CUBO, A Mathematical Journal Vol. 23, no. 01, pp. 121–144, April 2021 DOI: 10.4067/S0719-06462021000100121 Energy transfer in open quantum systems weakly coupled with two reservoirs Franco Fagnola1 Damiano Poletti2 Emanuela Sasso3 1,2 Dipartimento di Matematica, Politecnico di Milano, Piazza Leonardo da Vinci 32, I-20133 Milano, Italy. franco.fagnola@polimi.it; damiano.poletti@polimi.it 3 Dipartimento di Matematica, Università di Genova, Via Dodecaneso 35, I - 16146 Genova, Italy. sasso@dima.unige.it ABSTRACT We show that the energy transfer through an open quantum system with non-degenerate Hamiltonian weakly coupled with two reservoirs in equilibrium is approximately proportional to the difference of their temperatures unless both temperatures are small. RESUMEN Mostramos que la transferencia de enerǵıa a través de un sistema cuántico abierto con Hamiltoniano no-degenerado débilmente acoplado con dos reservorios en equilibrio es aprox- imadamente proporcional a la diferencia de sus temperaturas a menos que ambas temperaturas sean pequeñas. Keywords and Phrases: weak-coupling, quantum Markov semigroup, quantum transport, energy current. 2020 AMS Mathematics Subject Classification: 81S22, 82C10, 80A19. Accepted: 09 February, 2021 Received: 21 September, 2020 ©2021 F. Fagnola et al. This open access article is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. http://cubo.ufro.cl/ http://dx.doi.org/10.4067/S0719-06462021000100121 https://orcid.org/0000-0002-1661-0992 https://orcid.org/0000-0002-2333-4252 https://orcid.org/0000-0003-0994-2400 122 F. Fagnola, D. Poletti, E. Sasso CUBO 23, 1 (2021) 1 Introduction Energy transfer in classical and quantum systems and the validity of Fourier’s law of heat conduction have been a hot topic for many years (see [3, 4, 6, 7, 12, 18, 25, 26] and the references therein). For quantum systems, in particular, after experimental evidence of effective quantum energy transfer in photosynthesis in some biological systems has been found (see [14, 24]), investi- gations have focused on understanding to what extent quantum mechanics contributes to transport efficiency. Several models have been proposed involving open quantum systems (see e.g. [5, 6, 27]), mostly phenomenological, and also numerical simulations have been done showing different behaviours. The interaction of the open quantum system with reservoirs is described through interaction oper- ators that appear in the dissipative part of the Gorini-Kossakowski-Sudharshan-Lindblad (GKSL) [17, 22] generator L of the dynamics, while the Hamiltonian part is given by the commutator with the system Hamiltonian HS. However, when the GKSL generator is rigorously deduced from some scaling (weak coupling or low density limit) both the system Hamiltonian and the interaction op- erators appear in the GKSL generator L after non-trivial transformations (see [1, 2, 9, 10, 13, 19]). In this paper we study models of open quantum systems rigorously deduced from the weak coupling limit. We consider a quantum system with non-degenerate Hamiltonian HS coupled with two reservoirs in equilibrium at inverse temperatures β1 ≤ β2 and study variation of energy due to couplings with each reservoir. It is well-known (see Lebowitz and Spohn [25] (V.28)) that, by the second law of thermodynamics, energy (heat) flows from the hotter to the cooler reservoir. The energy flow, in general, is not proportional to the difference of temperature because of the nonlinear dependence of susceptibilities on temperature, namely an exact Fourier’s law does not hold. However, we rigorously prove that it holds in an approximate way when the temperatures of reservoirs are not too small or, as an alternative, differences between nearest energy levels are small. More precisely, we show that the amount of energy flowing through the system, Theorem 4.2, formula (4.5), is approximately proportional to the product of the temperature differences and a constant (conductivity) which can be interpreted as the average energy needed to jump from a level to the following higher level. The paper is organised as follows. In Section 2 we introduce quantum Markov semigroups (QMS) arising from the weak coupling limit of a non-degenerate system with two Boson reservoirs. The energy flow is computed explicitly in Section 3, Theorem 3.3, formula (3.7). The dependence of the energy flow on temperatures is studied in Section 4. Moreover, we also study (Theorem 4.3) the asymptotic behaviour of the invariant state when the eigenvalues of HS increase in number and form a set more and more packed. It turns out that the invariant state converges towards a CUBO 23, 1 (2021) Energy transfer in open quantum systems weakly coupled . . . 