Non-Markovian Quantum Dynamics from Environmental Relaxation

Transcription

1 Non-Markovian Quantum Dynamics from Environmental Relaxation S. Possanner 1, and B.. Stickler 2, 1 Institute of Theoretical and Computational Physics, Graz University of Technology, -81 Graz, ustria 2 Institute of Physics, Karl-Franzens Universität Graz, -81 Graz, ustria Dated: March 29, 212 We consider the dynamics of composite quantum systems in the particular case that the state operator relaxes towards the Born approximation. For this we augment the von Neumann equation by a relaxation operator imposing a finite relaxation time τ r. Under the premise that the relaxation is the dominant process we obtain a hierarchy of non-markovian master equations. The latter arises from an expansion of the total state operator in powers of the relaxation time τ r. In the Born- Markov limit τ r the Lindblad master equation is recovered. Higher order contributions enable a systematic treatment of correlations and non-markovian dynamics in a recursive manner. PCS numbers: 3.65.Yz, 3.67.Pp I. INTRODUCTION The notion of quantum dissipation and decoherence arising from system-environment coupling is becoming increasingly important in many branches of physics such as quantum computation [1], quantum optics [2], or semiconductor spintronics [3]. The progress in atomic and molecular interferometry made over the last decade [4 6] enables the testing of these important concepts of the theory of open quantum systems [7 9]. The latter is the most prominent tool for tackling such fundamental problems as the collapse of the wave function during measurement [1, 11] or the transition between the microand the macroscopic world in general [12]. The peculiar nature of quantum states coherent, delocalized, correlated, entangled makes the treatment of non-equilibrium processes considerably more complicated than in the classical case. The usual approach is to start from a closed quantum system consisting of interacting degrees of freedom and B. The state operator ρ of the composite system B undergoes unitary Hamiltonian time evolution, t ρ = i[h, ρ], 1 where H denotes the system s Hamiltonian, the square brackets [, ] stand for the commutator and we set the reduced Planck constant = 1. In the composite state space H = H H B, the most general form of the Hamiltonian H reads H = H 1 B + 1 H B + H I, 2 where the operator subscript B indicates an operator acting in the subspace H B H, 1 B denotes the respective identity and the operator H I accounts for the interactions between and B. Taking the partial possanner@tugraz.at benjamin.stickler@uni-graz.at trace,, over the subsystem B in the von Neumann equation 1 yields the exact equation of motion for the relevant degrees of freedom, i.e. t ρ = i[h, ρ ] i[h I, ρ], 3 where we introduced the reduced state operator ρ via ρ := ρ. 4 In general, the reduced equation of motion 3 is an integro-differential equation, featuring memory effects in B that cause the second term on the right-hand-side to be non-local in time. It describes the subsystem as an open quantum system that exchanges energy with the environment B. In the special case of Markovian time evolution, memory effects become negligible and equation 3 takes on the form t ρ = Lρ. 5 Here, the operator L is the infinitesimal generator of a dynamical semigroup [13 15]. In its most general form L is given by [16] Lρ = i[h eff, ρ ] K 1 + Γ k L k ρ L k 1 2 L k L kρ 1 2 ρ L k L k. k= 6 where H eff is an effective Hamiltonian, Γ k are transition rates channels and L k is an operator basis in the K-dimensional space [17] of hermitian operators in H. Equation 5 is commonly referred to as a master equation of Lindblad form, or Lindblad master equation. The second term on the right hand side of the generator 6 may account for quantum decoherence as well as dissipation in due to interactions with its environment B. Master equations of the Lindblad form 5 are frequently encountered in various fields of quantum physics, in particular in the context of quantum Brownian motion or quantum optics [18 26].

