally intense white-noise sourcesat the same temperature, there is no entanglement in the stea-dy state. However, if only one qubit is subjected to a finitetemperature source, some entanglement can be achieved.They found better but still small degrees of entanglement
228 J. Opt. Soc. Am. B / Vol. 28, No. 2 / February 2011 Elena del Valle
(<4%) and did not deepen on its origin, but their results showthat an asymmetric flow of excitation through the qubits isbeneficial for entanglement. Other authors, who did not con-sider unequal sources of excitation (having to recur to othermechanisms for entanglement generation), explored, on theother hand, configurations that are out of thermal equilibrium,where the excitation of the qubits is not necessarily producedby thermal sources but by more general processes able of in-verting the qubit populations [31,32,35]. Two qubits may un-dergo dissipation and pure dephasing but also an externallycontrollable and independent (in general) continuous pump-ing that can greatly affect their effective coupling [50,51].
In the present text, we put together different elements thathave been addressed separately in previous studies of envir-onmentally induced entanglement: direct coupling betweenthe qubits and independent and different kinds of reservoirsthat are not necessarily of a thermal nature. We give a com-plete picture of entanglement and its origin in the steady stateof such a general system. As a result, we find a configurationwhere entanglement is significantly enhanced (31%), that is,more than three times as compared to the best thermal caseand with a much better purity. We show that it can be evi-denced by the antibunching of the two qubits cross emission.
The rest of this paper is organized as follows. In Section 2,we introduce a theoretical model to describe two coupled qu-bits with decay, incoherent pumping and pure dephasing, andthe concurrence, which quantifies the degree of entanglementbetween them. In Section 3, we discuss different entangledconfigurations and optimize the concurrence for the most sui-table one: one qubit is excited while the other dissipates theexcitation. This is compared with the cases of thermal andcoherent excitation. In Section 4, we show how a strong anti-bunching between the two qubits emissions is linked with highdegrees of entanglement and we propose this effect as its in-dication of entanglement. In Section 5, we study the effect ofpure dephasing. In Section 6, we revise our claims when thecoupling appears effectively through the mediation of acavity mode in the dispersive limit. In Section 7, we presentthe conclusions.
2. THEORETICAL MODELLet us consider two qubits or two-level systems (i ¼ 1; 2), withlowering operators σi, frequencies ωi, and coupled withstrength g. Without loss of generality, we take the energyof the first qubit as a reference (ω1 ¼ 0), from which the otherone is detuned by a small quantity Δ ¼ ω1 − ω2. The corre-sponding Hamiltonian reads
H ¼ −Δσ†2σ2 þ gðσ†1σ2 þ σ†2σ1Þ; ð1Þ
Each qubit can be in the ground (jgi) or excited (jei) states,whose direct product produces a Hilbert space of dimensionfour (see Fig. 1: fj0i ¼ jggi; j1i ¼ jegi; j2i ¼ jgei; j3i ¼ jeeig).The qubits are in contact with different kinds of environmentsthat provide or dissipate excitation (at rates Pi and γi, respec-tively, with i=1,2) in an incoherent continuous way. Other in-teractions may purely bring dephasing to the coherentdynamics (at rates γdi ). These processes eventually driveany pure state into a statistical mixture of all possible states.A density matrix, ρ, properly describes the evolution of such asystem. The general master equation we consider has the stan-dard Liouvillian form [52,53]:
∂tρ ¼ i½ρ; H� þX2i¼1
�γi2Lσi þ
Pi
2Lσ†
i
þ γdi2Lσ†
iσi
�ρ; ð2Þ
with the corresponding Lindblad terms for the incoherentprocesses (LOρ≡ 2OρO† − O†Oρ − ρO†O). If the two qubitsshared a common environment, the Lindblad terms wouldshare a single expression LJ in terms of the collective opera-tor: J ¼ σ1 þ σ2. Such collective terms are sources of entan-glement, as explained in the Introduction. In this text, weinvestigate the steady state of a system where they are notpresent, solving exactly the equation ∂tρ ¼ 0.