123 Gibbs state with temperature equal to the mean temperatures of the two baths. Finally, in Section 5, we consider as system the Ising model Hamiltonian and show that the energy flow in this case is zero. We have not been able to extend our analysis to quantum spin chains because their Hamiltonians are highly degenerate and the GKSL generator arising from the weak coupling limit, albeit explicit, is not easily treatable. In particular, we could not extract the relevant information on invariant states. 2 Semigroups of weak coupling limit type We consider an open quantum system with Hamiltonian HS acting on a complex separable Hilbert space h with discrete spectral decomposition HS = ∑ m≥0 εmPεm (2.1) where εm, with εm < εn for m < n, are the eigenvalues of HS and Pεm are the corresponding eigenprojectors. The system is coupled with two reservoirs each one in equilibrium with inverse temperatures β1 ≤ β2 with interaction Hamiltonians H1 = D1 ⊗A+(φ1) + D∗1 ⊗A −(φ1), H2 = D2 ⊗A+(φ2) + D∗2 ⊗A −(φ2), where D1,D2 are bounded operators on h and A +(φj),A −(φj) creation and annihilation operators, in the Fock space of the reservoir j, with test function φj . It is well-known (see [2, 9, 13, 25]) that, in the weak coupling limit, the evolution of the system observables is governed by a quantum Markov semigroup (QMS) on B(h), the algebra of all bounded operators in h, with generator of the form L = ∑ j=1,2, ω∈B Lj,ω (2.2) where B is the set of all Bohr frequencies B := {ω | ∃εn,εm s.t. ω = εn −εm > 0}. (2.3) For every Bohr frequency ω, Lj,ω is a generator with the Gorini-Kossakowski-Sudharshan-Lindblad (GKSL) structure (see [17, 22]) Lj,ω(x) = i[Hj,ω,x] − Γ−j,ω 2 ( D∗j,ωDj,ωx− 2D ∗ j,ωxDj,ω + xDj,ωD ∗ j,ω ) − Γ+j,ω 2 ( Dj,ωD ∗ j,ωx− 2Dj,ωxD ∗ j,ω + xDj,ωD ∗ j,ω ) (2.4) for all x ∈B(h), with Kraus operators Dj,ω defined by Dj,ω = ∑ (εn,εm)∈Bω PεmDjPεn (2.5) 124 F. Fagnola, D. Poletti, E. Sasso CUBO 23, 1 (2021) where Bω = {(εn,εm) | εn −εm = ω}, Γ±j,ω = fj,ωγ ± j,ω γ−j,ω = eβjω eβjω − 1 , γ+j,ω = 1 eβjω − 1 , fj,ω = ∫ {y∈R3 | |y|=ω} |φj(y)|2dsy (ds denotes the surface integral) and Hj,ω are bounded self-adjoint operators on h commuting with HS of the form Hj,ω = κ − j,ωD ∗ j,ωDj,ω + κ + j,ωDj,ωD ∗ j,ω for some real constants κ±j,ω. In the sequel, following a customary convention to simplify the notation, we also denote D−j,ω := Dj,ω and D + j,ω := D ∗ j,ω and write Q±j,ω(x) = − 1 2 D∓j,ωD ± j,ωx + D ∓ j,ωxD ± j,ω − 1 2 xD∓j,ωD ± j,ω (2.6) the term of the GKSL generator arising from the interaction with the bath j due the Bohr frequency ω is Lj,ω = Γ−j,ωQ − j,ω + Γ + j,ωQ + j,ω + i[Hj,ω, · ] and the term arising from the interaction with the reservoir j is Lj = ∑ ω∈B Lj,ω. We now make some assumptions on constants in such a way as to ensure boundedness of operators Lj. First of all note that the series ∑ ω D ∗ j,ωDj,ω is strongly convergent. Indeed, for all vector u = ∑ n≥0 Pεnu in h, we have∑ ω 〈 u,D∗j,ωDj,ωu 〉 = ∑ ω ∑ n,m≥0 〈Pεm−ωDjPεmu,Pεn−ωDjPεnu〉 = ∑ ω ∑ n≥0 〈DjPεnu,Pεn−ωDjPεnu〉 ≤ ∑ n≥0 ‖DjPεnu‖ 2 = ‖Dj‖ 2 ‖u‖2 . As a consequence, if we assume sup ω∈B Γ±j,ω < +∞, sup ω∈B ∣∣κ±j,ω∣∣ < +∞, for j = 1, 2 GKSL generators Lj turn out to be bounded. The above condition will be assumed to be in force throughout the paper. Remark. Note that Lj depends on the inverse temperature βj only through the constants γ±j,ω. The above notation follows that of [1]. CUBO 23, 1 (2021) Energy transfer in open quantum systems weakly coupled . . . 125 For all normal linear operator S on B(h) we denote by S∗ the predual operator acting on the Banach space of trace class operators on h. Therefore, we denote by T = (Tt)t≥0 the QMS on B(h) generated by L and by T∗ = (T∗t)t≥0 the predual semigroup acting on trace class operators. In the same way, T j (resp. T j,ω and T j,ω∗ ) stand for the QMS generated by Lj (resp. Lj,ω and its predual semigroup). In this paper we are concerned with normal states, therefore we shall identify them with their densities which are positive operators on h with unit trace. We end this section by checking that, if reservoirs have the same temperature β1 = β2 = β and Zβ := tr ( e−βHS ) < +∞, then the Gibbs state has density ρβ = Z −1 β e −βHS (2.7) and is stationary. Proposition 2.1. If β1 = β2 = β and Zβ := tr ( e−βHS ) = ∑ n≥0 e−βεn dim(Pεn ) < +∞ then the Gibbs state (2.7) is invariant for all QMSs generated by L, L1, L2. Proof. We begin by observing that for (εn + ω,εn), (εn,εn −ω) ∈ Bω, we can compute directly (Lj,ω)∗(Pεn ) =Γ − j,ω(Pεn−ωDjPεnD ∗ jPεn−ω −PεnD ∗ i Pεn−ωDjPεn )+ Γ+j,ω(Pεn+ωD ∗ jPεnDjPεn+ω −PεnDjPεn+ωD ∗ jPεn ). A state of the form ρ = ∑ n ρεnPεn , which is a function of the system Hamiltonian HS (also called a diagonal state), satisfies L∗j(ρ) = ∑ ω ∑ n (Lj,ω)∗(ρεnPεn ) = ∑ ω ∑ (εn+ω,εn)∈Bω (ρεn+ωΓ − j,ω −ρεn Γ + j,ω)PεnDjPεn+ωD ∗ jPεn + ∑ ω ∑ (εn,εn−ω)∈Bω (ρεn−ωΓ + j,ω −ρεn Γ − j,ω)PεnD ∗ jPεn−ωDjPεn. Now if β1 = β2 = β and ρεn = e −βεn as in (2.7), we have Γ+j,ω Γ−j,ω = γ+j,ω γ−j,ω = e−βω = ρεn+ω ρεn , for all j = 1, 2, so that L∗j(ρ) = 0 and ρ = e−βHS/Zβ is an invariant state for the QMS generated by Lj. Since L = L1 + L2 it is an invariant state also for the QMS generated by L. 126 F. Fagnola, D. Poletti, E. Sasso CUBO 23, 1 (2021) 3 Energy current The rate of energy variation in the system, in a state ρ, due to interaction with the reservoir j is tr (ρLj(HS)) (see [25] (V.28)). Therefore tr (ρL1(HS)) − tr (ρL2(HS)) (3.1) is twice the rate at which the energy flows through the system from the hotter bath to the colder bath, namely, the energy current through the system. Adapting a result by Lebowitz and Spohn [25] Theorem 2 and Corollary 1, it is possible to prove that the energy current is non-negative for finite dimensional systems. Theorem 3.1. Suppose that h is finite dimensional and let ρ be a faithful invariant state, then the energy current (3.1) is non-negative. Proof. If a system is weakly coupled to a single bath j at inverse temperature βj, it is well-known that the Gibbs state ρβj = Z −1 βj e−βjHS , with Zβj = tr ( e−βjHS ) , is invariant. Consider the relative entropy of ρ with respect to ρβj defined by S(ρ|ρβj ) = tr ( ρ(log(ρ− log ρβj ) ) which is a notoriously non-increasing function (see [23], Theorem 1.5), i.e. S ( T j∗t(ρ)|T j ∗t(ρβj ) ) ≤ S(ρ|ρβj ), for all ρ and t ≥ 0. States T j∗t(ρ), j = 1, 2 will still be faithful for small t, therefore no problem arises when considering logarithms. Since ρβj is invariant, denoting ρt := T j ∗t(ρ), and differentiating we find d dt S(ρt|ρβj ) = d dt tr ( ρt(log ρt − log ρβj ) ) = tr ( ρ′t(log ρt − log ρβj ) ) + tr ( ρt d dt log ρt ) . Since for every x > 0, log x = ∫ +∞ 0 ( 1 1+s − 1 x+s ) ds, d dt log ρt = ∫ +∞ 0 (s + ρt) −1ρ′t(s + ρt) −1ds so that tr ( ρt d dt log ρt ) = tr ( ρ′t ∫ +∞ 0 ρt(s + ρt) −2ds ) = tr (ρ′t) = 0. By imposing ρβj = Z −1 βj e−βjHS , and recalling that ρ′t = L∗j(ρt), tr (ρ′t) = 0 by trace preservation, we obtain d dt S(ρt|ρβj ) = tr ( ρ′t(log ρt − log ρβj ) ) = tr ( ρ′t(log ρt + βjHS − log Z −1 βj ) ) = tr (ρ′t log ρt) + βjtr (ρtLj(HS)) . CUBO 23, 1 (2021) Energy transfer in open quantum systems weakly coupled . . . 127 In particular tr (ρ′t(log ρt)) + βjtr (ρtL(HS)) ≤ 0 by monotonicity of the relative entropy, namely −tr (L∗j(ρt) log ρt) −βjtr (ρtLj(HS)) ≥ 0. In our context, the entropy production of the system due to interaction with the bath at inverse temperature βj is − tr (L∗j(ρt) log ρt) −βjtr (ρtLj(HS)) ≥ 0. (3.2) Now, for all β,β1,β2 and ρ stationary state for the system S interacting with both baths, By taking a sum over j of the inequality before (3.2), we obtain β1tr (ρL1(HS)) + β2tr (ρL2(HS)) ≤ 0. Moreover, tr (ρL1(HS)) = −tr (ρL2(HS)) and so (β2 −β1)tr (ρL2(HS)) ≥ 0 In view β1 ≥ β2, we have tr (ρL1(HS)) = −tr (ρL2(HS)) ≥ 0 and the proof is complete. In this section we prove a general explicit formula for the energy current in a stationary state ρ which is a function of the system Hamiltonian HS. This not only confirms that it is positive also for possibly infinite dimensional systems if the eigenvalues of stationary state are a monotone system (i.e. there are no population inversions), but it allows us to establish proportionality to the difference of bath temperatures when they are not too small, namely an approximate Fourier law. Lemma 3.2. For all ω ∈ B and j = 1, 2 we have Q−j,ω(HS) = −ω D ∗ j,ωDj,ω Q + j,ω(HS) = ω Dj,ωD ∗ j,ω (3.3) and Lj(HS) = ∑ ω∈B ω ( Γ+j,ωDj,ωD ∗ j,ω − Γ − j,ωD ∗ j,ωDj,ω ) . (3.4) Proof. Writing HS as in (2.1) we compute Q−j,ω(HS) = − 1 2 D∗j,ωDj,ωHS + D ∗ j,ωHSDj,ω − 1 2 HSD ∗ j,ωDj,ω = ∑ (εn,εm)∈Bω ( εm PεnD ∗ jPεmDjPεn −εn PεnD ∗ jPεmDjPεn ) = − ∑ (εn,εm)∈Bω ωPεnD ∗ jPεmDjPεn = −ω D∗j,ωDj,ω. The proof of the other identity (3.3) is similar. Since [Hj,ω,HS] = 0 for all j,ω, (3.4) follows immediately. 