Perturbation Theory for Lindblad Superoperators for Interacting Open Quantum Systems

PHYSICAL REVIEW A 102, 032207 (2020)

Perturbation theory for Lindblad superoperators for interacting open quantum systems

V. Yu. Shishkov,1,2,3 E. S. Andrianov,1,2,3 A. A. Pukhov,1,2,3 A. P. Vinogradov ,1,2,3 and A. A. Lisyansky 4,5 1Dukhov Research Institute of Automatics, 22 Sushchevskaya, Moscow 127055, Moscow Region, Russia 2Moscow Institute of Physics and Technology, 9 Institutskiy Pereulok, Dolgoprudny 141700, Moscow Region, Russia 3Institute for Theoretical and Applied Electromagnetics, 13 Izhorskaya, Moscow 125412, Moscow Region, Russia 4Department of Physics, Queens College of the City University of New York, Flushing, New York 11367, USA 5The Graduate Center, City University of New York, New York, New York 10016, USA

(Received 4 December 2019; revised 2 May 2020; accepted 10 August 2020; published 8 September 2020)

We study the dynamics of interacting open quantum subsystems. Interactions merge these subsystems into a unified system. It is often believed that the Lindblad superoperator of the unified system is the sum of the Lindblad superoperators of the separate subsystems. Such an approach, however, results in a violation of the second law of thermodynamics. To avoid a cumbersome direct derivation of the correct superoperator from first principles, we develop a perturbation theory based on using Lindblad superoperators of separate subsystems, which are assumed to be known. Using interacting two-level systems as an example, we show that, starting with a certain order of the perturbation theory, the second law of thermodynamics holds. We demonstrate that the theory developed can be applied to the problems of quantum transport.

DOI: 10.1103/PhysRevA.102.032207

I. INTRODUCTION unified system grows exponentially with the number of subsystems. Therefore, for a unified system, the direct approach Recently, the study of open quantum systems has experi- is impractical. enced rapid growth connected to studies of interacting arrays It is desirable to express the Lindblad superoperator of of qubits [1], ensembles of quantum dots and molecules [2], the unified system through the Lindblad superoperators of and systems of atoms coupled with resonators [3–5]. Such subsystems. The key issue is that the eigenstates of the uni- systems are open in the sense that they interact with the fied system are no longer a direct product of the eigenstates environment. of isolated subsystems. Consequently, the transition opera- The exact dynamics of an open quantum system can tors between eigenstates of the unified system that appear only be obtained for a small number of problems [6–10]. in the Lindblad superoperator are no longer equal to direct Therefore, various approximate methods are used [11–13]. products of the transition operators of subsystems, and the One of the most common is the method of the Lindblad- Lindblad superoperator of the unified system is not equal Gorini-Kossakowski-Sudarshan (LGKS) equation [14,15]. to the sum of the Lindblad superoperators for isolated sub- This method implies that both the system and the reservoir systems [17–20]. However, in many cases it is believed that have separated Hamiltonians and that the coupling between even if the whole Hamiltonian includes the interactions be- them can be expressed as the interaction Hamiltonian. The tween subsystems, all the relaxation processes in the unified derivation of the LGKS equation involves the use of the system are the same as for isolated subsystems [2,21–32]. von Neumann equation for the system-reservoir density ma- Such an approximation is called the local approach. Math- trix and the subsequent elimination of the reservoir degrees ematically, it implies that the Lindblad superoperator of the of freedom. The states of the latter are assumed to be un- unified system is the sum of Lindblad superoperators of the changeable during the evolution of the system. After such isolated subsystems. One might suppose that the local ap- elimination, the evolution operator of the system’s density proach will only lead to minor quantitative differences from matrix includes the Hamiltonian and Lindblad superoperators the exact solution. Unfortunately, this is not the case. In [16]. These Lindblad superoperators are constructed from particular, the local approach may result in the violation of the product of the transition operators between the system the second law of thermodynamics [20,33]. Recently, consid- eigenstates. The transitions occur under the influence of the erable attention has been devoted to a detailed comparison reservoirs. The transition rates are calculated according to the between the dynamics predicted by the local approach and Fermi golden rule [17]. the exact Lindblad superoperators for a number of systems Although the general theory of deducing the LGKS equa- [34–41], including interacting two-level systems [42], har- tion is well developed [16], a direct way of constructing monic oscillators [43], and optomechanical systems [44]. In Lindblad superoperators requires knowledge of the exact [45] it was shown that the second law of thermodynamics system eigenstates and eigenenergies [18,19]. For a unified can be restored if the environment is divided into an ensem- system containing interacting subsystems, the complexity of ble of identically prepared auxiliary systems, which interact the calculation of the eigenstates and eigenenergies of the

2469-9926/2020/102(3)/032207(17) 032207-1 ©2020 American Physical Society V. YU. SHISHKOV et al. PHYSICAL REVIEW A 102, 032207 (2020) sequentially with an individual subsystem for an infinitesi- frequency of the atom, ωk is the frequency of a photon with mally short time. However, for the permanent interaction of the wave vector k and the polarization η, and kη is the the system and reservoirs the fulfillment of the second law is coupling constant between the atom and the photon with not guaranteed. the wave number k and polarization η. The operatorσ ˆ is the In this paper we develop a perturbative method for building transition operator from the excited state to the ground state the Lindblad superoperator of a system comprised of inter- of the two-level atom; the operatora ˆkη is the annihilation acting subsystems. The perturbation theory developed for the operator of the photon with the wave vector k and polarization η ˆ σ † + σ λ Lindblad superoperator is based on the construction of the . In this example the operator Sα is ( ˆ ˆ ), α is k0η0 , and † transition operators that appear in this Lindblad superoperator. ˆα η + η / η η R is k (âkη aˆk ) k0 0 , where k0 and 0 are the wave We demonstrate that the transition operators can be found vector and the polarization of the free-space mode for which = from the interaction of the Hamiltonian of the reservoir and k0η0 0, respectively. the system by employing the Heisenberg equation of motion. The von Neumann equation for the density matrixρ ˆ (t )of At each step of the perturbation theory the system of the the unified system and the reservoir follows from the Hamil- Heisenberg equations for the transition operator is linear and tonian (1), closed, can be solved exactly. The developed method ensures the fulfillment of the principles of thermodynamics. This ap- ∂ρˆ (t ) i = [ˆρ(t ), Hˆ + Hˆ + Hˆ ], (3) proach permits us to express the Lindblad operator of the ∂t h¯ S SR R unified system through the linear combination of products of the transition operators of the separate subsystems. Using in- where we introduced the Hamiltonian of the unified system teracting two-level systems as an example, we find the lowest ˆ = ˆ (is) + ε ˆ HS HS W . The exclusion of the reservoir degrees of order of the perturbation series that ensures the fulfillment of freedom according to the standard procedure [16,20,46,47] the second law of thermodynamics. demands the transition of the equation into the interaction representation, with the new density matrix defined as II. PERTURBATION THEORY APPROACH FOR OBTAINING THE LINDBLAD SUPEROPERATORS Hˆ + Hˆ Hˆ + Hˆ ρ˜ˆ (t ) = exp i S R t ρˆ (t )exp −i S R t . (4) We consider a system described by the Hamiltonian h¯ h¯ Hˆ = Hˆ (is) + εWˆ + Hˆ + Hˆ , S SR R (1) The von Neumann equation for the new density matrix ρ˜ˆ (t ) ˆ (is) = ˆ (is) can be obtained from Eq. (3), where HS α HSα is the Hamiltonian of the system comprised of isolated subsystems, which are enumerated by the α ˆ (is) ∂ρˆ subscript and described by the Hamiltonians HSα ; each ˜ (t ) i = ρˆ t , H˜ˆ α t , αth subsystem interacts with its own reservoir described by ∂ ˜ ( ) SR ( ) (5) t h¯ α the operator HˆRα, so the Hamiltonian for the reservoir of the whole system is Hˆ = Hˆ . The Hamiltonians Hˆ (is) and R α Rα Sα ˆ ˆ where the operator H˜ α (t ) is the Hamiltonian Hˆ α, given by HRα depend only on the subsystem and reservoir degrees of SR SR freedom, respectively. The interaction of the αth subsystem Eq. (2), in the interaction representation ˆ and its reservoir is described by the operator HSRα;soforthe whole system Hamiltonian for the interaction with reservoirs ˆ + ˆ ˆ + ˆ ˆ = HS HR − HS HR ˆ = ˆ H˜ SRα (t ) exp i t HˆSRα exp i t is HSR α HSRα. The interaction between the subsystems is h¯ h¯ governed by the Hamiltonian εWˆ = ε α α Wˆα α , where ε is 1 2 1 2 = λ ˆ / ˆ − ˆ / ˆ / ˆ a small dimensionless parameter. For example, in the case of h¯ α exp(iHSt h¯)Sα exp( iHSt h¯)exp(iHRt h¯)Rα ε interacting two-level systems, is the dimensionless coupling × exp(−iHˆRt/h¯). (6) constant between them. For simplicity, we assume that ε is the same for all subsystems. We assume that the Hamiltonian The standard procedure implies the averaging over the ˆ (is) α HSα of the th isolated subsystem has known nondegenerate reservoir degrees of freedom in Eq. (5)[18,19]. This aver- | (is) ω(is) eigenstates kαi and eigenfrequencies αi enumerated by aging procedure is similar to the well-known rotating-wave the subscript i. approximation in quantum optics [48,49]. The averaging over Usually, it is assumed that the Hamiltonian of the interac- the reservoir degrees of freedom in Eq. (5) leads to the master tion between a system and a reservoir has the form [16,17] equation for the system density matrix of the unified system ρ˜ˆS (t ) = TrR[ρ˜ˆ (t )] [18,19]:

Hˆ α = h¯λαSˆαRˆα, (2) SR ρˆ d ˜ S (t ) = ρ . where λα is the interaction constant and Sˆα and Rˆα are the Lêx[˜ˆS (t )] (7) dt operators of the subsystem and its reservoir, respectively. α = For example, for a single two-level atom, 1, interacting The influence of the reservoir on the unified system in with the electromagnetic field of free space, the Hamiltoni- Eq. (7) reduces to the Lindblad superoperator Lˆ [ρ˜ˆ (t )]. To (is) † † ex S ans are Hˆ α = h¯ω0σˆ σˆ , HˆRα = ,η h¯ωkaˆ aˆkη, and Hˆ α = S k kη SR derive this Lindblad superoperator Lêx[ρ˜ˆS (t )], according to † + σ † + σ ω k,η h¯ kη(âkη aˆkη )( ˆ ˆ ), where 0 is the transition the standard algorithm [18,19], we should find the explicit

