Non-additive dissipation in open quantum networks out of equilibrium Mark T. Mitchisona) and Martin B. Plenio Institut für Theoretische Physik, Albert-Einstein Allee 11, Universität Ulm, D-89069 Ulm, Germany We theoretically study a simple non-equilibrium quantum network whose dynamics can be expressed and exactly solved in terms of a time-local master equation. Specifically, we consider a pair of coupled fermionic modes, each one locally exchanging energy and particles with an independent, macroscopic thermal reservoir. We show that the generator of the asymptotic master equation is not additive, i.e. it cannot be expressed as a sum of contributions describing the action of each reservoir alone. Instead, we identify an additional interference term that generates coherences in the energy eigenbasis, associated with the current of conserved particles flowing in the steady state. Notably, non-additivity arises even for wide-band reservoirs coupled arbitrarily weakly to the system. Our results shed light on the non-trivial interplay between multiple thermal noise sources in modular open quantum systems.
I.
INTRODUCTION
An improved understanding of the dynamics of open quantum networks is desirable for research fields including quantum thermodynamics,1–4 quantum biology,5 mesoscopic electronics6 and the theory of non-equilibrium phase transitions.7–12 The Lindblad master equation13,14 is a popular and powerful tool for modelling such systems, which approximates the long-time dynamics under the assumption of weak coupling to memoryless environments, i.e. the Born-Markov approximation (BMA). Additional, uncontrolled approximations are typically required in order to obtain a completely positive evolution, such as the secular,15 singular-couplinglimit16 or wide-band-limit17,18 approximations. However, in composite open systems, distinct approximations may lead to different and sometimes drastically incorrect predictions. In particular, standard master equations derived under the BMA describe dissipation in terms of either local processes on a small number of sites or global transitions between energy eigenstates of the entire network.19 Unfortunately, the local approach appears to violate thermodynamic laws in certain cases,20,21 while the global approach fails to capture the coherences necessary to properly describe the non-equilibrium steady state (NESS).22 This poses a particular problem for the ongoing study of quantum thermal machines, where the multifaceted role played by coherence— acting variously as a performance-enhancing resource,23–30 a useful output31–33 or an unavoidable hindrance34,35 —remains incompletely understood. Here, we aim to elucidate these issues by studying an exactly solvable model of a non-equilibrium quantum network. More precisely, we derive a time-local master equation describing a pair of coupled, localised fermionic modes. Each mode exchanges particles and energy with a macroscopic reservoir represented by a semi-infinite, uniform tightbinding chain, such that the total Hamiltonian is quadratic in fermionic ladder operators. This represents a prototypical quantum thermal machine, which could be realised by electrons flowing through a serial double quantum dot36 or cold fermionic atoms confined by an optical lattice.37,38 Note that other authors have already derived and extensively studied exact time-local master equations describing
a)
[email protected]
general quadratic Fermi systems.39–42 In contrast to these previous approaches, the simplicity and symmetry of our specific set-up enables compact analytical solutions to be obtained using elementary methods and without needing the Born-Markov, secular, singular-coupling-limit or wide-bandlimit approximations. Instead, we use an alternative, perturbative approximation scheme, valid when the systemenvironment coupling is much smaller than the bandwidth of the reservoir vacuum noise, and explicitly confirm its accuracy using the exact solution. This simplification allows us to clearly identify and distinguish the different physical processes governing the dynamics. Our analysis challenges a central tenet of standard opensystems theory: namely, that two initially uncorrelated environments should give rise to two independent, additive contributions to the master equation in the weak-coupling limit. Instead, we distinguish three contributions to the generator of asymptotic time evolution. Two of these describe the individual thermalising effect of each reservoir, while the third is an interference term arising from the combined action of both reservoirs whenever they are initially out of equilibrium with each other. This interference gives rise to coherence in the energy eigenbasis of the network, a feature which reflects the conserved fermion current flowing in the NESS. These findings connect several previous theoretical studies on composite open quantum systems. In particular, a significant body of research has sought to assess the validity of approximate, additive master equations by comparing them to each other or to exact numerical results.19–22,43–49 Viewed broadly, these investigations indicate that the local and global Lindblad equations are only accurate in limited, complementary parameter regimes, and only for certain observables, even though the requisite conditions for the BMA might hold for each bath individually. On the other hand, a few recent papers have shown that the additivity assumption fails in the presence of multiple strong or structured noise sources.50–52 We extend these results by demonstrating that interference between different thermal baths out of equilibrium gives rise to non-additive dynamics, even when the open system couples arbitrarily weakly to spectrally unstructured reservoirs. The interference contribution is not in Lindblad form and thus underlies the occurrence of asymptotic non-Markovianity found by Ribeiro et al.,42 where the longtime dynamics cannot be described by a Markovian master equation even when the open system has lost all mem-
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Non-additive dissipation in open qantum networks
ory of its initial state.53–55 Moreover, we show that our nonadditive master equation interpolates between the local and global Lindblad equations and recovers them in different limits. Since additivity is a necessary consequence of the BornMarkov approximation,52 this implies the failure of the BMA for describing steady-state transport away from these limits. Nevertheless, we find that certain observables—such as the steady-state currents flowing into the baths—are accurately predicted by an additive master equation across a relatively broad range of parameters. Our work thus helps to clarify the validity of existing Lindblad models, while providing a reference point for future research aimed at moving systematically beyond the standard approximations. The plan of the paper is as follows. Section II introduces some preliminary concepts and overviews our main results. The microscopic model that forms the core of this work is defined and exactly solved in Section III. We derive the exact master equation describing the system and develop a weak-coupling approximation scheme in Section IV. Nonadditivity of the asymptotic dynamics is explored in Section V. We discuss our results and conclude in Section VI.
II.
PRELIMINARIES
To motivate the problem at hand, consider an open quantum system S comprising a network of coupled sites. This network is coupled weakly to multiple thermal reservoirs which may exchange particles and energy with S, as illustrated in Fig. 1. Each reservoir Bα is characterised by a temperature T α = 1/βα and chemical potential µα (we work in units where kB = 1 and ~ = 1). Let Hˆ S and Nˆ S respectively be the Hamiltonian and particle number operators on S, where [Hˆ S , Nˆ S ] = 0 unless µα = 0. We assume for simplicity that Hˆ S has a non-degenerate spectrum. The quantum state of the network at time t is denoted ρˆ S (t). We suppose that initially each of the baths is not correlated with the others nor with the open system, and that S asymptotically approaches a stationary state ρˆ ∞ ˆ S (t). We now summarise the S = limt→∞ ρ behaviour that we expect in general. One fundamental property of a thermal reservoir is that a small system weakly coupled to it should eventually equilibrate to the same temperature and chemical potential. In addition, the composition of several independent thermal reservoirs in equilibrium itself constitutes a thermal reservoir. Hence, for equal reservoir temperatures and chemical potentials the system should equilibrate, i.e.
Figure 1. An open quantum system S comprising a network of interacting modules connected to different thermal reservoirs Bα .
