Implications of non-Markovian dynamics on information-driven engine
IImplications of non-Markovian dynamics on information-driven engine
Obinna Abah ∗ and Mauro Paternostro † Centre for Theoretical Atomic, Molecular and Optical Physics,School of Mathematics and Physics, Queen’s University Belfast, Belfast BT7 1NN, United Kingdom
The understanding of memory e ff ects arising from the interaction between system and environment is a keyfor engineering quantum thermodynamic devices beyond the standard Markovian limit. We study the perfor-mance of measurement-based thermal machine whose working medium dynamics is subject to backflow ofinformation from the reservoir via collision based model. In this study, the non-Markovian e ff ect is introducedby allowing for additional unitary interactions between the environments. We present two strategies of realizingnon-Markovian dynamics and study their influence on the performance of the engine. Moreover, the role ofsystem-environment memory e ff ects on the engine work extraction and information gain through measurementcan be beneficial in short time. I. INTRODUCTION
The second law of thermodynamics is ubiquitous in nature:it stipulates that heat always flows from hot place to cold one.However, in 1867 Maxwell proposes the opposite with hisidea of an intelligent demon to illustrate the statistical natureof the second law of thermodynamics [1]. The demon, withsu ffi ciently information about the microscopic motions of in-dividual atoms and molecules, is capable to separate the fast-moving (“hot”) ones from the slow-moving (“cold”) ones andinduce the heat to flow from cold to hot, in apparent contra-diction with the second law of thermodynamics. It took nearlya century to resolve this apparent paradox following a seriesof works, starting from Szilard’s engine [2] through Landauer[3], Bennett [4] and others to clarify the link between the in-formation recorded by the demon and the thermodynamic en-tropy, see [5]. The advances in nanotechnology have made therealization of Maxwell’s thought experiment, Szilard’s enginepossible in recent time [6–9].In addition to this, there has been a parallel line of devel-opment in the non-Markovian dynamic behavior of systeminteracting with reservoir. Theoretical advances have beenmade on its characterization [10–12] as well as verifications[13–15] in various experimental setup. The role of memory(non-Markovian) e ff ects in understanding of information pro-cessing at both the classical and quantum level is currentlyattracting research interest [16–19]. Likewise, over the lastfew years , there has been an increase on the studies to under-stand or harness the non-Markovian e ff ect on quantum ther-modynamic machines [20–22]. Recently, studying the non-Markovian dynamic of a system has shed more light into theunderstanding of the Landauer principle [18].Over the past few years, great e ff ort has been devoted onstudying the interplay between thermodynamics and quan-tum mechanics [23–28]. Remarkable progress has been madein understanding the non-equilibrium processes in thermody-namics [29] as well as extending / generalizing the second lawof thermodynamics to incorporates measurement and feed-back driven processes [30–37]. Recently, the role of feed- ∗ [email protected] † [email protected] back control on information thermodynamic engine has beenexperimentally studied in di ff erent platform [38–43]. How-ever, the understanding of the machine performance whenthe feedback engine protocol is performed by system exhibit-ing non-Markovian dynamics is still lacking. Although theself-consistent formulation of an interpretation of thermody-namic laws in the presence of measurements and feedback isstill work in progress, and is attracting much attention, morepractical issues such as the enhancement of the performanceof cooling algorithms by feedback-based mechanisms are al-ready under investigation and exploitation [44–47].In this paper, we investigate the implications of non-Markovian dynamics on feedback information-driven ma-chines. Our findings show that memory e ff ect can enhancethe overall performance — work extraction and informationgain of the engine in a short time. The rest of the paper isorganized as follows. In the next section, Section II, we firstpresent the description of the measurement-based engine andthen briefly discuss its thermodynamic analysis. In SectionIII we introduce the collision based model of realizing non-Markovian dynamics and outline example of two di ff erentstrategies. Then, the characterization of the non-Markovianfeatures is numerical analyze in Section IV A, while the analy-sis of the feedback-driven engine in both Markovian and non-Markovian situation is devoted to Section IV B. Finally, Sec-tion V draws our conclusions. II. MEASUREMENT-BASED THERMO-MACHINE
The system initially prepared in given state is brought tocontact with a heat reservoir. Then, the system is decoupledand attached to a measuring apparatus initially prepared in agiven state. The apparatus determine the state of the systemand depending on the result of the measurement, a feedbackoperation is performed on the system. The setup consists ofthree components; system, reservoir and an ancilla.
