Temperature effects on quantum non-Markovianity via collision models
TTemperature effects on quantum non-Markovianity via collision models
Zhong-Xiao Man, ∗ Yun-Jie Xia, † and Rosario Lo Franco
2, 3, ‡ School of Physics and Physical Engineering, Shandong Provincial Key Laboratory of LaserPolarization and Information Technology, Qufu Normal University, 273165, Qufu, China Dipartimento di Energia, Ingegneria dell’Informazione e Modelli Matematici,Universit`a di Palermo, Viale delle Scienze, Edificio 9, 90128 Palermo, Italy Dipartimento di Fisica e Chimica, Universit`a di Palermo, via Archirafi 36, 90123 Palermo, Italy
Quantum non-Markovianity represents memory during the system dynamics, which is typically weakened bythe temperature. We here study the effects of environmental temperature on the non-Markovianity of an openquantum system by virtue of collision models. The environment is simulated by a chain of ancillary qubits thatare prepared in thermal states with a finite temperature T . Two distinct non-Markovian mechanisms are consid-ered via two types of collision models, one where the system S consecutively interacts with the ancillas and asecond where S collides only with an intermediate system S (cid:48) which in turn interacts with the ancillas. We showthat in both models the relation between non-Markovianity and temperature is non-monotonic. In particular,revivals of non-Markovianity may occur as temperature increases. We find that the physical reason behind thisbehavior can be revealed by examining a peculiar system-environment coherence exchange, leading to ancillaryqubit coherence larger than system coherence which triggers information backflow from the environment tothe system. These results provide insights on the mechanisms underlying the counterintuitive phenomenon oftemperature-enhanced quantum memory effects. I. INTRODUCTION
In most practical situations a quantum system is open, be-ing coupled to an environment that induces decoherence anddissipation of the system quantum properties [1]. The dynam-ics of an open quantum system is usually described with aMarkov approximation through a family of completely pos-itive trace-preserving reduced dynamical maps and a corre-sponding quantum master equation with a Lindblad genera-tor [2, 3]. In this case, the memoryless environment is as-sumed to be able to recover instantly from the interaction,which induces a monotonic one-way flow of information fromthe system to the environment. However, due to the increas-ing capability to manipulate quantum systems, in many sce-narios the Markov approximation is no longer valid leadingto the occurrence of non-Markovian dynamics [4, 5] and abackflow of information from the environment to the system.The non-Markovian dynamics not only embodies an impor-tant physical phenomenon linked to dynamical memory ef-fects but also proves useful to enhance practical procedures,such as quantum-state engineering and quantum control [6–13].Non-Markovianity has recently attracted considerable at-tention, particularly concerning the formulation of its quanti-tative measures [14–20], its experimental demonstration [21–25] and the exploration of its origin [26, 27]. Nevertheless, therole of non-Markovianity for the assessment of the propertiesof non-equilibrium quantum systems has remained so far lit-tle explored [28–32]. Non-Markovian dynamics can lead to anew type of entropy production term which is indispensable torecover the fluctuation relations for entropy [28]. In a bipar-tite system interacting dissipatively with a thermal reservoir in ∗ [email protected] † [email protected] ‡ [email protected] a cascaded model, the emerging non-Markovianity of one ofthe subsystems enables a heat flow with non-exponential timebehavior [29]. By means of Landauer’s principle, it has beenalso shown that memory effects are strategical in maintain-ing work extraction by erasure in realistic environments [30].Moreover, non-Markovian dynamics can induce the break-down of the validity of Landauer’s principle [31, 32].An efficient tool that makes the study of quantum thermo-dynamics in the non-Markovian regime possible [29, 31, 32]is the collision model [33–56]. In the collision model, the en-vironment is taken as a collection of N