Full decoherence induced by local fields in open spin chains with strong boundary couplings
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] D ec Full decoherence induced by local fields in open spin chains with strong boundary couplings
Vladislav Popkov , , Mario Salerno , and Roberto Livi , , , Dipartimento di Fisica e Astronomia, Universit`a di Firenze, via G. Sansone 1, 50019 Sesto Fiorentino, Italy Dipartimento di Fisica ”E. R. Caianiello”, CNISM and INFN Gruppo Collegato di Salerno,Universit`a di Salerno, via Giovanni Paolo II Stecca 8-9, I-84084, Fisciano (SA), Italy. Max Planck Institute for the Physics of Complex Systems, N¨othnitzer Straße 38, 01187 Dresden Germany INFN, Sezione di Firenze, and CSDC Universit`a di Firenze, via G.Sansone 1, 50019 Sesto Fiorentino, Italy ISC-CNR, via Madonna del Piano 10, 50019 Sesto Fiorentino, Italy Institut f¨ur Theoretische Physik, Universit¨at zu K¨oln, Z¨ulpicher Str. 77, 50937 Cologne, Germany
We investigate an open
XY Z spin / chain driven out of equilibrium by boundary reservoirs targetingdifferent spin orientations, aligned along the principal axes of anisotropy. We show that by tuning local magneticfields, applied to spins at sites near the boundaries, one can change any nonequilibrium steady state to a fullyuncorrelated Gibbsian state at infinite temperature. This phenomenon occurs for strong boundary coupling andon a critical manifold in the space of the fields amplitudes. The structure of this manifold depends on theanisotropy degree of the model and on the parity of the chain size. PACS numbers: 75.10.Pq, 03.65.Yz, 05.60.Gg, 05.70.Ln
I. INTRODUCTION
Manipulating a quantum system in non–equilibrium condi-tions appears nowadays one of the most promising perspec-tives for proceeding our exploration of the intrinsic richnessof quantum physics and for obtaining an insight on its po-tential applications [1–3]. In particular, much attention hasbeen devoted to the study of the nonequilibrium steady state(NESS) in quantum spin chains, coupled to an environment, ora measuring apparatus. This is described, under Markovian-ity assumptions [4–6], in the framework of a Lindblad Masterequation (LME) for a reduced density matrix, where a unitaryevolution, described via Hamiltonian dynamics, is competingwith a Lindblad dissipative action. Under these conditions,quantum spin chains subject to a gradient evolve towards aNESS, where spin and energy currents set in. In quasi one-dimensional systems, such currents exhibit quite exceptionalproperties like scalings, ballisticity and integrability [7–13].Many of these unexpected features stem from the fact thatthe NESS, corresponding to a fixed point of the LME dissi-pative dynamics with a gradient applied at the chain bound-aries, are not standard Gibbs-states. Moreover, further pecu-liar regimes appear when the time lapse between two succes-sive interactions of the quantum chain with the Lindblad reser-voir becomes infinitely small, while the interaction amplitudeis properly rescaled. In the framework of projective measure-ments, this kind of experimental protocol corresponds to theso-called Zeno effect, that determines how frequent projectivemeasurements on a quantum system have to be performed inorder to freeze it in a given state [14, 15].In this paper we shall rather focus on a Zeno regime fornon-projective measurements, that has been found to describenew counterintuitive scenarios for NESS. In particular, in [17]it was shown that in a boundary driven XXZ spin chain, forsuitable values of the spin anisotropy the NESS is a pure state.We want to point out the importance of this result in the per-spective of engineering dark states, that have the advantageto be more stable against decoherence, than isolated quantumsystems and, therefore, better candidates for technological ap- plications [3, 16]. Here we investigate how this nonprojectiveZeno regime can be manipulated by the action of a strictlylocal magnetic field, whose strength is of the order of the ex-change interaction energy of the XYZ Heisenberg spin chainmodel. The main result of our investigations is that, by sucha local effect, one can kill any coherence of the NESS andturn it into a mixed state at infinite temperature. More gener-ally, the von Neumann entropy of the NESS can be changedfrom its minimum value to its maximum one just by tuningthe local magnetic field, provided the coupling with the bathsis sufficiently strong.The paper is organized as follows. In Section II we describethe general properties of the nonprojectiveZeno setup and theway the spin XYZ chain is coupled to the Lindblad reservoirs.The effect of complete decoherence induced by the additionof a fine-tuned local magnetic field acting on the spins closeto the boundaries are discussed in Section III. A short accountof the symmetries characterizing the NESS in the special caseof a XXZ spin chain is reported in Section V. In Section VIwe investigate the non–commutativity of the different limits tobe performed in the model and the presence of correspondinghierarchical singularities. We conclude with a discussion onthe perspectives of our investigations (see Section VII).Appendices A,B, C and D contain some relevant technicalaspects.