2 2 The Lindblad master equation is usually obtained by performing Markovian approximations to the exact dynamics 3. This usually means that a typical parameter α of the composite system such as the correlation time, mass ratio or timescale ratio, tends towards zero or infinity [7 9]. The dynamics 5 are, therefore, only exact in the respective limiting case, which might not necessarily be a good approximation of the physical system considered. It is, thus, desirable to study the corrections to the Markovian case 5 which arise when the limiting parameter mentioned above is small, but not zero or large, but still finite. One expects to obtain non-markovian corrections which account for correlations between system and environment B. The enhanced model will be more difficult to treat, but it should still be much less involved than a full treatment of the composite system B. Over the last decade, considerable effort has been put into the derivation of non-markovian corrections to the Lindblad master equation 5 [27 34]. Two wellestablished approaches proved to be particularly fruitful, i.e. the projection operator technique and the timeconvolutionless projection operator method. The projection operator technique results in the Nakajima-Zwanzig equation [35 37] which is an exact equation for open quantum systems and its solution is comparably difficult to the solution of Eq. 1. series expansion of the Nakajima-Zwanzig integral kernel yields non-markovian evolution equations which are non-local in time [7]. This drawback is remedied by elimination of the non-locality in time with the help of a back propagator as developed by Shibata et al. [38, 39]. The resulting equation is referred to as the time-convolutionless master equation and it provides the means for the derivation of time-local, non-markovian contributions to Eq. 5 in ascending orders of the coupling strength between degrees of freedom and B. It might be interesting to note that the projection techniques described above have been motivated by a technical point of view. The aim is to eliminate from the von Neumann equation the irrelevant degrees of freedom B without employing any further assumptions on the dynamics of the physical system. Subsystem B is usually described by an arbitrary reference state χ B, which is why a physical interpretation of the results obtained appears to be difficult. Nevertheless, these methods are exact, but difficult to treat in the general case. In this work we present an alternative approach towards non-markovian contributions to Eq. 5. This approach is based on a particular physical picture and is closely related to the diffusion limit of the linear Boltzmann equation in classical kinetic theory [4 42]. In our approach, the non-markovicity arises from the relaxation of parts of the environment B towards an equilibrium state χ B on a finite timescale τ r. By explicitly accounting for this relaxation process by means of a relaxation operator Q in Eq. 1, we use a Hilbert expansion technique to derive a hierarchy of master equations for subsystem. In the limit τ r, we retrieve the Lindblad master equation 5. It has to be emphasized that by introducing the operator Q we depart from the exact description of the system s dynamics. However, this approach as well as the resulting equations of motion follow a clear physical picture and, therefore, allow for an easy interpretation. The paper is organized as follows. In section II we specify the physical picture of our approach. Moreover, we introduce the relaxation operator Q and a scaled version of the resulting equation of motion for the state operator ρ of the composite system B. In section III we employ a Hilbert expansion of ρ and derive a hierarchy of master equations for the reduced state operator ρ. Section IV contains a discussion of the results obtained. The paper is summarized in section V and a short outlook for possible future work is presented. mathematical analysis of the relaxation operator Q as well as the proof of existence and uniqueness of solutions of the equation of motion for ρ can be found in the appendices and B, respectively. In pp. C we explicitly compute the second order contribution to the hierarchy of master equations obtained. II. PHYSICL MODEL ND SCLING common approximation to the state operator ρ of a composite system B in which subsystem obeys Markovian dynamics is ρt = ρ t χ B, 7 where ρ t is the solution of Eq. 5 and χ B is some reference state in the environment B. This approximation is known as the Born approximation. It clearly depends on the physical system whether or not the state 7 represents a good approximation to the exact solution of Eq. 1. For systems where this is not the case, it might be desirable to have corrections to the Born approximation that can be expanded in orders of a typical parameter α which is zero in the Markovian limit. In order to achieve this, let us regard the Born approximation as a sort of equilibrium state of the composite system B and let τ r denote the corresponding relaxation time. We shall explicitly account for the relaxation of ρ towards the Born approximation by rewriting the equation of motion 1 as t ρ = i[h, ρ] + 1 τ r Qρ. 8 Here we introduced the relaxation operator Q as Qρ := ρ χ B ρ, 9 where χ B = 1 and we remark that Qρ = ρ. 1 In what follows the limit τ r in Eq. 8 will be denoted as the Born-Markov limit. Hence, taking in Eq.

