Optimal operation of a three-level quantum heat engine and universal nature of efficiency

We present a detailed study of a three-level quantum heat engine operating at maximum efficient power function, a trade-off objective function defined by the product of the efficiency and power output of the engine. First, for near equilibrium conditions, we find general expression for the efficiency and establish universal nature of efficiency at maximum power and maximum efficient power. Then in the high temperature limit, optimizing with respect to one parameter while constraining the other one, we obtain the lower and upper bounds on the efficiency for both strong as well as weak matter-field coupling conditions. Except for the weak matter-field coupling condition, the obtained bounds on the efficiency exactly match with the bounds already known for some models of classical heat engines. Further for weak matter-field coupling, we derive some new bounds on the the efficiency of the the engine which lie beyond the range covered by bounds obtained for strong matter-field coupling. We conclude by comparing the performance of our three-level quantum heat engine in maximum power and maximum efficient power regimes and show that the engine operating at maximum efficient power produces at least $88.89\%$ of the maximum power output while considerably reducing the power loss due to entropy production.


I. INTRODUCTION
The study of quantum heat engines (QHEs) started with the seminal work of Scovil and Schulz-DuBois (SSD) [1]. In their work, they investigated the thermodynamics of a three-level maser and showed that its limiting efficiency is given by Carnot efficiency [2]. Since then, three level systems have been employed to study various models of quantum heat engines (refrigerators)  and quantum absorption refrigerators [25][26][27][28][29][30][31][32].
Here, we specifically mention the work of Geva and Kosloff [4][5][6] on three-level amplifier. They studied the SSD engine in the spirit of finite-time thermodynamics using Alicki's definition of heat and work [33], and optimized its performance with respect to different control parameters. They showed that in the presence of external electromagnetic field, one has to incorporate the effect of the field on the dissipation superoperators in order to satisfy the second law of thermodynamics. Going one step further, Tannor and Boukobza formulated a new way of partitioning energy into heat and work [10][11][12]. They applied their formulation to a three-level system simultaneously coupled to two thermal baths at different temperatures and to a single mode of classical electromagnetic field, and showed that the second law of thermodynamics is always satisfied without incorporating the effect of the field on the dissipators [12]. Recently, their formalism has been used to study the phenomenon of noise-induced coherence [22] and quantum synchronization [34] in nanoscale engines .
In this work, we use Tannor and Boukobza's formalism to analyze the optimal performance of the SSD engine and set up its correspondence with some classical models of heat engines. At optimal performance, QHEs * vsingh@ku.edu.tr operating at finite power, show remarkable similarity to classical macroscopic heat engines. For instance, in hightemperature limit, many models of QHEs [22,[35][36][37][38][39][40] operate at Curzon-Ahlborn (CA) efficiency, a well known result in the field of finite-time thermodynamics [41][42][43], first obtained for a macroscopic model of heat engine known as endoreversible engine [44,45]. Similarly, in the low-dissipation regime [46], the behavior of quantum and classical heat engines are quite similar [47,48].
One another feature common in the operation of classical and QHEs is universal nature of efficiency [49]. Many models of classical and QHEs show universality of efficiency at maximum power (EMP) upto quadratic order in η C , i.e., η M P = η C /2+η 2 C /8+O(η 3 C ). Van den Broeck proved that in the linear response regime, η C /2 is universal for tight-coupling heat engines [50]. Further, Esposito and coauthors established the universality of the second term η 2 C /8 by invoking the symmetry of Onsager coefficients on the nonlinear level [49].
The universal features of efficiency are not unique to the EMP, two other optimization functions: Omega (Ω) function (or ecological function) [51,52] and efficient power (EP) function [53,54], P η = ηP (product of the efficiency and power of the engine), also exhibit this behavior [55,56]. Here, we will discuss universal character of efficiency at maximum efficient power (EMEP) only. The formal proof of universality of EMEP was established in Ref. [56]. It was shown that the first two universal terms are 2η C /3 and 2η 2 C /27. In this paper, we study the optimal performance of the SSD engine operating at maximum efficient power (MEP), a trade-off optimization function representing a trade-off between the power output and efficiency of a heat engine, in different operational regimes and compare its performance with the engine operating at maximum power (MP). The study of such objective function is important from the environmental and ecological point of view. It is already known that engines operating at max-imum power regime also waste a lot of power due to large entropy production [57,58]. Therefore, rather than operating in MP regime, the real heat engines should operate near MP regime, where they produce slightly smaller power output with appreciable larger efficiency, which makes them cost effective too [58]. The EP function was introduced by Stucki [53] in the context of biochemical energy conversion process. Extending Stucki's idea, Yan and Chen (YC) treated EP function as their objective function to investigate the performance of an endoreversible heat engine [54]. Recently, EP function has attracted considerable interest and have been employed to study the energy conversion process in low-dissipation heat engines [59,60], thermionic generators [61], biological systems [53,62], chemical reactions [63,64], Feynman's ratchet and pawl model [65] and in a quantum Otto engine [66].
The paper is organized as follows. Sec. II, we discuss the model of SSD engine. In Sec. III, we obtain analytic expression for the efficiency of the SSD engine operating near equilibrium conditions, and show universality of the EMP and EMEP. In Subsecs. IV A and IV B, we optimize engine's performance, operating in two different operational regimes (strong and weak matter-field coupling regimes), with respect to one parameter only, and obtain the lower and upper bounds on the EMEP for each case. Subsec. IV C is devoted to the discussion of universality of efficiency for one parameter optimization scheme under the effect of some symmetric constraints imposed on the control parameters of the engine. In Secs. V and VI, we we compare the performance of the SSD engine operating at MEP to the engine operating at MP. We conclude in section VII.

