The role of environmental correlations in the non-Markovian dynamics of a spin system
aa r X i v : . [ qu a n t - ph ] J un The role of environmental correlations in the non-Markovian dynamics of a spin system
Salvatore Lorenzo , , , Francesco Plastina , , and Mauro Paternostro Dipartimento di Fisica, Universit`a della Calabria,87036 Arcavacata di Rende (CS), Italy INFN - Gruppo collegato di Cosenza Centre for Theoretical Atomic,Molecular and Optical Physics,School of Mathematics and Physics,Queen’s University, Belfast BT7 1NN, United Kingdom (Dated: August 20, 2018)We put forward a framework to study the dynamics of a chain of interacting quantum particles affected byindividual or collective multi-mode environment, focussing on the role played by the environmental quantumcorrelations over the evolution of the chain. The presence of entanglement in the state of the environmentalsystem magnifies the non-Markovian nature of the chain’s dynamics, giving rise to structures in figures ofmerit such as entanglement and purity that are not observed under a separable multi-mode environment. Ouranalysis can be relevant to problems tackling the open-system dynamics of biological complexes of strongcurrent interest.
PACS numbers: 05.30.-d, 03.65.Yz, 03.67.-a, 05.60.Gg
The investigation on the interaction between a spin-liketwo-level system (TLS) and an environment embodied by anensemble of harmonic oscillators, the so-called spin-bosonmodel [1], has gained renewed momentum due to the cur-rent strong interest in the quantum-mechanical description ofexcitation-transfer processes in biological organisms or thedynamics of protein-solvent systems in photosynthetic com-plexes [2–6]. The interest in such a celebrated model istwofold, in this context. On one side, it is employed to un-derstand the active role of an environment in assisting trans-port across such structures [4]. On the other hand, the modelis extremely useful to characterize the extent to which non-classical behaviors survive in systems affected by environ-mental worlds that are strictly adherent to the paradigm ofbeing wet and warm . In this respect, we aim at gathering adescription of the TLS-environment evolution that is as com-plete as possible and able to account for subtle yet potentiallykey features of such dynamics emerging from the finitenessthe environment itself, its capability to retain information onthe system it is coupled to (and kick it back at later times) andits inherent structure. For, instance, Rebentrost et al. haveshown in Ref. [7] that environmental memory-keeping effectsare crucial for preserving the coherent behavior of excitonicenergy transfer processes in biological complexes.Stepping aside from interesting yet still controversial bio-physical scenarios, the general model addressed above playsan important role in various contexts of applied quantumphysics, from cavity quantum electrodynamics to its solid-state analogues involving defects in diamond coupled to semi-conducting microcavities [8] or superconducting devices andplanar strip-line resonators [9]. In the latter case, very in-teresting regimes are quickly becoming available where thecoupling strength of the coherent part of an evolution can bemade comparable to the one responsible for the open-systemdynamics due to the presence of the environment [10].All such considerations and physical configurations de-mand the development of a framework apt to encompassand harness features of non-Markovianity in various coupling regimes. Formally, one is required to retain, to some extent,the degrees of freedom of the environment so as to allow for itspossibly-non-trivial structure to influence the dynamics of thesystem of interest. This can be computationally very demand-ing and while sophisticated techniques have been put forwardto determine the quantum statistical properties of interactingquantum many-body systems, they are either hardly suitablefor non-linear configurations or face some difficulties in track-ing the temporal dynamics of the system itself.In this paper we contribute to the investigation on the in-teraction between a system of interacting TLS’s coupled to abosonic environment by studying the effect that environmen-tal entanglement has on the dynamics of a chain of interactingTLS’s. We build up a mathematical apparatus that, inspired byand significantly extending the approach in Ref. [11], solvesexactly the local interaction between a single TLS and itsmulti-mode