Quantum Signatures in the Quantum Carnot Cycle

The Carnot cycle combines reversible isothermal and adiabatic strokes to obtain optimal efficiency, at the expense of a vanishing power output. Here, we construct quantum Carnot-analog cycles, operating irreversibly at non-vanishing power. Swift thermalization is obtained utilizing shortcut to equilibrium protocols and the isolated strokes employ frictionless shortcut to adiabaticity protocols. We solve the dynamics for a working medium composed of a particle in a driven Harmonic trap. A complete description of the state is obtained, incorporating both changes in energy and coherence. In the limit of finite cycle-time, coherence disappears and the efficiency converges to the ideal Carnot efficiency. Thus, demonstrating the trade-off between power and efficiency. At short cycle-times, generation of coherence is necessary to achieve power. To evaluate the importance of quantum coherence, we compare three types of cycles, Carnot-shortcut, Endo-shortcut and Endo-global. In the first two, the coherence is limited to the interior of the strokes, while for the last cycle the coherence never vanishes. This allows the Endo-global engine to operate at shorter cycle-times relative to the shortcut cycles. Introducing pure-dephasing to the Endo-global engine terminates the quantum coherence, and with it, the power output. This phenomena can be identified by evaluating the cycle performance, therefore indicating a quantum signature.


I. INTRODUCTION
In 1824 Sadi Carnot envisioned a reversible reciprocating heat engine, composed of two adiabatic and two isothermal strokes. He argued that such a reversible engine is universal, and produces optimal work, depending only on the hot and cold bath temperatures, T h and T c [1]. The Carnot cycle serves as a template for all reversible engines. A heat engine transforms heat extracted from a hot bath, Q h , to Work W . To comply with the second law of thermodynamics, work is necessarily accompanied by heat flow to the cold bath Q c = |W | − Q h . The efficiency of the energy transformation is defined by η = −W/Q h , which is bounded by the Carnot efficiency η < η C = 1 − T c /T h , for any heat cycle. This universal bound depends only on bath temperatures, irrespective of the specific working medium. It translates a fundamental limitation, the second law of thermodynamics, to a practical operational limit applying to all engines. Optimal efficiency is achieved only for a reversible operation.
Practically, any finite power engine operates under irreversible conditions, which leads to a trade-off between efficiency and power. This relation has been extensively studied in the framework of finite-time thermodynamics [2,3]. A prominent emerging result states that under endoreversible conditions the efficiency at maximum power is given by the Curzon-Ahlborn-Novikov efficiency η CA = 1 − T c /T h [4,5]. Which has been generalized for weak dissipation [6,7].
In the quantum regime energy transfer is constrained * roie.dann@mail.huji.ac.il † ronnie@fh.huji.ac.il as well by the second law of thermodynamics [8][9][10][11]. Implying that quantum heat engines are also bounded by the Carnot efficiency. Reversibility and optimal efficiency is obtained in the quantum adiabatic limit, requiring an infinite cycle-time τ cycle .
In this paper we explore two quantum approaches to obtain non-vanishing power for a 'Carnot-analog' engine. The first type is termed Carnot-shortcut, defined by the cycle parameters of the reversible Carnot cycle (bath temperatures and external parameters). At the switching corners between strokes, the working medium is in equilibrium with the baths at the four corners of the cycle ( Fig. 1 Panel (a)). Finite power is achieved by shortcut protocols to each stroke: Shortcuts To Adiabaticity (STA) [12] on the unitaries and Shortcut To Equilibrium (STE) [13] for the thermalization strokes. The cycle is performed by external driving of the system and coupling/decoupling the Working Medium (WM) from the hot and cold baths. Typically, the WM Hamiltonian does not commute with itself at different times Ĥ S (t) ,Ĥ S (t ) = 0, leading to generation of coherence and an accompanied cost in work [14][15][16]. An alternative approach to obtain finite power is a Quantum Endoreversible cycle, for which the WM is in a non-equilibrium state throughout the cycle. In the class of endoreversible cycles, we construct the Endoshortcut and the Endo-global cycles, which are characterized by Branch and Global coherence operations, respectively. Branch coherence is restricted to the interior of the stroke, while at the cycle's four corners the coherence vanishes and the WM state is of a Gibbs form, with an internal temperature T = T bath . In the global coherence operation the WM state exhibits coherence throughout the whole cycle. As a result, coherence generated in one arXiv:1906.06946v1 [quant-ph] 17 Jun 2019 stroke continues to play a role in the subsequent strokes.
