Shortcut-to-adiabaticity quantum tripartite Otto cycle

For an Otto cycle there always exists a trade-off between the cycle efficiency and the output power due to the requirement of cycle length. The shortcut to adiabatic (STA) technology provides an effective way to deal with the difficulty of zero-output power in conventional Otto cycle. In this paper, the Otto cycle of three-qubit system as the working substance with counterdiabatic driving has been investigated. It is demonstrated that the tripartite Otto cycle as a universal machine, in the suitable regimes of external control parameter, could work as a quantum heat engine (QHE), refrigerator or heat pump. And, the performances of QHE and refrigerator with and without STA, such as the power and efficiency of QHE and the coefficient of performance (COP) and figure of merit (FOM) of refrigerator, have been investigated. It shows the application of STA scheme can lead to an effective enhancement in the performances of Otto cycle, including achievements of a high QHE’s/refrigerator’s power associated with a moderate QHE’s efficiency/COP of refrigerator. Especially, it is interesting that even in a short-time cycle the optimization of control parameters could arise a remarkable improvement in the efficiency (or COP) of STA QHE (refrigerator), approaching the ideal efficiency or COP of conventional Otto cycle with quasi-static process. Finally, with the aid of parameter optimization the trade-off regions between the efficiency and the power (the COP and the FOM) of STA Otto engine (refrigerator) have been advised.


Introduction
Thermodynamics has always been a fascinating field to investigate heat phenomenon and energy conversion [1]. Heat engine * Authors to whom any correspondence should be addressed.
Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. and refrigerator as two types of vital platforms of heat-to-work conversion, up to now, have been playing an irreplaceable role in thermodynamics. With the rapid development of nanotechnology and quantum information processing technology, the research domain in thermodynamics has been extended widely from the macro-scale to the micro-scale. Being different from the classical thermal machine (heat engine or refrigerator), the quantum machine utilizing the quantum system as working substance (WS) could manipulate the quantum state to do work or exchange heat, and sometimes show the superior performance (such as, higher efficiency or power) beyond the ones in the classical heat engine [2]. Currently, the explorations of quantum machine whether in theory [3][4][5][6][7][8][9] or in experiment [5][6][7][8][9][10][11][12][13][14][15], have become one of the most active branches in the field of quantum thermodynamics due to its wide potential application ranging from heat transport in nanodevices [16,17] to biological process control [18,19], among others [20].
Quantum Otto cycle as an effective scheme of thermodynamic cycle has been widely studied [18,[21][22][23][24][25][26][27][28][29]. And, the various quantum WS, such as harmonic oscillator [25,28], spin coupling atoms [21,24] and the hybrid system of atoms plus cavity [29], are considered. A full quantum Otto cycle usually consists of four strokes: two heat exchange strokes and two work exchange strokes [30]. The former occurs in the isochoric process where the WS alternatively is coupled to two baths at different temperatures. And the latter is implemented in the isentropic process where the WS evolves adiabatically from its initial thermal state to a final state by adjusting the controllable time-dependent parameter of system. In the conventional Otto cycle, the quasi-static isentropic stokes with an infinite time ensure the machine capturing high efficiency, but leads to a zero-output power. To overcome this drawback, it is necessary to consider the finite-time quantum thermodynamics [31][32][33][34][35][36] instead of quasi-static thermodynamics [37][38][39][40]. Therefore, understanding the finite-time effect of thermodynamic processes is crucial to the optimization of the finitetime heat machines [35,[41][42][43][44][45]. For a quantum Otto cycle, considering the finite-time adiabatic processes is the key to deliver more output for the same input without sacrificing power. One of the well-suited strategies to address this issue is the application of shortcut to adiabatic (STA) technology [46,47].
STA technology can simulate the perfect adiabatic process by a reasonable acceleration manipulation for the system. Under a suitable external driving, the system can be brought into the desired final state in a finite time, and in which any final state excitation can also be fully suppressed. At present, different STA technologies have been developed, such as the counterdiabatic driving (CD) [46][47][48][49][50][51][52][53][54], local counterdiabatic driving [55,56], methods based on invariants [57][58][59][60] and fast-forward technique [61][62][63]. These methods have also been verified in experiments [64][65][66][67]. In particular, CD is designed to keep the time-evolving quantum state on the adiabatic manifold at all times. By introducing an external Hamiltonian, the adiabatic eigenstate of the original Hamiltonian is the exact solution of the total Hamiltonian. Notably, recent theoretical studies have shown that they are useful for the enhancement of thermodynamic machine performances.