second law of thermodynamics requires that the total rate of entropy production in the reservoirs is non-negative: X − βα JαQ ≥ 0. (2) α
Assuming that different reservoirs couple to different sites of the network, basic conservation laws imply that the transfer of energy or particles between the reservoirs can only occur via commensurate energy or particle currents flowing within the system. Therefore, there exist one or more current observables on S, here denoted schematically by JˆS , having non-zero expectation value in the NESS:
(3) JˆS ∞ , 0 unless JαE,P = 0. (We denote expectation values at time t by h•it , with h•i∞ the limiting value as t → ∞.) In a network with open (i.e. nonperiodic) boundary conditions, the eigenstates |Ei of Hˆ S generally do not support internal currents, so that hE| JˆS |Ei = 0, or more generally h JˆS it = 0 for any energy-diagonal state satisfying [ρˆ S (t), Hˆ S ] = 0. In such cases, the non-equilibrium steady state must exhibit coherence in the eigenbasis of Hˆ S .22 A widely used dynamical model of the situation depicted in Fig. 1 is the quantum master equation ∂t ρˆ S (t) = −i Hˆ S , ρˆ S (t) + Lρˆ S (t), (4)
(1)
valid for times t much greater than the environment memory time. In order for Eq. (4) to generate a completely positive and trace-preserving (CPTP) evolution for any state P ρˆ S (t), the dissipator L must be in Lindblad form13,14 L = j γ j D[Lˆ j ], ˆ ρˆ S = Lˆ ρˆ S Lˆ † − 1 {Lˆ † L, ˆ ρˆ S } and Lˆ j is a jump operator where D[L] 2 describing an incoherent transition occurring at a rate γ j . Since the reservoirs are assumed to initially be statistically independent, the standard construction of the dissipator is a sum of generators Lα representing each bath Bα , i.e. X L= Lα . (5)
ˆ Nˆ ) ˆ µ, Nˆ = Tr[e−β(H−µ Here, Z β, H, ] is the partition function. On the other hand, if the reservoirs are initially out of equilibrium with each other, the imbalance in temperature or chemical potential sets up a current of energy or particles flowing into S from Bα . These energy and particle currents are respectively denoted JαE and JαP , while JαQ = JαE − µα JαP is the corresponding heat current. In the stationary state, the
This constitutes our definition of additivity, as studied previously in Refs. 50–52. This is distinct from the concept of additive decoherence rates explored, for example, in Refs. 56 and 57. The additivity assumption permits one to unambiguously identify the particle current JαP (t) and energy current JαE (t) entering the system from Bα as
JαP (t) = L†α Nˆ S t , JαE (t) = L†α Hˆ S t . (6)
ˆ
ρˆ ∞ S =
ˆ
e−β(HS −µNS ) Z β, Hˆ S , µ, Nˆ S
if βα = β and µα = µ.
α
Mitchison & Plenio
3
Here, L†α is the adjoint generator describing the Heisenbergˆ †α A] ˆ = picture evolution of observables, defined by Tr[ BL ˆ ˆ ˆ ˆ Tr[ALα B] for arbitrary operators A and B. Regarding the specific form of the generators Lα , various inequivalent approaches are commonly employed. These can be broadly classified into two groups according to the fixed point of each generator, i.e. the state rˆα satisfying Lα rˆα = 0. The first is the “global” approach, where each generator drives the entire network towards the corresponding equilibrium state, i.e. ˆ
Lα rˆα = 0 ⇐⇒ rˆα =
ˆ
(7)
ˆ
e−βα (Hsα −µα Nsα ) Oˆ s¯α , Z βα , Hˆ sα , µα , Nˆ sα
(8)
where Hˆ sα and Nˆ sα are respectively the Hamiltonian and particle number operator of sα , while Oˆ s¯α is an arbitrary density operator with support on the complement s¯α . Clearly, rˆα is not unique in this case. Generators satisfying Eq. (8) may either be derived microscopically using various approximations,19,22,43,45 or directly postulated on phenomenological grounds.59–62 In the local approach, ρˆ ∞ S is not necessarily diagonal in the energy eigenbasis, and therefore provides a consistent model for the internal current dynamics as required by Eq. (3). However, neither Eq. (1) nor Eq. (2) hold, in general, leading to potential violations of thermodynamic laws.20,21 In what follows, we consider the simplest case of a twosite network with two independent reservoirs labelled by the index α = L, R. We show that, in the limit of weak systemreservoir coupling, the asymptotic dynamics is governed by a master equation whose dissipator takes the form L = LL + LR + Lint .
(b)
ˆ
e−βα (HS −µα NS ) . Z βα , Hˆ S , µα , Nˆ S
Note that here we assume the fixed point rˆα to be unique. Lindblad generators satisfying Eq. (7) can be derived from a microscopic model under the Born-Markov and secular approximations.15 The latter approximation is assumed to be justified in the quantum-optical regime, where the separation between energy levels of Hˆ S is much larger than their environment-induced broadening. Eq. (7) directly implies the correct equilibration behaviour (1), and also ensures—via the Spohn inequality58 —that the second law (2) holds for the heat currents from the baths. However, since the unique fixed point of each generator satisfies [ˆrα , Hˆ S ] = 0, adding together generators according to Eq. (5) leads to a master equation whose steady state is diagonal in the eigenbasis of Hˆ S ,15 which is generally inconsistent with the condition (3).22 Alternatively, one can use a “local” approach, where each generator Lα acts non-trivially only on one part of the network sα ⊂ S, driving it towards thermal equilibrium while leaving its complement s¯α unaffected. That is, Lα rˆα = 0 ⇐⇒ rˆα =
(a)
(9)
In the quantum-optical limit, LL and LR determine the currents from the baths according to Eq. (6) and induce thermalisation according to Eq. (7). Outside of the quantum-optical
Figure 2. (a) The open system S comprises two fermionic modes with local energies hL,R and tunnel coupling g. Each mode can exchange fermions with an independent reservoir BL,R at temperature T L,R and chemical potential µL,R , thus establishing stationary energy and parE,P ticle currents JL,R flowing into S. (b) The reservoirs are explicitly √ modelled as 1D tight-binding chains, with Ω and ΓΩ respectively the intra-reservoir and system-reservoir tunnelling energies.
limit, these generators drive S towards the reduction of a global thermal state (i.e. including the baths and the systembath interaction) that accounts for system-reservoir correlations.63 The interference term Lint , which appears whenever BL and BR are not in equilibrium with each other, generates coherence in eigenbasis of Hˆ S as required by condition (3). In this way, all three properties (1)–(3) can be satisfied, but only by abandoning the additivity assumption (5). Physically, non-additivity stems from correlations between the two reservoirs.50 Out of equilibrium, such correlations grow steadily in time due to the particle current flowing between the baths. Hence, Eq. (5), which is justified by the initial statistical independence of the reservoirs, fails to hold as t → ∞, even if the reservoirs are spectrally unstructured and coupled arbitrarily weakly to the system.
III.
EXACTLY SOLVABLE MODEL
A.
Description of the model
Let us now detail our specific set-up, which belongs to the well-known family of resonant-level transport models.64,65 The total Hamiltonian takes the form Hˆ = Hˆ S + Hˆ B + Hˆ SB , describing a central open system S sandwiched between two fermionic particle reservoirs BL and BR , as illustrated in Fig. 2. This models a thermoelectric tunnel junction4 or entangler,31,32 which channels a current of fermions between two conducting leads due to a temperature or chemicalpotential gradient. Specifically, S comprises two localised fermionic modes with Hamiltonian X g † Hˆ S = hα nˆ α − cˆ L cˆ R + cˆ †R cˆ L , (10) 2 α=L,R
4
Non-additive dissipation in open qantum networks
where cˆ α annihilates a fermion on site α = L, R and satisfies the anti-commutation relations {ˆcα , cˆ †α0 } = δαα0 and {ˆcα , cˆ α0 } = 0, while nˆ α = cˆ †α cˆ α . We parametrise the local energies hα by their mean h = 12 (hL +hR ) and detuning δ = hL −hR , with g the tunnel coupling between the sites. The baths are particle reservoirs described by the Hamiltonian Hˆ B = Hˆ L + Hˆ R . Each Hamiltonian Hˆ α describes a uniform chain, i.e. a one-dimensional (1D) tight-binding model on M sites with hopping amplitude Ω > 0, given explicitly by M−1 Ω X ˆ† ˆ Hˆ α = − Am,α Am+1,α + Aˆ †m+1,α Aˆ m,α 2 m=1 X = ωq aˆ †q,α aˆ q,α .
