Finite-time two-spin quantum Otto engines: shortcuts to adiabaticity vs. irreversibility
FFinite-time two-spin quantum Otto engines: shortcuts to adiabaticity vs. irreversibility
Barıs¸ C¸ akmak College of Engineering and Natural Sciences, Bah¸ce¸sehir University, Be¸sikta¸s, Istanbul 34353, Turkey ∗ (Dated: March 3, 2021)We propose a quantum Otto cycle in a two spin-1 / ffi ciency but di ff erent parameters of the working medium. We observe that, for certainparameter regimes, the irreversibility, as measured by the e ffi ciency lags, due to finite-time driving is so lowthat non-adiabatic engine performs quite close to the adiabatic engine, leaving the STA engine only marginallybetter than the non-adiabatic one. This suggests that by designing the working medium Hamiltonian one mayspare the di ffi culty of dealing with an external control protocol. I. INTRODUCTION
Recent years have witnessed an ever increasing interest and a rapid development in the quest to understand the thermodynam-ics of out-of-equilibrium quantum systems [1–3]. This field of research, widely known as the quantum thermodynamics, blendsvarious branches of physics such as quantum information science, quantum optics, condensed matter physics and quantumcontrol, to name a few. In addition to its contribution to our fundamental understanding of how the concepts of classical thermo-dynamics can be generalized to quantum domain, quantum thermodynamics also significantly contributes in the development ofnew quantum technologies that take advantage of quantum systems by actively manipulating them [4–6].One of the major problems within the realm of quantum thermodynamics stems from this active manipulation. Traditionally,thermodynamic transformations are made quasi-statically, such that they are slow enough so that the subject system is kept atequilibrium at all times. An arbitrary finite time transformation would then require some thermodynamic control [7], so thatone could avoid any form of irreversibility (such as entropy production) originating from fast manipulation of the system [8, 9].Techniques of shortcuts to adiabaticity (STA) are perfectly suitable for such control purposes which make sure that the subjectsystem ends up at the adiabatic final state of the desired transformation at a finite time [10]. Among various STA techniques,counterdiabatic driving (CD) is probably the one that attracted the highest attention [11–14]. CD not only ensures that the systemends up at the adiabatic final state but also ensures that it follows the adiabatic eigenstates at all times, by introducing an externalcontrol Hamiltonian. Nevertheless, this property makes CD the STA method that is energetically most costly [15].One particular sub-field in quantum thermodynamics that quantum control proves to be useful is quantum heat engines, whichare generalizations of the corresponding classical engine cycles to cases that have quantum systems as their working medi-ums [16–21]. In addition to theoretical proposals of possible implementations [22–24], there are actual experiments demonstrat-ing operational quantum heat engines [25–29]. However, presence of adiabatic processes in the engine cycle forces them to beperformed slowly, resulting in a vanishingly small power output. STA techniques have shown to be quite useful in speeding upthese adiabatic processes [29–39], yet they come at the expense of a certain energetic cost (see also [9, 21] for an alternativeview). In this work, we consider a quantum Otto engine that has two coupled spins as a working medium. Following its char-acterization in the adiabatic limit, we investigate its finite time behavior with and without utilizing a specific STA scheme, andcompare their performances in di ff erent parameter regimes. We observe that, for a fixed adiabatic e ffi ciency, it is possible to finda set of parameters for which the engine cycle without any external control perform very close to the one with STA. We thenargue that such increase in the performance is due to the reduced irreversibility due to finite time driving, as quantified by thee ffi ciency lags (based on non-equilibrium lags) [8, 26], in these parameter regimes.This paper is organized as follows. In Sec. II we introduce the concepts that are central to this work such as the CD scheme,details of the quantum Otto cycle and how to characterize its performance with and without the presence of a STA scheme.We present our model