environment, so as to build an eigenbasis thatis then used in order to tackle the inter-particle coupling bykeeping track of the environmental properties. This helps usin incorporating the full quantum features of the environmen-tal state, therefore enabling a quantitative study of the role ofentanglement in the evolution of the chain under investigationand its relation to the non-Markovian nature of the dynamics.We illustrate our techniques by addressing the cases of a dimerand a trimer, which are the smallest non-trivial configurationsallowing us to study the most crucial attributes of such a com-plex problem. Our investigation opens up the possibility tostudy full-scale networks of coupled TLS’s in an open-systemfashion, covering the full range of coupling strengths with theenvironments, from the weak to the strong one. Although in-evitably leaving a few questions open, we believe such an en-deavor to be helpful in the contexts described above.The remainder of this paper is organized as follows: inSec. I we describe the method put forward in order to ana-lyze the dynamics of an N -site chain interacting with indi-vidual multi-mode bosonic environments without resorting tothe weak-coupling regime and thus allowing for a more com-plete overview of the effects of non-Markovianity on the evo-lution of the chain. Sec. II is devoted to the explicit study ona dimer of interacting particles affected by entangled environ-ments. The key features in the dynamics of such configura-tion are compared to what is achieved under the assumptionof independent environments so as to show that entanglementappear to enhance the non-Markovian aspects of the time evo-lution. Sec. III addresses the case of a common environmentaffecting the sites of our dimer, showing that the manifesta-tions mentioned above are insensitive to the non-local natureof the environment, but rather depend on the properties of itsstate such as the entanglement-sharing structure, as studied inSec. IV for the case of a trimer. Finally, in Sec. V we drawour conclusions and briefly discuss the questions opened byour investigation. I. THE MODEL AND THE ANALYTICAL METHOD TOITS APPROACH
The model under consideration is described by an Hamilto-nian of the form ˆH tot = ˆH S + ˆH SS + ˆH B + ˆS SB . (1)Here ˆH S = − ~ P Nj =1 ǫ j ˆ σ zj gives the free dynamics ofthe elements of the chain, ǫ j being the energy split be-tween the levels of each TLS, while { ˆ σ xj , ˆ σ yj , ˆ σ zj } arethe Pauli matrices for particle j =1 , .., N . Analogously, ˆH B = P Nj =1 P n j k =1 ~ ω j,k ˆ a † j,k ˆ a j,k is the free energy of the en-vironmental harmonic oscillators. The pedex k labels the n j independent modes interacting with the TLS at site j . Eachof such modes has frequency ω j,k and is described by thebosonic annihilation [creation] operator ˆ a j,k [ ˆ a † j,k ]. On theother hand, ˆH SS models the inter-particle coupling within thechain, while ˆH SB describes the TLS-environment interaction.We take these two terms as ˆH SB = N X j =1 ˆ σ zj ⊗ n j X k =1 g j,k (ˆ a † j,k + ˆ a j,k ) , ˆH SS = − ~ N − X n = j J j,j +1 (ˆ σ xj ˆ σ xj +1 + ˆ σ yj ˆ σ yj +1 ) (2)with J j,j +1 being the nearest-neighbor coupling strength be-tween the particles occupying sites j and j + 1 on the chainand g j,k the analogous parameter associated with the interac-tion of the j th TLS and its k th environmental mode.The first of Eq. (2) generates a displacement of the environ-mental oscillators, conditioned on the state of the correspond-ing TLS. Our approach to the description of the dynamics is tosolve the problems embodied by each the subsystem made outof a single TLS and the ensemble of local modes it interactswith. We thus consider ˆH j = ˆH j S + ˆH j B + ˆH j SB and look foreigenstates of the form | a i j ⊗ | φ i i j with a = ± and {|±i j } denoting the eigenbasis of ˆ σ zj . States | φ i i j are the eigenstatesof the total Hamiltonian of an ensemble of oscillators (per as- signed state of the corresponding TLS), that is n j X k =1 [ ± g j,k ( a † j,k + a j,k ) + ~ ω ˆ a † j,k ˆ a j,k ] | φ ± i j =( E j ± ǫ j ) | φ ± i j , (3)where E j is the total energy of the j th subsystem. A detailedcalculation shows that such energy depends on the number ofexcitations n j,k in the oscillators coupled to the TLS at the j th site according to the expression E jn = ± ǫ j + n j X k =1 ~ ω j,k ( n j,k − α j,k ) , (4)where α j,k = g j,k / ~ ω j,k . The corresponding eigenstatescan be