128 F. Fagnola, D. Poletti, E. Sasso CUBO 23, 1 (2021) We can now prove our formula for the energy current in a stationary state ρ which is a function of the system Hamiltonian HS. We suppose that the interaction of the system with both reservoirs is similar; this property is reflected by the assumptions on tr ( PεnD ∗ jPεmDj ) and f1,ω. In the sequel, to simplify the notation we also write ρn instead of ρεn . Theorem 3.3. For any state ρ which is a function of the system Hamiltonian HS, i.e. ρ = ∑ n≥0 ρnPεn (3.5) we have tr (ρLj(HS)) = ∑ ω∈B ω ∑ (εn,εm)∈Bω ( Γ+j,ωρm − Γ − j,ωρn ) tr ( PεnD ∗ jPεmDj ) . (3.6) If the state ρ is also stationary and, moreover, (1) tr (PεnD ∗ 1PεmD1) = tr (PεnD ∗ 2PεmD2) for all n,m, (2) f1,ω = f2,ω for all ω, then tr (ρL1(HS)) = 1 2 ∑ ω∈B ω f1,ω ( γ+1,ω −γ + 2,ω ) ∑ (εn,εm)∈Bω (ρm −ρn) tr (PεnD ∗ 1PεmD1) . (3.7) Proof. The proof of (3.6) is immediate from (3.4) and the following identities (cyclic property of the trace) tr ( PεmDj,ωPεnD ∗ j,ω ) = tr ( (PεmDj,ω)PεmD ∗ j,ω ) = tr ( PεnD ∗ j,ωPεmDj,ω ) . If the state ρ is stationary, then tr (ρL1(HS)) = tr (ρL(HS)) − tr (ρL2(HS)) = −tr (ρL2(HS)), so that tr (ρL1(HS)) = (tr (ρL1(HS)) − tr (ρL2(HS))) /2. Computing the right-hand side difference by means of (3.6) with j = 1, 2 we can write 2tr (ρL1(HS)) as∑ ω∈B ω f1,ω ∑ (εn,εm)∈Bω ( γ+1,ωρm −γ − 1,ωρn −γ + 2,ωρm + γ − 2,ωρn ) tr (PεnD ∗ 1PεmD1) = ∑ ω∈B ω f1,ω ∑ (εn,εm)∈Bω ( (γ+1,ω −γ + 2,ω)ρm − (γ − 1,ω −γ − 2,ω)ρn ) tr (PεnD ∗ 1PεmD1) . Since γ−j,ω = γ + j,ω + 1 for all j,ω, then γ + 1,ω −γ + 2,ω = γ − 1,ω −γ − 2,ω and (3.7) follows. Remark. Note that the above identity tr (PεnD ∗ 1PεmD1) = tr (PεnD ∗ 2PεmD2) holds whenever there exists an isometry R on h, commuting with HS, such that D2 = RD1R ∗. Indeed, in this case, R commutes with all spectral projections of HS and tr (PεnD ∗ 2PεmD2) = tr (PεnRD ∗ 1R ∗PεmRD1R ∗) = tr (PεnD ∗ 1PεmD1R ∗R) = tr (PεnD ∗ 1PεmD1) . CUBO 23, 1 (2021) Energy transfer in open quantum systems weakly coupled . . . 129 We will see later (Section 5) that this happens when the system interacts in the same way with the two baths. Formula (3.7) can be applied to effectively compute the energy current in several models highlighting the dependence on the difference of temperatures. Indeed, one readily sees that, for β1,β2 very close the term ω ( γ+1,ω −γ + 2,ω ) is an infinitesimum of order β−11 −β −1 2 while the other terms are close to some nonzero values. Moreover, it is also clear from (3.7) that the energy current is non-negative whenever the invariant state satisfies ρm > ρn for all n,m such that εm < εn i.e. population inversion does not occur. However, in order to find more explicit formulae we need additional information on the invari- ant state. This problem will be studied in the next section. We end this section by the following example Example 3.4. Let h = Cn+1 with orthonormal basis (ek)0≤k≤n. Consider an n-level system with Hamiltonian HS = n∑ k=0 k|ek〉〈ek| and interaction operators D1,D2 acting as Djek = ek−1 for k = 1, . . . ,n Dje0 = 0. Clearly B = {1, 2, . . . ,n} but the only nonzero Dj,ω are those corresponding to the frequency ω = 1 and D1,1 = D1, D2,1 = D2. Moreover, since εk = k, tr ( PεkD ∗ 1Pεk−1D1 ) = tr ( PεkD ∗ 2Pεk−1D2 ) = 1 for k = 1, . . . ,n. By Theorem 3.3 formula (3.6) we have tr (ρLj(HS)) = n−1∑ k=0 ( Γ+j,1ρk − Γ − j,1ρk+1 ) . If all Γ±j,1 (j = 1, 2) are nonzero, a straightforward computation shows that the unique stationary state is ρ = 1 −ν 1 −νn+1 n∑ k=0 νk|ek〉〈ek|, ν := Γ+1,1 + Γ + 2,1 Γ−1,1 + Γ − 2,1 and the energy current due to interaction with reservoir j is tr (ρLj(HS)) = 1 −ν 1 −νn+1 n−1∑ k=0 ( Γ+j,1ν k − Γ−j,1ν k+1 ) = 1 −νn 1 −νn+1 ( Γ+j,1 −νΓ − j,1 ) . Note that, dropping the index 1 corresponding to the unique effective frequency ω to simplify the 130 F. Fagnola, D. Poletti, E. Sasso CUBO 23, 1 (2021) notation, we have Γ+j −νΓ − j = Γ − j ( Γ+j Γ−j − Γ+1 + Γ + 2 Γ−1 + Γ − 2 ) = Γ−j ( γ+j γ−j − f1 γ + 1 + f2 γ + 2 f1 γ − 1 + f2 γ − 2 ) = Γ−j ( e−βj − f1 (e β2 − 1) + f2 (eβ1 − 1) f1 eβ1 (eβ2 − 1) + f2 eβ2 (eβ1 − 1) ) = Γ−j ( e−βj − f1 e −β1 (1 − e−β2 ) + f2e−β2 (1 − e−β1 ) f1(1 − e−β2 ) + f2(1 − e−β1 ) ) . For j = 1 we find Γ+j,1 −νΓ − j,1 = Γ − j f2(1 − e −β1 ) e−β1 − e−β2 f1(1 − e−β2 ) + f2(1 − e−β1 ) and so tr (ρL1(HS)) = 1 − ((Γ+1 + Γ + 2 )/(Γ − 1 + Γ − 2 )) n 1 − ((Γ+1 + Γ + 2 )/(Γ − 1 + Γ − 2 )) n+1 Γ−1 f2(1 − e −β1 )(e−β1 − e−β2 ) f1(1 − e−β2 ) + f2(1 − e−β1 ) Since Γ+j < Γ − j , this formula, for n big and β1,β2 small becomes tr (ρL1(HS)) ≈ f1f2(e −β1 − e−β2 ) f1(1 − e−β2 ) + f2(1 − e−β1 ) ≈ f1f2(β2 −β1) f2β1 + f1β2 = f1f2 ( 1 β1 − 1 β2 ) f1 β1 + f2 β2 showing that, in a certain regime of high temperature a Fourier law holds for all choices f1,f2 of the interactions strength. 4 Dependence of the energy current from temperature dif- ference and conductivity In this section we consider systems whose Hamiltonian HS has simple spectrum, namely each spectral projection Pεn is one-dimensional, and make explicit the dependence of the energy current on the difference of temperatures 1/β1 and 1/β2. We begin by noting that, if spectral projections Pεn are one-dimensional one can associate with the open quantum system a classical (time continuous) Markov chain with state space V the CUBO 23, 1 (2021) Energy transfer in open quantum systems weakly coupled . . . 