Exploiting coherence for quantum thermodynamic advantage

The introduction of the quantum analogue of a Carnot engine based on a bath comprising of particles with a small amount of coherence initiated an active line of research on the harnessing of different quantum resources for the enhancement of thermal machines beyond the standard reversible limit, with an emphasis on non-thermal baths containing quantum coherence. In our work, we investigate the impact of coherence on the thermodynamic tasks of a collision model which is composed of a system interacting, in the continuous time limit, with a series of coherent ancillas of two baths at different temperatures. Our results show the advantages of utilising coherence as a resource in the operation of the machine, and allows it: (i) to exhibit unconventional behaviour such as the appearance of a hybrid refrigerator, capable of simultaneous refrigeration and generation of work, and (ii) to function as an engine or a refrigerator with efficiencies larger than the Carnot bound. Moreover, we find an effective upper bound to the efficiency of the thermal machine operating as an engine in the presence of a coherent reservoir.


Introduction
Due to the nanotechnological progress and the increasing interest in quantum systems, heat engines are no longer limited to the size of steam engines from the industrial revolution [1].
In recent years, quantum resource theories have been established to study and characterise the practicality of quantum coherence effects [37]. In the context of thermodynamics, coherence was shown to be an essential ingredient for optimal charging of quantum batteries [38,39,40,41,42,43,44,45], it helps the transfer of energy in photosynthetic complexes [46,47] and leads to the emergence of interesting phenomena like heat flow reversals without reversing the arrow of time [48]. Coherence has been investigated as a promising candidate for designing thermal machines in which the working medium is a quantum system in contact with reservoirs at different temperatures and as a resource for exploring the fundamental limit of their efficiencies [49,28,50]. As a matter of fact, it is of great importance to consider scenarios where one can model and manipulate non-equilibrium reservoirs. By employing them, one expects to overcome the standard thermodynamic limits as proposed in the case of quantum measurement induced [51,52], squeezed [53,27,54,55,56,57] and coherent [17,58,59,24,60,61,62,63,64,65] engineered reservoirs. Analysing the machine's operation in both cases of thermal and coherent reservoirs represents the most genuine way to explore novel aspects that may be induced by coherence. For instance, it was reported in Ref. [66] that coherence injected in a finite sized reservoir interacting with a system can quantitatively modify the second law of thermodynamics and serves as a useful supply for enhancing the output efficiency of heat engines functioning between two coherent reservoirs.
For conventional system-reservoir models with a large continuum of modes, injecting coherence may become technically onerous [34]. A more versatile alternative that may lower these practical limitations is offered by quantum collision models (CM) [67,68,69,70,71,72,73,11,74,75]. Because of the simplicity of their mechanism, they are well-qualified to address the thermodynamics of coherent engineered reservoirs [76,77,78]. The basic formulation of CMs consists of depicting the environment as a large ensemble of elementary components or sub-units, often called ancillas, that interact sequentially with a system S. The system-ancilla joint dynamics are described by collisions that may or may not preserve energy. The conditions for a memoryless Markovian behaviour of the open quantum system evolution imply that the ancillas are initially prepared in a product state (uncorrelated) and that every ancilla collides only once with S. In addition to being an interesting asset to study non-thermal environments, this microscopic formalism is highly suited to model specific experiments based on micromasers [79,80], nuclear magnetic resonance NMR [81], superconducting quantum computers [82,83,84,85] and all-optical settings [86,87,88].
Although there is an active line of ongoing research on quantum CMs, the different effects of exploiting coherence in the quest of enhanced quantum thermal machines are little explored so far. On the other hand, it was a motivation for a recent derivation of the thermodynamic behaviour of an autonomous quantum thermal machine [89] that operates via collisions with a series of qubits which are initially prepared with energetic coherence, as well as a basic setup of a system interacting with coherent environmental ancillas [78].
In this work, we study a thermal machine consisting of a single qubit as a working substance and two reservoirs modelled by CMs. We carefully analyse the thermodynamics of the system's evolution in the presence of an infinitesimal amount of coherence in the reservoirs and consider the non-equilibrium steady state of the system [90,91]. We split the heat flows into a coherent contribution, arising because of the quantum coherence in the baths, and an incoherent contribution corresponding to the typical dissipation term. As a consequence of having non-resonant collisions, power can instead be divided into a term originating from coherence and a term that stems from the collisions. Furthermore, consuming coherences from the reservoirs allows the machine to perform thermodynamic tasks in classically forbidden regimes with efficiencies beyond the Carnot bound of the corresponding thermal reservoirs without coherence. We find the effective bound on the efficiency of the machine operating as an engine in the presence of quantum coherence. For a cold coherent reservoir, the device is capable of simultaneous refrigeration and generation of work, providing a kind of device dubbed hybrid refrigerator [92,59].
The paper is structured as follows: After introducing the coherent CM setup and its dynamics in Sec. 2, we lay out the different results in the case of harnessing coherence in the cold and the hot bath in Sec. 3 as well as its influence on the operational regimes of the machine and its efficiency at maximum power. Finally, we provide some concluding remarks in Sec. 4.