However, most of works mainly focus on STA Otto cycle with a single-or double-body WS, such as a harmonic oscillator [68][69][70][71][72], a single spin qubit [73,74] and a two-qubit [74][75][76], and only a single thermal function of the Otto cycle serving as a heat engine or a refrigerator has been considered. For the STA many-body Otto cycle as a universal machine integrating several thermal functions of heat machine, refrigerators and heat pump there has not been addressed enough. How the application of STA technology [68][69][70][77][78][79][80][81][82] in a quantum many-body Otto cycle affects the performances of machine is still an interesting and open question.
In this work, we consider an Otto cycle of three-qubit system as the WS, and investigate the performances of the machine assisted with the STA technology at length. Firstly, in terms of the characteristics of the work and heat exchange of the cycle, we demonstrate that the three-qubit system could serve as a universal thermal machine [83,84], i.e., the machine can perform a multi-functional device as a quantum Otto heat engine, refrigerator, unidirectional and bidirectional heat pumps which depend on the external control parameter. Secondly, we study the thermodynamics characteristics of STA Otto engine/refrigerator with CD scheme. Compared with the conventional adiabatic/nonadiabatic (free evolution in the expansion and compression processes) Otto cycle, the STA machine can exhibit several prominent advantages. For example, the STA Otto engine can get a high output power with a moderate efficiency, and both engine's efficiency and output power in CD scheme can be larger than that in free (nonadiabatic) evolution scheme; and, for the Otto refrigerator though the coefficient of performance (COP) of STA refrigerator is slightly less than that of the ideal adiabatic Otto refrigerator it could remain a significant cooling power under some regimes of control parameters, i.e., a significant figure of merit (FOM) can be obtained. Finally, the optimization of control parameter of Otto cycle has been considered. It is found that the optimization of control parameter can effectively reduce the STA cost and result in a high efficiency (COP and FOM) of STA engine (refrigerator) even in a short-time cycle.
The remainder of this paper is organized as follows. In section 2, we introduce the model of three-qubit quantum Otto cycle, and present the different thermal functions of three-qubit Otto cycle as a universal machine working as a heat engine, refrigerator and heat pump in terms of control parameters. In section 3, we mainly investigate the performances of the QHE/refrigerator with CD, and illustrate the effect of CD term on relevant thermodynamic quantities. In section 4, the optimization of control parameters of Otto cycle has been analyzed at length. Finally, we conclude the work in section 5. The computation of eigenvalues and eigenstates of the Hamiltonian are given in appendixes A and B.

Quantum Otto cycle and universal heat machine
Next, we consider a coupled three-qubit system as WS to implement an Otto cycle. The Hamiltonian of the system at any time t is denoted as H 0 (t) and expressed by where b j (t) ( j = a, b, c) is the time-dependent external magnetic field acting on the j th qubit of three-qubit system in z direction, and for simplicity we set b x,z represent the spin-1/2 Pauli matrices of the jth qubit, and g is the coupling strength among three qubits. For brevity, the energy eigenvalues (eigenstates) E 0 n (t) (|ψ n (t) ) (n = 0, 1, . . . , 7) of Hamiltonian H 0 (t) are analytically given in appendix A. The The works (heat exchange) implemented between the work substance (WS) and external agent (heat baths) in different processes, and the cycle is completed in time τ cycle = τ 1 + τ 2 + τ 3 + τ 4 . Under the suitable control parameters, b i,f (two external magnetic field endpoints of isentropic process), the Otto cycle can work as a quantum heat engine (QHE) (heat flowing from the hot bath (T h ) to the cold bath (T c ) as indicated by the red arrows) or a quantum refrigerator (heat flowing from the cold bath to the hot bath as indicated by the blue arrows). One part of the amount of heat extraction Q h from the hot bath is delivered to the cold one Q c , and the left contributes to the net output work W exp + W comp for the engine. And, the heat extraction Q c from the cold bath is implemented by external work cost |W exp + W comp | for the refrigerator.