Our aim in this study is to explore the performance and characteristics of the Quantum Carnot-analog engines, i.e., the trade-off between power and efficiency and the role of quantum coherence in the engine operation. For this purpose, we employed an Harmonic oscillator WM for which the cycle dynamics can be solved explicitly.
Previous studies of the Carnot cycle lacked coherence. These studies consider a Hamiltonian that commutes with itself at all times [17,18], or cycles performed in the adiabatic [19] or stochastic [6] limits. The relatively few studies on the quantum Carnot-analog cycle is in contrast to the popularity of the quantum Otto cycle. Analysis of the Otto cycle has been a major source of insight on quantum reciprocating engines [14,[20][21][22][23][24][25][26][27][28][29][30][31][32][33]. The vast difference in popularity between the quantum Otto and Carnot cycles arises from the difficulty to describe the open-system dynamics of non-adiabatically driven systems. Development of the Non-Adiabatic Master Equation (NAME) [34] and the Inertial Theorem [35] enable this study.
The analysis of the quantum Carnot-analog engine, demonstrates two roles of coherence in the energy representation. First, the control of coherence allows optimizing the efficiency for a finite cycle-time. Moreover, for very short cycle-times global coherence becomes crucial to obtain non-vanishing power. This phenomena is a quantum signature [36][37][38], which can be unraveled from thermodynamics observable.

II. DYNAMICS OF THE CARNOT-ANALOG CYCLE
We choose a particle of mass m confined by a varying Harmonic potential as the engine's working medium. The associated Hamiltonian readŝ whereQ andP are the position and momentum operators, and ω (t) is the oscillator frequency. The explicit time-dependence of ω (t) defines the cycle's protocol. The dynamics during the cycle strokes is associated with a time-dependent completely positive map [39,40], generated by where L (t)ρ (t) = −i Ĥ (t) ,ρ (t) for the adiabatic strokes, and L (t)ρ (t) = −i Ĥ (t) ,ρ (t) + L D (t)ρ (t) during the isothermal-type strokes, where L D (t) is the Lindblidian. For the Adiabats, the unitary maps U ch and U hc , are generated by Eq. (1). Isotherms dynamical maps, U h and U c , are generated by the NAME, Appendix A Eq. (A1), which are valid for weak system-bath coupling, and when the bath dynamics is fast relative to the system and driving [34]. We explore driving protocols that satisfy the Inertial Theorem [35]. This theorem provides an analytical solution, termed 'inertial solutions', for the (isolated) system dynamics in the limit of slow 'acceleration' of the driving. For the harmonic oscillator model, the inertial theorem is satisfied in the limit of a slow change in the inertial parameter, dµ/dt → 0, where µ =ω/ω 2 .
The cycle is constructed by combining four strokes: (i) An expansion stroke ( Fig. 1 Panel (a) corners 1 → 2), decreasing the oscillator frequency, while the system is coupled to a hot bath of a temperature T h . This stroke is termed open-expansion, referring to the fact that during this segment the working medium is an open quantum system, exchanging energy and entropy with the bath. (ii) Adiabatic expansion stroke (2 → 3) in which the system is isolated from the baths and the oscillator frequency is decreased. (iii) Open-compression (3 → 4), the particle is brought in contact with the cold bath of temperature T c and 'compressed' towards a higher frequency;(iv) Adiabatic compression (4 → 1), the final stroke restores the system to its initial state while keeping the particle isolated from the baths. Formally, the complete cycle propagator can be decomposed to stroke propagators: Each propagator U i is a completely positive map, associated with the cycle stroke. Work is defined in terms of the instantaneous power, tr Ḣρ , [41] and heat is obtained from the first law of thermodynamics E = W + Q.

III. CARNOT-SHORTCUT CYCLE
The Carnot-shortcut cycle is characterized by the four corners of the ideal Carnot cycle. These switching points between strokes are defined in terms of the bath temperatures T c , T h and oscillator frequencies ω 1 − ω 4 . Defining the compression ratio C = ω max /ω min = ω 1 /ω 3 , the cycle is completely determined by ω min = ω 3 , C and bath temperatures.