II. MODEL OF THREE LEVEL QUANTUM LASER HEAT ENGINE
SSD engine [1] is one of the simplest QHEs. Using the concept of stimulated emission in a population inverted medium, it converts the incoherent thermal energy of heat reservoirs to a coherent laser output. The model consists of a three-level system simultaneously coupled to two thermal reservoirs at temperatures T h and T c (T c < T h ), and to a single mode classical electromagnetic field (see Fig. 1). The hot reservoir at temperature T h induces the transition between the ground state |g and the upper state |1 , whereas the transition between the middle state |0 and the ground state |g is constantly de-excited by the cold reservoir at temperature T c . For power output mechanism, states |0 and |1 are coupled to a classical single mode field. The bare Hamiltonian of the three-level system is given by: H 0 = ω k |k k| where the sum runs over all three states and ω k 's represent the corresponding atomic frequencies. Under the rotating wave approximation, the following semiclassical Hamiltonian describes the interaction of the system with the classical field of frequency Model of the SSD engine simultaneously coupled to two thermal reservoirs at temperatures Tc and T h with coupling constants Γc Γ h , respectively. The interaction of the system with a classical single-mode field is represented by λ, the matter-field coupling constant.
ω: V (t) = λ(e iωt |1 0| + e −iωt |0 1|); λ is the fieldmatter coupling constant. The reduced dynamics of the matter-field system under the effect of the heat reservoirs is described by the following form of Lindblad master equation:ρ where L h and L c are the dissipation Lindblad superoperator describing the interaction of the system with the hot and cold reservoirs, respectively: where Γ c and Γ h are Weisskopf-Wigner decay constants, and n h(c) = 1/(exp[ ω h(c) /k B T h(c) ] − 1) is average number of photons in the mode of frequency ω h(c) in hot (cold) reservoir satisfying the relations ω c = ω 0 − ω g , In order to solve the density matrix equations, it is convenient to transform to a rotating frame in which semiclassical interaction Hamiltonian and the steadystate density matrix ρ R become time-independent [12]. DefiningH = (ω g |g g| + ω 2 |1 1| − ω 2 |0 0|), an arbitrary operator B in the rotating frame is given by B R = e iHt/ Be −iHt/ . It can be seen that superoperators L c [ρ] and L h [ρ] remain unchanged under this transformation. Finally, time evolution of the system density operator in this rotating frame is given by: where V R = λ(|1 0| + |0 1|). For a weak system-bath coupling, the heat flux, output power and efficiency of the SSD engine can be defined, using the formalism of Ref. [12], as follows: Here, we have used the sign convention in which all three energy fluxes: heat flux extracted from the hot bath, heat flux rejected to the cold bath and the power output are positive. Substituting the expressions for V R , H 0 and L h [ρ R ], and calculating the traces appearing in Eqs. (5) and (6)[see Appendix A], the heat flux and power output can be written as:Q where ρ 01 = 0|ρ R |1 and ρ 10 = 1|ρ R |0 . Using Eqs. (8) and (9) in Eq. (7), the efficiency of the engine is given by The positive power production condition [see Eq.(A11)] implies that ω c /ω h ≥ T c /T h , which in turn implies that η ≤ η C .