II. THE MODEL
We study an open chain of N quantum spins, representedby the Hamiltonian operator H , in contact with boundaryreservoirs. The time evolution of the reduced density matrix ρ is described by a quantum Master equation in the Lindbladform [4–6] (we set ~ = 1 ) ∂ρ∂t = − i [ H, ρ ] + Γ( L L [ ρ ] + L R [ ρ ]) , (1)where L L [ ρ ] , L R [ ρ ] are Lindblad dissipators acting on spinsat the left and right boundaries of the chain, respectively. Thisis an usual setup for studying transport in quantum spatiallyextended systems, where the explicit choice of L L and L R issuggested by the kind of application one has in mind. In thisway, one describes an effective coupling of the chain, or a partof it, with baths or environments. Within the quantum proto-col of repeated interactions, Eq.(1) describes an exact timeevolution of the extended quantum system, provided the cou-pling with the Lindblad reservoirs is suitably rescaled [6].Here we are interested to explore the strong coupling con-dition, i.e. Γ → ∞ , that corresponds to the so–called Zenoregime. In this case one can obtain the stationary solution ofEq.(1) in the form of the perturbative expansion ρ NESS ( ξ, Γ) = ∞ X k =0 (cid:18) (cid:19) k ρ k ( ξ ) , (2)where ρ NESS ( ξ, Γ) is the density matrix of the non equilib-rium steady–state and ξ is a symbol epitomizing the modelparameters (e.g. bulk anisotropy, exchange energy, magneticfield, etc.).Suppose that the stationary solution ρ NESS ( ξ, Γ) is unique.This fact will be validated further for all our examples.Moreover, the first term of expansion (2), i.e. ρ =lim Γ →∞ lim t →∞ ρ (Γ , ξ, t ) , satisfies the stationarity condition L LR [ ρ ] = 0 , where L LR = L L + L R is the sum of the Lind-blad actions in (1) . This suggests that ρ can be representedin a factorized form ρ = ρ L ⊗ (cid:18) I (cid:19) ⊗ N − + M ( ξ ) ! ⊗ ρ R , (3)where ρ L and ρ R are the one-site density matrices at the chainboundaries, satisfying L L [ ρ L ] = 0 and L R [ ρ R ] = 0 , and M is a matrix to be determined self-consistently. It is convenientto separate explicitly the identity matrix (cid:0) I (cid:1) ⊗ N − from M , insuch a way that M is a traceless operator, due to the condition T r ( ρ ) = 1 .By substituting the perturbative expansion (2) into Eq.(1)and by equating terms of the order Γ − k , one can easily obtainthe recurrence relation i [ H, ρ k ] = 12 L LR ρ k +1 , k = 0 , , , · · · (4)whose general solution has the form ρ k +1 = 2 L − LR ( i [ H, ρ k ]) + ρ L ⊗ M k +1 ⊗ ρ R , k = 0 , , , · · · (5)provided the following secular conditions (for more details see[25]) are satisfied [ H, ρ k ] ∩ ker( L LR ) = ∅ , (6)where ker( L LR ) denotes the nucleus of the operator L LR .Notice that, in order to obtain an explicit solution, one hasto compute the inverse operator L − LR , that appears in Eq.(5).In summary, Eqs (3), (5) and (6) define a general perturba-tive approach, that applies in the Zeno (i.e., strong coupling)regime. We consider the Hamiltonian H = H XY Z + V + V N − where H XY Z = N − X j =1 (cid:0) J x σ xj σ xj +1 + J y σ yj σ yj +1 + ∆ σ zj σ zj +1 (cid:1) , (7)is the Hamiltonian of an open XY Z
Heisenberg spin chainand V l is a local inhomogeneity field acting on spin l to bespecified later on (see Eqs 15-16). Moreover, we considerLindblad dissipators, L L and L R , favouring a relaxation ofboundary spins at k = 1 and k = N towards states describedby one-site density matrices ρ L and ρ R , i.e. L L [ ρ L ] = 0 and L R [ ρ R ] = 0 . In particular, we choose boundary reservoirsthat tend to align the spins at the left and right edges along thedirections ~l L and ~l R , respectively. These directions are identi-fied by the longitudinal and azimuthal coordinates as follows: ~l L,R = (sin θ L,R cos ϕ L,R , sin θ L,R sin ϕ L,R , cos θ L,R ) . Such a setting is achieved by choosing the Lindblad actionin the form L [ ρ ] = L L [ ρ ] + L R [ ρ ] , where L A [ ρ ] = − n ρ, S † A S A o + S A ρ S † A , A = L , R (8)and S L = [(cos θ L cos ϕ L ) σ x + (cos θ L sin ϕ L ) σ y − (sin θ L ) σ z + iσ x ( − sin ϕ L ) + iσ y (cos ϕ L )] / , (9) S R = [(cos θ R cos ϕ R ) σ xN + (cos θ R sin ϕ R ) σ yN − (sin θ R ) σ zN + iσ xN ( − sin ϕ R ) + iσ yN (cos ϕ R )] / . (10)In the absence of the unitary term in (1), the boundary spins re-lax with a characteristic time Γ − to specific states describedvia the one-site density matrices ρ L = 12 (cid:16) I + ~l L ~σ (cid:17) (11) ρ R = 12 (cid:16) I + ~l R ~σ N (cid:17) . (12)The condition L A [ ρ A ] = 0 follows from definition (8), whilethe relations S A S † A = ρ R and ( S A ) = ( S † A ) = 0 can beeasily checked.In analogy with [18], it can be easily shown that, for thechosen boundary dissipation setup described by Eqs (8), (9)and (10), the NESS is unique. By applying the perturbativeapproach in the Zeno regime, one finds that the unknown ma-trices M k (∆) are fully determined by secular conditions (6).As shown in Appendix A, for the specific choice (8) of theLindblad operators, they are equivalent to the requirement ofa null partial trace T r ,N ([ H, ρ k ]) = 0 , k = 0 , , , · · · . (13)We want to point out that the computation of the full setof matrices { M k (∆) } for any ∆ = 0 is quite a nontrivialtask. However, in the Zeno limit, Γ → ∞ , we are just in-terested in computing the zero–th and the first order contribu-tions M , M , which can be completely determined by solv-ing the set of secular equations (13) for k = 0 , , . III. MANIPULATIONS OF NESS BY NON UNIFORMEXTERNAL FIELDS
The properties of the model introduced in the previous sec-tion have been widely investigated for V l = 0 and ϕ = π/ in[25]. Here we are interested in studying how the properties ofthe NESS can be modified when V l is an additional local field,that corrupts the homogeneity of the XYZ spin chain.Notice first that a local field applied to the boundary spins atpositions k = 1 and k = N does not affect the strong couplinglimit ρ = lim Γ →∞ ρ NESS (Γ) . On the other hand, applyinga local field to the spins at positions k = 2 and k = N − canmodify ρ in a nontrivial way. The Hamiltonian reads H = H XY Z + V + V N − (14)where V = ~h~σ j = h x σ x + h y σ y + h z σ z (15) V N − = ~g~σ N − = g x σ xN − + g N − σ yj + g z σ zN − (16)Carrying out the procedure outlined in the previous sec-tion, we can find the form of the density matrix of the NESSin the Zeno regime, ρ . This is a function of the angles θ L , ϕ L , θ R , ϕ R , of the anisotropy parameter ∆ and of the lo-cal fields ~h, ~g . One can argue that, in general, the NESSshould be an entangled state, depending in a nontrivial wayon the local fields. Due to the boundary drive, the NESS typ-ically exhibits nonzero currents (magnetization current, heatcurrent, etc.), irrespectively of the presence of