3 3 FIG. 1. Color online Schematic representation of an observed System that interacts with an Environment that consists of two parts B and C, respectively. Subsystem B interacts with on a time scale τ I. Moreover, on a timescale τ r, B may exchange energy information with subsystem C, which is assumed to be completely isolated from the observed system. 8 the partial traces over degrees of freedom and B, respectively, yields t ρ = i[h, ρ ] i[h I, ρ], 11 t ρ B = i[h B, ρ B ] tr i[h I, ρ] + χ B ρ B τ r, 12 where tr ρ = ρ B is the reduced state operator of the environment B. lthough Eqs. 3 and 11 might seem to be identical on a first glance, the total state operator ρ will be different in these two equations, because of the introduction of the relaxation operator Q in Eq. 8. It depends on the particular situation whether the Hamiltonian H B contains interactions within the environment B or whether these interactions have been absorbed into the relaxation term Eq. 12. We point out that Eq. 8 does not conserve the total energy of the composite system B, which is, consequently, not a closed quantum system. Eq. 8 rather resembles a configuration in which subsystem B is coupled to a third subsystem C, which can be regarded as isolated from. This situation is sketched in Figure 1. Hence, the environment of is a composite system BC. In this case Eq. 8 results from tracing out the degrees of freedom C from the total equation of motion for the composite system BC. The remaining effect of subsystem C is that it relaxes the state operator ρ B to a particular equilibrium state χ B on a timescale τ r. In the case that C is a reservoir, i.e. features an infinite number of degrees of freedom, χ B could be the minimizer of a certain entropy functional in B. For instance, system could contain the conduction band electrons in a semiconductor, whereas system B describes the lattice phonons coupled to an external heat bath C. On the other hand, one could imagine that a probe C prepares the state χ B with a mean frequency 1/τ r. Such a scenario could be realized by two interacting spins, where one of the two spins is constantly monitored and prepared to be in state χ B. nother possible scenario could be a composite quantum system, where subsystem interacts solely with a part of the total environment due to short range iteractions. If the state χ B is a pure state the corresponding state of the composite system B must be uncorrelated [43], i.e. of the form 7. In writing Eq. 8, we presuppose that even for a mixed state χ B, the coupling of C to B leads to decorrelations in B. Thus, correlations between and B due to the interaction H I are gradually destroyed on a timescale τ r by the coupling of B to C. The aim of the following sections is to find approximate solutions to Eq. 8 in cases where the time scale τ r is small compared to all other relevant timescales of the system. For technical reasons which will become clear in the next section, let us introduce the mean-field operator H mf acting in H, We define furthermore, H mf := H I χ B. 13 H I := H I H mf 1 B, H := H + H mf, H B := H 1 B + 1 H B, 14 and rewrite Eq. 8 with the help of the definitions 14: t ρ = i[ H B, ρ] i[ H I, ρ] + 1 τ r Qρ. 15 s a next step we present a scaled version of Eq. 15 which is appropriately suited for the Born-Markov limit. For this suppose one can define a timescale τ B induced by H B as well as a timescale τ I induced by H I. The former timescale is a characteristic for the evolution of the isolated, mean-field-corrected subsystems and B, respectively, whereas the latter is a characteristic for the mean-field-corrected interaction between and B. The introduction of a typical parameter α 1 via τ I τ B = Oα, τ r τ B = Oα 2, 16 and of the timescale τ B to describe the dynamics, yields t = t τ B, 17 t ρ = i[ H B, ρ] i α [ H I, ρ] + 1 Qρ. 18 α2 Equation 18 corresponds to the equation of motion 15 for the composite system B in the Born-Markov scaling. We remark that since α 1, Eq. 18 implies strong interactions between system and environment B while the relaxation towards the Born approximation is the dominant process. III. DERIVTION OF MSTER EQUTIONS. Hilbert expansion of the state operator It is the aim of this section to search for an approximate solution ρ α t of Eq. 18 with initial condition

4 4 ρ α = ρ α i. For small values of α this approximate solution is supposed to be close to the exact solution ρ. In what follows we write t instead of t for the scaled time 17. Thus, we consider the following initial value problem, t ρ α = i[ H B, ρ α ] i α [ H I, ρ α ] + 1 α 2 Qρα, ρ α = ρ α i. 19 The first question of interest is whether or not the initial value problem 19 has a unique solution. We stress that this will not trivially be the case because Q is a nonlocal operator which contains the trace over the degrees of freedom B. The proof of existence and uniqueness of a solution ρ α t on a finite time interval [, T ] to the initial value problem 19 is given in pp. B. Let us proceed with the approximate solution of Eq. 19. We shall employ a series expansion of the solution in powers of α, thus assuming ρ α to be analytic in α within a certain radius around α =. By inserting the Hilbert expansion into Eq. 19, ρ α = α n ρ n, 2 n= subsequently multiplying by α 2 and sorting the terms in orders of α, one obtains the following system of equations Qρ =, Qρ 1 = i[ H I, ρ ], Qρ 2 = t ρ + i[ H B, ρ ] + i[ H I, ρ 1 ], Qρ 3 = t ρ 1 + i[ H B, ρ 1 ] + i[ H I, ρ 2 ], Qρ n = t ρ n 2 + i[ H B, ρ n 2 ] + i[ H I, ρ n 1 ], for n 4. 