Non-equilibrium entangled steady state of two independent two-level systems
aa r X i v : . [ qu a n t - ph ] J un Non-equilibrium entangled steady state of two independent two-level systems
S. Camalet Laboratoire de Physique Th´eorique de la Mati`ere Condens´ee, UMR 7600,Universit´e Pierre et Marie Curie, Jussieu, Paris-75005, France (Dated: Received: date / Revised version: date)We determine and study the steady state of two independent two-level systems weakly coupled toa stationary non-equilibrium environment. Whereas this bipartite state is necessarily uncorrelatedif the splitting energies of the two-level systems are different from each other, it can be entangled ifthey are equal. For identical two-level systems interacting with two bosonic heat baths at differenttemperatures, we discuss the influence of the baths temperatures and coupling parameters on theirentanglement. Geometric properties, such as the baths dimensionalities and the distance betweenthe two-level systems, are relevant. A regime is found where the steady state is a statistical mixtureof the product ground state and of the entangled singlet state with respective weights 2/3 and 1/3.
PACS numbers: 03.67.Bg,03.65.Yz,05.70.Ln
I. INTRODUCTION
For a quantum system, the influence of the surround-ings plays a role at a fundamental level. When the envi-ronment is taken into consideration, the system dynam-ics can no longer be described in terms of pure quantumstates and unitary evolution. An open quantum systemis generally in a statistical mixture of pure states. Thishas an important consequence for multipartite systems.As is well known, correlations between quantum systemscannot be completely understood in classical terms [1].There exist states which are not classically correlatedand lead to correlations with no classical counterpart,as clearly shown by violations of Bell inequalities for in-stance [2]. They are said to be entangled. Whereas al-most all pure states are entangled, this is not the casefor mixed states. In the space of mixed states, the setof non-entangled, or separable, states has a finite volume[3]. An interesting consequence of the geometrical prop-erties of this set is that the state of a multipartite opensystem can be entangled for finite periods of time, in thecourse of its evolution, and separable at infinite time orvice versa [4].The most common environment is a heat reservoir. Ifthe considered system is weakly coupled to an infinitenumber of degrees of freedom, initially in thermal equi-librium, it relaxes, in general, to a thermal state withthe temperature of its surroundings. In such an environ-ment, it is clear that, in the absence of direct interac-tions between the subsystems of a multipartite system,these subsystems are uncorrelated at long times. In otherwords, any initial correlation, quantum or classical, be-tween independent subsystems is generically destroyedby a thermal bath. Moreover, for the geometric reasonsmentioned above, quantum disentanglement can occurin a finite time [5, 6]. Furthermore, the disentanglinginfluence of the environment also exists when no energyis exchanged between the system and its surroundings,whereas, in this particular case, classical correlations canpersist [7, 8]. However, when independent systems interact with acommon environment, the indirect interaction betweenthem, mediated by this environment, may have a positiveimpact on their entanglement. Recent results evidencethe existence of this influence. It has been shown thata transient entanglement, between initially uncorrelatedsystems, can be induced by a thermal bath, for both non-dissipative [9] and dissipative [10–12] couplings. It hasalso been obtained that, in the limit of infinitely closenon-interacting systems, some special entangled statesare not affected by the environment [13]. In this limitingcase, the considered multipartite open system has not aunique steady state, which is exceptional, and hence theentanglement evolution depends on the system’s initialstate.In the above cited dynamical studies, the environmentis in thermal equilibrium and thus a relaxation dynamicstowards a unique steady state necessarily means decay ofcorrelations, both quantum and classical, between non-interacting systems. This may not be the case for a non-equilibrium surroundings. Stationary entanglement hasbeen found in the presence of particle [14, 15] or energyflow [16, 17]. However, in these studies, entanglementoccurs between systems that interact with each other di-rectly, via a two-level system, or via strong coupling to aheat bath, and this interaction plays an essential part inthe development of entanglement. Such a strong interac-tion has been shown to be unnecessary for a different kindof non-equilibrium environment [18]. In the presence of aclassical oscillating field, the steady state of two two-levelatoms, interacting with each other only via weak couplingto the electromagnetic vacuum, can be entangled.In this paper, we consider two independent two-levelsystems (TLS) coupled to a steady non-equilibrium envi-ronment. Examples of such surroundings are illustratedin Fig.1. They consist of several heat baths at differenttemperatures. These are not the only possible examplesand the two following sections are relevant to other en-vironments. In section II, we present the model used todescribe two non-interacting TLS in a stationary environ-ment. In section III, we first study the steady state of a
T T T
T T FIG. 1: Schematic representation of non-equilibrium environ-ments of two independent TLS. The depicted environmentsconsist of several heat baths at different temperatures. general system weakly coupled to its surroundings, andwe then apply our approach to the case of a system con-sisting of two independent TLS. The system steady stateis obtained, in the weak coupling limit, by solving per-turbatively an eigenvalue problem, which is derived fromthe system dynamics for arbitrary coupling strength. Asfar as one is interested only in the stationary state, noother approximation, such as a Markovian assumption,or elaborate method, such as a projection superoperatortechnique, are needed [19, 20]. In section IV, we focuson the special case of an environment that consists ofbosonic heat baths at different temperatures. It is shownthat two baths are enough to induce stationnary entan-glement of two identical TLS. The influence of the twobaths temperatures and of the coupling parameters is dis-cussed in some detail. Finally, we summarize our resultsin the last section.