At the four corners the WM is in a thermal state: Z is the partition function (see Fig.  1 Panel (a)).
The adiabatic strokes conserve the working medium's entropy, therefore, the populations satisfy n 2 = 1/ exp ω2 k B T h − 1 = n 3 = 1/ exp ω3 k B Tc − 1 and n 1 = n 4 , where n i is the population on the i'th corner. This leads the condition From the cycle definition ω 1 > ω 2 and Eq. (5) we obtain a lower bound for the compression ratio In general, quantum Carnot-analog cycles execute finite-time strokes, obtaining positive power P = −W/τ cycle . Specifically, the Carnot-shortcut cycle utilizes Shortcut To Equillibration (STE) protocols [13] during the open-expansion and open-compression strokes (see Appendix A), and Shortcuts To Adiabaticity (STA) protocols for the adiabatic expansion and compression (see Appendix B). These finite-time protocols begin and end in a thermal state, while generating coherence at intermediate times.
Shortcut to equilibrium protocols are designed to manipulate an open quantum system between thermal states. STE is based on the inertial theorem [35] and the NAME [34]. In the present open-expansion process we modify the frequency from ω 1 to ω 2 while the system interacts with a bath at temperature T h (or in the opencompression from ω 3 to ω 4 at bath temperature T c ). This protocol balances coherence generation and dissipation to achieve a target diagonal state.
The protocol duration can be varied within the framework of the inertial approximation with negligible deviations from optimal fidelity [13]. Once the protocol duration is cut short, the STE requires a rapid change of the Hamiltonian, generating more intermediate coherence. This leads to larger dissipation, resulting in an increase of the entropy production and work cost.
In addition, Carnot-shortcut cycle protocols require a stationary oscillator frequency at the four corners with no coherence. Hence, starting from a stationary state, to comply with the inertial condition, the driving is slowly accelerated and then decelerated, leading to the target stationary thermal state.
Frictionless protocols are constructed employing the Lewis-Riesenfeld invariant [12,42,43]. These protocols vary the oscillator frequency non-adiabatically, generating coherence at intermediate times and storing energy in the WM. At the end of the stroke, the coherence is fully extracted, leading to a total zero work cost. Since all stored energy is retrieved when the protocol is completed, we consider these processes as an analog to catalysis. In principle, these protocols can be achieved almost instantaneously . Nevertheless, this implies a temporary storage of an infinite amount of energy in the WM [12,43,44]. To comply with practical physical considerations, we choose a constant stroke duration for the adiabats, which is consistent with the inertial theorem. Moreover, the cycle-time is dominated by the open-strokes, therefore the time allocated to the adiabats does not alter the performance qualitatively.
Correspondence between the Carnot-shortcut cycle and the ideal classical result is obtained in the quantum adiabatic limit (diverging stroke times). In this limit, the WM state remains on the energy shell along the whole cycle and the STE and STA strokes converge to reversible isothermals and adiabats. The optimal work extraction becomes W C = ∆ω 32 (n 2 + 1) + ∆ω 14 (n 1 + 1) where ∆ω ij = ω i − ω j . In the high temperature limit Eq. (7) simplifies to

A. Performance of the Carnot-shortcut cycle
Performance of the Carnot-shortcut cycle is analysed by varying the duration of the open strokes, while keeping the time allocation of the adiabatic strokes fixed. The cycle is characterised by two operational modes: engine and dissipator. These are associated with different driving speed regimes. For 'slow' to 'moderate' driving speeds ('long' or 'medium' cycle-times) the cycle operates as an engine (negative work output, using the convention that outgoing energy is negative). For decreasing cycletimes the work output of the open-expansion decreases (in absolute value) and the work required to perform the open-compression stroke increases. This leads to a reduced efficiency. Once the cycle-time is reduced below τ trans = 23.87 (2π/ω min ), where ω min = 5 a.u, the cycle operates as a dissipator, converting net positive work to heat, that is dissipated to the cold bath.