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expression for the operators exp(iHˆSt/h¯)Sˆα exp(−iHˆSt/h¯)in time, the near-field interaction between dipole moments of the form two TLSs, which is the strongest electromagnetic interaction between subsystems at the nanoscale, is ∼ (ˆσ †σˆ + σˆ †σˆ ), ˆ / ˆ − ˆ / 1 2 2 1 exp(iHSt h¯)Sα exp( iHSt h¯) i.e., can be presented in the form (13). In Appendix A we discuss the extension of the theory developed for the general = k |Sˆα|k |k k | exp(i ω t ), (8) i j i j ij case. ij To develop the perturbation approach for determination where ωij = ω j − ωi and ω j are eigenfrequencies corre- of the Lindblad superoperators we represent the Hamiltonian sponding to the eigenstates |k of the Hamiltonian Hˆ . Once ˜ˆ j S HSRα (t )[Eq.(6)] as the series (8) is found, one can apply the standard procedure ˜ˆ = λ ˆ ˆ . and obtain the Lindblad superoperator HSRα (t ) h¯ α Sαij(t )Rα (t ) (14) ij ˆ ρˆ = λ2 ω ˆ , ρˆ ˆ† Lex[˜S (t )] αGα ( αij) S ω ˜ S(t )S ω αij αij ˆ = α ij In Eq. (14) we introduce the operators Rα (t ) (is) exp(iHˆ t/h¯)Rˆα exp(−iHˆ t/h¯) and Sˆ α (t ) = exp(iHˆ t/h¯)Sˆ † R R ij S αij + Sˆ ω ρ˜ˆ (t ), Sˆ , (9) (is) α S ωα − ˆ / ˆ ij ij exp( iHSt h¯), with Sαij defined by Eq. (12), and | (is) we use the fact that the eigenvectors kαi of the where we introduced the operators ˆ (is) Hamiltonian HS of isolated subsystems form the complete basis. The perturbation theory for determination of the Sˆ ω =k |Sˆα|k |k k | (10) αij i j i j Lindblad superoperators developed here is based on searching for the approximate expansion for the operators ω ˆ = ˆ / ˆ(is) − ˆ / and the function Gα ( αij) is defined as Sαij(t ) exp(iHSt h¯)Sαij exp( iHSt h¯) in the form +∞ ˆ = εn ˆ (n) . ω = ˆ† ˆ / ˆ − ˆ / Sαij(t ) Sαij(t ) (15) Gα ( αij) TrR[Rα exp(iHRt h¯)Rα exp( iHRt h¯)] 0 n

× exp(i ωα t )dt. (11) To construct the Lindblad equation it is neces- ij (n) sary to represent the operator Sˆ α (t ) in the form Hereafter we call the Lindblad superoperator (9) the exact ij Sˆ (n) exp(−i ω(n)t ), where the operators Sˆ (n) Lindblad superoperator. Generally, finding the exact Lindblad ω(n) ω(n) ω(n) ω(n) superoperator (9) analytically or even numerically is a very do not depend on time and are some yet unknown difficult problem because the exact Lindblad superoperator frequencies. Once such a representation is found, by applying the standard procedure we can arrive at the series expansion Lêx[ρ˜ˆS (t )] includes the product of the transition operators Sˆ ω Sˆ ω of the Lindblad superoperator αij. Finding the transition superoperators αij [Eq. (10)] is a very cumbersome problem because it demands the so- +∞ ˆ ρ = εn (n) ρ . lution of the eigenproblem for the unified Hamiltonian HS. Lˆ[˜ˆS (t )] Lˆ [˜ˆ S (t )] (16) At the same time, the complexity of this eigenproblem grows n=0 exponentially with the number of subsystems. To find the coefficients Sˆ (n) and frequencies ω(n),we As an alternative, we develop a perturbation approach for ω(n) ˆ the analytical determination of the Lindblad superoperators. solve the Heisenberg equation for Sαij(t ): For future derivations, we introduce the operators dSˆ α (t ) i ij = ˆ , ˆ . ˆ(is) = (is) ˆ (is) (is) (is). [HS Sαij(t )] (17) Sαij kαi Sα kα j kαi kα j (12) dt h¯ We assume that there is only pair interaction between sub- The initial condition for Eq. (17) follows from the defini- ˆ systems (which is true for the vast majority of problems in tion of Sαij(t ): quantum optics [49]) such that the interaction Hamiltonian Wˆ (is) Sˆ αij(0) = Sˆα . (18) between subsystems has the form ij We solve Eq. (17) using the perturbation method with the (is) (is) Wˆ = h¯ wα α Sˆ Sˆ . (13) ε ˆ 1 2i1 j1i2 j2 α1i1 j1 α2i2 j2 small parameter . To do this we expand the operator Sαij(t ) α =α 1 2 i1 j1i2 j2 in a power series of ε [Eq. (15)]. We substitute this series into Eq. (17) and collect all the terms with the same order of ε.As For the simpler and clearer presentation we assume that ε the interaction between the subsystems (13) is defined by a result, the terms proportional to the zeroth order of give the same operators that introduce each system-environment the equation for the zeroth order of the perturbation theory interaction in Eq. (2). This assumption is true for the most (see Appendix A), optical systems. For example, the interactions of each two- dSˆ (0) (t ) αij =− ω(is) ˆ (0) , level system (TLS) with dissipative and dephasing reservoirs i αijSαij(t ) (19) † † † † dt are ∼ (ˆσ , bˆk + bˆ σˆ1,2 ) and ∼ σˆ , σˆ1,2(bˆk + bˆ ), re- k 1 2 k k 1 2 k (is) (is) (is) † ω = ω − ω σ σ where αij α j αi . We obtain the initial condition spectively ( 1,2 and 1,2 are lowering and raising operators ˆ ˆ† for Eq. (19) from Eq. (18): between TLS states and bk and bk are annihilation and creation ˆ (0) = ˆ(is). operators of kth reservoir mode, respectively). At the same Sαij(0) Sαij (20)

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The solution to Eq. (19) with the initial condition (20) is the zeroth-order Lindblad superoperator. Using the Lindblad superoperator (22), we can write the LGKS equation for the ˆ (0) = ˆ(is) − ω(is) . ρˆ Sαij(t ) Sαij exp i αijt (21) density matrix of the system ˜S (t ), dρ˜ˆ (t ) The direct application of the standard procedure for S = Lˆ (0)[ρ˜ˆ (t )]. (23) ˆ ≈ ˆ (0) dt S Sαij(t ) Sαij(t ) leads to the Lindblad superoperator It can be seen from Eqs. (22) and (9) that the Lindblad (0) ˆ (0) ρˆ = λ2 ω(is) ˆ(0) , ρˆ ˆ(0)† superoperator Lˆ [ρ˜ˆS (t )] coincides with the one obtained with L [˜S (t )] αGα αij Sαij ˜ S (t )Sαij α ij the standard procedure for the isolated subsystems in the absence of the interaction between the subsystems (ε = 0). + ˆ(0) ρˆ , ˆ(0)† , Sαij ˜ S (t ) Sαij (22) The ﬁrst order of the perturbation theory for Eq. (17) ˆ = may be obtained by the substitution of the series Sαij(t ) ˆ(0) = ˆ(is) λ2 ω(is) εn ˆ (n) where Sαij Sαij and αGα ( αij) is deﬁned by (11). We n Sαij(t ) into Eq. (17) and selecting the terms that are call the Lindblad superoperator (22) the Lindblad superoper- proportional to ε. As a result, we obtain the equation for ˆ (1) ator of the zeroth order of the perturbation theory, or in short Sαij(t ) (see Appendix A), ⎡ ⎤ dSˆ (1) (t ) αij (is) (1) ⎣ (0) (0) (0) ⎦ =−i ωα Sˆ α (t ) + i wα α i j i j Sˆ α (t )Sˆ α (t ), Sˆ α (t ) . (24) dt ij ij 1 2 1 1 2 2 1i1 j1 2i2 j2 ij α1=α2 i1 j1i2 j2

The initial condition for Eq. (24) follows from the initial condition (18) for the operator Sˆ αij(t ) and takes the form ˆ (1) = . Sαij(0) 0 (25) The solution of Eq. (24) with the initial condition (25) is (see Appendix A) (0) (0) (0) wα α i j i j Sˆα Sˆα , Sˆα Sˆ (1) (t ) = 1 2 1 1 2 2 1i1 j1 2i2 j2 ij 1 − exp − i ω(is) + ω(is) t exp − i ω(is)t . (26) αij (is) (is) α1i1 j1 α2i2 j2 αij ωα + ωα α1=α2 i1 j1i2 j2 1i1 j1 2i2 j2 Combining the solutions (21) and (26), we obtain the solution to Eq. (17) up to the ﬁrst order of the perturbation theory (0) (1) Sˆ α (t ) ≈ Sˆ α (t ) + εSˆ α (t ), ij ij ij (0) (0) (0) wα α i j i j Sˆα Sˆα , Sˆα ˆ ≈ ˆ(0) + ε ˆ(1) − ω(is) − ε 1 2 1 1 2 2 1i1 j1 2i2 j2 ij Sαij(t ) Sαij Sαij exp i αijt (is) (is) ωα + ωα α1=α2 i1 j1i2 j2 1i1 j1 2i2 j2 × exp − i ω(is) + ω(is) + ω(is) t , (27) α1i1 j1 α2i2 j2 αij