complete Otto cycle includes two isochoric thermalization processes and two isentropic adiabatic processes shown in figure 1 are given as following: Step 1. Adiabatic expansion stroke (A → B). In this process, WS is isolated from the hot bath at inverse temperature β h = 1/T h (we have set k B = 1), and the system, after time τ 1 , is brought from its initial thermal state ρ A , to ρ B by adjusting the control parameter (external magnetic field) changing from b i to b f very slowly. According to the adiabatic theorem, there is no transition among the system's levels when a quantum system evolves slowly enough, i.e., the population of each energy level of the system remains unchanged, and it always kept in the instantaneous energy eigenstate of the initial Hamiltonian. Thus, in this process, there is only the work exchange between the system and the external agent, and can be written as where Step 2. Cold isochoric stroke (B → C). The WS is put in contact with the cold bath β c = 1/T c . For a time interval τ 2 the system will be thermalized from the state ρ B to a thermal state ρ C , ρ C = 7 n=0 e −β c E 0 n (b f ) |ψ n (t) ψ n (t)| /Z c with the partition function Z c = 7 n=0 e −β c E 0 n (b f ) . In this process, the control parameter remains constant, i.e., b(t) = b f with τ 1 t τ 1 + τ 2 , and the Hamiltonian of system is time-independent H(b f ). Therefore, only the heat exchange occurs between the system and the cold bath and the amount of heat exchange is where Λ γ α = [g sinh(g/T α ) + 2δ f sinh(δ γ /T α ) + ε f sinh (ε γ /T α )]/Ω γ α with Ω γ α = cosh(g/T α ) + 2 cosh(δ γ /T α ) + cosh (ε γ /T α ), γ = i, f and α = h, c.
Step 3. Adiabatic compression stroke (C → D). Similar to the adiabatic expansion stroke in step 1, the WS is first isolated from the cold bath, then it, after a time interval τ 3 , evolves from the initial thermal state ρ C to the final state ρ D adiabatically via changing the control parameter from b f to b i very slowly. In this adiabatic process, the energy change of WS completely contributes to the work to the external and reads Step 4. Hot isochoric stroke (D → A). Fixing the control parameter b i , i.e., the system's Hamiltonian remains in H(b i ), the WS is coupled to the hot bath lasting a time length τ 4 which brings the system into the initial state ρ A in step 1 from ρ D , and the Otto cycle ends. Similar to the process of B → C the heat exchange in this stroke becomes In a complete cycle, the total cycle time is the sum of the different processes τ cycle = τ 1 + τ 2 + τ 3 + τ 4 . In general, the Unidirectional heat pump thermalization processes take much less time than isentropic processes [6,[85][86][87], i.e., τ 1,3 τ 2,4 , the total cycle time can be approximately taken as τ cycle ≈ τ 1 + τ 3 = 2τ where for simplicity τ 1 = τ 3 = τ has been considered throughout this paper. Here, it is noted that the cycle time τ cycle characterizes the speed of quantum machine running (or the quick/slow evolution of the system fulfilling a complete cycle) which is closely related to performances of quantum machine, such as the power and efficiency of engine and FOM of refrigerator. Therefore, it can be regarded as an important parameter to be considered in our model. For the conventional Otto cycle with ideal quasi-static expansion and compression processes, the system's dynamic becomes reversible and the work done by the system completely comes from the adiabatic processes as given in equations (2) and (4). And, the total work output (or network) denoted as W AD in the adiabatic cycle is To choose the suitable working points (b f , b i ), the net work W AD , and the heat exchanges Q h and Q c in the threequbit Otto cycle can demonstrate different characteristics, and the Otto cycle can perform a universal machine working as a heat engine, refrigerator, bidirectional-or unidirectional-heat pumps as shown in table 1. Here, the positive (negative) work, W AD > 0 ( W AD < 0), in table 1 represents the work done by WS (external agent) to the external agent (WS) in a full cycle, and the positive (negative) heat exchange, Q h,c > 0 ( Q h,c < 0) corresponds to the amount of heat flowing out of (into) hot/cold baths in the isochoric thermalization processes. No matter which type of I-IV in table 1, the first law of thermodynamics always holds: The different thermal functional regions of Otto cycle with type I-IV listed in table 1 have been shown in the control parametric space of 0 b i,f 1 in figure 2 where T h = 5, T c = 1 (the temperatures of hot/cold baths), and g = 0.2 (the coupling strength) are fixed. From figure 2, one can see that the functions of the three-qubit Otto cycle strongly depends on the control parameters b i and b f , and each thermal functional regime corresponds to a certain continuous parametric space of (b f , b i ) which is convenience to switch the different functions orderly. We, in this paper, are interested in the heat engine and refrigerator and discuss them in the subsequent subsections at length.