The transition between dissipator and engine operational modes is caused by the large energy dissipation, which accompanies fast driving. Fig. 1 displays a map of the cycle in the energy frequency plane. For slow driving, Panel (a), the open-strokes follow the isotherms, leading to a Carnot-analog engine that obtains a close to optimal efficiency η C = 0.375. When the cycle-time shortens, the energy deviates significantly from the isotherms, Panel (c), and the cycle operates as a dissipator. Decrease in efficiency is attributed to fast driving, that requires large generation of coherence. Consequently, dissipation is increased, eventually canceling the useful extracted work. This is related to a reduction in the area confined by the cycle, in the Ĥ , ω plane.
The cycle's operational mode is dictated by the open strokes. Decreasing stroke times costs work, asymptotically, this cost scales as 1/t [13]. In the asymptotic limit the cycle power can be estimated as where W C is the ideal work, Eq. (7), and F is the friction action (see Appendix C). The three terms of Eq. (9) are positive for an engine operation, inferring that a maximum in power at a finite cycle-time τ cycle . Furthermore, for sufficiently small τ cycle Eq. (9) implies that the power output is negative and the cycle becomes a dissipator. This leads to the approximate relation between the cycle-time at maximum power τ * and the transition time τ trans : τ * = 2τ trans . For increased cycle-times, the efficiency of the cycle improves, obtaining the Carnot bound asymptotically (see Fig. 2 Panel (a)). Optimal power is achieved for relative short cycle-times τ maxP = 43 (2π/ω min ), (see Fig. 2 Panel (b)). We have also analyzed the performance at elevated temperatures, maintaining a constant temperature ratio. Asymptotically, the power increased linearly, Eq. (8), and the efficiency at maximum power converged to the value η * maxP slightly larger than η CA . This implies that cycle does not operate in the weak dissipation limit [7].

IV. ENDOREVERSIBLE SHORTCUT AND GLOBAL CARNOT CYCLES
Endoreversible cycles are defined by four corners for which the WM is in a Gibbs state with temperatures T = T bath . As a result, finite heat flow between the engine and bath occurs and the WM is never in equilibrium with the heat bath.
The Endo-shortcut cycle is constructed in a similar fashion to the Carnot-shortcut cycle Sec. III. The four corners are maintained in a Gibbs stateρ (ω, T ) = Alternatively, the Endo-global cycle operates as a Carnot-analog cycle where the strokes are performed with a pre-defined constant adiabatic speed |µ| = |ω/ω 2 | = const. The strokes are defined by four frequencies ω g 1 − ω g 4 , and µ. These determine the stroke duration t f and protocols ω (t) = ω i / (1 − µω i t) for the strokes starting at ω i and ending at ω (t f ), Cf. Appendix D.
Propagators of the free dynamics, Eq. (1), can be obtained explicitly in terms of an operator basis in Liouville space Cf. Appendix D. These propagators con-struct the WM unitary transformations of the adiabats. Next, the same free dynamics solution is used to derive the NAME, employed to generate the map of the openexpansion and open-compression strokes [34]. Combining the four strokes forms the Global cycle. von Neumann Entropy Sv.n as a function of the oscillator frequency ω (t) for the Endo-global cycle (thick continuous lines) and Carnot-shortcut cycle (transparent long dashed), in the slow driving regime τ cycle = 250 (2π/ωmin). The Endoreversible cycle operates between the hot (shortdashed red) and cold (short-dashed blue) isotherm lines of temperatures T h = 8 and Tc = 5.
In following, we analyze an Endo-global Carnot cycle operating between baths of temperatures T c and T h , with frequencies ω g where T g c = 5.25 and T g h = 7.75. The frequencies are chosen so as to comply with an ideal Carnot cycle (or Carnot-shortcut cycle) operating between temperatures T g c and T g h , in the long cycle-time limit. In this limit, the state at the four corners becomes isoentropic with the states of the Carnotshortcut cycle. Thermodynamic analysis is carried out on the limit-cycle [45], defined by the cycle parameters above.

A. Cycle performance comparison
We compare the performance of the three cycles with an emphasis on the role of coherence. The cycle output is determined by the operational mode and the cycle parameters: cycle frequencies, bath temperatures and cycle-time, Cf. Table I. The Shortcut cycles (Carnot and endoreversible) are characterized by a Branchedcoherence operation, where coherence is created 'locally' during each stroke, with initial and final diagonal Gibbs states. Conversely, in the limit-cycle of the Endo-global engine, coherence is maintained throughout the cycle (globally). As a result, coherence generated in one stroke can be utilized along the subsequent strokes, see Appendix E.