131 spectrum sp(HS) of HS in a canonical way. Indeed, for every bounded function f on V , we have L(f(HS)) = ∑ n≥0 f(εn)L(Pεn ) = ∑ ω∈B, (εn,εm)∈Bω  ∑ j Γ−j,ωPεnD ∗ jPεmDjPεn   (f(εm) −f(εn)) + ∑ ω∈B, (εn,εm)∈Bω  ∑ j Γ+j,ωPεmDjPεnD ∗ jPεm   (f(εn) −f(εm)) and we find a classical Markov chain with transition rate matrix Q = (qnm) qnm =   ∑ j Γ − j,εn−εm tr ( D∗jPεmDjPεn ) , if εn > εm,∑ j Γ + j,εm−εn tr ( DjPεmD ∗ jPεn ) , if εn < εm, − ∑ m 6=n qnm, if n = m. Now, if we consider the conditional expectation E : B(h) → `∞(V ; C), E(x) = ∑ m≥0 PεmxPεm, where `∞(V ; C) is the abelian algebra of bounded functions on V , we have that E ◦L = L◦E. (4.1) Therefore, by defining the predual map E∗ such that tr (E∗(ρ)x) = tr (ρE(x)), if ρ is an invariant state, we have also 0 = E∗(L∗(ρ)) = L∗(E∗(ρ)) and (πn) 7→ ∑ n≥0 πnPεn gives a one-to-one correspondence between diagonal invariant states of the open quantum system and invariant measures of the associated Markov chain. In the following, in order to have at hand an explicit formula for the invariant measure, we suppose, for simplicity, that the graph associated with the above Markov chain is a path graph and jumps can occur only to nearest neighbour levels, namely qnm = 0 for |n − m| ≥ 2. This assumption may hold, for instance, if the Hamiltonian HS is generic in the sense of [8], namely it is not only non-degenerate but also if εn−εm = εn′ −εm′ then εn = εn′ and εm = εm′. Moreover, we assume that qnm 6= 0 for |n−m| ≤ 1. In this case the associated classical Markov chain has a simpler structure allowing one to make explicit computations and describe explicitly the structure of invariant states (see also [11] in a more general situation). The explicit expression for the invariant state is ρ = ∑ n ρnPεn where ρn = ∏ 0≤k 0. Proof. Note that 1/(eβ1ω − 1) − 1/(eβ2ω − 1) ≤ 1/(β1ω) − 1/(β2ω) because the function x 7→ 1/(exω − 1) − 1/(ωx) is increasing on ]0, +∞[ since d dx ( 1 exω − 1 − 1 ωx ) = 1 ωx2 − ω( eωx/2 − e−ωx/2 )2 ≥ 0 by the elementary inequality eωx/2 − e−ωx/2 ≥ ωx. Moreover, by another elementary inequality 1 − e−βjω ≤ βjω, we have eβ1ω eβ1ω − 1 + eβ2ω eβ2ω − 1 = 1 1 − e−β1ω + 1 1 − e−β2ω ≥ 1 β1ω + 1 β2ω and the second inequality (4.4) follows. In order to prove the first inequality we first write the right-hand side as (eβ1ω − 1)−1 − (eβ2ω − 1)−1 eβ1ω(eβ1ω − 1)−1 + eβ2ω(eβ2ω − 1)−1 = eβ2ω − eβ1ω eβ1ωeβ2ω/2(eβ2ω/2 − e−β2ω/2) + eβ2ωeβ1ω/2(eβ1ω/2 − e−β1ω/2) = e−(β1+β2)ω/2 e(β2−β1)ω/2 − e−(β2−β1)ω/2 (1 − e−β2ω) + (1 − e−β1ω) CUBO 23, 1 (2021) Energy transfer in open quantum systems weakly coupled . . . 133 Noting that e(β2−β1)ω/2 − e−(β2−β1)ω/2 ≥ 1 + (β2 −β1)ω 2 − ( 1 − (β2 −β1)ω 2 ) (1 − e−β2ω) + (1 − e−β1ω) ≤ (β1 + β2) ω we find (eβ1ω − 1)−1 − (eβ2ω − 1)−1 eβ1ω(eβ1ω − 1)−1 + eβ2ω(eβ2ω − 1)−1 ≥ e−(β1+β2)ω/2 (β2 −β1) ω (β1 + β2) ω . This completes the proof. Remark. Note that the inequalities of Lemma 4.1 provide a sharp estimate in terms of the inverse temperature difference β2−β1 for small β1,β2, i.e. when the average of temperatures T1,T2 is big. Indeed, the difference of the right-hand side and left-hand side is equal to ( 1 − e−(β1+β2)ω/2 ) β2 −β1 β1 + β2 and for temperatures Tj > kB ·180 K= 2.49·10−21 J (approximately the lowest natural temperature ever recorded at ground level) we have βj < 1/(kB · 180 K) = 4.02 · 1020 J−1 so that the quantity that multiplies β2 −β1 is 1 β1 + β2 < 1.24 · 10−21J. Theorem 4.2. Suppose that (1) tr ( PεnD ∗ jPεmDj ) = 1 for all n,m and all j = 1, 2, (2) fj,ω = 1 for all ω and all j = 1, 2, (3) Jumps can occur only to nearest neighbour levels, (4) Formula (4.3) holds so that the state ρ defined by (4.2) with ρ0 determined by the normal- ization condition is invariant. Then κm 1 β1 − 1 β2 1 β1 + 1 β2 κ(ρ,HS) ≤ tr (ρL1(HS)) ≤ 1 β1 − 1 β2 1 β1 + 1 β2 κ(ρ,HS) (4.5) where κm = infm≥0 e −(β1+β2)(εm+1−εm)/2 and ĤS = ∑ m≥0 εm+1Pεm, κ(ρ,HS) = tr ( ρ(ĤS −HS) ) . 