for the working substance of our Otto engine in Sec. III, which is two-spin anisotropic XY model intransverse magnetic field. We continue this section by identifying the parameter region in which the working substance operatesas an engine and compare its performance in adiabatic, non-adiabatic and STA cases. We conclude in Sec. IV. ∗ [email protected] a r X i v : . [ qu a n t - ph ] M a r FIG. 1. Schematic representation of an ideal quantum Otto cycle in entropy (S) vs. volume (V) plane. Compression and expansion strokes arerealized by changing the control parameter in the Hamiltonian of the working medium λ . When performed adiabatically, the entropy remainsconstant during these processes, as depicted by constant populations of the energy levels. The working medium is put in contact with a hotand cold bath during heating and cooling strokes, respectively, resulting a change in the populations, and hence in the entropy. Further detailsof each stroke are explained in the text. II. PRELIMINARIESA. Counterdiabatic driving
Assume that we have a system that is described by a time-dependent Hamiltonian H ( t ). In order for this system to followthe adiabatic eigenstates of its bare Hamiltonian, any change in H ( t ) must be made very slowly, in a scale set by the energygap of the Hamiltonian. Any fast driving will induce transitions between its energy levels, resulting in the deviation from theadiabatic path. Utilizing the CD scheme [11, 14], it is possible to mimic the adiabatic evolution at a finite time by introducingan additional Hamiltonian, H CD ( t ), such that the system evolving with H ( t ) = H ( t ) + H CD ( t ) follows the adiabatic eigenstates atall times. Exact form of this Hamiltonian is given as [14] H CD ( t ) = i (cid:126) (cid:88) n ( ∂ t | n ( t ) (cid:105)(cid:104) n ( t ) | − (cid:104) n ( t ) | ∂ t n ( t ) (cid:105)| n ( t ) (cid:105)(cid:104) n ( t ) | ) , (1)where | n ( t ) (cid:105) is the n th eigenstate of the bare Hamiltonian H ( t ). The necessity to diagonalize H ( t ) to determine the H CD ( t ) isa demanding task for systems with large Hilbert space. Therefore, alternative approaches such as the ones that do not requirethe knowledge of full spectrum [40, 41] or approximate driving schemes [42] have been developed and utilized in many-bodysystems [43]. B. Quantum Otto Cycle
The quantum Otto cycle is a four-stroke process, which is simply generalization of the corresponding classical cycle toquantum systems [16–20]. Two out of the four-strokes are adiabatic compression and expansion branches and performed byvarying a control parameter, λ t , in the Hamiltonian of the working substance, H ( λ t ). Remaining two are isochoric heating andcooling branches which involves thermalization of the working substance to the corresponding bath temperature. The cycle isschematically depicted in Fig. 1, and below we briefly present the details of each stroke of the cycle:1. Compression:
The working substance is in a thermal state at the temperature of the cold bath ρ β = exp( − β H ( λ )) / Z ( β , λ ),where Z ( β , λ ) is the partition function with β = / k B T , and completely isolated from any heat bath. The time-dependent parameter in its Hamiltonian is increased from λ to λ in a time interval of τ , resulting an increase in theinternal energy of the system. During this stroke the time evolution of the working substance is unitary and ideallyassumed to be slow enough so that there are no unwanted transitions between the energy levels of the system so thattheir populations, therefore the entropy, remain constant, in accordance with the quantum adiabatic theorem. Suchan evolution guarantees that all the internal energy change of the working substance is due to the work done on it, (cid:104) W (cid:105) = tr[ H ( λ ) ρ com − H ( λ ) ρ β ], where ρ com denotes the state of the system at the end of the compression stroke.2. Heating:
The working substance is brought in contact with a hot bath at temperature β = / k B T during a time in-terval τ that is su ffi cient for the system to thermalize to the bath temperature. No work is done in this branch andthe change in the internal energy of the working substance is only due to the heat absorbed throughout the process, (cid:104) Q (cid:105) = tr[ H ( λ ) ρ β − H ( λ ) ρ com ], where ρ β = exp( − β H ( λ )) / Z ( β , λ ) with Z ( β , λ ) being the partition function.3. Expansion:
