Optimal refrigerator
Abstract
We study a refrigerator model which consists of two level systems interacting via a pulsed external field. Each system couples to its own thermal bath at temperatures and , respectively (). The refrigerator functions in two steps: thermally isolated interaction between the systems driven by the external field and isothermal relaxation back to equilibrium. There is a complementarity between the power of heat transfer from the cold bath and the efficiency: the latter nullifies when the former is maximized and vice versa. A reasonable compromise is achieved by optimizing the product of the heatpower and efficiency over the Hamiltonian of the two system. The efficiency is then found to be bounded from below by (an analogue of the CurzonAhlborn efficiency), besides being bound from above by the Carnot efficiency . The lower bound is reached in the equilibrium limit . The Carnot bound is reached (for a finite power and a finite amount of heat transferred per cycle) for . If the above maximization is constrained by assuming homogeneous energy spectra for both systems, the efficiency is bounded from above by and converges to it for .
pacs:
05.70.Ln, 05.30.d, 07.20.Mc, 84.60.hI Introduction
Thermodynamics studies principal limitations imposed on the performance of thermal machines, be they macroscopic heat engines or refrigerators lindblad ; callen ; landau , or small devices in nanophysics q_t and biology venturi . Taking as an example a refrigerator driven by a source of work, we recall three basic characteristics applicable to any thermal machine:

Heat transferred per cycle of operation from a cold body at temperature to a hot body at temperature ().

Power, which is the transferred heat divided over the cycle duration .

Efficiency (or performance coefficient) , which quantifies the useful output over the work consumed from the worksource for making the cycle. Note that workconsumption is obligatory, since the heat is transferred from cold to hot, i.e., against its natural gradient.
The second law imposes the Carnot bound
on the efficiency of refrigeration callen . Within the usual thermodynamics the Carnot bound (both for heatengines and refrigerators) is reached only for a reversible, i.e., an infinitely slow process, which means it is reached at zero power callen ; landau . The practical value of the Carnot bound is frequently questioned on this ground.
The drawback of zero power is partially cured within finitetime thermodynamics (FTT), which is still based on the quasiequilibrium concepts ftt . For heatengines FTT gives an upper bound , where is the efficiency at the maximal power of workextraction ca . Naturally, is smaller than the Carnot upper bound for heatengines.
Heat engines have recently been studied within microscopic theories, where one is easily able to go beyond the quasiequilibrium regime armen ; tu ; izumida_okuda ; udo ; esposito ; jmod ; domi ; henrich . For certain classes of heatengines the CA efficiency is a lower bound for the efficiency at the maximal power of work armen ; tu ; izumida_okuda . This bound is reached at the quasiequilibrium situation in agreement with the finding of FTT. The result is consistent with other studies udo ; esposito .
The interest in smallscale refrigerators is triggered by the importance of cooling processes for functioning of small devices and for displaying quantum features of matter q_t ; henrich ; kosloff_jap ; feldman ; kosloff_minimal_temperature ; segal ; rezek . In particular, the theory of these refrigerators can provide answers to several basic questions such as how the third law limits the performance of a cooling machine at low temperatures kosloff_jap , and how small are the temperatures reachable within a finite working time and under a reasonable amount of resource. Naturally, the smallscale refrigerators should also operate at a finite power. Note that the mirror symmetry between heatengines and refrigerators, which is wellknown for the zeropower case callen , does not hold more generally yan_chen .
The present situation with finitepower refrigerators is somewhat unclear yan_chen ; velasco ; jimenez ; unified . Here maximizing the power of cooling does not lead to reasonable results, since there is an additional complementarity (not present for heat engines) yan_chen ; velasco ; feldman ; kosloff_minimal_temperature : when maximizing the heattransfer power one simultaneously minimizes the efficiency to zero, and vice versa.