In order to spell out the nature of the reservoirs that are incontact with the qubits, we express the pumping and decayrates in terms of new parameters Γi and ri [54]:
γi ¼ Γið1 − riÞ; Pi ¼ Γiri ði ¼ 1; 2Þ: ð3Þ
The range 0 ≤ ri ≤ 1 includes a medium that only absorbs ex-citation (decay, ri ¼ 0), one which only provides it (pump,ri ¼ 1), as well as the most common assumption of a thermalbath with finite temperature (white noise, ri < 1=2). The para-meter Γi ¼ γi þ Pi quantifies the interaction of each qubitwith its reservoir as well as its effective spectral broadening.
Taking into account the evolution of the density matrix ρ ofour bipartite system, one can conclude that, in the steady state,it has the general block diagonal form:
in termsof the operatorsni ¼ σ†iσi andn12 ¼ σ†1σ2. The averagevalue hnii is the probability for qubit i to be excited, regardlessof the other one. hn12i accounts for the population transferbetween the qubits and hn1n2i for their effective coupling, inthe sense that, independent qubits would lead to hn1n2i ¼hn1ihn2i. The general expressions for the steady state oftwo coupled qubits can be written as hnii ¼ Peff
i =Γeffi ,
Elena del Valle Vol. 28, No. 2 / February 2011 / J. Opt. Soc. Am. B 229
hn1n2i ¼ ðP1hn2i þ P2hn1iÞ=ðΓ1 þ Γ2Þ, hn12i ¼ 2gðhn1i − hn2iÞ=ð2Δþ iΓtotÞ, where the effective parameters Peff
i ¼Pi þ ðP1 þ P2ÞXi and Γeff
i ¼ Γi þ ðΓ1 þ Γ2ÞXi are expressedin terms of an effective coherent exchange factor Xi ≡
4g2=ðΓ3−iΓtot½1þ ð2Δ=ΓtotÞ2�Þ related to the Purcell rates[51]. Xi quantifies how efficiently the external inputs and out-puts are distributed among the qubits thanks to the coherentcoupling and despite the total decoherence, Γtot ¼ Γ1 þΓ2 þ γd1 þ γd2 .
3. ENTANGLEMENT AND LINEAR ENTROPYAmong the four Bell states, jϕ�i and jψ�i, only the last two areachievable in the present configuration (since ρ03 ¼ 0). Let us,therefore, write the entangled state that will be achieved inthis system in its most general form: jψi≡ ðj1i þ eiβj2iÞ= ffiffiffi
2p
.Logically, the larger the probability to find the qubits in suchstate (the closer ρ is to jψihψ j), the larger the degree of entan-glement is in the mixture represented by ρ. In order to makethis statement mathematically precise, we can make explicitthe entangled contribution to ρ by expressing it as
where R1 ¼ ρ11 − jρ12j, R2 ¼ ρ22 − jρ12j, andRψ ¼ 2jρ12j. Riði ¼1; 2;ψÞ are not probabilities (R1, R2 may be negative) but,when they are normalized as
~Ri ¼jRij
ρ00 þ ρ33 þ jR1j þ jR2j þ Rψ; ð6Þ
they represent the contribution of the pure states jii to themixture where the entangled state has been identified andset apart. In order to enhance entanglement, we must maxi-mize ~Rψ ðthat is; ρ12Þ while minimizing the populations ρ00and ρ33, and the differences ~R1 and ~R2. The nonentangled con-tributions can be put together in a single expression to beminimized: ~R ¼ 1 − ~Rψ .
The degree of entanglement can be quantified by theconcurrence (C) [55], which ranges from 0 (separable states)to 1 (maximally entangled states). It is defined as C≡
½maxf0; ffiffiffiffiffiλ1
p−
ffiffiffiffiffiλ2
p−
ffiffiffiffiffiλ3
p−
ffiffiffiffiffiλ4
p g�, where fλ1; λ2; λ3; λ4g are theeigenvalues in decreasing order of the matrix ρTρ�T , with T
being an antidiagonal matrix with elements f−1; 1; 1;−1g. Theconcurrence in this system is given by C ¼ 2Max½f0; jρ12j − ffiffiffiffiffiffiffiffiffiffiffiffiffiρ00ρ33
p g�, which shows a threshold behavior that
we anticipated above: the coherence between the intermedi-ate states, jρ12j, must overcome the population of the spuriousstates ρ00, ρ33. This is related to the fact that high values ofconcurrence require high degrees of purity in the system[56]. The degree of purity of a density matrix ρ is measuredthrough the linear entropy, SL ≡
43 ½1 − Trðρ2Þ�, which is 0 for
a pure state, and 1 for a maximally mixed state (where all fourstates occur with the same probability 1=4).
