Non-Markovian dynamics of interacting qubit pair coupled to two independent bosonic baths
I. Sinayskiy, E. Ferraro, A. Napoli, A. Messina, F. Petruccione
aa r X i v : . [ qu a n t - ph ] J a n Non-Markovian dynamics of interacting qubit paircoupled to two independent bosonic baths
I. Sinayskiy
Quantum Research Group, School of Physics and National Institute for TheoreticalPhysics, University of KwaZulu-Natal, Durban, 4001, South AfricaE-mail: [email protected]
E. Ferraro
MIUR and Dipartimento di Scienze Fisiche ed Astronomiche dell’Universit`a diPalermo, via Archirafi 36, I-90123 Palermo, ItalyE-mail: [email protected]
A. Napoli
MIUR and Dipartimento di Scienze Fisiche ed Astronomiche dell’Universit`a diPalermo, via Archirafi 36, I-90123 Palermo, ItalyE-mail: [email protected]
A. Messina
MIUR and Dipartimento di Scienze Fisiche ed Astronomiche dell’Universit`a diPalermo, via Archirafi 36, I-90123 Palermo, ItalyE-mail: [email protected]
F. Petruccione
Quantum Research Group, School of Physics and National Institute for TheoreticalPhysics, University of KwaZulu-Natal, Durban, 4001, South AfricaE-mail: [email protected]
Abstract.
The dynamics of two weakly interacting spins coupled to separate bosonicbaths is studied. An analytical solution in Born approximation for arbitrary spectraldensity functions of the bosonic environments is found. It is shown that in the non-Markovian cases concurrence “lives” longer or reaches greater values.PACS numbers: 03.67.Mn, 03.65.Yz on-Markovian dynamics of interacting qubit . . .
1. Introduction
The implementation of more and more efficient nanodevices exploitable in applicativecontexts like quantum computers, often requires a highly challenging miniaturizationprocess aimed at packing a huge number of point-like basic elements, whose dynamicsmimics indeed that of a qubit. Stimulated by such a requirement, over the last few yearstheoretical schemes have been investigated in the language of spin models [1]. Apartfrom the simple dynamical behaviour of each elementary constituent these Hamiltonianmodels do indeed capture basic ingredients of several physical situations. In addition,spin models allow for the description of the effective interactions in a variety of differentphysical contexts ranging from high energy to nuclear physics [2, 3]. In condensed matterphysics they capture several aspects of high-temperature superconductors, quantum Hallsystems, and heavy fermions [4, 5, 6]. We point out that Hamiltonians for interactingspins can be realized artificially in Josephson junctions arrays [7] or with neutral atomsloaded in optical lattices [8, 9, 10] or else with electrons in quantum dots [11].In this context, a subject deserving a particular interest is the entanglementdynamics. In view of possible applications it is important to understand the extentat which quantum coherences may be protected against the unavoidable degradation ofthe purity of the state, in particular in the presence of many-body interactions.In this paper we focus our attention on a spin model recently introduced by Quiroga[12] and successively analyzed by other authors [13]-[16]. It consists of two interactingspins , each one coupled to a separate bosonic bath [12, 13]. Our aim is to study theentanglement dynamics of the two spins in the non-Markovian regime. Many authorshave addressed the question of the dynamics of the entanglement between qubits in thenon-Markovian environments. However, usually a system of non-interacting qubits incontact with separate bosonic baths is considered. Either, entanglement is introducedin the initial preparation [17, 18] or created by the interaction of qubits with a commonenvironment [19]. The focus of this paper is to study a system of directly interactingqubits. This is a typical situation in solid-state systems. For example, double quantumdots can be modeled as coupled qubit systems in contact with separate bosonic baths.For the demonstration of the dynamical properties of the system, in this paper we willconsider Lorentz spectral density and Ohmic spectral density with Lorentz-Drude cut-off. For a different model it has been shown that entanglement of qubits can occur insuper-Ohmic environments even at non-vanishinig temperature [20, 21].The paper is structured as follows. In Section 2 we describe in detail the model.In Section 3 we present the analytical solution of the non-Markovian master equationfor the reduced system constituted by the two interacting spins in the zero temperaturelimit. In Section 4 we analyze the entanglement dynamics of the two spins assumingfor the environment a Lorentz spectral density and an Ohmic spectral density with aLorentz-Drude cut-off function. Finally, conclusive remarks are given in Section 5. on-Markovian dynamics of interacting qubit . . .
