Implications of Coupling in Quantum Thermodynamic Machines
We study coupled quantum systems as the working media of thermodynamic machines. Under a suitable phase-space transformation, the coupled systems can be expressed as a composition of independent subsystems. We find that for the coupled systems, the figures of merit, that is the efficiency for engine and the coefficient of performance for refrigerator, are bounded (both from above and from below) by the corresponding figures of merit of the independent subsystems. We also show that the optimum work extractable from a coupled system is upper bounded by the optimum work obtained from the uncoupled system, thereby showing that the quantum correlations do not help in optimal work extraction. Further, we study two explicit examples; coupled spin- systems and coupled quantum oscillators with analogous interactions. Interestingly, for particular kind of interactions, the efficiency of the coupled oscillators outperforms that of the coupled spin- systems when they work as heat engines. However, for the same interaction, the coefficient of performance behaves in a reverse manner, while the systems work as the refrigerator. Thus the same coupling can cause opposite effects in the figures of merit of heat engine and refrigerator.
Study of thermodynamics in quantum regime can reveal new aspects of fundamental interests. As an example, the statement of the second law of thermodynamics in the presence of an ancilla (1); (2) or, when the system has coherence (3); (4), has been established in great details, from where the classical version of the second law emerges under appropriate limits. The study of thermodynamics in quantum domain can be approached from different directions such as information-theoretic point of view (5); (6); (7); (8); (9); (10) or resource-theoretic aspect (11); (12); (13). Another important constituent, in this area of study, is the work extraction from quantum systems (14); (15); (16); (17); (18). Besides these, analyzing different models of thermodynamic machines in the quantum domain can also provide new insight. Such a study helps us to understand the special behavior of thermodynamic quantities like work, heat, and efficiency in the quantum regime due to the presence non-classical features such as entanglement, quantum superposition, squeezing, etc.(19); (20); (21). The quantum heat devices can show interesting atypical behaviors such as exceeding Carnot limit (19); (21) when they act as heat engines. But these apparent behaviors are found to be compatible with the second law of thermodynamics when all the preparation costs are considered (22). Such machines also have practical importance in the realm of quantum computation and refrigeration of small systems (23).
The performance of coupled quantum systems as heat engines have been studied widely in recent past (24); (25); (26); (27); (28); (29); (30). Work and efficiency are two important quantities to characterize the performance of a heat engine. It has been shown that appropriate coupling can increase the efficiency of the system compared to the uncoupled one (27). In this work, we find an upper as well as a lower bound of the efficiency of the coupled system. We also show that the coupling and quantum correlations give no advantage to obtain optimum work when the working medium consists of quantum systems with quadratic coupling (to be specified later). The generality of these results are shown by considering different heat cycles.
Further, we compare the performances of different coupled quantum systems when used as the working medium of a heat device. For this purpose, we consider two extreme cases: coupled spin- systems (finite dimension) and coupled quantum oscillators (infinite dimension) as the working media of heat devices. In the case of Otto cycle, when there is no coupling, both the system has the same efficiency but the work output is higher for oscillator model. To compare the performance of the coupled systems, an analogous coupling for both the systems is taken. We consider that two spin- systems are coupled via Heisenberg XX or XY model of interaction (31); (32). For harmonic oscillators, we take quadratic interaction in both positions and momenta, which is analogous to the Heisenberg exchange interaction in spin systems. For this interaction, the Hamiltonians for coupled spins and coupled oscillators are similar in terms of ladder operators. The similar form of the free Hamiltonians for both of the systems is also ensured.
Here it is important to note that very recently, the performance of coupled harmonic oscillators as a heat engine is studied (30). But, in contrast to the present work, there the interaction has been considered only between the position degrees of freedom of two oscillators. Furthermore, the authors in (30) have done the efficiency analysis for two different modes separately. But, the actual efficiency of the system has to be defined by the ratio of total work (done by both the modes) to the total heat which we analyze here. Therefore, our analysis provides a comprehensive picture of the efficiency of the coupled system. The main results of our work are as follows:
When the Hamiltonian of the coupled system (at all stages of the cycle) can be decoupled (as two independent modes) in some suitably chosen co-ordinate system, then the efficiency of the coupled system is bounded (both from above and below) by the efficiencies of the independent modes, provided both the modes work as engines (Section III.1).
