Employing coherence to improve the performance of a quantum heat engine
The working substance fueling a quantum heat engine may contain prominent quantum properties such as coherence and nonclassical correlations. In particular, coherence has been somewhat related to quantum friction, whose origin has been mostly associated with the noncommutativity of a driven Hamiltonian at different times. Here, we consider a quantum Otto heat engine operated at finite time and undergoing incomplete thermalization with a heat source. We introduce analytical expressions relating the engine efficiency and extractable power to the coherence measured in the energy basis of the working substance during the engine cycle. We show that coherence, which has been related to an increase in entropy production and irreversibility, can in also be employed to improve the quantum engine performance, mitigating quantum friction. Such an improvement may be achieved by carefully tuning cycle parameters. To illustrate this coherent improvement we performed a numerical analysis in an experimentally feasible example.
One of the aims of quantum thermodynamics is to describe, at a fundamental level, the energy and entropy exchange among systems Esposito2009 (); Kosloff2013 (); Vinjanampathy2016 (); Millen2016 (); Alicki2018 (). The focus on the description and control of small quantum systems greatly spurred the thermodynamics of quantum heat engines and refrigerators Alicki2018 (); Gelbwaser-Klimovsky2015 (). Experimentally, a single-ion heat engine Ro=0000DFnagel2016 (), a three-ion refrigerator Maslennikov2017 (), and an Otto cycle exploring the harmonic oscillations of a nanobeam have been recently implemented Klaers2017 (). Even more recently, a quantum Otto heat engine using an ensemble of nitrogen-vacancy centers in diamonds Klatzow2018 () and a spin quantum heat engine Peterson2018 () have been reported. On the other hand, coherence is one of the fundamental properties of nature, setting apart the quantum from the classical descriptions of reality. Measures to quantify coherence have been recently proposed Aberg2006 (); Baumgratz2014 (); Streltsov2017 (), applying similar methods used to quantify entanglement. In particular, some measures have operational meaning, quantifying the distillation Winter2016 () and the erasing cost of quantum coherence Singh2017 ().
The role of coherence was theoretically addressed employing the photo-Carnot engine Scully2003 (), which models the working substance as a four-level system. The photo-Carnot engine is an extension of the model employed to thermodynamically describe the laser Scovil1959 (); Geusic1967 (), which is fueled by a three-level working substance. These models employ what could be called a “partial-spectrum thermalization” (PST), in which the heat source interacts only with a subset of the energy states, thus thermalizing part of the spectrum.
The PST approach to quantum machines has been one of the major frameworks to analyze the role of coherences in quantum machines Scully2003 (); Scully2010 (); Scully2011 (); Dorfman2011 (); Rahav2012 (); Dorfman2013 (); Goswami2013 (); T=0000FCrkpen=0000E7e2016 (); Uzdin2016 (); Dorfman2018 (). The role of coherence has also been addressed in other approaches Brandner2017 (); Dodonov2018 (); Marvian2018 (). Here, we focus on stroke-driven engine cycles; in particular, the quantum Otto heat engine (QOHE) Kosloff2017 ().
The heat engine efficiency is given by the ratio of the mean net extracted work and the mean absorbed heat from the hot heat reservoir. As evidenced by the Kelvin statement of the second law, even though the efficiency is defined in terms of two energetic quantities, it is fundamentally related to entropy production, irreversibility, and the second law of thermodynamics =0000C7engelbook2015 (). In classical thermodynamics, two processes are responsible for the irreversibility of engines. The external friction, or simply friction, is associated with the exchange of energy at the system boundary due to sliding. The internal friction Rezek2010 () is associated with the finite-time engine operation. It is manifested by the disparity between the internal dynamics and operation timescales. In order to achieve the best engine efficiency, the engine should operate quasistatically and be frictionless, in which case no entropy is produced during the cycle.