(11)
The effect of the reservoirs on the system is determined by the spectral density J(ω) =
This is a smooth function of ω in the limit M → ∞, where the spacing between adjacent wave vectors ∆q = π/(M +1) tends to zero and q becomes a continuous variable taking values in the firstRBrillouin zone q ∈ [0, π]. Using the prescription P dq, we obtain q ∆q → Γ JN (ω) = 2π
q
The ladder operators aˆ q,α describe quasi-free fermionic modes indexed by a dimensionless wave number q = πk/(M + 1), for k = 1, 2, . . . , M, with dispersion relation ωq = −Ω cos(q). The system and bath interact via tunnelling of fermions between the terminal site of each reservoir and the √ adjacent site of S. We parametrise the tunnelling energy as ΓΩ, where Γ sets the overall frequency scale of the dissipative dynamics. Explicitly, the interaction Hamiltonian reads as √ ΓΩ X † ˆ ˆ HSB = cˆ α A1,α + Aˆ †1,α cˆ α (14) 2 α=L,R XX = τq cˆ †α aˆ q,α + τ∗q aˆ †q,α cˆ α , (15) α=L,R
q
√ with tunnel couplings τq = ΓΩ/(2M + 2) sin(q). It is convenient to analyse the problem in a basis that diagonalises Hˆ S . To do this, we collect the ladder operators into column vectors cˆ = (ˆcL , cˆ R )T and aˆ q = (ˆaq,L , aˆ q,R )T . We then define a new canonical set of ladder operators dˆ = (dˆ1 , dˆ2 )T = Rcˆ and bˆ q = (bˆ q,1 , bˆ q,2 )T = Raˆ q , related by the orthogonal rotation matrix √ √ 1 √∆ + δ −√ ∆ − δ , R= √ (16) ∆−δ ∆+δ 2∆ p where ∆ = g2 + δ2 . The Hamiltonian hence splits into two independent pieces Hˆ = Hˆ 1 + Hˆ 2 , with X Hˆ j = E j dˆ†j dˆ j + ωq bˆ †q, j bˆ q, j + τq dˆ†j bˆ q, j + τ∗q bˆ †q, j dˆ j , (17) q
where E1 = h + 21 ∆ and E2 = h − 12 ∆ are the single-particle energy eigenvalues of Hˆ S . For concreteness, we √ assume that E j > 0, or equivalently that hα > 0 and g < 2 hL hR .
(18)
q
(12)
Here, Aˆ m,α annihilates a fermion localised on site m of bath Bα and satisfies {Aˆ m,α , Aˆ †m0 ,α0 } = δmm0 δαα0 and {Aˆ m,α , Aˆ m0 ,α0 } = 0. On the second line, the Hamiltonian is diagonalised by the canonical transformation r M 2 X sin(qm)Aˆ m,α . (13) aˆ q,α = M + 1 m=1
X |τq |2 δ(ω − ωq ).
r 1−
ω2 Θ(Ω − |ω|), Ω2
(19)
where Θ(x) is the Heaviside unit step function. We label this spectral density with a subscript N after Newns, who (to our knowledge) introduced it.66 According to Eq. (19), each environment is characterised by a spectral bandwidth Ω, leading to a vacuum correlation time of order Ω−1 . This is rather intuitive, since Ω sets the rate at which an excitation created at the boundary of Bα propagates irreversibly along the chain and away from the central system’s domain of influence. Note that choosing a 1D geometry for each environment is not as restrictive an assumption as it may appear. This is because an environment with arbitrary geometry can be mapped onto a 1D tight-binding model coupled to the system at a single boundary site, so long as the reservoir and interaction Hamiltonians take the generic forms (12) and (15).67–69 In general, the resulting 1D chain is described by inhomogeneous inter-site couplings and local site energies. This leads to scattering of excitations back towards the system, potentially giving rise to recurrences or other non-Markovian effects. In contrast, such backscattering is absent in the uniform chain considered here, which is characterised completely by just two frequencies Γ and Ω. Directly setting Ω−1 = 0 corresponds to the wideband-limit approximation,18 which leads to a frequencyindependent spectral density. In the following, we compute the solutions for finite Ω, which enables us to retain energydependent damping rates even for weak coupling, Γ Ω, since we need not assume that E j Ω.
B.
Formal solution
In this section, we provide the exact solution for the opensystem density matrix ρˆ S (t) = TrB [ρ(t)], ˆ where ρ(t) ˆ is the global quantum state at time t. The same solution formally applies to the general scenario depicted in Fig. 2 (a), i.e. any pair of environments described by Hamiltonians of the form (12) and (15), corresponding to a spectral density (18). We first describe the formal solution for this general case, before specialising to the Newns spectral density (19) of a uniform chain in later sections. Details of the calculation are presented in Appendix A. We consider factorised initial conditions of the form ρ(0) ˆ = ρˆ S (0)ρˆ L ρˆ R , with reservoir Bα initialised in the Gibbs
Mitchison & Plenio
5
state
With this notation, the correlation matrix reads as ˆ
ρˆ α =
C(t) = G† (t)C(0)G(t) + Z(t),
ˆ
e−βα (Hα −µα Nα ) . Z βα , Hˆ α , µα , Nˆ α
(20)
Z(t) =
Here, Nˆ α = q aˆ †q,α aˆ q,α is the number of particles in Bα . The dynamics preserves the total number of fermions ˆ N] ˆ = 0, where Nˆ = Nˆ L + Nˆ R + Nˆ S due to the relation [H, and Nˆ S = nˆ L + nˆ R . For any physical initial state satisfying ˆ ρ(0)] ˆ ρ(t)] [N, ˆ = 0, we thus have [Nˆ S , ρˆ S (t)] = TrB [N, ˆ = 0. Therefore, ρˆ S (t) is characterised by five independent real numbers, in general. Four of these are encapsulated by the Hermitian correlation matrix i h (21) C jk (t) = Tr ρˆ S (t)dˆ†j dˆk ,
We compute Eqs. (21) and (22) by solving the equations of motion for dˆ j (t) and bˆ q, j (t) in the Laplace domain. Here, ˆ ˆ −iHt ˆ ˆ O(t) = eiHt Oe denotes the Heisenberg-picture time deˆ The solution for t > 0 can be pendence of an operator O. completely expressed in terms of the propagator G jk (t) = h0| dˆ j (t)dˆk† |0i ,
(23)
where |0i is the vacuum state, i.e. Nˆ |0i = 0. According to Eq. (17), the particle number in each j sector is separately P ˆ Nˆ j ] = 0, with Nˆ j = dˆ†j dˆ j + q bˆ †q, j bˆ q, j . It thus conserved: [H, follows from the definition (23) that G jk (t) = δ jk G j (t), with Z G j (t) = dω e−iωt ϕ j (ω). (24) The function ϕ j (ω) is the probability distribution of excitation energies associated with the state dˆ†j |0i, i.e. X ϕ j (ω) = |hn| dˆ†j |0i|2 δ(ω − εn ), (25) n
where |ni is a single-particle eigenstate of Hˆ with energy εn , i.e. Hˆ |ni = εn |ni (note that εn may be negative70 ). Therefore, Z dω ϕ j (ω) = 1, (26) lim ϕ j (ω) = δ(ω − E j ).
Γ→0
(27)
(For general spectral densities, the limit Γ → 0 here means that all tunnel couplings τq → 0.) The solution can be compactly represented in terms of the matrix F(ω) = Rf(ω)RT , where f(ω) = diag[ fL (ω), fR (ω)] with fα (ω) = (eβα (ω−µα ) + 1)−1 the Fermi-Dirac function of reservoir Bα , and the noise kernel Z Φ(t) = dω eiωt J(ω)F(ω). (28)
t
dt0
0
P
which satisfies 0 ≤ C(t) ≤ 1. (Here, and throughout this document, the elements of a matrix A are denoted by A jk .) The fifth degree of freedom is the double-occupancy probability h i D(t) = Tr ρˆ S (t)dˆ1† dˆ2† dˆ2 dˆ1 . (22)
Z
Z
(29)
t
dt00 G† (t0 )Φ(t0 − t00 )G(t00 ).
(30)
0
The double-occupancy probability is (31) D(t) = |det G(t)|2 D(0) + det Z(t) X 2 ∗ + |G j (t)| Zkk (t)C j j (0) − G j (t)Gk (t)Zk j (t)C jk (0) . j,k
We see that, so long as limt→∞ G j (t) = 0, the system relaxes to a unique stationary state ρˆ ∞ S that is independent of 71 the initial conditions. Moreover, this ρˆ ∞ beS is Gaussian cause D(∞) = det C(∞). The stationary state is therefore determined completely by its correlation matrix Z C jk (∞) = dω J(ω)F jk (ω) ϑ j (ω) + iπϕ j (ω) × ϑk (ω) − iπϕk (ω) . (32) Here, we defined the Hilbert transform of ϕ j (ω), Z ϕ j (ω0 ) , ϑ j (ω) = P dω0 ω − ω0
(33)
where P denotes the Cauchy principal value. The foregoing equations, which hold for any spectral density (18), constitute the complete formal solution given knowledge of the propagator G(t). However, in general, computing the propagator is a challenging problem.