II. MODEL
The total Hamiltonian of two independent TLS andtheir environment E can be written as H = X i =1 (cid:20) − ∆ i σ ( i ) z + v i σ ( i ) z + w i σ ( i )+ + w † i σ ( i ) − (cid:21) + H E (1)where ∆ i are the level splittings of the TLS, v i = v † i and w i are operators of E and H E is the self-Hamiltonianof E . The Pauli operator σ ( i ) z has eigenvalues ± |±i i . Theoperators σ ( i ) ± then read σ ( i )+ = [ σ ( i ) − ] † = | + i ii h−| . Weintroduce, for further use, the following notations : | i = | + i | + i , | i = | + i |−i , | i = |−i | + i , | i = |−i |−i . (2)Two TLS interacting with their environment but notdirectly with each other can always be described by aHamiltonian of the form (1). The system E is assumedto consist of an infinite number of degrees of freedom and to lead to a decohering and dissipative reduced dynamicsof the TLS.As the initial state of the complete system, we considerΩ = X k,l r kl | k ih l | ⊗ ρ E (3)where ρ E commutes with H E . The two-TLS system and E are initially uncorrelated. As we will see below, the condi-tion [ ρ E , H E ] = 0 implies the stationarity of relevant cor-relation functions of E . Typical environments we are in-terested in are made up of several heat baths at differenttemperatures T n , as sketched in Fig.1. In this case, theenvironment Hamiltonian and initial state read, respec-tively, as H E = P n H E n where n runs over the heat reser-voirs and [ H E n , H E n ′ ] = 0, and ρ E ∝ Q n exp( − H E n /T n ),and commute with each other. Throughout this paper,we use units in which ~ = k B = 1. III. NON-EQUILIBRIUM STEADY STATE
In this section, we first derive a matrix equation for thesteady state of a generic open system S initially uncor-related with its environment E . More explicit equationsare then obtained for a steady environment and in thelimit of weak coupling between S and E . This weak cou-pling approach is applied to the two TLS described bythe Hamiltonian (1). In this case, the steady state equa-tion can be solved. The result is radically different for∆ = ∆ and ∆ = ∆ . A. General case
In general, under the influence of its environment E ,a system S relaxes to a steady state determined by itsself-Hamiltonian and by its interaction with E . If E isin thermal equilibrium and S interacts with it weakly,this state does not depend on any detail of the intrinsicdynamics of E or of the coupling between S and E . But,as we will see, this is a very particular case. To determinethe steady state of S , we first write its reduced densitymatrix, at positive times t , as ρ ( t ) = i π Z R + iη dze − izt Tr E h ( z − L ) − Ω i (4)where Tr E denotes the partial trace over E , η is a positivereal number, and Ω is the initial state of the total system S + E . The Liouvillian L is defined by L . . . = [ H, . . . ]where H is the Hamiltonian of S + E . This Hamiltoniancan be decomposed as H = H S + H int + H E where H S and H E are the self-Hamiltonians of S and E , respec-tively, and H int accounts for the interaction between S and E . The condition Tr E ( ρ E H int ) = 0 can be assumedwithout loss of generality. It can always be satisfied byappropriately redefining H S and H int . The eigenstatesand eigenenergies of H S will be denoted by | k i and ǫ k inthe following.