Here we intend to study optimal regimes of finitepower refrigeration via a model which can be optimized over almost all of its parameters. The model represents a junction immersed between two thermal baths at different temperatures and driven by an external worksource. This type of models is frequently studied for modelling heat transport; see, e.g., q_t ; segal ; dhar . Our model is quantum, but it admits a classical interpretation, because all the involved density matrices will be diagonal [in the energy representation] at initial and final moments of studied processes ^{1}^{1}1This aspect is similar to the Ising model. This is a model for quantummechanical spin, but it can be given a classical interpretation via an overdamped particle moving in a asymmetric doublewell potential. If the transversal components of the quantum spin are excited, this analogy breaks down. However, it still holds for the spinflip process, where the transversal components are absent both initially and finally. In fact, the dynamics of the Ising model is introduced via such spinflip processes, and this dynamics admits a classical interpretation..
This paper is organized as follows. The model is introduced in section II. Here we also show that the efficiency of the model is bounded by the Carnot value, and provide a general discussion of the refrigeration power. We confirm the heatpowerefficiency complementarity in section III and conclude that the most meaningful way of optimizing its functioning is to maximize the product of efficiency and the heat power. The optimization procedure is reported in section IV. We discuss the quasiequilibrium limit of our model in section V. There we show that there is a lower bound () for the efficiency, in addition to the upper Carnot bound . The same expression was obtained within finitetime thermodynamics as an upper bound when optimizing the product of heatpower and efficiency or the ratio of the efficiency over the cycle time yan_chen ; velasco . Section VI discusses the attainability of the Carnot efficiency at a finite power. Entropy production inherent in the functioning of the model refrigerator is studied in section VII, while in section VIII we outline consequences of constraining features of the model to the quasiclassical domain. This constraint allows to reproduce the prediction of FTT on the upper bound of . We summarize in section IX. Some technical questions are relegated to Appendix.
Ii The model
Consider two quantum systems and with Hamiltonians and , respectively. Each system has energy levels. and constitute the working medium of our refrigerator; see Fig. 1.
Initially, and do not interact and are in equilibrium at temperatures [we set ]:
(1) 
where and are the initial Gibbsian density matrices of and , respectively. We write
(2)  
(3) 
where is a diagonal matrix with entries , and where without loss of generality we have nullified the lowest energy level of both and . Thus the overall initial density matrix is
(4) 
and the initial Hamiltonian .
The goal of any refrigerator is to transfer heat from the cooler bath to the hotter one at the expense of consuming work from an external source. The present refrigerator model functions in two steps: thermally isolated workconsumption and isothermal relaxation; see Fig. 1. Let us describe these steps in detail.
1. and interact with each other and with the external sources of work. The overall interaction is described via a timedependent potential in the total Hamiltonian
(5) 
of . The interaction process is thermally isolated: is nonzero only in a short timewindow and is so large there that the influence of all other couplings [e.g., couplings to the baths] can be neglected [pulsed regime]. The timedependent potential may explicitly depend on the coupling time .
Thus the dynamics of is unitary for :
(6) 
where is the initial state defined in (1), is the final density matrix, is the unitary evolution operator, and where is the timeordering operator. The work put into reads lindblad ; callen
(7) 
where and are initial and final energies of .
2. Once the overall system arrives at the final state , is switched off, and and (within some relaxation time) return back to their initial states (1) under influence of the hot and cold thermal baths, respectively. Thus the cycle is complete and can be repeated again. Because the energy is conserved during the relaxation, the hot bath gets an amount of heat , while the cold bath gives up the amount of heat :
(8) 
where and are the partial traces. Eq. (1) and the unitarity of lead to
(9) 
where is the relative entropy, which employed in deriving thermodynamic bounds since lindblad ; partovi .
nullifies if and only if ; otherwise it is positive. Eq. (9) is the Clausius inequality, with quantifying the entropy production. This point will be readdressed and confirmed in section VII.
Eqs. (7–9) and the energy conservation imply
(10) 
meaning that in the refrigeration regime we have and thus . Thus within the step 1 the work source transfers some energy from to , while in the step 2, recovers this energy from the cold bath thereby cooling it and closing the cycle.
Eq. (9) leads to the Carnot bound for the efficiency [we denote ]
(11) 
We note from (11) that the deviation from the Carnot bound is controlled by the ratio of the entropy production to the work .