21a 21b 21c 21d 21e We remark that even though Eqs. 21c and 21d are of the general form 21e, they have been written explicitly for the purpose of a better understanding of the concepts elaborated in this section. Regarding Eqs. 21, the question immediately arises whether or not the system is well-posed, i.e. whether or not the right-hand-sides of Eqs. 21 lie in the image of the operator Q, such that a solution ρ α of the form 2 can be obtained, at least in principle. It is therefore necessary to investigate the operator Q, defined in Eq. 9, in more detail. We note in passing that Q is very similar to one of the projection operators used in the projection operator techniques mentioned in the introduction [35, 36, 38, 39]. However, strictly speaking it is not a projection operator since Q 2 = Q. For the subsequent analysis, let us introduce the following notations: H : space of hermitian operators in H. H,B : space of hermitian operators in H,B. Moreover, let DQ H stand for the domain of Q, thus the operator Q is a mapping Q : DQ H. 22 We assume that DQ is a linear space a detailed analysis of the operator Q can be found in pp.. Here, we briefly repeat the main results of pp. needed in what follows: i Let Ker Q denote the kernel of Q. One has DQ = Ker Q Ker Q, 23 where Ker Q denotes the space orthogonal to the kernel of Q. Hence any X DQ can be decomposed into X = X Ker + X, 24 where X Ker Ker Q and X Ker Q. ii For X Ker Ker Q one has X Ker = X χ B, X H. 25 iii For X Ker Q one has X =. 26 iv Let Im Q denote the image of Q. One has Im Q = Ker Q. 27 v The equation QX = Y is well-posed and thus has a solution if Y Ker Q. Moreover, it has a unique solution in Ker Q denoted X. It follows immediately from Eq. 26 that this solution is given by X = Y. 28 We begin now with the investigation of well-posedness of Eqs. 21. We use Eqs. 24 and 25 to decompose each term ρ n of the Hilbert expansion, ρ n = ρ n χ B + ρ n. 29 Moreover, we note the important property i[ H I, ρ n χ B] = ρ n H, 3 which is a consequence of the introduction of the meanfield operator, c.f. Eq. 14. Let us take the trace over

5 5 the degrees of freedom B in Eqs. 21 and let us, furthermore, use the property 3 to obtain = i[ H I, ρ ], 31a = t ρ + i[ H, ρ ] + i[ H I, ρ 1 ], 31b = t ρ 1 + i[ H, ρ 1 ] + i[ H I, ρ 2 ], 31c = t ρ n 2 + i[ H, ρ n 2 ] + i[ H I, ρ n 1], for n 4, 31d where we omitted the result = obtained from Eq. 21a. From property v of the relaxation operator Q it is clear that the system 21 is well-posed if and only if Eqs. 31 are fulfilled. In what follows we shall present an inductive proof that this can be indeed achieved. Furthermore, we shall prove that a system consisting of the first N N equations 31 is closed and that its solution can be computed recursively from Eq. 31b. We know a priori that Eq. 21a is well-posed and that its solution is obtained as ρ = ρ χ B, ρ =. 32 Supposed Eqs. 21b to 21e are also well-posed, we can employ Eq. 28 to determine their unique solutions ρ n Ker Q as ρ 1 = i[ H I, ρ ], 33a ρ 2 = t ρ i[ H B, ρ ] i[ H I, ρ 1 ], 33b ρ 3 = t ρ 1 i[ H B, ρ 1 ] i[ H I, ρ 2 ], 33c ρ n = t ρ n 2 i[ H B, ρ n 2 ] i[ H I, ρ n 1 ], for n 4. 33d From Eq. 32 we deduce that Eq. 31a is fulfilled trivially and, thus, Eq. 21b is well-posed. This enables us to insert the result 33a into Eq. 31b to obtain t ρ = i[ H, ρ ] [ H I, [ H I, ρ χ B]]. 34 Equation 34 is a master equation of Lindblad form, as will be elaborated later in more detail in subsection III B. The second term on the right-hand-side of Eq. 34 is the dissipative part; thus, let us define the dissipator D : DD H H, DX := [ H I, [ H I, X χ B ]]. 35 Since X χ B = X Ker Ker Q, the operator D can also be viewed as a mapping from Ker Q to H. Using the short notation 35, equation 34 reads t ρ = i[ H, ρ ] + Dρ. 36 For now we suppose the Lindblad master equation 36 to have a unique solution. This assumption is sufficient for completing the inductive proof of well-posedness of Eqs. 21, as will become transparent in the remainder of this subsection. first consequence of well-posedness of the Lindblad equation 36 is that Eq. 21c is also well-posed and, thus, that its unique solution ρ 2 Ker Q given in Eq. 33b is valid. Inserting this into Eq. 31c results in t ρ 1 = i[ H, ρ 1 ] 37 + i[ H I, t ρ + i[ H B, ρ ] + i[ H I, ρ 1 ]]. This equation can be simplified by use of Eq. 32, the property 3, and the result 33a which yields with t ρ 1 = i[ H, ρ 1 ] + Dρ1 + S 1, 38 S 1 = i[ H I, ρ [H B, χ B ] + i[ H I, [ H I, [ H I, ρ χ B]]]. 39 We remark that the first two terms on the right-handside of Eq. 38 form exactly the Lindblad generator from equation 36. The additional term S 1 does not depend on ρ 1 and, thus, can be viewed as a well-defined, local source term. Hence, Eq. 38 has a unique solution. We deduce that Eq. 21d is well-posed and its unique solution ρ 3 Ker Q given in Eq. 33c is valid. One can already see the evolving pattern that will result in the well-posedness of the entire system 21. In order to complete the proof we shall proceed by induction. Therefore, suppose that Eqs. 21e are well-posed up to order n 1. The solution to the n 1-th order equation is then written as ρ n 1 = ρ n 1 χ B + ρ n 1. 4 Due to Eq. 33d, ρ n 1 is given by ρ n 1 = t ρ n 3 i[ H B, ρ n 3 ] i[ H I, ρ n 2 ]. 41 The aim is now to specify under which condition the n-th order Eq. 21e is also well-posed. From Eq. 31d one deduces that this condition reads t ρ n 2 = i[ H, ρ n 2 ] i[ H I, ρ n 1]. 42 Inserting Eq. 41 into Eq. 42 yields t ρ n 2 = i[ H, ρ n 2 ] + i[ H I, t ρ n 3 ] [ H I, [ H B, ρ n 3 ]] 43 [ H I, [ H I, ρ n 2 ]].

