Heat currents in qubit systems

There is a current interest in quantum thermodynamics in the context of open quantum systems. An important issue is the consistency of quantum thermodynamics, in particular the second law of thermodynamics, i.e., the flow of heat from a hot reservoir to a cold reservoir. Here recent emphasis has been on composite system and in particular the issue regarding the application of local or global master equations. In order to contribute to this discussion we discuss two cases, namely as an example a single qubit and as a simple composite system two coupled qubits driven by two heat reservoirs at different temperatures, respectively. Applying a global Lindblad master equation approach we present explicit expressions for the heat currents in agreement with the second law of thermodynamics. The analysis is carried out in the Born-Markov approximation. We also discuss issues regarding the possible presence of coherences in the steady state.


I. INTRODUCTION
There is a current interest in quantum thermodynamics [1][2][3][4][5][6][7][8][9].Thermodynamics is a universal theory and must from a fundamental point of view emerge from a microscopic point of view.In this context the emergence of in particular the second law of thermodynamics.
In the Clausius formulation the second law implies that heat flows from a hot reservoir to a cold reservoir and this essential property should also hold in the quantum regime.In non equilibrium quantum statistical mechanics the instantaneous heat current is subject to both statistical and quantum fluctuations, however, the mean heat current must obey the second law of thermodynamics.
The natural framework for discussing these issues is the theory of open quantum systems [10][11][12].Here the focus is on a small controllable quantum system interacting with an environment which can be either radiation fields or heat baths.In order to obtain a steady heat current we must drive the quantum system by several heat reservoirs, typically two reservoirs, maintained at different temperatures.
Whereas the case of a single qubit (two-level atomic system) driven by two heat bath is easily analysed [13] and discussed in the present paper, the discussion in the literature is usually focussed on composite systems [4,14].Here a recent discussion has focussed on the connection between a global master equation approach as applied in the present paper and a local master equation approach in which the interactions in the composite system are ignored in the dissipative kernels.It has been shown that in the case of a local master equation approach the apparent violation of the second law can be remedied by using a protocol of repeated interactions; this scheme accounts for a work contribution which together with the heat contribution restores the validity of the second law [2,[15][16][17].In the case of coupled quantum oscillators driven by two heat reservoirs a global approach was discussed in [18] resolving the issue regarding the second law.We also note recent work [19] dealing with the general issues regarding the consistency of quantum master equations in the analysis of quantum thermodynamics.
In the present work, motivated by the study in [18], we discuss a composite system of two coupled qubits driven by two heat reservoirs.Coupled qubit systems has been studied extensively but mainly in the context of quantum entanglement [20][21][22][23][24][25][26][27][28][29][30][31]; in the present paper, however, the focus is on quantum thermodynamics and the second law of thermodynamics.
To the best of our knowledge explicit expressions for the heat currents in a coupled twoqubit systems exhibiting consistency with the second law of thermodynamics are, however, not available.We therefore list our findings below.In the simple case of a single qubit driven by two reservoirs we find for the currents q A from reservoir A to the qubit the expression where ω 0 is the energy splitting, g A,B ≡ g A,B (ω 0 ) the spectral functions, and n A,B ≡ 1/(exp(ω 0 /T A,B ) − 1) the Planck distributions.