written as | ψ n j ± i j = |±i j ⊗| α n j ± i j = |±i j ⊗ n j k =1 | α n k ± i j ,where n j = { n , n , .., n n j } is the vector specified by thenumber of excitations populating the n j oscillators interact-ing with the j th TLS and we have introduced the excitation-added coherent states | α n k ± i =[( a † j ± α j,k ) n k / √ n k !] | ± α j,k i .Although it is straightforward to check that h α m + | α n + i = δ mn ( h α m − | α n − i = δ mn ) and the set of states {| + , α n + i} ( {|− , α n − i} )form an orthonormal basis, states | α n + i and | α n − i are notmutually orthogonal. A treatment with an arbitrary numberof modes per site, performed along the lines of the discus-sion above, is certainly possible although computationally de-manding. Therefore, from now on, we will focus our attentionto the case of a single mode per site and drop the mode-label.In the oscillator basis discussed above, the Hamiltonian (1)takes the form ˆH tot = N X j =1 ˆH j + ˆH SS , (5)each ˆH j being now diagonal. The remaining term ˆH SS ,which is defined in the second line of Eq. (2), only containsthe degrees of freedom of the TLS chain. In the following,we consider a homogeneous chain with J j,j +1 = J, ∀ j =1 , .., N . By introducing the raising and lowering operators ˆ σ ± j = (ˆ σ xj ± i ˆ σ yj ) / , it is convenient to rewrite it as ˆH SS = − ~ J N − X n =1 (ˆ σ + j ˆ σ − j +1 + ˆ σ − j ˆ σ + j +1 ) . (6)In the interaction picture defined with respect to the unper-turbed Hamiltonian ˆH = P j ˆH j , we get ˆH I ( t )= − ~ J N Y j =1 ˆΘ ± j N − X n =1 ( σ + n σ − n +1 + h.c. ) N Y k =1 ˆΘ ± k ! † (7)with ˆΘ ± j = P ∞ n =0 e iE n ± t |± , α n ± ih± , α n ± | .After elaborating on this expression, we obtain ˆH I ( t ) = − ~ J N − X n =1 (ˆ σ + n ˆ σ − n +1 ˆΘ + n ˆΘ − n +1 ˆΘ −† n ˆΘ + † n +1 + h.c. ) . (8)By expanding the ˆΘ ± n ’s in a coherent-state basis, we finallyarrive at the effective expression for the intra-chain Hamilto-nian ˆH I ( t ) = − ~ J N − X n =1 [ˆΓ + n ˆΓ − n +1 + ˆΓ − n ˆΓ + n +1 ] , (9)where ˆΓ ± j = ˆ σ ± j ⊗ ˆ D j [ ± α ( t ))] with ˆ D ( ξ ) the displacementoperator of amplitude ξ ∈ C [12] and α j ( t ) = 2 α j (1 − e iω j t ) .While the analysis above explicitly consider individual envi-ronments affecting the elements of the chain, one can alsostudy the complementary situation where all the TLS’s col-lectively interact with a single (and common) ensemble of os-cillators. In this case, [ ˆH tot , P j ˆ σ jz ]=0 , which in turn impliesthat H SS can be easily diagonalized. We tackle an explicit ex-ample of this situation in Sec. III.In order to obtain information on the full dynamics of theTLS subsystem, we write the density matrix ρ ( t ) of the chain-oscillators system in the excitation-added displaced basis. Forall our simulations, we assume the factorized initial state ρ (0) = ρ chain (0) ⊗ ρ ho (0) (10)with ρ chain ( t ) [ ρ ho ( t ) ] describing the state of the chain (oscil-lator environment) at time t . In what follows we consider twosignificant instances of ρ ho (0) : the tensor product of individ-ual thermal states and the case of entangled oscillators pre-pared in N -mode squeezed states [12]. As it will be discussedlater on, such three paradigmatic cases allow us to well iden-tify the effect of the entanglement among the environments onthe dynamics of the TLS’s.Let us start addressing the case of a thermal state ρ th = N O j =1 ∞ X l =0 n lj (1 + n j ) l +1 | l i j h l | , (11)where n j =( e β ~ ω j − − is the average number of quanta inthe oscillator at temperature T (we have taken the inverse tem-perature β = 1 /k b T with k b the Boltzmann constant) and | l i is the state with l excitations. In the excitation-added dis-placed basis of the oscillators, we can recast ρ th into the form ρ th = N O j =1 ∞ X n,m,l =0 n lj (1 + n j ) l +1 | α n ± i j h α n ± | l i j h l | α m ± i j h α m ± | , (12)where the + or − sign for the j th oscillator has to be chosenaccording to the state of the corresponding element in ρ chain .The scalar products h α n ± | l i needed for the evaluation of theexpression above are given in terms of Charlier polynomials C n ( m ; α ) as [13, 14] h α n − | m i = α n + m e − α / √ n ! √ m ! C n ( m ; α ) , h α n + | m i = α n + m e − α / √ n ! √ m ! C m ( n ; α ) , (13) where we have assumed α ∈ R . In the remainder of themanuscript we take n j = n, ∀ j . We will also consider an N -mode a pure squeezed-vacuum state for the harmonic oscil-lators | sq N i =(1 − λ ) − / P l λ l ⊗ Nj =1 | l i j with λ = tanh r and r the squeezing parameter. This state can be expressedin the chosen oscillator basis following the very same lineshighlighted above and using again the Charlier polynomials.It is worth emphasizing that these two environmental statesare locally indistinguishable as a squeezed vacuum is locallyequivalent to a thermal state. II. CHAIN DYNAMICS UNDER INDEPENDENTENVIRONMENTS