We note in passing that all quantities introduced so far are meaningful also without the stage 2. Then the problem reduces to cooling the initially equilibrium system with help of the worksource and the system . Both the worksource and are clearly necessary to achieve cooling ^{2}^{2}2Indeed, if and the worksource form a closed system, no cooling is possible due to the Thomson’s formulation of the second law jmod (cyclic processes cannot lead to workextraction). If and form a closed system, then and no cooling is possible due to (10).. quantifies the amount of cooling, while accounts for the relative effort of cooling.
ii.1 Power
Recall that the power of refrigeration is defined as the ratio of the transferred heat to the cycle duration . For our model is limited mainly by the duration of the second stage, i.e., should be larger than the relaxation time , which depends on the concrete physics of the systembath coupling.
Though some aspects of the following discussion are rather general, it will be useful to have in mind a concrete relaxation scenario. Consider the collisional relaxation scenario, where the target system interacts with independent bath particles via successive collisions; see partovi ; mityugov and Appendix A. For our purpose the target system is or that interact with, respectively, the hot and cold bath. Each collision lasts a time , which is much smaller than the characteristic time between two collisions. The interaction Hamiltonian between the target system and a bath particle is conserved, so that no work is done in switching the systembath interaction on and off; see partovi ; mityugov and Appendix A.1.
The relaxation process is typically (but not always) exponential with the characteristic relaxation time depending on the collisional interaction; see Appendix A.2. This time can be much smaller than any characteristic time of or . Since the two baths act on and independently, the overall relaxation process drives to the initial state (1, 4).
If the interaction time of [see (6)] is also much smaller than , one realizes a thermallyisolated process, because the overlap between the pulse and a collision can be neglected ^{3}^{3}3 Analogous conclusion on the irrelevance of the systembath interaction during the pulse action is obtained when this interaction is always on, but its magnitude is small [weakcoupling]. Now the relaxation time is much larger than the internal characteristic time of and . Because the systembath interaction is always on, there will be a contribution in the work (7) coming from the systembath interaction Hamiltonian ruben . This contribution arises even when the conditions for the pulsed regime hold ruben . However, within the weak coupling assumption this additional contribution is proportional to the square of the systembath interaction constant and can be neglected ruben . We stress that this additional contribution does not arise within the collisional relaxation scenario, because the pulse and collisions are wellseparated in time. .
To achieve a cyclic process within the exponential relaxation with the relaxation time , the cycle time should be larger than . For each cycle the deviation of the postrelaxation state from the exact equilibrium state (1, 4) will be of order . Thus if the ratio is simply large, but finite, one can perform roughly number of cycles at a finite power, before deviations from cyclicity would accumulate and the refrigerator will need resetting.
Though, as we stressed above, the relaxation process is normally exponential ^{4}^{4}4 More generally, the relaxation need not be exponential, but it still can be such that although the difference between the system densitymatrix at time and the corresponding Gibbsian density matrix goes to zero for , this difference does not turn to zero after any finite . One of referees of this paper pointed out to us that i) the latter feature holds for a rather general class of relaxation processes taking place under a constant Hamiltonian; ii) it is rooted in the KuboMartinSchwinger (KMS) condition kms for correlation function evaluated over an equilibrium state; see landau for a heuristic version of this argument; iii) the collisional relaxation is different in this respect, because the Hamiltonian is not constant. As we stress in Appendix A, a general point of the collisional relaxation is that no work is involved in this timedependence., there are also situations within the collision relaxation scenario, where the system settles in the equilibrium state after just one intercollision time ; see section VI.1 and Appendix A.2 for details. The above limitations on the number of cycles does not apply to this relaxation scenario.
ii.1.1 Comparing with the power of the Carnot cycle
The above situation does differ from the power consideration of usual (reversible) thermodynamic cycles, e.g., the Carnot cycle callen ; landau ; sekimoto ; sekimoto_sasa . There the external fields driving the working medium through various stages have to be much slower than the relaxation to the momentary equilibrium. The latter means that the working medium is described by its equilibrium Gibbs distribution with timedependent parameters. The condition of momentary equilibrium for the working medium is necessary for the Carnot cycle to reach the Carnot efficiency callen ; landau .