6 6 gain we employ the decompositions ρ n 2 = ρ n 2 χ B + ρ n 2, 44 ρ n 3 = ρ n 3 χ B + ρ n 3, 45 and profit from the fact that ρ n 2 Ker Q is uniquely defined by Eq. 28, with the result ρ n 2 = t ρ n 4 i[ H B, ρ n 4 ] i[ H I, ρ n 3 ]. 46 We note that we were able to obtain Eqs. 41, 44 and 45 because we supposed Eq. 21e to be well-posed up to order n 1. Moreover, t ρ n 3 = t ρ n 3 χ B + t ρ n 3, 47 and thus property 3 yields i[ H I, t ρ n 3 ] = i[ H I, t ρ n 3]. 48 The decompositions 44 and 45 are applied to Eq. 43 and one obtains, also using Eq. 48, t ρ n 2 = i[ H, ρ n 2 ] + i[ H I, t ρ n 3] [ H I, [ H B, ρ n 3 χ B + ρ n 3]] [ H I, [ H I, ρ n 2]] 49 [ H I, [ H I, ρ n 2 χ B ]]. In the last term on the right-hand-side one can introduce the definition 35 of the dissipator D in order to obtain, finally t ρ n 2 = i[ H, ρ n 2 ] + Dρ n 2 + S n 2, 5 where S n 2, n 4, is given by S n 2 = i[ H I, t ρ n 3] [ H I, ρ n 3 [H B, χ B ]] [ H I, [ H B, ρ n 3]] + [ H I, [ H I, t ρ n 4 ]] + i[ H I, [ H I, [ H B, ρ n 4 ]]] + i[ H I, [ H I, [ H I, ρ n 3 ]]]. 51 Here ρ n 2 was expressed with the help of Eq. 46. Equation 5 is called the n 2-th order master equation for n 4. It arises solely from the requirement that Eq. 21e of order n is well-posed. Besides the Lindblad generator, which has already been found in the zeroth and first order master equations 36 and 38, respectively, Eq. 5 comprises the additional source term S n 2. This term depends solely on operators ρ k obtained from Eqs. 21e of order k < n. Therefore, under the premise that the Lindblad master equation 36 yields sufficiently well-behaved solutions, Eqs. 5 are solvable up to arbitrary order n, which proves the well-posedness of Eqs. 21. It follows, moreover, from the particular form of the source term S n 2 that the first N N equations 5 form a closed system of equations, in which solutions can be computed recursively. For n = 4, the source term 51 is evaluated in pp. C. B. Lindblad master equation We shall briefly elaborate on the Lindblad master equation 36. This equation will also be called zeroth order master equation. Recalling that H I = H I H mf 1 B, a straightforward calculation results in the following form of the dissipator 35, Dρ = tr B 2HI ρ H I HI 2 ρ ρ HI 2 2H mf ρ Hmf + H mf 2 ρ + ρ H mf We note that the interaction Hamiltonian H I can be written in the form [7] H I = i i B i, 53 where i H and B i H B. Therefore, the mean-field operator reads H mf = i i B i χ B. 54 By inserting relations 53 and 54 into Eq. 52, one obtains Dρ = ij Γ ij 2 j ρ i i j ρ ρ i j, where the coefficients Γ ij are defined as 55 Γ ij = B i B j χb B i χb B j χb. 56 Here we made use of the standard definition of correlation functions and expectation values, B i B j χb = B i B j χ B, 57 B i χb = B i χ B. 58 In fact, the coefficients 56 stand for the covariance of B i and B j in the state χ B. In summary, the zeroth order master equation for the reduced system can be written as t ρ = i [ + ij H, ρ ] i i [ ] B i χb i, ρ 59 Γ ij 2 j ρ i i j ρ ρ i j.

7 7 This equation can be transformed into the Lindblad form 5 by expanding the operators i, j in an appropriate basis L k H. The second term on the right-handside of Eq. 59 represents an energy shift induced by the mean-field approximation of the interaction between system and environment B. We emphasize that this energy shift has to occur in the zeroth order equation of the reduced system, because otherwise the Hilbert expansion 2 would result in an ill-posed equation 21b. For the same reason the coefficients 56 stand for the covariance of B i and B j, rather than their correlation. IV. DISCUSSION Let us briefly summarize what has been accomplished so far. The goal of the present work was to find approximate solutions ρ α to Eq. 19 in the case that the parameter α is small but not zero. For this, we invoked a Hilbert expansion of the form ρ α = n= α n ρ n χ B + ρ n, 6 where ρ n χ B Ker Q and ρ n Ker Q. We note that in this representation, the reduced state operator reads ρ α := ρ α = n= α n ρ n. 61 fter inserting the ansatz 6 into Eq. 19 we required equality of the left- and the right-hand-side of the equation in each power α n. The further requirement of wellposedness of the resulting equations 21 gave rise to the following hierarchy of master equations for the ρ n : t ρ t ρ 1 t ρ n 2 = Lρ, 62a = Lρ1 + S 1, 62b = Lρ n 2 + S n 2 for n 4. 