1. Steady state equation
For an initial state Ω of the form (3), the matrix ele-ments ˜ r kl ( z ) = h k | Tr E [( z −L ) − Ω] | l i of the Laplace trans-form of ρ , are given by˜ r kl ( z ) = X k ′ ,l ′ Γ kl,k ′ l ′ ( z ) r k ′ l ′ (5)where the functionsΓ kl,k ′ l ′ ( z ) = h k | Tr E h ( z − L ) − | k ′ ih l ′ | ⊗ ρ E i | l i (6)depend only on the environment part of the initial state(3). Equation (5) can be read as a matrix relation be-tween two column vectors r and ˜ r ( z ) with elements r kl and ˜ r kl ( z ), respectively, and a square matrix Γ ( z ) whoseelements are given by (6). An important feature ofthis matrix is that the column vector v with elements v kl = δ kl is always left eigenvector of Γ ( z ) with eigen-value z − , i.e., P k Γ kk,k ′ l ′ ( z ) = δ k ′ l ′ /z [21]. This equal-ity ensures the conservation of the trace of the densitymatrix ρ , and follows simply from P k h k | Tr E ( . . . ) | k i =Tr( . . . ). The matrix Γ ( z ) can thus be written as Γ ( z ) = z − u ( z ) v T + Γ ′ ( z ) where v T Γ ′ ( z ) = 0 and v T u ( z ) = 1.The column vector u ( z ) is right eigenvector of Γ ( z ) witheigenvalue z − . Provided it has no pole on the real axis,the corresponding term of Γ ( z ) can be analytically con-tinued in the lower half plane and gives a constant con-tribution to the time-evolved density matrix (4). Since v T r = P k r kk = 1 for any density matrix ρ (0), thiscontribution does not depend on the initial state of S .In summary, the steady state of the open system S is P k,l u kl | k ih l | where u kl are the elements of the columnvector u determined bylim η → + { iη Γ ( iη ) } u = u . (7)Note that the condition [ ρ E , H E ] = 0 was not used toderive this equation.
2. Weak coupling limit
To determine the steady state of S in the limit of weakcoupling to E , we first expand the matrix elements (6) inpowers of the Liouvillian L int . . . = [ H int , . . . ]. We obtainΓ kl,k ′ l ′ ( z ) = 1 z − ǫ k + ǫ l (cid:26) δ k ′ k δ l ′ l + i γ kl,k ′ l ′ ( z ) z − ǫ k ′ + ǫ l ′ (cid:27) (8)up to second order, where γ kl,k ′ l ′ ( z ) can be ex-pressed in terms of the correlation functions C klk ′ l ′ ( t ) =Tr[ ρ E exp( itH E ) h kl exp( − itH E ) h k ′ l ′ ] of the environmentoperators h kl = h k | H int | l i = h † lk , as γ kl,k ′ l ′ ( z ) = Z ∞ dte izt n e itω l ′ k C l ′ lkk ′ ( t ) + e itω lk ′ C l ′ lkk ′ ( − t ) − X j (cid:2) δ ll ′ e itω lj C kjjk ′ ( t ) + δ kk ′ e itω jk C l ′ jjl ( − t ) (cid:3)o . (9) In this expression, we have used the notation ω kl = ǫ k − ǫ l . The stationarity of the correlation functions C klk ′ l ′ stems directly from the steady environment assumption[ ρ E , H E ] = 0. For the Hamiltonian (1) and with the defi-nitions (2), h = − h = v + v , h = − h = v − v , h = h = w , h = h = w and h = h = 0.In the absence of interaction between S and E , theeigenvalue problem (7) reduces to ( ǫ k − ǫ l ) u kl = 0. Conse-quently, the only matrix elements u kl with nonvanishingzeroth-order approximations are that for which ǫ k = ǫ l .Thus, if the energy spectrum { ǫ k } is nondegenerate, thecorresponding steady density matrix is diagonal in thebasis {| k i} . In the opposite case, there can exist co-herences between states | k i of equal energy. The matrixelements u kl to zeroth order, are determined by the equa-tions X ǫ k ′ = ǫ l ′ γ kl,k ′ l ′ ( i + ) u k ′ l ′ = 0 (10)where k and l satisfy ǫ k = ǫ l . The remaining coherences u kl are at least of first order in H int . By writing explic-itly the coefficients γ kl,k ′ k ′ ( i + ), it can be shown that,for an environment in thermal equilibrium, i.e., ρ E ∝ exp( − H E /T ), the thermal state u kl ∝ δ kl exp( − ǫ k /T ) issolution of (10), even in the presence of degeneracy inthe spectrum of H S , see Appendix. B. Different splitting energies