C.
Propagator for uniform-chain environments
In order to find an explicit expression for the propagator, we now specialise to environments modelled by uniform chains with spectral density (19). Relaxation to a unique steady state is guaranteed in this case if we make the additional, technical assumption that q E 2j + ΓΩ < Ω. (34) Physically, this inequality requires that the energy levels of the open system lie well within the reservoirs’ energy bands, so that any initial excitation can eventually be absorbed. Under this condition, we show in Appendix A that ϕ j (ω) =
1 JN (ω) , 1 − Γ/Ω (ω − E 0j )2 + Γ2j /4
where we defined shifted energies and decay rates q 1 − Γ/Ω − E 2j /Ω2 1 − Γ/2Ω 0 E j, Γj = Γ. Ej = 1 − Γ/Ω 1 − Γ/Ω
(35)
(36)
The propagator is then evaluated using Eq. (24). We also prove in Appendix A that ϕ j (ω) Γ ϑ j (ω) = 1− ω − Ej . (37) 2Ω JN (ω)
6
Non-additive dissipation in open qantum networks 0.10
D. Steady-state observables for uniform-chain environments
Γ=0.02Δ Γ=0.02Δ
0.08
Γ=0.2Δ
We now present solutions for some interesting observables in the asymptotic stationary state, using the explicit formulae quoted in the previous section. If the baths are initially in equilibrium, i.e. βα = β, µα = µ and fα (ω) = f (ω), then Z C jk (∞) = δ jk dω ϕ j (ω) f (ω), (38) which differs from the strict equilibrium value f (E j ) due to the finite width of the energy distribution ϕ j (ω). Using Eqs. (25) and (38), we show in Appendix B that, for equilibrium baths, ρˆ ∞ S is the reduced state of the global Gibbs ensemble " # ˆ Nˆ ) −β(H−µ e . ρˆ ∞ (39) S = Tr B ˆ µ, Nˆ Z β, H, We thus infer that the energy-level broadening represented by Eq. (38) arises from the system-bath correlations reflected in Eq. (39), both being related to a finite system-reservoir coupling Γ. With the help of Eq. (27), we recover thermalisation in the strict weak-coupling sense (1) in the limit Γ → 0. Out of equilibrium, particle and energy currents flow from each bath Bα into S. These currents are defined respectively ˆ Nˆ α ] and JˆαE = −i[H, ˆ Hˆ α ]. Fermion conservaas JˆαP = −i[H, tion demands that a corresponding particle current JˆSP flows between the two sites of the system, which is defined to satisfy the continuity equations, e.g. ∂t nˆ L (t) = JˆLP (t) − JˆSP (t). Explicitly, we have X JˆαP = i τ∗q aˆ †q,α cˆ α − τq cˆ †α aˆ q,α , (40) q
JˆαE = i
X
ωq τ∗q aˆ †q,α cˆ α − τq cˆ †α aˆ q,α ,
g † = cˆ L cˆ R − cˆ †R cˆ L . 2i
(42)
Note that the mean intra-system current h JˆSP it = g Im [C12 (t)] vanishes identically if [Hˆ S , ρˆ S (t)] = 0. It follows from the definitions that, in the stationary state, the particle currents are homogeneous throughout the system, i.e. JLP = JSP = −JRP , where JσP = h JˆσP i∞ , for σ = S, L, R. Likewise, the asymptotic mean energy currents JαE = h JˆαE i∞ satisfy JLE = −JRE . The asymptotic currents are calculated in Appendix C. For the Newns spectral density (19), we obtain the standard Landauer formulae6 Z JLP = dω T (ω) fL (ω) − fR (ω) , (43) Z JLE = dω ωT (ω) fL (ω) − fR (ω) , (44)
πg2 ϕ1 (ω)ϕ2 (ω). 2
(ω) 0.02
0.00 0.04
0.06
0.08
0.10
0.12
0.14
0.16
0.18
ω/Ω
Figure 3. Transmission function versus frequency, with h = 1, g = δ = 0.5 and Ω = 10. The approximation (46) for Γ ∆ is also shown by the green dotted line.
The transmission probability is proportional to the overlap between the energy distributions of the two eigenmodes of Hˆ S [see Eq. (25)]. Note that the Landauer formulae imply the second law (2) for any transmission function T (ω) ≥ 0.4 We plot the transmission function in Fig. 3. For Γ ∆, T (ω) is a bimodal distribution that is well approximated by T (ω) ≈
2 g2 X Γ j ϕ j (ω). 4∆2 j=1
(46)
As Γ is increased, the two peaks at ω = E1,2 broaden and ultimately merge into a single maximum for Γ ∆. IV.
MASTER EQUATION
A.
Exact master equation
In order to analyse the non-additive properties of the dynamics, we first write Eqs. (29) and (31) in differential form, corresponding to an exact time-local master equation for the system density operator (see Refs. 40 and 41 for alternative derivations). All equations presented in this section hold for an arbitrary spectral density (18). Assuming that G(t) is non-singular, we define Hermitian matrices H(t) and Γ(t) such that iH(t) + 12 Γ(t) = −G−1 ∂t G. We also define rate matrices Λ± (t) by Λ+ = ∂t Z − i[H, Z] + 21 {Γ, Z},
Λ− = Γ − Λ+ ,
(47)
where time arguments are suppressed. Upon differentiating Eqs. (29) and (31), some tedious algebra reveals that ∂t C = i[H, C] − 21 {Γ, C} + Λ+ , X ∂t D = −Tr[Γ]D + Λ+j jCkk − Λ+jk Ck j .
(48) (49)
j,k
Direct comparison confirms that these equations of motion are equivalent to the master equation
with the transmission function T (ω) =
0.04
(41)
q
JˆSP
Γ=2Δ
0.06
(45)
∂t ρˆ S (t) = − i[Hˆ S0 (t), ρˆ S (t)] + L(t)ρˆ S (t),
(50)
Mitchison & Plenio
7
where Hˆ S0 (t) = dˆ † HT dˆ and we defined the dissipator L(t)ρˆ S =
2 X
(b) 0.30 0.25
Λ−jk (t)
dˆ j ρˆ S dˆk† − 12 {dˆk† dˆ j , ρˆ S }
0.3
2 X
C11(t) 0.20
C22(t)
j,k=1
+
(a) 0.4
Im[C12(t)]
0.2
0.15
Re[C12(t)] 0.10
Λ+jk (t) dˆk† ρˆ S dˆ j − 12 {dˆ j dˆk† , ρˆ S } .
0.1
(51)
j,k=1
0.05
0.0 0
After diagonalising the rate matrices Λ± (t) by a unitary rotation [U± (t)]† Λ± (t)U± (t) = diag[λ±1 (t), λ±2 (t)], we cast the dissipator into canonical form53 L(t) =
2 X X
λ sj (t)D[Lˆ sj (t)].
(52)
j=1 s=±
Here, the time-dependent jump operators are defined as P P Lˆ −j (t) = k U −jk (t)dˆk and Lˆ +j (t) = k [U +jk (t)]∗ dˆk† . These describe loss and gain of excitations from and into modes determined by the eigenbases of the matrices Λ± (t). Note that only if [Λ+ (t), Λ− (t)] = 0 can we choose Lˆ +j (t) = [Lˆ −j (t)]† , i.e. the loss and gain modes differ, in general.
1
2
3
4
5
0.00 0
1
2
3
4
5
Γt
Γt
Figure 4. Example evolution of the system correlation matrix with a vacuum initial condition C jk (0) = 0, comparing exact values (points) with the EPA (lines). Parameters: Γ = 0.2, Ω = 100, δ = 0.1, g = 0.5; (a) h = 1, µL = 1, µR = 0, T L = T R = 0.1; (b) h = 50, µL = µR = 49, T L = 10, T R = 0.01;
The master equation (50) takes a simple form under the EPA. In particular, we have that H = diag[E1 , E2 ], i.e. Hˆ S0 (t) = Hˆ S , Γ = diag[Γ1 , Γ2 ] and +
Λ (t) =
Z
t
dt0 Φ(t0 )G(t0 ) + h.c.