The global efficiency (i.e. efficiency of coupled system) reaches the lower bound (mentioned in (i)) when the upper bound (mentioned in (i)) of the efficiency achieves Carnot efficiency. When one of the modes is not working as an engine, the global efficiency is upper bounded by the efficiency of the other mode (Section IV.1).
We have also shown that the optimal work extractable from a coupled system is upper bounded by the optimal work extractable from the uncoupled systems (Section III.3).
For meticulous comparison, we also consider coupled oscillators and coupled spins as the working medium of a refrigerator. The refrigeration cycle is similar to that of the heat engine. We find that:
Like the efficiency, the global coefficient of performance (COP) is bounded (both from above and below) by the COPs of the independent modes (Section V).
Organization of the paper goes as follows: In Section-II, we introduce the Otto cycle and illustrate the performance of uncoupled spins and oscillators as the working substance. In Section-III, a general form of quadratic coupling in harmonic oscillators and their characteristics are discussed when they work as a heat engine. Further, we discuss similar forms of coupling found in spin systems, widely known as Heisenberg XY model. In Section-IV, we describe the performance of the engine for special cases. Performances of the systems as refrigerators is discussed in Section-V. Section-VI is devoted to discussions and future possibilities.
Ii Quantum Otto cycle
Quantum Otto cycles are analogous to the classical Otto cycle, and the latter consists of two isochoric processes (work, ) and two adiabatic processes (heat, ). When the working medium of the Otto cycle is classical ideal gas, the efficiency of the system is written as , where and are initial and final volumes () of the adiabatic expansion process and , is the ratio of the specific heats (33). Similar to the classical cycle, the quantum Otto cycle consists of two adiabatic processes and two thermalization processes (34); (35). The system exchanges heat with the bath during the thermalization processes and the work is done when the system undergoes adiabatic processes. Work and heat are calculated from the change in mean energies, where mean energy, for a system represented by the state and the Hamiltonian , is defined as .
Here, we consider a four-staged Otto cycle. As an example, harmonic oscillator as the working medium of a quantum Otto cycle is pictorially described in Fig. 1. The four stages of the cycle are:
Stage 1: In this stage, the system represented by the density matrix (defined in Stage 4) and the Hamiltonian , is attached to a hot bath at temperature . During the process, the Hamiltonian is kept fixed. At the end of this stage, the system reaches equilibrium with the bath. Therefore, the final state is given as , where , with being the Boltzmann constant. Hence the amount of heat absorbed by the system from the hot bath is .
Stage 2: The system is decoupled from the bath and the Hamiltonian is changed from to slowly enough so that the quantum adiabatic theorem holds. Since there is no heat exchange between the system and the bath, the change in mean energy is equal to the work. The work done in this process is , where and is the unitary associated with the adiabatic process, defined as . Here is the time ordering operator, and .
Stage 3: The system is attached to the cold bath at inverse temperature . The system reaches equilibrium with the cold bath at the end of the process and the state of the system becomes . Therefore the heat rejected to the cold bath is given as .
Stage 4: The system is detached from the cold bath and the Hamiltonian is slowly varied from to . The work done in this process is equal to the change in the mean energy, which is given as , where is the density matrix at the end of the adiabatic process, defined as and is given by , so that and . Finally, the cycle is completed by attaching the system with the hot bath.
The net work done by the system is and the efficiency is defined as .
In Ref. (36); (37), a two-staged cycle is considered with two n-level systems. There, the coupling between two n-level systems is only for a short time to perform SWAP like operations. This cycle is equivalent to a cycle with single n-level system undergoing a four-staged Otto cycle. This type of cycle is considered in Section II.1. Our cycle is a four staged cycle, where the working medium consists of two oscillators (or spins) coupled to each other throughout the cycle. Both the subsystems are connected to one and the same bath at a time. Our primary aim is to find the effect of this coupling in the performance of the engine or refrigerator. We analyze the performance of the system using internal parameters such as bare mode frequencies and coupling strengths. Later in this paper, we consider mean energy preserving co-ordinate transformations. We refer the subsystems after the co-ordinate transformations as ’independent subsystems’ since they appear to be independent but they may not represent the actual subsystems. Before discussing the coupled systems, we briefly review the case of a single system as a heat engine.