A new kind of (internal) friction in microscopic engines with quantum working substances, intrinsically non-classical in nature, has been studied in the past decades Rezek2010 (); Kosloff2002 (); Feldmann2003 (); Feldmann2004 (); Feldmann2006 (); Rezek2006 (); Feldmann2012 (); Zagoskin2012 (); Thomas2014 (); Plastina2014 (); Alecce2015 (); Correa2015 (); Campo2018 (). The origin of such a quantum friction is attributed to the noncommutativity of the driving Hamiltonian at different times Kosloff2002 (); Feldmann2003 (); Feldmann2004 (); Feldmann2006 (); Rezek2006 (); Rezek2010 (); Feldmann2012 (); Plastina2014 (); Thomas2014 (); Alecce2015 (), which induces transitions among the instantaneous energy eigenstates. Furthermore, when operating in the quasistatic regime (transitionless regime), the quantum friction becomes zero Kosloff2002 (); Feldmann2004 (); Feldmann2006 (); Rezek2006 (); Rezek2010 (); Feldmann2012 (); Plastina2014 (); Thomas2014 (); Alecce2015 (), just as the internal friction of classical engines would. This research avenue spurred the idea of quantum lubrication, which seeks to render the effects of quantum friction negligible while operating the quantum engine at finite-time. One of the most commonly employed strategy is to perform shortcuts to quantum adiabaticity Campo2018 (); Torrontegui2013 (). This method adds so-called counteradiabatic driving fields that makes the working substance evolve in a transitionless dynamics Campo2014 (); Funo2017 (). How costly is such an additional control remains an open question Zheng2016 (); Campbell2017 (); Calzetta2018 ().
Some investigations have connected the coherence in quantum engines to quantum friction Feldmann2006 (); Rezek2006 (); Feldmann2012 (); Thomas2014 (); Correa2015 (). However, no simple expression explicitly relating them has been obtained so far. Here, we present analytical expressions that shows how the engine efficiency and power output depend explicitly on the quantum friction generated in the engine cycle. Furthermore, our formulas easily show how quantum friction is connected to the coherence in the energy basis, henceforth called energy coherence or just coherence.
Coherence has been related to an additional contribution to entropy production, thus making an irreversible process even more irreversible Santos2017 (); Francica2017 (). Hence, for a quantum heat engine, coherence could be expected to degrade the engine performance. Nevertheless, our results elucidates the subtle coherence effects by showing that it can in some regimes reduce quantum friction, enhancing the quantum engine performance. This may be achieved by carefully tuning the cycle parameters, in which case coherence essentially acts as a coherent lubricant. To illustrate the coherent lubrication of a quantum engine, we employed the numerical analyzes of a single-qubit-working-substance performing an Otto cycle undergoing an incomplete thermalization with the hot source.
Ii The Engine Cycle
Let us consider a single-qubit working substance which fuels a QOHE similar to the system employed in the experimental implementation of Ref. Peterson2018 (). The stroke-driven engine cycle is comprised by two Hamiltonian driven protocols (energy gap expansion and compression) and two undriven thermalization strokes, which are depicted in Fig. 1. In Otto engines, the work and heat exchanges are separated among the strokes: work is only exchanged in the two Hamiltonian driven strokes and heat is only exchanged in the two undriven thermalization strokes.
The working substance begins in the cold Gibbs state , where is the cold inverse temperature, is the initial Hamiltonian (“exp” stands for expansion), and is the associated partition function. The initial Hamiltonian is given by , where is the initial transition frequency, and denote the Pauli matrices.
In the first stroke, the energy gap of the working substance is increased by the driven Hamiltonian in a unitary dynamics. The working substance is assumed to be disconnected from the heat sources so that no energy is exchanged with them. We can also consider, in a realistic scenario, that the time scale of such an evolution is fast enough so that the energy exchanged between system and environment can be neglected Peterson2018 (). Hence, the state after the expansion stroke is given by , where , is the time-ordering operator, , and
In the second stroke, the working substance interacts with a hot heat reservoir at inverse temperature and it undergoes a hot thermalization. The Hamiltonian is kept fixed at spanning the time interval . Some stroke-driven models of quantum heat engines assume that this thermalization stroke is complete so that the system reaches thermal equilibrium state at the end of the stroke Wang2009 (); Altintas2015 (). More precisely, in order to achieve this complete thermalization the condition should be satisfied, where is the thermalization time and is the relaxation time of the working substance with the hot heat reservoir. In general, the first stroke generates coherence in the energy basis, all of which would be erased if such a complete thermalization was performed. Therefore, we consider a hot incomplete thermalization stroke, in which the thermalization time is on the order of . Performing an incomplete thermalization in our QOHE model will allow coherence to be transferred from the first to the third engine stroke.
In the third stroke, the working substance energy gap is decreased to its original value during a unitarily driven dynamics. The compression Hamiltonian drives the qubit according to the condition for the time interval , with (“com” stands for compression). This condition guarantees that takes the same values that did in the expansion stroke, but in inverse order (see Appendix A for a detailed explanation). Denoting by the final state of the second stroke, the state after the compression stroke is given by , where .
