Feedback of Non-Markovian Quantum Dynamics in Dimer System: the Effect of Correlated Environments and Temperature

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1 Commun. Theor. Phys. 64 (05) Vol. 64, o. 6, December, 05 Feedback of on-markovian Quantum Dynamics in Dimer System: the Effect of Correlated Environments and Temperature ZHU Qin-Sheng (ý â), DIG Chang-Chun (òđë),, WU Shao-Yi ( Â), and LAI Wei ( Ú) School of Physical Electronics, University of Electronic Science and Technology of China, Chengdu 60054, China Experimental Middle School of Chengdu Economic and Technology Development Zone, Chengdu 60054, China (Received April 7, 05; revised manuscript received uly 6, 05) Abstract In this work, we research the non-markovian dynamical process of the dimer system and the effect of the interactional environments for the information feedback under different temperature T. ot only the functional relation of the trace distance and the fidelity are obtained, but also the changing properties of the fidelity and the measure quantity (φ) which are used to quantify the degree of the non-markovian process are discussed as a function of the interactional strength q between the environments. These results show a possible method which can preserve the information and enhance the distinguishability of the pair of states in decohering environments. PACS numbers: Y, Ta, 4.50.Lc, Ud Key words: non-markovian process, measure, decoherence Introduction For a realistic physical system, because of the inevitable interaction between the environment and systems, the decoherence and dissipative phenomena always exist and make the system information loss to environment. [ 3] Hence, the research about the closed systems needs to be extended to the open systems. Among the open systems, some quantum systems show a Markovian behavior in which the system information continues lossing to the environment, namely neglecting quantum memory effects and feedback effect of environments which have been reported by many researches in the past years. [4 8] evertheless, some quantum systems exhibit a pronounced non-markovian behavior with a feedback flow of information from the environment to the system when the environment has a nontrivial structure, signifying the quantum memory effects. In quantum information and quantum computation fields, many significant theoretical concept and experimental works have been carried out, e.g. sudden death and birth of entanglement, [9 4] the non-markovian character of the noise leads to the system arriving steady state entanglement in dimer system, [5] and so on. [6 0] Recently, some results about the controlling methods of the non-markovian process, e.g. optimal control, [] optimal feedback control, [] and so on, [3 4] have been obtained with the development of quantum engineering technology. In theory, the trace distance [] which introduced by Breuer et al. [5 6] had been used to quantify the degree of non-markovian process or the feedback of the information (realied in Refs. [7 8]). Simultaneously, Xue et al. [9] realied the coherent feedback control by applying a quantum control field for an open system and succussed suppressing decoherence. Therefore researching the flow of information between the system and environments is important and maybe have a paramount relevance to a number of applications in physics. Based on the results which aroused by the interactions within the environment, [30 3] in this work, firstly, it is shown how the interactional strength parameter q between the two different Fermi-spin environments affects the degree of memory effects or the flow of information between the system and its environment. Secondly, by choosing a fixed pure states, we obtain the function relation of the trace distance and fidelity of random initial states pairs. Finally, the changing behaviors of the measure quantity (φ) [8,5] had been discussed as a function of q for different temperature T. This paper is organied as follows. In Sec., the model Hamiltonian of the total system is given. In Sec. 3, the exact solution of reduce density matrix of the dimer system is obtained. In Sec. 4, the properties of the trace distance and fidelity are researched. In Sec. 5, the measure quantity (φ) and the flow of system information are discussed. Conclusion is given in the final section. The Model Hamiltonian of the Total System Considering the simplest electronic energy transfer system formed by a dimer system, the Hamiltonian of the dimer systems is given as [5,9,33 34] H d = ε + ε + ( + ), () Supported by the Fundamental Research Funds for the Central Universities under Grants o. ZYGX ccding66@63.com c 05 Chinese Physical Society and IOP Publishing Ltd