where superoperator. Using the Lindblad superoperator (29), we can (0) (0) (0) write the LGKS equation for the density matrix of the system wα α i j i j Sˆα Sˆα , Sˆα Sˆ(1) = 1 2 1 1 2 2 1i1 j1 2i2 j2 ij . (28) as follows: αij ω(is) + ω(is) α =α α α 1 2 i1 j1i2 j2 1i1 j1 2i2 j2 ρˆ d ˜ S (t ) (0) (1) = Lˆ [ρ˜ˆS (t )] + εLˆ [ρ˜ˆS (t )]. (30) The direct application of the standard procedure for the dt ˆ ≈ ˆ (0) + εˆ (1) operator Sαij(t ) Sαij(t ) Sαij(t ) leads to the Lindblad It is important to note that the Lindblad superoperator (0) (1) (0) (0) ρ + superoperator Lˆ [ρ˜ˆS(t )] + εLˆ [ρ˜ˆS(t )], with Lˆ [ρ˜ˆS (t )] de- of the first order of the perturbation theory Lˆ [˜ˆS (t )] (1) ε (1) ρ fined by Eq. (22) and Lˆ [ρ˜ˆS (t )] equal to Lˆ [˜ˆS (t )] is nontrivial in the sense that, unlike the Lindblad superoperator (22), it cannot be represented as a sum of the ˆ (1) ρˆ = λ2 ω(is) ˆ(1) , ρˆ ˆ(0)† L [˜ S (t )] αGα αij Sαij ˜ S (t )Sαij Lindblad superoperators of isolated subsystems. Nevertheless, α (0) (1) ij the Lindblad superoperator Lˆ [ρ˜ˆS (t )] + εLˆ [ρ˜ˆS (t )] is con- + ˆ(1) ρˆ , ˆ(0)† + λ2 ω(is) structed from the product of the transition operators of the Sαij ˜ S (t ) Sαij αGα αij (0) subsystems Sˆα that can be seen from (28) and (29). Together, α ij ij these two facts indicate that the first-order Lindblad superop- × ˆ(0) , ρˆ ˆ(1)† + ˆ(0) ρˆ , ˆ(1)† . Sαij ˜ S (t )Sαij Sαij ˜ S (t ) Sαij erators of the perturbation theory takes into account the cross (29) relaxation of the subsystems. For practical applications it is more convenient to write the (0) We call the Lindblad superoperator Lˆ [ρ˜ˆS (t )] + LGKS equation for the density matrixρ ˆS (t ) rather than for the (1) εLˆ [ρ˜ˆS (t )] the Lindblad superoperator of the first order density matrix in the interaction representation ρ˜ˆS (t ). These of the perturbation theory, or in short the first-order Lindblad two density matrices for the unified system are connected by

032207-4 PERTURBATION THEORY FOR LINDBLAD … PHYSICAL REVIEW A 102, 032207 (2020) the equation that follows from Eq. (4): The relaxation dynamics, described by the LGKS equation (23), may violate the second law of thermodynamics and ˆ ˆ HS HS incorrectly predicts the dynamics of the system [20,33]. ρ˜ˆS (t ) = exp i t ρˆS (t )exp −i t . (31) h¯ h¯ In the next section we show that although the Lindblad superoperator of the lower orders of the perturbation theory This transition from the density matrix ρ˜ˆ (t ) to the density S may violate the second law of thermodynamics, the Lindblad matrixρ ˆ (t ) in the LGKS equation (7) can be easily done S superoperator of the higher order of the perturbation theory because restores the fulfillment of the second law of thermodynamics. HˆS HˆS exp −i t Lêx[ρ˜ˆS (t )] exp i t = Lêx[ˆρS (t )], (32) h¯ h¯ III. FULFILLMENT OF THE SECOND LAW OF THERMODYNAMICS which follows from the definitions of the exact Lindblad superoperator Lêx[ρ˜ˆS (t )] [see Eq. (9)] and the transition op- Below, for simplicity, we consider all the reservoirs to Sˆ ω have zero temperature Tα = 0, which is a good approxima- erators αij [see Eq. (10)]. As a result, the LGKS equation with the exact Lindblad superoperator for the density matrix tion for quantum optics. In this limit, the second law of the ρˆS (t ) takes the form thermodynamics is equivalent to the demand that the energy flow between the system and reservoir is directed from the dρˆS (t ) i = [ˆρS (t ), HˆS] + Lêx[ˆρS (t )]. (33) system to the reservoir. The energy flow in our case (when dt h¯ ˆ the Hamiltonian of the unified system HS does not depend on The same transition can be done in the LGKS equations time) is (23) and (30) with the zeroth-order Lindblad superoper- dρˆ (t ) ˆ (0) ρˆ = ˆ S . ator L [˜S (t )] and the first-order Lindblad superoperator J Tr HS (39) (0) (1) dt Lˆ [ρ˜ˆS (t )] + εLˆ [ρ˜ˆS (t )]. To do this transition we use the approximate equalities The positive sign of the energy flow J indicates that the energy flows from the reservoir to the system. The negative Hˆ Hˆ − S ˆ (0) ρ S ≈ ˆ (0) ρ , sign of the energy flow J indicates that the energy flows exp i t L [˜ˆ S(t )] exp i t L [ˆS(t )] (34) h¯ h¯ from the system to the reservoir. Therefore, the second law ˆ ˆ HS (0) (1) HS of thermodynamics at zero temperature takes the form exp −i t {Lˆ [ρ˜ˆS (t )] + εLˆ [ρ˜ˆS (t )]} exp i t h¯ h¯ J 0. (40) ≈ ˆ (0) ρ + ε ˆ (1) ρ . L [ˆS (t )] L [ˆS (t )] (35) It is worth emphasizing that if there is no interaction between subsystems then, as was pointed out after Eq. (37), the The difference between the left-hand side and the right- LGKS equation (33) with the exact Lindblad superoperator hand side of the approximate equality (34) is proportional to takes the form ε, which follows directly from Eq. (22) and initial condition dρˆS (t ) i = ρˆ (t ), Hˆ (is) + Lˆ (0)[ˆρ (t )]. (41) (20) (see Appendix B). The difference between the left-hand dt h¯ S S S side and the right-hand side of the approximate equality (35)is This means that in the case of ε = 0 the Hamiltonian of proportional to ε2, which follows directly from Eqs. (29) and (is) the unified system is HˆS = Hˆ and the exact Lindblad su- (27) (see Appendix B). Therefore, we can omit these terms S peroperator Lêx[ˆρS (t )] is equal to the zeroth-order Lindblad on the right-hand sides of Eqs. (34) and (35). As a result, we ˆ (0) ρ ρ superoperator L [ˆS (t )]. In this case, at zero temperature obtain the LGKS equation for the density matrix ˆS (t ) with it is easy to prove that the second law of thermodynamics the zeroth-order Lindblad superoperator holds. Indeed, the substitution of Eq. (41) and the equality (is) dρˆ (t ) i HˆS = Hˆ into Eq. (39) and the use of the commutation S = ρ , ˆ + ˆ (0) ρ S [ˆS (t ) HS] L [ˆS (t )] (36) ˆ (is), ˆ(0) =− ω(is) ˆ(0) dt h¯ relation [HαS Sαij] h¯ αijSαij leads to and the LGKS equation for the density matrixρ ˆ (t ) with the (is) 2 (is) S J =−2¯h ω λ Gα ω first-order Lindblad superoperator 0 αij α αij α ωαij0 ρ d ˆS (t ) i (0) (1) (0)† (0) = [ˆρ (t ), Hˆ ] + Lˆ [ˆρ (t )] + εLˆ [ˆρ (t )]. (37) × Tr Sˆα Sˆα ρˆS (t ) 0. (42) dt h¯ S S S S ij ij In Eq. (42) the subscript 0 in J means that the energy We note that the LGKS equation (36) is equal to one used 0 flow (39) is calculated when there is no interaction between in the local approach [17–20]. Indeed, the Hamiltonian of (is) (0) (is) subsystems (ε = 0). The definitions of ω and Sˆ are the unified system, Hˆ = Hˆ + εWˆ , contains the interaction αij αij S S given after Eq. (22). The sum in Eq. (42) includes only non- between the subsystems εWˆ . At the same time the Lindblad (is) negative frequencies ω because at zero temperature and superoperator Lˆ (0)[ˆρ (t )] coincides with the one obtained with αij S ω(is) ω(is) the exact standard procedure, as it would be in the absence of negative αij, Gα ( αij) is equal to zero (this means that the interaction between the subsystems, only transitions with the energy flowing out of the system are possible) [17]. Thus, in this case the second law of thermo- (0) Lˆ [ˆρS (t )] = Lˆsubsystem[ˆρS (t )]. (38) dynamics holds (for a rigorous proof of the second law for subsystem arbitrary temperature see [18,19,46]).