For a heat engine, the major quantities to characterize the performances of engine are the efficiency and the power. And, the efficiency can be defined as the ratio of the total output work W tot to the total input energy from the hot bath Q h and external agent denoted as W dri , i.e., η = W tot /( Q h + W dri ), and the power is described as the average net output work per unit time, i.e., P = W net /τ cycle with W net = W tot − W dri (noted that W dri is defined as the extra energy cost of external agent to sustain adiabatic evolution in a finitetime expansion/compression processes, such as the STA cost introduced in the subsequent sections in this paper, and in the ideal adiabatic Otto cycle W dri is zero). For the three-qubit ideal adiabatic heat engine, the total output work equals the net output work with zero driven energy cost, W tot = W net = W AD , and the efficiency, in terms of equations (5) and (6), is expressed as with . The power becomes zero, i.e., due to the requirement of infinity time to implement the quasistatic adiabatic expansion/compression process.
To obtain the nonzero output power the cycle of engine can be fulfilled in a finite time. We suppose that the system (WS) performs the free evolution with Hamiltonian H 0 (t) in the expansion/compression strokes. The dynamics of WS is governed by the von-Neumann equationρ(t) = −i [H 0 (t), ρ(t) ] /h (seth = 1, hereafter). However, the finite power output in this scheme is not free where partial work will be dissipated to overcome the internal quantum friction. Usually, the work deficit caused by quantum friction can also be taken as irreversible work W irr and defined as [88] where S(ρ σ) = Tr(ρ ln ρ − ρ ln σ) represents the quantum relative entropy between the two states ρ and σ, and ρ t (ρ ad t ) is the instantaneous state of H 0 (t) (the corresponding adiabatic evolution state with fixed populations distribution being the same as that of the thermal state with inverse temperature β at the beginning of expansion or compression processes) in the process with (without) internal friction at time t. Thus, in the nonadiabatic cycle the total work done by WS W NA tot can be divided into two parts: the dissipative irreversible work W irr caused by the internal friction in the expansion/compression processes and the output work W AD in the adiabatic cycle, net the net output work. Therefore, the power of the nonadiabatic heat engine reads and the corresponding efficiency is where Q NA h represents heat input from hot bath to WS and no external driven energy is injected in this nonadiabatic cycle. Here, it is noticed that although the WS remains the same energy spectrum with Hamiltonian H(b i ) at the beginning of hot isochoric stroke in the nonadiabatic and adiabatic cycle schemes the heat input from the hot bath Q NA h in the nonadiabatic cycle scheme and Q h (as given in equation (5)) in adiabatic cycle scheme are not equal, i.e., Q NA h = Q h , due to the different population distributions of WS with H(b i ) at the beginning of heating process in the two schemes.
For a refrigerator, the COP, cooling power and FOM independently defined as ε = Q h / W tot (the ratio of heat extraction by WS from the cold bath Q c to the input total work W tot by the external agent), J C = Q h /τ cycle , (the average amount of heat flowing out the cold bath in unit time), and χ = εJ C (COP multiplied by cooling power) are regarded as the major indicators to characterize its performances. For the conventional Otto refrigerator the input total work by external agent into WS is W tot = − W AD where no extra energy costs except for the work done by external agent in the ideal adiabatic expansion/compression processes. According to equations (2)-(4) the COP, cooling power and FOM of adiabatic three-qubit Otto refrigerator can be expressed as and respectively. Similarly, for a nonadiabatic Otto refrigerator with a finite-time cycle due to the quantum friction in the expansion/compression processes the external agent could consume an extra work W irr (irreversible work) beyond the work − W AD in the adiabatic cycle scheme, i.e., W NA tot = − W AD − W irr . To denote Q NA c as the heat extraction from the cold bath, the nonadiabatic COP, cooling power and FOM can be independently given as and Here, the heat extractions from the cold bath Q NA c in the nonadiabatic cycle scheme does not equal Q c (given in equation (3)) in adiabatic cycle scheme, i.e., Q NA c = Q c . The same reason is for that Q NA h = Q h .