For unequal nonzero ∆ and ∆ , the spectrum of theHamiltonian H S = − P i ∆ i σ ( i ) z / − ˜ γ − − ˜ γ − ˜ γ +1 + ˜ γ +2 − ˜ γ − γ +1 + ˜ γ − − ˜ γ +2 − ˜ γ − ˜ γ − + ˜ γ +2 − ˜ γ +1 ˜ γ − + ˜ γ − − ˜ γ +2 − ˜ γ +1 p p p p = 0(11)where p k = u kk . The elements of the above matrix canbe written as˜ γ + / − i = 2 π X A,B P A/B |h B | w i | A i| δ ( E A − E B + ∆ i ) (12)where E A and | A i denote the eigenenergies and eigen-states of H E , and P A are the eigenvalues of ρ E . Thecoefficients ˜ γ + i and ˜ γ − i are the Fermi golden rule rates ofthe TLS i [20].The solution of (11) leads to a product steady state ρ = ρ ⊗ ρ where ρ i = (˜ γ + i + ˜ γ − i ) − (cid:2) ˜ γ + i | + i ii h + | + ˜ γ − i |−i ii h−| (cid:3) . (13)The two TLS are uncorrelated, to lowest order in H int ,when their splitting energies are different from eachother. Moreover, the steady state ρ i of TLS i is thesame in the presence or absence of the other TLS. Inthe special case ∆ = 0, the zeroth-order coherences u and u are a priori different from zero since ǫ = ǫ and ǫ = ǫ . But, for an environment E consisting of severalheat baths, it is shown in the Appendix that ρ = ρ ⊗ I/ ρ is given by (13) and I is the 2 × C. Identical splitting energies
For ∆ = ∆ = ∆ = 0, the states | i and | i havethe same energy ǫ = ǫ = 0. The other energies are ǫ = − ǫ = ∆. Here, equation (10) takes the form (cid:18) ˜ γ − β ∗ − β − β T α (cid:19) p cc ∗ = 0 (14)where c = u = u ∗ , ˜ γ is the 4 × p T = ( p p p p ). The coefficient α reads as α = 12 (cid:0) ˜ γ − + ˜ γ +1 + ˜ γ − + ˜ γ +2 (cid:1) + X A,B P A |h A | v | B i| δ ( ω AB )+ i X A,B (cid:2) |h A | w | B i| − |h A | w | B i| (cid:3) P A + P B ω AB − ∆ (15)where v = 2 √ π ( v − v ) and ω AB = E A − E B . Theelements of β T = ( β β β β ) are given by β / = 2 π X A,B P A/B h A | w | B ih B | w † | A i δ ( ω AB − ∆) , (16) β = − ( β + β ) / i ˜ β and β = − ( β + β ) / − i ˜ β , where˜ β = P A,B h A | w † | B ih B | w | A i ( P A − P B ) / ( ω AB + ∆). Fortwo identical two-level atoms coupled to the electomag-netic vacuum, β and ˜ β are, respectively, the collectivedecay rate and the dipole-dipole interaction energy of theatoms [18, 22].It is instructive, for the following, to relate the co-efficients (16) to Fermi golden rule rates. Instead ofanalysing the influence of E on the two TLS in the basisof product states (2), the basis made up of the states | i , | i and the entangled Bell states | ψ ± i = | i ± | i√ √ (cid:0) | + i |−i ± |−i | + i (cid:1) , (17)can be used. Both bases correspond to the same en-ergy spectrum {± ∆ , } . The Fermi golden rule ratesfor the downward transitions | i → | ψ ± i are given by2 π P A,B P A |h B |h ψ ± | H int | i| A i| δ ( ω BA − ∆) = (˜ γ +1 +˜ γ +2 ) / ± Re β . This last expression is also valid for | ψ ± i → | i . For the upward transitions | i → | ψ ± i and | ψ ± i → | i , the rates are (˜ γ − + ˜ γ − ) / ± Re β .Equation (14) can be solved by diagonalizing ˜ γ . Theeigenvalues of ˜ γ are λ = 0, λ = ˜ γ − + ˜ γ +1 , λ = ˜ γ − + ˜ γ +2 and λ = λ + λ . We denote by ψ n and φ n thecorresponding right and left eigenvectors. Since ψ is theonly right eigenvector for which the sum of its elementsdoes not vanish, p = ψ + P n> λ − n ψ n φ Tn ( c β ∗ + c ∗ β ),and the coherence c is solution of αc − β T ψ − X n> λ − n β T ψ n φ Tn ( β ∗ c + β c ∗ ) = 0 . (18)We will see in the next section that c can be nonzeroand lead to stationary entanglement of the TLS. In thespecial case ∆ = ∆ = 0, it can be shown that c = 0and p k = 1 / E is made up ofheat baths, see Appendix.The treatment of section III B applies when the differ-ence δ = ∆ − ∆ is large enough that it can be consideredfinite in the expansion in terms of the interaction Hamil-tonian H int . In (14), this difference is exactly zero. Apossible approach to understand the influence of a small δ , consists in expanding the coefficients (6) both in H int and δ . This gives equation (14) with α + iδ in place of α ,which reduces to (14) for δ much smaller than the othermatrix elements, and leads to the uncorrelated state (13)with ∆ = ∆ , in the opposite limit. IV. MULTIPLE HEAT BATHS ENVIRONMENT