the local fields.However, in the Zeno limit, there are critical values of thelocal fields for which a complete decoherence of the NESSoccurs.More precisely, we formulate our results under the follow-ing boundary condition assumptions: • targeted boundary polarizations are neither collinearnor anti-collinear ( ~l L = ± ~l R ); • at least one of the polarizations (e.g. the left targetedpolarization) is directed along one of the anisotropy axis X, Y, or Z ; • the corresponding local fields ( ~h at site 2 and ~g at site N − ) are collinear to the respective targeted boundarypolarizations ~h = h~l L , ~g = g~l R .Then, there exists a zero-dimensional or a one dimensionalcritical manifold in the h, g –plane ( h cr , g cr ) , such that, in theZeno limit , the NESS on this manifold becomes ρ NESS (∆) | ( h cr ,g cr ) = ρ L ⊗ (cid:18) I (cid:19) ⊗ N − ! ⊗ ρ R , (17)Notice that this a peculiar state: apart the frozen boundaryspins, all the internal spins are at infinite temperature. Indeed,tracing out the boundary spins, one obtains the Gibbs state atinfinite temperature T r ,N ρ L ⊗ (cid:18) I (cid:19) ⊗ N − ! ⊗ ρ R ! = (cid:18) I (cid:19) ⊗ N − . (18) Also notice that on the critical manifold the Von-Neumannentropy of the NESS, S V NE = − T r ( ρ NESS log ρ NESS ) ,in the Zeno limit attains its maximum value given by lim Γ →∞ max( − T r ( ρ NESS log ρ NESS )) = N − , since ρ L , ρ R are pure states. In the following, we also refer tostate (17) as the stateofmaximaldecoherence.We have performed explicit calculations (see below) thatconfirm the above statement for different spin chains up to N = 8 . The particular form of the NESS assumed in thesecases, however, strongly suggests that the above results maybeof general validity and the critical manifold ( h cr , g cr ) inde-pendent on N .The critical manifold has been fully identified for the fol-lowing cases.- XYZ chain : J x = J y = ∆ . If the left, ~l L , and the right, ~l R , polarizations point in directions of different principal axes ~l L = e α , ~l R = e β α = β α, β = X, Y, Z (19)where e X = (1 , , , e Y = (0 , , , e Z = (0 , , , thenfor chains with an even number, N , of spins, the manifold ( h cr , g cr ) consists of three critical points: P α = ( − J α , , P β = (0 , − J β ) and P α,β = ( − J α , − J β ) . For odd N , thecritical point P α,β is missing and the critical manifold reducesonly to the points P α , P β , above. If only one of the two bound-ary driving points in the direction of a principal axis, the crit-ical manifold reduces to a single point, either P α or P β , forboth even and odd N .- XXZ chain : J x = J y = J = ∆ . If both ~l L and ~l R lay onto the XY –plane, we can parametrize the targetedboundary polarizations via a twisting angle in the XY –plane ϕ as θ = θ = π , ϕ = ϕ, ϕ = 0 , corresponding to ~l L = (cos φ, sin φ, and ~l R = (1 , , . The critical fieldsare aligned parallel to the targeted boundary magnetization,i.e. ~h cr = ( h cr cos φ, h cr sin φ, , ~g cr = ( g cr , , , and wefind the one–dimensional critical manifold h cr + g cr = − J, h cr = − J (20)Notice that this expression is independent of system size N ,of the anisotropy ∆ and of the twisting angle ϕ . If one of thetwo targeted polarizations points out of the XY –plane, thecritical manifold becomes zero-dimensional and consists ofone, two or three critical points (depending on the polarizationdirection and on N being even or odd) as discussed for the fullanisotropic case.- XXX chain : J x = J y = ∆ ≡ J . The critical manifold forarbitrary non-collinear boundary drivings is one-dimensionaland it is given by Eq.(20).The above statements are illustrated in Figs. 1 and 2 forthe case of a chain of N=4 spins. In particular, in Fig. 1 weshow a contour plot of the VNE surface as a function of theapplied fields for the XYZ case with left and right boundarypolarizations fixed along the X and Z directions, respectively. Figure 1: (Color online) Contour plot of the Von-Neumann entropy S V NE in the Zeno limit, as a function of the local fields for anopen XYZ chain of N = 4 spins with exchange parameters J x =1 . J y = 0 . , ∆ = 2 . Green, white and green dots denote the criticalpoints P X = ( − J X , , P XZ = ( − J X , − ∆) , P Z = (0 , − ,where the VNE reaches its maximum value S V NE = 2 and theNESS becomes a completely mixed state, respectively. Other pa-rameters are fixed as ~l L = e X , ~l R = e Z . Green, yellow, pink,orange, brown, red and blue contour lines refer to S V NE values: , . , . , . , . , . , . , respectively. Notice the presence ofthe narrow corridors around P X and P Z in which the deviation, S V NE − , of the VNE from its maximum value becomes very small. The three critical points P X , P Z , P X,Z mentioned above cor-respond to the green, red, and white dots shown in the toppanel of the Figure. Notice the presence of narrow corri-dors (blue shaded) around the P X and P Z critical points, in-side which the VNE keeps very close to the maximal value S V NE = 2 but never reach it, except at the critical points.This is quite different from the partially anisotropic XXZ caseshown in Fig. 2, where the existence of the critical line (blueline) is quite evident.Similar results are found also for longer chains. In particu-lar, in Fig.3 we show a cut of the VNE surface for a partiallyanisotropic XXZ chain of N = 5 spins. For the sake of sim-plicity we have set J x = J y = 1 and considered the cut at h = 0 so that the VNE of the NESS, in the Zeno limit, be-comes a function of g only. We see that for g = g cr = − ,the VNE reaches the maximum value N − indicating thatthe corresponding NESS has the form (17).As to the dependence of the critical manifold on parity of N , we find that while for odd sizes N = 3 , , and XY Z
Hamiltonian ( see Fig. 4, top panel for an illustration) thereare only two critical points (the critical point P α,β is missing),for even sizes N = 4 , , cases there are three critical points.These observations strongly suggest a qualitative differencebetween even and odd N in the model, which is manifested inother NESS properties as well, see e.g. (22),(23).It is worth to note here that for h = g = − J , i.e. the caseexcluded in (20), the NESS behaves non-analytically in theZeno limit Γ → ∞ . As we are going to discuss in Sec.VI, thisnon-analyticity is a consequence of the non-commutativity of Figure 2: (Color online) Contour plot of S V NE in the Zeno limit, asa function of the local fields for the XXZ chain with N = 4 spins.Parameters are J x = J y ≡ J = 1 . , ∆ = 1 ,~l L = − e Y , ~l R = e X .The green, yellow, pink, orange, brown, red, blue contour lines referto S V NE values: . , . , . , . , . , . , , respectively. The bluecontour is in full overlap with the critical line h Y + g X = − J = − . the