62c Here, L is the generator of a dynamical semigroup of the Lindblad form 6, specified in Eq. 59. Moreover, D is the dissipator defined in Eq. 35 and the source terms S 1 to S n 2 are given by Eqs. 39 and 51, respectively. Under the initial conditions ρ = ρ,i H, 63a ρ 1 = ρ,i H, 63b ρ n 2 = ρ n 2,i H for n 4, 63c the formal solution of Eqs. Duhamel s formula ρ t = elt ρ,i, ρ 1 t = elt ρ 1,i + ρ n 2 t = e Lt ρ n 2,i + for n 4, 62 can be obtained via ds e Lt s S 1 s, ds e Lt s S n 2 s 64a 64b 64c where e Lt with t denotes the dynamical semigroup generated by L. Under the assumption that the power series 61 converges for all t which is reasonable for small α, one can perform the sum in the results 64 in order to obtain ρ α t = elt ρ α,i + ds e Lt s S α s. 65 Here we defined the initial values ρ α,i and the operator S α as ρ α,i := S α := n= α n ρ n,i, 66 α n S n, 67 n=1 where we made use of S =. The integral on the right-hand-side of Eq. 65 makes the non-markovicity of the time evolution of the reduced state 61 transparent, since S α depends on ρ α in a rather complicated way. In the Born-Markov limit α, the term S α vanishes and one recovers the Markovian dynamics for the reduced system induced by the Lindblad generator L. In writing the formal solution 65, we note that the total state operator ρ α of the composite system has been determined entirely. This follows from the fact that the terms ρ n in Eq. 6 are uniquely defined by Eqs. 33. Therefore, if the power series 6 is convergent for all t which is reasonable for small α, it represents the unique solution of the initial value problem 19. Let us focus briefly on the orthogonal terms ρ n. With the definition ρ := α n ρ n, 68 n=1 where we used that ρ =, we point out that the contribution 68 to the solution 6 is traceless, tr ρ =. 69 Therefore, it solely describes correlations between system and environment B. Moreover, it is obvious that the power series 68 vanishes in the Born-Markov limit α

8 8, thereby confirming the absence of correlations in the Markovian time evolution. t first glance it might seem that nothing has been gained because the evaluation of Eqs. 61 and 68 requires the calculation of an infinite number of terms. However, provided that these power series converge, their benefits can be found in the fact that one can successively approach the exact solution ρ α of Eq. 19 until a desired accuracy has been reached. For instance, truncating the series 6 after two terms appears to be a valid approximation in the case that α 1. In general, once the Lindblad master equation 59 has been solved for ρ, the source terms S n and thus the higher order corrections ρ n and ρ n can be computed recursively in order to achieve the desired accuracy. In this way, correlations between system and environment B can be incorporated rather easily in the reduced dynamics for. V. CONCLUSION In this work we employed a Hilbert expansion in order to obtain approximate solutions of a von Neumann equation which was augmented by a relaxation operator Q. This operator relaxes the state operator of a composite quantum system B towards the Born approximation on a timescale τ r. This approach resulted in the hierarchy of master equations 62 for the reduced state operator ρ. In zeroth order, which accounts for the exact dynamics in the Born-Markov limit τ r, a master equation of Lindblad form was recovered. The transition rates derived are exactly the covariance functions between different components of the environmental part of the interaction Hamiltonian H I in the reference state χ B. Moreover, the discussed approach allows to systematically incorporate correlations and non-markovian effects in the reduced system dynamics. These effects can be calculated recursively from the solution to the Lindblad master equation, as was achieved in Eqs. 64. Such an approach might be advantageous in physical systems for which the Born approximation is nearly justified or for which a full treatment on the basis of projection techniques is far to complex. We point out that the non-markovian quantum dynamics derived here follow from a transparent physical picture, namely the relaxation of the total state operator towards the Born approximation. This appears to be a reasonable scenario, for instance, if the environment contains degrees of freedom that are completely isolated from the observed system, c.f. Fig. 1. Nevertheless, the results obtained are merely valid under three assumptions: the coupling of different degrees of freedom B and C of the environment diminish correlations between observed system and environment, the Born relaxation is by far the fastest process