The working substance is again isolated from any bath and and the time-dependent parameter in its Hamil-tonian is now decreased from λ to λ in a time interval of τ , resulting an decrease in the internal energy of the system.Similar to the compression stroke, the evolution during this stroke is unitary and again should be performed slow enoughto make sure the process remains adiabatic and entropy remains constant. The decrease in the working substance energy isextracted as work, (cid:104) W (cid:105) = tr[ H ( λ ) ρ exp − H ( λ ) ρ β ], where ρ exp is the state of the system at the end of the expansion stroke.4. Cooling:
In this final branch of the cycle, the working substance is brought in contact with a cold bath at temperature β = / k B T during a time interval τ that is su ffi cient for the system to thermalize to the bath temperature. No workis done in this branch and the change in the internal energy of the working substance is only due to the released heatthroughout the process, (cid:104) Q (cid:105) = tr[ H ( λ ) ρ β − H ( λ ) ρ exp ].The unitary evolution in the compression and expansion strokes are governed by the von Neumann equation ˙ ρ ( t ) = − i (cid:126) [ H ( λ t ) , ρ ( t )].Note that negative (positive) values of work or heat correspond to the case of these quantities being extracted from (absorbedby) the system. Therefore, in the engine cycle described above we have (cid:104) W (cid:105) , (cid:104) Q (cid:105) > (cid:104) W (cid:105) , (cid:104) Q (cid:105) <
0, such that the totalwork in the adiabatic case (cid:104) W A (cid:105) = (cid:104) W (cid:105) + (cid:104) W (cid:105) < ffi ciency, canonly be obtained if the compression and expansion strokes are performed adiabatically, as described. However, this condition canonly be met if λ t is varied extremely slowly, which in turn results in a vanishing power output from the engine due to very longcycle times. One central aim of this manuscript is to present a way to overcome this di ffi culty by introducing a STA scheme, sothat one can mimic the adiabatic dynamics of the working substance at a finite time, and thus yielding finite power. Furthermore,we will also consider finite-time driving without any control applied on the system, which will result in non-adiabatic excitationsbetween energy levels, leading to a irreversible loss in the work output of the working substance. C. Performance of the engine
Here, we would like to present the figures of merit of the engine and how we account for the energetic cost of applying theSTA together with how we include them into these figures of merit. Considering a true adiabatic cycle, e ffi ciency and power ofan Otto engine are given by the usual expressions η A = − (cid:104) W (cid:105) + (cid:104) W (cid:105)(cid:104) Q (cid:105) , P A = − (cid:104) W (cid:105) + (cid:104) W (cid:105) τ cycle . (2)On the other hand, if an STA scheme is employed to fasten the adiabatic strokes of the Otto cycle, due to the energetic cost ofthe external control, the e ffi ciency and power of the engine are modified as follows [33–35, 37] η STA = − (cid:104) W STA1 (cid:105) + (cid:104) W STA3 (cid:105)(cid:104) Q (cid:105) + V CD1 + V CD3 (3)and P STA = (cid:104) W STA1 (cid:105) + (cid:104) W STA3 (cid:105) − V CD1 − V CD3 τ cycle , (4)where τ cycle is the total cycle time of the engine and we characterize the cost as [36] V CD i = (cid:90) τ (cid:104) ˙ H CD ( t ) (cid:105) i dt (5)with i = ,
3, which is the sum of the average of the time derivative of the CD Hamiltonian over the driving time and theexpectation value is calculated using the state of the system driven by the bare Hamiltonian of the system. The reasoning andthe details of the derivation of Eq. 5 can be found in [36] (especially Appendix A of the mentioned reference). We wouldlike to note that the debate on quantifying the costs of STA schemes is still an ongoing one and the above definition is notunique [33–35, 44–47] (see [10] for a comprehensive review).Since the STA scheme enables us to mimic the adiabatic evolution, the total work output in these equations is equal to thatof the adiabatic cycle (cid:104) W A (cid:105) = (cid:104) W STA1 (cid:105) + (cid:104) W STA3 (cid:105) . Therefore, in the absence of the CD Hamiltonian and the adiabatic evolutionof the system, η STA reduces to η A . As it is in general done in the literature [35], we assume that the thermalization times areshorter than the times spent in the adiabatic strokes, and thus the total cycle time τ cycle = τ + τ = τ for equal compression andexpansion stroke times. III. TWO-SPIN ENGINE