limits Γ → ∞ and h = g → − J .Conversely, for any finite boundary coupling Γ , i.e. far fromthe Zeno limit, the NESS is analytic for arbitrary amplitudesof the local fields (the first order correction to the NESS forlarge Γ is proportional to Γ − as shown in the App.C). This isalso seen from Fig.5 where the VNE of the NESS is reportedas a function of the local field g for different values of theboundary coupling Γ and same parameters as in Fig.3 (seecurve ∆ = 0 . ). Notice that the thin black line obtained for Γ = 10 , is already in full overlap with the Zeno limitingcurve depicted in Fig.3 for ∆ = 0 . . Also note the persistenceof the peak at g = − even for relatively small values of Γ away from the Zeno limit.Similar behaviors are observed for different choices ofboundary polarizations and of local fields (not shown forbrevity), thus opening the possibility to detect the signatureof the above phenomena in real experiments. In this respectwe remark that the near–boundary magnetic field h and theanisotropy ∆ as suitable parameters for controlling the dissi-pative state of the system in a NESS. Thus, if g = 0 , the NESScan be made a pure state by tuning the anisotropy ∆ to a spe-cific value ∆ ∗ ( ϕ, N ) . For instance, we find that for g = 0 and ∆ ∗± ( π/ ,
5) = q ± √ the NESS is a pure state [26],while for g cr = − , the NESS in the bulk becomes an infinitetemperature state (17), i.e. a maximally mixed state. Thus,by suitably tuning the anisotropy and the local magnetic fieldone can pass from minimally mixed (pure) NESS state to amaximally mixed one.It should be emphasized at this point that general thermo-dynamic equilibrium quantities, e.g. the temperature, are notwell-defined for a generic NESS. In fact, pure states allowedby Liouvillian dynamics are not ground states of the Hamil-tonian, but are characterized by a property of being commoneigenvectors of a modified Hamiltonian and of all Lindblad - - - - - g S VNE
Figure 3: (Color online) Von-Neumann entropy of the NESS S V NE = − T r ( ρ NESS log ρ NESS ) in the Zeno limit, asfunction of local field g , for different values of spin exchangeanisotropy. Thick,thin,dashed and dotted curves correspond to ∆ =0 . , . , . , . , respectively. For g = − the NESS is acompletely mixed state for which VNE reaches its upper limit. Pa-rameters: h = 0 , N = 5 , θ L = θ R = π , ϕ L = − π/ , ϕ R = 0 . - - - - - h cr S VN E - - - - - h cr S VN E Figure 4: (Color online) Cuts of the Von-Neumann entropy surface ofthe NESS in the Zeno limit, as function of critical field for the XYZchains with N = 5 (top panel) and N = 6 (bottom panel) spins.The red, blue and black lines refer to cuts made at g Z = 0 , h X = 0 and g Z = − , respectively. Other parameters are fixed as in Fig. 1.Notice that for N odd the VNE reaches its maximum value N − only at points P X = ( − J x , and P Z = (0 , − while for N even the maximum is reached also at the point P XZ = ( − J x , − ∆) . operators [16, 21]. Likewise, an absence of currents in theNESS does not necessarily imply a thermalization of the sys-tem: in fact also for fully matching boundary conditions theNESS is not a Gibbs state at some temperature, so that correla-tion functions remain far from those of an equilibrium system.From this point of view, the decoherence effect described inpresent paper can be viewed as a reaction of a nonequilibriumsystem on a local perturbation (the local magnetic field): as iswell-known, a local perturbation in nonequilibrium can leadto global changes of a steady state.On the other hand, a fully mixed state as such has appearedalready in the context of driven spin chains: if both Lind-blad boundary reservoirs target trivial states with zero polar-ization ( ρ R = ρ L = I/ ), the NESS is maximally mixed ρ NESS = (cid:0) I (cid:1) ⊗ N , which is a trivial solution of the steadyLindblad equation for any value of boundary coupling. Therespective NESS is often being referred to as a state with in-finite temperature [22]. Note that our case is drastically dif-ferent from the latter: the maximally mixed state (17) appearsonly in the bulk, after tracing the boundary spins, in a systemwith generically strong boundary gradients, and under strongboundary coupling.A few more remarks are in order: (i) the amplitudes of thecritical local fields scale with the amplitude of the Hamilto-nian exchange interaction, i.e. h cr → γh cr if H XXZ → γH XXZ (this is a consequence of the linearity of the recur-rence relations (5) and (13) ); (ii) the NESS may take the form(17) only in the Zeno limit Γ → ∞ ; in fact, the first ordercorrection to the NESS is proportional to Γ − and does notvanish (see Appendix C); the fully decoherent state (17) is in-trinsic to nonequilibrium conditions and, strikingly enough, itpersists even for nearly matching or fully matching boundarydriving, as we are going to discuss in Sec.IV.We want to conclude this Section by pointing out that afully analytic treatment of the problem for arbitrary large val-ues of N should encounter serious technical difficulties. Themain one concerns the solution of the consistency relationsdetermined by the secular conditions (6) for the perturbativeexpansion (2), with the zero-order term given by (17). More-over, finding the first order correction to NESS, proportionalto Γ − , amounts to solve a system of equations, whose num-ber grows exponentially with N . With Matematica we wereable to solve that system of equations analytically for N ≤ and numerically up to N ≤ . IV. MATCHING AND QUASI–MATCHING BOUNDARYDRIVINGS
In the previous Section we have discussed the case wherethe complete alignment of the boundary Lindblad baths wasexcluded. In this Section we want to analyze the specific casewhere they are aligned (or quasi–aligned) in the same direc-tion on the XY –plane.A complete alignment, i.e. ~l L = ~l R , corresponds to a per-fect matching between the left and right boundary Lindbladbaths, that yields a total absence of boundary gradients, so thatany current of the NESS vanishes. Also in this case the Gibbs - - - - - g S VNE