The fourth stroke is a undriven thermalization with a cold heat reservoir at inverse temperature . The stroke spans the time interval and the driving Hamiltonian is given by . In order to close the engine cycle, i.e., , we consider complete thermalization in this stroke. Therefore, the cold thermalization time must satisfies the condition .
The four relevant energetic quantities to analyze the thermodynamic of the engine are the following. The first- and third-stroke works and , respectively, where denotes the mean instantaneous internal energy; and the hot and cold heats and , which are the energy absorbed by and released from the working substance during the interaction with the hot and cold heat reservoirs, respectively.
The dynamics of a qubit with a Hamiltonian with energy gap interacting with a Markovian heat reservoir at inverse temperature can be described by the master equation Breuerbook2003 (); Chakraborty2016 ()
where , , is the vacuum decay rate, is the Bose-Einstein distribution, and () is the ladder operator in the energy eigenbasis that takes the excited (ground) state and transforms it into the ground (excited) state. The analytical solution of this equation is used to obtain the state after the hot incomplete thermalization stroke, and hence the thermodynamic relations with incomplete thermalization (for details see Appendix A).
The residual coherence that is transferred from the expansion to the compression strokes due to incomplete thermalization affects the thermodynamic quantities. In order to pinpoint the effects of the coherence along the cycle, we consider an alternative engine cycle in which a dephasing operation in the energy basis is performed at the end of the first stroke. Hence, the dephased engine may generate coherence during the expansion and compression strokes but this coherence is not transferred along the cycle, even with incomplete thermalization. We note that the dephasing operation in the energy basis has no energetic cost, since it does not change the working substance mean internal energy. Further details are discussed in the next section.
Iii The role of quantum coherence in efficiency and power
The four relevant states , , , and (related to the four strokes) are the key states of the engine cycle that will be employed to completely analyze the performance of the proposed engine. For further reference, we call this set of states the key-working-substance states.
Before we proceed, it is convenient to establish a few important quantities that are going to be important throughout our analyzes. The (Kullback-Leibler) divergence between an arbitrary state and a reference state is given by , where is the (von Neumann) entropy Vedral2002 (). We conventionally write the instantaneous Gibbs equilibrium state with Hamiltonian and inverse temperature as , where is the partition function and denoting the cold and hot thermal states, respectively. When the reference state of the divergence is some thermal state , we will call the thermal divergence.
For the driving strokes, we define the states , with , as the states that would have been obtained if the driving was performed quasistatically (without transition among the instantaneous eigenstates) and if the initial state was the thermal state , where () for the expansion (compression) stroke. More explicitly, denoting by the instantaneous energy eigenstates, the two states associated with the end of the expansion and compression strokes are and , where , with , are the Boltzmann weights calculated with inverse temperature and Hamiltonian . When the reference state of the divergence is the quasistatically evolved state , we call the quasistatic divergence.
The efficiency of the quantum heat engine is given by the ratio of the net extracted work over the heat absorbed from the hot source, i.e., , with and . For a QOHE, we show that this efficiency can be written in two distinct ways, each of which showing the role of coherence in a slightly different manner. The first way is quite general, applying to QOHEs fueled by any quantum working substance, while the second expression applies to a qubit working substance.
The lag-from-Carnot efficiency expresses the efficiency lag between the QOHE and the Carnot efficiency, and it is given by (see Appendix C)
where is the Carnot efficiency and
is the thermal efficiency lag Peterson2018 (). The thermal efficiency lag is comprised by three thermal divergences, each of which measuring the divergence from the key-working-substance states to the corresponding thermal reference state. Equation (3) is a generalization of the efficiency expression recently presented in Ref. Peterson2018 (), which considered a hot complete-thermalization stroke instead of an incomplete one. The lag introduced in Eq. (4) encompasses both the finite-time effects of the driven and thermalization dynamics of the engine cycle. It is related to the irreversibility of the engine cycle and implies the well-known result that the Otto efficiency is upper bounded by the Carnot efficiency.