2 o. 6 Communications in Theoretical Physics 677 where ε and ε are the energy levels of the dimer system, is the amplitude of the transition. and are the level states of the dimer system. After some calculations, we can obtain the eigenvalues and eigenvectors of the H d. E = + +, [ + ψ = + ] +, E = +, [ ψ = + ] +, () where = (ε + ε )/ and = (ε ε )/. The paper concentrates on, in general conditions ε ε, the relationship between the degree of the non- Markovian process and the interactional strength parameter q between the environments, namely how the parameter q affects the flow of information between the dimer system and the environments. The Hamiltonian of the total system has the following form: [,5,9,33 34] H = H d + (H Bi + H dbi ) + q i=, k = σ k, σ k,. (3) Here, each environment B i consists of i particles (i =, ) with spin / are the Pauli matrices and α i denotes the fre- where σ k,i quency of σ k,i i H Bi = α i σ k,i, (4). Because a dimer system is made of two subsystems (for example, two spin-like particles or two qubits), which bounding to each other, we consider two subsystems (each with a pseudo-spin-/ particles) in different environments, [35] respectively. The interaction between the subsystem and the environment can be described as γ(σ /) i (σk,i /)[] (Here γ is the coupling constant and σ is pseudo-spin Pauli operator for the pseudo-spin- / particles). Here we only consider the interaction between the upper level and the environment, [5,8 3] so the interaction between the dimer and the environment is described by [5,9,33 36] H db = γ σk,, ( ), H db = γ σk,, ( ). (5) Here the dimer system only interacts with part of the particles in the environment, so and satisfy and, respectively. The last term of Eq. (3) describes an Ising-type interaction between the environments and interactional strength q. The cases q = 0 and q 0, describe independent and interactional spin bath, respectively. 3 The Exact Solution of Reduced Density Matrix of the Dimer System In order to obtain the exact solution of the system, we define collective spin operators S i = i σ k,i, S i,0 = i σ k,i, S i, = S i S i,0. Thus, the total Hamiltonian can be written as: H = H d + α i Si + q(s,0 + S,)(S,0 + S,), i=, H d = (ε + γ S,0) + (ε + γ S,0) + ( + ). (6) Because the dimer system interacts with only part of the particles in the environment, we introduce an orthonormal basis in the bath Hilbert space H B in terms of states j 0, m 0, j, m. [] These states are defined as the eigenstates of S 0, (S 0), S and (S ) i.e., (S l ) j 0, m 0, j, m = j l (j l + ) j 0, m 0, j, m, S l j 0, m 0, j, m = m l j 0, m 0, j, m, (l = 0, ), (S l ) = (S x l ) + (S y l ) + (S l ), (7) with j 0 = 0,..., /, m 0 = j 0,..., j 0, j = 0,..., ( )/ and m = j,..., j. The formal solution of the von eumann equation can be written as d ρ(t) = Lρ(t) = i[h, ρ(t)], (8) dt ρ(t) = e Lt ρ(0), (9) where ρ(t) denotes the density matrix of the total system. Our main goal is to derive the dynamics of the reduced density matrix ρ d (t). ρ d (t) = tr B (e Lt ρ(0)), (0) where tr B denotes the partial trace taken over the Hilbert space of the spin bath. For the initial state ρ(0) = ρ d (0) ρ B (0), the reduced density matrices ρ d (t) of the dimer system is ρ d (t) = Tr B [U(t)ρ d (0)ρ B (0)U (t)] = C [ ]Q [ a a a a ] Q [ ], () where and are the conjugate of the level states and, respectively. The matrix Q and parameter C are given in Appendix A. Here, the initial state ρ d (0) of the dimer system is [ ] a a ρ d (0) = a, () a