The Carnot-shortcut cycle shows a superior performance at moderate and long cycle-times, Figs. 2 and 3. In this operational regime, the efficiency, power, maximum power, and efficiency at maximum power exceed both the Endo-global and Endo-shortcut cycles. However, for short cycle-times, both shortcut cycles become dissipators, producing negative power, while the Endoglobal cycle continues to operate as an engine. This comparison is unbiased, since the same bath temperatures are considered for the Carnot-shortcut and Endo-global cycles. We find that the optimal performance of the Carnot-shortcut engine is superior to the Endo-global engine. Quantitatively, the maximum power and optimal efficiency the Carnot-shortcut engine are greater Fig. 3. The role of coherence in the Endo-global cycle is illustrated by adding pure-dephasing to the adiabats, Appendix F, see Fig. 6. For long cycle-times the influence of dephasing is minor. Conversely, for short cycle-times the engine operation requires coherence and the dephasing nulls the power output.

V. DISCUSSION
A family of finite power quantum Carnot-analog engines is investigated to determine the role of coherence. In the construction of the models, we have overcome the difficulty of describing thermalization in the presence of non-adiabatic driving [34]. Including an explicit description of the working medium dynamics.
Three types of quantum Carnot-analog engines are compared: The Carnot-shortcut, Endo-shortcut and Endo-global cycles. The Carnot-shortcut cycle achieves finite power by employing frictionless protocols during the unitary strokes and accelerating the thermalization. This acceleration is accompanied by increased dissipation to the bath, which eventually dominates the performance. At this threshold, the cycle transforms from an engine to a dissipator. The performance characteristics of this model demonstrate the universal trade-off between power and efficiency [46][47][48][49].
Traditionally, the scheme to obtain finite power is to operate in the endoreversible regime. In this regime, the working medium remains in a non-equilibrium state throughout the whole cycle, allowing finite heat transport between the bath and engine [4,50,51]. We compare two endoreversible engines that differ by their coherence properties. The Endo-shortcut cycle utilizes coherence locally on each stroke to achieve finite stroke duration. The second endoreversible engine, the Endo-global cycle, is based on constant adiabatic speed protocols. This cycle exhibits 'global coherence', which is incorporated in the power generation, Appendix E. Global coherence enables engine operation at short cycle-times, where the Endo-Shortcut and Carnot-Shorcut become dissipators.
In the presence of pure-dephasing, the coherence vanishes, reducing power and efficiency. Eventually, strong dephasing transforms the cycle from an engine to a dissipator, a signature of a pure quantum operational mode, see Fig. 6.
Carnot and Otto engines have been the keystones in the study of quantum heat devices. It is illuminating to compare between the two quantum cycles [20]. To this end we assume the same working medium and bath temperatures. The Otto and Carnot cycles differ by the permissible compression ratio range. The Otto cycle is constrained by T h Tc ≥ C > 1, while the compression ratio of the Carnot cycle obeys C > T h Tc . Optimal work for the Otto cycle is achieved when C = T h Tc and vanishes in limits C = 1 and C = T h Tc . On the other hand, the power of the Carnot-analog has a local optimum, and the work output diverges with C, Eq. (8).
A major performance measure of engines is the efficiency. Otto and the Carnot cycles have different efficiency characteristics, in particular, their dependence on the compression ratio and bath temperatures. The Otto efficiency η O = 1 − C −1 converges to η C in the limit of zero work. Moreover, optimal efficiency increases conjointly with the work, achieving η O = η CA at optimum in the high temperature limit [46]. Conversely, for the Carnot-shortcut cycle, efficiency is constant η c , work monotonically increases with C. Unlike the power which obtains its optimum at a finite compression ratio. In the present study, the efficiency at maximum power is found to be greater than the Curzon-Ahlborn efficiency η maxP > η CA .
Carnot and Otto engines differ by their thermalization strokes and share their unitary adiabatic strokes. For the unitaries, shortcut to adiabaticity protocols have been applied to achieve frictionless operation [21,27,30].