In the case of coupled qubits we obtain [n A (ω + ) − n B (ω + )], (1.3) where In discussing open quantum systems described by a master equation the focus is on the density matrix ρ pp ′ (t), where the index p refers to an energy basis.Here we distinguish between populations characterised by the diagonal density matrix elements ρ nn (t) and coherences described by the off-diagonal matrix elements ρ nm (t), n ̸ = m.For physical reasons the populations must be positive, ρ nn ≥ 0 and, moreover, satisfy the trace condition n ρ nn = 1 (conservation of probability).Such a requirement does not apply to the coherences ρ n̸ =m which in general are complex numbers.For a single qubit driven by one or several heat reservoirs the coherences do not couple to the populations.The populations settle to their steady state values and the coherences die out.For a composite system the situation is different.Here coherences can in principle couple to populations and influence the steady state.This is for example the case in [32][33][34].Here we address this issue in the case of driven coupled qubits, It is well-established that the proper master equation to be used in dealing with open quantum systems is the Lindblad form [10,[35][36][37] which in the Markov limit ensures that the density matrix evolves in time according to a quantum dynamical semigroup defining a so-called quantum channel [38].The Lindblad master equation thus ensures that the populations stay positive and that the trace condition n ρ nn (t) = 0 (conservation of probability) is satisfied.The Redfield master equation [39] introduced prior to the Lindblad form preserves the trace but does not represent a quantum channel and can lead to unphysical negative populations; note that the trace condition does not in itself ensure positive populations.
In recent work we applied a field theoretical condensed matter approach to open quantum system [40].Choosing a Caldeira-Leggett multi oscillator heat bath [41,42] and invoking Wick's theorem we derived a Dyson equation for the transmission operator T propagating the density operator according to ρ(t) = T (t, 0)ρ(0).The Dyson equation incorporating secular effects has the schematic form T = T 0 + T 0 KT , where the kernel K can be determined by a an expansion in powers of the system-bath coupling in terms of Feynman diagrams.
Finally, applying a so-called quasi particle approximation, well-known in condensed matter many body theory, we derived a master equation in the Markov approximation.In the field theoretical approach the Dyson equation replaces some of the physical assumptions made in the standard derivation of a master equation.In the Born approximation the field theoretical method basically corresponds to the Redfield equation [10,39] in an energy basis.
Whereas the trace condition is automatically satisfied, the field theoretical approach does not define a quantum channel, unless a further selection rule is imposed.The selection rule corresponds to the rotating wave approximation used in turning the Redfield equation into the Lindblad equation in the standard derivation [10].In more detail, the system operator S α in an energy basis, S α nn ′ , appears in a bilinear combination S α nn ′ S β pp ′ in both the Redfield and Lindblad cases.As discussed in [40], the selection rule enters as a constraint on the energy transitions according to S α nn ′ S β pp ′ δ(E nn ′ + E pp ′ ), where E nn ′ = E n − E n ′ is the energy associated with the transition n → n ′ .In a physical interpretation the selection rule implies that the transition energies in the bilinear terms have to match.It is also clear that whether the selection rule imposes a constraint depends on the matrix elements S α nn ′ .In cases where the selection rule is trivially satisfied the Redfield and Lindblad approaches yield identical density matrices.
In the present paper we find that a Redfield approach to the case of coupled qubits does yield coherence contribution to the steady state and, moreover, gives rise to unphysical negative populations.In a proper Lindblad approach, corresponding to a quantum channel, the populations are positive and the heat currents are in accordance with the second law of thermodynamics.We note that the present general analysis included for the sake of completion has a strong overlap with the analysis in [19].The paper is organised as follows.
In Sec.II we discuss i) the master equation, ii) the heat currents, and iii) the Redfield -Lindblad approaches and coherences.In Sec.III we consider i) the single qubit case and ii) the coupled-qubit case.In Sec.III C we show that a Redfield approach yields unphysical negative populations.In Sec.IV we give a brief discussion and summary.Details regarding the Redfield approach is deferred to an Appendix