ancillas organized ina chain and the system of interest S interacts, or collides, ateach time step with an ancilla. It has been shown that when theancillas are initially uncorrelated and no correlations are cre-ated among them along the process, a Lindblad master equa-tion can be derived [34, 35]. By introducing either correla-tions in the initial state of the ancillas or inter-ancilla colli-sions, one can then recover the dynamics of any indivisible,and thus non-Markovian, channel [36–39]. In other words,the non-Makovian dynamics can be achieved in the collisionmodel when the system-environment interaction is mediatedby the ancillary degrees of freedom. In analogy to the well-known situation where the non-Markovian dynamics of a sys-tem arises when it is coherently coupled to an auxiliary systemin contact with a Markovian bath, a class of Lindblad-typemaster equations for a bipartite system has been also foundthrough collision models such that the reduced master equa-tion of the system of interest is derived exactly [40]. By con-structing such composite collision models, one can simulatea lot of known instances of quantum non-Markovian dynam-ics, such as the emission of an atom into a reservoir with aLorentzian, or multi-Lorentzian, spectral density or a qubitsubject to random telegraph noise [49].Albeit it is generally believed that quantum memory effectsare more important at low temperatures [57], the way temper-ature influences non-Markovianity depends on both quantumthermodynamics and open quantum system dynamics. For a a r X i v : . [ qu a n t - ph ] A p r qubit subject to a dephasing bath with an Ohmic class spec-trum, there exists a temperature-dependent critical value of theOhmicity parameter for the onset of non-Markovianity whichincreases for high temperatures [58]. For a qubit in contactwith a critical Ising spin thermal bath it has been then shownthat the non-Markovianity decreases close to the critical pointof the system in such a way that the higher the temperature, thehigher the decrease [59]. Moreover, it is known that the non-Markovianity of a chromophore qubit in a super-Ohmic bathis reduced when the temperature increases [60]. However,temperature may also enhance the non-Markovianity in somesituations. For an inhomogeneous bosonic finite-chain envi-ronment, temperature has been shown to be a crucial factor indetermining the character of the evolution and for certain pa-rameter values non-Markovianity can increase with the tem-perature [61]. In a spin-boson model made of a two-level sys-tem which is linearly coupled to an environment of harmonicoscillators, a non-monotonic behavior of non-Markovianity asa function of temperature has been reported, with the sys-tem dynamics being strongly non-Markovian at low temper-atures [62]. Another analysis, studying both entanglementand non-Markovianity measures to reveal how second-orderweak-coupling master equations either overestimate or under-estimate memory effects, suggests that non-Markonivity canbe enriched by temperature [63].The above results, limited to specific situations, alreadyshow how subtle the effect of temperature on quantum non-Markovianity can be during an open system dynamics. Inparticular, the occurrence of temperature-enhanced memoryeffects remains counterintuitive and requires further studieswhich can unveil the underlying mechanisms. In this workwe address this issue by means of suitable collision mod-els, which reveal themselves specially advantageous to unveilthe role of environmental elements in ruling the temperature-dependent non-Markovian dynamics of the system. Weconsider two types of collision models with different non-Markovian mechanisms, finding that in both models the vari-ation of non-Markovianity as a function of temperature is notmonotonic and providing the possible physical reason behindthis phenomenon. II. MEASURE OF NON-MARKOVIANITY