In this section, we consider an environment E made upof several heat baths, as sketched in Fig.1, each consist-ing of an infinite number of harmonic degrees of freedomwhich are coupled linearly to the TLS. In other words, thespin-boson model [23], which appropriately describes var-ious physical environments [19], is generalized to two TLSand several heat reservoirs. We show that two bosonicbaths can induce stationary entanglement of two identi-cal TLS. A. Environment model
We write the Hamiltonian of E as H E = P n H E n where n runs over the heat baths and H E n = X q ω nq a † nq a nq . (19)In this expression, the sum runs over the harmonic modesof the bath n . The annihilation operators a nq satisfy thebosonic commutation relation [ a nq , a † n ′ q ′ ] = δ nn ′ δ qq ′ . Forthe coupling operators, we consider w i = X n,q k ( i ) nq (cid:0) a † nq + a nq (cid:1) , (20)and a similar expression for v i . The coupling parameters k ( i ) nq are assumed to be real. The environment is initiallyin the state ρ E ∝ Q n exp( − H E n /T n ) where T n is thetemperature of bath n .Here, the rates (12) can be written as˜ γ ± i = ± X n J ( i ) n − e ∓ ∆ i /T n (21)where J ( i ) n = 2 π P q [ k ( i ) nq ] δ ( ω nq − ∆ i ), and the coeffi-cients (16), which are relevant only in the case ∆ i = ∆,are given by similar expressions with J ( i ) n replaced by K n = 2 π P q k (1) nq k (2) nq δ ( ω nq − ∆). Clearly, J ( i ) n is neces-sarily positive but not K n , and | K n | < J (1) n J (2) n . Themain difference between J ( i ) n and K n is that the formerdepends only on the coupling of TLS i to E , whereas thelatter is determined by both coupling operators w and w . An important physical parameter that controls theratio | K n | /J (1) n J (2) n is the distance d n between the TLScoupling points to bath n . This ratio reaches its maxi-mum value of 1 when the two TLS interact in exactly thesame way with bath n , which necessarily means d n = 0[13]. This spatial dependence is discussed more fully atthe end of section IV C 2.Finally, we comment on the second term in (15), whichplays a role in the following. It can be cast into theform P n R ∞ dωδ ( ω ) L n ( ω ) / tanh( ω/ T n ) where the spec-tral functions L n are defined similarly to J ( i ) n with ω in place of ∆ i . The function L n vanishes for frequen-cies ω higher than a cut-off frequency [19]. Its low-frequency behavior leads to various possibilities. First, α is finite only if, for any n , L n /ω does not diverge for ω →
0. If, in this limit, this ratio goes to zero for any n , then the second term in (15) vanishes. This termreads as α n T n + α n ′ T n ′ + . . . for Ohmic spectral densities L m ∼ ω , m = n , n ′ , . . . [19]. For a bath consisting of a D -dimensional continuous field, L n ∼ ω D for ω →
0, andhence is Ohmic for D = 1. However, note that, whereas J ( i ) n and K n are determined by the transverse couplingoperators w i , the functions L n depend on the longitudi-nal coupling. Consequently, L n can in principle be madeas small as we wish, irrespective of the transverse cou-pling strength. B. Steady state for identical two-level systems
From now on, we consider the case of identical TLSsplitting energies ∆ i = ∆, for which, as seen above, sta-tionary TLS entanglement may exist. We further assumethat the two TLS are coupled identically to the heatbaths, i.e., J (1) n = J (2) n = J n . This can hold for w = w if the two TLS are connected to different points of bath n . As a consequence of these assumptions, ˜ γ ± = ˜ γ ± , see(21). To simplify the following expressions, we introducethe notations :˜ γ +1 = ˜ γ +2 = γ , ˜ γ − = ˜ γ − = γη (22) β = β , β = βη ′ , α = γ (1 + η + ξ ) where β , η ′ and ξ are real for the coupling operators (20)and with the above assumptions. As discussed above, ξ is determined by the longitudinal coupling, whereas allthe other parameters are related to the lateral couplingoperators w i . The coefficient ˜ β , defined right after (16),is also real and does not contribute to the TLS steadystate.Under the assumption of real β , η ′ , ξ and ˜ β , we find,from (18), a real coherence c = γβ ( η ′ − η )(1 − η ) n η ′ − η ) + γ β (1 + η ) (1 + η + ξ ) − (1 + η ′ )(1 + 3 η + 3 η ′ + η ′ η ) o − . (23)The populations of the TLS steady state ρ can be writtenin terms of c as p p = p p = 1 z ηη + cβγz ( z ( η + η ′ ) − + ( η ′ η + η ′ − η − − − η η ) (24)where z = 1 + η . Note that c = 0 and ρ is uncorrelatedfor η ′ = η or η = 1. When this last equality is satisfied, ρ is proportional to the identity matrix, as expected fromthe case ∆ = 0, see Appendix. The equality η ′ = η holds, for instance, when E is in thermal equilibrium.The denominator in (23) vanishes for γ = β , η ′ = η and ξ = 0. These three conditions are fulfilled for w = w and v = v . There is not a unique steady state whenthe two TLS interact with E in exactly the same way [13].We also remark that, since c is real and p = p , the TLSsteady state can be written as ρ = p | ih | + p | ih | +( p + c ) | ψ + ih ψ + | + ( p − c ) | ψ − ih ψ − | with the Bell states | ψ ± i given by (17). We wil see below that, though | ψ + i and | ψ − i have the same energy ǫ = ǫ = 0, there existsa parameter regime in which p = 0 and p = | c | , and ρ is hence entangled. C. Entanglement induced by two heat baths
We now study the entanglement of ρ for an environ-ment E that consists of two heat baths of temperatures T and T . The steady state ρ is entangled if and only ifits partial transpose ρ Γ = P k p k | k ih k | + c ( | ih | + | ih | )has negative eigenvalues [24, 25]. The eigenvalues of ρ Γ are p = p and λ ± = ( p + p ) / ± [( p − p ) +4 c ] / / λ − can be negative.
1. Low-temperature entanglement region
As an interesting example, we consider the case ξ = 0and K = 0. This last condition means that the indi-rect interaction between the TLS is mediated only by
0 4 8T (units of )0 0.2 0.4 T ( un i t s o f ) D D FIG. 2: Entanglement region in the ( T , T ) plane for K = 0.The TLS steady state is entangled for temperatures below thedrawn line. The solid lines correspond to K = J , ξ = 0,and J /J = 5, 10 and 50. The size of the entanglementregion increases with J . For the dashed and dotted lines, thecoupling parameters are ξ = 0, and, respectively, K = 0 . J and J = 500 J , and K = 0 . J and J = 150 J . The short-dashed line is obtained for an Ohmic ξ = 0 . T / ∆ and for K = J = 50 J . bath 2. With this value of K , the results discussed herehold also for the three bath setup depicted in Fig.1 when T = T . We find that there can be a low-temperatureregion, determined by J /J and K /J , in which ρ isentangled, see Fig.2. We remark that the line delimitingthis entanglement region in the ( T , T ) plane, is tangentto the equilibrium line T = T for T , T →
0, and isessentially vertical at its other end for T ≪ ∆. Thesetwo behaviors come from the fact that the temperaturescontribute to ρ only via Boltzmann factors exp( − ∆ /T n ).Analytical results can be obtained by expanding theeigenvalue λ − to lowest order in these factors. It as-sumes negative values in the vicinity of T = T = 0, for | K | > J / √ J > J [ √ | K | /J − − . These re-quirements are the same in the Ohmic case discussed atend of IV A, for which ξ = ¯ ξ T + ¯ ξ T vanishes in the lim-its T , T →
0. For given coupling parameters satisfyingthe above conditions, ρ is not entangled if the tempera-tures T and T are too high. However, for ξ = 0, themaximum possible value of T is proportional to ∆ J /J in the large J limit, see Fig.2. Consequently, in thiscase, entangled states exist for any temperature T . For T , in contrast, our numerical results suggest that thesteady state is always separable for T greater than avalue of about 0 . J . For ξ = ¯ ξ T + ¯ ξ T , ρ is necessarily separable for T higherthan a temperature that diverges for ¯ ξ → -1 0 1K /J 0.5 1 K / J FIG. 3: Region of the parameter plane ( K /J , K /J ), K >
0, where entangled steady states can be found, for J /J =2 .
4, 2 .
7, 3, 5, 10, 100 and 1000. The entanglement region isabove the drawn line. Its size increases with J , to a maximumasymptotic value which is practically reached for J = 1000 J .