A. Description of the protocol
We now introduce and illustrate the protocol that we aimat studying for the investigation of the e ff ects that a process a r X i v : . [ qu a n t - ph ] J a n U SR U SM | | R ( θ , ϕ ) SR R RS S MM
Step 1& 2 Step 3 Step 4 & 5
FIG. 1. The various steps of the general protocol that we consider.The jagged light-blue area stands for the tracing-out of the environ-mental system. The information-gathering process in Step 3 consistsof a projective measurement performed on the state of the ancilla M ,which is projected onto the elements of its computational basis, suchas {| (cid:105) M , | (cid:105) M } in the case of a qubit. of information-gathering and feedback have on the capabilityof the system to perform work. We proceed step by step, asfollows: Step 1: Initial preparation.–
System S and thermal reser-voir(s) R are prepared in their respective equilibrium statesat inverse temperature β i = / k B T i and frequency ω i , where i = S , R . The initial system-reservoir state is described by thedensity matrix ρ S R = (cid:79) i = R , S ρ i = (cid:79) i = R , S e − β i H i Z i , (1)where H i denotes the Hamiltonian of element i and Z i = tr (cid:104) e − β i H i (cid:105) is the corresponding partition function. For simplic-ity, we will consider the case in which the system and thereservoir are made of two-level systems. Step 2: System-environment coupling.–
System and reservoirinteract unitarily. In line with the usual formalism used in col-lisional models for quantum open-system dynamics [37, 48–53], in what follows we will concentrate on a time-evolutionoperator of the partial-SWAP form such as U S R = e − i τ [cos(2 τ ) + i sin(2 τ ) U sw ] , (2)where τ is a dimensionless interaction time and U sw is thetwo-particle SWAP transformation | i , j (cid:105) S R U sw −→ | j , i (cid:105) S R with | i (cid:105) S [ | j (cid:105) R ] a state of the computational basis chosen for S [ R ].The S - R state after such unitary evolution is thus ρ uS R = U S R ( ρ S ⊗ ρ R ) U † S R . (3)In general, the joint dynamics embodied by U S R gives rise toquantum correlations between system and environment. Theenvironment is then discarded, leaving us with the reducedstate of the system only ρ uS = tr R (cid:104) ρ uS R (cid:105) . (4) Step 3: Pre-measurement.–
The system is then brought intocontact with a measuring apparatus, i.e. an ancillary qubit M prepared in state ρ M . The S - M coupling takes place accord-ing to the unitary transformation U S M , which gives the jointdensity matrix ρ pmS M = U S M ( ρ uS ⊗ ρ M ) U † S M . (5)We assume that U S M takes place over a dimensionless system-probe interaction time τ m and that the corresponding couplingHamiltonian H S M such that U S M = e − i τ m ( H S M + H S ) takes the formof a spin-spin coupling H S M = σ ( j ) S ⊗ σ ( j ) M , whose form willbe specified later on. Here, σ ( j ) i is the j = x , y , z Pauli spinoperator of particle i = S , M . Step 4: Measurement.–
This is the actual information-gathering step where the information on S acquired by the an-cilla during Step 3 through their mutual interaction is inferredvia an actual measurement process. The latter is described bythe complete set of projective operators (cid:110) M ( k ) M (cid:111) , defined in theHilbert space of the ancilla M . Let us assume that the ancillais initially prepared in one of its computational-basis states,i.e. ρ Mp = | p (cid:105)(cid:104) p | M . The probability that outcome k is obtainedas a result of such measurement is given by P k = tr SM (cid:104) M ( k ) M ρ pmS M M ( k ) M (cid:105) = tr S (cid:104) F k ρ uS (cid:105) (6)with F k = E † k E k and E k = M (cid:104) k | U S M | p (cid:105) M an element of thepositive-operator value measure (POVM) induced on the sys-tem. The corresponding post-measurement state of the systemreads ρ kS = E k ρ uS E † k P k . (7) Step 5: Feedback control operation.–