2. The model
Our analysis is focused on the dynamics of a composite system coupled to bosonicenvironments. Parts of the dynamical system are weakly interacting. The totalHamiltonian can be written as: H = H S + λ H I + H B + λH SB , (1)where ( H S + λ H I ) is the Hamiltonian describing the dynamics of the composite system, H S is the Hamiltonian of the free components of the system, H I is the Hamiltonianof interaction between the parts of the system. The operator H B describes bosonicenvironments, the Hamiltonian H SB denotes the Hamiltonian of the interaction betweensystem and environment. The parameter λ is a dimensionless expansion parameter. Thenon-Markovian dynamics of the reduced system will be described by a Master Equationcontaining the terms not higher than the square of the expansion parameter λ .The second-order time-convolutionless form of the Master equation is given by [22]: ddt ρ IS ( t ) = − λ Z t dτ tr B [ H SB ( t ) , [ H SB ( τ ) , ρ IS ( t ) ⊗ ρ B ]] , (2)where H SB ( t ) denotes the Hamiltonian H SB and ρ IS ( t ) denotes the density matrix of thereduced system in the interaction picture by the Hamiltonian ( H S + λ H I + H B ). Thedensity matrix ρ B = e − βH B / tr[ e − βH B ] describes the state of the environment.The present general approach is applied to a system consisting of a pair of weaklyinteracting spins, each one coupled to a bosonic bath. The total Hamiltonian is givenby Eq. (1). The Hamiltonian of the two free spins characterized by the same energy ǫ reads H S = ǫ σ z + ǫ σ z . (3)As usual σ zi and σ ± i are the Pauli operators describing the i − th spin ( i = 1 , λ H I = K (cid:0) σ +1 σ − + σ − σ +2 (cid:1) , (4)where K is a constant defining the strength of the spin-spin interaction. TheHamiltonian of the bosonic baths characterized by the annihilation and creationoperators b ni and b † ni ( i = 1 ,
2) reads H B = X n ω n, b † n, b n, + X m ω m, b † m, b m, . (5)The coupling of each spin to the separate bosonic baths is described by H SB = σ +1 X n g n, b n, + σ +2 X m g m, b m, + h . c ., (6)where g n, and g m, denote the coupling between the spin and its corresponding bosonicbaths. In this paper units are chosen such that k B = ~ = 1. The Hamiltonian λH SB inthe interaction picture defined by the Hamiltonian ( H S + λ H I + H B ) is given by λH SB ( t ) = σ +1 X n g n, b n, e i ( ǫ − ω n, ) t + σ +2 X n g n, b n, e i ( ǫ − ω n, ) t + h . c .. (7) on-Markovian dynamics of interacting qubit . . . λ and higher. Bydirect calculation we show that − λ Z t dτ tr B [ H SB ( t ) , [ H SB ( τ ) , ρ IS ( t ) ⊗ ρ B ]] = X j =1 L ( Dj ) ( t ) ρ IS ( t ) , (8)where L ( Dj ) ( t ) is the Liouville superoperator defined by L ( Dj ) ρ IS ( t ) = B ( j ) ( t ) (cid:2) σ − j ρ S ( t ) , σ + j (cid:3) + ¯ B ( j ) ( t ) (cid:2) σ − j , ρ S ( t ) σ + j (cid:3) (9)+ ¯ A ( j ) ( t ) (cid:2) σ + j ρ S ( t ) , σ − j (cid:3) + A ( j ) ( t ) (cid:2) σ + j , ρ S ( t ) σ − j (cid:3) . The quantities A ( j ) ( t ) and B ( j ) ( t ) appearing in the previous expression are the so-calledcorrelation functions, whose explicit form is given by A ( j ) ( t ) = Z t dτ X n | g n,j | h b † n,j b n,j i Bj e i ( ǫ − ω n,j )( t − τ ) (10)= i X n | g n,j | h b † n,j b n,j i Bj − e i ( ǫ − ω n,j ) t ǫ − ω n,j ,B ( j ) ( t ) = Z t dτ X n | g n,j | h b n,j b † n,j i Bj e i ( ǫ − ω n,j )( t − τ ) (11)= i X n | g n,j | h b n,j b † n,j i Bj − e i ( ǫ − ω n,j ) t ǫ − ω n,j , where h O i Bj ≡ tr B j { Oρ Bj } , ¯ A ( j ) ( t ) and ¯ B ( j ) ( t ) being the complex conjugate of A ( j ) ( t ) and B ( j ) ( t ), respectively. To obtain expression (8) we used the fact that thebosonic environments assumed in this article are uncorrelated with each other and h b n,j b † n,j i Bj , h b † n,j b n,j i Bj are the only non-zero second-order correlations in the bath, allthe other vanish.Transforming back to the Schr¨odinger picture we obtain the following MasterEquation ddt ρ S ( t ) = − i [ ǫ σ z + ǫ σ z + K (cid:0) σ +1 σ − + σ − σ +2 (cid:1) , ρ S ( t )]+ X j =1 L ( Dj ) ( t ) ρ S ( t ) . (12)It is easy to see that the superoperator L defined as L ρ S ( t ) = − i [ ǫ σ z + ǫ σ z , ρ S ( t )] (13)commutes with the superoperator L ME ( t ) given by L ME ( t ) ρ S ( t ) = − i [ K (cid:0) σ +1 σ − + σ − σ +2 (cid:1) , ρ S ( t )] + X j =1 L ( Dj ) ( t ) ρ S ( t ) , (14)and can be neglected as it is irrelevant for the dynamics of the expectation values definedby the density matrix ρ S ( t ). So, the final form of the Master Equation which is goingto be studied in this article reads ddt ρ S ( t ) = − i [ K (cid:0) σ +1 σ − + σ − σ +2 (cid:1) , ρ S ( t )] + X j =1 L ( Dj ) ( t ) ρ S ( t ) . (15) on-Markovian dynamics of interacting qubit . . .
3. Exact solution of the Master equation
In order to solve the Master equation (15), it is useful to separate the equations ofmotion for the diagonal elements of the density operator ρ S ( t ) from those relative to theoff-diagonal elements. We have indeed proved that the diagonal and two non-diagonalelements of ρ S ( t ) have to satisfy the following system of the equations ddt ρ S ( t ) ρ S ( t ) ρ S ( t ) ρ S ( t ) ρ S ( t ) ρ S ( t ) = Λ ( t ) ρ S ( t ) ρ S ( t ) ρ S ( t ) ρ S ( t ) ρ S ( t ) ρ S ( t ) , (16)where Λ ( t ) = − β − β α α β − α − β α iK − iKβ − α − β α − iK iK β β − α − α iK − iK ξ − iK iK ξ (17)and α j = A ( j ) ( t ) + ¯ A ( j ) ( t ) , β j = B ( j ) ( t ) + ¯ B ( j ) ( t ) , (18) ξ = − A (1) ( t ) − ¯ A (2) ( t ) − B (1) ( t ) − ¯ B (2) ( t ) . In what follows we will consider the case in which the two bosonic baths are bothprepared in a thermal state with T = 0. This assumption in turn implies that thecorrelation functions reduce to A ( j ) ( t ) ≡ , B ( j ) ( t ) ≡ B ( t ) = i X n | g n | − e i ( ǫ − ω n ) t ǫ − ω n . (19)Under these hypotheses it is possible to rewrite Λ ( t ) in the following way, Λ ( t ) =( B ( t ) + ¯ B ( t )) L + iKL , where L and L are 6 × ρ S ( t ) ρ S ( t ) ρ S ( t ) ρ S ( t ) ρ S ( t ) ρ S ( t ) = U ( t ) ρ S (0) ρ S (0) ρ S (0) ρ S (0) ρ S (0) ρ S (0) , (20)where U ( t ) = T e R t dτ Λ ( τ ) = e G ( t ) L e ( iKt ) L (21) on-Markovian dynamics of interacting qubit . . . G ( t ) appearing in the expression for the matrix U (6) ( t ) is defined as G ( t ) = Φ( t ) + ¯Φ( t ) , (22)with Φ( t ) = Z t dτ B ( τ ) = X n | g n | − e i ( ǫ − ω n ) t + i ( ǫ − ω n ) t ( ǫ − ω n ) . (23)The time dependence of the off-diagonal element ρ S ( t ) is trivial, namely ρ S ( t ) =exp ( − t )) ρ S (0). For the other off-diagonal elements we get the following systemof equations: ddt ρ S ( t ) ρ S ( t ) ρ S ( t ) ρ S ( t ) = Λ ( t ) ρ S ( t ) ρ S ( t ) ρ S ( t ) ρ S ( t ) , (24)where Λ ( t ) = − β − B ( t ) iK iK β − B ( t ) 0 00 β − B ( t ) − iKβ − iK − B ( t ) . (25)One can check that the solution for the above equation has the following form: ρ S ( t ) ρ S ( t ) ! = e − G ( t ) − Φ( t ) e iKtσ x ρ S (0) ρ S (0) ! (26)and ρ S ( t ) ρ S ( t ) ! = U ( t ) ρ S (0) ρ S (0) ! + U ( t ) Z t dτ U − ( τ ) ρ S ( τ ) ρ S ( τ ) ! , (27)where the operator U ( t ) is defined by U ( t ) = e − Φ( t ) e − iKtσ x . (28)At this point we are in the position to explicitly write the density matrix of thetwo coupled spins at a generic time t starting from an arbitrary initial condition. Forsimplicity, we report on such a solution in the Appendix. In what follows, instead, wefocus on the cases in which the initial state of the pair of coupled spins is the Bellstate | Ψ − i = √ ( | i − | i ) or the factorized state | Ψ i = | i . Exploiting the resultspresented in the Appendix it is possible to demonstrate that the state of the reducedsystem at a generic time instant t can be written in the simple form ρ S (Bell) ( t ) = e − G ( t ) | Ψ − ih Ψ − | + (cid:0) − e − G ( t ) (cid:1) | ih | (29)and ρ S (0) ( t ) = e − G ( t ) | Ψ( t ) ih Ψ( t ) | + (cid:0) − e − G ( t ) (cid:1) | ih | , (30) on-Markovian dynamics of interacting qubit . . . | Ψ( t ) i = cos( Kt ) | i − i sin( Kt ) | i . (31)Another point which we would like to mention here is the connection between thenon-Markovian Master Equation (15) and the Markovian one. The Markovian limit ofthe Master Equation (15) can be constructed by taking the limit t → ∞ in the set ofcorrelation functions A ( j ) ( t ) and B ( j ) ( t ). The solution of the corresponding MarkovianMaster Equation for the system at hand can be constructed from non-Markovian onesby replacing functions Φ( t ) and G ( t ) with the corresponding Markovian onesΦ( t ) ⇒ Φ M ( t ) = tB M , (32)where B M = lim t →∞ B ( t ) . (33)In particular, for the function G ( t ) we have G ( t ) ⇒ G M ( t ) = t ( B M + ¯ B M ) = t πJ ( ω ) , (34)where J ( ω ) is the bath spectral density and ω = ǫ .
4. Entanglement dynamics
As emphasized before, the solution we have found has been obtained without specifyingthe spectral properties of the bath. The density matrix ρ S ( t ) describing the pair of thecoupled spins, however, depends on the bath spectral density through the function G ( t ).In this section, exploiting our results, we will analyze some dynamical properties ofthe central system for different spectral distributions of the environment. In particular,we will examine how the entanglement evolution is affected by the choice of the reservoirspectral density. Let us start by considering as a first case the Lorentzian distribution J ( ω ) = γ π γ ( ω − ǫ ) + γ , (35)where γ and γ are the reservoir and the system decay rate respectively. This choice inturn implies that the correlation function B ( t ), as given in the previous section, is B ( t ) = γ − e − γt ) (36)and consequently G ( t ) = γ t + γ γ ( e − γt − . (37)We have already demonstrated that starting from the Bell state | ψ − i = 1 √ | i − | i ) (38)at a generic time t the density operator describing our system can be written as inEq.(29). It is interesting to analyze how the interaction of the two coupled spins withthe environments modifies the entanglement initially present in the system. To this end on-Markovian dynamics of interacting qubit . . . Figure 1.