ii.1 Single system as a heat engine
Otto cycle with a single harmonic oscillator (or a spin system) constituting the working medium of the engine is studied in different works (27); (30); (34); (35); (38); (39). Here we briefly review this. Consider a harmonic oscillator with Hamiltonian
where is the mass and is the frequency of the oscillator, and and are the creation and the annihilation operators respectively. We set and to unity. The cycle is constructed such that in Stage 2, the frequency is changed from to and in Stage 4, is changed to . In Stages 1 and 3, thermalization occurs with the respective heat baths as discussed above. The mean population of the thermal state of a harmonic oscillator with frequency and inverse temperature is . We also assume that the adiabatic processes are slow enough ( in Stages 2 and 4), so that coherence is not created between the eigenstates of the final Hamiltonian. Therefore, the mean population in the initial and the final states of the adiabatic process are same. Under these assumptions, the heat absorbed from hot reservoir is given by
where is obtained by substituting in Eq. (1). Here and are respectively the initial and the final density matrices in Stage 1. Similarly, the heat rejected to the cold reservoir is
where is calculated by substituting in Eq. (1) and and are the initial and final density matrices respectively for the thermalization process described in Stage 3. The net work done by the system is given as :
The efficiency of the system is then given as
The condition for the system to work as an engine is (with ) is satisfied when and , so that we have , where is the Carnot efficiency. The work output is zero when the system operates at Carnot value.
Now, consider a single spin- system, placed under a magnetic field applied along the z-direction. To get a similar form like oscillator, we need to add a term of the form , where is the identity matrix and is a constant. Adding a constant term () with each energy eigenvalue does not alter the characteristics of the engine. Thus we can write the corresponding Hamiltonian as,
where , and and are raising and lowering operators respectively. This Hamiltonian has a similar structure as the oscillator Hamiltonian given in Eq. (1). Now consider a cycle constructed such that frequency varies from to in Stage 2 and returns to the initial value () in Stage 4. The heat absorbed from the hot reservoir is
where is obtained by substituting in Eq. (6). We also have and , since in Stage 4 and in Stage 2. Similar as above, the net work done by the system is given by
So we can calculate the efficiency of system as
Even though the dimensionality of spin and harmonic oscillators are different, we kept the same energy level spacings in both the cases. Hence, both the cycles have the same efficiencies as shown in Eqs. (5) and (9). From Eqs. (4) and (8), we have . This inequality is true, because, for positive real values of and (), we have .
Consider two single systems (oscillators or spins), which are uncoupled and undergoing the cycle as discussed above. Then the work done by uncoupled oscillators is is greater than the work done by the spins . But the efficiency of the uncoupled oscillators is equal to that of uncoupled spins, . Now, our interest is to compare the performances of spins and oscillators when the analogous type of coupling is introduced.
Iii Performance of coupled system
In this section, we study the effect of coupling in the performance of joint systems as the working media of Otto cycle. First, we consider two identical oscillators coupled via positions and momenta and then we consider spins coupled though Heisenberg XY model.
iii.1 Coupled oscillators
Consider two oscillators (labeled as 1 and 2) with same mass and frequency, and consider that they are coupled through their positions and momenta. The total Hamiltonian of the composite system reads (41); (42); (43):
where and are the coupling strengths with same units as that of . We can write this Hamiltonian in terms of ladder operators ( and , where ) as,
where (). For the quadratic coupling given in Eq. (11), let us consider the following co-ordinate transformation,
The Hamiltonian in terms of new coordinates reads,
where and , where , are the creation and annihilation operators for the oscillators and . Here and are eigenmode frequencies and and are the effective masses in the new co-ordinate frame and the explicit expressions are given as,
Note that, in the new frame, the modes ( and ) are uncoupled. Now consider the above mentioned Otto cycle in which is changed from its initial value to in the first adiabatic process. Correspondingly, the eigenfrequency for the oscillator changes from to and similarly, frequency changes from to in the case of oscillator . The eigenfrequencies return to the respective initial values in the second adiabatic process. Here one can consider that the working medium consists of two independent oscillators. Hence the total work done by the system can be considered as the sum of the contributions from independent oscillators. Note that and ( and ) are the effective frequencies of the subsystems after the co-ordinate transformation. Therefore they are functions of actual frequencies () and coupling strengths. Hence and ( and ) are controlled by changing the frequencies of the actual subsystems. It can be done by changing potential in the case of oscillators or by changing the external magnetic field for the spins (see Section III.2). We assume that there is no cross over of energy levels of the total Hamiltonian during the adiabatic process. The density matrix may change during the adiabatic process. But the process is slow enough such that the populations at the instantaneous eigenstates of the Hamiltonian remain same (quantum adiabatic theorem). As discussed in Section II.1, the total amount of heat absorbed by the system from hot reservoir is given by
The first term denotes the heat absorbed by the system () and the second term represents the heat absorbed by the system (). Similarly, the total work is the sum of the work done by the independent systems, . Thus here
The efficiency of the individual system is given as , where . But the actual efficiency of the coupled system is defined as the ratio of total work over the total heat absorbed by the system. So we can write
When both the systems are working in engine mode (i.e., and ), we can write the above equation as
where . Therefore we can write
Therefore, when both independent oscillators work as an engine, the actual efficiency of the engine is bounded above and below by the efficiencies of the independent oscillators. For certain parameter values (see Section IV.1), one of the independent oscillators can work as refrigerator. In that case, the efficiency of the system is upper bounded by the efficiency of the other independent subsystem working as the engine.