Evolution
We consider a system described by a Hamiltonian H S and interacting with an environment modelled with the collision model. The environment consists of N B baths each containing an infinite ensemble of identically prepared ancillas which interact sequentially with the system for a time τ and then are discarded, see Fig. 1. We assume that there are no initial system-environment correlations. The reduced dynamics of the system after one collision can be written as ( = 1): where ρ S [ρ S ]is the state of the system before [after] the collision, is the state of the environment and we do not assume any initial correlations between the baths, and is the total system-environment Hamiltonian. The total Hamiltonian of the environment is the sum of the corresponding Hamiltonian H E,i for each bath: We assume the system-environment interaction to be of the general form H SE = where g i,k are the coupling constants between the system and the ancilla from bath i. The index k lists the different system's operators interacting with a corresponding ancilla. The operators S i,k and A i,k pertain to the system and ancilla's Hilbert space, respectively. Moreover, we also assume the operators A i,k to be eigenoperators of the ancilla's Hamiltonian, such that: As we will see, the factor √ τ in Eq. (5), though not necessary, ensures consistency when taking the continuous limit τ → 0.
We now assume that the ancillas are prepared in a thermal state at temperature T i with a small coherence term ( where ρ th E,i = e −β i H E,i /tr e −β i H E,i and i quantifies the ancilla's quantum coherence. In the continuous limit, τ → 0, following Ref. [78,93], the evolution of the system's reduced density matrix is ruled by the following Markovian master equation: where the effective Hamiltonian correction is given by The dissipators are defined by: where the Lindbladian is defined as: The dissipation rates where the averages · are taken over the thermal part of the environmental state ρ th E,i , fulfil the local detailed balance condition