(54)
0
B.
Exponential-propagator approximation
Throughout the rest of the paper, we specialise to the Newns spectral density (19) and assume that Γ Ω. Although the latter condition is usually deemed necessary for the Born-Markov approximation to hold, we shall see that it is by no means sufficient. In order to simplify the subsequent discussion, we introduce an approximation scheme where terms of order O(Γ/Ω) are neglected. This amounts to the replacement 0
G j (t) ≈ e−iE j t−Γ j t/2 .
(53)
For the sake of clarity and concision, we henceforth refer to this as the exponential-propagator approximation (EPA). The EPA is justified in Appendix D, where we give an explicit expression for the error in G j (t) thus incurred. We also derive a rigorous upper bound on the magnitude of this error that is proportional to Γ/Ω and decays to zero as t → ∞. To the same order of approximation, we write E 0j ≈ E j and Γ j = 2πJN (E j ). Unfortunately, we have not been able to derive a bound on the error induced by calculating general expectation values such as the correlation matrix (29) within the EPA. Nevertheless, a direct numerical comparison shows that, for sufficiently small Γ/Ω, the EPA gives an excellent approximation to both the transient and steady-state dynamics, even if E j is comparable to Ω. We illustrate the agreement between the approximation and the exact solution in Fig. 4 for a few example parameters. We emphasise that the EPA only requires that the coupling Γ is weak in comparison to the environment’s energy scale Ω, so that the approximation (53) becomes exact in the wideband limit Ω → ∞. On the other hand, the relation between Γ and the system energy scales E j is not restricted.
The loss and gain modes defined by Eq. (52) can be identified by inspection of Eq. (54) in two special cases. First, assume that the two baths are in thermal equilibrium with each other, so that fL (ω) = fR (ω) = f (ω). In that case, F(ω) = f (ω)1 is proportional to the identity, and Λ± (t) are both diagonal. Hence, Lˆ −j = (Lˆ +j )† = dˆ j , i.e. excitations are pumped into eigenmodes of Hˆ S , driving the system towards a state that is diagonal in the energy eigenbasis of S. If the system-environment coupling is sufficiently weak, this ensures the proper thermalisation behaviour (see Section V A). Second, consider the white-noise limit, with Ω → ∞ and F(ω) = F a constant matrix. We then have that Φ(t) = ΓFδ(t) and Γ = Γ1, and since G(0) = 1 it follows that Λ+ (t) = ΓF and Λ− (t) = Γ(1 − F). Therefore, the rate matrices Λ± can be diagonalised by the rotation RT , leading to Lindblad opera− tors Lˆ −j = (Lˆ +j )† with Lˆ 1,2 = cˆ L,R , i.e. energy is pumped into localised modes. Of course, this only holds exactly in the unphysical scenario of infinite energy density in the baths, corresponding to β j → 0 or β j , |µ j | → ∞. Nevertheless, the local master equation may be an excellent approximation for sufficiently large chemical-potential bias or temperature. According to Eq. (54), the rate matrices are determined by the product of the propagator G(t)—which is diagonal in the energy eigenbasis—and the noise kernel Φ(t)—which is diagonal in the site basis. Hence, in general, the gain and loss modes correspond neither to the energy eigenbasis dˆ j nor the site basis cˆ α , but instead lie somewhere in between. In particular, whenever the baths are not in equilibrium with each other, the rate matrices Λ± are non-diagonal and the dissipation generates some coherence in the eigenbasis of Hˆ S . This can be seen, for example, in Fig. 4, where the coherence can be comparable in magnitude to the populations and has a large imaginary part, reflecting the current flowing through the system.
8
Non-additive dissipation in open qantum networks
V.
ASYMPTOTIC NON-ADDITIVITY
A.
Non-additivity in the energy eigenbasis
steady-state currents according to Eq. (6). Indeed, a direct computation yields
Now we demonstrate that the asymptotic dynamics as t → ∞ is not additive. We write the rate matrices in the limit simply as Λ± = limt→∞ Λ± (t). Explicitly, these have the components Z Λ+jk = i dω JN (ω)F jk (ω) 1 1 × , (55) − E j − ω + iΓ j /2 Ek − ω − iΓk /2 and Λ−jk = Γ j δ jk − Λ+jk . It follows that the asymptotic dissipator L = limt→∞ L(t) admits the decomposition L = LL + LR + Lint .
(56)
Here, the generators Lα represent the thermalising effect of each individual bath α = L, R on the populations, while the interference term Lint describes the generation of coherence due to the combined effect of the non-equilibrium baths, as described below. First, let us examine the generators Lα , which take the form Lα =
2 X
Z 2 D E X g2 † ˆ dω Γ j ϕ j (ω) fL (ω) − fR (ω) , LL NS = 2 ∞ 4∆ j=1
Z 2 D E X g2 L†L Hˆ S = dω E j Γ j ϕ j (ω) fL (ω) − fR (ω) , (62) 2 ∞ 4∆ j=1 P P † ˆ † ˆ with α hLα NS i∞ = 0 = α hLα HS i∞ . These expressions can be shown to be equivalent to the exact Landauer formulae (43) and (44) using Eq. (46) and writing E j ϕ j (ω) ≈ ωϕ j (ω), which is a valid approximation in the quantum-optical regime. Since the generators Lα are in Lindblad form, they obey the Spohn inequality58 Tr ln rˆα − ln ρˆ S (t) Lα ρˆ S (t) ≥ 0. (63) However, Lint does not, by itself, generate a positive evolution, and thus does not necessarily satisfy such an inequality.72 We nonetheless show in Appendix E that, in the weakcoupling limit, from the Spohn inequality one may recover the second law of thermodynamics in the form − lim
Γ→0
− + ˆ ˆ† γα, j D[d j ] + γα, j D[d j ] .
(57)
j=1
The decay and gain rates are defined by Z − γα, = Γ dω ϕ j (ω) 1 − fα (ω) , α, j j Z + γα, = Γ dω ϕ j (ω) fα (ω), α, j j
(58)
where ΓL, j = Γ j R21 j and ΓR, j = Γ j R22 j . One readily verifies that the unique fixed point rˆα satisfying Lα rˆα = 0 is Gaussian, with the correlation matrix Z i h Tr rˆα dˆ†j dˆk = δ jk dω ϕ j (ω) fα (ω). (59) In the quantum-optical limit where Γ ∆, this describes an approximately thermal distribution, up to corrections due to level broadening as discussed in Section III D. In fact, using the arguments given in Appendix B, one can show that rˆα is the reduction of a global Gibbs state in equilibrium with the corresponding bath, i.e. # " ˆ ˆ e−βα (H−µα N ) . (60) rˆα = TrB ˆ µα , Nˆ Z βα , H, It follows immediately that Eq. (7) is recovered as Γ → 0. Another interesting property of the generators Lα in the quantum-optical limit is that they accurately reproduce the
(61)
1X βα JαQ ≥ 0, Γ α
(64)
with the heat current defined by JαQ = hL†α (Hˆ S − µα Nˆ S )i∞ . Here, it is necessary to first divide by Γ before taking the limit P in order to avoid recovering the trivial equality limΓ→0 α βα JαQ = 0 implied by limΓ→0 JαQ = 0. Corrections to the left-hand side of inequality (64) for finite Γ are quoted explicitly in Appendix E. Now we turn to the interference contribution, defined by h X Lint ρˆ S = Λ+jk dˆk† ρˆ S dˆ j − 12 {dˆ j dˆk† , ρˆ S } j,k
i − dˆ j ρˆ S dˆk† − 12 {dˆk† dˆ j , ρˆ S } . Here, Λ+12 = Λ+21
∗
(65)
= ξ + iη, with
Z 2 g X Γ j dω ϕ j (ω) fL (ω) − fR (ω) , 2∆ j=1 Z g η= dω (E1 − ω)ϕ1 (ω) − (E2 − ω)ϕ2 (ω) 2∆ × fL (ω) − fR (ω) . ξ=
(66)
(67)
Therefore, Lint is associated with the difference in distribution functions fα (ω). In particular, Lint is non-negligible unless fL (ω) ≈ fR (ω) over the entire frequency range in which ϕ j (ω) differs appreciably from zero. The interference term satisfies the properties L†int dˆ†j dˆ j = 0,
L†int dˆ1† dˆ2 = Λ+12 .