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When the subsystems interact with each other, it turns out ˆ deph = σ †σ ˆ Hamiltonian HSR ˆ ˆ Rdeph, is dephasing. A reservoir that in the lower order of perturbation theory the second law of ˆ dis = which interacts with a TLS through the Hamiltonian HSR thermodynamics may be violated, whereas the higher order of deph (ˆσ † + σˆ )Rˆ is dissipative. Indeed, the Hamiltonian Hˆ = the perturbation theory restores its fulfillment. Indeed, let us dis SR σ †σ ˆ ˆ (is) = consider the zeroth order of perturbation theory. To obtain the ˆ ˆ Rdeph commutes with the TLS Hamiltonian HS ω σ †σ ˆ dis = σ † + σ ˆ energy flow between the unified system and the reservoirs, one h¯ TLS ˆ ˆ , while the Hamiltonian HSR (ˆ ˆ )Rdis does should substitute Eq. (36) into Eq. (39). As a result we obtain not. (local approach) When the interaction of an isolated subsystem with a dissipative reservoir permits transitions between system states and (0) (is) (0) (is) J = Tr Hˆ + εWˆ Lˆ [ˆρS (t )] = J0 + ε J0, (43) ω = < S at least some αij 0, then according to Eq. (8), J0 0, (0) and the second law of thermodynamics holds until ε| J0| where the (0) in J means that the energy flow (39)is | | calculated with the zeroth-order Lindblad superoperator. The J0 . This puts a limit on the perturbation parameter. In this energy flow J on the right-hand side of Eq. (43) is defined case, even using local Lindblad superoperators does not cause 0 a violation of the second law of thermodynamics. by Eq. (42). The energy flow J0 on the right-hand side of Eq. (43)is The situation changes when a subsystem interacts only with a dephasing reservoir [20,33]. In this case, all the fre- = { ˆ ˆ (0) ρ }. ω(is) = J0 Tr W L [ˆS (t )] (44) quencies αij 0 and consequently J0 is equal to zero [see Eq. (42)]. Then the second law of thermodynamics in As mentioned in the preceding section, the zeroth-order zeroth order (local approach) holds only if ε = 0orε J < Lindblad superoperator Lˆ (0)[ˆρ (t )] does not take into account 0 S 0. In a general case, ε J may be positive and the second the cross-relaxation processes between the subsystems. This 0 law of thermodynamics would be violated. The addition of means that the application of the zeroth-order Lindblad super- a dissipative reservoir may not help because now we re- operator Lˆ (0)[ˆρ (t )] leads to the additional “wrong” term ε J S 0 ε| deph + dis| | dis| in the energy flow (43) between the system and reservoirs. We quire that J0 J0 J0 . Thus, if the subsystems interact with each other, ε = 0, the local Lindblad superop- showed after Eq. (42) that the term J0, is nonpositive, but the erator, which does not take into account the cross-relaxation wrong term ε J0 may have a positive or a negative sign. As processes, may lead to a violation of the second law of ther- a result, when ε J0 > 0 and |J0| <ε| J0| the second law of thermodynamics is violated, because the energy starts to flow modynamics [20,33] even if the system interacts with both from the reservoir to the system, and J (0) > 0. dephasing and dissipative reservoirs. For example, the local Let us discuss how to determine the lowest order of the Lindblad superoperator may lead to the pumping of the system perturbation series that ensures the fulfillment of the second by the dephasing reservoir with zero temperature [20,33]. law of thermodynamics. To be more concrete, we consider Application of the first-order Lindblad superoperator ˆ (0) ρ + ε ˆ (1) ρ two types of reservoirs: a dephasing reservoirs that does not L [ˆS (t )] L [ˆS (t )] may fix this problem. Indeed, the substitution of Eq. (37) into the energy flow (39)givesthe change the energies of the isolated subsystems but affects (1) the phases of the nondiagonal elements of the each of the energy flow J calculated with the zeroth-order Lindblad subsystem density matrices and a dissipative reservoirs that superoperator drains the energy from the isolated subsystems. For example, (1) = ˆ (is) + ε ˆ { ˆ (0) ρ + ε ˆ (1) ρ } the reservoir Rˆ interacting with a two-level system described J Tr HS W L [ˆS (t )] L [ˆS (t )] ˆ (is) = ω σ †σ by the Hamiltonian HS h¯ TLS ˆ ˆ (where the operator = J + ε2 J , (45) σˆ =|ge| is the transition operator between the ground |e 1 1 | and the excited g states of the TLS), through the coupling where we introduce the energy flows

=− ω(is)λ2 ω(is) ˆ(0) + ε ˆ(1) † ˆ(0) + ε ˆ(1) ρ , J1 2¯h αij αGα αij Tr Sαij Sαij Sαij Sαij ˆS(t ) (46) α ω αij 0 = ω(is)λ2 ω(is) ˆ(1)† ˆ(1) ρ − ω(is)λ2 ω(is) ˆ(0)† ˆ , ˆ(1) ρ J1 2¯h αij αGα αij Tr Sαij Sαij ˆS(t ) 2¯h αij αGα αij Tr Sαij W Sαij ˆS(t ) α ωαij0 α ωαij0 + ˆ(0)†, ˆ ˆ(1) ρ − ω(is)λ2 ω(is) ˆ(1)† ˆ , ˆ(0) ρ + ˆ(1)†, ˆ ˆ(0) ρ . Sαij W Sαij ˆS(t ) 2¯h αij αGα αij Tr Sαij W Sαij ˆS(t ) Sαij W Sαij ˆS(t ) (47) α ωαij0

(0) (1) (0) (1) ˆ + ε ˆ † ˆ + ε ˆ with the zeroth-order Lindblad superoperator (ε| J0| |J0|), In Eq. (12) the operator (Sαij Sαij) (Sαij Sαij) and the density matrixρ ˆS (t ) are non-negative, so the energy we conclude that the first order of the perturbation theory 2 flow J1 is negative or zero. The wrong term ε J1 on the preserves the second law of thermodynamics in the wider right-hand side of Eq. (45) may have an arbitrary sign. range of parameters than the zeroth order of the perturbation If J1 < 0[seeEq.(45)], then the second law of thermo- theory. 2 dynamics is not violated until ε | J1| |J1|. Comparing In a general case the correct description of the energy flow 2 (n) this inequality (ε | J1| |J1|) with one that we obtained J in the nth order demands knowledge of the nth order of the

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(n) Lindblad superoperator Lˆ [ˆρS (t )]. In the previous example, To develop the perturbation theory for N interacting TLSs, this means that to obtain the correct energy flow in the first we use the Hamiltonian order of perturbation theory it is necessary to use the first- N N N order Lindblad superoperator Lˆ (1)[ˆρ (t )]. We also note that S Hˆ = h¯ω σˆ †σˆ + hg¯ σˆ †σˆ , (49) the nth order of the perturbation theory leads to the wrong S j j j jk j k = = = term in the energy flow J (n) that is proportional to εn+1. j 1 j 1 k j The higher orders of the perturbation theory expand the ∗ where the coupling constants obey the relationship g = g range of parameters for which the second law of thermo- jk kj ˆ dynamics holds, because the nth order of the perturbation because the Hamiltonian HS is Hermitian. We consider a non- |ω − ω | | | ε =| |/|ω − ω | theory for the Lindblad superoperator describes the energy resonant case j k g jk . Then g jk j k flow correctly with a precision up to εn+1. As was pointed can be used as a small parameter. We discuss the degenerate |ω − ω | | | out above, when the system interacts with dissipative and case ( j k g jk ) for the problem of quantum transport dephasing reservoirs, the main reason for the violation of the in Appendix C. second law of thermodynamics is the interaction of the system The Hamiltonian of the reservoirs has the form with the dephasing reservoir. The reason is that the dephasing ˆ = ω(a) † + ω(b) ˆ† ˆ , reservoir gives zero energy flow at ε = 0 and may give the HR h¯ jn aˆ jnaˆ jn h¯ jn b jnb jn (50) energy flow from the reservoir to the system for ε = 0atlower j,n j,n orders of the perturbation theory. The energy flow to the dissi- a bˆ pative reservoir may be estimated as γ Tr[HˆSρˆS (t )], where γ is where the annihilation operators ˆ jn and jn belong to dissi- the dissipation rate of the unperturbed system. In accordance pative and dephasing reservoirs, respectively. The interaction with Eqs. (43) and (45), in the nth order of the perturbation between the system and the reservoirs is described by the theory, possible energy flow from the dephasing reservoir to Hamiltonian the system may be estimated as εn+1 Tr[Hˆ ρˆ (t )], where S S ˆ = σ † + σ † + is the dephasing rate at ε = 0. The fulfillment of the second HSR h¯v jn(ˆj ˆ j )(â jn aˆ jn) j,n law of thermodynamics demands that the energy flow from the system to the dissipative reservoir is greater than the wrong † † + h¯w σˆ σˆ (bˆ + bˆ ). (51) energy flow from the dephasing reservoir to the system. Thus, jn j j jn jn j,n the order of the perturbation theory n required for the second law of thermodynamic to hold can be estimated as If we neglect an interaction between the TLSs (g jk = 0), the Lindblad equation for the density matrix of the whole + systemρ ˆ (t ) can be easily obtained [16] εn 1 <γ/ . (48) S ∂ i ρˆ (t ) = ρˆ (t ), Hˆ (0) + Lˆ (0)[ˆρ (t )], (52) ∂ S S S S Below, by using specific examples, we show that the cri- t h¯ terion (48) not only ensures the fulfillment of the second law where of thermodynamics, but also defines the minimal necessary order of the perturbation theory that guarantees the correct N prediction of the dynamics of an open quantum system. ˆ (0) = ω σ †σ , HS h¯ j ˆ j ˆ j (53) j=1 N Lˆ (0)[ˆρ ] = γ n¯(−ω )(2σ ˆ ρˆ σˆ † − σˆ †σˆ ρˆ − ρˆ σˆ †σˆ ) IV. EXAMPLE: THE LINDBLAD SUPEROPERATOR S j j j S j j j S S j j = FOR N COUPLED TWO-LEVEL SYSTEMS j 1 N The local Lindblad superoperator, which, as was shown in + γ n¯(ω )(2σ ˆ †ρˆ σˆ − σˆ σˆ †ρˆ − ρˆ σˆ σˆ †) the Sec. II, is equal to the zeroth-order Lindblad superoperator j j j S j j j S S j j j=1 (36), is widely used for the description of interacting quantum dots [2], qubits [1], and molecules [24], because the local N + σ †σ ρσ †σ − σ †σ ρ − ρσ †σ , Lindblad superoperator is easy to obtain. As we discussed j (2 ˆ j ˆ j ˆ j ˆ j ˆ j ˆ j ˆ j ˆ j ) (54) j=1 above, the main drawback of the local Lindblad superoperator is that it may lead to a violation of the second law of ther- (exp(¯hω/T ) − 1)−1,ω>0 n¯(ω) = (55) modynamics [20,33]. In such a case, the dynamics of an open 1 + (exp(¯h|ω|/T ) − 1)−1,ω<0, quantum system may differ from the exact dynamics not only γ = π | |2δ ω(a) − ω , quantitatively but also qualitatively. In particular, the local j h¯ v jn jn j (56) Lindblad superoperator may give unphysical results. Below n we consider the dynamics of several interacting TLSs and = π | |2δ ω(a) − ω ω − −ω . j h¯ lim w jn jn [¯n( ) n¯( )] (57) ω→0 show when a description based on local Lindblad superopera- n tor fails and how the higher orders of the perturbation theory developed for the Lindblad superoperator restore the correct Note that the dephasing rates j are mainly determined by dynamics of an open quantum system. the low-frequency behavior of the reservoirs.