Thermodynamic quantities
Thermodynamic quantities can be calculated following Ref. [93,78,11]. We define work in the usual way as arising from the time-dependence of the total Hamiltonian, see below. Other contributions to the change of internal energy come from energy exchanges between the system and the environment and will be collectively categorised as heat. Therefore, the heat current flowing from bath i is obtained from the energy change of the corresponding ancilla before and after the collision: where we have defined the change in the total density matrix of the system plus environment: Expanding Eq. (13), and following the detailed derivation presented in Appendix A, we may split the heat flow from bath i into coherent and incoherent contributions: The coherent contribution of the heat arises because of the initial coherence in the ancilla and reads: where, in analogy to the operator G S , we have defined where tr E i is the partial trace on the system and all the environments except the ith.
The heat current's incoherent contribution depends on the typical double-commutator structure:Q Notice, however, that the last expression does not necessarily reduce to the common expression involving the system's expectation value of the dissipator applied to the system's Hamiltonian tr[D i (H S )ρ S ] unless the ancillas are resonant with the system's transition they are coupled to. In the next section, we will consider a situation in which this resonant condition is not met. The total power that needs to be injected or extracted from the system iṡ Notice that if we define the rate of change of the internal energy: then the first law of thermodynamics is automatically satisfied: The total power can be split into a coherent and a collisional contribution: The coherent contribution for the power arises only when the ancilla has some initial coherence and its expression reads: The power due to the collisioṅ arises for local (but not global) master equations when the system-environment coupling does not satisfy local energy conservation, in other words, when [H SE , H S + H E ] = 0 [94]. The splitting of heat and work considered here is in line with other studies on quantum thermodynamics with non-equilibrium reservoirs, see for instance Ref. [78]. However, we remark that this splitting is different, although physically equivalent, to the one considered in Ref. [78] in which the condition [H SE , H S + H E ] = 0 was assumed. As a consequence, the term that we dubQ i,coh was interpreted there as a power term arising from coherence while the power terms we dubẆ coh andẆ coll were identically zero.
In this paper, all terms arising from the energy change of the environment have been identified as heat terms, coherent and incoherent. The remaining terms in the energy balance arise from the time-dependence of the total Hamiltonian have been identified as power contributions which include a termẆ coh only arising because of the ancilla's coherence and a termẆ col only arising because of the "locality" of the master equation as discussed in [93].
If the system's Hamiltonian is explicitly time-dependent there will also be an associated power term proportional to ∂H S /∂t. We will not consider this additional term in this work. We are assuming the sign convention such that power or heat currents are positive when energy flows into the system. Depending on the signs of these quantities the device exhibits different functionings as discussed in the next section.

Entropic quantities
We now pass to discuss the change in entropic quantities and the second law of thermodynamics. During a collision, the ancillas may lose or gain quantum coherence related to the off-diagonal entries of their density matrices in their energy eigenbasis. By using the definition of the relative entropy of coherence [37,95] where S(ρ) = − tr ρ log ρ is the von Neumann entropy and ρ d is the diagonal part of ρ in a given basis, we can measure the rate of change in relative entropy of coherence of the ancillas before and after the collision: Using perturbation theory and taking the limit of τ → 0, we find the following relation between the relative entropy, coherent heat flow and change in relative entropy of coherence of an ancillȧ where we have defined the relative entropy as S(ρ ||ρ) = Tr[ρ log ρ ] − Tr[ρ log ρ]. This relation extends the result found in Ref. [78] to a more general scenario: there the relative entropy was related to the coherent power and not to the coherent heat. Since the relative entropy in Eq. (27) cannot be negative, we obtain a lower bound for the coherent heat flow from each bath, which can be interpreted as a modified local second law of thermodynamics While the previous expression concerns each bath individually, we can also obtain a modified second law for the whole environment. Because the initial state of the ancilla's, ρ E , is a product state, we may write the total relative entropy of the bath as where we have defined the state of the environment after each collision and the total mutual information as while I(ρ E ) = 0. We thus find: which represents a global modified second law. Classes of modified versions of the second law of thermodynamics stemming from coherence in a system interacting with a large environment have been investigated in the literature, see e.g. [96,97,98] whereas, for coherent reservoirs, they remain not widely explored [78,75]. Our modified second law demonstrates that the coherent heat contribution is constrained by the loss of coherence in the auxiliary's state and that clearly shows that coherence is a resource to be harnessed to perform thermodynamic tasks in non-equilibrium processes. Ref. [66], obtained similar results but for a generic finite size reservoir with coherence.
The results of this section are quite general and only depend on a few assumptions, chiefly no initial system-environment correlations or within the environment. In the next section we showcase them for the specific case of a one-qubit system coupled to two environments at different temperatures.