134 F. Fagnola, D. Poletti, E. Sasso CUBO 23, 1 (2021) Proof. By applying (3.7) in this context, we have tr (ρL1(HS)) = 1 2 ∑ n≥0 (εn+1 −εn)(ρn −ρn+1) ( Γ+1,εn+1−εn − Γ + 2,εn+1−εn ) = 1 2 ∑ n≥0 (εn+1 −εn) ( 1 − qn,n+1 qn+1,n ) ρn ( Γ+1,εn+1−εn − Γ + 2,εn+1−εn ) = ∑ n≥0 (εn+1 −εn)ρn Γ+1,εn+1−εn − Γ + 2,εn+1−εn Γ−1,εn+1−εn + Γ − 2,εn+1−εn . Now the proof follows applying Lemma 4.1 with ω = εn+1 − εn to estimate the right-hand side ratio. Remark. Formula (4.5) shows that the energy current tr (ρL1(HS)) has an explicit dependence on the difference β−11 −β −1 2 of the reservoirs’ temperatures. This dependence holds only through two inequalities, but it suggests the existence of an “approximate” Fourier law (see [4, 21]) for the current. Clearly there can be further dependecies through the term κ(ρ,HS), however it holds inf k (εk+1 −εk) ≤ κ(ρ,HS) ≤ sup k (εk+1 −εk) . Therefore the energy current depends on the temperature difference mainly through the explicit term and one could say that there really is an “approximate” Fourier Law. Furthermore it is worth noticing that, for β1,β2 fixed, the inequality (4.5) is better the smaller is supm≥0(εm+1 − εm) so that κm is close to 1 and the inequalities are approximately equalities. However, it should also be noted that, in this case, κ(ρ,HS) becomes small as well. Eventually note that, due to the nature of our system, we cannot investigate spatial properties of energy flow. Therefore our discussion of the Fourier’s law is concerned with proportionality to temperature difference and not with dependency on size. Remark. Since the above QMS are of weak coupling limit type, one can write explicitly the entropy production (in the sense of [15, 16]). It is tempting to study in detail what happens when supm≥0(εm+1 − εm) tends to 0 so that the eigenvalues of HS increase in number and form a set more and more packed. In a more precise way, for all n ≥ 1 we assume that the system Hamiltonian is a self-adjoint operator H(n)S on an (n + 1)-dimensional Hilbert space h with simple pure point spectrum ( ε (n) k ) 0≤k≤n with ε0 = 0 and, for all a,b with 0 ≤ a < b ≤ +∞, we have lim n→∞ card { k | a < ε(n)k ≤ b } n = µ(]a,b]) (4.6) for some continuous probability density µ on [0, +∞[. In other words, the empirical distribution of eigenvalues of H (n) S converges weakly to a probability distribution on [0, +∞[ . Suppose, for simplicity, that µ has no atoms, i.e. µ({r}) = 0 for all r ≥ 0. CUBO 23, 1 (2021) Energy transfer in open quantum systems weakly coupled . . . 135 We can now prove the following result on the distribution of eigenvalues of the stationary state and energy in stationary conditions. Theorem 4.3. Under the assumptions of Theorem 4.2, for all n ≥ 1, let H(n)S be as above and suppose that (4.6) holds. Let ρ(n) be the invariant state (4.2) and let β̃ = 2 (1/β1 + 1/β2) −1 be the harmonic mean of the inverse temperatures (i.e. β̃−1 arithmetic mean of β−11 ,β −1 2 ). (i) Eigenvectors ρ (n) k of ρ (n) satisfy lim n→∞ ∑ {k |a<εk≤b} ρ (n) k = ∫ b a e−β̃rdµ(r)∫ ∞ 0 e−β̃rdµ(r) (ii) The average energy in the system satisfies lim n→∞ tr ( ρ(n)H (n) S ) = ∫ ∞ 0 e−β̃r rdµ(r)∫ ∞ 0 e−β̃rdµ(r) . This result reminds the one in [20] where the steady state can be described by a general- ized Gibbs state and the steady-state current is proportional to the difference in the reservoirs’ magnetizations. In the proof we need the following Lemma. Lemma 4.4. Let β̃ = 2/ ( β−11 + β −1 2 ) be the harmonic mean of inverse temperatures (i.e. β̃−1 is the arithmetic mean of β−11 and β −1 2 ). For all 1 ≤ k ≤ n and for supj ωj < 1/(3β2), 1 − β̃ ωk ≤ qk,k+1 qk+1,k ≤ 1 − β̃ ωk + ( β̃ ωk )2 (4.7) where ωk = εk+1 −εk and e−β̃εk(1+β̃ supj ωj ) ≤ k−1∏ j=0 qj,j+1 qj+1,j ≤ e−β̃εk(1−β̃ supj ωj ) (4.8) Proof. By the elementary inequality 1 − e−βjωk ≤ βjωk we have qk,k+1 qk+1,k = 1 eβ1ωk − 1 + 1 eβ2ωk − 1 eβ1ωk eβ1ωk − 1 + eβ2ωk eβ2ωk − 1 = 1 − 2 (1 − e−β1ωk )−1 + (1 − e−β2ωk )−1 ≥ 1 − 2ωk 1/β1 + 1/β2 136 F. Fagnola, D. Poletti, E. Sasso CUBO 23, 1 (2021) In the same way, by the elementary inequalities 1−e−βjωk ≥ βjωk−(βjωk) 2 /2 and 1/(1 −βjωk/2) ≤ 1 + βjωk, we find for βjωk < 1 qk,k+1 qk+1,k ≤ 1 − 2ωk 1/ (β1 (1 −β1ωk/2)) + 1/ (β2 (1 −β2ωk/2)) ≤ 1 − 2ωk 1/β1 (1 + β1ωk/2) + 1/β2 (1 + β2ωk/2) ≤ 1 − 2ωk 1/β1 + 1/β2 + 2ωk = 1 − β̃ ωk 1 + β̃ ωk and so (4.7) follows. In order to prove the upper bound in (4.8), note that, since log(1 −x) ≤−x log  k−1∏ j=0 qj,j+1 qj+1,j   ≤ k−1∑ j=0 log ( 1 − β̃ ωj ( 1 − β̃ ωj )) ≤− k−1∑ j=0 β̃ωj ( 1 − β̃ωj ) , as a consequence log  k−1∏ j=0 qj,j+1 qj+1,j   ≤−k−1∑ j=0 β̃ωj ( 1 − β̃ sup l ωl ) = −β̃εk ( 1 − β̃ sup l ωl ) . For the lower bound, we begin by the inequality log  k−1∏ j=0 qj,j+1 qj+1,j   = k−1∑ j=0 log ( qj,j+1 qj+1,j ) ≥ k−1∑ j=0 log ( 1 − β̃ωj ) . Note that log(1 −x) + x + x2 ≥ 0 for 0 ≤ x ≤ 2/3 and, since β̃ωj < 2/3 by our assumption, we have log  k−1∏ j=0 qj,j+1 qj+1,j   ≥−k−1∑ j=0 β̃ωj ( 1 + β̃ sup l ωl ) = −β̃�k ( 1 + β̃ sup l ωl ) . This completes the proof. Proof of Theorem 4.3. Let µn be the empirical distribution of the eigenvalues of H (n) S i.e. µn = 1 n + 1 n∑ k=0 δεk and note that ∑ {k |a<εk≤b} ρ (n) k = 1 n + 1 ∑ {k |a<εk≤b} k−1∏ j=0 qj,j+1 qj+1,j 1 n + 1 n∑ k=0 k−1∏ j=0 qj,j+1 qj+1,j . (4.9) CUBO 23, 1 (2021) Energy transfer in open quantum systems weakly coupled . . . 