There has been a controversy regarding the cost in work in executing these shortcuts [20,30,52]. We advocate the idea that the temporary energy storage in the working medium can be recovered without additional cost. The trade-off between energy and coherence supplies the recovery mechanism, employed to retrieve all the stored energy at the end of the stroke [20,53]. To account for the protocol cost one has to incorporate the dissipation in the controller per se. Here, we assume this cost is negligible [52].
A unique feature of this study is the incorporation of shortcuts in the thermalization strokes. These protocols require active driving, accompanied by direct dissipation of heat to the baths. Such a protocol can also be used to accelerate the themalization in the Otto cycle. The work cost and entropy production, associated with these shortcuts, are an intrinsic characteristic of the dynamics. This is a direct manifestation of the efficiency and power trade-off.
The quantum signature of these engines emerges when global coherence is maintained. In the limit of short-cycle times, significant coherence is generated and dominates the power output. Both the Endo-global, the sudden Otto cycle and the two-stroke NV engines, share a quantum mode of operation [20,37,54]. The STE protocol rapidly transfers the working medium between two Gibbs states with different entropies. Construction of the protocol relies on the Non-Adiabatic Master Equation (NAME) [34] and the Inertial Theorem [35]. Derived from first principles, the NAME describes the reduced dynamics of nonadiabatically driven open quantum system in the weak coupling Markovian regime. The NAME incorporates, as a limit, both the adiabatic and Floquet master equations.
To derive the NAME requires an explicit solution of the closed system dynamics. Utilizing the inertial theorem, we obtain the free propagatorÛ S (t) for slowly 'accelerated' external control. For this study, we consider a harmonic oscillator working medium, where the control is achieved by varying the potential. In this case, the inertial condition is associated with a slow change in the adiabatic parameter µ =ω/ω 2 . The explicit solution of free dynamics propagator allows to transform the systembath coupling to the interaction representation. Using the Born-Markov approximation we derive the generator of the open system dynamics. For this case, we consider an Ohmic Boson bath with Markovian properties. The derivation leads to the reduced system dynamics in the interaction representation Here, the density operator in the interaction picture reads where N is the occupation number of the Bose-Einstein distribution, κ = 4 − µ 2 , and α is a modified frequency, determined by the non-adiabatic driving protocol [34]. In terms of the oscillator frequency, the modified frequency is given by The Lindblad jump operators becomeb ≡b ≡b (0) = mω(0) κ κ+iµ 2 Q (0) + µ+iκ 2mω(0)P (0) . In the adiabatic driving limit, µ → 0, the Lindblad jump operators converge to the adiabatic creation and annihilation operatorsb † ,b →â † ,â. Therefore, in this limit, Eq. (A1) reduces to the adiabatic master equation.
The completely positive map generated by Eq. (A1) preserves the Gaussian form. This property is termed canonical invariance [46,55,56]. Formally, the Gaussian state can be also expressed in a product form where Z = tr e γ(t)b 2 e β(t)b †b e γ * (t)b †2 , the operatorŝ b †b ,b 2 ,b †2 vary with µ. Parameters γ and β are timedependent functions.
For an initial thermal state, the construction is simplified and the reduced system remains in the following form throughout the entire evolutioñ with initial conditions β (0) = − ω(0) k B T and µ (0) = 0. Substituting Eq. (A5) into Eq. (A1) leads to a nonlinear differential equation for β (t) Equation (A6) constitutes the basis for the STE control scheme. We define y (t) ≡ e β(t) and guess a polynomial solution for y. This solution should transfer an initial thermal state of frequency ω (0) to a final thermal state with a frequency ω (t f ), which leads to the boundary conditions and µ (0) = µ (t f ) = 0. Furthermore, the condition on µ implies that the state and protocol are stationary at initial and final times, leading toβ (0) =β (t f ) =β (0) =β (t f ).