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As discussed in the preceding section, the master equa- The solution to Eq. (61)is tion (52) preserves the second law of thermodynamics. When = N g g jk 0, the zeroth-order term of the perturbation theory coin- σ (1) =− σ †σ − jk σ − ω ˆ j (t ) (2 ˆ j ˆ j 1) ˆk exp( i jt ) cides with the local approach for the Lindblad superoperators ω j − ωk k=1 and leads to the master equation k= j ∂ i N ρ t = ρ t , Hˆ + Lˆ (0) ρ t , g ∂ ˆS ( ) [ˆS ( ) S] [ˆS ( )] (58) + σ †σ − jk σ − ω . t h¯ (2 ˆ j ˆ j 1) ˆk exp( i kt ) (62) ω j − ωk (0) k=1 where HˆS is defined by Eq. (49) and Lˆ [ˆρS (t )] is defined by k= j Eq. (54). In this case, the interaction between the TLSs is taken Employing Eqs. (60) and (61), we can obtain the solution ˆ to Eq. (59) in the first order with respect to ε: into account in the Hamiltonian HS but not in the Lindblad ⎛ ⎞ superoperator Lˆ (0)[ˆρ (t )]. This superoperator describes the S N σ †σ − σ relaxation in the case of noninteracting TLSs [see Eq. (52)]. ⎜ g jk(2 ˆ j ˆ j 1)ˆk ⎟ σˆ j (t ) ≈ ⎝σˆ j − ⎠ exp(−iω jt ) According to the discussion in the preceding section, it is not ω j − ωk k=1 surprising that the master equation (58) may lead to a violation k= j of the second law of the thermodynamics. N σ †σ − σ g jk(2 ˆ j ˆ j 1)ˆk To obtain the Lindblad superoperators in the first order + exp(−iωkt ). (63) ω j − ωk of the perturbation theory, one should solve the Heisenberg k=1 k= j equation forσ ˆ j (t ), σ / =−ω σ + σ † σ − In the same order, from Eq. (63) one can obtain d ˆ j (t ) dt i j ˆ j (t ) i[2 ˆ j (t )ˆj (t ) 1] ⎛ ⎞ N N † † ⎜ gkjσˆ jσˆ + g jkσˆ σˆk ⎟ σ †σ ≈ σ †σ + k j × g jkσˆk (t ), σˆ j (0) = σˆ j. (59) ˆ j ˆ j (t ) ⎝ ˆ j ˆ j ⎠ ω j − ωk k=1 k=1 k= j k= j N † To solve Eq. (59), we expand the operatorσ ˆ j (t )intoa g σˆ σˆ − kj j k ω − ω seriesσ ˆ (t ) = σˆ (0)(t ) + εσˆ (1)(t ) +··· by using the pertur- exp[i( k j )t] j j j ω j − ωk k=1 bation theory with the small parameter ε =|g jk|/|ω j − ωk|. k= j σ (0) The-zeroth order approximation ˆ j (t ) satisfies the equation N † gkjσˆkσˆ (0) (0) (0) j σ / =−ω σ , σ = σ , + exp[i(ω j − ωk )t]. (64) d ˆ j (t ) dt i j ˆ j (t ) ˆ j (0) ˆ j (60) ω j − ωk k=1 σ (0) = σ − ω k= j and has the form ˆ j (t ) ˆ j exp( i jt ). σ (1) Now we use approximate the expressions (63) and (64) The first-order approximation term ˆ j (t ) is governed by the equation for deriving the Lindblad superoperator in the first order. As a result, we obtain the master equation with the Lindblad σ (1) / =−ω σ (1) + σ (0)† σ (0) − d ˆ j (t ) dt i j ˆ j (t ) i 2ˆj (t )ˆj (t ) 1 superoperators in the first order of the perturbation theory N ∂ i (0) (1) × σ (0) , σ (1) = . ρˆS (t ) =− [HˆS, ρˆS] + Lˆ [ˆρS (t )] + Lˆ [ˆρS (t )], (65) g jk ˆk (t ) ˆ j (0) 0 (61) ∂t h¯ k=1 k= j (0) where Lˆ [ˆρS (t )] is defined by Eq. (54) and

⎛⎡ ⎤ ⎡ ⎤⎞ N N g (2σ ˆ †σˆ − 1)ˆσ † N g (2σ ˆ †σˆ − 1)ˆσ † (1) ⎜⎢ jk j j k ⎥ ⎢ jk j j k ⎥⎟ Lˆ [ˆρS (t )] =− γ jn¯(−ω j )⎝⎣σˆ j, ρˆS ⎦ + ⎣σˆ jρˆS, ⎦⎠ ω j − ωk ω j − ωk j=1 k=1 k=1 k= j k= j ⎛⎡ ⎤ ⎡ ⎤⎞ N N † N † ⎜⎢ g jk(2σ ˆ σˆ j − 1)ˆσk ⎥ ⎢ g jk(2σ ˆ σˆ j − 1)ˆσk ⎥⎟ − γ −ω j , ρ σ † + j ρ , σ † jn¯( j )⎝⎣ ˆS ˆ j ⎦ ⎣ ˆS ˆ j ⎦⎠ ω j − ωk ω j − ωk j=1 k=1 k=1 k= j k= j ⎛⎡ ⎤ ⎡ ⎤⎞ N N σ †σ − σ † N σ †σ − σ † ⎜⎢ g jk(2 ˆ j ˆ j 1)ˆk ⎥ ⎢ g jk(2 ˆ j ˆ j 1)ˆk ⎥⎟ − γ jn¯(ω j )⎝⎣ , ρˆSσˆ j⎦ + ⎣ ρˆS, σˆ j⎦⎠ ω j − ωk ω j − ωk j=1 k=1 k=1 k= j k= j ⎛⎡ ⎤ ⎡ ⎤⎞ N N † N † ⎜⎢ g jk(2σ ˆ σˆ j − 1)ˆσk ⎥ ⎢ g jk(2σ ˆ σˆ j − 1)ˆσk ⎥⎟ + γ ω σ †, ρ j + σ †ρ , j jn¯( j )⎝⎣ ˆ j ˆS ⎦ ⎣ ˆ j ˆS ⎦⎠ ω j − ωk ω j − ωk j=1 k=1 k=1 k= j k= j

032207-8 PERTURBATION THEORY FOR LINDBLAD … PHYSICAL REVIEW A 102, 032207 (2020) ⎛⎡ ⎤ ⎡ ⎤⎞ N N † † N † † ⎜⎢ gkjσˆ jσˆ + g jkσˆ σˆk ⎥ ⎢ gkjσˆ jσˆ + g jkσˆ σˆk ⎥⎟ + σ †σ , ρ k j + σ †σ ρ , k j j⎝⎣ ˆ j ˆ j ˆS ⎦ ⎣ ˆ j ˆ j ˆS ⎦⎠ ω j − ωk ω j − ωk j=1 k=1 k=1 k= j k= j ⎛⎡ ⎤ ⎡ ⎤⎞ N N † † N † † ⎜⎢ gkjσˆ jσˆ + g jkσˆ σˆk ⎥ ⎢ gkjσˆ jσˆ + g jkσˆ σˆk ⎥⎟ + k j , ρ σ †σ + k j ρ , σ †σ . j⎝⎣ ˆS ˆ j ˆ j⎦ ⎣ ˆS ˆ j ˆ j⎦⎠ (66) ω j − ωk ω j − ωk j=1 k=1 k=1 k= j k= j

Below, to illustrate how the theory works, we discuss importantly this energy flow exceeds the energy flow from the examples with a minimum number of subsystems. For this system to the dissipative reservoirs (see Figs. 1 and 2). purpose, we consider several complex open quantum systems At the same time, the first order of the perturbation theory and compare the dynamics predicted by the master equa- satisfies the criterion (48) and, as Figs. 1 and 2 show, the tions with the following superoperators: (i) the exact Lindblad total energy flow is directed from the system to the reservoirs. superoperators [see (33)], (ii) the Lindblad superoperators ob- Therefore, the dynamics of the system predicted by the first tained by the perturbation theory, and (iii) the local Lindblad order of the perturbation theory and by the exact Lindblad superoperators. As was discussed in Sec. II [see Eq. (9)], to superoperators are in good agreement. obtain the exact Lindblad superoperators we use computer In the next example, we show how cross-relaxation pro- simulations to solve the eigenproblem for the Hamiltonian cesses affect the quantum transport through a chain of three (49) and implement the standard procedure. TLSs with the frequencies ω1, ω2, and ω3 such that ω1 <ω2, We start with a system containing two TLSs and having ω3 <ω2, and ω1 <ω3 (Fig. 3). The Hamiltonian of the sys- two different transition frequencies ω1 and ω2 = 1.2ω1, de- tem has the form (49) with the coupling constants g13 = g31 = −3 −3 phasing rates 1 = 10 ω1 and 2 = 10 ω1, and dissipation 0 and g12 = g23 = g21 = g32 = g = 0: rates γ = 10−7ω and γ = 10−6ω . We assume that the tem- 1 1 2 1 ˆ = ω σ †σ + ω σ †σ + ω σ †σ + σ †σ + σ †σ perature is zero T = 0 and the coupling constants are the HS h¯ 1 ˆ1 ˆ1 h¯ 2 ˆ2 ˆ2 h¯ 3 ˆ3 ˆ3 hg¯ (ˆ1 ˆ2 ˆ2 ˆ1) = = = . ω |ω − ω | same g12 g21 g 0 01 1. In this case g 2 1 . + hg¯ (ˆσ †σˆ + σˆ †σˆ ). (67) To emphasize the necessity of the perturbation approach, we 2 3 2 3 consider the case when the dephasing rates are much greater Each of these TLSs interacts with dissipative and de- , γ ,γ than the dissipative rates 1 2 1 2. This relationship phasing reservoirs, which have Hamiltonians (50). In the between the dephasing and the dissipative rates is typical interaction Hamiltonians (51) j is equal to 1, 2, or 3 for the for quantum dots [50–52] and dye molecules [53,54]. For first, second, or third TLS, respectively. We assume that all certainty, we assume that initially the first TLS is in the excited the dissipation rates (56) are the same, γ1 = γ2 = γ3 = γ , and state, while the second TLS is in the ground state. The time that the dephasing rates (57) are also identical, 1 = 2 = = ρ ˆ dependence of the system energy E(t ) Tr[ ˆ S (t )HS] when 3 = . In a quantum transport problem, it is common to calculated with the local approach (zeroth order of the perturbation theory) (58), the first-order master equation (65), and the exact master equation is shown in Fig. 1. One can see that the first-order perturbation theory is in good agreement with the solution of the exact Lindblad equation. However, not only does the local Lindblad equation give dynamics that differs from that given by the exact Lindblad equation, it also predicts that the energy of the system rises with time and becomes greater than the initial system energy (Fig. 1). This violates the second law of thermodynamics in the Clausius form [16,17] because this additional energy originates from the energy flow from the reservoir to the system at zero temperature. From Fig. 2 one can see that the local Lindblad equation predicts energy transfers from the low-energy TLS to the high-energy TLS in such a way that the total energy of the system rises. At the same time, both the first-order perturbation theory and the exact Lindblad equation show that this transition is suppressed FIG. 1. System energy normalized by its initial energy. The pa- (Fig. 2). rameters of the system are ω2 = 1.2ω1, g = 0.01ω1, 1 = 2 = In this example, the criterion (48) is violated in the zeroth −3 −7 −6 10 ω1, γ1 = 10 ω1, γ2 = 10 ω1, and temperature T = 0. Ini- order of the perturbation theory for the Lindblad superopera- tially, the first and the second TLSs are in the excited and ground ε = / ω − tor (the local approach) with the small parameter g ( 2 states, respectively. Blue dashed, red dash-dotted, and green solid ω 1). As expected, the local approach leads to a violation of the lines are obtained by solving the local Lindblad equation, by using second law of thermodynamics. Indeed, the net energy flow is the perturbation theory, and by solving the exact Lindblad equation, directed from the dephasing reservoir to the system and more respectively.