We are now in a position to start addressing the effective dy-namics of the TLS chain under the influences of the various in-stances of environment introduced in the previous Section. Inorder to strip down our discussion from unnecessary compli-cations, we study here the smallest configuration of the modelat hand, i.e. a two-TLS chain (a dimer ) affected by two indi-vidual harmonic oscillators, one per site. The two TLS willbe considered as identical. While embodying an interestingenough situation [2], this case allows for an agile descriptionof the physics involved.In the dimer basis, the interaction hamiltonian in Eq. (9)takes the form ˆH I ( t ) = D ( α ) ˆ D † ( α ) 00 ˆ D † ( α ) ˆ D ( α ) 0 00 0 0 0 (14)In order to simplify the notation, we take identically coupledTLS-oscillator subsystems, so that α j = α ( j =1 , (thisassumption, as well as the one of identical TLS’s, does notlimit the generality of our results and can be relaxed with onlymild complications). The corresponding time-evolution op-erator ˆ U ( t ) is then calculated as a time-ordered Dyson series ˆ U ( t ) = P m ˆ U n ( t ) with ˆ U m ( t )= ( − i ) m m ! Z t dt Z t dt ·· Z t dt m ˆ T ˆH I ( t ) ˆH I ( t ) ·· ˆH I ( t m ) , (15)where ˆ T is the time-ordering operator. The anti-diagonal formof ˆH I ( t ) helps considerably in working out a manageableexpression or the time-evolution operator. In fact, the even(odd) terms in the expansion of ˆ U ( t ) involve diagonal (anti- (a) (b) p p Coherences p p FIG. 1: (color online): Coherences of the dimer’s density matrixfor a chain affected by an environment prepared in a thermal and asqueezed-vacuum state [panel (a) and (b) , respectively]. We havetaken identical parameters for the two TLS-oscillator subsystemswith ~ J = 0 . eV, ~ ω = 0 . eV, g = 0 . eV at room tempera-ture ( T = 293 K). In this case α = 1 . As explained in the text, thesqueezing factor for the results in panel (b) has been taken so that thereduced single-oscillator state has the same effective temperature asin the thermal state case [i.e. panel (a) ]. Quantitatively, r = 0 . .A coherence revival at t = π/ω is present for a squeezed-vacuumenvironment: the presence of quantum correlations between the en-vironmental modes fundamentally affects the dynamics of the TLS’s. diagonal) operators, in the dimer basis. In details, we get m Y k =1 ˆH I ( t k )= | + −i h + − |⊗ m Y k =1 ˆ D + − ( α ( t k − )) ˆ D † + − ( α ( t k ))+ | − + i h− + |⊗ m Y k =1 ˆ D †− + ( α ( t k − )) ˆ D − + ( α ( t k )) , m +1 Y k =1 ˆH I ( t k )= | + −i h− + |⊗ m Y k =0 ˆ D + − ( α ( t k )) ˆ D †− + ( α ( t k +1 ))+ | − + i h + − |⊗ m Y k =0 ˆ D †− + ( α ( t k )) ˆ D + − ( α ( t k +1 )) (16)where ˆ D + − ( α ( t k ))= ˆ D (+ α ( t k )) ˆ D † ( − α ( t k )) [with an anal-ogous definition for ˆ D − + ( α ( t k )) ].By initializing the first (second) TLS in the coherent super-position | in i = a |−i + b | + i with | a | + | b | =1 ( | in i = |−i ) and considering that [ H tot , P j =1 ˆ σ zj ] = 0 , the state | + , + i will stays unpopulated, while the probability to pop-ulate |− , −i remains constant. In Fig. 1 we plot the co-herences of the density matrix of the two TLS’s affected byboson environments prepared in either uncorrelated thermalstates or a squeezed-vacuum one. In order to perform a faith-ful comparison between such two cases, we have taken thesqueezing parameter according to tanh r = e − β ~ ω with thevalue of β provided by the thermal-state environment. Thisensures that the reduced single-oscillator state obtained fromthe squeezed-vacuum environment is a thermal state havingthe same temperature as the one considered in the genuinelythermal instance. In this way, we can nicely isolate the effectof the entanglement shared by the environmental oscillators.Clearly, the revival of coherences at t = π/ω is only dueto the presence of correlations in the environment and an in-dication of a significant deviation of the state properties in such two different environmental preparations. Fig. 2 showsthe purity P ( t ) = Tr ( ρ ) of the chain’s density matrix ρ in both the thermal and squeezed case. We plot such a fig-ure of merit against the intra-site coupling strength J and there-scaled interaction time ωt . Quite reasonably, at values of J that are comparable with the TLS-oscillator coupling, thedifferences between the thermal and squeezed-vacuum casesbecomes very small. We can understand this by consideringthat, for decreasing values of J , the non-local interaction be-tween the sites of the chain becomes less relevant than thelocal TLS-oscillator coupling. This implies that each TLS isstrongly tied to its associated mode so that the local featuresof the oscillator environment become more preponderant thanthe non-local ones and the differences with a configurationof independent environmental states fades away. On the con-trary, the inter-site tunnelling arising from a large value of J makes the inter-mode entanglement relevant. Such differ-ences manifest themselves in a rather distinct behavior of theproperties of the TLS state.As a special but very interesting case, we consider thechain to be initialized in the maximally entangled state (1 / √ | − −i + | + + i ) and wonder about the behaviorthat entanglement has under the interaction model we arestudying. Such an initial state is interesting on its own as,remarkably, it is relatively straightforward to find closed ana-lytical expressions for the elements of the reduced TLS den-sity matrix ρ . While the populations of such matrix remainconstant, the interesting element turns out to be the coherence h + + | ρ ( t ) | − −i , for which we find h + + | ρ ( t ) | − −i = 12 e iǫt +8 α λ th +1) λ th − [1 − cos( ωt )] t (17)for an oscillator environment prepared in the tensor product oftwo individual thermal states, while h + + | ρ ( t ) | − −i = 12 e iǫt +8 α λ sq +2 λ sq cos( ωt ) λ sq − [1 − cos( ωt )] t (18) FIG. 2: (Color online): Purity of dimer’s density matrix affected byindividual environments prepared in a thermal state (red surface) anda squeezed-vacuum one (green surface, staying above the previousone). P ( t ) is plotted against the chain inter-site coupling energy ~ J for hω = 0 . eV, g = 0 . eV and T = 293 K, which correspondsto α = 1 / . We have taken a squeezing factor r = 0 . (see textfor further details on this choice). (a) (b) FIG. 3: (Color online): Purity [panel (a) ] and logarithmic negativity[panel (b) ] of the state of a two-site chain initially prepared in themaximally entangled state (1 / √ | − −i + | + + i ) and affectedby either thermal states of the oscillator environment (red dashedcurve) or an entangled squeezed-vacuum state (solid dark one) for ( ~ J, ~ ω, g ) = (0 . , . , . eV and r = 1 . , correspondingto α = 1 / . The revival at t = π/ω associated with the squeezed-vacuum case is very pronounced. is valid for a squeezed-vacuum environment with λ th = λ sq = n/ ( n + 1) .The additional oscillatory term proportional to λ sq cos ωt appearing in Eq. (18) is responsible for the differences in thetrend followed by the coherences corresponding to such twocases. The state purity is given by P ( t ) = 12 + 2 |h + + | ρ S ( t ) | − −i| , (19)which is reported in Fig. 3 (a) for the two different typesof environmental states considered in this study, highlightingin a rather striking way how strongly the entanglement be-tween the environmental oscillators affects the TLS’s proper-ties. The purity revival at t = π/ω is also reflected in the be-havior of the entanglement shared by the TLS’s. We use log-arithmic negativity [15] as our entanglement measure. This isdefined as N ( t ) = log || ρ P T || with ρ P T the partially trans-posed density matrix of the two TLS’s and || ˆ A || = Tr p ˆ A † ˆ A the trace norm of any operator ˆ A . At t = π/ω , N ( t ) goesfrom zero (the system experiences a sudden death of entan-glement) to a rather large value.The trend illustrated so far has features of independencefrom the specific instance of initial state considered for thedimer. We have studied the evolution of N ( t ) when the sys-tem is initialized in a separable state, finding qualitativelyanalogous behaviors of such figure of merit. For instance,Fig. 4 reports the results corresponding to a chain preparedas (1 / √ |−i + | + i ) ⊗ |−i : entanglement builds up in thesystem (both the curves in Fig. 4 are such that N (0) = 0 ) andnever disappears (we have considered a relatively small cou-pling between environments and chain). However, while thecase