Based on the outcomeof the measurement at Step 4, the controller performs a condi-tional operation on the state of the system [30, 34]. The mostgeneral unitary transformation on a single-qubit state is a ro-tation by an angle α about an arbitrary axis identified by theunit vector n = (sin θ cos φ, sin θ sin φ, cos θ ), which has beenwritten in polar coordinates specified by the polar angle θ andazimuthal one φ . By including a general global phase γ , suchrotation reads R ( v ) = e i γ (cid:32) cos α − i cos θ sin α − i sin α sin θ e − i φ − i sin α sin θ e i φ cos α + i cos θ sin α (cid:33) (8)with v : = ( γ, α, θ, φ ). In our case, the set of parameters uponwhich such rotation depends should be interpreted as condi-tioned on the outcome of the measurement performed, at Step4, on the ancilla M . That is v −→ v k : = ( γ k , α k , θ k , φ k ) . (9)The use of such conditioned rotation, which embodies oursimple feedback control operation, delivers the state of thesystem ρ f bS , k = R ( v k ) ρ kS R † ( v k ) . (10) Step 6: The reset.–
The system evolves independently and afresh ancilla is made available to the next iteration of the pro-tocol, which proceeds again from Step 1 onwards. This stagehas no e ff ect on the analysis that follows. B. Thermodynamics of the machine
We proceed with the thermodynamic analysis of the pro-tocol presented above, by calculating the changes in internalenergy E [ ρ ] ≡ tr (cid:2) H ρ (cid:3) and entropy S [ ρ ] ≡ − k B tr (cid:2) ρ ln ρ (cid:3) ofthe system associated with the preparation, measurement andfeedback-control protocols.First, after the system preparation (interaction with thereservoir), the change in the system internal energy is ∆ E u = E [ ρ uS ] − E [ ρ S ] = tr (cid:104) H S ρ uS (cid:105) − tr (cid:2) H S ρ S (cid:3) , (11)and the change in system entropy reads ∆ S u = S [ ρ uS ] − S [ ρ S ] = − k B (cid:16) tr (cid:104) ρ uS ln ρ uS (cid:105) − tr (cid:2) ρ S ln ρ S (cid:3)(cid:17) . (12)From the first law of thermodynamics, ∆ E = ∆ W +∆ Q , and as-suming that the heat exchange between the system and reser-voir is governed by ∆ Q uS = − ∆ Q uR , the work done on / by thesystem can be written as ∆ W u = ∆ E u + ∆ Q uR , where the lastterm associates the exchange of heat ∆ Q uR = tr (cid:104) H R (cid:16) ρ uR − ρ R (cid:17)(cid:105) with its environment and ρ uR = tr S (cid:104) ρ uS R (cid:105) is the marginal state ofthe reservoir after interaction.For the measurement step, the thermodynamic quantitiesare as follows. The variation of internal energy of the systemreads ∆ E pm = E [ ρ pmS ] − E [ ρ uS ] = tr (cid:104) H S ρ pmS (cid:105) − tr (cid:104) H S ρ uS (cid:105) , (13)where ρ pmS = tr M (cid:104) ρ pmS M (cid:105) is the reduced state of the system afterthe pre-measurement step (cf. Step 3). The correspondingchange in entropy of the state of the system is ∆ S pm = S [ ρ pmS ] − S [ ρ uS ] = − k B (cid:16) tr (cid:104) ρ pmS ln ρ pmS (cid:105) − tr (cid:104) ρ uS ln ρ uS (cid:105)(cid:17) . (14)Based on the second law of phenomenological non-equilibrium thermodynamics, the entropy production char-acterizing the irreversibility of the measurement processreads [37] Σ m = ∆ S mS M = ∆ S mS + ∆ S mM − I mS : M = . (15)The first two terms in the right-hand side Eq. (15) correspondto the change in entropy of the system and the ancilla, whilethe third term is the quantum mutual information betweenthem. As the state of the measurement apparatus is unchange,we have ∆ S mM =
0, in this particular case. The gain of infor-mation about the system achieved through the measurement is I mS : M = S ( ρ kS ) − S ( ρ uS ) ≡ I qm . On the other hand, from the firstlaw, we have that the work done by the measurement reads ∆ W m = ∆ E m = E [ ρ kS ] − E [ ρ uS ].Then, during the feedback step, the variation of system en-ergy and entropy are ∆ E f bk = E [ ρ f bS , k ] − E [ ρ kS ] = tr (cid:104) H S ρ f bS , k (cid:105) − tr (cid:104) H S ρ kS (cid:105) , ∆ S f bk = S [ ρ f bS , k ] − S [ ρ kS ] , (16) respectively. Here definitions analogous to those used abovehold for both S [ ρ f bS , k ] and S [ ρ uS ]. Using again the first andsecond laws, we have ∆ E f bk = ∆ W f bk + ∆ Q f bk , Σ f b = ∆ S f bk + ∆ S f bc ≥ , (17)where ∆ S f bk and ∆ S f bc are the entropy change associated withthe system and feedback controller respectively. Thus, an up-per