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FIG. 4. Dependence of the quantum transport efficiency η on the dephasing rate . The parameters of the system are ω2 = 2ω1, −10 −6 g = 0.01ω1, γ = 10 ω1, γdrain = 10 ω1,and(a)ω3 = 0.8ω1 and (b) ω3 = 1.2ω1. The blue dashed line corresponds to master equation (69) with the local Lindblad superoperator Lˆ[ˆρS (t )], the red dash- dotted line corresponds to master equation (69) with the Lindblad superoperator Lˆ[ˆρS (t )] obtained in the first order of the perturbation theory, the purple dotted line corresponds to master equation (69) with the Lindblad superoperator Lˆ[ˆρS (t )] obtained in the second order of the perturbation theory, and the green solid line corresponds to master equation (69) with the exact Lindblad superoperator Lˆ[ˆρ (t )]. FIG. 2. Time dependence of the probability of (a) the first TLS S Arrows indicate the dephasing rate above which the criterion (48) and (b) the second TLS to be in the exited state. The inset shows the is violated with ε = g/|ω − ω | and n = 0 (blue arrow), n = 1(red probability of the second TLS to be in the exited state at a smaller 2 1 arrow), and n = 2 (purple arrow). scale in time. The parameters and the lines are described in Fig. 1.

Lindblad superoperator Lˆ (0)[ˆρ (t )] defined by Eq. (54) coin- introduce an additional relaxation channel that drains excita- S cides with the Lindblad superoperator obtained with the local tions from the end of the TLS chain [30]. In our case, this is approach [2,21–32]. In the first order of the perturbation the- the third TLS, which has additional relaxation described by (0) ory, the Lindblad superoperator takes the form Lˆ [ˆρS (t )] + the Lindblad superoperator [30] (1) (1) Lˆ [ˆρS (t )], where Lˆ [ˆρS (t )] is defined by Eq. (66). † † † At the initial moment, only the first TLS is excited. The Ldrain(ˆρS ) = γdrain(2σ ˆ3ρˆSσˆ − σˆ σˆ3ρˆS − ρˆSσˆ σˆ3). (68) 3 3 3 capability of the system to conduct the excitation from the We assume that the rate of the excitation drain γdrain is first TLS to the third is characterized by the quantum transport faster than the dissipation rate of this TLS, γdrain γ . efficiency η defined as [30] Thus, the master equation for the density matrix of the +∞ † systemρ ˆS takes the form η =− σ σ ρ . tr[ ˆ3 ˆ3Ldrain(ˆS )]dt (70) 0 ∂ i ρˆ (t ) =− [Hˆ , ρˆ ] + Lˆ[ˆρ (t )] + Lˆ [ˆρ (t )], (69) From a physical point of view, the quantum transport ef- ∂t S h S S S drain S ¯ ficiency η of the chain of the TLSs is the probability of the where the Hamiltonian HˆS and the Lindblad superoperator initial excitation of the first TLS to be transmitted to the drain. Lˆdrain[ˆρS (t )] are defined by Eqs. (67) and (68), respectively, The dependence of the quantum transport efficiency on and the Lindblad superoperator Lˆ[ˆρS (t )] that arises due to the dephasing rate is shown in Fig. 4. The master equation interactions of the TLSs with dissipating and dephasing (69) with the exact Lindblad superoperator Lêx[ˆρS (t )] [Eq. (9)] reservoirs is described by Eqs. (50) and (51), respectively. predicts that the quantum transport efficiency η is very small Note that in the zeroth order of the perturbation theory, the and is independent of the dephasing rate (the green solid line in Fig. 4). As one can see from Fig. 4, the dynamics predicted with (0) the local Lindblad superoperator Lˆ[ˆρS (t )] = Lˆ [ˆρS (t )] fails to describe the quantum transport when the dephasing rate is greater than the value

>γ(ω2 − ω1)/g. (71) The reason for this is that the local Lindblad equation does FIG. 3. Schematic of the TLS chain. At the initial moment, only not include cross relaxations. It is important to note that the the ﬁrst TLS is excited. inequality (71) coincides with the criterion (48), which is

032207-10 PERTURBATION THEORY FOR LINDBLAD … PHYSICAL REVIEW A 102, 032207 (2020) needed for the second law of thermodynamics to hold. To The reason for this is that the first-order perturbation theory compare Eqs. (71) and (48)wehavetosetε = g/|ω2 − ω1| describes the cross relaxation only for the neighboring TLSs. and n = 0inEq.(48). However, at high dephasing rates, the main role of the quan- The master equation (69) with the Lindblad superoperator tum efficiency is played by the cross relaxation of the first and (0) (1) Lˆ [ˆρS (t )] + Lˆ [ˆρS (t )] obtained in the first order of pertur- third TLSs. Note that the inequality (72) is in agreement with bation theory describes the cross relaxation of the neighboring the criterion (48)forn = 1. TLSs. The quantum transport efficiency calculated with the Now we consider the second order of the perturbation perturbation theory is close to the one calculated with the theory. As discussed at the beginning of this section, to ob- exact Lindblad superoperators with a wide range of dephas- tain the Lindblad superoperator Lˆ[ˆρS (t )] one should solve the ing rates (Fig. 4). The first-order perturbation theory fails to Heisenberg equation (59) within the perturbation theory up describe the quantum transport efficiency when to the second order of the small parameter g/|ω2 − ω1|.The derivation of this solution is long but straightforward. The >γ[(ω − ω )/g]2. (72) 2 1 result of the solution is

2 − ω g − ω g − ω σˆ (t ) = ςˆ e i 1t + (2ˆς †ςˆ − 1)ˆς e i 2t + (2ˆς †ςˆ − 1)(2ˆς †ςˆ − 1)ˆς e i 3t , (73) 1 1 ω − ω 1 1 2 (ω − ω )(ω − ω ) 1 1 2 2 3 1 2 1 3 2 3 2 − ω g − ω g − ω g − ω +ω −ω σ = ς i 2t + ς †ς − ς i 3t + ς i 1t + ς ς ς † i( 1 2 3 )t ˆ2(t ) ˆ2e (2ˆ2 ˆ2 1) ˆ3e ˆ1e 2 ˆ1 ˆ2 ˆ3 e ω2 − ω3 ω2 − ω1 (ω1 − ω3)(ω2 − ω3) 2 g − −ω +ω +ω g g g − ω −ω +ω + ς †ς ς i( 1 2 3 )t + + ς ς †ς i( 1 2 3 )t , 2 ˆ1 ˆ2 ˆ3e 2 ˆ1 ˆ2 ˆ3e (ω3 − ω1)(ω2 − ω1) ω1 + ω3 − 2ω2 ω2 − ω3 ω2 − ω1 (74) 2 − ω g − ω g − ω σ = ς i 3t + ς †ς − ς i 2t + ς †ς − ς †ς − ς i 1t , ˆ3(t ) ˆ3e (2ˆ3 ˆ3 1)ˆ2e (2ˆ3 ˆ3 1)(2ˆ2 ˆ2 1)ˆ1e (75) ω3 − ω2 (ω3 − ω1)(ω2 − ω1)

where the operatorsς ˆ1,ˆς2, andς ˆ3 can be derived from the Each subsystem also interacts with its own reservoirs. When conditionsσ ˆ1(0) = σˆ1,ˆσ2(0) = σˆ2, andσ ˆ3(0) = σˆ3. the local Lindblad superoperators are applied, the second law The master equation (69) with the Lindblad superoperator of thermodynamics may be violated and the prediction of the obtained in the second order of the perturbation theory pre- dynamics of the system may be incorrect. We have shown dicts the quantum efficiency, which is in good agreement with that the main reason for this is the interaction of the system the one calculated with the exact Lindblad superoperators in a with dephasing reservoirs. The perturbation theory that we wide range of the dephasing rate (Fig. 4). The second order developed can restore the correct dynamics of an open quan- of the perturbation theory describes the cross correlations of tum system and provide the fulfillment of the second law of all three TLSs but does not correctly predict the quantum thermodynamics. The perturbation theory developed for the transport efficiency when the dephasing rate is as high as Lindblad superoperators can be applied to various open quantum system problems. The theory does not require additional complicated calculations, such as solving the eigenvalue prob- >γ[(ω − ω )/g]3. (76) 2 1 lem, and it takes into account cross relaxation that may play The inequality (76) is the same as the criterion (48)for an important role in the dynamics of an open quantum system. n = 2. Thus, in the case of the TLS chain shown in Fig. 3, ACKNOWLEDGMENTS the transport efficiency cannot be significantly improved by increasing the dephasing rate of each TLS (Fig. 4). This E.S.A. and V.Y.S. are grateful to the Foundation for the Ad- contradicts the local theory that leads to incorrect system vancement of Theoretical Physics and Mathematics BASIS. dynamics. The perturbation theory takes into account the A.A.L. acknowledges the support of the ONR under Grant No. cross-relaxation processes between TLSs and restores the cor- N00014-20-1-2198. rect dynamics of the system (Fig. 4). At the same time, the correct description of the system dynamics at higher dephas- APPENDIX A: PERTURBATION THEORY FOR ing rates requires higher orders of the perturbation theory. TRANSITION OPERATORS In this Appendix we present the details of the derivation V. CONCLUSION of perturbation series for the Lindblad superoperator. We con- In this paper we have analyzed the applicability of Lind- sider the system with the Hamiltonian (1). blad superoperators obtained within the local theory to the First, we consider the case ε = 0 in the Hamiltonian (1), problems of open quantum systems and developed a per- which corresponds to the separate subsystems, and discuss turbation theory for the Lindblad superoperators for open the well-known results. According to the well-known standard quantum systems containing several interacting subsystems. algorithm [19] for obtaining the Lindblad equation in the case