II. GENERAL ANALYSIS
Here we set up the general scheme for the evaluation of the density matrix and heat currents for an open system driven by two heat reservoirs.

A. Master equation
The density operator for an open quantum system S characterised by the Hamiltonian H interacting with a single or several heat reservoirs is in the Markov approximation governed by a master equation of the form [10,11,40] The von Neumann term −i[H, ρ(t)] yields the reversible unitary time evolution [43].The second term driven by the super operator L characterises the coupling to the environment.
This term has the structure L = K + i∆, where ∆ is a Lamb type correction to the energy levels.The dissipative kernel K characterises the relaxation of the system.In the following we assume that the shift ∆ is incorporated in H and focus on the dissipative kernel K.
In an energy basis, which we shall use in the following, we have H|n⟩ = E n |n⟩, where |n⟩ denotes the wavefunction and E n the associated energy.Expanding the von Neumann term the master equation in (2.1) takes the form where the energy shift is given by E pp ′ = E p −E p ′ , the density matrix by ρ(t) pp ′ = ⟨p|ρ(t)|p ′ ⟩, and the kernel super matrix by K pp ′ ,qq ′ = ⟨p|⟨q ′ |K|q⟩|p ′ ⟩.We are, moreover, in the following assuming that the Markov approximation, i.e., the separation of time scales, applies and that the kernel K is time independent.
Consequently, the steady state density matrix denoted by ρ pp ′ is determined by a set of linear equations note that the trace condition n ρ nn (t) = 1 implies n dρ nn (t)/dt = 0 and thus 3) with respect to populations and coherences, respectively, we obtain the coupled linear equations where in (2.5) the populations ρ qq are coupled to the coherences ρ q̸ =q ′ driving the populations in (2.6).Thus, in order for coherences to influence the population we must have K pp,q̸ =q ′ ̸ = 0.

B. Currents
The mean energy of the system is given by E(t) = Tr(Hρ(t)).For the energy or heat flux we thus have q(t) = Tr(Hdρ(t)/dt) or inserting (2.3) For a single heat reservoir the system equilibrates and it it follows directly from (2.3) that the heat current q vanishes.However, in the presence of several heat reservoirs maintained at different temperatures a steady state heat current will be generated.
In the case of two uncorrelated heat reservoirs A and B the kernel K in (2.3) is composed of two parts associated with each reservoir and we have note that in the case where the heat bath are correlated there will an additional term K A,B pp ′ ,qq ′ depending on both heat baths.
As a result the total current is determined by (2.9) The vanishing total current has components associated with each bath, i.e., q = q A + q B = 0, (2.10) ) (2.12) We infer that the heat flux from the reservoir A to the system S given by q A equals the heat flux −q B = q A flowing from the system S into the reservoir B, expressing energy conservation, i.e. the first law of thermodynamics.Regarding the second law of thermodynamics the issue is to show that a positive heat current flows from the hot heat reservoir to the cold heat reservoir.From the above it follows that in order to evaluate the heat currents for a given open quantum system driven by two reservoirs we need three components: The energy spectrum of the system E n , the kernel elements K A,B nn,qq ′ , and the density operator ρ qq ′ .

C. Redfield -Lindblad -Coherences
Then issue of coherences in the steady state and currents is subtle [32][33][34].In recent work [40] mentioned in the introduction we presented a detailed discussion of the application of field theoretical methods to open quantum systems.The field theoretical approach is based on a systematic expansion in terms of diagrams and applies a condensed matter quasi-particle approximation in order to implement the Markov limit.The field theoretical method is the basis for the present analysis.
Assuming a system -bath coupling of the form α S α B α , where S α are the system operators and B α the bath operators, the dissipative kernel K pp ′ ,qq ′ in the Born approximation and in an energy basis is given by Here D αβ (ω) is the bath correlation function given by where By inspection of (2.13) it is easily seen that the trace condition in (2.4) is satisfied.
However, the field theoretical approach which in the Born approximation is equivalent to the Redfield equation [39] does not represent the generator of a quantum dynamical semigroup and thus does not guarantee that the populations are positive.In order to implement the semigroup properties and thus guarantee positive populations one must impose a further secular rotating wave approximation (RWA) yielding the Lindblad equation [10].
In operator form the Lindblad master equation has the general form However, as shown in [40] the introduction of the RWA amounts to introducing delta function constraints in the kernel (2.13) yielding the expression (2.17) In the interaction representation the system operator S α pq (t) ∝ exp(−iE pq t).Consequently, the product S α pq (t)S β p ′ q ′ (t) ∝ exp(−i(E pq + E p ′ q ′ )t).For E pq + E p ′ q ′ ̸ = 0 this term will oscillate and is discarded in the RWA.As a result, whether the delta function constraint eliminates terms in the super operator K pp ′ ,qq ′ depends on the matrix elements S α pq and thus on the specific model under consideration.
We stress that a proper analysis satisfying the correct physics requires the Lindblad equation.Whether the Lindblad equation then generates coherences in the steady state will, as mentioned above, depend on the specific model [32][33][34].In the case of a single driven qubit it follows immediately that coherences are absent in the steady state.In the coupled qubit case we find that a Redfield treatment according to (2.13) yield coherences in the steady state and unphysical negative populations.On the other hand, for coupled qubits a Lindblad treatment according to (2.17) yields vanishing coherences, positive populations and simple expressions for the currents.