The degree of non-Markovianity in a dynamical process canbe quantified by different measures, such as the BLP measurebased on the distinguishability between the evolutions of twodifferent initial states of the system [14], the LPP measurebased on the volume of accessible states of the system [15],the RHP measure [16] and the ACHL measure [20] based onthe time-behavior of the master equation decay rates.The trace distance between the evolutions of two differentinitial states ρ (0) and ρ (0) of an open system is one of themost employed quantifiers. Since a Markovian evolution cannever increase the trace distance, when this happens it is asignature of non-Markovian dynamics of the system. Basedon this concept, the non-Markovianity can be quantified by a FIG. 1. (Color online) Schematic diagram of the direct collisionmodel. (a) The system S collides with the ancilla qubit R and theybecome correlated, as denoted by the dashed line in panel (b). (b)The intracollision between R and R takes place and the tripartitecorrelation SR R may be generated. (c) The ancilla R is tracedout and the process is iterated: namely, the collision of S - R in panel(c) is followed by that of R - R in panel (d) and so on. the BLP measure N defined as [14] N = max ρ (0) ,ρ (0) (cid:90) σ> σ [ t, ρ (0) , ρ (0)] dt, (1)where σ [ t, ρ (0) , ρ (0)] = dD [ ρ ( t ) , ρ ( t )] /dt is the rate ofchange of the trace distance given by D [ ρ ( t ) , ρ ( t )] = 12 Tr | ρ ( t ) − ρ ( t ) | , (2)with | A | = √ A † A . To evaluate the non-Markovianity N , onethen has to find a specific pair of optimal initial states to max-imize the time derivative of the trace distance. In Ref. [64], itis proved that the pair of optimal states is associated with twoantipodal pure states on the surface of the Bloch sphere. Wethus adopt, as usual, the pair of optimal initial states ρ , (0) = | ψ , (0) (cid:105) (cid:104) ψ , (0) | with | ψ , (0) (cid:105) = ( | (cid:105) ± | (cid:105) ) / √ .Since the dynamics of the system in the collision modelis implemented via N equal discrete time steps, in thefollowing the measure N shall be computed by substi-tuting σ [ t, ρ (0) , ρ (0)] dt with the difference ∆ D [ n ] = D [ ρ ,n , ρ ,n ] − D [ ρ ,n − , ρ ,n − ] between the trace distancesat steps n and n − and then summing up all the positivecontributions, that is N = max ρ (0) ,ρ (0) N (cid:88) n, ∆ D [ n ] > ∆ D [ n ] . ( n = 1 , , . . . , N ) (3)The value of the final collision step N is taken such as to coverall the oscillations of the trace distance during the evolution. III. NON-MARKOVIANITY IN THE DIRECT COLLISIONMODEL
In the first model, illustrated in Fig. 1, the system qubit S directly interacts with the environment R which comprises N identical qubits R , R , . . . , R N . The system qubit and ageneric environment qubit are described, respectively, by theHamiltonians ( (cid:126) = 1 ) ˆ H S = ω S ˆ σ Sz / , ˆ H R ≡ ˆ H R n = ω R ˆ σ R n z / , (4)where ˆ σ µz = | (cid:105) µ (cid:104) | − | (cid:105) µ (cid:104) | is the Pauli operator and {| (cid:105) µ , | (cid:105) µ } are the logical states of the qubit µ = S, R n ( n = 1 , , ..., N ) with transition frequency ω µ (hereafter, forsimplicity, we take ω R n = ω R = ω S = ω ). The system-bathcoupling is assumed to be “white-noise” (very large environ-ment) so that the system never collides twice with the samequbit [56]. As a consequence, at each collision step n the sys-tem S collides with a “fresh” R n . Such a model can emulate,for a suitable combination of parameters and interactions, anatom coupled to a lossy cavity [38].Among the possible choices for the interaction between S and environment qubit R n , here we focus on a Heisenberg-like coherent interaction described by the Hamiltonian ˆ H int = g (ˆ σ Sx ⊗ ˆ σ R n x + ˆ σ Sy ⊗ ˆ σ R n y + ˆ σ Sz ⊗ ˆ σ R n z ) , (5)where ˆ σ µj ( j = x, y, z ) is the Pauli operator, g denotes a cou-pling constant and each collision is described by a unitaryoperator ˆ U S,R n = e − i ˆ H int τ , τ being the collision time. Bymeans of the equality e i φ (ˆ σ x ⊗ ˆ σ x +ˆ σ y ⊗ ˆ σ y +ˆ σ z ⊗ ˆ σ z ) = e − i φ (cos φ ˆ I + i sin φ ˆ S ) (6)with ˆ I the identity operator and ˆ S the two qubit swap operatorwith the action | ψ (cid:105) ⊗ | ψ (cid:105) → | ψ (cid:105) ⊗ | ψ (cid:105) for all | ψ (cid:105) , | ψ (cid:105) ∈ C , the unitary time evolution operator can be written as ˆ U SR n = (cos J ) ˆ I SR n + i (sin J ) ˆ S SR n , (7)where J = 2 gτ is a dimensionless interaction strength be-tween S and R n which is supposed to be the same for any n = 1 , , . . . , N . It is immediate to see that J = π/ inducesa complete swap between the state of S and that of R n . Thus, < J < π/ means a partial swap conveying the intuitiveidea that, at each collision, part of the information containedin the state of S is transferred into R n . In the ordered basis {| (cid:105) SR n , | (cid:105) SR n , | (cid:105) SR n , | (cid:105) SR n } , ˆ U SR n reads ˆ U SR n = e iJ J i sin J i sin J cos J