Shortcut-to-adiabaticity quantum Otto cycle
To ensure adiabatic evolution the conventional Otto cycle requires expansion/compression processes implemented slowly enough which also leads to the zero-output power. In order to achieve an output power as large as possible it is necessary to speed up the expansion/compression evolutions and suppress the transitions between the energy levels of WS as possible.
For the above purpose, the STA technology has become an effective way. Specially, one could introduce a CD Hamiltonian H CD (t) to the original Hamiltonian H 0 (t). The form of H CD (t) can be exactly expressed as [50] where |n(t) is the n th eigenstate of H 0 (t), which ensures that the adiabatic eigenstates of the original Hamiltonian H 0 (t) are the same as that of the total Hamiltonian H(t) = H 0 (t) + H CD (t). Substituting the energy eigenstates of Hamiltonian H 0 (t), i.e., |ψ n (t) (eigenvalues E 0 n (t)), given in appendix A, into (19) one can obtain the exact form of CD driving in our three-qubit model as where α(t) =˙b (t)g 4b(t) 2 +g 2 , β(t) = 2ḃ(t)g 16b(t) 2 +g 2 . In addition, for simplicity, the energy eigenvalues E n (t) and eigenstates |Φ n (t) of total Hamiltonian H(t) can be evaluated and presented in appendix B. For a complete Otto cycle, the boundary conditions of H(t) at the beginning/end of cycle become And the smoothness of the external field trajectory requiresb(0) = 0 andb(τ ) = 0. There are many driving functions that satisfy the above requirements [57,70,74]. Here we choose a simple interpolating ansatz b(t) = 5 j=0 a j t j and substitute the polynomial into six equations. One suitable driving function for working parameter b(t) satisfying the above requirements can be written as with γ = t/τ . It is pointed out that the form of driving function b(t) satisfying the above six boundary conditions is not unique, such as b(t) = b i + (b f − b i )sin 2 [π sin 2 (πγ/2)/2] being alternative expression [89], but it has not significant difference from the one in equation (21) for improvement of machine's performance in this work. Thus, we take the form of b(t) in equation (21). Currently, to achieve the general optimal driving function in the STA thermodynamics cycle is a challenging task and still remains an open question. However, the application of machine learning on quantum thermodynamics might provide a potential way to this issue [90]. Here, it is noted that for the three-qubit Otto cycle without and with CD the work performed by WS in the expansion/compression processes become the same.

Energy cost of CD driving
STA technology allows expansion or compression process to become adiabatic in a finite time. However, it is not free to implemented. Here, it is noticed that the application of CD scheme is required to consume extra energy to suppress the coherence generated by system evolving in a finite time, i.e., the average work produced by introducing the CD with Hamiltonian H CD (t). Explicitly, the energy cost of CD can be written as [74] where Ḣ l CD (t) = Tr[ρ 0 (t)Ḣ l CD (t)] with l = exp, comp, CD occurring in the expansion/compression processes, the energy cost of CD can be divided into two parts: H exp CD τ and H comp  driving time τ , such as τ < 5, the CD cost in the expansion process is obviously higher than that in compression process, i.e., H exp CD τ > H comp CD τ . It can be understood that the large populations at high energy levels of WS in the expansion process require more energy consumed by the external agent to suppress the transition of energy levels than that in compression process. And, as the driving time τ grows (the transition rate among energy levels becomes weak) the CD cost H exp CD τ in the expansion decreases rapidly and closes to the one H comp CD τ in the compression process, such as when τ 10, H exp CD τ ≈ H comp CD τ , and in the limit τ → ∞ the ideal adiabatic process is reached H exp,comp CD τ → ∞.