The concurrence C ( t ) for a Lorentz bath distribution for different valuesof the ratio γ/γ ( γ/γ = 0 . γ/γ = 1 (dashed line), γ/γ = 10 (dottedline), Markovian case (dot-and-dash line)). The initial state is the Bell state (38). we consider the time behavior of the concurrence [23] of the two spins. Using Eq.(29)it is easy to demonstrate that in correspondence to any environmental spectral density,the concurrence is given by C ( t ) = e − G ( t ) . (39)Thus, when J ( ω ) assumes the form (35) we have C ( t ) = exp (cid:18) − ( γ t + γ γ ( e − γt − (cid:19) . (40)In Figure 1 we plot C ( t ) against γ t for different values of the ratio γ/γ in the non-Markovian case. For comparison with the Markovian limit (34) we include also theMarkovian case ( G M ( t ) = γ t ). As expected, in the presence of the two baths theconcurrence function, starting from its maximum value, decreases as time elapses.However, in the non-Markovian regime, corresponding to γ/γ <
2, the entanglementin the two spins persists for a longer time with respect to the Markovian case.Suppose now that the two environments are characterized by an Ohmic spectraldensity with a Lorentz-Drude cut-off function [19, 24] J ( ω ) = 2 ωπ ω c ω c + ω , (41)where ω is the frequency of the bath and ω c is the cut-off frequency. Under this on-Markovian dynamics of interacting qubit . . . Figure 2.
The concurrence C ( t ) for a Lorentz-Drude bath distribution for differentvalues of the ratio ω c /ω ( ω c /ω = 0 . ω c /ω = 1 (dashed line), ω c /ω = 10(dotted line), Markovian case (dot-and-dash line)). The initial state is the Bell state(38). hypothesis, putting ω = ǫ/
2, the correlation function becomes B ( t ) = − i ω c ω c − iω (1 − e − ( ω c − iω ) t ) (42)and thus G ( t ) = 4 ω c ω ω c + ω t + 4 ω c ( ω c + ω ) ( ω c − ω ) e − ω c t sin( ω t ) ++ 8 ω c ω ( ω c + ω ) e − ω c t cos( ω t ) − ω c ω ( ω c + ω ) . (43)The corresponding Markovian function reads G M ( t ) = 2 πJ ( ω ) t = 4 ω c ω ω c + ω t. (44)Using Eq.(39) it is possible to analyze the evolution of the degree of entanglementof the two spins starting from the Bell state (38). The results we have obtained arereported in Figure 2 for different values of the ratio ω c /ω . Comparing the four plots,we may observe that when the spectrum of the reservoir does not completely overlapwith the frequency of the system, that is ω c ≪ ω , the concurrence decreases to zeromore slowly than in the opposite case, ω c ≫ ω . The results we have obtained, reportedin Figures 1 and 2, indicate that when the baths are characterized by Ohmic spectraldensities with a Lorentz-Drude cut-off function, as given in Eq.(41), the entanglement on-Markovian dynamics of interacting qubit . . . ω c /ω .Following the analysis developed in this section it is also interesting to examinethe behavior of the system starting from a factorized initial condition instead of anentangled one. In what follows, in particular, we suppose that at t = 0 the two spinsare in the separable state | , i and we study the time behaviour of the concurrence.We find that in this case C ( t ) = e − G ( t ) | sin(2 Kt ) | . (45)The interaction between the two spins, as expressed by the effective Hamiltonian (4),enables the generation of entanglement starting from the factorized initial conditiongiven before. On the other hand, in view of the fact that the two spins are coupled totwo different baths, the quantum correlations that are established in the pair of spins willbe destroyed. In the non-Markovian regime, however, we expect that the entanglementwill be preserved for a longer time with respect to the Markovian one. This is confirmedby the time behaviour of the concurrence function of the two spins for the Lorentzianspectral density of the baths (Figure 3) and for the Ohmic spectral density of the baths(Figure 4). Looking at these figures we also observe that the degree of entanglementthat we can realize in the system starting from the state | , i depends on the ratio γ/γ or ω c /ω . In particular, for the Lorentz spectral density, Figure 3, the maximumvalue of the concurrence function is reached in the highly non-Markovian case, that is, γ/γ = 0 .