Figure 5: (Color online) Von-Neumann entropy of the NESS as func-tion of the local field g and for different values of the coupling Γ .Other parameters are fixed as: N = 5 , ∆ = 0 . , J x = J y = 1 , h = 0 , θ L = θ R = π , ϕ L = − π/ , ϕ R = 0 . The thin (black),red (dashed), dotted (green), dot-dashed (blue) curves refer to values Γ = 10 , Γ = 10 , Γ = 50 , Γ = 25 , respectively. state at infinite temperature can be achieved by suitably tun-ing the values of the near–boundary fields, but for even-sizedchains, only.Let us first illustrate this finding for the XYZ case. With noloss of generality, we can set ~l L = ~l R = e Z = (0 , , . Thebehavior of the driven chain with local fields depends drasti-cally on whether the size of the chain N is an even or an oddnumber: in the former case we find the critical one dimen-sional manifold, defined by h cr + g cr = − , h cr = − ∆ ; (21)in the latter case N = 3 , , .. , we do not find any critical point.This result has been found explicitly for ≤ N ≤ , but,since it depends on the effect of local perturbations, it seemsreasonable to conjecture that it should hold for larger finitevalues of N . This result holds as long as the Heisenberg ex-change interaction in the plane perpendicular to the targeteddirection (the XY –plane in this example) is anisotropic, i.e. J x = J y . Conversely, for J x = J y , the infinite temperaturestate (17) cannot be reached for any value of the local fields h and g . There is a delicate point to be taken into accountwhen we fix h = h cr and we perform the limit J y → J x , i.e.we reestablish the model isotropy: for complete alignment, ~l L = ~l R = e Z , the NESS is singular. The way this singularitysets in is shown in Fig. 6. In the limit when the anisotropyin the direction transversal to the targeted direction becomesinfinitesimally small | J y − J x | → the NESS is a pure statewith minimal possible S V NE → for any amplitude of thelocal field values, except at a critical point where S V NE ismaximal.The noncommutativity of similar limits and the dependenceof the NESS properties on the parity of system size N inLindblad–driven Heisenberg chains, have been observed inprevious studies [23],[24]. Also in these cases, the origin ofnoncommutativity is a consequence of global symmetries ofthe NESS, that, for our model, are discussed in Section V. - - - - - g S VNE
Figure 6: (Color online) Von-Neumann entropy of the NESS S V NE = − T r ( ρ NESS log ρ NESS ) in the Zeno limit, as functionof local field g , for XY Z model with matching boundary driving ~l L = ~l R = (0 , , , for different values of spin exchange anisotropydifference J y − J x . Thin, thick, dashed and dotted curves correspondto J y − J x = 0 . , . , . , . . Parameters: J x = 1 . , ∆ = 2 , N = 4 . In the isotropic case, as long as the local fields are paral-lel to the targeted spin polarization, the NESS does not de-pend on them: it is a trivial factorized homogeneous statewith a maximal polarization matching the boundaries, i.e. ρ NESS = ( ρ L ) ⊗ N . This can be easily verified by a straight-forward calculation.Another kind of NESS singularity can be found in the par-tiallyanisotropiccase, with quasi–matching boundary drivingin the XY isotropy plane. As a mismatch parameter we in-roduce the angular difference between the targeted polariza-tions at the left and the right boundaries, ϕ = ϕ L − ϕ R . For ϕ = 0 and in the absence of local fields, we have found thatthe spin polarization at each site of the chain is parallel tothe targeted polarization; on the other hand, even in the Zenolimit, it does not saturate to the value imposed at the bound-aries j = 1 , N . In general, this is not an equilibrium Gibbsstate, even in the Zeno limit and for any finite boundary cou-pling Γ . However, if the near–boundary fields are switched onand tuned to their critical values, the coherence of this stateis destroyed and the NESS becomes an infinite temperatureGibbs state. On the other hand, we have found that there isa relevant difference between quasi–matching and mismatch-ing conditions for even and odd values of N (notice that theisotropic and the free fermion cases, ∆ = 1 , ∆ = 0 , are spe-cial and should be considered separately). Our results can besummarized as follows:- N odd. We can fix the boundary mismatch by choosing ϕ L = ϕ, ϕ R = 0 , the left local field h = 0 , and study theNESS as a function of the right local field g . At g = g cr = − , the NESS becomes trivial (maximally mixed); however,as shown in panel (a) of Fig.7, for small mismatch we finda singular behavior of the NESS close to g = g cr . Analyticcalculations (not reported here) show that for ϕ = 0 there is asingularity at g = g cr , as a result of the non-commutativity ofthe limits ϕ → and g → g cr .- N even. Unlike the previous case, the NESS is analytic for - - - - - - g S VNE (a) - - - g S VNE (b)
Figure 7: (Color online) Von Neumann entropy of an internal block,(sites , .., N − ), for N = 5 (Panel a), and N = 4 (Panel b),versus the local field g , for different ϕ . Parameters: ∆ = 0 . . Panel(a): Thick and dotted curve correspond to ϕ = π/ and ϕ = π/ respectively. Panel (b): Thick and dotted curve correspond to ϕ = π/ and ϕ = 0 respectively. small and zero mismatch (see panel (b) of Fig.7). For g = g cr the NESS becomes trivial (maximally mixed), also for ϕ = 0 .Finally, let us comment about two special cases, for ”equi-librium” boundary driving conditions, i.e. ϕ L = ϕ R . For ∆ = 0 (free fermion case), the NESS is a fully mixed state(apart from the boundaries) for all values of g . For ∆ = 1 (isotropic Heisenberg Hamiltonian), the NESS is a trivial fac-torized state, fully polarized along the axis of the boundarydriving, for any value of g . Both statements can be straight-forwardly verified.NESS singularities, onset of which can be recognized inFig. 6 and Fig. 7a, appear because of non-commutativity oflimits. Noncommutativity of various limits, implying singu-larities and nonergodicity, which are due to global symme-tries is a well-established phenomenon and occurs already inKubo linear response theory describing fluctuations of a ther-malized background. In nonequilibrium open quantum sys-tems, however, the presence of NESS symmetries at specialvalue of parameters is manifested much strongly, due to richerphase space which includes both bulk parameters (such asanisotropy and external field amplitudes) and boundary pa- rameters (such as coupling strength). As a result, noncommu-tativity of the limits and consequent NESS singularities seemsto be a rather common NESS feature. In the next two sectionswe reveal some of NESS symmetries and show that the re-spective singularities, connected with them, can be observedalready in a finite system consisting of a few qubits. V. SYMMETRIES OF NESS
Symmetries of the LME are powerful tools that reveal gen-eral, system size-independent properties of the Liouvilleandynamics (1). In the case of multiple steady states, symme-try based analysis allows one to predict different basins of at-traction of the density matrix for different initial conditions[19]. For a unique steady state, symmetry analysis provides aqualitative description of the Liouvillean spectrum [20] or theformulation of selection rules for steady state spin and heatcurrents [23]. It is instructive to list several general NESSsymmetries valid for our setup. We restrict to
XXZ