Results
We now specialise our problem to that of a single qubit in contact with two baths (N B = 2). We assume the qubit's Hamiltonian to be: We also assume the ancillas to be qubits such that their Hamiltonians read: Notice that the ancillas are not resonant with the qubit. This causes extra terms to appear in the power and heat expressions proportional to the non vanishing gap [99].
The system-environment interaction Hamiltonian is assumed of the rotating-wave type with S i = σ − S and A i = σ − E,i (where we have therefore dropped the index k as there is only one value). For the state of the ancilla before the collision we choose: where the angle φ i can be interpreted as the azimuth of the ancilla's Bloch vector. Under these assumptions, the master equation for the system's qubit becomes: where the effective Hamiltonian correction is: and we have assumed equal rates γ for the two baths such that: g 2 E,i = 2γ(2n i + 1), where n i = [exp(2β i B) − 1] −1 is the thermal occupation in each ancilla. Notice that in the absence of environmental coherence ( i = 0), the master equation would simply correspond to that of a qubit in contact with a single effective bath with an average thermal occupation n = (n 1 + n 2 )/2 and in the long-time limit will equilibrate with this average thermal occupation.
The steady state of the system's qubit can be found by solving the equationρ S = 0 and its analytical expression is reported in Appendix B. This shows that the steady state density matrix elements are affected by the presence of coherence, resulting in terms proportional to 1 and 2 .
We now pass to the thermodynamic quantities, heat currents and power. To this end, to simplify the results we assume coherence only in one bath and set 2 = 0. The case with coherence in both baths does not lead to additional qualitatively different scenarios considered here as proven in Appendix D.
Detailed expressions for the coherent, incoherent and collisions contributions to heat and work for any state of the system can be found in Appendix C. Substituting the expressions for the steady state, found in Appendix B, in Eqs. (13)- (19), we obtain for the heat currents and power (totalling their coherent, incoherent and collisional contributions):Q where the common factor is given by: V ( 1 ) = 2γ B 2 (n 1 − n 2 ) + γ(1 + n 1 + n 2 ) [(n 1 − n 2 )(1 + n 1 + n 2 )γ + (2n 1 + 1) 2 1 ] (1 + n 1 + n 2 ) [B 2 + (1 + n 1 + n 2 ) 2 γ 2 + (2n 1 + 1)γ 2 1 ] . (42) The fact that the thermodynamic quantities contain a common factor is a consequence of the system-environment exchange interaction Hamiltonian that we have assumed and which was found in related models, e.g. in Refs. [93,11,99]. An immediate consequence is that the ratios of these quantities, linked to the efficiency and coefficient of performance (COP), only depend on the ratios of the ancillas magnetic fields yielding Otto-like expressions. For instance, we obtain the efficiency where the input heatQ in is the sum of all positive heat contributions. Notice that the factor V ( 1 ) contains explicitly the strength of coherence 1 but not its phase. In the case of two baths with coherence, only their relative phase would appear in the expression, see Appendix D. The factor V ( 1 ) also contains the system's magnetic field B and would appear also in the absence of bath coherence [99]. In this case, this field plays the role of an effective detuning which increases the magnitude of the heat currents and power but does not alter their signs and therefore the type of operating machine.
In the next two subsections we illustrate our results for the case in which the cold environment produces coherent ancillas (T 1 < T 2 ) and the opposite case in which the hot environment is coherent (T 2 < T 1 ).