Without loss of generality, we can analyze the entanglementand linear entropy in the steady state by considering the para-meters Δ ≥ 0, on the one hand, 0 ≤ r2 ≤ r1 ≤ 1 on the otherhand, and arbitrary Γ1, Γ2 ≥ 0. This simply implies that we la-bel as 2 the qubit that is in contact with the medium which has
Fig. 1. Scheme of the two coupled qubits or two-level systems (a)and their energy levels (b), with coupling (g), pumping (Pi), and decayparameters (γi).
Fig. 2. (Color online) (a) Distribution in the C − SL plane of all thepossible two-qubit configurations (shaded region). The thin solid linecorresponds to the maximum C for a given SL in a general bipartitesystem. The dashed blue line corresponds to the optimal configuration(r1 ¼ 1, r2 ¼ 0, Γ1 ¼ Γ2 ¼ Γ, see Eq. (8), a good approximation to themaximal C versus SL in our system (with the exception of the darkblue region above). Below, in dark purple, the particular case of ther-mal baths. (b) C (solid black) and SL (dashed red) for the optimal caseas a function of α ¼
tributions to the steady state: ~R1 (dotted brown), ~R2 (dashed purple)and ~R (solid blue). The shaded area represents ~Rψ (as ~Rψ þ ~R ¼ 1).Idem in (d) and (e), but for two thermal baths at infinite and zero tem-peratures: r1 ¼ 1=2, r2 ¼ 0. The vertical guide lines mark the pointswhere entanglement appears and where it is maximum, close tothe points where ~R is minimum and ~R approaches ~R1, respectively.
230 J. Opt. Soc. Am. B / Vol. 28, No. 2 / February 2011 Elena del Valle
the most dissipative nature. Let us ignore dephasing effectsfor the moment (we bring them back in Section 5).
We start by noting that if r1 ¼ r2 ¼ r, that is, if the reser-voirs are of the same nature, there is no entanglement inthe system (C ¼ 0), regardless of all the other parameters,since, in this case, the density matrix is diagonal with ele-ments fð1 − rÞ2; rð1 − rÞ; rð1 − rÞ; r2g, that is, a mixture of se-parable states. This result has already been pointed out in theliterature [28,31]; however, let us insist on the fact that it is notthe amount of decoherence induced on the qubits by their en-vironments that destroys entanglement, but their similarity innature (or temperatures in the case of thermal baths). Let usthen consider the cases with r2 < r1 in the rest of this section.
As in [56], we examine the region of the concurrence-linearentropy plane that our system can access in Fig. 2(a). Theshaded region is reconstructed by randomly choosing allthe parameters and computing their C and SL. The accessibleregion is well below the black thin line for the maximally en-tangled mixed states [56,57] that provides the maximumconcurrence achievable for a given linear entropy. More inter-estingly, the points are bounded in good approximation by asecond (dashed blue) line specific to our system. This line cor-responds to the extreme case of reservoirs with exactly oppo-site natures, r1 ¼ 1 and r2 ¼ 0, but equally strong influence onthe qubits, Γ1 ¼ Γ2 ¼ Γ (that is, P1 ¼ γ2 ¼ Γ, P2 ¼ γ1 ¼ 0).The steady state can be written in terms of a single unitlesscomplex number,
and ρ12 ¼ αe−iβ4þα2. The two qubits are sharing a single excitation
hn1i þ hn2i ¼ 1. Note that both detuning (Δ) and the averagedecoherence (Γ) contribute symmetrically to α and have thesame effect on the steady state: to make the coherent couplingless effective. The phase β ¼ − arctanðΓ=ΔÞ, which is thesame as that of the entangled state jψi formed in the steadystate, can be rotated by changing these two parameters. Thisis a way to phase shift the entangled state obtained in the sys-tem. Concurrence and linear entropy read
We plot them in Fig. 2(b), as a function of α moving leftwardsalong the dashed blue line of Fig. 2(a). The contributions ~R1
(dotted brown), ~R2 (dashed purple), ~R (solid blue), and ~Rψ(shaded area) are also presented in the inset, Fig. 2(c), fora better understanding of the origin of entanglement. At van-ishing α (or large effective coupling), the excitation is equallyshared among the states and the system is maximally mixed(with ~Rψ ¼ 0). Concurrence becomes different from zero atα ¼ 1, which is close to the point where the nonentangled con-tribution to the density matrix, ~R, reaches its minimum (and~Rψ its maximum). The contributions of the spurious states ρ00,ρ33 have been considerably reduced while the coherence jρ12jis sufficiently large to overcome them.