Let denote the spectral decomposition of some reference state used to compute the divergence, where are the eigenvalues and the eigenprojectors of . We show in Appendix B that the divergence of an arbitrary state with respect to can always be decomposed as
where is the full dephasing map and is the relative entropy of coherence (in the reference state basis) Aberg2006 (); Baumgratz2014 (); Streltsov2017 (); Winter2016 (); Singh2017 (). This is an extension of the results previously obtained in Refs. Santos2017 (); Francica2017 (), where the authors have considered a thermal divergence (). From now on, in or notation, we conveniently assume that the energy basis of the instantaneous Hamiltonian of the working substance as the relevant basis where the full dephasing and the relative entropy of coherence are computed. Applying the decomposition in Eq. (5) to Eq. (4), the thermal efficiency lag can be written as
quantify the contribution of the populations and coherences of the key-working-substance states, respectively. These terms show explicitly how the coherence of the key-working-substance states of the QOHE contributes to the engine efficiency. Equation (3) was obtained assuming a quantum Otto cycle (as described in the preceding section), without assuming any particular model for the quantum working substance. Before we discuss the consequences of Eq. (3), we present a similar relation that was obtained for a single-qubit working substance.
When all strokes occur quasistatically, the engine achieves its maximum efficiency, i.e., minimum irreversibility, namely the quantum Otto efficiency . For a single-qubit working substance, we obtained a lag-from-Otto efficiency which is given by (see Appendix D and Appendix E)
quantifying the contribution of the populations and coherences of the key-working-substance states, respectively. The efficiency lags introduced in Eqs. (6) and (11) have been decomposed into a diagonal and a coherent part with respect to the relevant instantaneous energy basis.
The engine average power output per cycle is given by , where is the cycle time duration. The relation between the efficiency and power is given by . Writing the efficiency in terms of the thermal efficiency lag in Eq. (6) , the power can be similarly divided into a diagonal and a coherent contribution
A similar expression can be obtained if one writes the power in terms of the lag-from-Otto (11) for the efficiency, in which case the quasistatic efficiency lag will appear.
The two expressions for the finite-time efficiency, given by Eqs. (3) and (9), explicitly show how the energy coherence of the key-working-substance states contribute to the engine performance and irreversibility. Moreover, we note that in both efficiency lags, the incomplete thermalization stroke contributes with a negative sign in the therm . Since the efficiency lags may decreases due to the incomplete thermalization, the thermalization time can be used to tune the engine efficiency. From Eq. (15), we can see that the coherence of states and decreases the power output, whereas the coherence of enhances it. Again, a fine tuning of incomplete thermalization may enhance the power output by the engine due to the residual coherence .
Let us now investigate the effects of the coherence in the key-working-substance states during the cycle employing numerical simulations. In our calculations, we consider energy scales compatible with quantum thermodynamics experiments in nuclear magnetic resonance (NMR) setups Peterson2018 (); Batalh=0000E3o2014 (); Batalh=0000E3o2015 (); Camati2016 (); Micadei2017 (). The initial and final frequency gaps of the expansion stroke will be chosen as and , respectively. The chosen temperatures are such that the thermal energy scale of the cold (hot) heat reservoir is half (double) the energy gap of the working substance at the time of the interaction with the heat source. More precisely, the cold and hot inverse temperatures will be chosen as and , respectively.
We assume that a complete thermalization with the cold environment is approximately achieved at a finite time satisfying the condition . We also consider that the interaction time with the hot reservoir is smaller or of the same order of the relaxation time of the hot environment , which results in an incomplete thermalization with the hot environment in the engine cycle. The strength of the interaction of the working substance with the hot and cold heat reservoirs, namely the vacuum decay rate, will be assumed as , which implies the relaxation time and . In order to reach approximately a complete thermalization at the end of the fourth stroke, we considered in the numerical simulations the cold thermalization time as . In this case, the working substance approximately return to the cold Gibbs state, being the trace distance of the with respect to the final state of the fourth stroke is approximately .
Figures. 2(a) and 2(b) display the relative entropy of coherence as a function of the driving time and the thermalization time , respectively. In Fig. 2(a) one can see that the relative entropy of coherence at the end of the first and second strokes are qualitatively similar. The relative entropy of coherence is smaller due to the incomplete thermalization that partially erased the coherence as measured by . The coherence at the end of the third stroke of the dephased engine cycle behaves qualitatively as and , even being smaller than . Note how the behavior of the coherence at the end of the third stroke in the original (not dephased) engine cycle is different, due to the coherence transferred from the first to the second stroke. The coherence generated by the compression stroke (third stroke) interferes with the residual coherence at the end of the second stroke generating the different structure for the oscillations, see Fig. 2(a).