There are a number of e ff orts in the literature that considers coupled spin systems as working mediums of a quantum Ottoengine or refrigerator [48–54]. However, there are only a few works considering a STA scheme in working substances with suchcomposite spin systems [36, 43, 55]. In what follows, we will first identify the parameter regions for which our system operatesas a quantum Otto engine and then apply a STA protocol using the parameter set for which we have the maximum work output.Our working medium is composed of two-spin-1 / H ( t ) = [1 + γ ( t )] σ ax σ bx + [1 − γ ( t )] σ ay σ by + h ( t )( σ az + σ bz ) , (6)where γ ( t ) ∈ [0 ,
1] is the anisotropy parameter, h ( t ) ∈ [0 ,
1] is the external magnetic field, { σ x , σ y , σ z } are the usual Pauli matricesand superscripts a and b are the labels for two spins. In what follows, we will suppress the explicit time-dependence of γ and h for the sake of the simplicity of the notation. The energy spectrum of Eq. 6 is given as (cid:110) − , − (cid:112) h + γ , (cid:112) h + γ , (cid:111) . Theinternal energy change in the adiabatic branches of the cycle is only due to the change of the two energy levels in the middle ofthe spectrum. It is possible to analytically calculate the work output and the e ffi ciency of the two-spin engine in the adiabaticregime as follows [36] (cid:104) W A (cid:105) = − ( (cid:104) W (cid:105) + (cid:104) W (cid:105) ) = C ( λ − λ ) (cid:104) cosh (cid:16) λ T (cid:17) + cosh (cid:16) T (cid:17)(cid:105) (cid:104) cosh (cid:16) λ T (cid:17) + cosh (cid:16) T (cid:17)(cid:105) , (7) η A = C ( λ − λ ) C λ + sinh (cid:16) T (cid:17) cosh (cid:16) λ T (cid:17) − sinh (cid:16) T (cid:17) cosh (cid:16) λ T (cid:17) + sinh (cid:16) T − T (cid:17) , (8)where C = (cid:110) sinh (cid:16) λ T (cid:17) (cid:104) cosh (cid:16) λ T (cid:17) + cosh (cid:16) T (cid:17)(cid:105) − sinh (cid:16) λ T (cid:17) (cid:104) cosh (cid:16) λ T (cid:17) + cosh (cid:16) T (cid:17)(cid:105)(cid:111) , λ x = (cid:112) h x + γ x with x = , / or the external field in steps ( i ) and ( iii ) of the cycle described in Sec. II B, T is the temperature of the cold bath and T is the temperature of the hot bath. It is important to note that both the extracted workand e ffi ciency is not individually dependent on h of γ , but a combination of them (cid:112) h + γ . In Fig. 2, we plot these quantitiesas a function of λ and λ and observe that it is only possible to get a working quantum Otto engine for a specific parameterregime. Naturally, one has to have λ < λ so that we increase the internal energy in the compression stroke and, together withthe energy absorbed during the heating, extract it in the expansion stroke. Interestingly however, there is also a lower bound to λ . Below the line λ ≈ . λ , marked by the lower thick blue line in Fig. 2, the working substance again ceases to operate asan Otto engine, i.e. do not generate a net work output. Within the region that one has an operating engine, the maximum workoutput, and the corresponding e ffi ciency η = . λ = . λ =
1, which is marked with a “ (cid:63) ” in Fig.s 2 (a) and (b) . A. Finite-time operation with and without CD
We will now attempt to accelerate adiabatic cycle with maximum work output and e ffi ciency utilizing a STA scheme, specif-ically through a CD, and see if and how much we can improve the power output of the engine. For comparison, we will also (a) (b) FIG. 2. The net work output (a) and the e ffi ciency (b) of the two-spin engine with anisotropic XY interaction in transverse magnetic fieldoperating between the bath temperatures T = T =
10. Solid blue lines denote the boundaries of the operating regimes of the engine,namely λ = λ and λ = . λ . The “ (cid:63) ” signs in both figures denote the maximum work output and the e ffi ciency which is obtained at λ = . λ = analyze the performance of the engine without any control, i.e. its true finite-time, non-adiabatic behavior. Our aim in makingthis comparison is to see, despite the STA costs, is it meaningful, and if so, how much advantageous it is to apply a CD schemeto mimic adiabatic dynamics at finite-time considering the performance of the non-adiabatic engine.Our analysis will be focused around the system parameters that generate the maximum e ffi ciency, i.e. we will fix λ = . λ =