bound to the amount of thermodynamic work extractedby the feedback protocol is ∆ W f bk ≥ ∆ E f bk − T c ∆ S f bk , wherewe employ the relation ∆ S = − ∆ Q / T c and T c is the controllertemperature. The work extracted by the system is beyond thesecond law due to the correlation between the system and thememory. The form of such bound was first given in Ref. [30],more details on the subject can be found in Ref. [37]. We notethat the feedback protocol can be engineered so as to changeonly the system density matrix and leave that of the ancillauna ff ected. This is possible, for instance, by post-selectingonly the measurement events where the state of the ancilla isfound to be the initially prepared one | p (cid:105) M . Therefore, ne-glecting − ∆ S f bk which is usually non-negative, we defined thetotal work done on / by the system through the measurementand feedback protocol as ∆ W t = ∆ W m + ∆ W f b = E [ ρ f bS , k ] − E [ ρ uS ] . (18) III. NON-MARKOVIAN DYNAMICS OF THE SYSTEM -COLLISIONAL BASED MODEL
Here, we consider a situation where the system undergoesnon-Markovian dynamics as a result of its interaction with theenvironment (taking place at steps 1 and 2 of our protocol).The realization of the dynamics that we decide to consideris that of collisional models, which o ff er great flexibility andrichness of phenomenology [52].In particular, we consider the case in which the reservoir’smemory mechanism arises from collisions between di ff erentelements of a structured, multi-party environment, followingan interaction with the system. This scenario has been suc-cessfully used in the past to model memory-bearing mecha-nisms able to propagate to the environment information ac-quired on the state of the system [54]. More recently, thisrealization of memory-bearing e ff ects has been used to assessthe performance of a quantum Otto cycle having a harmonicsystem as a working medium [55]. Collisional models allowfor the tracking of the dynamics of both system and environ-ments, which in turn makes it possible to follow the ensuingemergence of the system-environment correlations responsi-ble for memory e ff ects [48, 50–52, 56, 57]. They are thusinvaluable methodological tools to assess the back-action ofmemory-bearing environments on the information-driven en-gine at the core of our study.As anticipated above, we assume an environment R made up of a large number of elements, which we label { E , E , .., E n } and assume, for the sake of simplicity, to bemutually identical. The total state of system and environ-ment is initially factorized and the dynamics proceeds through R ( θ , ϕ ) SE …… E E N SE …… E E N SE …… E E N SE …… E E N (a) (b)(c) (d) FIG. 2. Schematic of non-Markovian dynamics via collision modelfor nearest sub-environment collisions. The system and the sub-environment particles are initially uncorrelated. In the first step (a),the system S interacts with E . The next step, (b) E interacts with E and thereby correlating the system and particles E and E . Thenstep (c), E is traced away. After which the system interacts with E before isolating the system for measurement and feedback processesin strategy 1. For the strategy 2, the system and sub-environment par-ticles collisional iterations are performed up to E , (a) - (d), beforethe measurement and feedback. as sequential collisions (interaction process) between S andan element E n of the environment. These are followed bypairwise collisions / interactions between the elements of the-environment, as illustrated in Fig. 2. In Ref. [54], it has beenshown that the degree of non-Markovianity of the reducedsystem dynamics depends on how the erasure of system-environment correlations is performed.Here, we will consider two inequivalent schemes of tracingout the degree of freedom of the environment. The first sce-nario that we consider to compute the reduced dynamics of S requires the environmental particle E n to be traced out when ithas interacted with S and E n + but before the system interactswith E n + . In the second scenario, the reduced dynamics of thesystem is obtained by tracing out the environmental