The maximum concurrence in the absolute for this system,
Cmax ¼� ffiffiffi
5p
− 1
�=4 ≈ 31%; ð10Þ
is reached at α ¼ 1þ ffiffiffi5
p. This is the region where a large
contribution Rψ is combined with low SL. Moreover, the none-ntangled contribution ~R becomes similar to ~R1 meaning thatthe steady state is close to the mixture Mψ ¼ Rψ jψihψ j þð1 − Rψ Þj1ih1j, only between the entangled state jψi and j1i.The large contribution of j1i to the steady state is expectedsince the first qubit is pumped and the second decays. Whatis less expected is that, by populating this state, we are pur-ifying the total mixture and enhancing the presence of the en-tangled state. Increasing α further leads to the saturation ofthe system into state Mψ and eventually to self-quenching ofcoherence [58]. Note, however, that concurrence decreasesslowly and never becomes strictly zero again, due to the factthat ρ → Mψ and, therefore, C → Rψ .
The small region in dark blue in Fig. 2(a), to the right andabove the line in Eq. (8), corresponds to cases more entangledthan the configuration previously discussed for the sameentropy. Relaxing the previous conditions to Γ1 ≠ Γ2, for in-stance, is enough to fill this area. In any case, configurationsabove the dashed line exist only for very mixed states, withSL > ð17 − 3
ffiffiffi5
p Þ=30 ≈ 0:34, being less appealing for applica-tions. In Fig. 3(a), we plot the corresponding concurrenceas a function of Γ1 and Γ2 in order to show that it is robustto their difference: C > 0 as long as Γ1 þ Γ2 > 2, and C >
Cmax=2 ≈ 15% in most of the area shown.
Fig. 3. Contour plots of concurrence C as a function of Γ1=g and Γ2=g for (a) r1 ¼ 1, r2 ¼ 0 and (b) r1 ¼ 1=2, r2 ¼ 0. To the left of the white lines,C ¼ 0. The maximum value achieved with this system is in (a), Cmax ≈ 31%. In (b), with thermal baths, concurrence rises up to C ≈ 10% for anasymmetric configuration. (c) Idem but for the case of coherent excitation of the first qubit (with strength Ω1), Γ1 ¼ 0 and r2 ¼ 0.
Elena del Valle Vol. 28, No. 2 / February 2011 / J. Opt. Soc. Am. B 231
To conclude this analysis, one can check that if onemedium provides an overall dissipation and the other onean overall gain (0 < r2 < 1=2 < r1 < 1), then C can reach non-negligible values (above 10%). This is one of the important re-sults in this text; the opposite nature of the reservoirs can leadthe steady state close to Mψ , allowing for the highest degreesof entanglement in the system.
On the other hand, if we keep equal interactions with thereservoirs, Γ1 ¼ Γ2 ¼ Γ, but they are not restricted in theirnatures, the values of the concurrence decreases. Let us con-sider the case that has been previously studied in the litera-ture, two thermal reservoirs in contact with two qubits. In ournotation: 0 < r2 < r1 < 1=2. In this case, the concurrencedoes not grow higher than 4% (as noted in the Introduction),which is reached for the extreme case, r1 ¼ 1=2, r2 ¼ 0.Again, within the new restrictions, it is favorable for entangle-ment that excitation is provided through one qubit while theother only dissipates. The equivalent expressions to Eq. (8)read
This case is featured in Figs. 2(d) and 2(e). We observe that Cbecomes different from zero again at the minimal ~R (maximalentangled contribution ~Rψ ) and that its maximum value isreached when ~R1 approaches ~R. However, the concurrenceremains 1 order of magnitude smaller than Cmax and the linearentropy does not drop to 0. With thermal excitation, ~Rψ is al-ways too small and the steady state is not close enough toMψto exhibit a high degree of entanglement. However, in contrastwith the optimally pumped case, one can increase entangle-ment from these figures by allowing Γ1 ≠ Γ2. Concurrenceis increased, filling the purple darker shaded region inFig. 2(a). The maximum concurrence here is C ≈ 10% at Γ1 ≈
1:24 and Γ2 ≈ 6:45. This is shown in Fig. 3(b) where the highestvalues of C appear in light gray around those rates.