III. QUBIT SYSTEMS DRIVEN BY TWO HEAT RESERVOIRS
In this section we turn to the case of qubit systems driven by two heat reservoirs generating a non equilibrium heat current.As an example and illustration we first consider the case of a driven single qubit.The second case, constituting the main issue in the paper, is a coupled qubit system.

A. Single qubit
Here we consider the case of a single qubit driven by two heat reservoirs A and B maintained at temperatures T A and T B , respectively.In this case the energy or heat flows from the hot reservoir to the cold reservoir via the qubit; the configuration is depicted in Fig. 1.

Model
In a Pauli matrix basis [44] the Hamiltonian and the coupling to the reservoirs are given by For the bath correlations we find with spectral functions and Planck distributions ) ) we note that the delta function constraints in (2.17) are trivially satisfied showing that the Redfield and Lindblad schemes yield identical kernel elements.Note also that there is no coupling between populations and coherences, i.e.K nn,q̸ =q ′ = 0.

Density matrix
Expressing the kernel K in matrix form we have (suppressing indices) forming a "population" block P and a "coherence" block C. For the determinant of K we thus have det K = det P det C, where det P = 0 and det C = (g(1 + 2n)) 2 /4 ̸ = 0.
Consequently, the populations ρ nn determined by the linear equations P ρ = 0, requiring a vanishing determinant, yield finite populations, whereas the coherences determined by Cρ = 0 vanish in the steady state since the determinant is non vanishing.Applying the scheme in Sec.II we readily obtain the populations or diagonal density matrix elements A few comments regarding the density matrix.Introducing the ratio r = ρ ++ /ρ −− we have from (3.18) and (3.19) In general the ratio r depends on both the spectral densities g A,B , i.e., the structure of the reservoirs, and the temperatures T A,B .Turning off for example reservoir B by setting g B = 0 we obtain the well-studied case of a single qubit driven by a single reservoir [10].

Currents
Noting that E + = ω 0 /2 and E − = −ω 0 /2 the scheme in Sec.II likewise yields expressions for the currents.The steady state heat currents from reservoir A and from reservoir B to the qubit are thus given by First we note that by construction q A + q B = 0 showing that energy is conserved, i.e., the first law of thermodynamics.Secondly, since n A > n B for T A > T B energy flows from the hot heat reservoir with temperature T A to the cold heat reservoir maintained at temperature T B , demonstrating that the second law of thermodynamics holds.
Regarding the currents given by (3.21) and (3.22) we note that they depend on both spectral representations g A and g B .Removing for example reservoir B by setting g B = 0 the currents vanish and the qubit coupled only to reservoir A equilibrates with temperature T A .In the high temperature limit for T A,B ≫ ω 0 we obtain expanding (3.21) and (3.22) the classical results at low temperatures for T A,B ≪ ω 0 the vanishing currents

B. Coupled qubits
This is the main section in the paper.We discuss in detail the case of two coupled qubits driven by two reservoirs as a model of a simple composite system.Heat reservoir A drives qubit 1 and heat reservoir B drives qubit 2. In this case heat flows between the reservoirs and the qubits transmitted across the system by the qubit coupling; the configuration is depicted in Fig. 2.