The amount of coherence as measured by decays exponentially with the thermalization time , see Fig. 2(b). On the other hand, the coherence oscillates quickly due to the interference of the residual coherence and the coherence generated by the driven dynamics in the compression stroke. As the thermalization time increases, the oscillating amplitudes of become less pronounced, going asymptotically to zero; in which case, the coherence approaches because the coherence is increasingly erased. In the expressions for the efficiency and power output, the term explicitly appears, suggesting that whenever efficiency and power output can be enhanced. From Fig. 2(b) we can see that the rapid oscillations make this inequality be satisfied for very narrow time intervals. Such a behavior will be present in the efficiency and power output as will be seen shortly.
In Figs. 3(a) and 3(b) we plot the thermal and quasistatic efficiency lags as a function of the driving time and the thermalization time , respectively. The two efficiency lags differ by the constant amount as can easily be obtained from the two expressions of the efficiency (3) and (9). By measuring the departure of the maximum achievable efficiency, the quasistatic efficiency lag is a direct quantification of the amount of quantum friction. From these plots we see that the quantum friction is quite small, meaning that the engine efficiency comes very close to the Otto efficiency.
Our results generalize and qualitatively explain some previous findings in the literature. For instance, in Refs. Rezek2010 (); Feldmann2006 () noise has been used to improve the quantum engine efficiency. Here we observe that, if the noise is such that it decreases the contribution of either or , the efficiency can be enhanced. Moreover, quite a few papers have considered the so-called “energy entropy” of the working substance as a measure of quantum friction Feldmann2004 (); Rezek2010 (); Feldmann2012 (); Zagoskin2012 (). Although not directly connected in these works, the difference between what these authors called energy entropy and the von Neumann entropy is nothing but the relative entropy of coherence. The present work elucidates why this energy entropy was related to quantum friction in Refs. Feldmann2004 (); Rezek2010 (); Feldmann2012 (); Zagoskin2012 (). We note that the so-called energy entropy is in fact associated with a measure of coherence in the working-substance. In Refs. Feldmann2006 (); Rezek2006 (); Feldmann2012 (); Thomas2014 (); Correa2015 (), quantum friction has been somewhat related to the presence of coherence. Ref. Feldmann2012 () for instance, employed the -norm of coherence to quantify quantum friction. Our results complement these findings by providing a concrete relation that elucidates how energy coherence is linked to quantum friction by means of the quasistatic efficiency lag .
We have seen some effects of the transferred coherence in the previous discussion. We further analyze this phenomenon by showing exactly how it contributes energetically to the thermodynamic quantities in the quantum cycle. In order to obtain the relations shown, hereafter, we assumed a single qubit as working substance, as described in Sec. II.
Let us denote by and the instantaneous eigenenergies and eigenstates of the engine Hamiltonian, respectively, where the index () stands for the ground (excited) state. The energy transition probability in the first-stroke is given by
Evaluating the first-stroke work and the second-stroke heat one obtains and , respectively, where and is the component of the qubit Bloch vector associated with . In particular, for a complete thermalization, this Bloch vector component will be .
The third-stroke work can be evaluated as
where is the third-stroke energy transition probability, is the energy probability amplitude, and
is the transferred-coherence energy contribution (see Appendix F). In Eq. (18), is the total decay rate of the qubit after the interaction with the hot heat reservoir, and is one of the coherence elements of the qubit state in the instantaneous energy basis at the beginning of the second stroke.
The contribution in Eq. (17) comes exclusively from the residual coherence at the end of the second stroke, i.e., the coherence that was not completely erased by incomplete thermalization. If the thermalization was complete () then . Since the third-stroke work is present in the engine efficiency and power output, the above expression quantifies how the transference of coherence affects these thermodynamic quantities. In particular, they will oscillate due to the complex exponential.
The internal energies of the original and dephased QOHEs are related as: , , , and , where is given in Eq. (18). From these relations we can readily obtain the efficiency and power , which precisely quantify the effects of the transferred coherence from first to the third stroke.
In Fig. 4(a) we compare the efficiency of the original QOHE, which transfers coherence, and the dephased QOHE, which does not transfer coherence, as a function of the driving time for a fixed incomplete thermalization time (). Observing Fig 4(a), we can note that the original QOHE may perform better or worse than the dephased QOHE. In this parameter regime, the transferred coherence can make the original QOHE perform about times more efficiently than the dephased QOHE in the intermediate region of the plot. Furthermore, even for such a small thermalization time (about 1/3 of the relaxation time), the efficiency of the original QOHE reaches values very close to Otto’s efficiency. This is a consequence of the small quantum friction generated by the engine cycle as seen in Fig. 3(a). A similar behavior can be observed in the power output as can be seen in Fig. 4(b). The power output of the original QOHE can also be greater or smaller than the power of the dephased QOHE. Note that the efficiency and power of both the original and dephased QOHE oscillate. Since, by construction, there is no transference of coherence in the dephased QOHE, these oscillations are not a manifestation of the transference of coherence. They arise from the choice of the driving Hamiltonian.