Quantum Otto heat engine with CD
Since the STA heat engine can perform the adiabatic evolutions in the expansion/compression processes the total output work is the same as that in the conventional adiabatic cycle, i.e., W STA tot = W AD . However, the finite-time STA cycle requires an extra energy input by the external agent, i.e., the CD cost H CD τ cycle , which also results in the total energy input and the net output work being Q STA h + H CD τ cycle and W STA net = W AD − H CD τ cycle . Therefore, the efficiency and the power of STA heat engine are given as [69,70,77,78] where Q STA h = Q h and Under the assumption of ideal STA with zero CD-cost, H CD τ cycle = 0, the power of engine in equation (24) reduces to which corresponds to the ideal STA power of engine. We, in figure 4, plot the variations of the efficiency and power of the Otto engine operating at the adiabatic, nonadiabatic and STA schemes with cycle time τ cycle (refer to the total time in a complete cycle, i.e., τ cycle = 2τ ). From figure 4(a), it can be seen that as expected both the STA efficiency, η STA in equation (23) and the nonadiabatic efficiency η NA in equation (12) monotonically increase with τ cycle , and approach the ideal adiabatic efficiency η AD in equation (8) for a long-time cycle corresponding to a slow evolution. However, before η STA and η NA reach the steady value η AD the STA efficiency is always larger than the nonadiabatic efficiency, i.e., η STA > η NA with τ cycle < 60. Especially, for a rapid cycle, the nonadiabatic engine cannot operate, such as about τ cycle 9, η NA 0 due to the work output being suppressed completely by the irreversible work from the friction i.e., W NA net = W AD − W irr 0, while the STA engine can achieve a remarkable efficiency even for a very short-time cycle, such as when τ cycle = 2 the STA efficiency η STA ≈ 0.35. Similarly, the STA power P STA in equation (24) is also larger than the nonadiabatic power P NA in equation (11) except for a long cycle time the powers P STA and P NA tend to the same value and reach the ideal STA efficiency with zero CD-cost P ISTA defined in equation (25), that is, P STA > P NA with τ cycle 50 and P NA ≈ P STA ≈ P ISTA for τ cycle > 50 shown in figure 4(b). In addition, due to the introduction of CD cost the STA power P STA behaves a non-monotonic function of cycle time τ cycle where P STA increases first and then decreases as increasing τ cycle . And, when τ cycle ≈ 9 the STA power P STA = 1.23 × 10 −3 is approximately 96% of its maximum value at τ cycle ≈ 12, while the nonadiabatic efficiency becomes P NA ≈ 0 at τ cycle ≈ 9.

Quantum Otto refrigerator with CD
As shown in figure 2, to adjust the work points (b f , b i ) of Otto cycle the machine can perform different thermal functions. Here, to set control parameters: b i = 0.8 and b f = 0 in refrigeration regime II we study the performances of STA refrigerator. Like the heat engine, the COP, cooling power and FOM of the refrigerator under the application of CD scheme can be modified as follows [72] (27) and where W STA tot = − W AD . Similar to P ISTA , we also introduce ideal STA cooling power J C STA and FOM χ ISTA , which is expressed as and Since the cooling power of STA refrigerator is independent of the CD cost the two qualities J C STA and J C ISTA respectively defined in (27) and (29) are the same, i.e., J C STA = J C ISTA . In figures 5(a)-(c), we plot the variations of the COP, cooling power and FOM of the STA Otto refrigerator with respect to τ cycle (τ cycle = 2τ ), respectively, and make comparisons with the results in the adiabatic and nonadiabatic schemes. Figure 5(a) shows that the variation of COP of refrigerator with three different evolution schemes: the adiabatic COP ε AD in equation (13), the STA COP ε STA in equation (26) and nonadiabatic COP ε NA in equation (16) with cycle time τ cycle . One can see that for any time τ cycle an obvious hierarchical relationship exists, i.e., ε NA ε STA ε AD (the equal sign holds only for long enough cycle time), which is similar to the feature of engine's efficiencies shown in figure 4(a). Meanwhile, due to the quantum friction the non-negative COP can appear only when the nonadiabatic refrigerator performs a slow evolution, such as for τ cycle 44, ε NA 0. While, the STA refrigerator can achieve a high COP even for a rapid evolution (i.e., a short cycle period), for example, when τ cycle = 2, ε STA ≈ 0.035, it is significantly more than half of adiabatic COP ε AD (ε AD ≈ 0.057) of ideal adiabatic refrigerator. For a slightly longer time, such as τ cycle = 60 STA COP has approached the adiabatic COP ε AD and is about 2.5 times nonadiabatic COP, i.e., ε STA ≈ ε AD ≈ 2.5ε NA . The similar superiorities of STA refrigerator also exist in cooling power and FOM as independently demonstrated in figures 5(b) and (c). And, one can see that only when the cycle time is approximately larger than a certain value, i.e., τ cycle 44, the nonadiabatic refrigerator can operate, that is, non-negative cooling power and FOM emerging J C NA 0 and χ NA 0 for τ cycle 44. Meanwhile, we also notice that in the working regime of τ cycle 44 though J C NA and χ NA of nonadiabatic refrigerator exhibit the increasing behaviors with increasing τ cycle they are smaller than J C STA and χ STA of STA refrigerator, respectively, i.e., J C NA < J C STA and χ NA < χ STA which for the long time cycle is clearly shown in inset of figure 5(c). Especially, the FOM χ STA decreasing with τ cycle (seen in figure 5(c)) demonstrates that the shorter of the cycle time the better of the overall performance of STA refrigerator.