III. UNIVERSAL NATURE OF THE EFFICIENCY
In this section, we will explicitly show the universal nature of both EMP and EMEP. The expressions for the power output and EP are derived in Appendix A [Eqs.
(A11) and (A12)]. Optimization of these equations with respect to control parameters ω h and ω c yields very complex equations, which cannot be solved analytically under the general conditions. However, close to the equilibrium, they can be solved to give analytic expression for the efficiency upto second order term in η C , which is sufficient for our purpose as we want to focus only on the universal nature of the EMP and EMEP.
As mentioned in the Introduction, the appearance of the first two universal terms in the Taylor series of the EMP was first proven by Esposito and coauthors for tight-coupling heat engines possessing a left-right symmetry in the system. We briefly outline the algorithm followed in Ref. [49]. The following formalism is valid for the engines obeying tight-coupling condition between the energy flux I E and matter flux I: where x and y are dimensionless scaled energies (to be explained later). The above equation implies that the energy is exported by the particles of a given energy . The general formula for the EMP is given by where L = −I 1 (x, x) and M = I 11 (x, x)/2. Further, for the systems possessing a left-right symmetry, the inversion of flux, leads to the condition 2M = −∂ x L, which reduces the second term in Eq. (12) to η 2 C /8, thus establishing the universality of the coefficient 1/8 under the symmetry specified by Eq. (13).
In order to inquire the universal character of efficiency in the SSD model, we have to first identify the flux term. In our model, energy is transported from hot to cold reservoir by the flux of photons. Comparing Eq. (A11) with Eq. (11), we identify I as follows In the above equation, we have put x ≡ ω c /k B T c and y ≡ ω h /k B T h , and used the expressions n h = 1/(e y − 1) and n c = 1/(e x − 1). By inspecting Eq. (14), we can see that symmetry criterion (13) is satisfied for Γ h = Γ c . Under this condition, we should observe the universality of efficiency for the SSD model. We confirm this observation by explicit calculating the form of efficiency in Eq. (12). By evaluating expressions for L, M and ∂ x L for the current I given in Eq. (14), and substituting in Eq. (12), we find where α is the solution of the transcendental equation [67], which can be solved by specifying the numerical values of λ, Γ h and Γ c . For λ 2 = Γ h Γ c , solution of above equation yields α = 2.9327. For the symmetric dissipation, Γ c = Γ h , the term inside the square bracket in Eq. (15) becomes equal to 1/2, yielding the coefficient of second term as 1/8, and hence proving our assertion.