00 0 0 e iJ . (8)In the present model, the non-Markovian dynamics of the sys-tem is introduced via the interactions between two nearest-neighbor qubits R n and R n +1 . Such interactions are de-scribed by an operation similar to that of Eq. (8), namely ˆ V R n R n +1 = e i Ω i sin Ω 00 i sin Ω cos Ω 00 0 0 e i Ω , (9) FIG. 2. (Color online) (a)
Non-Markovianity N vs. the collisionstrength Ω between the environment qubits for different temperatures T of the environment. (b) Non-Markovianity N vs. the temperature T for different Ω . In both plots, the remaining parameters are givenby ω = 5 ω and J = 0 . . ω/ω (small collision time andweak interaction between system and environment qubits). where ≤ Ω ≤ π/ is the dimensionless R n - R n +1 interac-tion strength which is taken to be the same for any n .As illustrated in Fig. 1 exemplifying the first two stepsof collisions, in each step we consider the ordered triplet ( S, R n − , R n ) in such a way that after the collision between S and R n − via the unitary operation ˆ U SR n − , the systemshifts by one site while R n − collides with R n via ˆ V R n − R n .Notice that R n − - R n collision occurs before S - R n collisionso that S and R n are already correlated before they collidewith each other. The three qubits after the two collisions cannow be all correlated with the total state ρ SR n − R n (the cor-relations are labeled by the dashed lines in Fig. 1). Then, wetrace out the qubit R n − giving rise to the reduced state ρ SR n of S − R n and proceed to the next step with the new orderedtriplet ( S, R n , R n +1 ) . Under the actions of ˆ U SR n of Eq. (8)and ˆ V R n R n +1 of Eq. (9), the total state of SR n R n +1 at thestep n is obtained from the step n − as ρ SR n R n +1 =ˆ V R n R n +1 ˆ U SR n (cid:0) ρ SR n ⊗ ρ R n +1 (cid:1) ˆ U † SR n ˆ V † R n R n +1 , (10)where ρ R n +1 ≡ ρ R is the pre-collision state of the environ- FIG. 3. (Color online) Contour plot of the non-Markovianity N fordifferent T and Ω . The non-Markovian regime is colored while theMarkovian regime is white. The blue lines are the contour lines of N . The dotted red line is the curve of the thresholds of Ω triggeringnon-Markovian dynamics. The remaining parameters are given by ω = 5 ω and J = 0 . . ω/ω (small collision time and weakinteraction between system and environment qubits). mental qubit. Here, to reveal the effect of environmental tem-perature on the non-Markovianity, we assume the environ-mental qubits are initially prepared in the same thermal states ρ R = e − β ˆ H R /Z at temperature T R , where β = 1 /k B T R ( k B being the Boltzmann constant) and Z is the partition function.In our analysis, we consider a dimensionless temperature T defined by T ≡ k B T R / ( (cid:126) ω ) , where ω is a reference fre-quency. We also take values of J (cid:28) ω/ω so to have smallcollision times and a weak interaction between the system andthe environment qubits.In such a model, the system experiences a homogeniza-tion process and reaches asymptotically the very same state ρ R [32]. The forward transfer of the lost information of thesystem S via intracollisions of environment qubits triggersdynamical memory effects of the system, so that the non-Markovianity is closely related to the intracollision strength Ω . Fig. 2(a) shows the dependence of non-Markovianity N on Ω for different temperatures T of the environment. Inboth zero temperature ( T = 0 ) and thermal environments( T > ), the non-Markovianity is activated when Ω exceedsa given threshold (see the inset of Fig. 2(a) for a more evi-dent demonstration) and then monotonically increases with Ω .From this first analysis, it emerges that the thermal environ-ment does not affect the monotonic relation between N and Ω , while the thresholds of Ω triggering the non-Markovianitydepend on the temperature. On the other hand, the variationsof non-Markovianity N with respect