1. For the Ohmic spectral density, Figure 4, the highly non-Markovian case( ω c /ω = 0 .
1) corresponds to the presence of the quantum correlation in the system forthe longest time.Before concluding we wish to compare our results with the ones obtained in theMarkovian [13, 16] and post-Markovian [25, 26] regimes relatively to the same physicalsystem. In order to do this, we concentrate our attention on the temporal behavior ofthe probability P ( t ) of finding the qubit pair in the state | , i supposing that at time t = 0 the system is prepared in the state | , i . In Figure 5, where we show P ( t ) inthe three different regimes, time is scaled in units of the strength K of the spin-spininteraction. As shown, when we are in the non-Markovian regime, P ( t ) reaches amaximum value that is greater than the one characterizing the Markovian and post-Markovian cases. Moreover, as expected in view of the presence of the two baths, in allthe regimes the probability P ( t ) decays toward zero after reaching its maximum value.
5. Conclusions
In this paper we have analyzed the non-Markovian dynamics of a pair of weaklyinteracting spins coupled to two separate bosonic baths. After deriving the second-order master equation, that is local in time, we have given an exact solved with theassumption that the two bosonic environments are both prepared in a thermal statewith T = 0. It is important to emphasize that our solution is valid whatever the initial on-Markovian dynamics of interacting qubit . . . Figure 3.
The concurrence C ( t ) for a Lorentz bath distribution for different valuesof the ratio γ/γ ( γ/γ = 0 . γ/γ = 1 (dashed line), γ/γ = 10 (dottedline), Markovian case (dot-and-dash line)). The initial state is | , i . Figure 4.
The concurrence C ( t ) for a Lorentz-Drude bath distribution for differentvalues of the ratio ω c /ω ( ω c /ω = 0 . ω c /ω = 1 (dashed line), ω c /ω = 10(dotted line), Markovian case (dot-and-dash line)). The initial state is | , i . on-Markovian dynamics of interacting qubit . . . Figure 5.
Dynamics of the probability to find the system in the state | , i (Markovian regime (solid line), post-Markovian regime (dashed line), non-Markovianregime (dotted line) for a Lorentz bath distribution with γ/γ = 4). The initial stateis the separable state | , i . conditions of the system or the spectral properties of the two baths may be. From thesolution of the non-Markovian Master Equation obtained we construct a solution of thecorresponding Master Equation in the Markovian limit. Starting from the knowledgeof the solution of master equation we have studied the temporal behaviour of theentanglement established in the pair of interacting spins for different spectral densities.The results show that in the non-Markovian case the concurrence, that is a measure ofentanglement, of the system of two spins “lives” longer or reaches greater values withrespect to the Markovian regime. We wish to stress that the results presented in thepresent paper are not directly connected to the so-called “entanglement sudden death”[27] because the concurrence does not vanish for a certain finite instant of time andhas “infinite” tails (39), (45). Our results motivate further studies on stronger couplingconstants and non-zero temperatures. Acknowledgements