The precise meaning of the external fields being slow is important here. If is the characteristic time of the fields, then the deviations from the momentary equilibrium are of order landau ; sekimoto ; sekimoto_sasa . This fact is rather general and does not depend on details of the system and of the studied process, e.g., it does not depend whether the process is thermally isolated or adiabatic ^{5}^{5}5If a slow thermally isolated process is performed on a finite system, there are additional limitations in achieving the momentary equilibrium; see sekimoto ; minima for more details. These limitations are however not essential for the present argument.. In particular, it is this deviation of the state from the momentary equilibrium that brings in the entropy production (or work dissipation) of order of landau ; sekimoto ; sekimoto_sasa .
Thus performing the reversible Carnot cycle with (approximately) the Carnot efficiency means keeping the ratio very small.
Now there are two basic differences between the Carnot cycle and our situation:

In our case we do not require the working medium to be close to its momentary equilibrium state during the whole process. It suffices that the medium gets enough time to relax to its final equilibrium.

A small, but finite for the Carnot cycle situation means that deviations from the momentary equilibrium are visible already within one cycle. In contrast, a small, but finite for our situation means that we can perform an exponentially large number of cycles before deviations from the cyclicity will be sizable. Here is a numerical example. Assume that . In our situation the same amount of deviation from the cyclicity will come after cycles. This is a large number, especially taking into account that no realistic machine is supposed to work indefinitely long. Such machines do need resetting or repairing. The point is that our machine can perform many cycles at a finite power before any resetting is necessary. . For the standard Carnot cycle already within one cycle the deviation from the momentary equilibrium will amount to
Iii Complementarity between the transferred heat and efficiency
We now proceed to optimizing the functioning of the refrigerator over the three sets of available parameters: the energy spacings , , and the unitary operators . It should be evident from (5, 1) that optimizing over these parameters is equivalent to optimizing over the full timedependent Hamiltonian of . We stress in this context that no limitations on the magnitude of are imposed. This means that the unitary operator can in principle be generated in an arbitrary short coupling time .
We start by maximizing the transferred heat , which is the main characteristics of the refrigerator. Since depends only on , we choose and so that the final energy attains its minimal value zero. Then we maximize over . Note from (2)
It is clear that goes to zero when, e.g., (), while amounts to the SWAP operation . It is checked by a direct inspection that the maximization of the initial energy over produces the same structure of times degenerate upper energy levels . Denoting
(12) 
we obtain for
(13) 
where according to the above discussion, is maximized for , and where is to be found from maximizing in (13) over , i.e., is determined via
(14) 
For the efficiency we get for the present situation ( and have times degenerate upper levels, while amounts to the SWAP operation):
(15) 
The maximization of led us to , which then means that in (15) goes to zero.
Thus can be cooled down to its ground state (), but at a vanishing efficiency, i.e., at expense of an infinite work. To make this result consistent with the classic message of the third law klein , we should slightly adjust the latter: one cannot reach the zero temperature [of an initially equilibrium system] in a finite time and with finite resources [infinite work is not a finite resource]. At any rate, one should note that the classic formulation of the third law motivates its operational statement using exclusively equilibrium concepts. Modern perspectives on the third law are discussed in kosloff_jap ; kosloff_minimal_temperature ; rezek ; grif ; scully .
Note that the efficiency in (15) reaches its maximal Carnot value for
(16) 
which nullifies the transferred heat ; see (13).
Now we have to show that tends to zero upon maximizing over all free parameters , and . Denoting and for the eigenvectors of and , respectively, we note from (7, 8) that and feel only via the matrix
(17) 
This matrix is doublestochastic olkin :
(18) 
Conversely, for any doublestochastic matrix there is some unitary matrix with matrix elements , so that olkin . Thus, when maximizing various functions of and over the unitary , we can directly maximize over the independent elements of double stochastic matrix .
We did not find an analytic way of carrying out the complete maximization of over all free parameters. Thus we had to rely on numerical recipes of Mathematica 7, which for confirmed that nullifies whenever reaches (along any path) its maximal Carnot value. We believe this holds for an arbitrary , though we lack any rigorous proof of this assertion.