A. Universality of the EMEP
The general expression for the EMEP analogous to Eq.
Comparing Eqs. (12) and (17), we can see that in order to obtain the explicit expression for the EMEP, we just have to replace η 2 C /4 by 4η 2 C /27 in Eq. (15); everything else remains the same. Since, for Γ h = Γ c , the term inside square bracket in Eq. (15) is 1/2, we get second term as 2η 2 C /27. From the above procedure, we can conclude that if universal nature of the EMP can be established for a model under consideration, the universality of the EMEP automatically follows. The universal character of the EMEP has already been established for the low-dissipation [59], endoreversible [54,68] and nonlinear irreversible [56] models of classical heat engine. Ours is the first study of a QHE in which universality of the EMEP is explored and explicitly shown.
B. Global optimization of efficient power function in low-temperature limit Now, we study the operation of the SSD engine in the low-temperature regime. In the low-temperature limit, we assume in this case is given by In our previous work [23], we have proven the equivalence of the SSD engine operating in the low-temperature limit to Feynman's ratchet and pawl engine [69][70][71], a classical heat engine based on the principle of Brownian fluctuations. Hence, the analysis of this section is also valid for Feynman's model. Maximization of Eq. (18) with respect to ω h and ω c , and a little simplification yields the following equations: It is not possible to obtain analytic solution of these two equations for ω h and ω c . However combining Eqs. (10), (19) and (20), and writing in terms of η C = 1 − T c /T h , we obtain following transcendental equation, It is clear from the Eq. (21) that the efficiency does not depend on the system-parameters and depends on η C only. We plot Eq. (21) in Fig. 2 and compare the EMEP (EMP) of the cold SSD engine with the corresponding EMEP (EMP) of endoreversible or low-dissipation heat engines. Near equilibrium, a perturbative solution for the Eq. (21) is available and is given by [65]: Hence, again in this regime, we are able to show the existence of the first two universal terms (2η C /3 and 2η 2 C /27) for a two-parameter optimization scheme.

IV. LOCAL OPTIMIZATION
Since for the unconstrained regime, general solution for the two-parameter optimization of the SSD engine is not available, in the following, we will optimize the performance of the engine with respect to one control parameter only while keeping the other one fixed at a constant value. In the high-temperature limit, it is possible to obtain analytic expressions for the EMP and EMEP. In this limit, we put n h ≈ k B T h / ω h and n c ≈ k B T c / ω c .

A. High temperature and strong coupling regime
Assuming the strong matter-field coupling (λ Γ h,c ), the expression for EP function in the high-temperature limit can be written as It is important to mention that two parameter optimization of EP function given in Eq. (23) is not possible. Such two parameter optimization scheme leads to the trivial solution, ω c = ω h = 0, which is not a useful result. It indicates that a unique maximum of P η with respect to both ω c and ω h cannot exist. It can be argued as follows. For the given values of the bath temperatures and the coupling constants, under the scaling (ω c , ω h ) → (βω c , βω h ), where β is a certain positive number, the EP function also scales as P η → βP η . Hence, there cannot exist a unique optimal solution (ω * h , ω * c ) that yields a unique maximum for EP function. The same is also true for the optimization of power output of the engine with respect to ω c and ω h . Therefore we will perform optimization with respect to one parameter only while keeping the other one fixed. First we keep ω h fixed, then optimizing EP [Eq. (23)] with respect to ω c , i.e., setting ∂P η /∂ω c = 0, we evaluate EMEP as where γ = Γ h /Γ c . For a given value of τ , η P η ω h is a monotonically increasing function of γ. Hence, we can obtain the lower and upper bounds of EMEP by setting γ → 0 and γ → ∞, respectively. Writing in terms of η C = 1−τ , we have (25) Recently, η Pη − ≡ 2η C /3, has also been obtained as the lower bound of the low-dissipation heat engines operating at MEP [59,60]. The upper bound η Y C obtained here, was first obtained by Yan and Chen for an endoreversible heat engine [54]. Hence, we name it after them. η Y C can also be obtained for symmetric low-dissipation heat engines [59].
Alternately, we may fix the value of ω c and optimize EP function with respect to ω h . In this case, we get following equation: Upper bound η Pη + obtained here also serves as the upper bound of the low-dissipation model of heat engine [59,60]. The same expression also appears in the optimization of Feynman's model operating at MEP in hightemperature regime [65].
The corresponding efficiency bounds for the optimization of power output of the SSD engine is given by [22] Comparing Eqs. (25) and (27) with Eqs. (28)-(29), we can conclude that the SSD engine operating under MEP is far more efficient than the engine operating at MP (see Fig. 3).