(68)
Mitchison & Plenio
9
Hence, L†int generates coherence while leaving the populations unaffected. This is to be expected, since energyeigenbasis coherence is an intrinsic property of the NESS of a quantum network, as discussed in Section II. It is noteworthy that, in the Schrödinger picture, Lint couples populations and coherences. Such a contribution is therefore precluded in the standard global Lindblad approach due to the secular approximation, which enforces decoupling of populations and coherences.15 Nevertheless, Lint is generally a significant contribution even in the quantum-optical limit where the secular approximation is widely believed to be valid. We note also that Lint can never be written in Lindblad form. Therefore, departures from Markovian evolution can be used to detect non-additivity, as discussed in Section V C.
B.
Non-additivity in the local basis
It is also possible to investigate the violation of additivity in a local, rather than global, picture of dissipation. We transform to the site basis cˆ α and decompose the asymptotic dissipator as L = L¯ L + L¯ R + L¯ LR .
(69)
The local dissipators are Lindblad generators that act only on a single site, given by L¯ α = γ¯ α− D[ˆcα ] + γ¯ α+ D[ˆc†α ],
(70)
P ± where γ¯ α± = 2j=1 γα, j . The remaining contribution to L describes delocalised, incoherent processes acting on both sites together: Xh ¯ +αα0 cˆ †α0 ρˆ S cˆ α − 1 {ˆcα cˆ †α0 , ρˆ S } L¯ LR ρˆ S = Λ 2 α,α0
i ¯ −αα0 cˆ α ρˆ S cˆ †α0 − 1 {ˆc†α0 cˆ α , ρˆ S } , +Λ 2
(71)
¯ ±RL )∗ , with ¯ ±LR = (Λ where Λ Z g + ¯ Re [ΛLR ] = dω Γ2 ϕ2 (ω) − Γ1 ϕ1 (ω) 4∆ × fL (ω) + fR (ω) , Z ¯ +LR ] = g Im [Λ dω (E1 − ω)ϕ1 (ω) − (E2 − ω)ϕ2 (ω) 2∆ × fL (ω) − fR (ω) , Z g − ¯ Re [ΛLR ] = dω Γ2 ϕ2 (ω) − Γ1 ϕ1 (ω) 4∆ × 2 − fL (ω) − fR (ω) , ¯ −LR ] = − Im [Λ ¯ +LR ]. Im [Λ (72) The cross-term L¯ LR is associated with the difference between the frequency distributions ϕ j (ω) of the open system’s energy levels. That is, L¯ LR reflects the extent to which the occupation numbers fα (ω) of reservoir states differ between the distinct frequency ranges sampled by the two distributions
ϕ j (ω). In particular, L¯ LR is negligible only in the white-noise limit, where ∆ Ω and the distribution functions fL,R (ω) are essentially constant over the range in which ϕ j (ω) are non-zero. Note that here the local generators L¯ α can be meaningfully associated with bath Bα only in the sense that each one depends only on the variables of Bα and acts non-trivially only on site α of the system. Even in the weak-coupling limit, L¯ L,R do not obey the properties (6) and (8) expected of additive, thermal dissipators unless δ ≈ 0, as discussed below. Let us first examine the validity of Eq. (8). The fixed point rˆα satisfying L¯ α rˆα = 0 is of the form rˆα =
ˆ e−βα (Hsα −µα nˆ α ) Oˆ α0 , Z βα , Hˆ sα , µα , nˆ α
(73)
where, by analogy with Eq. (8), we defined an effective local Hamiltonian acting on site α, − γ¯ α Hˆ sα = T α ln + µ ˆ α, (74) α n γ¯ α+ while Oˆ α0 is an arbitrary density operator with support on the other site α0 , α. Using Eq. (27), we find βα (h−µα ) e + cosh(βα ∆/2) γ¯ α− lim + = eβα (h−µα ) . (75) Γ→0 γ ¯α 1 + eβα (h−µα ) cosh(βα ∆/2) This only has the detailed-balance form required for thermalisation if βα ∆ 1 or if |βα (h − µα )| 1, which corresponds to the white-noise limit. In that case, we have that limΓ→0 ln(¯γα− /¯γα+ ) ≈ βα (h − µα ), and the weak-coupling fixed point (73) can be approximated by Eq. (8) with Hˆ sα = hˆnα and Nˆ α = nˆ α . However, this only corresponds to the “true” site Hamiltonian hα nˆ α [cf. Eq. (10)] if δ = 0. Second, let us discuss the definition of the currents via Eq. (6). For the particle current, we have, for example, E D E D E D = L† nˆ L − L¯ †LR nˆ L = JLP − L¯ †LR nˆ L , (76) L¯ †L Nˆ S ∞
∞
∞
where we identified hL† nˆ L i∞ = JLP using the exact master equation (50). Hence, Eq. (76) differs from the true particle current by an amount D E D E g (Γ2 − Γ1 ) Re cˆ †L cˆ R . − L¯ †LR nˆ L = (77) ∞ ∞ 2∆ Clearly, this correction is negligible if L¯ LR ≈ 0 or if ∆/Ω → 0 so that Γ1 ≈ Γ2 . In addition, Eq. (77) vanishes as δ → 0, since one can easily show that Re hˆc†L cˆ R i∞ ∝ δ/∆. On the other hand, we find for the energy current 2 D D E E D E gX 2 L¯ †L Hˆ S = hL L¯ †L Nˆ S − R1 j Γ j Re cˆ †L cˆ R . (78) ∞ ∞ 2 ∞ j=1
This expression is only correct when L¯ LR ≈ 0 and δ h, g, such that hL¯ †L Hˆ S i∞ ≈ hJLP . This approximately agrees with the exact Landauer formulae (43) and (44) in the white-noise limit, where Ω → ∞, T (h + ω) = T (h − ω) and fα (ω) ≈ const.
10
Non-additive dissipation in open qantum networks T L = T R = 0.02∆ 2.0
0.5
0.5
1.0 μL /h
1.5
2.0
μR /h
μR /h
0.25 0.20 0.15 0.10 0.05
1.0
0.4
ν∞/Γ
1.5
ν∞/Γ
0.06 0.05 0.04 0.03 0.02 0.01
1.0
0.5
0.0 0.0
µL = µR = 0
(c)
2.0
1.5
0.0 0.0
T L = T R = 0.2∆
(b)
0.5
1.0 μL /h
1.5
2.0
0.3 TR /h
(a)
ν∞/Γ 0.0025 0.0020 0.0015 0.0010 0.0005
0.2
0.1
0.0 0.0
0.1
0.2 TL /h
0.3
0.4
Figure 5. Asymptotic non-Markovianity u∞ in the limit Γ ∆ Ω, with (a), (b) variable chemical potential with fixed temperatures, or (c) variable temperature with fixed chemical potentials. White dotted lines in (a) and (b) indicate loci where the chemical potentials µL,R equal the system energies E1,2 . Parameters: h = 1, δ = 0, g = 0.5, Γ = 0.01 and Ω = 10.
C.
Non-Markovianity witnesses non-additivity
Since neither Lint nor L¯ LR are in Lindblad form, it may not be possible to cast the overall dissipator L into Lindblad form for certain values of the parameters. Therefore, its canonical representation (52) may exhibit one or more negative rates λ±j (t) even as t → ∞. Negativity of the rates λ sj (t) signals the onset of non-Markovian evolution, according to the measure of non-Markovianity based on completely-positive divisible maps proposed by Rivas et al.73 In this framework, the degree of instantaneous non-Markovianity is quantified by53 XX ν(t) = max[0, −λ sj (t)]. (79)
the violation of additivity in either the global or local pictures of dissipation. Indeed, in Fig. 5 we observe that nonMarkovianity is associated with parameters for which the master equation is not additive, in neither the local nor energy eigenbases. As discussed in previous sections, this occurs when the baths are far from equilibrium, yet the chemical-potential bias and the temperature are not so large that the white-noise limit is recovered. Since ANM is, in principle, experimentally accessible via quantum process tomography, it represents a measurable signature of non-additivity.