3 678 Communications in Theoretical Physics Vol. 64 and the bath is given as the canonical distribution [,3 3] ρ B (0) = Z e βq(s,0 +S, )(S,0 +S, ) e βαis i, (3) i= β = /K B T (T is temperature and K B is Boltmann constant). The parameters of Eq. () and the partition function Z [,36 37] of the bath in Eq. (3) are given in Appendix A. 4 The Fidelity and Trace Distance of the Dimer System An important feature of the non-markovianity dynamical process is the flow of information between the environment and the system. For a non-markovianity dynamical process, it is important to know the degree of the feedback information from the environment to the system. The fidelity and trace distance which describe a measurement of a distance between quantum states can clarify the above problems. [,5] For a pair of quantum states ρ and ρ, the fidelity and trace distance are defined: F(ρ, ρ ) = Tr ρ / ρ ρ /, D(ρ, ρ ) = Tr ρ ρ. (4) If the value of fidelity between two states is close to one, it means that the states are also close in trace distance. [] For a pair of quantum states ρ and ρ of the dimer system, [ ] [ b b c c ρ = b, ρ = b c c ], (c + c = ). (5) Choosing a fix point in quantum state space of dimer system, one can obtain the function relationship between fidelity and trace distance. F(ρ, ρ ) = ((q + q ) + ( ) i (q q ) + 4q q ). (6) i=0, The computational process and the related parameters of Eq. (6) are shown in appendix B. Fig. (Color online) The change of the trace distance D(ρ, ρ ) as a function of time t and the coupling parameter q for the initial pair of states ρ (0) = (/) + (/) + (/) + (/) and ρ (0) = (/) (/) (/) + (/). In Figs. (a) and (b), the temperatures of the baths are 77 K and 300 K, respectively. For both figures, the parameters are γ = 0. ps, γ = 0.3 ps, = 0 ps, = 4, =, = 8, = 6, α = 50 ps, α = 00 ps, = 0 ps, = 0 ps. Because the fidelity and trace distance can describe the information flow between the system and environments, [,5] we obtain the changing properties of the trace distance as function of time t and the parameter q for the different pair of initial states, as shown in Fig.. From Fig., for small parameter q (0 < q < 0) at low temperature T, we can find a sudden increase for the value of the trace distance D at some time intervals, which is strengthened with the decrease of the interacting particles number i (i =, ) of the environments. With the increasing of the parameter q (0 < q < 70), the trace distance D decreases first and then presents the nonperiodic oscillation under low temperature T. Especially, when q exceeds 70, the non-periodic oscillation of the trace distance D disappears and D approximately remains unchanged. This nature may provide promising prospect for the study of quantum information and the measurement of the states. For higher temperature T, the trace distance D does not exhibit sudden increase for smaller parameter q (0 < q < 50) but still presents the smaller non-periodic oscillation when q arrives a threshold value. Finally, we know that the change of the trace distance D depends on the initial state from Fig. and Eqs. () and (4). 5 The Measure Quantity (φ) of the on- Markovianity Process To research the flow of information between the environment and the dimer system, a positive function (φ) (φ represents a pair of states), which describes a measure about the total increase of the distinguishability of pair states over the whole time-evolution, [5] is defined as fol-

4 o. 6 Communications in Theoretical Physics 679 lows: (φ) max dtσ(t, ρ, (0)). (7) ρ,(0) σ>0 Here, the rate σ of the trace distance D(ρ, ρ ) is defined by σ(t, ρ, ρ ) = d dt D(ρ, ρ ). (8) Physically, σ > 0 (σ < 0) is the intensity of the flow of information from the environment feedback (the system losing) to the system (the environment). Based on the previous studies, [8,5] there are two results about the distinguishability of the pair states and the information exchange between the environment and the system. One is that a nonero value of (φ) means that there is a pair of initial states for which the trace distance or the distinguishability increases over a certain time interval and can be interpreted as the feedback of information from the environment to the system (i.e., so called the phenomena of the memory effect). The other is that (φ) is equal to ero [8,5] if and only if the trace distance of any pair initial states is a monotonically decreasing function of time t, indicating a continuous lossing of information from the system. Since the interesting phenomenon of the concurrence aroused by the coupling parameter q has been studied in our recent paper, [9] further research on