in the composite system, strong system-environment interaction singular coupling scaling. The derived model could be enhanced rather easily by replacing the relaxation operator Q, introduced in Eq. 9, by a more sophisticated dissipative term for subsystem B; for instance by a Lindblad dissipator. However, this would result in minor changes only, because the hierarchy 31 is a general result which does not depend on the particular form of the relaxation operator Q. This hierarchy is a mere consequence of the complete isolation of the observed system from the environmental reservoir or probe C, which manifests itself in the relation Qρ =. ppendix : nalysis of the operator Q Defining for X, Y H the scalar product X, Y := tr χ 1/2 Xχ 1/2 Y, 1 where χ H satisfies χ = χ χ B, tr χ = 1, χv, v H > v H, 2 the space H becomes a Hilbert space with the associated norm X = X, X X H. 3 In 2,, H denotes the scalar product in H. Note that X, Y = Y, X, X, X, 4 X, Y H. The positive-definiteness of 1 follows from X, X = tr χ 1/4 Xχ 1/4 χ 1/4 Xχ 1/4 = tr Y 2 5, where Y = χ 1/4 Xχ 1/4 H. Moreover, 1 is linear in both arguments. We shall further define a scalar product in the subspace H H, X, Y := tr χ 1/2 X χ 1/2 Y, 6 and denote the corresponding norm by X = X, X, X H. For the operator Q defined in Eq. 9, 7 QX = X χ B X, X DQ H, 8 one has the following properties:

9 9 i Q is linear, bounded, self-adjoint and non-positive. ii For X Ker Q the kernel of Q we have X = X χ B, X H. 9 Moreover, we have Ker Q = {X DQ X = }. 1 iii Let P : H Ker Q denote the orthogonal projection operator on Ker Q. Then there exists d > such that X DQ. QX, X d X PX 2 11 iv The image of Q, denoted by Im Q, is closed and we have Im Q = Ker Q. 12 Further, the equation QX = Y has a solution in DQ if and only if Y Im Q. The solution is moreover unique in Ker Q. In the following we denote, and X := X H. 13 Q X = QX =, X DQ. 14 Moreover, in what follows we shall frequently make use of the identity X χ B, Y = X, Y. 15 We shall now prove the above statements. i The linearity of Q is obvious. Let us show that Q is a well-defined, bounded operator. For this one needs a constant c > such that QX 2 c X 2, X DQ. 16 Using the identities 14 and 15 one obtains QX 2 = QX, QX = X χ B, QX X, QX = X, Q X X, QX = X, X χ B + X, X = X 2 + X 2 X 2, 17 which proves 16. Therefore Q is bounded and thus also continous. The self-adjointness follows from QX, Y = X χ B, Y X, Y = X, Y X, Y = Y, X Y, X = QY, X = X, QY. 18 In order to prove the non-positivity of Q we estimate the term X 2 using Eq. 15 together with the Cauchy- Schwarz inequality, X 2 = X, X χ B X X χ = X X X 2. It follows the non-positivity of Q, 19 QX, X = X 2 X 2 X 2 X 2 =. 2 ii For X Ker Q we have QX, X = which can be written as = X 2 X 2 = X, X χ B X, X. 21 The solutions of Eq. 21 are given by X = X χ B, X H arbitrary. Conversely, QX χ B = X χ B X χ B =. 22 Moreover, for Y Ker Q, we have X, Y = for X Ker Q and thus X χ B, Y = X, Y = X H. 23 Since Eq. 23 must hold for arbitrary X H we conclude Y = Y = Y Ker Q. 24 iii We now prove the coercitivity relation 11. For X Ker Q this relation is fulfilled trivially because QX = and PX = X. Now suppose X Ker Q. Then, according to Eq. 24, QX, X = X 2 + X 2 = X 2, 25 which completes the coercitivity proof. iv First we show that the image of Q is closed. Let X n be a sequence in DQ and let J n be a sequence in Im Q such that QX n = J n. Moreover, let J n J as n. We have to prove that J Im Q, i.e. that there exists X DQ such that QX = J. To any sequence X n DQ one can construct a corresponding sequence Y n Ker Q by setting Y n = X n PX n and one has QX n = QY n = J n. The coercitivity relation then yields QY n Y m, Y n Y m d Y n Y m 2 n, m N. 26 In addition, from the Cauchy-Schwarz inequality we have QY n QY m Y n Y m QY n Y m, Y n Y m, 27 and consequently 1 d QY n QY m = 1 d J n J m Y n Y m. 28