VI.
CONCLUSION
s=± j=1,2
Asymptotic non-Markovianity (ANM) corresponds to limt→∞ ν(t) = ν∞ > 0. In a previous work, Ribeiro et al.42 demonstrated that ANM arises in a class of spin and fermionic transport models, which includes the set-up considered here within the wide-band-limit approximation Ω−1 = 0. ANM has also been recently identified in several other open-system scenarios.53–55 We plot ν∞ in Fig. 5, as calculated using the EPA rate matrices defined by Eq. (55). At low temperatures, we find ν∞ > 0 whenever the chemical potential of one reservoir lies in between the two energy eigenlevels of the system while that of the other reservoir lies outside. This effect is significantly reduced once the thermal energy grows comparable to the splitting ∆. It is remarkable that a highly non-Markovian evolution can be induced merely by tuning a macroscopic potential difference, without any engineering or microscopic understanding of the environment Hamiltonian. Indeed, the ANM effect survives even in the wide-band limit Ω → ∞ where each environment is completely unstructured. In Fig. 5(c) we also show that some non-Markovianity may be generated by purely thermal driving, i.e. where the reservoirs have identical chemical potentials and different temperatures. Since the generators LL,R (L¯ L,R ) are in Lindblad form, the appearance of asymptotic non-Markovianity is associated with the cross-term Lint (L¯ LR ). Hence, ANM witnesses
In this work, we have presented a study of an exactly solvable microscopic system-environment model, featuring a two-site fermionic network driven out of equilibrium by two independent thermal baths. We have derived the exact master equation describing the open-system density matrix and developed a simplifying approximation scheme valid when the system-reservoir coupling Γ is much smaller than the reservoir bandwidth Ω. Our results demonstrate that the asymptotic master equation is not additive, i.e. the dissipator cannot be written as a sum of the form (5) where each term pertains to a single bath, even if the system-reservoir coupling is vanishingly small in comparison to every other relevant energy scale. Examining the master equation in the energy eigenbasis of the open system, we identify a generator Lα associated to each bath, which drives the open network towards thermal equilibrium according to Eq. (7) [or, more generally, Eq. (60)]. Furthermore, Lα determines the currents via Eq. (6) in the quantumoptical limit. This generator can therefore be interpreted as an individual contribution describing the effect of bath Bα acting in isolation. Nevertheless, additivity is violated away from equilibrium due to an additional interference term Lint that generates coherence, as required for a current-carrying NESS in an extended system. In extreme cases, this non-additivity can lead to an asymptotically non-Markovian master equation. Despite this, the interference term Lint leaves the energy-
Mitchison & Plenio
11
eigenbasis populations invariant and therefore the global, additive Lindblad model obtained by setting Lint = 0 still leads to accurate approximations for the boundary currents in the quantum-optical regime. Such an additive model coincides in the limit Γ → 0 with the standard master equation derived under the BMA and secular approximation. We have also examined the violation of additivity in the local basis. However, here it is not always possible to identify thermal generators satisfying the properties (6) and (8), even in the weak-coupling limit. Thus, the identification of L¯ α as an “individual” bath contribution is not fully justified, since it does not accurately represent the effect of a single bath acting in isolation. Nevertheless, an additive, local Lindblad model is recovered when the modes are near-resonant, i.e. δ h, g, and when the noise correlation time is much smaller than the inverse frequency splitting ∆−1 . This conclusion is entirely consistent with recent studies showing that local additive Lindblad models may perform well in non-equilibrium scenarios when the network nodes are nearly degenerate,48,49 but fail for large detunings.20 It has been shown52 that additive open-system dynamics is obtained whenever the BMA (or, more generally, the secondorder time-convolutionless projection operator method15 ) holds. Hence, our results indicate a breakdown of these assumptions at asymptotically large times when multiple thermal reservoirs act in competition. Note that, while the BMA is commonly cast in terms of an assumption that the systemreservoir density matrix factorises, it is more accurately described as a projection P onto product states of the form Pρ(t) ˆ = ρˆ S (t)ρˆ B ,
(80)
where ρˆ B is the initial state of the environment, in combination with a low-order perturbation expansion in the systemreservoir interaction.19 The breakdown of additivity for arbitrarily weak systembath coupling Γ indicates that Eq. (80) does not serve as a good reference point for a perturbative expansion in powers of Γ. This can be understood as a consequence of the correlations that build up between the two parts of the environment due to the current flowing between them. As time increases, these correlations cause the joint state ρ(t) ˆ to move arbitrarily far away from the subspace spanned by states of the form (80) with ρˆ B = ρˆ L ρˆ R . A very interesting open question is whether a projection of the type (80) can still facilitate a good perturbative description of the long-time dynamics, but with a correlated reservoir state ρˆ B . Although the evidence presented here pertains only to the specific system considered, we expect the qualitative conclusions regarding non-additivity to hold for other extended open systems, where currents and energy-eigenbasis coherence are inextricably linked. Indeed, previous results demonstrating asymptotic non-Markovianity in larger 1D fermionic and spin networks driven out of equilibrium42 already support this conclusion, because ANM witnesses non-additivity, as we have shown. Furthermore, our model can be readily generalised to a bosonic setting where the ladder operators obey commutation, rather than anti-commutation, relations. Since the Heisenberg equations are essentially identical in
this case, one expects the same conclusions regarding additivity to hold, although we leave a detailed analysis of this problem to future work (see also Refs. 74 and 75). We note also a very recent study of a different bosonic model of coupled mechanical oscillators that displays coherences in the NESS,76 which are naturally explained by the interference between non-equilibrium baths. Looking ahead, it will be interesting to investigate the consequences of the results presented here for the thermodynamics of small heat machines running between multiple thermal reservoirs. For example, it remains to be seen how non-additive noise may affect the thermodynamic power and efficiency of such machines, or their ability to generate quantum resources such as coherence and entanglement. One may also ask whether the interference between the reservoirs is manifested in the fluctuations of the energy and particle currents. Finally, the framework presented herein appears to be an ideal setting to explore strong-coupling effects in thermodynamics,77–82 since system-environment correlations in both equilibrium and far-from-equilibrium states may be taken into account.
ACKNOWLEDGMENTS
We gratefully acknowledge the stimulating comments and conversation of Jonatan Brask, Nicolas Brunner, Luis Correa, Géraldine Haack, Patrick Hofer, Susana Huelga, Andreas Lemmer, Fabio Mascherpa, Martí Perarnau-Llobet, Ralph Silva and Andrea Smirne. This work was funded by the ERC Synergy grant BioQ and the EU project QUCHIP.
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APPENDICES Appendix A: Solution of the equations of motion
Here, we provide details of the exact solution presented in Section III. The equations of motion read as X i∂t dˆ j (t) = E j dˆ j (t) + τq bˆ q, j (t), (A1) q
i∂t bˆ q, j (t) = ωq bˆ q, j (t) + τ∗q dˆ j (t).