By comparisons of the performances (COP, cooling power and FOM) of refrigerator with the CD and the nonadiabatic schemes, it demonstrates that the STA refrigerator can exhibit the prominent superiorities over that of nonadiabatic refrigerator, especially in the situation of short-time cycle.

Effects of control parameters on STA Otto heat engine and refrigerator
The application of STA scheme on the Otto cycle can effectively enhance the performance of machine such as the nonzero-output power in a finite cycle time. However, the power and the efficiency (the COP and the FOM) of the STA heat engine (refrigerator) strongly depend on the external control parameters (here, they refer to the initial and the final magnetic fields). Especially, for a short-time cycle the influence of control parameters on the machine's performances would become more obvious than that in a long-time cycle due to the large CD cost injected the cycle by the external agent. Thus, to manipulate the machine working in the optimal parameters becomes significant. Next, we mainly focus on the effects of control parameters (initial and final magnetic fields b i and b f ) on the STA heat engine and refrigerator in the short-time cycle, and show the characteristics of STA machine via optimizing the control parameters (or working point (b f , b i )).
Based on equations (22)-(24), we plot the variations of STA efficiency η STA , STA power P STA and CD cost H CD τ cycle with control parameters b i and b f (0 b i,f 1), which is corresponding to the red area of figure 2, for short-time cycle τ cycle = 2 in figures 6(a)-(c) where we set: T h = 5, T c = 1 and g = 0.2. From figure 6(a) it can be seen that the distribution of STA power is approximately symmetrical about red straight line with b i ≈ 1.8b f being the fitting function of black dotted curve of optimal control parameter for STA power. Here, it is noted that the optimal control parameter refers to the working points (b f , b i ) which ensures that the largest power/efficiency (FOM/COP) of STA heat engine (refrigerator) occurs when the final magnetic field b f , under the fixed initial magnetic field b i , is taken. Meanwhile, the STA power increases monotonically along the optimal parameter curve, and the largest STA power P A STA ≈ 0.32 is achieved at working point A (b f = 0.53, b i = 1). However, figure 6(b) shows that the STA efficiency is not symmetrical about the optimal parameter curve (the dotted curve or its fitting line shown as the red straight line satisfied b i ≈ 3.94b f + 0.04 with b f > 0.03) which is close to the upper boundary of positive work region. It also implies that at fixed b i the optimal final magnetic field b f for STA efficiency becomes smaller than that for STA power. In addition, similar to the STA power the STA efficiency is also an increasing function of initial or final magnetic field (b i or b f ) in the optimal parameter region (i.e., the black dotted line), and the STA efficiency reaches the maximum η B STA ≈ 0.69 at the working point B (b f = 0.26, b i = 1). From figure 6(c) one can see that the CD cost is positively (negatively) correlated with the initial magnetic field b i (the final magnetic field b f ) for fixed b f (b i ) in the regime of heat engine. Meanwhile, the CD cost at the optimal working points of efficiency (including point B) is always higher than that at the working points of optimal power (including point A) when the same initial magnetic field b i is considered. In figure 6(d) the variations of the STA power and STA efficiency of heat engine independently operating at optimal working point A of power and B of efficiency with the cycle time have been shown. We can see that at the optimal working point of power, i.e., point A (b f = 0.53, b i = 1), the STA power basically equals the ideal STA power P ISTA given in equation (25), i.e., P A STA ≈ P A ISTA which is larger than that at the optimal working point B (b f = 0.26, b i = 1) of efficiency, i.e., P B STA , and the shorter the cycle time becomes the more obvious the advantage is. However, the STA efficiency at point B, η B STA , is absolutely superior to that at point A. Even in a short-time cycle the efficiency η B STA can also achieve remarkable value, such as for τ cycle = 2, η B STA is only about 6% efficiency loss comparing ideal STA (or adiabatic) efficiency η B AD , i.e., η B STA ≈ 0.94η B AD which is larger than that at point A, η A STA , though less CD cost occurs at point A. In addition, one can see that the optimal curves of efficiency and power of STA engine do not overlap, which also indicates that the region between the two optimal curves (see figure 6(c)) becomes the suitable region of control parameters for the trade-off between the STA power and efficiency.