Hamil-tonian with J x = J y = 1 , and perpendicular targeted polar-izations in the XY –plane, i.e. ~l L = (0 , − , , ~l R = (1 , , .The LME has a symmetry, depending on parity of N , whichconnects the NESS for positive and negative ∆ . Let us denoteby ρ NESS ( N, ∆ , h, g, Γ) the nonequilibrium steady state so-lution of the Lindblad master equation (see (1) and (8) ) forthe Hamiltonian (B1) reported in Appendix B. It is known thatthis NESS is unique[18] for any set of its parameters; more-over, one can easily check that ρ NESS ( N, − ∆ , h, g, Γ) =
U ρ ∗ NESS ( N, ∆ , h, g, Γ) U (22) ρ NESS ( N, − ∆ , h, g, Γ) = Σ y U ρ ∗ NESS ( N, ∆ , h, g, Γ) U Σ y (23)These relations hold for even and odd values of N , respec-tively; here Σ y = ( σ y ) ⊗ N , U = Q n odd ⊗ σ zn and the asteriskon the r.h.s. of both equations denotes complex conjugation inthe basis where σ z is diagonal. Eqs (22) and (23) hold for anyvalue of the local fields h, g and for any coupling Γ , includ-ing the Zeno limit Γ → ∞ . Due to properties (22) and (23),we can restrict to the case ∆ ≥ further on. For g = − h , ρ NESS ( N, ∆ , h, g, Γ) has the automorphic symmetry, ρ NESS ( N, ∆ , h, − h, Γ) =Σ x U rot Rρ NESS ( N, ∆ , h, − h, Γ) RU + rot Σ x , (24)where R ( A ⊗ B ⊗ ... ⊗ C ) = ( C ⊗ .... ⊗ B ⊗ A ) R is a left-rightreflection, U rot = diag (1 , i ) ⊗ N is a rotation in XY plane, U rot σ xn U + rot = σ yn , U rot σ yn U + rot = − σ xn , and Σ x = ( σ x ) ⊗ N . VI. NON-COMMUTATIVITY OF THE LIMITS Γ → ∞ AND h → h crit , ∆ → ∆ crit . HIERARCHICALSINGULARITIES. Here we consider the
XXZ
Hamiltonian and a perpendic-ular targeted polarizations in the XY –plane ~l L = (0 , − , , ~l R = (1 , , ; the near–boundary fields are taken on the crit-ical manifold, i.e. h + g = − . For N = 3 , and ∆ > wehave found the noncommutativity conditon lim Γ →∞ lim h → ρ NESS ( N, h, − h − , ∆ , Γ) =lim h → lim Γ →∞ ρ NESS ( N, h, − h − , ∆ , Γ) . (25)Making use of (17), the r.h.s. of (25) can be rewritten lim h → lim Γ →∞ ρ NESS ( N, h, − h − , ∆ , Γ) = ρ L (cid:18) I (cid:19) ⊗ N − ρ R . (26)For the simplest nontrivial case N = 3 , the validity of thesenoncommutativity relations is verified by the calculations re-ported in Appendix B (see (B4)). On top of (25), we find anadditional singularity at the isotropic point ∆ = 1 for N > Γ →∞ lim ∆ → lim h → ρ NESS ( N, h, − h − , ∆ , Γ) =lim Γ →∞ lim h → lim ∆ → ρ NESS ( N, h, − h − , ∆ , Γ) . (27)Due to the symmetry conditions (22) and (23), the singularityis present also for ∆ = − . Eqs (25) and (27) entail the pres-ence in our model of a hierarchical singularity. Namely, thefull parameter space of a model is a four dimensional one andconsists of the parameters { ∆ , Γ − , h, g } . As a consequenceof (25), a NESS is singular on a critical one-dimensional man-ifold { any ∆ , Γ − = 0 , h = − , g = − } . According to(27), further singularities appear for two special values of theanisotropy, inside the critical manifold { ∆ = ± , Γ − =0 , h = − , g = − } , engendering a zero-dimensional sub-manifold of the critical manifold. Thus, a hierarchy of singu-larities is formed. It is quite remarkable that such hierarchicalsingularities emerge without performing the thermodynamiclimit N → ∞ . In fact, as shown in Appendix D, they can beexplicitly detected already for N = 4 . For N = 5 we havefound other singular manifolds, parametrized by h, g, and ∆ .For the sake of space, details of this case will be reported in afuture publication.The appearance of the singularity at h → − , g → − is a consequence of the additional symmetry (24) at thispoint. By direct inspection of the analytic formulae obtainedfor N = 3 , , , we can guess the form of the limit state lim Γ →∞ lim h → lim ∆ → as a fully factorized one, namely lim Γ →∞ lim h →− lim ∆ → ρ NESS ( N, h, ∆ , Γ) = (28) ρ L (cid:18) σ x − σ y + 12 I (cid:19) ⊗ N − ρ R . Conversely, for generic ∆ and odd N ≥ , the limit state lim Γ →∞ lim h →− ρ NESS ( N, h, ∆ , Γ) does not take a factor-ized form. Notice also that from making use of Eqs (22) and(23), we readily obtain also the NESS limit state for ∆ → − : lim Γ →∞ lim h →− lim ∆ →− ρ NESS ( N, h, ∆ , Γ) = (29) ρ L N − Y i =2 ⊗ (cid:18) ( − i (cid:0) ( − N σ x + σ y (cid:1) + 12 I (cid:19) ρ R . VII. CONCLUSION.
In this paper we extensively analyzed the properties of theNESS of open Heisenberg spin chains, subject to the action ofLME at their boundaries and of perturbing magnetic fields atthe near-boundary sites. The setup we deal with operates inthe Zeno regime, i.e. in the strong coupling limit, Γ → + ∞ (see Eq. (1) ). Most of our analytic and numeric calculationshave been performed for relatively small values of the chainsize N . On the other hand, as a consequence of the local na-ture of the reservoirs and of the perturbing magnetic fields,we conjecture that many of these results could be extendedto large finite values of N : the delicate question of how theymight be modified in the thermodynamic limit is still open. Atthe present level of standard computational power, the strategyof performing large scale calculations to get any inference onsuch a limit is impractical, because the number of equationsto be solved grows exponentially with N .Despite all of these limitations, the main outcome of ourstudy is quite unexpected: by tuning the near–boundary mag-netic fields we can manipulate the NESS, making it pass froma dark pure state (for a suitable choice of the value of theanisotropy parameter ∆ ), to a fully uncorrelated mixed stateat infinite temperature.We have also discussed how this general scenario emergesin the anisotropic, partially anisotropic and isotropic cases.The influence of different alignment conditions imposed bythe Lindblad reservoirs has been extensively explored, to-gether with the symmetries of the NESS and their importancefor engendering hierarchical singularities due to the noncom-mutativity of different limits, performed on the model param-eters.A physically relevant point in our discussion concerns thepossibility of performing such a manipulation of the NESSalso for large but finite values of Γ : numerical investigationsconfirm this expectation, thus opening interesting perspectivesof experimental investigations. Acknowledgements
VP acknowledges the Dipartimento diFisica e Astronomia, Universit`a di Firenze, for partial sup-port through a FIRB initiative. M.S. acknowledges supportfrom the Ministero dell’ Istruzione, dell’ Universit´a e dellaRicerca (MIUR) through a
Programma di Ricerca Scientificadi Rilevante Interesse Nazionale (PRIN)-2010 initiative. Asubstantial part of the manuscript was written during a work-shop in Galileo Galilei Institute in Florence. We thank DavidMukamel for useful discussions. RL acknowledges the sup-port and the kind hospitality of MPIPKS in Dresden, wherepart of this manuscript was written.