Finally, if instead of incoherently, the first qubit is coher-ently excited (with a Hamiltonian term of the form HL ¼Ω1ðσ1 þ σ†1Þ, Ω1 being the coupling strength with a resonantlaser field), the steady-state entanglement mechanism pre-sented in this section applies similarly. One can obtain analy-tically the steady-state properties, concurrence, and linearentropy and find a situation very closed to the one we ana-lyzed: High degrees of entanglement are found for intermedi-ate values of Ω1 and Γ2 when the first qubit receives theexcitation (setting Γ1 ¼ 0) and the second qubit loses it(r2 ¼ 0). This is shown in Fig. 3(c).
4. ANTIBUNCHINGIs there an experimental observable that can evidence thehigh degrees of entanglement that we have analyzed here?One possibility is to reconstruct the steady-state density ma-trix through quantum tomography, but this method involvescomplicated setups and numerous and repeated measure-ments [59]. Here, we propose an alternative method basedon photon counting, that is, on the quantity
δ≡ hn1ihn2i − hn1n2i ¼ ρ11ρ22 − ρ00ρ33: ð13Þ
hn1i and hn2i are proportional to the intensity of the lightemitted from each qubit, obtained by counting photons fromeach source, while hn1n2i is obtained by counting simulta-neous photon detections. δ is directly linked to the second-order cross correlation function [51] at zero delay, gð2Þ12 ð0Þ ¼1 − δ=ðhn1ihn2iÞ. Although these photon counting experimentsmust be performed repeatedly in order to gather enough sta-tistical information to obtain reliable average values, δ is stillconsiderably less demanding to measure than the full densitymatrix.
δ is zero if the qubits, acting as two random photon sources,
are independent (gð2Þ12 ð0Þ ¼ gð2Þ12 ð∞Þ ¼ 1), and different from
zero if one qubit’s emission is conditional to the other qubit’sstate. δ < 0 implies that the simultaneous emission from bothqubits is enhanced in the system as compared to the indepen-dent emissions. This is a necessary condition for photon
bunching, although bunching also requires gð2Þ12 ð0Þ > g
ð2Þ12 ðτÞ.
An example where δ < 0 is the cross simultaneous emissionof two coupled harmonic oscillators [50]. On the other hand,δ > 0 implies that simultaneous emission from both qubits isless likely than in the uncoupled situation. Again, this is ne-
cessary for photon antibunching (gð2Þ12 ð0Þ < gð2Þ12 ðτÞ).
In the steady state of our system, the emission from one ofthe qubits is always antibunched (gð2Þii ð0Þ ¼ 0 < g
ð2Þii ðτÞ,
i ¼ 1; 2, as it corresponds to a two-level system), and the crossemission from both qubits fits 0 ≤ δ ≤ 1=4. One can checkthese limits from gathering δs from many randomly generatedconfigurations. It cannot go below zero because the only co-herence and entanglement in the system comes from the statejψi (that gives the maximum value δ ¼ 1=4) and not jϕi≡ðj0i þ eiβ
0 j3iÞ= ffiffiffi2
p(that would give δ ¼ −1=4). The sign of δ
is linked to the type of entangled state realized in the system,also when there is superposition or mixture with other statesand C < 1. For instance, if we plot C versus δ for the maxi-mally entangled mixed states [56] with entanglement provided
Fig. 4. (Color online) Distribution in the C − δ plane of all the pos-sible qubit configurations (shaded region). The thin solid and dottedlines correspond to different examples of entangled mixed states, withjψi (see the main text). The pure state jψi corresponds to the extremepoint ð1=4; 1Þ. The dashed blue line corresponds to the configuration(r1 ¼ 1, r2 ¼ 0, Γ1 ¼ Γ2 ¼ Γ, increasing Γ anticlockwise) which en-closes all possible realizations. Inside, in dark purple, the particularcase of thermal reservoirs. Inset, C versus hn1i for thermal reservoirs(where hn1i < 0:5, in dark purple) and for those configurations whereδ > 0:04 (in blue). The dashed blue line (r1 ¼ 1, r2 ¼ 0, Γ1 ¼ Γ2 ¼ Γ)goes clockwise with increasing Γ.