Similarly, according to equations (22), (26) and (28), we plot the variations of STA FOM χ STA , STA COP ε STA and Owing to the optimal curve for STA FOM lies at the left side of that for STA COP, the CD cost in the optimal region for STA FOM is more than that for STA COP at the fixed b i . Meanwhile, by comparisons of the performances (COP and FOM) of STA refrigerator at the optimal working point C for FOM and point D for COP in figure 7(d), it can be seen that at the optimal working point C the STA FOM χ C STA basically coincide with the ideal STA FOM χ C ISTA (defined in equation (30)), i.e., χ C STA ≈ χ C ISTA , beyond χ D STA at optimal working point D for COP in the short-time cycle, and the advantage disappears rapidly as the cycle time τ cycle increases and tends to the same value for a slight long τ cycle . While, the STA COP at optimal working point D ε D STA exhibits obvious advantage over STA FOM ε C STA at point C. Though both ε D STA and ε C STA increase as the cycle time τ cycle increases due to the reduction of CD cost, ε D STA is always larger than ε C STA for any τ cycle , and ε D STA reaches the ideal STA (or adiabatic) COP ε D AD given in equation (13) for a long cycle time τ cycle and ε C STA remains at a lower value. Finally, similar to the situation of STA engine a trade-off regime between the FOM and COP of STA refrigerator is indicated corresponding to the region sandwiched between two lines optimal curves of the FOM and COP shown in figure 7(c).

Conclusions
By the construction of three-qubit Otto cycle we demonstrate that the three-qubit Otto cycle can perform a universal machine to implement the functions of a quantum engine, refrigerator, and heat pumps by adjusting the working parameter. We have investigated the performances of heat engine's (refrigerator's) with CD scheme at length, such as the STA engine's efficiency and power (refrigerator's COP, cooling power and FOM). It demonstrates that both the STA engine (or refrigerator) can exhibit the prominent superiorities in the engine's efficiency and power (or the refrigerator's COP, cooling power and FOM) over the corresponding nonadiabatic engine and the refrigerator. Especially, for the short-time cycle the STA engine (or refrigerator) still can achieve a remarkable efficiency and output power (or COP, cooling power and FOM) while the nonadiabatic engine and refrigerator cannot operate due to the large irreversible work caused by the friction in the rapid expansion and compression processes of the cycle. In addition, we also investigated the effects of control parameters in the performances of the STA Otto engine and refrigerator. It has revealed that by optimizing the control parameters of the machine, the cycle time of the Otto cycle achieving the optimal performances, such as the adiabatic efficiency of engine and COP of refrigerator, can be effectively shortened, which significantly enhances the performance of quantum machine. Here, it is noted that the CD driving protocol described in equation (19) in principle can be applied to arbitrary system (i.e., few-and many-body systems) to fulfill a finite-time adiabatic evolution (i.e., STA). However, it requires the exact knowledge of eigenspectrum of system and is usually suitable for the fewbody system with exact solvable eneryspectrum, but, in general, will be failed in the many-body system. Most recently, to achieve the approximate CD terms via using a type of variational method become a potential way to deal with the issue of STA of many-body system [54,89,91,92]. It is possible to extend the three-qubit system in this paper to multi-qubit system utilizing the variational method, which will be considered in the future work. Anyway, this work might be beneficial for the design of high-performance many-body quantum machine.