particleonce it has interacted with system S . The remaining environ-mental particle interacts with the next homonimous particlebefore the latter subsequent collides with the system. We alsoassume that the environment-environment interaction evolu-tion is described by the unitary operator [50–52] U EE = e − i τ e [cos(2 τ e ) + i sin(2 τ e ) U sw ] , (19)which describes another partial-SWAP gate between two con-secutive elements of the environment, parameterized by thedimensionless interaction time τ e .The first scenario (which we term strategy-1 ) that we con-sider involves tracing out the particle E n after it has collidedwith E n + , as exemplified in Fig. 2 (a) - (c) . It starts witha collision between S and E n , modelled through the unitaryoperation U S R in Eq. (2), which delivers the joint state ρ S E n = U S R ( ρ S ⊗ ρ E n ) U † S R . (20) The three particles S , E n and E n + then become correlatedthrough the intra-environment interaction U E n E n + in Eq. (19),after which particle E n is traced out. This results in the bipar-tite S - E n + state ρ S E n + = tr E n (cid:104) U EE ( ρ S E n ⊗ ρ E n + ) U † EE (cid:105) . (21)The marginal state of the system is computed after the inter-action with E n + . Thus, strategy-1 prepare the system in state ρ uS = tr E n + (cid:104) U S R ρ S E n + U † S R (cid:105) . (22)We remark that retaining the correlations up to the third envi-ronment – which corresponds to the systematic collision withthe environmental components E n , E n + , and E n + as in Fig. 2– does not change the resulting dynamics [56]. At the end ofthe system-environment interaction, the engine-protocol steps[ step 3 - 6 ] are performed before the system collides with an-other fresh environment.In the second scenario, dubbed strategy-2 , the correlationestablished between S and E n is removed before the intra-environment interaction E n − E n + . The states achieved at eachstage of strategy-2 are thus as follows. First, the collision be-tween system and E n occurs, which gives the state ρ S E n = U S R ( ρ S ⊗ ρ E n ) U † S R , (23)and their resulting marginals ρ S (cid:48) = tr E n [ ρ S E n ] and ρ E (cid:48) n = tr S [ ρ S E n ] for the system and E n respectively. Then, themarginal state of the E n + sub-environment component afterthe intra-environment collision is ρ E (cid:48) n + = tr E (cid:48) n [ U EE ( ρ E (cid:48) n ⊗ ρ E n + ) U † EE ] . (24)The resulting state of the system prepared by strategy-2 be-comes ρ uS = tr E (cid:48) n + [ U S R ( ρ S (cid:48) ⊗ ρ E (cid:48) n + ) U † S R ] . (25)This scenario clearly di ff ers from the first one in both the num-ber of particles being involved, and the amount of correlationsthat are retained as a result of the system-environment inter-action. In turn, this influences the non-Markovian features ofthe dynamical maps applied to S and arising from the imple-mentation of such strategies.To quantify the degree of non-Markovianity of the reducedsystem dynamics undergone by S , we employ the measure fornon-Markovianity proposed in Ref. [10] which is associatedwith back-flow of information from the environment to thesystem. This is based on the time behavior of the trace dis-tance between two di ff erent initial quantum states of S , thatis D ( ρ , ρ ) = || ρ − ρ || , (26)where || ρ || = tr (cid:104) (cid:112) ρ † ρ (cid:105) is the trace norm of operator ρ and ρ , are two density matrices of S . For Markovian dynamics, D ( ρ , ρ ) monotonically decreases with time for any pair ofinitial states ρ , (0). On the contrary, a dynamical process issignalled as non-Markovian if there is a pair of such states forwhich this quantity exhibits a non-monotonic behaviour. (a) (b) n D n D FIG. 3. The trace distance D between evolved system states as a function of the number of collision iteration n with the environment forboth strategies. Upper (a) [lower (b) ] panel are the results for the strategy-1 [ strategy-2 ]. We have considered the initial states ρ S ( σ zS ) and ρ S ( σ yS ), while the sub-environments are prepared in ρ R ( σ zR ). The red dotted curve corresponds to the Markovian situation, τ e = .