corresponding to a thermal-state environment has a peri-odicity of π/ω , when a squeezing is present, a resurgence ofentanglement occurs at half the thermal-state period.Such effects can be linked to an enhanced non-Markoviannature of the TLS dynamics arising when considering entan-gled environmental oscillators. The analysis of this connec-tion is the main aim of the following section. FIG. 4: (Color online): Logarithmic negativity between the two sitesof a chain affected by either thermal states of the oscillator environ-ment (green curve) or an entangled squeezed-vacuum state (red one)for r = 0 . and α = 1 / . At t = π/ω , N ( t ) has a revival, forthe squeezed-vacuum case, which is absent from the thermal state-affected system. A. Characterizing the non-Markovian nature of the dynamics
The quantification of the degree of non-Markovianity ofadynamical evolution has been the focus of a significant re-cent research activity [16–19]. The main goal of such investi-gation has been the identification of appropriate tools for thecharacterization of the many facets of non-Markovianity andthe proposal of operational ways to quantify it. More recently,such tools have found application in the characterization ofthe memory-keeping evolution of systems of physical rele-vance [20, 21]. In this Subsection we move along such linesand employ one of the above-mentioned measures to deter-mine the features of the dynamics under study here and thedifferences between the two environmental configurations ad-dressed so far.Quantitatively, we take inspiration from the measure ofnon-Markovianity proposed by Rivas et al. in Ref. [17], whichis based on the following approach: consider a bipartite en-tangled state such that only one component of the biparti-tion is affected by the dynamics we would like to charac-terize. Then, by computing the amount of entanglement be-tween the two parties at different instants of times within aselected interval [ t , t max ] we can detect non-markovianityby checking for a non-monotonic behavior of the quantumcorrelations. That is, for ρ SA (0) = | Φ ih Φ | with | Φ i , somemaximally entangled pure bipartite state, we define ∆ E = E [ ρ SA ( t )] − E [ ρ SA ( t max )] (where E denotes a legitimateentanglement monotone). We thus take I = Z t max (cid:12)(cid:12)(cid:12)(cid:12) dE [ ρ SA ( t )] dt (cid:12)(cid:12)(cid:12)(cid:12) dt − ∆ E (20)If the evolution of the system is Markovian, the derivative of E [ ρ SA ( t )] is negative and I ( E ) = 0 .While the measure in [17] has been explicitly formulatedfor two qubits, so as to characterize the nature of a single-qubitchannel, one can certainly think about extending the argumentso as to address the case of a larger system. In particular,here we would like to be able to qualitatively understand the FIG. 5: (Color online): Witness of non-Markovianity for the thermal(red dashed line) and squeezed blue solid line) case for ~ ω = 0 . and r = 1 . against the coupling strength α . reasons behind the occurrence of the resurgence peak when anentangled environment is at hand.Without embarking into the challenge of a rigorous formu-lation of the measure for non-Markovianity, we believe it isreasonable to consider the following situation: we take a gen-uinely tripartite entangled state of the two sites of the dimerat hand and an additional, ancillary TLS, which is exposedto a perfectly unitary dynamics (i.e. no environmental bosonis interacting with the ancilla). More specifically, we con-sider the W state (1 / √ | + − −i + |− + −i + |− − + i ) a ,which also ensures the presence of entanglement in any two-TLS reduction. While, as stated above, the ancilla a under-goes a unitary evolution, particles and are exposed to theeffective quantum channel described so far. We then applyEq. (20), where E is taken to be the logarithmic negativitybetween a and the dimer, to the cases where the environmentis prepared in a squeezed-vacuum and a two-mode thermalstate. The results are shown in Fig. 5 against the effectivecoupling rate α = g/ ~ ω for t max = 2 π/ω . While in the caseof a thermal environment, I grows monotonically and satu-rates for α & . , for a squeezed-vacuum state it peaks atabout this value of α , signaling an increased non-Markovianbehavior associated with such an environmental state. Thismagnified non-Markovianity is at the origin of the recurrencepeaks exhibited in the analyses of the previous Sections. Forlager coupling strengths α , each TLS-environment interactionbecomes so strong with respect to the inter-TLS one that thesystem fragments into a series of local sub-systems, each be-ing basically oblivious to any non-local nature of the environ-mental state. III. CHAIN DYNAMICS UNDER A COMMONENVIRONMENT