Model
In a Pauli matrix basis [44] the Hamiltonian and coupling to the reservoirs are given by

Kernel
In order to evaluate the dissipative kernel according to the scheme in Sec.II we require the matrix elements of the system operators σ ± 1,2 in the energy basis |n⟩, n = 1 • • • 4 of the system Hamiltonian H in (3.27).With the notation (σ ± 1,2 ) pq = ⟨p|σ ± 1,2 |q⟩ we find the non-vanishing matrix elements and associated energy transitions We note that reservoir A drives the σ 1 transitions, whereas reservoir B drives the σ 2 transitions.Moreover, the system operators only appear in the combinations σ + 1 σ − 1 and σ + 2 σ − 2 in the kernel.We also note that the transitions fall in two groups.The transitions 1 ↔ 3 and 2 ↔ 4 are associated with ω + , whereas the transitions 1 ↔ 2 and 3 ↔ 4 are associated with ω − .
In order to identify nonvanishing kernel elements we inspect the expressions (2.13) in the Redfield case and (2.17) in the case.We introduce the positive parameters a n , b n depending on the coupling strength, spectral densities, Planck distributions and temperatures according to the scheme Using the assignment , we obtain, since the delta function constraints are trivially satisfied, in both the Redfield and Lindblad cases the kernels K A and K B : and In the Lindblad case, due to the delta function constraints, the above expressions for K A,B , yielding populations and currents, exhaust the class of non vanishing kernel elements.
In the Redfield case, relaxing the delta function selection rules, we find non vanishing expressions for kernel elements of the form K A,B pp,q̸ =q ′ , K A,B p̸ =p ′ ,qq , and K A,B p̸ =p ′ ,q̸ =q ′ indicating that the populations couple to the coherences.The Redfield case will be discussed in the next section.However, since the Lindblad equation represent a proper quantum channel we proceed with the evaluation of populations and currents.

Density matrix
Introducing the total kernel K = K A + K B , where K A and K B are given by (3.48) and (3.49), respectively, the populations ρ nn are determined by the linear equations q K pp,qq ρ qq = 0.By inspection we note incidentally that det K = 0 allowing for a solution.In terms of the parameters we find the normalised populations, i.e., n ρ nn = 1, In the high temperature limit T A , T B ≫ ω + , ω − we have n A,B (ω ± ) → T A,B /ω ± and a simple calculation yields ρ nn → 1/4, i.e., equal population of the four energy levels.Likewise, for T A , T B ≪ ω + , ω − we have ρ 11 → 1 and ρ 22 , ρ 33 , ρ 44 → 0, i.e., occupation of the lowest energy level.
Finally, for λ = 0 we have By inspection we conclude that the qubits decouple and we obtain for the density operator the direct product ρ = ρ A ⊗ ρ B , where and are the equilibrium density operators for qubit 1 coupled to reservoir A and qubit 2 coupled to reservoir B, respectively.
In order to illustrate the Lindblad case we have in Fig. 3 depicted the populations ρ nn as a function of the mean temperature T = (T A + T B )/2 for T A = T B , i.e. in the equilibrium case, with parameter choice ω 1 = 1.0, ω 2 = 2.0, g A (ω ± ) = g B (ω ± ) = 1.0, and λ = 0.5.

Currents
Applying the scheme in Sec.II the heat currents q A and q b are given by q A = n E n K A nn,pp ρ pp and q B = n E n K B nn,pp ρ pp and we obtain the expressions or by insertion of a n , b n , s n and By inspection we have q A + q B = 0 expressing energy conservation, i.e., the first law of thermodynamics.Moreover, and more importantly, since n A (ω ± ) > n B (ω ± ) for T A > T B the current expressions are in accordance with the second law of thermodynamics, i.e., heat flows from the hot reservoir to the cold reservoir; this result corroborates the analysis in [18] in the case of coupled oscillators driven by two reservoirs.
The heat currents q A and q B in (3.56 -3.61) fall in two parts associated with the energy shifts ω + and ω − .For vanishing coupling, λ = 0, we have α = 1 and β = 1, the qubits are uncoupled and the currents vanish.Likewise, quenching e.g.reservoir B by setting g B = 0 we obtain a vanishing current.In the high temperature limit for T A , T B ≫ ω + , ω − we have the classical results and At low temperatures T A , T B ≪ ω + , ω − , (3.66) (3.67) and In Fig. 4 we have depicted the currents q A and q B as a function of T A in the range 0.5 to 1.5 with T B = 1, with parameter choice ω 1 = 1.0, ω 2 = 2.0, g A (ω ± ) = g b (ω ± ) = 1, and λ = 0.5; note that the currents vanish in equilibrium for T A = T B .