Iv Maximizing the product of the transferred heat and efficiency.
We saw above that neither nor are good target quantities for determining an optimal regime of refrigeration. But
(19) 
is such a target quantity, as will be seen shortly. This is the most natural choice for our setup. This choice was also employed in yan_chen . Refs. unified ; feldman ; kosloff_minimal_temperature report on different approaches to defining refrigeration regimes.
The numerical maximization of over , and has been carried out for along the lines discussed around (17, 18). It produced the same structure: both and have times degenerate upper levels, see (12), and the optimal again corresponds to SWAP operation ^{6}^{6}6Let us recall how the SWAP is defined via a purestate base. Let be an orthonormal base in the Hilbert where lives. Let also be an orthonormal base in the Hilbert space where lives. Any unitary operator acting on the composite Hilbert space can be defined with respect to the orthonormal base , where . Let us now define for all pairs and : .:
(20) 
Recalling the expression (5) for the total Hamiltonian , we can state this result as follows: there exist a coupling potential that for a given coupling time generates the unitary SWAP operation following to (6). This operation does not explicitly depend on , because depends on ; see also our discussion in the beginning of section III. Note that both the initial and final states in (20) are diagonal in the energy representation. Evidently, the intermediate state for is not diagonal in this representation.
The efficiency and the transferred heat are given by, respectively, (13) and (15), where instead of and we should substitute and , respectively. The latter two quantities are obtained from maximizing ,
(21) 
where and are found from maximizing via . Note that and depend on . The efficiency and the transferred heat are given, respectively, by (15) and (13) with and .
Though we have numerically checked these results for only, we again trust that they hold for an arbitrary (one can, of course, always consider the above structure of energy spacings and as a useful ansatz).
SWAP is one of the basic gates of quantum information processing galindo ; see domi for an interesting discussion on the computational power of thermodynamic processes. SWAP is sometimes realized as a composition of more elementary unitary operations, but its direct realizations in realistic systems also attracted attention; see, e.g., swap for a direct implementation of SWAP in quantum optics. Note that for implementing the SWAP as in (20) the external agent need not have any information on the actual density matrices and .
iv.1 Effective temperatures
Since the state of after the action of is , and because in the optimal regime the upper level for both and is times degenerate, one can introduce nonequilibrium temperatures and for respectively and via [note (1)]
(22) 
where we recall that () is the state of () after applying the pulse. Using (12) we deduce
(23) 
where and ; see (12). This implies
(24) 
As expected, the refrigeration condition , see (13, 21), is equivalent to
(25) 
i.e., after the pulse the cold system gets colder, while the hot system gets hotter. Note that the existence of temperatures and was not imposed, they emerged out of optimization. In terms of these temperatures the efficiency (15) is conveniently written as
(26) 
We eventually focus on two important limits: quasiequilibrium , and the regime .
V Quasiequilibrium regime : a lower bound for the efficiency
In this regime the temperatures and are nearly equal to each other: .
First we note that sharply at , reads
(27) 
where
and where is given by :
(28) 
We now work out the optimal and for . It can be seen from (21) that the proper expansion parameter for is . We write
(29) 
We substitute (29) into and and expand them over . Both expansions start from terms of order . Now and are determined by equating to zero the terms in and . Thus the terms together with (28) define :
(30) 
which should be nonnegative due to . The terms together with (28) and (30) define and :
(31) 
and so on. Eqs. (29, 30, 31) imply for the efficiency at ()
(32) 
Note that the expansion (32) does not apply for , since in this limit ; see (28). Thus, in the limit , scales as ,
(33) 
while the consumed work is smaller and scales as .
Eq. (32) suggests that the maximization of imposes a lower bound on the efficiency:
(34) 
This is numerically checked to be the case for all and all ; see also Fig. 2.
The expression of was already obtained within finitetime thermodynamics—but as an upper bound on the efficiency—and argued to be an analogue of the CurzonAhlborn efficiency for refrigerators yan_chen ; velasco . Section VIII explains that also within the present microscopic approach can be an upper bound for provided that is maximized under certain constraints.