2. Requirements on the characteristics of the environment
Stationary entanglement can also be obtained for K =0. Since ρ is obviously invariant under the bath per-mutation ( J , K ) ↔ ( J , K ), it is enough to consider J > J . In this case, it can be shown that there existentangled steady states in the vicinity of T = T = 0 if( J + J ) − ( K + K ) − | K + K | (cid:12)(cid:12)(cid:12)(cid:12) K − K J J (cid:12)(cid:12)(cid:12)(cid:12) < . (25)This condition remains the same if the signs of both K and K are changed. Figure 3 shows, for K >
0, thecoupling parameter region where ρ can be found entan-gled. The following interesting conclusions can be drawnfrom these results. There is a particular value of J /J below which ρ is separable. In other words, the couplingsto the two heat baths must differ enough from each otherin order to observe stationary entanglement. For given J /J and K /J such that entangled steady states exist,these states are obtained for | K | /J not too far from 1.As mentioned above, the ratio | K | /J depends essen-tially on the distance d between the two points of bath 2where the TLS are connected. More precisely, it is deter-mined by a dimensionless parameter ¯ d = ∆ d /v where v is a characteristic field velocity of bath 2. The ratio | K | /J is small for large ¯ d . This imposes limitationson ∆ and on the temperature T to obtain an entangledsteady state. A distance d of 1 µ m and a low field ve-locity v of 10 m.s − , which is the order of magnitude ofthe sound velocity in solids, give a temperature of about10 mK, which is an experimentally accessible value. An-other important characteristic of bath 2 is its dimension-ality D . For example, for a continuous free field, K /J is equal to sin( ¯ d ) / ¯ d for D = 3, J ( ¯ d ) where J is the ze-roth order Bessel function of the first kind, for D = 2,and cos( ¯ d ) for D = 1. Thus, in this last case, stationary
0 0.08 0.16T (units of J /J )0 0.04 N ega t i v i t y
0 0.060 1 P opu l a t i on s D
11 2
FIG. 4: Negativity as a function of T in units of ∆ J /J for K = J = 100 J (short-dashed line), K = J = 1000 J (full line), K = 0 . J = 9 . J (dash-dotted line) and K =0 . J = 95 J (dashed line). The dotted lines correspond tothe large J approximation discussed in the text. The insetshows the populations of the ground and singlet states asfunctions of T in units of ∆ J /J for K = 0 . J = 95 J , K = J = 100 J and K = J = 1000 J . The two otherpopulations are small. The other parameters are ξ = 0, K =0 and T = 0 . entanglement can be obtained for large distances d andthe limitations discussed above do not apply.
3. Maximum attainable entanglement
Finally, we present quantitative results for the entan-glement of the steady state ρ . As a measure of entangle-ment, we use the negativity N ( ρ ) = ( k ρ Γ k − / k . k denotes the trace norm [3, 26]. Negativity rangesfrom 0 for separable states to 1 / − λ − when this eigenvalueis negative, and to 0 otherwise. The maximum value of N that we have found, is reached for the coupling param-eters K = 0, ξ = 0 and J ≫ J , and the temperatures T ≪ ∆ and T ≫ ∆, see Fig.4. In this regime, the TLSsteady state is given by p = σ (cid:2) (1 + θ ) (1 + 2 θ ) − κ (cid:3) , p = σθ (1 + θ )(1 + 2 θ ) p = σθ (1 + 2 θ ) , c = − σκθ (26)where σ = [(1 + 2 θ ) − κ ] − , κ = K /J and θ =( T / ∆)( J /J ). For | κ | 6 = 1, as θ increases from zero toinfinity, the ground state population p decreases from1 to 1 / p = p and p increases from 0 to 1 /
4, and | c | increases from zero to a maximum and then decaysback to zero. The low θ behavior is very different for | κ | = 1. In this case, c is finite in the limit θ → K = ± J and a temperature ∆ ≪ T ≪ ∆ J /J ,we find ρ = (2 / | + i h + | ⊗ | + i h + | + (1 / | ψ ∓ ih ψ ∓ | with the Bell states | ψ ∓ i given by (17), and a negativity N = ( √ − / ≃ .