Coherence in the cold bath
We start by assuming that the coldest bath has some initial coherence; therefore we assume that T 1 < T 2 . In this case, depending on the choice of the coherence strength 1 and the magnetic fields B 1 and B 2 , the setup can operate as different types of thermal machines, as summarised in Table 1.
The two conditions (44)-(45) determine the functioning diagram reported in Fig. 2(a). There, we see that all four regimes illustrated in Table 1 appear for certain values of the parameters. In the panels (b,c,d) of Fig. 2 we plot the thermodynamic quantitieṡ W ,Q 1 andQ 2 along three cuts of the functioning diagram. The crossing points that appear in panels (c) and (d) of Fig. 2 correspond to effective Carnot points determined by Eq. (45) where all thermodynamic quantities go to zero and change sign. Figure 3 shows how both the coherent and incoherent heat flows from bath 1 vary with 1 . We see that at 1 ≈ 0.2, the contribution to the overall heat flow (from bath 1) due to coherence surpasses the incoherent heat flow. The rate of change of coherencė C(ρ E,1 ) is also plotted in Fig. 3 providing evidence that the bound in Eq. (28) holds.
We now pass to discuss the efficiency and coefficient of performance of the thermal devices. As discussed in the previous section, these quantities are given by the Otto values:  when operating as an engine and when operating as a refrigerator. In absence of coherence and for thermal baths, these quantities are smaller or equal than the corresponding Carnot values: with equality obtained only when the heat currents and power drop to zero. This is because, in absence of coherence ( 1 = 0), the condition COP > COP C is equivalent to the condition n 1 < n 2 that corresponds to the functioning of the device as an engine. However, in the presence of coherence, we see in Fig. 2 that the refrigerator regime survives in a classically "forbidden" area for which n 1 < n 2 . In this area, labelled R in the diagram, the COP is indeed larger than the Carnot value COP C as evidenced in Fig. 4, where we plot the cooling powerQ 1 against the COP . We find that in the region R where COP > COP C , the cooling power is nonzero, in contrast to the Carnot point where power is strictly null. Moreover, Fig. 4 shows that the COP at maximum cooling power is larger in the presence of coherence. The COP is not bounded and diverges at the transition between the functioning as a refrigerator and as a hybrid refrigerator for Regarding the efficiency of the system as an engine, applying the same considerations as before, we find that the condition η > η C would correspond to n 2 < n 1 . In Fig. 2(a) however, we observe that the system never operates as an engine for n 2 < n 1 , even in the presence of coherence.
Two more functionings appear in Fig. 2(a): the accelerator and the hybrid  Figure 4. Cooling powerQ 1 against the COP for the device operating as a refrigerator. We compare the cases with no coherence (solid, 1 = 0) and with coherence (dashed, 1 = 0.6). Parameters as in Fig. 2 with 0 refrigerator. For the accelerator, heat flows in the spontaneous direction (hot to cold) but this process is accelerated by positive work injected into the system and transformed into heat that is dissipated in the cold environment. The hybrid refrigerator instead does the opposite: it extracts heat from the coldest bath, converts part of it into work which can be extracted and dumps the rest into the hot bath. This apparently paradoxical functioning is made possible necessarily by the presence of quantum coherence in the cold bath which acts as an extra source of work.
Summarising, when coherence is present in the cold bath, there is a region of parameters for which the system operating as a refrigerator has a larger, in principle unbounded, COP than the Carnot value. However, when the system operates as an engine, its efficiency is always smaller than the Carnot efficiency. As we will see in the next section, this situation will be reversed when the coherence is in the hot bath.

Coherence in the hot bath
We now consider the case where the coherence is in the hot bath. We thus assume T 1 > T 2 . The conditions for the different operating regimes can be found by looking again at the signs of Eqs. (39)-(41) as we did in the previous section and are reported in Table 2. Notice that the signs ofQ 1 andQ 2 are reversed compared to Table 1.
Equations (44)-(45) are still valid and give us the conditions at which the power is zero. Using these equations we find the functioning diagram shown in Fig. 5(a). In contrast to the case where the coherence is in the cold bath, we see that the device never operates as a hybrid refrigerator. Moreover, a coherence 1 > * 1 allows the system to operate as an engine even when B 2 < B 1 , where in absence of coherence a refrigerator would be expected. In this forbidden zone, the efficiency of the corresponding engine is larger than the Carnot value as shown in Fig. 6.
In Fig. 6(a), we show the power outputẆ against the efficiency when the system behaves as an engine. In absence of coherence, 1 = 0, the maximum achievable efficiency is the Carnot value η C (Eq. (48)) where however the power is zero. The value of the efficiency at maximum power is obtained at the Curzon-Ahlborn value: On the other hand, in the presence of coherence 1 = 0, the efficiency is much larger and surpasses both the Carnot value (at non zero power) and the Curzon-Ahlborn value. Fig. 6(b), shows that the efficiency at maximum power η M P grows quadratically for very small 1 and linearly for larger values. For a given 1 , the maximum value of the efficiency η max is obtained for the value of B 2 at the transition, in the diagram of Fig. 5(a), between the functioning as an engine and that as a refrigerator. The value of B 2 can be obtained by solving the equation: where * 1 is defined in Eq. (45). The value of η max , plotted in Fig. 6(b), is not universal, depending on all parameters, but represents an effective Carnot bound on the functioning of the machine as an engine.