Recalling (15), our discussion after (23–25) and (26), we can interpret the lower bound for the efficiency as a lower bound on the intermediate temperature of :
(35) 
i.e., the lowest temperature cannot be too low under optimal . Compare this with the fact that under vanishing efficiency (that is for very large amount of the consumed work), can be arbitrary low; see our discussion after (15). Thus a welldefined lowest (per cycle) temperature emerged once we restricted the resource of cooling (the consumed work).
Vi The manylevel regime: Reaching the Carnot limit at a finite power.
Now we turn to studying the regime
(36) 
First of all, let us introduce two new variables
(37) 
denote
(38) 
and rewrite in (21) as
(39) 
The expression in RHS of (39) is now to be optimized over and . We note that if these parameters stay finite in the limit , the value of is read off directly: . The finitness of and in the limit is confirmed by expanding the RHS of (39) over the small parameter , collecting terms , differentiating them over and , and equating the resulting expressions to zero. This produces:
(40) 
Substituting these into (15) and (21) we get
(41)  
(42) 
Note from (36, 38) that the dominant factor in the efficiency is the Carnot value , while the subleading term is naturally negative; see (41). Likewise, the dominant factor in is , while the subleading term is . We also see that the limit does not commute with the equilibrium limit , since the corrections in (41, 42) diverge for .
Thus in this regime the efficiency converges to the Carnot value; see Fig. 3.
Recalling (12) we see from (37, 40) that in the regime (36) the total occupation of the higher levels of is small, so that is predominantly in its ground state before applying the workconsuming external pulse. In contrast, is more probably in one of its excited states. These facts are expected, because has to give up some energy, while has to accept it.
We note that this regime resembles in several aspects the macroscopic regime of a particle system. Recall that for (weakly coupled) particles the number of energy levels scales as , while energy scales as . Now for the above situation (42) the transferred heat is (in the leading order) a product of the colder temperature and the ”number of particles” .
The effective temperatures and [see (23–25)] in this limit are close to their initial values:
(43)  
(44) 
where we employed (37, 40). Though during the refrigeration process the systems and are able to process large amounts of work and heat (), their temperatures are not perturbed strongly.
vi.1 Finiteness of power
It is important to note that the asymptotic attainability (41) of the Carnot bound for is related to a finite transferred heat , but it also can be related to a finite power , where in our model the cycle time basically coincides with the relaxation time; recall our discussion in section II.1. This appears to be unexpected, because within the standard thermodynamic analysis the Carnot efficiency is reached by the Carnot cycle at a vanishing power callen ; see section II.1.1 for a precise meaning of this statement. In any refrigerator model known to us—see, e.g., segal —approaching the Carnot limit means nullifying the power. See also in this context our discussion around equation (16); various reasons preventing the approach to the Carnot efficiency for thermal machines (even for small machines working at zero power) are analysed in sekimoto . Now we supplement our discussion in section II.1 with more specific arguments.
We already stressed in section II.1 that within the second stage of the refrigerator functioning, where both and relax to equilibrium under influence of the corresponding thermal baths, the relaxation mechanism can be associated with the collisional systembath interaction; see Appendix A for a detailed discussion of this mechanism. Here there are three characteristic times: the single collision duration time is much smaller than the intercollision time , while the relaxation of the system to its equilibrium state is governed by the time . The assumed condition allows implementing the thermally isolated workconsuming pulse, because if the pulse time is also much smaller than , the pulse does not overlap with collisions.
In Appendix A.2 we study the relaxation time of the system with fold degenerate upper energy levels and nondegenerate lowest energy level. We also account for the limit , where the Carnot efficiency is reached; see (41). It is shown that for such a system the relaxation time can—depending on the details of the thermal bath and its interaction with the system—range from few ’s to a very long times . The former relaxation time means a finite power, while the latter time implies vanishing power for .
These two extreme cases are easy to describe without addressing the formalism of Appendices A.1 and A.2.