Appendix A: Inverse of the Lindblad dissipators and secularconditions. L L and L R are linear super-operators acting on a ma-trix ρ as defined by Eqs (9) and (10). In our case,each super-operator act locally on a single qubit only.The eigen-basis { φ αR } α =1 of L R φ αR = λ α φ αR is φ R = { ρ R , ρ R − I, − sinϕ R σ x + cosϕ R σ y , cosθ R ( cosϕ R σ x + sinϕ R σ y ) − sinθ R σ z } , , with the respective eigenvalues { λ α } = { , − , − , − } . Here I is a 2 × σ x , σ y , σ z are Pauli matrices, and ρ R is targeted spin opien-tation at the right boundary. Analogously, the eigen-basis andeigenvalues of the eigenproblem L L φ βL = µ β φ βL are φ L = { ρ L , ρ L − I, − sinϕ L σ x + cosϕ L σ y , cosθ L ( cosϕ L σ x + sinϕ L σ y ) − sinθ L σ z } and { µ β } = { , − , − , − } , where ρ L is the targeted spin opientation at the left boundary. Sincethe bases φ R and φ L are complete, any matrix F acting in theappropriate Hilbert space is expanded as F = X α =1 4 X β =1 φ βL ⊗ F βα ⊗ φ αR , (A1)where F βα are N − × N − matrices. Indeed, let us intro-duce complementary bases ψ L , ψ R as ψ L,R = { I/ , ρ L,R − I, ( − sinϕ L,R σ x + cosϕ L,R σ y ) / , ( cosθ L,R ( cosϕ L,R σ x + sinϕ L,R σ y ) − sinθ L,R σ z ) / } , trace-orthonormal to the φ R , φ L respectively, T r ( ψ γR φ αR ) = δ αγ , T r ( ψ γL φ βL ) = δ βγ .Then, the coefficients of the expansion (A1) are given by F βα = T r ,N (( ψ βL ⊗ I ⊗ N − ) F ( I ⊗ N − ⊗ ψ αR )) . On the otherhand, in terms of the expansion (A1) the superoperator inverse ( L L + L R ) − is simply ( L L + L R ) − F = X α,β λ α + µ β φ βL ⊗ F βα ⊗ φ αR . (A2)The above sum contains a singular term with α = β = 1 ,because λ + µ = 0 . To eliminate the singularity, one mustrequire F = T r ,N F = 0 , which generates the secular con-ditions (13). Appendix B: Analytic treatment of N = 3 case Here we prove the property (17) for N = 3 , and demon-strate a singularity of the NESS at a fixed value of local fields h, g . Note that we treat case N = 3 for simplicity and fordemonstration purposes only; Also for simplicity, we consider XXZ
Hamiltonian and perpendicular targeted polarizationsin XY plane ~l L = (0 , − , , ~l R = (1 , , , H = H XXZ − hσ y + gσ xN − (B1)We have ρ = ρ L ⊗ (cid:0) I + M (cid:1) ⊗ ρ R and ρ =2 L − LR ( i [ H, ρ ]) + ρ L ⊗ M ⊗ ρ R , with ρ L , ρ R given by(11),(12), and M = P α k σ k , M = P β k σ k , where { σ k } k =1 is a set of Pauli matrices, and α k , β k are unknowns.Secular conditions (13) at zero-th order k = 0 give a set ofthree equations ( h + 1) α = 0( g + 1) α = 0( g + 1) α + (1 + h ) α = 0 , from which the ρ cannot be completely determined. The sec-ular conditions (13) for k = 1 provide missing relations, − ( h + 1) β − (cid:0) + 1 (cid:1) α + 2∆ = 0 − ( g + 1) β − (cid:0) + 1 (cid:1) α −
2∆ = 0( g + 1) β + ( h + 1) β − α = 0 from which ρ can be readily found. Namely, if h = − , g = − , then α = 0 α = ( g + 1) ∆( g + h + 2)(2∆ + 1) ( g + 2 g + h + 2 h + 2) (B2) α = ( − h −
1) ∆( g + h + 2)(2∆ + 1) ( g + 2 g + h + 2 h + 2) Observables of the system change nontrivially with h, g .In particular, the current-like two-point correlation function j z = 2 h σ x σ y − σ y σ x i NESS has the form j z = 4 α = 4( g + 1) ∆( g + h + 2)(2∆ + 1) ( g + 2 g + h + 2 h + 2) . (B3)Consequently, manipulating the h, g , one can change thesign of the above correlation or make it vanish for all ∆ , for g + h = 0 . Moreover, for h = h cr = − − g , all α k = 0 ,see (B2), and we recover (17). If, however, h = − , g = − ,then the solution for α k reads α = 0 α = − α = ∆2∆ + 1 , (B4)manifesting a singularity of the NESS at the point h = g = − for any nonzero ∆ = 0 , see also section VI. Appendix C: Corrections to (17) of the order / Γ Here we show that the perturbation theory (5) predicts M = 0 for arbitrary local fields g, h , if M = 0 . We re-strict to XXZ
Hamiltonian J x = J y = 1 , and perpendic-ular boundary twisting in the XY –plane, ~l L = (0 , − , , ~l R = (1 , , .Let us set ρ = ρ L (cid:0) I (cid:1) ⊗ N − ρ R as predicted by (17) forcritical values of the local field. We then obtain, in the zerothorder of perturbation Q = i [ H, ρ ] = i [ h , + h N − ,N , ρ ] = (C1) = 12 N − (cid:0) K XZ ⊗ I ⊗ N − ⊗ ρ R − ρ L ⊗ I ⊗ N − ⊗ K ZY (cid:1) , where K αβ = − ∆ σ α ⊗ σ β + σ β ⊗ σ α , and h k,k +1 is the localHamiltonian term, h k,k +1 = σ xk σ xk +1 + σ yk σ yk +1 + ∆ σ zk σ zk +1 .The secular conditions T r ,N Q = 0 are trivially satisfied.0Noting that Q has the property L LR Q = − Q , we obtainfrom (4) and (5) the first order correction to ρ ρ = − Q + ρ L ⊗ M ⊗ ρ R .Let us assume that M = 0 . Then, in the second order ofperturbation theory, we have i [ H, ρ ] = − i [ H, Q ] = (C2) − i [ h + h + hσ y + gσ xN − + h N − ,N − + h N − ,N , Q ] . After some calculations we obtain i [ H, ρ ] = R + const × (C3) ∆( − I ⊗ σ y ⊗ I ⊗ N − ⊗ ρ R + ρ L ⊗ I ⊗ N − ⊗ σ x ⊗ I ) , where the unwanted secular terms are written out explicitly,and T r ,N R = 0 . The unwanted terms proportional to ∆ do not depend on h, g . For any ∆ = 0 the secular condi-tions T r ,N [ H, ρ ] = 0 cannot be satisfied. This contradic-tion shows that M = 0 for any ∆ = 0 . Appendix D: Hierachical singularity in the NESS for N = 4 Here we restrict to