This work is based upon research supported by the South African Research ChairInitiative of the Department of Science and Technology and National ResearchFoundation. AM (AN) acknowledges partial support by MIUR project II04C0E3F3(II04C1AF4E)
Collaborazioni Interuniversitarie ed Internazionali tipologia C . on-Markovian dynamics of interacting qubit . . . Appendix
The full solution for the density matrix of the pair of spins for arbitrary initial conditionsreads: ρ S ( t ) = e − G ( t ) ρ S (0) , (A.1) ρ S ( t ) = e − G ( t ) (1 − e − G ( t ) ) ρ S (0) + e − G ( t ) cos ( Kt ) ρ S (0) (A.2)+ e − G ( t ) sin ( Kt ) ρ S (0) − e − G ( t ) sin(2 Kt )Im( ρ S (0)) ,ρ S ( t ) = e − G ( t ) (1 − e − G ( t ) ) ρ S (0) + e − G ( t ) sin ( Kt ) ρ S (0) (A.3)+ e − G ( t ) cos ( Kt ) ρ S (0) + e − G ( t ) sin(2 Kt )Im( ρ S (0)) ,ρ S ( t ) = 1 − ρ S ( t ) − ρ S ( t ) − ρ S ( t ) , (A.4) ρ S ( t ) = e − G ( t ) cos ( Kt ) ρ S (0) + e − G ( t ) sin ( Kt ) ρ S (0) (A.5)+ i e − G ( t ) sin(2 Kt )( ρ S (0) − ρ S (0)) ,ρ S ( t ) = e − t ) ρ S (0) , (A.6) ρ S ( t ) = e − G ( t ) − Φ( t ) cos( Kt ) ρ S (0) + ie − G ( t ) − Φ( t ) sin( Kt ) ρ S (0) , (A.7) ρ S ( t ) = e − G ( t ) − Φ( t ) cos( Kt ) ρ S (0) + ie − G ( t ) − Φ( t ) sin( Kt ) ρ S (0) , (A.8) ρ S ( t ) = e − Φ( t ) cos( Kt ) ρ S (0) − ie − Φ( t ) sin( Kt ) ρ S (0) (A.9)+ Z t dτ β ( τ ) e − G ( τ ) (cid:0) cos K ( t − τ ) ρ S ( τ ) − i sin K ( t − τ ) ρ S ( τ ) (cid:1) ,ρ S ( t ) = e − Φ( t ) cos( Kt ) ρ S (0) − ie − Φ( t ) sin( Kt ) ρ S (0) (A.10)+ Z t dτ β ( τ ) e − G ( τ ) (cid:0) cos K ( t − τ ) ρ S ( τ ) − i sin K ( t − τ ) ρ S ( τ ) (cid:1) . on-Markovian dynamics of interacting qubit . . . ρ S (0) = p | ih | + p | ih | + p | ih | + (1 − p − p − p ) | ih | (A.11)+ C | ih | + ¯ C | ih | + C | ih | + ¯ C | ih | . The function G ( t ) can be re-written in the following way G ( t ) = Φ( t ) + ¯Φ( t ) = 4 X n | g n | sin ǫ − ω n ) t ( ǫ − ω n ) ≥ . (A.12)After straightforward transformations we get ρ S ( t ) = (cid:0) − ρ S (0) − ρ S (0) − ρ S (0) (cid:1) + (cid:0) − e − G ( t ) (cid:1) ρ S (0) (A.13)+ (cid:0) − e − G ( t ) (cid:1) (cid:0) ρ S (0) + ρ S (0) (cid:1) , taking into account the above expression for ρ S ( t ) and the fact that G ( t ) ≥ ρ S ( t ) and ρ S ( t ) are nonnegative. To prove the positivity of the solutionwe need to show that ρ S ( t ) and ρ S ( t ) are nonnegative too. To this end we show thatcos ( Kt ) ρ S (0) + sin ( Kt ) ρ S (0) − sin(2 Kt )Im( ρ S (0)) ≥ . (A.14)Using the positivity condition for the initial density matrix ρ S (0) which implies that p p ≥ | C | or ρ S (0) ρ S (0) ≥ | ρ S (0) | we can strengthen the above inequality byreplacing sin(2 Kt )Im( ρ S (0)) by ± sin(2 Kt ) p ρ S (0) ρ S (0) and getcos ( Kt ) ρ S (0) + sin ( Kt ) ρ S (0) ± sin(2 Kt ) q ρ S (0) ρ S (0) (A.15)= (cid:18) cos( Kt ) q ρ S (0) ± sin( Kt ) q ρ S (0) (cid:19) ≥ . Thus, from the above inequality it follows that ρ S ( t ) ≥
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