Now consider an Otto cycle, where the global parameter ( or or both) is externally controlled and varied in the adiabatic branches, keeping fixed (see Fig. 2). Even in this case, using the following analysis, one can show the existence of the non-trivial bounds for the efficiency. In this paper, we consider those quadratic couplings for which we can write the total Hamiltonian as the sum of the Hamiltonians of two independent subsystems under some co-ordinate transformation. Let us consider the Hamiltonian at the end of Stage 1 and before the first adiabatic process (Stage 2) and be the Hamiltonian at the end of the first adiabatic process. Before the starting of the adiabatic process, the system is in a thermal state and hence we can write the initial state as in the eigenbasis of . Further, the adiabatic process ensures that the populations in the instantaneous eigenstates of the Hamiltonian remain unchanged during the process. Therefore, we can write the density matrix of the final state as in the basis of . For example, consider the coupled spin systems after the first thermalization process. Let us suppose and are the energy eigenvalues for the independent subsystem with the corresponding populations and () respectively. Similarly, and are the energy eigenvalues for the second spin with populations and () respectively. So the total energy can be written as
The terms and in the right hand side represent the energy eigenvalues of the composite system and the corresponding populations respectively. At the end of the adiabatic process, the energy eigenvalues of the composite system changes but the Hamiltonian structure allows us to write it as the sum of the contributions from two independent subsystems. Therefore the energy eigenvalues at the end of the adiabatic process are given as . But the corresponding populations remains fixed because of the quantum adiabatic theorem. Hence we can write
Therefore the populations of the eigenstates of the independent subsystems are also unchanged during the process. Similar characteristics can be observed in the second adiabatic process also. Hence the total system can be considered as composition of two independent subsystems in the beginning as well as at the end of each process in the cycle. So, the total heat (or work) is the sum of the heat (or work) obtained from its independent subsystems. Therefore, the efficiency of the system will be bounded above and below by the efficiencies of the independent systems [Eq. (22)].
An isothermal process can be simulated by an infinite number of infinitesimal adiabatic and isochoric processes (44). As discussed above, in the adiabatic and isochoric processes, the work and the heat contributions from individual subsystems can be identified separately. Therefore, the isothermal process of the total system can be considered equivalent to the isothermal processes of two independent subsystems taken together. To understand the generality of the bounds of the efficiency observed in Otto cycle, we can consider other cycles such as Carnot cycle consisting of two isothermal processes and two adiabatic processes, and Stirling cycle consisting of two isothermal processes and two isochoric processes (35); (39); (40) (see Fig. 3). Like the Otto cycle, in the Stirling cycle, the global efficiency is bounded by the efficiencies of independent subsystems because at any stage of the cycle, the total system can be decomposed into two independent subsystems. This shows that the analysis on the bounds of the efficiency represented in Eq. (22) is a generic one and applicable to some of the other cycles also. Using similar analysis, we can also show that for a quantum Carnot cycle, both the independent subsystems work at Carnot efficiency, which is same as the global efficiency, i.e., .