XXZ
Hamiltonian with J x = J y = 1 ,and perpendicular boundary twisting in the XY –plane ~l L =(0 , − , , ~l R = (1 , , . For N = 4 we have equations tosatisfy from the secular conditions (13) for k = 0 , , and theset of variables { α ki } , { β ki } to determine the matrices M = P ′ k,i =0 α ki σ k ⊗ σ i , M = P ′ β ki σ k ⊗ σ i . The ”prime” inthe sum denotes the absence of the terms α , β since thematrices M k are traceless. The matrices { σ , σ , σ , σ } = { I, σ x , σ y , σ z } are unit matrix and Pauli matrices. We do notlist here all equations but just their solutions for differentvalues of parameters, obtained using Matematica. For g = − h − we have, in agreement with (17), M = 0 , while, outof coefficients { β ki } , only six are determined, namely β = β = 1 ,β = β = 11 + h , (D1) β = β = 0 , while other β ki (and therefore, the M ) have to be determinedat the next order k = 2 of the perturbation theory. From (D1)it is clear that the case h = 0 has to be considered sepa-rately. In fact, for h = g = − we obtain a different solution: M = 0 , while the coefficients { β ki } are β = β = ∆ − ,β = β = ∆ − , (D2) β = β = β = β = 0 , thus at h = g = − we have a singularity in the first orderof perturbative expansion, in M . On the other hand, (D2) for ∆ = 1 there is a singularity in M : we have to treat this caseseparately. For ∆ = 1 we find M = (cid:0) σ x − σ y + I (cid:1) ⊗ ,in agreement with (29), while the set of β ki is β = β = 12 β = β = 1 β = β = 0 . So at ∆ = 1 , h = g = − we have a singularity in the zerothorder of the perturbative expansion, at the level of M . Sum-marizing, for N = 4 we have M = 0 on the two-dimensionalmanifold of the phase space characterized by { ∆ arbitrary, g = − h − } , except at the point { ∆ = 1 , h = g = − } ,where M = (cid:0) σ x − σ y + I (cid:1) ⊗ . On a one-dimensionalsubmanifold { ∆ = 1 , g = h = − } there is a singularity in M . [1] H. H¨affner, C.F.Roos and R. Blatt, Phys. Reports , 155(2008), and references therein.[2] P. Schindler, M. Muller,D. Nigg, J. T. Barreiro, E.A. MartinezM. Hennrich, T. Monz,S. Diehl, P. Zoller and R. Blatt, NaturePhysics , 361 (2013)[3] S. Diehl, A. Micheli, A. Kantian, B. Kraus, H. P. Buchler andP. Zoller, Nature Physics , 878 (2008)[4] H.-P. Breuer and F. Petruccione, The Theory of Open QuantumSystems , Oxford University Press, (2002).[5] M.B. Plenio and P.L Knight,
Rev. Mod. Phys. , 101 (1998).[6] S. R. Clark, J. Prior, M. J. Hartmann, D. Jaksch and M. B. Ple- nio, New J. of Phys. , 025005(2010).[7] F. Heidrich-Meisner, A. Honecker, and W. Brenig, Eur. Phys. J.Special Topics , 135 (2007), and references therein.[8] A. Klumper, Lect. Notes Phys. , 349 (2004).[9] Z. Cai and T. Barthel, Phys. Rev. Lett. , 150403 (2013)[10] J. J. Mendoza-Arenas, S. Al-Assam, S. R. Clark, D. Jaksch,
J.Stat. Mech. (2013) P07007[11] T. Prosen,
Phys. Rev. Lett. , 137201 (2011).[12] M. ˇZnidariˇc,
J. Stat. Mech.
P12008 (2011); M. ˇZnidariˇc,
Phys.Rev. Lett. , 220601 (2011).[13] D. Karevski, V. Popkov and G. M. Schtz,
Phys. Rev. Lett. , Phys.Rev. E , 062118 (2013)[14] P. Facchi and S. Pascazio, Phys. Rev. Lett. , 080401 (2002);P. Facchi and S. Pascazio, J. of Phys A , 493001 (2008)[15] P. Facchi, H. Nakazato and S. Pascazio, Phys. Rev. Lett. ,2699 (2001)[16] B. Kraus, H. P. Buchler, S. Diehl, A. Micheli and P. Zoller, Phys. Rev. A , 042307 (2008)[17] M. Salerno and V. Popkov, Phys. Rev. E , 022108 (2013)[18] T. Prosen, Phys. Scr. , 058511 (2012)[19] V. Albert and L. Jiang, Phys. Rev. A , 022118 (2014) [20] T. Prosen, Phys.Rev.Lett. , 090404 (2012); T. Prosen, Phys.Rev. A , 044103 (2012)[21] N. Yamamoto, Phys.Rev.A. , 024104 (2005)[22] M. ˇZnidariˇc, Phys. Rev.Lett. , 220601 (2011); M. ˇZnidariˇc,J. of Stat. Mech. P12008 (2011)[23] V. Popkov and R. Livi,
New J. Phys. (2013) 023030[24] Popkov and M. Salerno, J. Stat. Mech
P02040 (2013)[25] V. Popkov,