B. Weak coupling in high-temperature regime
In addition to the strong matter-field coupling regime (λ Γ h,c ), we can also find analytic expressions for the efficiency in weak matter-field coupling regime (λ Γ h,c ). In the high-temperature limit (n c,h 1), the first two terms in the denominator of Eq. (A12) can be ignored. Plus we need extreme dissipation conditions, i.e., Hence, for γ → 0 and γ → ∞, Eq. (A12) can be approximated by the following two equations, respectively: Optimization of Eqs. (30) and (31) with respect to ω c (ω h fixed) yields the following bounds on the efficiency (32) Note that the above bounds lie below the the parametric area bounded by the efficiency curves given in Eq. (25) (see Fig. 3). To the best of our knowledge, these are the new bounds which are not previously obtained for any model of classical or QHE. Similarly, for the optimization of Eqs. (30) and (31) with respect to ω h (ω c fixed), EMEP lies in the range Similar to the above case, the efficiency curves in Eq. (33) lie above the parametric area bounded by the efficiency curves given in Eq. (27) (see Fig. 3).
The corresponding expressions for the EMP show similar trend, and, are given by We summarize our findings in Table I. As can be seen from Table I, Taylor's series expansions for different expressions for the EMP and EMEP show very interesting behavior. For the EMEP, the first universal term 2η C /3 is present in all cases, and the second terms constitute an arithmetic series with common difference 2η 2 C /27. Similarly for the EMP, the first universal term η C /2 is present in all cases, and the second term increases by η 2 C /8 in going from the first case to the last case. Additionally, the third terms in the series expansion of various forms of the EMP also constitute an arithmetic series with common difference η 3 C /16.

C. Universality of efficiency in one-parameter optimization
Now, we explore the universal nature of efficiency for one-parameter optimization scheme. We notice that if we put Γ c = Γ h (γ = 1), in Eq. (24), the obtained form of the efficiency, does not include the second universal term 2η 2 C /27 unlike the case of global optimization over the two parameters as shown in section II. We attribute this to the nature of optimization scheme. The parametric space available to the control variables is different for two different optimization schemes, hence explaining the difference between them. However, one can still retain the second order universal term η 2 C /27 if one imposes an extra symmetric condition on the constraints of the optimization. The constraints are symmetric in the sense that under the exchange of the control variables, the constraint equation remains unchanged. The physics of such constraints is explored in the Ref. [72]. Here, we want to focus only on the universal character of the efficiency under such constraints. For instance, if we impose a symmetric constrain, viz, ω c + ω h = k, optimization of Eq. (23) with respect to ω h leads to the following equation: where we have put x = ω c /k for simplicity. The above equation is not solvable in terms of real radicals due to Casus irreducibilis (see Appendix C). However, the equation can be solved in terms of trigonometric functions [73], and the solution is given by where , and taking its series expansion, we have which clearly shows the presence of the first two universal terms. In the similar manner, for another symmetric constraint ω c ω h = k , the expression for efficiency, η = 1 − k /ω 2 h , turns out to be and we again retain the second universal term η 2 C /27. Here, M = √ kf (τ ), is function of τ only. We can also obtain the first two universal terms (η C /2 and η 2 C /8) in the series expansion of the EMP for the optimization under symmetric constraints. Thus for the SSD model, we have shown that in order to establish the universality of efficiency upto the quadratic order term in η C , we have to impose an additional symmetric condition in addition to the condition γ = 1. Similar is also true for the optimization of an ultra hot Otto engine [72] and Feynman's ratchet model [74] both of which possess a certain left-right symmetry in the system.

V. FRACTIONAL LOSS OF POWER AT MAXIMUM ECOLOGICAL FUNCTION AND MAXIMUM POWER OUTPUT
In this section, we make a comparison of the performance of the SSD engine operating at MEP to that of operating at MP. In both cases, we find the expressions for the fractional loss of power due to entropy production, S tot =Q c /T c −Q h /T h . Power loss due to entropy production is given by: P lost = T 2Ṡtot =Q c − (1 − η C )Q h . Further using the definitions of power output P =Q h −Q c and efficiency η = P/Q h , the ratio of power loss to power output can be derived as: We calculate the ratio R in four different cases: two for the optimization of EP with respect to ω c (ω h fixed) in the extreme dissipation limits when γ → 0 and γ → ∞, and similar two cases for optimization with respect to ω h (ω c fixed). Using Eqs. (25) and (41), we have Similar equations for the optimization of P η with respect to ω h for a fixed ω c can be obtained by using Eqs. (27) and (41): For near equilibrium conditions (η C → 0), all above equations approach the value 1/2 (also see Fig. 4). The frac-tional loss of power is lower for the case with fixed ω c R P ω c (γ→∞)  (42) and (43)] represent the case when EP function is optimized whereas the upper lying curves [Eqs. (44) and (45)] represent the corresponding case for the optimization of power output.
As can be seen from Fig. 4, the curves representing the optimal power case follow the same trend as noted for the optimal EP. More importantly, for small values of η C (near equilibrium), the curves (lower set) for optimal EP lie well below the curves (upper set) for optimal power.
We specifically mention the case where R P ω h (γ→0) = 1, which implies that in this case, power loss is equal to the power output. The corresponding case for the optimal EP presents us with much better scenario. In this case, R Pη ω h (γ→0) = 1/2, which implies that loss of power is half of the power output. It indicates that EP function is a good target function to optimize if our preference is fuel conservation.