to the temperature T canbe rich and non-monotonic, as shown in Fig. 2(b). For rel-atively small values of Ω (e.g., Ω = 0 . ), the increase of T can enable the non-Markovianity which maintains nonzerovalues within a finite region of T > . For larger values of Ω , the system dynamics exhibits non-Markovian character al-ready for a zero-temperature environment. In this case, the FIG. 4. (Color online) Coherences C S,n of the system S and C R,n of the environment qubit R n +1 after the n -th collision as a functionof n . The relevant parameters are Ω = 0 . , ω = 5 ω and J =0 . . ω/ω (small collision time and weak interaction betweensystem and environment qubits). non-Markovianity approximately exhibits a plateau for small T and then experiences successive decreasing and increasingbehaviors, eventually vanishing at high temperatures. For par-ticular values of the environment qubits interaction strength(e.g., Ω = 0 . , . ), when T increases we also observe thatthe non-Markovianity N may vanish within a finite interval of T and then revive again. In other words, manipulations of theenvironment temperature T can induce successive transitionsbetween non-Markovian and Markovian regimes for the sys-tem dynamics. A comprehensive picture for the dependenceof N on T and Ω is shown in Fig. 3, where we can see thenon-Markovianity thresholds of Ω (identified by the dotted redline) for a given T and the crossovers between non-Markovianand Markovian regimes as T increases for a given Ω .In order to gain a deeper understanding of the temperatureeffects on the non-Markovianity, we examine the coherence ofthe system, which is related to the trace distance of this modelas D [ ρ ,n , ρ ,n ] = 2 C S,n , where C S,n = | (cid:104) | ρ S,n | (cid:105) | = | (cid:104) | ρ S,n | (cid:105) | is the coherence degree of the state ρ S,n of S after the n -th collision, the initial state being prepared in | ψ (cid:105) S, = ( | (cid:105) S, ± | (cid:105) S, ) / √ . Notice that the latter is a bonafide quantifier of coherence, being the half of the so-called l -norm measure within a resource theory [65, 66]. Therefore,the initial coherence of S has the maximum value C S, = 0 . .The temporary growth of C S,n thus serves as a witness forthe onset of non-Markovian dynamics. Moreover, to assessthe role of the environmental constituents, we consider thecoherence C R,n of the environment qubit R n +1 transferredfrom S after the n -th collision of S - R n - R n +1 . In Fig. 4(a)-(f), we illustrate the evolution of C S,n and C R,n versus n fordifferent temperatures with Ω = 0 . , whose non-Markoviancharacter is plotted in Fig. 2(b) (blue dot-dashed curve). Anoverall comparison of the panels in Fig. 4(a)-(f), which in-dicate a temperature range from T = 1 to T = 7 , verifiesthe fact that the initial increase of temperature speeds up the FIG. 5. (Color online) Schematic diagram of the indirect collisionmodel. (a) The system S collides with the intermediate qubit S (cid:48) and they become correlated, as denoted by the dashed line in panel(b). (b) The qubit S (cid:48) interacts with R and the correlation among SS (cid:48) R is generated, as denoted by the dashed line in (c). (c) Theintracollision between R and R takes place and the correlation of SS (cid:48) R R is then established. (d) The ancilla qubit R is traced outand the process is iterated: namely, the collision of S - S (cid:48) in (d) isfollowed by the collision S (cid:48) - R in (e) and R - R in (f) and so on. decay of the system coherence C S,n . Therefore, on the onehand, the increase of temperature suppresses and eventuallyterminates the non-Markovianity, as seen in Fig. 4(a)-(b) for T = 1 and T = 2 and already confirmed in Fig. 2(b). On theother hand, however, the quick decay of C S,n can cause thecoherence C R,n of the environment qubit R n +1 to approach(see Fig. 4(c)) and even to exceed (see Fig. 4(d)-(e)-(f)) thecoherence C S,n of the system. This behavior in turn inducesthe information backflow from the environment to the system,namely, a revival of non-Markovian regime. In fact, C R,n overcomes C S,n in correspondence to the recovery of the non-Markovian character of the system dynamics from a Marko-vian one (compare Fig. 4(d)-(e)-(f) and the blue dot-dashedcurve of Fig. 2(b)). In the high temperature regime, the sys-tem coherence decays more quickly and the non-Markoviandynamics will cease if the intracollision strength Ω is not suf-ficiently large. For instance, from Fig. 2(b) one sees that at T = 10 the non-Markovianity vanishes if Ω = 0 . , whileit remains nonzero when Ω = 0 . . In other words, to getnon-Markovian dynamics (quantum memory effects) at hightemperatures, one has to increase the interaction strength Ω between environmental ancillary qubits, which allows a moreefficient transfer of environmental quantum coherence IV. NON-MARKOVIANITY IN THE INDIRECTCOLLISION MODEL