232 J. Opt. Soc. Am. B / Vol. 28, No. 2 / February 2011 Elena del Valle
by jψi, we obtain the black thin line in Fig. 4, that abruptly fallsat δ ¼ 1=9. This is because for this kind of states with C < 2=3,δ remains constant. One would obtain the symmetrical curveat negative δ, if the mixture was with jϕi. In this example, highdegrees of entanglement are related to large δ. The otherdotted lines appearing in Fig. 4 are more examples of this re-lationship between the type of entanglement and C with δ. Thecentral black dotted line corresponds to the superposition or amixture of jψi with jϕi, when jψi is the dominant state:C ¼ 4δ. There is a symmetric counterpart curve (not shown)in the opposite situation, where jϕi is dominant, with C ¼ 4jδjand δ < 0. The upper red dotted line corresponds to the super-position or mixture of jψi with j0i or with j3i : C ¼ 2
ffiffiffiδ
p. The
counterpart curve, with jϕi, is symmetrical. The space be-tween these two dotted lines could be filled with mixturesof jψi with both j0i and j3i. The lower blue dotted line corre-sponds to the mixture of jψi with j1i, the state Mψ : C ¼1 −
ffiffiffiffiffiffiffiffiffiffiffiffi1 − 4δ
p. In all these cases, large δ is correlated with large
C, although the connection is rather general and not exclusiveenough to define any entanglement witness in terms of δ.
Let us go back to our system and investigate how to usethese correlations to extract information about C from themeasured δ. The shaded region in Fig. 4 corresponds, as inFig. 2, to the situations realized in our system. It is completelyenclosed this time by the dashed blue line, which correspondsto reservoirs with opposite natures (as analyzed in the pre-vious section). In this limiting case, δ reads
δ ¼� α4þ α2
�2: ð14Þ
Thanks to this analytical boundary, we can turn the generalstatement that there is some correlation between δ and C intoa more accurate (mathematical) one:C−ðδÞ ≤ C ≤ CþðδÞwhere
C�ðδÞ ¼2
ffiffiffiffiffi2δ
p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
These inequalities become most stringent when δ > 0:04, forinstance δ > 0:061 implies 20% < C < 28:3%. More precise in-formation can be obtained if hn1i is included in the analysis,looking at the inset of Fig. 4. In blue, we see a cloud of numeri-cally generated points where δ > 0:04. There is also a clearcorrelation between large, unsaturated, population of the dots(0:8 < hn1i < 0:91) and large degrees of entanglement(10% < C < Cmax).
Such large δ and C cannot be obtained with thermal reser-voirs for the qubits. The small accessible area in that case isshaded in darker purple in Fig. 4 (within the dashed bluecurve) and in the inset. One cannot use δ or hn1i as entangle-ment indicators because, for all δ or hn1i, C can take a broadrange of values that always includes 0.
5. PURE DEPHASINGPure dephasing provides extra decay for the coherence in thesystem. It weakens the correlations established between thequbits and is, therefore, an enemy of entanglement. We cansee this in Fig. 5(a) where we plot the effect of increasing de-phasing (γd1 ¼ γd2 ¼ γd) on entanglement. The curve at the topis the same as in Fig. 2(b), with opposite kinds of reservoirs.The rest of the curves correspond to increasing values of the
Fig. 5. (Color online) Effect of dephasing on the results for opposite reservoirs (r1 ¼ 1, r2 ¼ 0, Γ1 ¼ Γ2 ¼ Γ), of C (a) and δ (b) as a function of Γ.The set of curves corresponds to values of γd from 0 to 20g, increasing in steps of 2g. Entanglement (a) is diminished by pure dephasing, going fromtop to bottom curves, and the maximum is reached at higher Γs (see the dashed line, joining the maxima of all curves). The maximum δ remains1=16 for all values of dephasing although this is reached for lower Γs.