iii.2 Coupled spin system
In order to compare the performance of the quantum Otto cycle with coupled oscillator and that with coupled spin-1/2 system, let us now consider two spin-1/2 systems coupled via Heisenberg exchange interaction, placed in a magnetic field applied along the -direction. The Hamiltonian in terms of spin operators are given by
where and are the interaction constants along and directions. This model is generally known as Heisenberg XY model. Adding an equal energy with each level, we can write the Hamiltonian in terms of raising and lowering operators ( and , where ) as
Equations (11) and (26) have the similar form. As we have seen in the case of oscillators, Eq. (26) can be written as the sum of the Hamiltonians of independent subsystems. Therefore, the efficiency of the system is bounded from above and below by the efficiencies of independent subsystems as given in Eq. (22). In the following section (Sec. IV), we compare the performances of coupled spins and coupled oscillators when they undergo separately the quantum Otto cycles for different values of , , , and .
iii.3 Optimal work
As we have seen in Eqs. (4) and (8), work is a function of and . The optimal work can be estimated by maximizing work with respect to and . Now suppose that work is maximum at and . Therefore, for two uncoupled oscillators, maximum work occurs when both the system work with and . Now consider and are frequencies of the independent modes and of the coupled oscillators respectively, before the first adiabatic process and and are the frequencies of the independent modes and respectively, after the first adiabatic process. Then, and ( and ) are functions of , and (, and ). Similar arguments can be made for coupled spins too. Therefore, if the subsystem provides optimal work, then the work obtained from the subsystem may not be optimal because and ( and ) are not independent. Therefore we have
and are the maximum values of work obtained from the coupled and uncoupled systems respectively. The equality holds for the case where both independent subsystems have the same frequency.
Now consider a cycle in which the global parameters and or and are varied in the adiabatic branches instead of varying . Because of quadratic coupling, we can show that the total system consists of two independent subsystems, throughout the cycle. Then the total work is the sum of the contributions from its independent subsystems and therefore Eq. (27) holds even when we change global parameters to extract work.
In Stirling cycle and in Carnot cycle, for a working medium with quadratic interaction, we can always write the total work as the sum of the contributions from its independent subsystems. Therefore, using the argument mentioned above, we can say that the Eq. (27) holds in general.
Iv Special cases
In this section, we discuss the performance of the Otto cycle when the coupled systems with specific values of interaction constants constitute as the working medium. In Section IV.1, we take in the case of spin, which is known as Heisenberg XX model. Analogous interaction in oscillators is achieved by setting . Another interesting model is obtained with values in spins and in oscillators, are discussed in Section IV.3.
iv.1 XX model
Let us consider the following case: . For coupled oscillators, we can write the Hamiltonian in Eq. (11) in terms of ladder operators as
From Eq. (17), we get the frequencies of the independent modes as and . Therefore, inside the Otto cycle where the value of is varied from to during the first adiabatic process, we have and . In the new co-ordinates, the oscillators are independent. Hence the total heat absorbed from the hot reservoir is the sum of the heat absorbed by the independent subsystems and . Substituting the values of , , and in the Eqs. (18) and (19), we get the expressions for the heat and the work respectively. The explicit expression for the work obtained from Eq. (19) is given as
where . The efficiency of the independent subsystems are obtained as
So we have . Interestingly, the upper bound is analogous to the upper bound of the efficiency obtained for the coupled spins with isotropic Heisenberg Hamiltonian (27). Now we calculate the global efficiency as the ratio of the total work () by the total heat (). We get the global efficiency by substituting the values of , , and in the Eq. (20). Now we expand this efficiency for small coupling constant up to the third order, and we get:
where and .