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ε = 0, the Schrödinger operator Sˆα should be expanded in the interact with each other, the interaction of each subsystem | (is) with its own reservoir remains the same, as in the case of basis of the eigenstates kαi of the subsystem Hamiltonian (is) noninteracting subsystems [see Eq. (1)]. HSα , The procedure described in the preceding section covers ˆ = (is) ˆ (is) (is) (is) = ˆ(is), ε = ε = Sα kαi Sα kα j kαi kα j Sαij (A1) the case 0. In the case 0, the Lindblad superoperator ij∈α ij (A5) is not correct despite the fact that the reservoirs and the interaction between reservoirs and systems remains un- where changed. The Lindblad superoperator (A5) has been obtained ˆ(is) = (is) ˆ (is) (is) (is), ω(is) = ω(is) − ω(is). | (is) (is)| Sαij kαi Sα kα j kαi kα j αij α j αi by employing the basis of the system eigenstates kαi kα j . If ε = 0, these basis states are not the eigenstates of the (A2) Hamiltonian HˆS and now Since all the eigenstates |k(is) and the eigenfrequencies αi ˆ / ˆ − ˆ / = ˆ(is) − ω(is) . (is) (is) exp(iHSt h¯)Sα exp( iHSt h¯) Sαij exp i αijt ωα of the Hamiltonians Hˆ α are assumed to be known, the i S ij interaction representation for the operators Sˆα takes the form (A6) ˆ (is) / ˆ − ˆ (is) / To employ the standard algorithm, we should find new exp iHS t h¯ Sα exp iHS t h¯ operators Sˆα ω =k |Sˆα|k |k k |, where |k are eigen- (is) (is) ij i j i j j = iHˆ t/h Sˆα − iHˆ t/h exp( Sα ¯) exp Sα ¯ ω Hˆ states, j are eigenfrequencies of the Hamiltonian S, and (is) (is) (is) (is) (is) ω = ω − ω = k Sˆα k k k exp − i ω t ij i j. However, now the subsystems are not iso- αi α j αi α j αij | ω ij lated; therefore, the eigenstates k j and eigenfrequencies j of the Hamiltonian Hˆ of interacting subsystems are unknown, = ˆ(is) − ω(is) , S Sαij exp i αijt (A3) so the operators Sˆ are also unknown. In the general case, α ωij ij this problem cannot be solved analytically. Moreover, the (is) numerical solution is usually difficult to implement. As an al- where Sˆα does not depend on time. Here we use ij ternative, we develop the perturbation theory for the Lindblad | (is) the fact that kα j is the eigenstate of the Hamilto- superoperator. ˆ (is) ˆ (is) | (is)= ω(is) | (is) nian Hα , and exp(iHSα t ) kα j exp(i α j t ) kα j and The perturbation theory for the Lindblad superoperator (is)| − ˆ (is) = (is)| − ω(is) developed here is based on searching for the approximate kα j exp( iHSα t ) kα j exp( i α j t ). It should be noted expansion for the operator exp(iHˆ t/h¯)Sˆα exp(−iHˆ t/h¯)in that the operator Sˆ(is) corresponds to transition of the subsys- S S αij the form tem α from the state |k(is) to the state |k(is). α j αi ˆ / ˆ − ˆ / ˆ (is) exp(iHSt h¯)Sα exp( iHSt h¯) If the unified system HS interacts only with one reservoir Hˆ then the standard procedure leads to the Lindblad super- = ˆ(0) − ω(0) + ε1 Rα S ω(0) exp( i t ) operator [16] ω(0) ˆ ρ ≡ λ2 ω(is) ˆ(is), ρ ˆ(is)† × ˆ(1) − ω(1) +··· , Lα[ˆS (t )] αGα αij Sαij ˆS (t )Sαij S ω(1) exp( i t ) (A7) αij ω(1) + ˆ(is)ρ , ˆ(is)† , ˆ(n) Sαij ˆS (t ) Sαij (A4) where the operators S ω(n) do not depend on time. The expansion (A7) is to be found by means of the perturbation theory whereρ ˆS(t ) is the density matrix of the big system and 2 (is) for a small parameter ε. Once the expansion (A7) is found we λ Gα ( ω ) is the relaxation rate. In the more general case α αij can directly apply the standard procedure using the expansion ˆ (is) when the system HS interacts with the set of the reservoirs (A7) to obtain the Lindblad superoperators in a power series ˆ = ˆ ˆ ˆ = HR α HRα and these reservoirs are uncorrelated RαRβ of ε, 0, the standard procedure leads to the Lindblad superoperator +∞ of separate subsystems [16,17] ˆ ρ = εn ˆ (n) ρ . L[ˆS (t )] L [ˆS (t )] (A8) ˆ ρ ≡ λ2 ω(is) ˆ(is), ρ ˆ(is)† n=0 L[ˆS (t )] αGα αij Sαij ˆS (t )Sαij α ij Lˆ (n) ρ t Belowweshowhow [ˆS ( )] can be obtained. The + ˆ(is)ρ , ˆ(is)† starting point for our perturbation theory is the expansion Sαij ˆS (t ) Sαij for the operators Sˆα (A1). The expansion (A1) is still valid | (is) = Lˆsubsystem[ˆρS (t )]. (A5) because the set of the states kαi is still the complete basis subsystem of the whole system. We substitute the expansion (A1)into the expression exp(iHˆSt/h¯)Sˆα exp(−iHˆSt/h¯) and arrive at the Thus, in the case when the subsystems do not interact, the expression Lindblad superoperator reduces to the sum of the superoperators for isolated subsystems [17,20]. exp(iHˆSt/h¯)Sˆα exp(−iHˆSt/h¯) Now we consider the case when the subsystems can inter- = ˆ / ˆ(is) − ˆ / = ˆ . act with each other, i.e., ε = 0 in the Hamiltonian (1), and exp(iHSt h¯)Sαij exp( iHSt h¯) Sαij(t ) develop the perturbation theory for the Lindblad superoper- ij ij ators. We suppose that despite the fact that the subsystems (A9)

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i For convenience, we introduce new operators ˆ (1) / =− ω(is) ˆ (1) + ˆ (0) , ˆ (0) , dSαij(t ) dt i αijSαij(t ) W (t ) Sαij(t ) ˆ = ˆ / ˆ(is) − ˆ / . h¯ Sαij(t ) exp(iHSt h¯)Sα exp( iHSt h¯) (A10) ij (1) Sˆ α (0) = 0, (A18) ˆ ij Now the standard procedure demands the operator Sαij(t ) i to be presented in the form dSˆ (2) (t )/dt =−i ω(is)Sˆ (2) (t ) + Wˆ (1)(t ), Sˆ (0) (t ) αij αij αij h¯ αij ˆ = ˆ − ω , Sαij(t ) S ω˜ exp( i ˜ t ) (A11) i (0) (1) (2) ω + Wˆ (t ), Sˆ (t ) , Sˆ (0) = 0. (A19) ˜ h¯ αij αij ˆ ω where the operators S ω˜ do not depend on time and ˜ are some yet unknown frequencies. The substitution of the The solution of Eq. (A17)is expansion (A11) into the expression (A9) allows us to directly (0) (0) (is) Sˆ α (t ) = Sˆα exp(−i ωα t ), (A20) apply the standard procedure to obtain the exact Lindblad ij ij ij Sˆ(0) = Sˆ(is) superoperators. Below we show how the expansion (A11)is where αij αij. Note that the expression (A20) has the obtained by means of the perturbation theory. To find the form (A11). Moreover, the expansion of (A11) in the zeroth expansion (A11) we take the time derivative from both sides order of the perturbation theory takes the form of (A10). As a result, we obtain the equation for the operators exp(iHˆSt/h¯)Sˆα exp(−iHˆSt/h¯) Sˆ α (t ), ij ≈ ˆ (0) = ˆ(0) − ω(is) . Sαij(t ) Sαij exp i αijt (A21) (is) i αij αij dSˆ α (t )/dt =−i ω Sˆ α (t ) + ε [Wˆ (t ), Sˆ α (t )], (A12) ij αij ij h¯ ij Since Sˆ(0) = Sˆ(is), the expansion (A21) is equal to the ex- ˆ αij αij where the operator W (t ) is the operator of subsystems inter- pansion (A3). If we now apply the standard procedure using action, taken in the Heisenberg representation as [55] the expansion (A21), we will obtain the Lindblad superoperator (22). The Lindblad superoperator (22) is equal to the Wˆ (t ) = exp(iHˆSt/h¯)Wˆ exp(−iHˆSt/h¯). (A13) Lindblad superoperator (A5); thus the Lindblad superopera- The initial condition for Eq. (A12) follows from the defini- tor in the zeroth order of the perturbation theory developed tion (A10)att = 0 and has the form coincides with the local Lindblad superoperator. Sˆ (0) = Sˆ(is). (A14) To solve the higher orders of the perturbation theory we αij αij use the explicit form (13) for the interaction operator Wˆ . To obtain the expansion (A11), one may solve Eq. (A12) Employing (A13) and (A10) we get with the initial condition (A14). It is easier to perform this by (0) (0) Wˆ (t ) = exp(iHˆ t/h¯)¯h wα α Sˆ Sˆ employing the perturbation theory. We expand the operators S 1 2i1 j1i2 j2 α1i1 j1 α2i2 j2 ˆ ε α1=α2 i1 j1i2 j2 Sαij(t ) in a series of the small parameter , − ˆ / ˆ = ˆ (0) + εˆ (1) +··· . exp( iHSt h¯) Sαij(t ) Sαij(t ) Sαij(t ) (A15) = h¯ wα α i j i j Sˆ α (t )Sˆ α (t ). (A22) To solve Eq. (A12) with the perturbation theory, we should 1 2 1 1 2 2 1i1 j1 2i2 j2 α =α expand Wˆ (t ) in a series of the small parameter ε as well, 1 2 i1 j1i2 j2 Substituting the expansions (A15) and (A16) into the ex- (n) (0) (1) pression (A22), we can express the terms W (t ) through the Wˆ (t ) = Wˆ (t ) + εWˆ (t ) +··· . (A16) ˆ (m) terms Sαij(t )asfollows: We substitute these expansions for Sˆ (t ) and Wˆ (t )in αij (n) (n ) (n ) W (t ) = h¯ wα α Sˆ 1 (t )Sˆ 2 (t ). Eq. (A12) and equate factors at the same powers of ε.Asa 1 2i1 j1i2 j2 α1i1 j1 α2i2 j2 + = α =α result, we obtain up to the second power n1 n2 n 1 2 i1 j1i2 j2 (A23) ˆ (0) / =− ω(is) ˆ (0) , ˆ (0) = ˆ(is) , dSαij(t ) dt i αijSαij(t ) Sαij(0) Sα ω(is) (A17) Returning to the first order of the perturbation theory, we αij should substitute W (0)(t ) and (A20)into(A18). As a result, we arrive at