(A2)
These are readily by transforming to Laplace space,83 R ∞ solved −zt ˆ ˜ e.g. d j (z) = 0 dt e d j (t). After rearranging the resulting linear, algebraic system of equations, we transform back to the time domain to obtain X dˆ j (t) = G j (t)dˆ j + τq K j (ωq , t)bˆ q, j , (A3) q
bˆ q, j (t) = e−iωq t bˆ q, j + τ∗q K j (ωq , t)dˆ j X + τ∗q τ p I j (ωq , ω p , t)bˆ p, j . p
(A4)
Mitchison & Plenio
13
(a)
(b)
Here, C1 denotes a Bromwich contour, while C2 is a closed contour encircling the branch cut in the clockwise sense. The two integration contours can be shown to give equal and opposite contributions using Cauchy’s theorem (see Fig. 6). After a change of variables to ω = iz, the integral along C2 takes the form (24), with ϕ j (ω) given by Eq. (35). It remains to justify Eqs. (32) and (37). By taking the Laplace transform of Eq. (24), we obtain Z ϕ j (ω) G˜ j (z) = dω . (A11) z + iω
Figure 6. Integration contours used to invert the Laplace transform G˜ j (z), assuming that the inequality (34) holds. (a) A Bromwich contour lies to the right of the branch cut (thick red line) and extends parallel to the entire imaginary axis. (b) A closed integration contour that avoids enclosing any singularities. Taking the limits R → ∞ and → 0, the integrals along C3 , C4 and C5 sum to zero, leaving only the contributions from C1 and C2 .
Here, G j (t) denotes the inverse Laplace transform of G˜ j (z), where −1 ˜ G˜ j (z) = z + iE j + W(z) , (A5) Z J(ω) ˜ W(z) = dω . (A6) z + iω We also introduced the functions Z t 0 K j (ω, t) = −i dt0 e−iω(t−t )G j (t0 ), 0 Z t 0 I j (ω, ω0 , t) = −i dt0 e−iω(t−t ) K j (ω0 , t0 ).
(A7) (A8)
0
By inspection of Eq. (A3) one sees that G(t) is indeed given by Eq. (23). Expressions (29) and (31) for the correlation matrix C(t) and double occupancy D(t) follow directly upon plugging Eq. (A3) into the definitions (21) and (22). Note that the above equations hold for an arbitrary spectral density J(ω), which determines the propagator via Eqs. (A5) and (A6). Let us now compute the propagator for the specific problem at hand, with spectral density (19). The integral (A6) evaluates to Γ p 2 ˜ W(z) = z + Ω2 − z , (A9) 2Ω where the positive (negative) sign for the square root is used for Re z > 0 ( Re z < 0). This implies that any poles of G˜ j (z) lie on the imaginary axis, which would correspond to undamped oscillations in the time domain. In order to avoid this behaviour, we assume that inequality (34) holds so that G˜ j (z) has no poles. As a result, G˜ j (z) is holomorphic everywhere in the complex z plane except for along a cut connecting the branch points at z = ±iΩ. Therefore, G j (t) → 0 as t → ∞ and S relaxes to a unique stationary state (see Appendix D). The propagator is now obtained by carrying out the inverse Laplace transform, for t > 0, Z Z 1 1 zt ˜ G j (t) = e G j (z) = − ezt G˜ j (z). (A10) 2πi C1 2πi C2
We therefore deduce that, in general, Z ∞ lim G˜ j ( − iω) = dt e−iωt G j (t) &0
0
= πϕ j (ω) + iϑ j (ω).
(A12)
Using the first line, we obtain Eq. (32) from Eq. (29), while comparison of the second line with Eqs. (A5) and (A9) yields Eq. (37) for the uniform-chain environment model.
Appendix B: Thermalisation at finite system-bath coupling
In this appendix, we prove Eq. (39). That is, when βα = β, µα = µ and fα (ω) = f (ω), the open system equilibrates to a state ρˆ ∞ S given by the reduction of a global Gibbs distribution. The following arguments can also be used to deduce Eq. (60) from Eq. (59). Since the marginal of a Gaussian state is itself Gaussian,84 it suffices to check that Eq. (39) yields the correct ˆ Nˆ j ] = 0 correlation matrix (38). Due to the symmetry [H, of the global Hamiltonian, it follows that its equilibrium ˆ ˆ correlation matrix is diagonal, i.e. Tr[e−β(H−µN) dˆ†j dˆk ] ∝ δ jk . We explicitly compute the diagonal elements using a baˆ given by18 sis transformation to the eigenmodes of H, P † ˆ ˆ d j = n h0| d j |ni eˆ n , where |ni = eˆ n |0i is a single-particle eigenstate of Hˆ [cf. Eq. (25)] and the ladder P operators eˆ n diagonalise the total Hamiltonian as Hˆ = n εn eˆ †n eˆ n . Substituting the above and using Eq. (25), straightforward manipulations lead to h i ˆ ˆ Z Tr e−β(H−µN ) dˆ†j dˆ j = dω ϕ j (ω) f (ω), (B1) ˆ µ, Nˆ Z β, H, in agreement with Eq. (39), which completes the proof.
Appendix C: Computing the currents
Due to the homogeneity of the steady-state currents, the particle current may be computed from the expectation value of either the intrasystem current (42) or the boundary current (40).
P The expected value of the intrasystem current is simply JˆS t = g Im [C12 (t)]. In the limit t → ∞, the particle current can thus be found directly from Eq. (32):
14
Non-additive dissipation in open qantum networks
JSP =
πg2 2∆
Z
dω J(ω) ϕ1 (ω)ϑ2 (ω) − ϕ2 (ω)ϑ1 (ω) fL (ω) − fR (ω) .
(C1)
To compute the expectation value of Eqs. (40) and (41), we make use of Eqs. (A3) and (A4) and obtain, for example Z Z
P T † −iωt 0 0 0 ˆ JL t = 2 Im R dω J(ω)K (ω, t)F(ω) e 1 + dω J(ω )I(ω , ω, t) R , Z 11 Z
E JˆL t = 2 Im RT dω J(ω)K† (ω, t)F(ω) ωe−iωt 1 + dω0 ω0 J(ω0 )I(ω0 , ω, t) R ,
(C2) (C3)
11
where K(ω, t) = diag[K1 (ω, t), K2 (ω, t)] and I(ω0 , ω, t) = diag[I1 (ω0 , ω, t), I2 (ω0 , ω, t)]. In the limit t → ∞, we find that Z X JLP = 2π Rk1 Rl1 dω J(ω)Fkl (ω) ϕk (ω) − J(ω) ϑk (ω)ϑl (ω) + π2 ϕk (ω)ϕl (ω) ,
(C4)
k,l
JLE = 2π
X
Z Rk1 Rl1
dω ωJ(ω)Fkl (ω) ϕk (ω) − J(ω) ϑk (ω)ϑl (ω) + π2 ϕk (ω)ϕl (ω) .
(C5)
k,l
The above equations hold for arbitrary spectral densities. The explicit expressions (43)–(45) for the Newns spectral density (19) can be derived by straightforward algebra with the help of Eq. (37).
Here, p j is the residue at the pole, s Ω2 − (E 0j − iΓ j /2)2 pj = , Ω2 − ΓΩ − E 2j while the error term is given by Z err j (t) = − dω e−iωt ϕ j (ω).
Appendix D: Exponential-propagator approximation
(D2)
(D3)
Cerr
In this appendix, we justify the exponential-propagator approximation (53) and prove that the error is of order O(Γ/Ω). The integral (24), with ϕ j (ω) given by Eq. (35), can be estimated with high accuracy for Γ Ω by analytically continuing the integrand into the complex ω plane. Choosing (arbitrarily) the branch of the square root function whose real part is positive for Im ω < 0, we integrate along a closed contour encircling the pole at ω = E 0j − iΓ j /2 in the anti-clockwise sense (see Fig. 7). This integral differs from the exact G j (t) by the contribution along the semi-circular arc Cerr . We therefore obtain 0
G j (t) = p j e−iE j t−Γ j t + err j (t).
(D1)
A trivial rearrangement of terms leads to 0 0 G j (t) = e−iE j t−Γ j t + p j − 1 e−iE j t−Γ j t + err j (t).
The first term above is the EPA contribution (53). The second term is a small relative correction to the leading-order result, which is clearly of first order in Γ/Ω and decays exponentially in time. Finally, the remaining absolute error |err j (t)| can be bounded as Z π e−i(θ+Ωte−iθ ) J (Ωe−iθ ) Ω N |err j (t)| ≤ dθ (Ωe−iθ − E 0j )2 + Γ2j /4 1 − Γ/Ω 0 Z π e−Ωt sin θ sin1/2 θ ΓΩ2 = √ dθ (Ωe−iθ − E 0j )2 + Γ2j /4 2π(Ω − Γ) 0