STE for non-thermal initial and final states
The Endo-shortcut cycle, Sec. IV, includes opencompression and open-expansion strokes between nonequilibrium states. During these strokes, the working medium state is of the Gibbs form where the internal temperatures T i and T f differs from the bath temperatures, i.e., The control protocol ω (t) for the open-strokes is obtained by a similar reverse-engineering method as in the case of an initial and final equilibrium states (T i , T f = T ). However, since the system-bath interaction leads to nonvanishing decay rates, the initial and final states are nonstationary, which impliesβ (0) ,β (t f ) = 0. As in the previous construction, we require a continuous change in the control protocol, associated with the restrictionω (0) = ω (t f ) = 0. Substituting the conditionω (0) = 0 into the decay rates of Eq. (A6) we obtain the initial value foṙ β (this is in accordance with the dynamics of the timeindependent master equation [57]). This leads to the boundary conditions for y: andÿ (0) =ÿ (t f ) = 0. Following a similar derivation as the previous section, we introduce a fifth order polynomial that satisfies the boundary conditions to obtain ω (t).
Appendix B: Adiabatic strokes -Shortcut to adiabaticity protocols utilizing the Lewis Riesenfeld invariant The (thermodynamic) adiabatic strokes are achieved utilizing shortcut to adiabaticity (STA) protocols. These transform a diagonal state in the energy basis, of a frequency ω i , to a state with the same population, with a final frequency ω f . These protocols are engineered utilizing the Lewis-Riesenfeld invariant [42]. We follow a similar procedure as presented in Refs. [12,42] to construct the STA protocols.
We introduce an Hermitian invariantÎ for the harmonic oscillator algebra, SU (1,1). Generally, such an invariant can be expressed as sum of the algebra operatorŝ and must satisfy the condition To obtain transition-less driving we desire a protocol for which the invariant commutes with the Hamiltonian at initial and final times: Since the operators share a common eigenstate basis at these times and the eigenvalues ofÎ are stationary, the engineered protocol induces transition between two diagonal states in the energy representation. Substituting Eq. (B1) into Eq. (B2) leads to three coupled differential equations for the time-dependent coefficientsα By defining β ≡ σ 2 and conducting some algebraic manipulations (see Ref [42]) Eqs. (B4) can be represented in terms of a single differential equation Solving for m 2 ω 2 σ + m 2σ and substituting the solution into Eq. (B5) gives where, c is an arbitrary real integration constant. Next, we define σ = c 1/4 ρ, and substitute the definition into Eq. (B6), to obtain with the subsidiary condition (Eq. (B6)) This equation introduces constraints on the protocol ω (t), which comply with the boundary conditions of ρ.
The strategy to engineer a transition-less control protocol is to choose ρ (t) such that relations Eq. (B3) are satisfied. This leads to the following boundary conditionṡ Equation (B8) is now solved by introducing a polynomial solution. A fifth order polynomial in t is sufficient to satisfy the conditions of Eq. (B9). We substitute the polynomial solution into Eq. (B8) and solve for ω (t).
To obtain the expectation values ofĤ,L andĈ, we introduce the eigenstates ofÎ (t). These obey the eigenvalue equationÎ with time-dependent eigenstates and time-independent eigenvalues λ. Next, we define the creation operatorâ = 2 −1 Q /ρ − i mρQ − ρP and matching annihilation operator. These operators satisfy: a |λ = √ λ |λ − 1 and a † |λ = √ 1 + λ |λ + 1 . By expressingĤ,L andĈ in terms of a and a † one obtains The final step is to sum over the contribution of each state |λ . For an initial Gibbs state the energy reads The derivation for L and Ĉ follows a similar procedure.

Appendix C: Friction action
Shortcut to equilibrium process rely on non-adiabatic driving of an open quantum system. The driving incorporates the dissipative and unitary dynamics to lead the system towards a target thermal state. These protocols accelerate the system thermalization rate by generating coherence at intermediate times (transforming energy to coherence), and terminating them at the end of the protocol. The protocol duration τ stroke of the STE can be varied within the inertial approximation, with a negligible influence on the final fidelity. For increasing protocol duration, the generation of coherence reduces, converging to the quantum-adiabatic result in the limit τ stroke → ∞. As a result, as the protocol duration increases less coherence dissipates to the bath and the work output improves (reduced in the convention of W < 0 for work extraction). Asymptotically, the work output scales as τ −1 stroke [13], therefore, one can introduce the 'friction action' F and express the total cycle work output W as a function of the ideal work W C and F In the quantum adiabatic limit the process is optimal and the work output converges to W C . Substituting Eq. (C1) into the expression for the power output P = −W/τ cycle , leads to the asymptotic relation (C2) Equation (C2) is consistent with the general argument that the frictional forces should be independent of the sign ofω. Hence, to lowest order the power against friction is proportional to µ 2 . Since µ ∝ 1/τ cycle we expect that the assymptotic power against friction scales as Eq. (C2) [15].