We now consider a mechanism of non-Markovian dynam-ics based on another collision model, where the interaction ofthe system qubit S with the environment qubit R n is medi-ated by an intermediate qubit S (cid:48) , as depicted in Fig. 5. Sucha scenario implies that the information contained in S is firsttransferred to S (cid:48) and then damped into R via the collisions be- tween S (cid:48) and R n . It is known that, in the absence of environ-mental intracollisions, this composite model can emulate (forshort collision times and Jaynes-Cummings-type interactions)a two-level atom in a lossy cavity, S (cid:48) playing the role of thecavity mode [49]. For straightforward extension, in the pres-ence of environmental intracollisions, this model may repre-sent a two-level atom in a reservoir with a photonic band gap[49, 67, 68].We choose the Heisenberg-type coherent interaction be-tween S and S (cid:48) , with interaction strength ≤ κ ≤ π/ ,represented by the unitary operator ˆ U SS (cid:48) , analogous to thatof Eq. (8), having the form ˆ U SS (cid:48) = e iκ κ i sin κ i sin κ cos κ
00 0 0 e iκ . (11)The unitary operators ˆ U S (cid:48) R n and ˆ V R n R n +1 representing, re-spectively, the S (cid:48) - R n interaction and the interaction betweenadjacent environment qubits are the same of Eqs. (8) and (9)with interaction strengths J and Ω .As shown in Fig. 5, in each round of collisions we deal withfour qubits ( S, S (cid:48) , R n − , R n ) in such a way that, after the col-lisions of S - S (cid:48) and S (cid:48) - R n − , the qubits S and S (cid:48) shift by onesite while R n − collides with R n , which results in the corre-lated total state ρ SS (cid:48) R n − R n (the correlations are indicated bydashed lines). Then, we trace out the qubit R n − obtainingthe reduced state ρ SS (cid:48) R n of SS (cid:48) R n and proceed to the nextstep with the new ordered group ( S, S (cid:48) , R n , R n +1 ) . As a con-sequence, the total state of SS (cid:48) R n R n +1 at the n -th collisionis determined from the ( n − -th collision as ρ SS (cid:48) R n R n +1 =ˆ V R n R n +1 ˆ U S (cid:48) R n ˆ U SS (cid:48) (cid:0) ρ SS (cid:48) R n ⊗ ρ R n +1 (cid:1) ˆ U † SS (cid:48) ˆ U † S (cid:48) R n ˆ V † R n R n +1 . (12)The temperature effects are included in this model by consid-ering the qubit S (cid:48) and all the environmental qubits prepared inthe same thermal states ρ R = e − β ˆ H R /Z with temperature T . A. Absence of collisions between environment qubits
In this subsection, we consider the non-Markovianity inthe absence of collisions between environment qubits, i.e., for
Ω = 0 . For this indirect collision model, the information ofthe system S is first transferred to the qubit S (cid:48) via the coherentinteraction and then dissipated to the environment through thecollisions between S (cid:48) and environment qubits. In this case, theintermediate qubit S (cid:48) can have the role of a quantum memoryleading to the non-Markovian dynamics even without colli-sions between environment qubits. The interaction strength κ between S and S (cid:48) is then crucial in activating the non-Markovianity, as verified in Fig. 6 where N increases with κ for a given J at a fixed temperature. Moreover, the non-Markovianity achieves a nonzero value only when κ is greaterthan a threshold and the larger the value of J , the larger the FIG. 6. (Color online) Non-Markovianity N versus κ (the interac-tion strength between S and S (cid:48) ) for different values of J (the inter-action strength between S (cid:48) and environment qubits) in the absenceof intra-interactions between environment qubits, that is Ω = 0 . Theremaining parameters are given as T = 1 and ω = 5 ω .FIG. 7. (Color online) Non-Markovianity N as a function of T fordifferent κ with Ω = 0 , ω = 5 ω and J = 0 . . ω/ω . threshold of κ required to trigger the non-Markovian regime.From Fig. 6 one also observes that, for a given κ , the non-Markovianity N decreases with J , which implies that a stronginteraction between S (cid:48) and the environment qubits weakensthe non-Markovianity of the system S .In Fig. 7, the effect of the temperature T on the non-Markovianity is taken into account for different values of κ .We notice that the non-Markovianity as a function of T is verysensitive to the value of κ , in that it can: decrease directlyto zero (e.g., for κ = 0 . ), disappear for a finite range oftemperature and then revive (e.g., for κ = 0 . ), or decreaseto a minimum value and then slowly grow for larger κ . Re-markably, non-Markovianity can persist at high temperaturesprovided that the values of κ are sufficiently large. FIG. 8. (Color online) Non-Markovianity N versus Ω for different κ at T = 0 , with ω = 5 ω and J = 0 . . ω/ω . B. Presence of collisions between environment qubits
Now we take the intracollisions between environmentqubits R n and R n +1 into account so that the two mecha-nisms of non-Markovian dynamics, namely the interaction S - S (cid:48) ruled by κ and the interaction R n - R n +1 ruled by Ω , coexistin one and the same model.We first explore the role of Ω in enhancing the non-Markovianity at zero temperature. When the coupling be-tween S and S (cid:48) is weak with relatively small κ , we know fromthe previous subsection that the dynamics of S is Markovian( N = 0 ) if Ω = 0 . As shown in Fig. 8(a), by introducing theinteractions R n - R n +1 a threshold of Ω exists which triggersa non-Markovian regime. Such a threshold increases with κ :namely, the smaller the value of κ , the larger the threshold of Ω . The subsequent variations of N with Ω are non-monotonic.In particular, we find that the activated non-Markovianity N can disappear within a finite interval of Ω and then reappear(e.g., for κ = 0 . , . ). When the coupling between S and S (cid:48) is strong with larger κ , the dynamics may be alreadynon-Markovian even for Ω = 0 , as seen in Fig. 7 at T = 0 and shown more in detail in Fig. 8(b). In this case, the non-Markovianity can be further enriched by introducing the inter-actions between environmental qubits R n - R n +1 .The effects of the environment temperature on non- FIG. 9. (Color online) (a) Non-Markovianity N versus Ω for differ-ent T . (b) Non-Markovianity N versus T for different Ω . The otherparameters are given by ω = 5 ω , κ = 0 . , J = 0 . . ω/ω . Markovianity are displayed in Fig. 9. In particular, N exhibitsa non-monotonic variation with respect to the environmentalqubit interaction strength Ω (see Fig. 9(a)), with a first de-scent and a successive ascent. Once again, we notice that thenon-Markovianity can completely disappear for a finite rangeof Ω and then revive. The non-Markovianity N as a func-tion of T is then shown in Fig. 9(b), where we observe thatthe non-Markovianity is unavoidably weakened by increasing T from zero for all the given Ω , but it does not necessarilyvanish for larger values of temperature. In fact, N can in-crease slowly (e.g., for Ω = 0 . ), disappear completely (e.g.,for Ω = 0 . ), collapse and revive (e.g., for Ω = 0 . ) and os-cillate (e.g., for Ω = 1 . ). A comprehensive picture of thevariation of N as a function of both Ω and T , for fixed κ and J , is given in Fig. 10, where the above detailed behav-iors can be retrieved. The values of κ and J are such that thesystem dynamics is non-Markovian for T = 0 and Ω = 0 .Such a plot is useful to immediately see how, in this compos-ite indirect collision model, the temperature affects the systemnon-Markovianity in a different way from the case of the di-rect collision model treated in Sec. III. As a matter of fact,Fig. 10 shows that the temperature has a general detrimen-tal effect on non-Markovianity, which can never overcome itsvalue at T = 0 for the higher values of T , as instead hap-pens for the direct collision model (see Fig. 2(b)). However, arange of values of Ω exists for which a temperature thresholdcan be found