0, whilethe blue dashed and green dot-dashed curves represent the non-Markovian dynamics with the dimensionless inter-environmental couplingtime τ e = π/
43 and τ e = π/ τ = π/
42 for weak coupling and the system andenvironment frequency parameters are ω S = ω R =
3, while their inverse temperature is fixed at β S = β R = . IV. ANALYSIS OF NON-MARKOVIANITY AND ITS ROLEIN THE PERFORMANCE OF THE ENGINE
Now we present the numerical analysis of the non-Markovian dynamics of the collision model for both strategiesdescribed above and then, their role on the thermodynamics ofthe engine. In the remainder of the paper, we will assume boththe system and reservoir to be two-level systems with Hamil-tonian H i = ω i σ ( j ) i / i = S , R ), with the thermal state densitymatrix of the form ρ i (cid:16) σ ( j ) i (cid:17) = exp( − β i H i ) / Z i , (27)where j = x , y , z is a label for the j -Pauli spin operator of par-ticle i = S , R , and β i is the corresponding inverse temperature. A. Non-Markovianity features from both strategies
We numerically analyze the behaviour of the trace dis-tance D ( ρ S , ρ S ) as the collision-based model for system-environment interactions are repeatedly executed. This anal-ysis elucidates how the system can be initializes in a stateresulting in dynamical signatures of non-Markovianity usingdi ff erent strategies described in Section III and correspondsto the first two steps of the engine protocol, see Section II A.We present the behaviour of the trace distance in Eq. (26) fortwo initial states prepared at ρ S ( σ zS ) and ρ S ( σ yS ). We haveassumed that all environmental particles / qubits are initializedin the state ρ R ( σ zR ). Figures 3(a) and 3(b) show the di ff er-ences between the two strategies addressed in this study. Forpurely Markovian dynamics ( τ e =
0, red dotted curves), thetrace distance decreases monotonously while switching on theinter-environment interaction times ( τ e (cid:44)
0, blue dashed andgreen dot-dashed curves) results in revivals that are evidenceof non-Markovianity. In fact, this system-environment inter-action produces a backflow mechanism - which is seen as os-cillations of the trace distance that fades out in the large num- ber of collisions with fresh ancilla. The strong environment-environment interaction time τ e = π/ strategy-1 (Fig. 3 (a) )but fades out to a non-zero value in the strategy-2 , see Fig.3 (b) . While the non-Markovian dynamics persists for bothstrategies in strong intra-environment interaction, the interme-diate coupling strength shows a clear dependence of the non-Markovian nature on the way information / correlation is devel-oped via collisions. For a weaker environment-environmentparticle interaction times τ e < π/
4, both strategies trace dis-tance decreases as the number of environmental collision in-creases, see blue dashed curves in Fig. 3. For more extensivediscussion on the way information is exchanged between thesystem and environment for the two strategies and their di ff er-ences / superiority, see Refs. [54, 56]. B. Feedback-driven engine analysis
Let now evaluate the influence of non-Markovianity on per-formance of the measurement-based machine described insection II above. We consider a two-level system initially pre-pared in the state ρ S ( σ zS ) and many identical subenvironmentprepared in the state ρ R ( σ zR ). The measurement ancilla is pre-pared in the state ρ M = | (cid:105)(cid:104) | with the system-measurementapparatus unitary evolution U S M characterized by the cou-pling of the form H S M = σ x ⊗ σ x . Here, we assumed the x -measurement direction but we note that the same optimalvalue is obtained for y -measurement direction considering theinitial state of the system. After a feedback operation is per-formed on the state of the system based on the outcome of themeasurement, the thermodynamic quantities, work and quan-tum mutual information are numerically calculated, see Fig.4. Note, in the numerics, the maximal values of the energy E [ ρ f bS ] and entropy S [ ρ f bS ] are used and obtained by sampling (a) n Δ W t n I q m - - - - - - n Δ S k f b (b) n Δ W t n I q m - - - - - - n Δ S k f b FIG. 4. Feedback driven engine performance: The total work extraction ∆ W t , the quantum mutual information I qm and the entropy changeduring feedback step ∆ S fbk as a function of number of collision n with the environment. The upper panel (a) corresponds to strategy-1 whilethe lower panel (b) is for strategy-2 . The red dotted curve corresponds to the Markovian dynamics, τ e = . τ e = π/