Summary and Conclusions
In this paper, we have demonstrated the role of non-thermal bath effects in nonequilibrium processes. We provide a thermodynamic analysis of a microscopic collision model wherein a single qubit is interacting repeatedly with two local reservoirs at different temperatures, which consist of a collection of initially prepared ancillas with an infinitesimally small amount of coherence.
A key insight is that the system-ancilla interaction does not satisfy local energy conservation, thus generating coherent and incoherent contributions in the heat flow and coherent and collisional terms in the power. In the continuous time limit, we have shown that the loss or gain of coherence in the state of the ancillas is given by a modified second law of thermodynamics which is described by a lower bound for the coherent heat current from one of the baths.
We have shown coherence to be a tuning parameter alongside the magnetic field of the baths for characterising the different operational regimes of the machine. Since including coherence in both baths can be reduced to the case with coherence in only one bath, to simplify the analysis, we studied the latter case. We found that injecting some amount of coherence into the cold bath allows the refrigerator to survive the classically forbidden regime. This implies that its coefficient of performance surpasses the Carnot limit of the corresponding equilibrium reservoirs without coherence. In addition, coherence can result in advantageous effects such as the appearance of a hybrid refrigerator that simultaneously produces work and refrigerates the cold reservoir. In contrast, when the hot bath contains coherence and the cold reservoir is an equilibrium bath, the machine never operates as a hybrid refrigerator. In the case of the heat engine regime, coherence acts as fuel that drives the efficiency beyond the Carnot value corresponding to incoherent baths. Similarly, coherence leads to efficiency at maximum power much larger than the classical Curzon-Ahlborn value.
The simplicity of our model provides a general insight into the advantages of employing coherent reservoirs in the performance of thermodynamic tasks. Our findings can be further explored in systems of higher dimensionality, including quantum harmonic oscillators. Implementing our proposal in current experiments could face critical technical challenges such as scaling, decoherence and manufacturing errors. Nonetheless, the first proof-of-principle demonstrations have already shown great potential for future thermodynamic applications [82,84,85].
We may write In the following we will ignore terms of order τ 2 and higher, as they will tend to zero when dividing by τ and taking the limit as τ → 0. Plugging Eq. (A.5) into (A.1), we findQ The first term represents the coherent heat flow, defined in Eq. (16): where G E,i is defined in Eq. (17). The second expression in Eq. (A.6) therefore gives rise to the incoherent heat floẇ Now let us pass to work. The power is defined aṡ Plugging in Eq. (A.5), we finḋ (A.14) The expression in Eq. (A.13) gives rise to coherent poweṙ The collisional power then arises from the second term in Eq. (A.14)

Appendix B. Steady state solution
In this section, we show the explicit expression of the system's steady state when the system is subject to the master equation Eq. (37). We write the density matrix ρ S in the basis of eigenstates of σ z S as: In this representation, the entries of the steady state density matrix read: where we have introduced the effective decay rate γ eff = 2γ and an effective coherence strength, expressed in complex polar form: with eff and φ, real parameters, denoting its magnitude and phase, respectively.

Appendix D. Quantum coherence in both baths
In this appendix we analyse what happens in the case in which coherence is present in both baths. We are going to show that the expressions of the heat currents and power can be also obtained with coherence only in one of the two baths and with an effective amplitude.