04. The same entangled state can be reached for K = 0, as it will be clear from the discussionbelow. Our numerical results suggest that finite values of N correspond generally to states ρ such that essentiallyonly the ground state and one of the Bell states (17) arepopulated, see inset of Fig.4.To better understand the above results, it is interestingto consider the rates discussed after (17). For T ≪ ∆and T ≫ ∆, the rates of the upward transitions | i →| ψ ± i and | ψ ± i → | i are r ± up = ( J ± K ) T / ∆, and thatof the downward transitions | i → | ψ ± i and | ψ ± i → | i are J ± K + r ± up . For K = J and T ≪ ∆ J /J ,the rate of | i → | ψ − i and | ψ − i → | i is equal to r − up ,whereas that of | i → | ψ + i and | ψ + i → | i is 2 J ≫ r ± up .Consequently, the states | i and | ψ + i are essentially notpopulated, and the transition rate from the state | ψ − i to the ground state | i is effectively twice that of thereverse transition, leading to a factor of two between thetwo corresponding populations. The situation is similarfo K = − J . If | K | is too far from J or if T is toohigh, the values of the different rates are comparable andso are the populations of the states | i , | ψ ± i and | i , andhence ρ is separable. V. CONCLUSION
In summary, we have studied a system of two indepen-dent TLS weakly coupled to a stationary non-equilibriumenvironment. Considering first a general open system,we have determined their steady state. Without speci-fiying any further the surroundings of the TLS, it canbe shown that their steady state is uncorrelated if theirsplitting energies are different from each other. More-over, the state of each TLS is the same wether or not theother TLS is present. Consequently, in this case, a finitestrength of the coupling to the environment is required topossibly generate stationary TLS entanglement. In theopposite case of identical splitting energies, on the con-trary, stationary correlations between the TLS can existfor extremely weak coupling to the environment.To determine wether these correlations can be quan-tum, we have considered the case of an environment con-sisting of several bosonic heat baths at different temper-atures. We have shown that, for TLS coupled similarlyto two baths, there are temperatures and coupling pa-rameters for which the TLS steady state is entangled.An important requirement is that, for at least one bath,the points to which the TLS are connected must be closeenough to each other. However, this condition can berelaxed when one of the bath is one-dimensional. In thiscase, the TLS can be as far apart as we like. There arealso requirements on the baths temperatures. Essentially,one of them must be sufficiently low, of the order of theTLS splitting energy. Depending on the characteristicsof the coupling, the other temperature can be unlimited.We have found a parameter regime where the TLSsteady state is a statistical mixture of the product groundstate and of the entangled singlet state with weights 2/3and 1/3, respectively. This mixed state is entangled andthe corresponding negativity is about 0.04 which is thelargest value we have obtained. Interestingly, this regimecan be fully understood in terms of Fermi golden ruletransitions between appropriate states. To conclude, ourresults show that a relatively simple non-equilibrium en-vironment can lead to stationary entanglement of twoTLS, but certainly do not exhaust all the possible effectsof stationary non-equilibrium surroundings on quantumcorrelations. Larger entanglement of independent TLS,as measured by negativity for instance, may be achievablewith other environments or for TLS coupled differentlyto the environment. Further studies in these directionswould be of interest.
Appendix A: Special uncorrelated states
In this appendix, our purpose is to show that, for somespecial cases, the solution of (10) is of the form u kl = p k δ kl . This is the case if the sums X k ′ γ kl,k ′ k ′ ( i + ) p k ′ = π X A,B,k ′ P A h A | h kk ′ | B ih B | h k ′ l | A i× (cid:20)(cid:18) p k + p l − P B P A p k ′ (cid:19) δ ( ω AB − ω k ′ k ) + iπ p l − p k ω AB − ω k ′ k (cid:21) (A1)vanish for k and l such that ǫ k = ǫ l .For an environment in thermal equilibrium, i.e., P A ∝ exp( − E A /T ) where T is its temperature, u kl ∝ exp( − ǫ k /T ) δ kl satisfies (10) since, in the sums (A1), p k = p l and P B p k ′ /P A p k = exp[ − ( ω BA + ω k ′ k ) /T ]. Thisproof applies to any system S .We now consider the case of zero splitting energy ∆ and of an environment E that consists of heat baths atdifferent temperatures T n . First, the populations p k ob-tained in section III B ensure the vanishing of (A1) for k = l . For k = l , we start by showing that ˜ γ +2 = ˜ γ − which implies p = p and p = p , see (13). The differ-ence of these rates reads as˜ γ +2 − ˜ γ − = 2 π X A,B ( P A − P B ) |h B | w | A i| δ ( ω AB ) . (A2)For the kind of environment considered, H E = P n H E n and hence its eigenstates and eigenenergies can be writ-ten as | A i = Q n | A ( n ) i and E A = P n E A ( n ) . The pop-ulations P A factorise as P A ∝ Q n exp( − E A ( n ) /T n ). TheTLS are coupled to each bath thus w = P n w n . Con-sequently, the difference (A2) satisfies˜ γ +2 − ˜ γ − ∝ X n X A ( n ) ,B ( n ) |h B ( n ) | w n | A ( n ) i| × (cid:16) e − E A ( n ) /T n − e − E B ( n ) /T n (cid:17) δ ( E A ( n ) − E B ( n ) ) (A3)and hence vanishes. For ∆ = 0, the sum (A1) must bezero for ( k, l ) = (1 , , ,
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