VI. RATIO OF POWER AT MAXIMUM EFFICIENT POWER TO MAXIMUM POWER
Since the fractional loss of power is less when our SSD engine operates at MEP as compared to the the case when engine is operating at MP, it is useful to calculate the ratio (R ) of power at MEP to optimal power. Dividing Eq. (B5) by Eq. (B6), and taking the limits γ → 0 and γ → ∞, we get following two expressions, respectively: Similar equations can be obtained for optimization over ω h (fixed ω c ), and are given by For very small temperature differences, i.e., η C → 0, the ratio R = 8/9, which shows that at least 88.89% of the MP is produced in the MEP regime, which is a considerable amount keeping in mind that the power loss in MEP regime is at least 1/2 of the case when engine operates at MP. It is clear from Fig. 5 that ratio R increases with increasing η C , which is expected behavior since the efficiency also increases, while the dissipation decreases.  46) and (47)].

VII. CONCLUDING REMARKS
We have thoroughly investigated the performance of the SSD engine operating under the conditions of MEP and side by side compared its performance with the engine operating at MP. First, for close to the equilibrium conditions, we found a analytic solution for the efficiency of the SSD engine and explicitly showed the universality of the first two terms of both EMP and EMEP under the symmetric dissipation (γ = 1) condition. Then, we carried out optimization of EP function alternatively with respect to one of the control frequencies ω c or ω h while keeping the other one fixed at a constant value. In the high-temperature limit, we were able to find lower and upper bounds on the EMEP for strong (λ Γ h,c ) as well as for weak (λ Γ h,c ) matter-field coupling conditions. Then, we showed that the obtained form of the EMEP in case of one parameter optimization shows universal features of efficiency only in the presence of an extra symmetry imposed on the control parameters of the engine. It is important to highlight that except for the weak matter-field coupling (λ Γ h,c ) condition, the obtained expressions of the EMEP and EMP in all cases discussed above are same as obtained for different models of classical heat engine. Specifically, in weak matter-field coupling regime, we obtained some new bounds on the efficiency of the SSD engine which lie beyond the area covered by bounds obtained for strong matter-field coupling.
Finally, we have compared the optimal performance of the SSD engine operating at MEP to that of operating at MP. It can be inferred that fraction loss of power due to entropy production is appreciably low in the case of heat engine operating at MEP while at the same time it produces at least 88.89% of the MP output. This indicates that EP function is a good optimization function and real engines should be designed to operate along the lines of maximizing EP function if our preference is environment and fuel conservation.
Evaluation of the trace in Eq. (6) leads to the following form of output power, Similarly calculating the trace in Eq. (5), heat fluxQ h from the hot reservoir can be written aṡ Substituting Eqs. (A6) and (A7) in Eq. (A8), we get final expression for the power output. Since EP function is just power output multiplied by the efficiency, we have following expressions for power output and EP, respectively: . (A12) Appendix B: Optimization of P and Pη in high-temperature and strong-coupling regime In the high temperatures limit, n h and n c can be approximated as Using Eq. (B1) in Eq. (A11) and (A12) and dropping the terms containing Γ c,h in comparison to λ, we can write P and P η in terms of γ = Γ h /Γ c and τ = T c /T h in the following form Expressions for power at MEP and MP