C. Redfield approach
In this section we discuss briefly the Redfield approach to the coupled qubit system.
Relaxing the delta function constraint or selection rule yielding the Lindblad equation it follows from Sec. II that the kernel elements K A,B nn,23 , K A,B nn,32 , K A,B 23,nn , K A,B 32,nn for n = 1, 4 and K A,B 23,23 , K A,B 32,32 are non vanishing indicating that the populations ρ nn couple to the coherences ρ n̸ =n ′ .The calculation is deferred to Appendix .
Considering the case g A,B (ω ± ) = g we find the populations and coherences and The trace condition n ρ nn = 1 yields the normalisation factor and we have, moreover, introduced the parameters shows that ρ 44 and ρ 44 assumes negative value in the range 0 < T ≲ 6.0, indicating that the Redfield approach is inadequate in analysing the coupled qubit case.

IV. DISCUSSION AND SUMMARY
In this paper we have discussed heat currents in two qubit systems driven by two heat reservoirs maintained at different temperatures in order to test the validity of the second law of thermodynamics in the microscopic domain, an important issue in quantum thermodynamics.The results for a single qubit system driven by two reservoirs are given in Sec.III A and are in accordance with the second law.The case of coupled qubits is more involved, the results are presented in Sec.III B and are likewise in agreement with the second law.We note that the current expressions in the single qubit and coupled qubit cases have a similar structure.In deriving these results it is essential to apply a global approach and operate in the basis of the full system Hamiltonian.Moreover, in order to avoid an erroneous result it is also essential to use the Lindblad master equation approach and in this way ensure a proper quantum channel with positive populations.
In order to discuss the possible role of coherences, we made use of a recent field theoretical approach to open quantum system.This method yields at a first stage the Redfield master equation and has to be supplemented with a selection rule in order to produce the Lindblad master equation.The Redfield equation applied directly to the coupled qubit case gives rise to a coupling between coherences and populations in the density matrix but yields negative populations and is thus inadequate is describing the proper physics.

Appendix: The Redfield case
Here we give details of the Redfield approach to the coupled qubit case.In the following we assume identical reservoirs regarding their structure and define g = g A,B (ω ± ).We introduce the parameters c n , d n according to the scheme With the kernel assigment  q A q B FIG. 4: We depict the heat currents q A and q B as functions of T A for T B = 1.
Hamiltonian, b k the Bose operator associated with the wavenumber k, and T the associated temperature.
) where the levels |+⟩ and |−⟩ have energies E + = ω 0 /2 and E − = −ω 0 /2, corresponding to the energy splitting ω 0 .The qubit is driven by two heat reservoirs in the rotating wave approximation (RWA).The bath operators A and B sample the bath modes with wavenumber k, frequencies ω A,B k , and strength λ A,B k ; a k and b k are Bose operators.Referring to Sec.II we choose the assignment

. 38 )E 4 =
ω m .(3.39)The expansion coefficients and relevant energy shifts ω + = E 42 = E 31 and ω − = E 43 = E 21 are given by α = cos θ/2, (3.40) β = sin θ/2, (3.41) .75) In equilibrium T A = T B we have k = 0.The populations are decoupled from the coherences and the Redfield and Lindblad expression for the populations are in agreement.In the non equilibrium case for T A ̸ = T B we have k ̸ = 0 and the populations are reduced due to the coherences.The coherences ρ 23 and ρ 32 are complex conjugate (the density matrix is hermitian) and their sign depends on k ∝ (n A (ω ± ) − n B (ω ± )), i.e., the sign of T A − T B .We have in Fig.5depicted the populations ρ nn as a function of the mean temperature T = (T A +T B )/2 in the range ∆T < T < 8 with non equilibrium bias ∆T = (T A −T B )/2 = 5.0 with parameter choice ω 1 = 1.0, ω 2 = 2.0, g A (ω ± ) = g b (ω ± ) = 1.0, and λ = 0.5.The plot