1. We mainly have three di ff erent options in driving λ between these values: ( i ) changing h while keeping γ constant, ( ii )changing γ while keeping h constant, ( iii ) changing both h and γ . We would like to note that the results obtained in cases ( i ) and( ii ) have no significant qualitative di ff erence from each other, therefore we continue with case ( i ) considering that varying theexternal field is simpler than controlling the interactions between the qubits. In addition, our results on case ( i ) below will provethat with the appropriate choice of system parameters it is possible operate near the highest e ffi ciency even without any STA. Asa result, we leave case ( iii ) aside since it presents an unnecessarily complicated scenario that requires simultaneous driving inboth interactions and external field.The CD term that generates the STA for the bare system Hamiltonian given in Eq. 6 has the following general form [56] (seealso [57]) H CD ( t ) = h ˙ γ − ˙ h γ h + γ ) ( σ ax σ by + σ ay σ bx ) . (9)Natural implication of case ( i ) is ˙ γ =
0, which is important in identifying the form of Eq. 9. We also require the CD drivingterm to vanish at the beginning and the end of the driving, i.e. H CD ( t = , τ ) = ff erence between them for our purposes in this work, and therefore continue with the simple choice below which ensurescontinuous first time derivatives at the boundaries h = h − h − h ) t τ (cid:32) − t τ (cid:33) , (10)where h and h are the initial and final external field strengths in the compression stroke.Since λ = (cid:112) h + γ and we are operating between λ = . λ = ffi ciency, the external fieldis varied between the following interval h = (cid:112) . − γ and h = (cid:112) − γ . Note also that the value of the parameter λ = . γ ≥ . h values, but otherwise we are free to chose γ in the closed interval[0 , .
6] while keeping the e ffi ciency same. Such freedom allows us to investigate di ff erent working medium Hamiltonians. (a) (b) η A η STA η NA τ η γ = P no cost P STA P NA τ γ = (c) (d) η A η STA η NA τ η γ = P no cost P STA P NA τ γ = FIG. 3. E ffi ciency ( a ) and power ( b ) of the two-spin engine with anisotropic XY interaction in transverse magnetic field operating between thebath temperatures T = T =
10 with γ = . h = h = .
8. ( c ) and ( d ) as for the previous panels but with γ = . h = .
52 to h = .
95. The inset in panel ( d ) displays a zoom into the curves for a better display of the di ff erences between the presented cases. Dashedhorizontal lines in ( a ) and ( c ) marks the adiabatic e ffi ciency. Even though we have fixed the adiabatic e ffi ciency, the short and intermediate time performance of the engine does dependon the choice of γ and the interval within which the external field is varied through unitary performing the adiabatic branches.In addition to the change in the entropy of the working medium in the thermalization strokes, it is known that finite-time drivingof a closed quantum system in the adiabatic branches results in an irreversibility in the system that can not be traced back toheat exchange and lead to the introduction of irreversible work, W irr [60–64] (especially see [8] for a comprehensive review). Inthe context of quantum heat engines any deviation from adiabaticity in the compression and expansion strokes results in the lossof useful work and thus called inner friction [62–66]. In order to quantify the overall e ff ect entropy production throughout thecycle, including those due to finite-time, non-equilibrium operation, on the e ffi ciency of our engine, we will adopt the so-callede ffi ciency lags [26], defined through the relation η = η Carnot − L , and the explicit form of the lag is given as L = D (cid:16) ρ exp ( t ) || ρ β (cid:17) + D (cid:16) ρ com ( t ) || ρ β (cid:17) β (cid:104) Q (cid:105) , (11)where D ( ρ || σ ) = tr[ ρ ln ρ − ρ ln σ ] is the relative entropy and η Carnot = − T / T . In fact, each term appearing in the numeratorof Eq. 11 are also called as non-equilibrium lags and actually equal to the β W irr in the compression and expansion processes,respectively. We would like to direct the interested reader to Ref. [8], especially Sec. III-F. A similar approach have been takenin quantifying the deviation from reversibility using e ffi ciency lags in an irreversible quantum Carnot cycle [67].We are now ready to compare the performances of the non-adiabatic and CD engines for two di ff erent values of γ , namely γ = . γ = .