Fig. 6. (Color online) (a) Maximum of the concurrence (solid black) and the corresponding linear entropy (dashed red) as a function of the cavitydecay rate γcav=Gcav in the case where the coupling between the qubits is mediated by a cavity mode. The qubits are coupled to the cavity mode withstrength Gcav and largely detuned from it by Δcav ¼ 10Gcav so that their effective coupling is geff ≈ 0:1Gcav. The qubits are optimally pumped withα ¼ ð1þ ffiffiffi
5p Þgeff . (b) The same quantities as in (a) are plotted, together with the cavity population ncav (dotted), as a function of the detuning
Δcav=Gcav for the case of γcav ¼ 2Gcav. The limit Cmax (horizontal guide line) is recovered at large detunings.
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dephasing rate in steps of 2g up to γd ¼ 20g. Entanglementdecreases, and its maximum value for a given γd requires high-er Γ. The set of maximum C and the corresponding required Γare plotted with a dashed thin line superimposed to the curvesfor clarity. Entanglement is quite robust in this configuration;it disappears but asymptotically and very slowly. Note that γdmust be 1 order of magnitude larger than g so that C is de-creased to the values obtained with thermal reservoirs (10%).
In Fig. 5(b), we plot the counterpart curves for δ that shrinkand move leftwards with dephasing. However, the maximum δremains 1=16 for all dephasing, taking place at lower Γs. Giventhat the tendency of the maximum δ is the opposite to that ofthe maximum C, the possibility of using δ as an indicator ofentanglement fails at large dephasing.
6. EFFECTIVE COUPLING THROUGH ACAVITY MODEIn this text, we have assumed the existence of some coupling gbetween the qubits. Let us analyze in this last section the si-tuation where the coupling is mediated by a cavity in the dis-persive limit [60], as explained in the Introduction. We nowsolve a master equation that includes, instead of the directcoupling g, the coupling of each qubit to a common cavitymode with decay rate γcav. The qubits are largely detuned fromthe cavity by Δcav ¼ 10Gcav so that the effective coupling iswell approximated by geff ≈ 0:1Gcav. Given that the photonsexchanged between the qubits are virtual, entanglement isonly slightly affected by the cavity losses. In Fig. 6(a), we seehow the maximum C (and SL, for the same α ¼ ð1þ ffiffiffi
5p Þgeff )
decreases (increases) slowly in a large range of γcav. The start-ing point for C, at very small γcav, is below the absolute max-imum of entanglement we found in previous sections, Cmax,because the detuning is not large enough. Cmax is recoveredonly at very large detunings Δcav ≫ Gcav, where the effectivecoupling between the qubits (although small) dominates fullythe dynamics. This limiting value is independent of γcav; how-ever, less detuning is required to achieve it in better cavities.The qubit parameters α, at which the maximum entanglementis obtained, also change with the detuning since they dependon geff . In Fig. 6(b), we illustrate these general results with theexample of a typical cavity γcav ¼ 2Gcav. Note that entangle-ment remains low until the cavity empties (ncav → 0) andits excitation becomes virtual.
7. CONCLUSIONSWe have computed the entanglement (C) and linear entropy(SL) for two coupled qubits in the steady state created by twoindependent sources of incoherent continuous excitation.Such a configuration is realizable for instance with semicon-ductors, e.g., with quantum dots coupled directly or through acavity mode. We have studied not only the case where the ex-citation is of a thermal origin, but also a more general out-of-equilibrium situation where populations can be inverted(hn1i > 0:5), for instance due to off-resonant laser excitation.We found and analyzed the properties of maximally entangledmixed states for such systems. This provides, for systems ex-cited independently and incoherently, the counterpart of theanalysis of Munro et al. [56]. Maximally entangled mixedstates correspond to a configuration where one qubit essen-tially dissipates excitation while the other essentially gains it.This allows entanglement to be greatly enhanced (C up to
31%) as compared to the best thermal values (C up to10%), with also a much higher purity of the state. We haveused both numerical results, in the most general case, andanalytical formulas, in this optimal case, to fully understandand characterize entanglement formation. Entanglement (pro-vided by the entangled state jψi) is enhanced in the steadystate when the pumped qubit approaches saturation, becausethis removes population from spurious states (j0i and j3i) andpurifies the statistical mixture, even in the presence of puredephasing. Finally, we have shown that the quantity δ ¼hn1ihn2i − hn1n2i, that can be measured experimentally byphoton counting, can be used as an indicator of high degreesof entanglement in this system and specially for the optimalconfiguration.
ACKNOWLEDGMENTSThe author thanks Prof. S. F. Huelga for useful comments.This research was supported by the Newton International Fel-lowship program.
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