Appendix D: Endo-global cycle construction
The Endo-global cycle is constructed by combining two open strokes and two adiabats, that operate at a constant adiabatic speed. During the open strokes, the working medium dynamics is generated by the non-adiabatic master equation Eq. (A1). This equation incorporates both the dissipative and unitary effects. It is exact for Markovian dynamics in the weak coupling regime and when the external driving is slow relative to the bath dynamics (see Ref. [34]). The condition of constant adiabatic speed, |µ| = const, leads to protocols of the following form ω (t) = ω i / (1 − ω i µt), with initial frequency ω i .
The unitary dynamics are given in terms of an operator basis in Liouville space, v (t) = {Ĥ (t),L (t) ,Ĉ (t) ,Î} T . Here,Ĥ (t) is the Hamiltonian, Eq. (1), is the Lagrangian, and is the position-momentum correlation operator andÎ is the identity operator. These operators completely determine the dynamics of the harmonic oscillator Gaussian state (A4), and give a clear physical interpretation of energy and coherence, The solution for v (t) is given by [20]: with (D3) where κ = 4 − µ 2 and c = cos (κθ (t)), s = sin (κθ (t)). The free propagation, governed by Eq. (D3), mixes coherence and populations due to non-adiabatic driving [20].
In order to judge the performance for varying cycletimes, we construct different cycles with the same cycle frequencies, and vary |µ|. The adiabatic parameter determines the stroke duration according to where ω f is the final frequency. The cycles are then propagated until convergence to the limit-cycle [45,58], where the performance is evaluated.

Appendix E: Branch and Global coherence operation
Coherence is associated with non-diagonal elements in the energy representation. This means that a state possessing coherence is non-stationary under the free Hamiltonian dynamics. There have been many proposals to quantify coherence [59]. In this context, two measures have been employed to analyze quantum heat engines: (i) Divergence, which becomes the difference between the energy and the von Neumann entropies [33,54,60,61], and (ii) The algebraic definition of coherence, utilized here, Eq. (D1). Fig. 7 compares the coherence during the Carnotshortcut and Endo-global cycles. The Carnot-shortcut is associated with branch coherence operation, for which the coherence vanishes between adjacent strokes. In comparison, the coherence of the Endo-global cycle oscillates, but never reaches zero. Thus, justifying the name 'global coherence operation'.
Appendix F: Addition of pure dephasing Pure dephasing is introduced during the adiabats of the Endo-global cycle to evaluate the importance of co-  Fig. 1. Endo-global engine is characterized by a global coherence operation, maintaining a nonvanishing coherence throughout the whole cycle. In contrast, the Carnot-shortcut exhibits branch coherence operation, coherence is created 'locally' during each stroke, with initial and final diagonal Gibbs states . Note, that the coherence of the Carnot-shortcut cycle vanishes at the intersections between adjacent strokes.
herence in the engine operation. Such dephasing can be caused by noise in the driving field [62,63]. A Lindbladian describing dephasing is added to the free dynamics.
In the Heisenberg picture it reads Here, the last term induces pure dephasing in the instantaneous energy basis, with a dephasing strength k d . We solve the dynamics, by representing the system in terms of the operator basis v (t) = {Ĥ (t),L (t) ,Ĉ (t) ,Î} T , Cf. Appendix D. Substituting the basis operators into Eq. (F1) we express the dynamics of v (t) in a matrix-vector notation where µ is the adiabatic parameter, I is the identity matrix and M is given by Equation (F2) is solved with a standard Runge-Kutta-Dormand-Prince propagator, leading to the system dynamics in the presence of pure-dephasing. The reconstruction of the density operator, Eq. (A4), from the {Ĥ (t) ,L (t) ,Ĉ (t) ,Î} operator basis assumes that the working medium is described by a generalized canonical state. Strictly, the dephasing dynamics, Eq. (F1), does not conserve the Gaussian structure. Nevertheless, for the cases studied, where the coherence is Stroke duration of the adiabatic compression of the shortcut cycles 5 relatively small, the Gaussian state is a valid representation.