the properties of the information flow of the dimer systems is worthwhile. Here, the change of (φ) is demonstrated by drawing a large sample of random pairs of initial states for different temperature T and shown in Fig.. At first glance, Fig. reveals that for different values of the parameter q, the values of (φ) are different associated with different pairs of initial states, and show different changing properties for different temperatures. At low temperature T, (φ) has some interesting behaviors, as shown in Fig. (a). For smaller parameter q (0 < q < 8), both (φ) and the feedback of the information gradually decrease with the increase of the parameter q. For 8 < q < 54, the information back from the environment to the dimer system is strengthened. As q [54, 0], (φ) gradually decreases and approaches to ero (0 < (φ) < 0.04). From Fig. (a), one can find as q exceeds the critical value (about 00), the fluctuation of the flow of information between the environment and system is depressed, and the system information is preserved or a steady distinguishability of the pair of states is achieved. In Fig. (b), at high temperature T, (φ) equals to ero when the parameter 0 < q < 00. According to Fig. (b), the information flow out from the dimer system to the environment and the changing behavior of (φ) is similar to Markov process. [8,5] With the increase of q (00 < q < 75), the feedback effect of the information from the environment to the dimer system, is strengthened and exhibits a non-markov process. [8,5] But (φ) gradually decreases with the further increase of q (q > 75) and approaches to ero (0 < (φ) < 0.05). This behavior stems from the depression of the fluctuation of the flow of information and the loss of the system information approaches to ero with the increase of q. The above changing properties can be understood from the information flow. At higher temperature T, it is shown that the system information flows to the environments when q is small (Fig. 3(a)). (i) The information of system H d flows to environments H B and H B. (ii) The exchange of the system information (mutual information [] ) between environments H B and H B is much less than the information of system flow to environments, resulting in the loss of the system information. When q is large (Fig. 3(b)), the exchange of the system information between environments is strengthen, which arouses the feed back of the system information. In other words, the feedback of system information can be affected by the new information transfer channels H d H B (H B ) H B (H B ) H d. Fig. (Color online) The non-markovianity (φ) as a function of the coupling parameter q for the dimer system. 000 randomly pairs of pure or mixed initial states are choosen. In Figs. (a) and (b), the temperatures of the baths are 77 K and 30 K, respectively. For both figures, the parameters are = = 4, = =, γ = 0. ps, γ = 0.3 ps, = 0 ps, α = 50 ps, α = 00 ps, = 0 ps, = 0 ps. The relationship of the information flow at low temperature is shown in Fig. 3(b). (i) The information of system flow out and feedback occur at the same time. (ii) The exchange of the system information (mutual information [] ) between environments H B and H B makes some system

5 680 Communications in Theoretical Physics Vol. 64 information preserved by means of the exchange interaction. (iii) For smaller q, this system information stored in the exchange interaction which flows from environments H B and H B affects the feedback of the environments and shows the depression of the feedback. (iv) Similar to case of the high temperature, because the change of environments (the exchange of the system information) between environments is strengthened with increasing q, the feedback aroused by the change of environments occurs. So, not only the system information feedback from the environments H B and H B to system, but also that of the system can been achieved by the new information transfer channels H d H B (H B ) H B (H B ) H d. The competitive relationship about the information flow between the system and environments is shown as a function of the interaction between environments in Fig. 3. Contrasting the results of Ref. [9], our results not only show that the feedback of the information can be controlled by changing the interaction