3. While the former results in changing h = h = .
8, the latter implies variation from h = .
52 to h = .
95. Note that for γ = . γ = . (a) (b) γ = γ = τ ℒ γ = γ = τ FIG. 4. E ffi ciency lag ( a ) and total STA cost ( b ) of the two-spin engine with anisotropic XY interaction in transverse magnetic field operatingbetween the bath temperatures T = T =
10 with γ = . γ = . In Fig. 3 present our results on e ffi ciency and power delivered by the quantum Otto engine for γ = . a ) − ( b ) and for γ = . c ) − ( d ), respectively, as a function of the driving time. To begin with, we observe a clear di ff erence between thee ffi ciencies of the engines. The non-adiabatic case for γ = . b ), the non-adiabatic engine for γ = . ffi ciency fairly close to that of the adiabatic enginein driving times as short as τ = . ffi ciency as compared to the non-adiabaticengines since the additional term H CD ( t ) ensures that the working medium evolution is adiabatic regardless of the driving time,suppressing irreversible entropy production originating from unwanted transitions between the energy levels. The deviation of η S TA from the adiabatic e ffi ciency, η A , stems from the energetic cost of applying the CD term that we presented in Sec. II C.Note that the STA costs are higher in the case of γ = . γ = .
3, which is related to the better performance ofthe latter even when there is not external control is applied. We discuss the reason behind this increased performance in detailbelow, in relation to our results presented in Fig. 4. Finally, despite the considerable di ff erences in the e ffi ciencies, we do notobserve much di ff erence between the non-adiabatic and CD engines for both γ , but much higher power values obtained in thecase of γ = .
3. Note that the inset in Fig. 3-( d ) displays how tiny the di ff erence is between the CD engine and the non-adiabaticone for the mentioned case, and together they are also very close to the power output of a hypothetical engine that achieves anadiabatic work output at a finite time with no cost.We now would like to elaborate on the superior non-adiabatic performance of the γ = . γ = . γ = . a ), we calculate the e ffi ciency lag given in Eq. 11 as a function of the driving time and clearlyobserve that L is much smaller for γ = .
3, which is in accordance with the di ff erence in the performance between the two. Theminimum of L = η Carnot − η A , which is denoted by the faint horizontal line in the figure, originates from the distance betweenthe true adiabatic states at the end of expansion and compression strokes to the hot and cold baths, respectively. Therefore, anyincrease in L above this value is due to the irreversible entropy production, which results in irreversible work, caused by thefast driving of the system. Note that L remains very close to this minimum value for γ = . W irr generatedin the unitary branches due to deviations from adiabatic evolution is quite small. Similarly, the minimum driving time requiredfor γ = . ffi ciency can be understood from this plot. The Carnot e ffi ciency forthe cycle presented here is η Carnot = . L < . τ ≈ . a ) and Fig. 4-( a ). In fact, any point in Fig. 3-( a ) and ( b ) for the non-adiabatic engine can begenerated by using η Carnot and Fig. 4-( a ) as pointed out above Eq. 11.As for the STA costs presented in Fig. 4-( b ), we again see that it is significantly reduced for γ = . γ = . ffi ciency and power calculations in both cases. With the help of L , it is possible to better understandthis behavior. As mentioned many times before, the CD scheme aims to suppress any unwanted transitions as one changes λ t away from the adiabatic limit, i.e. suppress irreversible entropy production along the unitary strokes. The higher this irreversibleentropy production the higher would the costs of applying the CD become. From Fig. 4-( a ), we know that working medium isdriven farther away from the adiabatic track in case of fast changes in h for γ = . γ = .
3, and thus, we have η A η NA at τ = γ η FIG. 5. E ffi ciency as a function of the anisotropy parameter of the two-spin engine with anisotropic XY interaction in transverse magneticfield operating between the bath temperatures T = T =
10 at driving time τ = .