(1) (is) (1) (0) (0) (0) (is) (is) (is) dSˆ (t )/dt =−i ω Sˆ (t ) + i wα α Sˆ Sˆ , Sˆ exp − i ω + ω + ω t , αij αij αij 1 2i1 j1i2 j2 α1i1 j1 α2i2 j2 αij α1i1 j1 α2i2 j2 αij α1=α2 i1 j1i2 j2 ˆ (1) = . Sαij(0) 0 (A24)

The solution to Eq. (A24)is (0) (0) (0) wα α i j i j Sˆα Sˆα , Sˆα Sˆ (1) (t ) = 1 2 1 1 2 2 1i1 j1 2i2 j2 ij 1 − exp − i ω(is) + ω(is) t exp − i ω(is)t . (A25) αij (is) (is) α1i1 j1 α2i2 j2 αij ωα + ωα α1=α2 i1 j1i2 j2 1i1 j1 2i2 j2

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Note that if some frequencies ω(is) + ω(is) are equal to zero, that is, secular terms appear, it is necessary to use the α1i1 j1 α2i2 j2 ω(is) method of multiple scales [56]. In this case one obtains the corrections to the frequencies αij. Below, for simplicity we assume (is) (is) that all the frequencies ω + ω in the sum (A25) are not equal to zero when wα α = 0. Thus, the solution of α1i1 j1 α2i2 j2 1 2i1 j1i2 j2 Eq. (A12) up to the ﬁrst order by ε takes the form

ˆ / ˆ(0) − ˆ / = ˆ ≈ ˆ (0) + εˆ (1) exp(iHSt h¯)Sαij exp( iHSt h¯) Sαij(t ) Sαij(t ) Sαij(t ) (0) (0) (0) wα α i j i j Sˆα Sˆα , Sˆα = ˆ(0) − ω(is) + ε 1 2 1 1 2 2 1i1 j1 2i2 j2 ij Sαij exp i αijt (is) (is) ωα + ωα α1=α2 i1 j1i2 j2 1i1 j1 2i2 j2 × 1 − exp − i ω(is) + ω(is) t exp − i ω(is)t . (A26) α1i1 j1 α2i2 j2 αij

The expansion (28) according to the general theory leads operators, which are not present in system-reservoir interac- to the Lindblad superoperator (29). The higher orders of the tion (2). Then the Heisenberg equation for the first-order ap- perturbation theory can be obtained in the same way. proximation (A18) will contain some new operators of zeroth ˆ (0) , ˆ (0) Above we assumed that the interaction between the sub- order, which appear due to the commutator [W (t ) Sαij(t )]. systems (13) is defined by the same operators that are To make the system of operator equations (A18) closed present in the system-environment interaction [Eq. (2)]. If one needs to write down the Heisenberg equation for new this assumption is not fulfilled, the theory developed can operators appearing in the zeroth order. Thus, for the finite- be straightforwardly generalized to be applied to such cases dimensional case, we obtain a finite-dimensional linear in finite-dimensional problems. Here we briefly discuss this system, which can be solved exactly. The same generalization generalization. Suppose the interaction (13) contains the can be done for higher orders of the perturbation theory.

APPENDIX B: PROOF OF EQS. (34) AND (35) We start with the proof of Eq. (34). The substitution of Eq. (22) into the left-hand side of Eq. (34) leads to ˆ ˆ HS (0) HS 2 (is) † † exp −i t Lˆ [ρ˜ˆ (t )] exp i t = λαGα ω {[Sˆ α (−t ), ρˆ (t )Sˆ (−t )] + [Sˆ α (−t )ˆρ (t ), Sˆ (−t )]}. (B1) h¯ S h¯ αij ij S αij ij S αij α ij

ˆ ˆ (0) ε ˆ The difference between the operator Sαij(t ) and the operator Sαij(t ) is equal to . Therefore, we replace the operators Sαij(t ) ˆ (0) with the operators Sαij(t )inEq.(B1) and obtain ˆ ˆ HS (0) HS 2 (is) (0) (0)† (0) (0)† exp −i t Lˆ [ρ˜ˆ (t )] exp i t ≈ λαGα ω Sˆ (−t ), ρˆ (t )Sˆ (−t ) + Sˆ (−t )ˆρ (t ), Sˆ (−t ) . h¯ S h¯ αij αij S αij αij S αij α ij (B2) The substitution of Eq. (21) into Eq. (B2) leads to Eq. (34). (0) Equation (35) can be proved in the same way. To prove Eq. (35) we rewrite the Lindblad superoperator Lˆ [ρ˜ˆS (t )] + (1) εLˆ [ρ˜ˆS (t )] in the form ˆ (0) ρˆ + ε ˆ (1) ρˆ ≈ λ2 ω(is) ˆ(0) + ε ˆ(1) , ρˆ ˆ(0) + ε ˆ(1) † L [˜ S (t )] L [˜ S (t )] αGα αij Sαij Sαij ˜ S (t )(Sαij Sαij) α ij + ˆ(0) + ε ˆ(1) ρˆ , ˆ(0) + ε ˆ(1) † . Sαij Sαij ˜ S (t ) Sαij Sαij (B3)

The substitution of Eq. (B3) into Eq. (35) leads to Hˆ Hˆ exp −i S t {Lˆ (0)[ρ˜ˆ (t )] + εLˆ (1)[ρ˜ˆ (t )]} exp i S t h¯ S S h¯ ˆ ˆ ˆ ˆ 2 (is) HS (0) (1) HS HS (0) (1) † HS = λαGα ω exp −i t Sˆ + εSˆ exp i t , ρˆ (t )exp −i t Sˆ + εSˆ exp i t αij h¯ αij αij h¯ S h¯ αij αij h¯ α ij Hˆ Hˆ Hˆ † Hˆ + exp −i S t Sˆ(0) + εSˆ(1) exp i S t ρˆ (t ), exp −i S t Sˆ(0) + εSˆ(1) exp i S t . (B4) h¯ αij αij h¯ S h¯ αij αij h¯

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γ +∞ σ †σ γ +∞ σ †σ FIG. 5. Occupancies integrated over time for the excitation transfer problem, (a) 1 0 ˆ1 ˆ1 dt and (b) 2 0 ˆ2 ˆ2 dt, plotted as functions of the ratio of the TLSs’ excitation frequencies ω2/ω 1. The parameters of the system are the same as in Fig. 1: g = 0.01ω1, 1 = −3 −6 −7 2 = 10 ω1, γ1 = 10 ω1, γ2 = 10 ω1, and temperature T = 0. Blue dashed lines represent the solutions of the local Lindblad equation, red solid lines represent the solution of the exact Lindblad equation, and vertical orange lines indicate the degenerate case ω2/ω 1 = 1.

− ˆ / ˆ(0) + ε ˆ(1) ˆ / Let us consider the operator exp( iHSt h¯)(Sαij Sαij)exp(iHSt h¯). We use the deﬁnitions (28) and (A10) and obtain ˆ ˆ wα α Sˆ (−t )Sˆ (−t ), Sˆ (−t ) HS (0) (1) HS 1 2i1 j1i2 j2 α1i1 j1 α2i2 j2 αij − ˆ + ε ˆ = ˆ α − + . exp i t Sαij Sαij exp i t S ij( t ) (is) (is) (B5) h¯ h¯ ωα + ωα α1=α2 i1 j1i2 j2 1i1 j1 2i2 j2 The substitution of Eq. (27) into Eq. (B5) leads to the approximate equality Hˆ Hˆ exp −i S t Sˆ(0) + εSˆ(1) exp i S t ≈ Sˆ(0) + εSˆ(1) exp −i ω(is)t . (B6) h¯ αij αij h¯ αij αij αij The difference between the left-hand side and the right-hand side of Eq. (B6) is proportional to ε2. The substitution of Eq. (B6) into Eq. (B4) leads to the approximate equality (35).

APPENDIX C: APPLICABILITY OF LOCAL LINDBLAD caused by application the local Lindblad equations, would not EQUATIONS FOR DEGENERATE SYSTEMS require the additional energy in the degenerate case. We illustrate the above arguments in Fig. 5, which shows In this appendix we consider the applicability of the local a comparison between the results obtained with the local Lindblad equations for the degenerate systems, i.e., when the Lindblad equation and exact Lindblad equation. As an exam- subsystems have the same excitation energies. ple we consider the excitation transfer problem between two The local Lindblad equation leads to erroneous energy ω ω transitions between the subsystems. These erroneous energy TLSs with transitions frequencies 1 and 2. The first TLS is transitions require additional energy, which is of the order of initially in the excited state and the second one is in the ground the energy mismatch between the subsystems. The reservoirs state. To characterize the excitation transfer from the first TLS provide the additional energy to the system. If the dissipa- to the second TLS we use their occupancies integrated over γ +∞ σ †σ γ +∞ σ †σ tions in the system do not compensate for this energy flow time, 1 0 ˆ1 ˆ1 dt and 2 0 ˆ2 ˆ2 dt. One can see ω /ω ≈ from the reservoirs, violation of the second law of thermody- that for the degenerate case ( 2 1 1) the local Lindblad namics occurs. Keeping in mind this mechanism, one should equations and exact Lindblad equations give the same results, expect that the local Lindblad equations for the degenerate i.e., the zeroth order of perturbation theory is quantitatively subsystems should be thermodynamically consistent. Indeed, correct. Thus, for the problem of quantum transport, it is the the erroneous excitation transitions between the subsystems, nondegenerate case that requires special attention.

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