137 Clearly, by Lemma 4.4, 1 n + 1 ∑ {k |a<εk≤b} k−1∏ j=0 qj,j+1 qj+1,j ≤ 1 n + 1 ∑ {k |a<εk≤b} e−β̃εk(1−β̃ supj ωj ) ≤ eβ̃ 2b supj ωj ∫ ]a,b] e−β̃εk dµn(r) and also 1 n + 1 ∑ {k |a<εk≤b} k−1∏ j=0 qj,j+1 qj+1,j ≥ e−β̃ 2a supj ωj n + 1 ∑ {k |a<εk≤b} e−β̃εk = e−β̃ 2a supj ωj ∫ ]a,b] e−β̃εk dµn(r). Since supj ωj goes to 0, probability measures µn converge weakly to µ and the function r → e−β̃r is bounded continuous on [0, +∞[, taking the limit as n →∞, we have lim n→∞ 1 n + 1 ∑ {k |a<εk≤b} k−1∏ j=0 qj,j+1 qj+1,j = ∫ ]a,b] e−β̃εk dµ(r). In the same way, taking a = 0 and b = +∞, we see that the denominator of (4.9) converges to∫ +∞ 0 e−β̃r dµ(r) and the proof of (i) is complete. The proof of (ii) is similar. � Remark. Theorem 4.3 (i) shows that, if µ has density µ′, then the asymptotic distribution of eigenvalues of the stationary state is λ 7→ e−β̃λµ′(λ)∫ +∞ 0 e−β̃rµ′(r)dr . The asymptotic average energy in the system can be easily computed in some remarkable cases noting that the integral of e−β̃r with respect to µ is the moment generating function φ of µ evaluated at −β̃ and so the asymptotic average energy in the system is − d dβ̃ φ(−β̃) φ(−β̃) = − d dβ̃ log ( φ(−β̃) ) . We can easily find an explicit result in two cases: µ normal distribution N(m,σ2) average energy m− β̃σ µ gamma distribution Γ(α,θ) average energy α/(β̃ + θ) The asymptotic average energy in the system is decreasing in β̃, i.e. increasing in the average temperature as expected, for all probability measure µ because the moment generating function of a probability distribution is log-convex and the derivative of a convex function is increasing. Remark. Note that, by choosing a suitable spacing of eigenvalues εn we can control the rate of convergence to 0 of κ ( ρ(n),H (n) S ) at will, as n tends to +∞. 138 F. Fagnola, D. Poletti, E. Sasso CUBO 23, 1 (2021) 5 One dimensional Ising chain In this section we consider a one-dimensional Ising chain with nearest neighbour interaction. We will show that, in this case, if the heat baths interact locally at both ends of the chain, then the energy current is zero. Spin interaction (see 5.1) occurs only in the z component. In the case where also the other components interact the derivation of the GKSL generator turns out to be really difficult (see [5]). Indeed, starting from the diagonalized HS, one finds a cumbersome expression for the operators Dω. In spite of the simple system Hamitonian HS (5.1) Theorems 4.2 and 4.3 do not apply to this model because its spectrum is degenerate. The system space is h = C2⊗N with N > 2. Define Pauli matrices σx =   0 1 1 0   σy =   0 −i i 0   σz =   1 0 0 −1   with respect to the orthonormal basis e+ = [1, 0] T, e− = [0, 1] T of C2. Consider the one dimensional Ising chain with Hamiltonian HS = Jz N−1∑ j=1 σzjσ z j+1, Jz > 0, N > 2 (5.1) Subsequently let us define eα := ⊗Nj=1eα(j), α ∈{−1, 1} N, as a basis of h, where e−1 := e− and e+1 := e+. Vectors {eα}α form an eigenbasis for HS and the spectrum is sp(HS) = {Jz (2k − (N − 1)) | k = 0, . . . ,N − 1}. The eigenspace associated with the eigenvalue εk = Jz(2k−(N −1)) is the linear span of the elements eα such that exactly k neighbouring elements in α have the same sign. Thus one can define the sets Ak := { α ∈{−1, 1}N ∣∣∣ ∑N−1j=1 α(j)α(j + 1) = 2k − (N − 1)} , and the spectral projection associated with the eigenvalue εk is given by Pk := ∑ α∈Ak |eα〉〈eα|. The system is coupled with two heat reservoirs at inverse temperature β1,β2 with β1 ≤ β2 through the interaction H1 = σ u 1 ⊗ (A −(φ1) + A +(φ1)), H2 = σ v N ⊗ (A −(φ2) + A +(φ2)), (5.2) CUBO 23, 1 (2021) Energy transfer in open quantum systems weakly coupled . . . 139 where u,v ∈ R3 and σui is defined as σui = u1σ x i + u2σ y i + u3σ z i . The set of positive Bohr frequencies is given by B := {2Jz(n−m) = εn −εm | n,m ∈{0, . . . ,N − 1}, n > m}, while the operators Dj,ω are given by (2.5). Thus one has D1,2Jz = (u1 − iu2) ∑ α∈Cl ++ σx1 |eα〉〈eα| + (u1 + iu2) ∑ α∈Cl−− σx1 |eα〉〈eα| where Cl++ (resp. C l −−) denotes the set of configurations α ∈{−1, +1}N with ++ (resp. −−) in the first two sites (l stands for left). While D1,ω = 0 for every ω ∈ B −{2Jz} because the Pauli matrices act only on the first site and so the number of neighbouring sites with the same sign can vary of at most one after the action of σu1 and for ω = 2Jz one has D1,2Jz = N−1∑ n=1 ∑ α∈An ∑ β∈An+1 〈eα,σx1eβ〉 |eα〉〈eβ|. With similar arguments one can see that D2,ω = 0 for every ω ∈ B−{2Jz}, while D2,2Jz = (v1 − iv2) ∑ α∈Cr ++ σxN|eα〉〈eα| + (v1 + iv2) ∑ α∈Cr−− σxN|eα〉〈eα| where