We now move to the study of a configuration where the TLSchain is affected by a common environment. The expressionfor the system-environment coupling given in the first line ofEq. (2) is thus modified into ˆH SB = g n o X k =1 (ˆ a k + ˆ a † k ) ⊗ N X j =1 ˆ σ zj (21) where we have assumed an oscillator environment consistingof n o modes, each identically coupled to all the TLS of and N -element chain. As remarked above, the symmetries in thepresent case are such that one can easily gather the exact an-alytical form of the evolved density matrix of the chain. Forillustrative purposes, and to make a clear comparison to thecase treated in previous Sections, we address a chain of twoidentical element, this time affected by a common environ-ment composed by two boson modes. In Ref. [2], an anal-ogous problem has been approached by restricting the anal-ysis to the single-excitation subspace and using a numericalmethod for the analysis of non-Markovian dynamics. Here,we shall pursue an analytic approach that is made possible bythe analysis conducted in Sec. I We start by considering anarbitrary pure state | η i =( a |−i + b | + i ) at site of the chain,while the second element is prepared in |−i . The time evo-lution operator in the excitation-added displaced basis of theenvironmental modes is given by ˆ U ( t ) = X n,m e iωnt (cid:0) f ( t ) ± | ± ±ih± ± | ⊗ | α ±± n,m ih α ±± n,m | +[ θ ( t ) | ∓ ±ih∓ ± | + h ( t ) | ∓ ±ih± ∓ | ] ⊗ | n, m ih n, m | ) (22) SqueezedThermal FIG. 6: (Color online) Purity against dimensionless interaction time ωt for thermal (red) and squeezed (green) environment states, with ~ J = 0 . , ~ ω = 0 . , g = 0 . , α = 1 / . In the two cases,squeezing parameter and (effective) temperature are r = 0 . and T = 293 K. The dashed curves refer to the case of common environ-ments. (a) (b) pp p p pp p p FIG. 7: (Color online): Logarithmic Negativity for ~ J/g = 0 . and α = 1 / , in the common- (dashed curve) and independent- (solidline) environment case. Panel (a) addresses the case of a thermalenvironment, while panel (b) is for a squeezed one. with ϕ ± ( t ) = e ± i (2 ǫ + ωα ) t , ϑ ( t ) = cos( Jt ) and κ ( t ) = i sin( Jt ) . Notice that, if α = 0 , then | α ± n i = | n i . With this expression at hand, the reduced density matrix of the chain is ρ ( t )= b ϑ ( t ) b ϑ ( t ) κ ∗ ( t ) ab ∗ ϕ + ( t ) ϑ ( t ) γ ( t )0 b ϑ ( t ) κ ( t ) b κ ( t ) ab ∗ ϕ + ( t ) κ ( t ) γ ( t )0 a ∗ bϕ − ( t ) ϑ ( t ) γ ∗ ( t ) a ∗ bϕ − ( t ) κ ∗ ( t ) γ ∗ ( t ) a (23)where we have introduced the decoherence factor γ ( t ) = exp[ − α n ( e iωt − e − iωt + βω − (24)for the thermal state case and γ ( t ) = exp[ − α (1 − e − iωt )(1 − e − iωt λ sq ) + [1 − ωt ) + cos(2 ωt )] λ sq λ sq − (25)for squeezed-vacuum one.Very similar features between the individual-environmentand the common-environment cases are found by studying thebehavior of purity. Fig. 6 shows that, regardless of the en-vironmental configuration (whether made out of a commonsystem or two individual oscillators), the squeezed-vacuumpreparation is responsible for the revival of purity discussedabove, a feature that is absent from the thermal case.It is interesting to notice that the central block of the densitymatrix in Eq. (23) does not contain any decoherence factor,thus being fully decoherence-free: entanglement and puritywill be preserved in time, in such a subspace, which is associ-ated with a single excitation. Clearly, this marks a net differ-ence with the case of two independent environments, whereno such special symmetry exists. Such a distinction is evi-dent from the inspection of Fig. 7, where we plot the loga-rithmic negativity in the common and individual-environmentcases, for both the thermal and squeezed-vaccum preparation:the long-time behavior in the individual-environment arrange-ment clearly manifests a decrease of N ( t ) that is absent fromthe common-environment configuration. IV. THREE-SITE CHAIN: THE ROLE OF THESHARING-STRUCTURE OF ENVIRONMENTALENTANGLEMENT