10 1 Since J n is a Cauchy sequence in Im Q we obtain that Y n is a Cauchy sequence in DQ. By assumption DQ is complete and therefore Y n Y DQ. We already proved that Q is continuous, i.e. QY n QY. One obtains QY = J with Y DQ. Thus, the image of Q is closed and we have Im Q = Ker Q. We finally prove that the equation QX = Y has a unique solution X Ker Q. It is obvious that there exists a solution if Y Im Q. Let X be such a solution, then X PX Ker Q is also a solution. ssume that there are two solutions X 1, X 2 Ker Q such that QX 1 = QX 2 = Y. Then QX 1 QX 2 = QX 1 X 2 =. 29 It follows that X 1 X 2 Ker Q Ker Q = {} and therefore X 1 = X 2. In the following, we denote S σ t = σt χ B. The formal solution of B6 reads Fσt = ρt = e T t ρ + ds e T t s S σ s, B7 where T := L 1. The proof is performed in two steps. First we demonstrate that the mapping B5-B7 is welldefined and we prove then that the mapping is a contraction, possessing thus a unique fixed point, solution of equation B2. From Eq. B1 it follows that e T t ρ 2 H = e 2t e Lt ρ 2 H C ρ 2 H t [, T ]. B8 Therefore, Eq. B7 yields ppendix B: Existence and uniqueness In this section we demonstrate the existence and uniqueness of a solution to the initial value problem 19 on the basis of a fixed point argument. Let < T < and α = 1 for convenience the proof holds for α >. The following proposition holds: Let Q denote the operator defined in Eq. 9. Furthermore, let H H such that the Liouville operator L = i[h, ] in H is the infinitesimal generator of a bounded one-parameter semigroup e Lt in H, i.e. one has C > such that e Lt ρ 2 H C ρ 2 H ρ H, t [, T ], B1 where H denotes the norm 3. Then, the intial value problem { t ρ Lρ = Qρ B2 ρ = ρ i H admits a unique solution ρ L 2 [, T ], H. In order to prove this claim we denote the norm and the corresponding scalar product in L 2 [, T ], H by and ρ L 2 = ρ, σ L 2 = dt ρt H, dt ρt, σt H, B3 B4 respectively. Let us now define the following fixed point map, F : L 2 [, T ], H L 2 [, T ], H, σ ρ, B5 where ρ is the solution of { t ρ Lρ + ρ = σ χ B ρ = ρ i H. B6 and ρt H ρt 2 H C ρ H + C C ρ H + C ds S σ s H, ds S σ s H, B9 2 T 2C 2 ρ 2 H + 2C 2 ds S σ s H, B1 where an estimate of the second term can be found using the Cauchy-Schwarz inequality, 2 T ds S σ s H = T T S σ 2 L 2. Inserting Eq. B11 into Eq. B1 yields ds S σ s H 1 T 2 B11 ρt 2 H 2C 2 ρ 2 H + 2C 2 T S σ 2 L 2. B12 Hence, ρt 2 H is bounded in [, T ] and, thus, belongs to the space L 2 [, T ], H which proves that the fixed point map B5 is well-defined. In order to show that B7 is a contraction we introduce the following norm in L 2 [, T ], H, ρ 2 δ := dt e δt ρt 2 H. For F to be contractive it is required that B13 Fσ 1 Fσ 2 2 δ k σ 1 σ 2 2 δ, k < 1. B14

11 11 One obtains Fσ 1 Fσ 2 2 δ = = C = C C δ dt e δt ds e T t s [S σ1 s S σ2 s] 2 H dt e δt ds S σ1 s S σ2 s 2 H ds s dt e δt S σ1 s S σ2 s 2 H ds e δs S σ1 s S σ2 s 2 H. Due to the inequality 19 one has S σ1 s S σ2 s 2 H and this results in B15 = σ 1 s σ 2 s χ B 2 H = σ 1 s σ 2 s 2 σ 1 s σ 2 s 2 H, Fσ 1 Fσ 2 2 δ C δ σ 1 σ 2 2 δ. B16 B17 Here, δ can be chosen in such a way that relation B14 is fulfilled. Thus F is a contraction, its unique fixed point is the solution of Eq. B2. ppendix C: Second-order contribution We compute the source term 51 for n = 4. For this, we need the solutions of Eqs. 21a and 21b, namely ρ and ρ 1, given by 32 and 33a, respectively. It follows that and t ρ = t ρ χ B C1 = i[ H, ρ ] χ B + Dρ χ B, t ρ 1 = i[ H I, t ρ ] C2 = [ H I, [ H, ρ ] χ B] i[ H I, Dρ χ B]. We now compute all the terms appearing on the righthand side of Eq. 51. Whenever possible, we make use of definition 35 of the operator D in order to simplify the notation. 1. With the help of Eq. C2 the first term on the right-hand-side of Eq. 51 i[ H I, t ρ 1 ] = [ H I, [ H I, i[ H, ρ ] χ B]] + [ H I, [ H I, Dρ χ B]] = Di[ H, ρ ] D2 ρ, C3 where D 2 = DD. 2. The second term results in [ H I, [ H B, ρ 1 ]] = i[ H I, [ H B, [ H I, ρ ]]]. C4 3. In the third term we set ρ n 3 ρ Equation C1 helps in evaluating term four to [ H I, [ H I, t ρ ]] = = [ H I, [ H I, i[ H, ρ ] χ B]] + [ H I, [ H I, Dρ χ B]] = Di[ H, ρ ] D2 ρ. 5. We get for term five i[ H I, [ H I, [ H B, ρ ]]] = = [ H I, [ H I, i[ H, ρ ] χ B]] + i[ H I, [ H I, ρ [H B, χ B ]]] = Di[ H, ρ ] + i[ H I, [ H I, ρ [H B, χ B ]]]. 6. Finally, term six becomes i[ H I, [ H I, [ H I, ρ 1 ]]] = = [ H I, [ H I, [ H I, [ H I, ρ ]]]] + [ H I, [ H I, [ H I, ρ 1 χ B]]]. C5 C6 C7 dding the results of Eqs. C3-C7 gives the desired result for S 2 : S 2 = 2D 2 ρ + Di[ H, ρ ] + i[ H I, [ H B, [ H I, ρ [ H I, ρ 1 [H B, χ B ] χ B]]] + i[ H I, [ H I, ρ [H B, χ B ]]] + [ H I, [ H I, [ H I, [ H I, ρ ]]]] + [ H I, [ H I, [ H I, ρ 1 χ B]]] CKNOWLEDGMENTS. C8 The authors are very grateful to E. Schachinger for carefully reading the manuscript. We also thank W. Pötz and C. Negulescu for precious input and fruitful discussions. The first author acknowledges the support from the ustrian Science Fund, Vienna, under the contract number P21326-N16; the second author was supported by the ustrian Science Fund FWF: P22129-N16.

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