Cr++ (resp. C r −−) denotes the set of configurations with ++ (resp. −−) in the last two sites (r stands for right). From now on we will drop the subscript 2Jz and only deal with operators related to that Bohr frequency, as the others vanish. Recalling the definition of linear maps (2.6) and the constants γ+i = 1/(e 2Jzβi − 1), γ−i = e 2Jzβi/(e2Jzβi − 1), we can write the GKSL generator of the evolution as follows L = ∑ i∈{1,N} γ−i Q − i + γ + i Q + i . A close scrutiny at the operators Di,D ∗ i shows that, for each fixed configuration α ∈{−1, +1} N−2 of the N − 2 inner sites of the chain the 4-dimensional projections pα on subspaces hα := span{eα | α(j) = α(j) for all 2 ≤ j ≤ N − 1; α(1),α(N) ∈{−1, 1}} commute with both Di and D ∗ i for i ∈ {1,N}, then subalgebras pα1B(h)pα2 are invariant for the semigroup T generated by L. This commutation allows us to restrict our study only to cases where 140 F. Fagnola, D. Poletti, E. Sasso CUBO 23, 1 (2021) the invariant state is of the form ρ = ∑ α∈{−1,1}N−2 pαρpα = ∑ α∈{−1,1}N−2 λαρα, (5.3) where ρα is an invariant state supported only on hα and λα are real constants that sum up to 1. Indeed the off diagonal terms, pα1ρpα2 with α1 6= α2, do not contribute to current flow, since tr (pα1ρpα2L1(HS)) = tr (pα1ρL1(HS)pα2 ) = 0. Moreover all the conditional expectations Eα(x) := pαxpα commute with L, ensuring that both∑ αEα,∗(ρ) and every Eα,∗(ρ) must also be invariant states on their own. As a further refinement we can repeat the same argument using the conditional expectation E(x) := ∑N−1 k=0 PkxPk. Indeed E commutes with the Lindbladian L and tr (Pk1ρPk2L1(HS)) = tr (Pk1ρL1(HS)Pk2 ) = 0 for k1 6= k2, since the spectral projections commute with DjD∗j , D ∗ jDj and L1(HS) is a linear combination of these operators by Lemma 3.2, equation (3.3). In this way we can focus our study on invariant states of the form (5.3) with pαρpα = ρα =   ρα11 0 0 0 0 ρα22 ρ α 23 0 0 ρα32 ρ α 33 0 0 0 0 ρα44   , where we expanded the state with respect to the basis of four vectors ecαc,edαc, ecαd,edαd defined as follows: ecαc is the vector eα(2),α(2),...,α(N−1),α(N−1), ecαd = eα(2),α(2),...,α(N−1),−α(N−1) and vectors edαc, edαd are defined in a similar way. Now we have reduced and simplified the class of states we want to use when looking for a invariant state, without, however, losing any contribution to the current flow. In order to find the invariant state, first of all it is not too difficult to show that L∗ leaves invariant the subspace of diagonal elements. Then compute L∗(ρα23|edαc〉〈ecαd|) = − 1 2 [ Γ+1 + Γ − 1 + Γ + N + Γ − N ] ρα23|edαc〉〈ecαd|, and similarly L∗(ρα32|ecαd〉〈edαc|) = − 1 2 [ Γ+1 + Γ − 1 + Γ + N + Γ − N ] ρα32|ecαd〉〈edαc|, where Γ±1 = ‖u1 + iu2‖ 2 γ±1 and Γ ± N = ‖v1 + iv2‖ 2 γ±N . (The above Γ ± i slightly differ from the constants in Section 2). Therefore the invariant state condition L∗(ρ) = 0 implies ρα23 = ρα32 = 0. CUBO 23, 1 (2021) Energy transfer in open quantum systems weakly coupled . . . 141 We can now just consider the reduced dynamics on diagonal elements of pαB(h)pα, given by L∗ =   −(Γ−1 + Γ − N ) Γ − 1 Γ − N 0 Γ+1 −(Γ + 1 + Γ − N ) 0 Γ − N Γ+N 0 −(Γ + N + Γ − 1 ) Γ − 1 0 Γ+N Γ + 1 −(Γ + N + Γ + 1 )   , The unique invariant law for the time-continuous Markov chain generated by the above matrix is π = Z−1 [ 1, e2Jzβ1, e2Jzβ2, e2Jz(β1+β2) ] , where Z−1 is a normalization constant that is independent of u,v and is the same for all α. Therefore the unique T -invariant state supported on hα is ρα = Z −1 ( |ecαc〉〈ecαc| + e2Jzβ1|edαc〉〈edαc| + e2Jzβ2|ecαd〉〈ecαd| + e2Jz(β1+β2)|edαd〉〈edαd| ) . Recalling (5.3) we can now write any invariant state for the semigroup T . We can now evaluate the energy flow tr (ρL1(HS)) via the expression L1(HS) = ∑ ω∈B+ ω ( γ+1,ωD1D ∗ 1 −γ − 1,ωD ∗ 1D1 ) = 2Jz ( γ+1 D1D ∗ 1 −γ − 1 D ∗ 1D1 ) that, together with the formula for ρα, yields Z tr (ρL1(HS)) = Z tr   ∑ α∈{−1,1}N−2 λαραL1(HS)   = ∑ α∈{−1,1}N−2 2Jzλα ( γ+1 e β1ω + γ+1 e (β1+β2)ω −γ−1 e β2ω −γ−1 ) = 0 Remark. For N = 2, it can be shown by direct computation that the energy current is strictly positive. Indeed, because of low dimensionality the ends of the chain can interact directly. Acknowledgement. The authors would like to thank Stefano Olla for drawing their attention to the problem of en- ergy transport in quantum systems and fruitful discussions at the workshop “Quantum Transport Equations and Applications” at Casa Matemática Oaxaca (México) in September 2018. They also gratefully thank CUBO’s anonymous referees whose remarks and comments to improve an earlier version of this paper. 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