After having analyzed in the previous Sections the basicbuilding-block for any complex structure of interacting TLS’s,we move to the study of an open three-site chain (a trimer) af-fected by individual environments. While, qualitatively, theresult gathered in Sec. II will be fully confirmed, such a sce-nario offers us an opportunity to investigate whether the shar-ing structure of the environmental entanglement plays any rolein the open-system dynamics undergone by the TLS-chain.To start with, we have considered the three-mode squeezedstate | sq i =(1 − λ ) − / P l =0 λ l | l, l, l i . This state is such that, upon tracing out one of the modes, the remain-ing two are fully separable thermal states, thus witnessing theGHZ-like nature of such state. We have found negligible dif-ferences in the dynamics of the trimer arising from the use ofsuch environmental state or the tensor product ⊗ j =1 ρ th ,j : asshown in Fig. 8, for instance, the purity function P ( t ) of thethree-TLS state corresponding to such cases is very similar[as seen by inspecting the two bottom curves in Fig. 8]. Theinsensitivity to squeezing is confirmed by looking at the bipar-tite entanglement between any pair of chain elements achievedafter tracing out one TLS-oscillator subsystem. Fig. 9 illus-trates the behavior of the logarithmic negativity for such bi-partite states; the results are valid, qualitatively, regardless ofthe pair being taken and show that there is no revival peak at = π/ω in the case of a squeezed environment. Moreover, asimilar behavior is observed when considering the tripartitenegativity , which is a legitimate entanglement monotone forthe tripartite case [22]. FIG. 8: (Color online): Purity of three sites chain for the cases ofthermal (green), generalized squeezed (red) and double split (blue)environment states. The nature of genuine tripartite entanglement ofthe generalized squeezed state is almost invisible, while some differ-ences in the time evolution can be pointed out for the split squeezedstate.
The situation however changes if we modify theentanglement-sharing structure among the environmental os-cillators. To show this fact, we have taken a three-mode stateobtained by superimposing at a beam-splitter one mode of asqueezed state | sq i to an ancilla prepared in the vacuumstate | i . More formally, we consider the state | sq ′ i = ( ⊗ ˆ B )( θ ) | sq i ⊗ | i (26)with ˆ B ( θ ) = exp[( θ/ a † ˆ a − ˆ a † ˆ a )] being the beam splitteroperator of transmittivity cos ( θ/ [12]. For θ = 0 ( θ = π ),a two-mode squeezed-vacuum state of modes and ( and ) is retrieved, while for intermediate values of such param-eter, an entangled three-mode state is achieved having non-separable two-mode reduced states. In this second case, theenvironmental entanglement visibly affects the global proper-ties of the system in a way consistent with what we have seenearlier in Sec. II, see Fig. 8, thus reinforcing the idea that thestructure of quantum correlations shared by the environmen-tal oscillators plays a key role in the detailed non-Markoviandynamics of the TLS subsystem. V. CONCLUSIONS
We have studied the full non-Markovian dynamics of a sys-tem of interacting TLS’s arranged in a linear configuration andaffected by a bosonic environment, considering both the indi-vidual and common-environment cases. By properly choos-ing the basis onto which each TLS-oscillator interaction is de-scribed, we have been able to push the analytic approach to such a problem up to the point of constructing a handy recipefor tracking any regime of interaction, from the weak to thestrong-coupling regime with, in principle, an arbitrary num-ber of environmental modes per site. This has paved the wayto the study of the effects that quantum entanglement sharedby the elements of the environment has on the dynamics ofthe TLS system. Entanglement appears to enhance the non-Markovian character of the evolution, giving rise to specificfeatures in relevant figures of merit of a TLS chain that arenot present in the case of uncorrelated environmental states.We have illustrated such effects by studying a dimer, whichis the basic building-block for the construction of interestinginteraction configurations that are usually considered for thedescription of biological complexes. Our investigation leavesopen a series of questions related, in particular, to the scalingof the effects revealed here with the degree of connectivity, thedepth and the geometry of the network of interacting TLS’s.We plan to address such issues by extending and improvingthe implementability of our method.
Acknowledgments
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