5. Dashed horizontal line marks the adiabatic e ffi ciency. higher STA cost. The main di ff erence between these two cases is the range within which we change the external field whichgets smaller as γ is reduced. Therefore, this lead us to conclude that the enhanced performance in the latter case stems fromsuch restricted variation in h , which is in accordance with previous works [36]. Note again that we are able to achieve the sameadiabatic e ffi ciency with a smaller change in h for small γ is by exploiting the dependence of the energy spectrum of the workingmedium to these two parameters.Finally, we would like to take a closer look on the dependence of the e ffi ciency in case of the non-adiabatic engine on theanisotropy parameter γ . To that end, we fix the driving time to be τ = . γ . Our result is presented in Fig. 5 and we observethat as γ is increased, e ffi ciency at the aforementioned driving time is quickly reduced due to irreversible entropy productioncaused by the fast driving. An interesting feature visible in this plot is the fact that the non-adiabatic engine e ffi ciency convergesto the adiabatic e ffi ciency as γ →
0, which is the isotropic limit of the model. In fact, right in this limit the CD Hamiltoniangiven in Eq. 9 goes to zero, and the total driving Hamiltonian becomes equal to the bare Hamiltonian, H ( t ) = H ( t ). Therefore,time evolution of the system governed by the von Neumann equation actually follows the adiabatic state and one can drive thesystem at an arbitrary speed without any additional control. This is called a fixed-point condition in [56] and shown to be trivialin case of a single spin but can have non-trivial consequences in more complicated systems such as the one considered in thiswork. In [36], the authors introduced the anisotropy parameter to avoid this fixed-point, in order to make a solid analysis of theSTA engine of two-spins. However, here we show that it is in fact possible to exploit this fixed-point condition in a quantumheat engine cycle to avoid irreversibility without going through the complications of the STA scheme. IV. CONCLUSION AND OUTLOOK
We consider a quantum Otto cycle with a working medium described by the two spin-1 / XY model in a transversemagnetic field. Following the full characterization of the parameter regime for which the coupled spin system operates as anengine in the adiabatic limit and identifying the maximum e ffi ciency, we focus on the finite-time behavior of the engine. Tomimic adiabatic dynamics at a finite-time we apply a STA scheme through CD taking the energetic cost of it fully into accountin the evaluation of the performance of the engine. In addition, we analyze the actual finite-time dynamics of the engine withoututilizing any external control protocol and compare it with the STA performance. We observe that when we fix the e ffi ciency ofthe engine to be maximal, as the anisotropy parameter is decreased the irreversibility of the non-adiabatic engine due to finite-time driving of the external field, as measured by the e ffi ciency lags, becomes very small. This results in a significant increase inboth e ffi ciency and power of the non-adiabatic engine as compared to higher γ , which makes its performance closer to that of theSTA engine. Our results suggest that for certain parameters of the Hamiltonian describing the working medium, implementing aSTA scheme is not necessary and the non-adiabatic engine can operate with a similar performance due to reduced irreversibility.Our results may have the potential to contribute to the quest of designing energy e ffi cient quantum thermal machines. Eventhough STA methods are to be perfectly suitable to fasten the adiabatic strokes in a quantum heat engine cycle, they are ingeneral resource intensive (especially CD) both on the control side and energetically [15]. An alternative approach was putforward in [68] in which the authors consider a two-level quantum Otto engine and refrigerator without any external control,and focus on identifying the e ffi ciency and power of the machine by optimizing the ecological function that takes the trade-o ff between increased power output and entropy production into account. Building on our results that demonstrate the presence ofreduced irreversibility in certain parameter regimes for two-spin quantum Otto engines, it is possible to make a more systematicanalysis based on the approach of [68], which we leave as a future work. Another interesting direction could be to utilizethe machine learning methods in improving the performance of quantum thermal machines. Recently in [69], a reinforcementlearning technique is introduced to reduce the entropy production in a closed quantum system due to a finite-time driving. Suchan approach is perfectly suitable to be utilized in the work strokes of a quantum heat engine cycle. Specifically in the model thatwe have considered, one can take advantage of this method and systematically study the whole parameter landscape, learningregions of reduced entropy production. ACKNOWLEDGMENT
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