which reactivate dynamical memory effects forthe system lost at lower temperatures. In analogy with the di- FIG. 10. (Color online) Contour plot of the non-Markovianity N fordifferent T and Ω . The non-Markovian regime is colored while theMarkovian regime is white. The alternate solid and dashed blue linesare the contour curves of N . The solid red line is the curve of thethresholds of Ω and T triggering non-Markovianity. The other pa-rameters are ω = 5 ω , κ = 0 . . ω/ω , J = 0 . . ω/ω . rect collision model, such a feature is to be related to peculiarcoherence exchanges from the system S to the environmentalcomponents. V. CONCLUSION
In conclusion, we have studied the effects of temperature onthe non-Markovian character of an open quantum system dy-namics by means of two types of collision models which entaildifferent mechanisms for the occurrence of non-Markovianity.In the first model, that is the direct collision model, the sys-tem S consecutively interacts with a chain of environmentqubits that are prepared in the same thermal states at tem-perature T , and the non-Markovianity N is induced by theintracollisions of environment qubits. As expected, the non-Markovian dynamics can be triggered when the intracolli-sion strength is greater than a temperature-dependent thresh-old. In striking contrast to the usual understanding of the ef-fect of the temperature on the non-Markovianity [58–61, 63],we have found that the behavior of N as a function of T is non-monotonic, exhibiting a process of reduction and en-hancement when temperature increases. In particular, we haveshown that the non-Markovianity can vanish within a finiteinterval of T and then reappear when T increases. We havegiven a possible interpretation of this counterintuitive revivalof dynamical memory effects by resorting to the exchangesof coherence between the system and environment qubits. Infact, albeit the temperature can accelerate the decay of coher-ence of the system and suppress the non-Markovianity untilcertain values, in the regime of high temperature this quickdecay of system coherence can cause the coherence trans-ferred to environment qubits to exceed that of the system. Thismechanism in turn induces a backflow of information from theenvironment to the system and thus non-Markovian dynamics.In the second model, that is the indirect collision model,the system S indirectly interacts with the environment qubitsthrough collisions with an intermediate qubit S (cid:48) . In thiscase, S (cid:48) serves as the memory for the transferred informationfrom S towards the environment, representing a distinct non-Markovian mechanism. Without intracollisions between envi-ronment qubits, the non-Markovian dynamics for the systemcan still arise provided that the interaction strength of S - S (cid:48) issufficiently large. Moreover, the non-monotonic relation be-tween the non-Markovianity measure N and T is once againobserved. When the environmental intracollisions are takeninto account, the two mentioned non-Markovian mechanismscoexist in the same model. In this case we have found thatthe presence of interactions between environmental qubits en-riches non-Markovianity. The temperature has now the gen-eral effect to reduce the degree of non-Markovianity with re-spect to its value at zero temperature. However, once againnon-Markovianity of the system can exhibit revivals as a func- tion of the temperature.Our findings within collision models are confirmed by somerealistic composite quantum systems which exhibit a non-monotonic relation between non-Markovianity and tempera-ture [61, 62]. More in general, our results contribute towardsthe capability of engineering suitable environments with opti-mal temperature conditions to exploit dynamical memory ef-fects of an open quantum system, which is strategic for noisyintermediate scale quantum information processing [69]. ACKNOWLEDGMENTS
R.L.F. acknowledges Francesco Ciccarello for fruitfuldiscussions and comments. This work is supported byNational Natural Science Foundation (China) under GrantNos. 11574178 and 61675115, and Shandong Provin-cial Natural Science Foundation (China) under GrantNo. ZR2016JL005. [1] H.-P. Breuer and F. Petruccione,
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