We need to compare the performance of the oscillator system with that of the coupled spin system as a heat engine. For that, we consider two spin-1/2 systems coupled via Heisenberg XX Hamiltonian (). Representing this Hamiltonian in terms of ladder operators, it takes similar form of the Hamiltonian that we considered in the case of oscillators (see below). Therefore we write,
Coefficient of in Eq. (28) is characteristically same as the coefficient of in Eq. (32). Same is true with the coefficient appeared in Eqs. (28) and (32). To compare the performance of coupled spins and oscillators, we can diagonalize the Hamiltonian for the coupled spins so that in the new basis, spins are uncoupled. So we can write
Therefore, the total heat exchanged between the system and the hot bath is the sum of contributions from spins A and B. So we get the heat exchanged between system ( or ) and the hot bath as (see Eq. (7))
The total heat exchange between the system and the hot bath is The total work done by the system is the sum of the contributions from the individual spins defined in the new basis. So we get
with . Here, and are same as that obtained in coupled oscillators given in Eq. (30). Therefore, the efficiency of the engine is given as
We can expand this efficiency for small values of as
The difference in the efficiencies obtained from coupled oscillators and coupled spins for small coupling is calculated from Eqs. (31) and (37). When the coupling , both oscillator as well as spin system yields the same efficiency as discussed in Section II. By introducing a small coupling between the systems, we can write the difference between the efficiencies of the oscillator model and the spin model as
since and . Hence in this model, for small values of , the efficiency achieved by coupled oscillators is higher than the efficiency obtained from coupled spin model. Now we can see the behavior of the efficiency as a function of (see Fig. 4). The characteristics of the efficiencies in Fig. 4 remain the same with the choice of different values of , , , and . When () and (), both the independent systems work as engines. It is interesting to note that when , the upper bound of the efficiency, which is the efficiency of oscillator , attains the Carnot value () with zero work output. At this point, the global efficiency of the system is equal to the efficiency of oscillator , . When , oscillator works as refrigerator, which in turn reduces the efficiency of the engine. Hence the efficiency of the total system lies below the efficiency of oscillator . Now we can compare the performance in terms of work given in Eqs. (29) and (35). The terms on the right-hand side of these equations are positive when both the independent systems work in the engine mode. In that case, using the analysis made in Sec. II for uncoupled systems, we can show that .
iv.2 Optimal work and correlations
n the thermodynamic cycle, the origin of any correlation between the actual subsystems is due to the coupling. When the system thermalizes at the end of stage 1 and stage 3, the state of the system becomes a product state irrespective of the initial state, if there is no coupling present in the system. The proof that the optimum work extractable from a coupled system is upper bounded by the optimum work obtained from the uncoupled system (Eq. (27)) implies that the presence of quantum correlations do not have any advantage in optimal work extraction. To illustrate this fact (see also Section III.3) with an example, we analyze the behavior of work versus concurrence for the coupled spin systems. The concurrence is a measure of entanglement of an arbitrary two qubit state (46); (45); (47). The concurrence has one to one correspondence with the entanglement of formation. The concurrence of a state is defined as
where , , and are the eigenvalues of the matrix written in descending order. The Matrix is defined as where is the Pauli matrix. We estimate the concurrences for the thermal states at the end of Stage 1 and Stage 3 denoted as and respectively (26). For the numerical analysis shown in Fig. 5, we fix the temperatures of the hot and the cold bath, then we randomly choose the values of , and such that system should work as an engine. The optimum work is obtained only for zero concurrence and hence the numerical analysis shows that our theoretically obtained bound for the work holds.
iv.3 XY model
Here we consider the case (say). The Hamiltonian corresponding to the coupled oscillators is now written as
In the new co-ordinate system (described by Eq. (17)), both the independent oscillators have the same frequency, which is given by . Therefore, in the cycle, we have and . Hence, we get and from Eq. (21), we obtain
On the other hand, in the spin case, the analogous Hamiltonian is an example of Heisenberg XY model. In this case, the interaction Hamiltonian has the following form , where is the interaction constant. This model is well studied as quantum Otto cycle (24); (25). In terms of raising and lowering operators, we can write the spin Hamiltonian as
where . So we have two independent spins with the same frequency. So one would expect the efficiency of this system is similar to that of a single system (or two uncoupled systems) given in Eq. (9). This is true only when Stage 2 and Stage 4 are done slow enough. Because, in this case, the eigenvectors of the Hamiltonian are functions of and . Hence, when the system works as Otto cycle, by changing the magnetic field associated with the system, internal friction appears to be depending upon the rate at which magnetic field is changed (24); (25). This is due to the non-commutativity of the Hamiltonian at different instances during the driving. Here the adiabatic processes are done slow enough so that quantum adiabatic theorem holds. Since the energy level spacings for both the independent subsystems are equal () in Eq. (42), these subsystems undergo identical cycles with the same efficiency. Therefore, the global efficiency is also equal to the efficiency of the subsystem, which is given by
Now comparing Eqs. (41) and (43), we get , for a range of parameter such that the systems work as engines (where may be also large). For a small coupling, we can expand the efficiencies of the coupled oscillators and coupled spin systems as