between the environments, but also reveals that the system information can be preserved. Simultaneously the distinguishability of the states had been enhanced with the increase the interactional strength q, especially for lower temperature T. Simultaneously, the behaviors of the feedback are more depended on the initial states. This system may be formed by doping the dimer system between two layer quantum dot array (similar doping atom between two layer graphene [38] ) and the interactional strength q may be changed with the distance between two layer quantum dot array and applied to the external field, for example the electric field and the magnetic fields (this is similar to the research of graphene. [39] This nature may provide promising prospect for the study of quantum information, the design of quantum devices, quantum measurement and entanglement preservation. [40 43] Fig. 3 The flow of the system information between the system and one of two environments as well as that between two environments. Figures 3(a) and 3(b) describe the different information flow for the high and low temperature, respectively. 6 Conclusions The non-markovian dynamical properties of the dimer system interacting with two interactional environments are investigated by use of trace distance and the measure quantity (φ) which quantify the degree of non- Markovian process. The results indicate that the interactional strength parameter q between the environments can effectively affect the information feedback. Especially, at low temperature, the feedback of the system information is first depressed and then strengthened with the increase of q. These results can be helpful to understand the mechanisms of depression of the entanglement decoherence and the preservation of quantum states distinguishability, and the quantum information of the biological systems. [40 43] Appendix A In this appendix, the detailed expressions of the parameters C, Q in Eq. () and the partition function Z [,3 3] of the bath in Eq. (3) are provided. ( Q = ) + ( ) + [ ( ) + + ( ) + + e ie t ] + ( ) e ie t ( ) + C = Z / j 0 j0 =0 m 0 = j 0 ( )/ j / j 0 m = j j0 =0 m 0 = j 0 ( )/ j =0 j m = j ν(, j 0 )ν(( ), j ), (A) ν(, j 0)ν(( ), j )(e βq(m +m 0 )(m +m 0 ) e βα(m +m 0 ) e βα(m +m 0 ) ). (A)

6 o. 6 Communications in Theoretical Physics 68 The partition function Z of the bath is Z = / j 0 j0 =0 m 0 = j 0 ( )/ j / j 0 m = j j0 =0 m 0 = j 0 ( )/ j =0 j m = j ν(, j 0 )ν(( ), j ) ν(, j 0 )ν(( ), j ) j, m, j 0, m 0 j, m, j 0, m 0 ( e βq(s,0 +S, )(S,0 +S, ) i= e βαis i ) m 0, j 0, m, j m 0, j 0, m, j = / j 0 j0 =0 m 0 = j 0 ( )/ j / j 0 m = j j0 =0 m 0 = j 0 ( )/ j =0 j m = j ν(, j 0 )ν(( ), j ) ν(, j 0 )ν(( ), j )(e βq(m +m 0 )(m +m 0 ) e βα(m +m 0 ) e βα(m +m 0 ) ), (A3) where the parameters of above expressions are = + γ m 0 γ m 0, = + γ m 0 + γ m 0, E = + ( ) +, E = ( ) +. (A4) Here ν(, ) =!/(/ j)!(/ + j)!!/(/ j )!(/ + j + )! denotes the degeneracy of the spin bath. [5,3] Obviously, the following equality holds: / j 0 =0 ( )/ / j 0 =0 ( )/ j =0 ν(, j 0)ν(( ), j )ν(, j 0)ν(( ), j ) (j 0 + )(j + )(j 0 + )(j + ) = +. (A5) Appendix B Firstly, a fixed pure state ρ B = has been chosen in order to calculate the function relationship between fidelity and trace distance. Using Eq. (4), we have D(ρ B, ρ ) = Tr ρ B ρ = b + (b ), of Eq. (5). [ ρ / (D D ) b (D 3 D 4 )b = (D 3 D 4 )b C where + D = ( + b ), D 3 = ], (A7) +, D(ρ B, ρ ) = Tr ρ B ρ = c + (c ), D(ρ, ρ ) = Tr ρ B ρ = c + b (b c + b c ) + (c b ), F(, ρ ) = b, F(, ρ ) = c. (A6) Secondly, we obtain the analytical expression of the fidelity in Eq. (4) for a pair of quantum states ρ and ρ where F(ρ, ρ ) = Tr ρ / ρ ρ / = Tr [ ] / q q = q q i=0, D = ( b ), D 4 = C = + ( + b ) + ( b ), = + 4 b 4b + 4b., (A8) Putting ρ / and ρ to the expression of the fidelity in Eq. (4), we further obtain ((q + q ) + ( ) i (q q ) + 4q q ), (A9) q = (D D ) b 4 c + (D D )(D 3 D 4 ) b (b c + b c ) + (D 3 D 4 ) b ( c ), q = (D 3 D 4 ) b c + C(D 3 D 4 )(b c + b c ) + C ( c ), q = (D D )(D 3 D 4 ) b c b + (D 3 D 4 ) b c + C(D D ) b c + C(D 3 D 4 )b ( c ). (A0) From Eq. (A9), the parameters q and q are the functions of b c + b c, b, c, and b. Simultaneously, q q = A b + A b 4 c + A 3 c + A 4 b + (A A b + A A 3 + A A 4 b

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