1. For simplicity let us focus on the relaxation of the system that after the workextracting pulse (20) is left in the state , and is now subjected to a stream of the bath particles (the situation with is very similar). Recall that each bath particle before colliding with is in the Gibbsian equilibrium state at the temperature . Now assume that each bath particle also has fold degenerate upper level, and one lowest energy level. Also, the nonzero energy spacing for the bath particles is equal to that of . Then the relaxation of is achieved just after one collision provided that the systembath interaction [during this collision] amounts to a SWAP operation. Note that the characteristic time of this relaxation is , and that this is a nonexponential scenario of relaxation, because the system exactly settles into its equilibrium Gibbsian state after the first collision.
No work is done during collisional relaxation; see Appendix A. Indeed, under above assumptions on the energy levels of the bath particles, the SWAP operation commutes with the free Hamiltonian (where is the Hamiltonian of the given bath particle), which implies that the final energy of plus the bath particle is equal to its initial value. Since each separate collision is a thermally isolated process, this means that no work is done; see (7).
2. If each collision is very weak and almost does not exchange heat with the system , the relaxation time becomes very long. Intermediate cases are discussed in Appendix A.2. These intermediate cases are relevant, since the power of refrigeration is finite even for long relaxation times . Indeed, we recall from (42) that .
vi.2 More realistic spectra still allowing to reach the Carnot bound
One can ask whether the convergence (41) to the Carnot bound is a unique feature of the spectra (12) in the limit , or whether there are other situations that still allow . The answer is positive as we now intend to show. For the energy spectra (3) we postulate []
(45) 
where is a parameter. Next, we assume that the following six conditions hold
(46)  
(47)  
(48) 
Under conditions (46, 47, 48)—and assuming the SWAP operation for the pulse—we show below that the results analogous to (41, 42) hold,
(49) 
where the role of a large parameter in (41, 42) is now played by ^{7}^{7}7In the definition (47) of one can as well employ instead of . This will not lead to serious changes, because we always assume that is fixed. . Note that (46) and (47) still imply that .
Vii Entropy production
Entropy production is an important characteristics of thermal machines, because it quantifies the irreversibility of their functioning lindblad ; callen . For our refrigerator model, no entropy is produced during the first stage, which is thermally isolated from the baths. However, a finite amount of entropy is produced during the second, relaxational stage. The overall entropy production reads
and controls the deviation of efficiency from its maximal Carnot value; see (11). In the optimal conditions (20, 12), we get
(50)  
(51) 
where and are the entropies produced in, respectively, cold and hot bath. Indeed, consider the system that after the external field action is left in the state with density matrix [see (20)], and now under influence of the thermal bath should return to its initial state . Now
(52) 
is the difference between the nonequilibrium free energy of in the state [and in contact to a thermal bath at temperature ] and the equilibrium free energy . Simultaneously, in (52) is the maximal work that can be extracted from the system (in state ) in contact with the bath armen . During relaxation this potential work is let to relax into the bath increasing its entropy by . Likewise, is the entropy production during the relaxation of the system in contact with the bath.
Now in the regime , amounts to , see (51), while the consumed work and the transferred heat scale as . In other words, the entropy production is much smaller than both and . This explains why for a large the Carnot efficiency is reached; see (11, 41).
In the equilibrium limit , reads
(53) 
where is given by (28), and where in deriving (53) we employed (51) and asymptotic expansions presented after (28). Note that now is smaller than , but has the same scale as the consumed work ; see our discussion after (32). Thus cannot be neglected, and this explains why the Carnot efficiency is not reached in the equilibrium limit ; see (11).
Viii Classical limit.
We saw above that the optimization of the target quantity produced an inhomogeneous type of spectrum, where a batch of (quasi)degenerate energy levels is separated from the ground state by a gap. It is meaningful to carry out the optimization of imposing a certain homogeneity in the spectra of and . The simplest situation of this type is the equidistant spectra
(54) 
for and ; recall (3). For and , these spectra correspond to the classical limit.
Thus, now we maximize imposing conditions (54). We found numerically that the optimal again corresponds to SWAP operation; see (20). For we get
where and are found from maximizing . The efficiency is still given by (15).
In the limit we get from maximizing :