43. The green dot-dashed curve represent the full swap non-Markovian dynamics, τ e = π/ τ = π/
42 for weak coupling and the system and environment frequencies parameters are ω S = ω R = τ m = π/
14 and β S = β R = . of the feedback rotation parameters R (0 , α, θ, φ ) from 0 − π .In Fig. 4, the feedback engine performance, work per-formed by the engine protocol and the corresponding quantummutual information associated with the measurement step,as a function of repeated collision are presented for the twodi ff erent non-Markovian strategies described above. For theMarkovian dynamics ( τ e =
0, red dotted curves in Fig. 4(a)and (b)), the work extraction and quantum mutual informa-tion increases as the system-environment interactions timesgrow until it they reach constant values many collision itera-tion. For the strategy-1 , Fig. 4(a), as the system dynamics isprepared to be non-Markovian, an oscillatory behaviour whichvanishes in the long collision time are observed for both en-gine performance quantities - work extraction and informationgain. The non-Markovian feature is strong at short collisiontimes and can exceed their Markovian counterpart. However,the intermediate system-environment iteration is marked withsuppression of the engine performance due to memory e ff ect.For the non-swap environment-environment interactions (e.g τ e = π/ ∆ E u , ∆ Q u and ∆ W u ) during the preparation step vanishes. In addi-tion, we remark that including the work done on / by the sys-tem during the preparation ( step -1 ) does not a ff ect our resultsqualitatively. Moreover, the system entropy change during thefeedback ∆ S f bk exhibit similar behaviour and always negative,see right panel of Fig. 4(a). Figure 4(b) shows the work extraction and informationgain through measurement resulting from implementation of strategy-2 . We observe that such non-Markovian dynamicsscenario ( τ e (cid:44)
0) gives rise to non oscillatory behaviour con-trary to strategy-1 and the amount of work extraction and in-formation gain quantities never exceed the Markovian one.This behaviour is akin to the observation in the trace dis-tance Fig. 3(b), in which the strategy-2 oscillation are shorttime leave. Interestingly, for strong environment-environmentinteraction time τ e = π/
4, the work extraction and informa-tion gain saturate to finite value that is lower than the Marko-vian case, see the green curves in Fig. 4(b). Likewise,the saturation occurs at a vanishing change in the systemwork done, ∆ W u =
0. For more iterations with fresh envi-ronments under weaker interaction environment-environmenttime τ e = π/
43, the quantities attain the Markovian values.However, it takes di ff erent amount of environment collisionsto achieve the Markovian conditions for both strategies. V. CONCLUSION
We have investigated the interplay between memory e ff ectsand performance of a feedback-driven quantum engine. Theengine setup consists of system, reservoir and measurementprobe which have modelled as set of two-level systems. Wehave employed the trace distance as a measure of memorye ff ects (non-Markovianity) to illustrate two strategies of real-izing non-Markovian dynamics. We have observed that thememory e ff ect can enhance the performance - work extraction and information gain of feedback driven engine in a system-environment interaction short time. However, the perfor-mance decreases during the intermediate interaction time andapproaches the Markovian value at very long time. Besidesshedding light on the interplay between non-Markovianity andmeasurement driven engine, this study suggest more theoret-ical e ff ort to understand the role of memory on informationthermodynamics. ACKNOWLEDGEMENT
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