Now let us consider a fixed driving time , in this case the efficiency and power of the original QOHE oscillate as function of the thermalization time as displayed in Figs. 4(c) and (d). Compared to the Figs. 2(a) and (b), the oscillations in Figs. 4(c) and (d) occur strictly due to the transference of the residual coherence after the incomplete thermalization with the hot reservoir [see Eq. (18)]. Therefore, these oscillations are a consequence of constructive and destructive interference between the residual coherence and the coherence generated by the third-stroke driving. These coherence-induced oscillations in the the efficiency and power are damped as the thermalization time increases.
We have seen that the residual coherence transferred between the expansion and compression strokes may or may not enhance the engine performance. A fine control over the driving and thermalization times is paramount to make the quantum engine run in a suitable parameter regime, thus taking full advantage of coherence transference.
Quantum lubrication is a method by which the engine efficiency can be enhanced through the reduction of quantum friction Feldmann2006 () controlling the entropy production and irreversibility along the quantum cycle. The typical method employed in the literature is to perform shortcuts to adiabaticity by means of counter-adiabatic driving fields Campo2018 (); Torrontegui2013 (); Campo2014 (); Funo2017 (); Zheng2016 (); Campbell2017 (); Calzetta2018 (). We have seen that the fine control over driving and thermalization times can also be employed to make a quantum engine almost frictionless due to coherence transfer over incomplete thermalization. Since this method does not rely on additional driving fields but by purely controlling the parameters of the engine cycle, we refer to it as a dynamical quantum lubrication strategy.
Our results elucidate the precise role of quantum coherence in QOHEs by relating the engine efficiency and power output to the relative entropy of coherence. In particular the quasistatic efficiency lag, Eq. (10), is a measure of quantum friction that unveils the finite-time irreversibility. On the other hand, the thermal efficiency lag, Eq. (4), encompasses the standard Carnot bound besides the finite-time irreversibility effects.
Coherence has been usually associated with an additional contribution to entropy production and irreversibility, which in principle would degrade the engine performance. We have shown, though, that by carefully controlling the time allocation in the different strokes of the quantum engine, one can use the interference of energy coherence as a dynamical quantum lubricant. This effect can reduce the protocol irreversibility, enhancing considerably the efficiency and power output. We note that, in general, the presence of coherence in the working substance makes the fine control over the driving and thermalization times paramount to operate the quantum engine in an optimal regime.
The coherent lubrication discussed here can be tested in an experimental scenario with current quantum technologies, as evidenced by the presented numerical example.
We thank C. Ivan Henao and M. Herrera for very helpful discussions concerning the relation between quantum friction and the generation energy coherence. We acknowledge financial support from UFABC, CNPq, CAPES, and FAPESP. This research was performed as part of the Brazilian National Institute of Science and Technology for Quantum Information (INCT-IQ).
Appendix A The Engine Cycle
In Sec. II we explained the QOHE cycle, however some important aspects were not thoroughly discussed. First, we show that the energy transition probability of the expansion stroke is the same as the energy transition probability of the compression stroke. Then, we discuss the relation between the expansion and compression driving fields, which are not the backward of one another as considered in some papers Peterson2018 (). Also, we show the expressions of the master equation for incomplete thermalization.
Relation between expansion and compression strokes
In Fig. A1 we show how the instantaneous eigenenergies change during one cycle. The Hamiltonian of the total engine is given by
where each of these Hamiltonians have been defined in the main text [see Eq. (1)].
The unitary evolution of the first and third strokes are
respectively, where is defined in the time interval and is defined in he time interval . Recall that , where , i.e., the expansion and compression strokes take the same amount of time to be performed. Changing variables in Eq. (A2) to one obtains
This seems a quite strange result. Even though the variation of the Hamiltonian is different, the time-evolution operator coincides. However, if we want to establish the physical condition that makes third-stroke driving Hamiltonian go back to the initial Hamiltonian of the first stroke passing through the same Hamiltonians in between, the Eq. (A4) is true as demonstrated above.
The definition of the transition probability between energy states of the expansion and compression strokes are
By definition, the energy transition probability of the expansion and compression strokes are
respectively. Using from Eq. (A4) and opening the modulus square one can easily show that
Incomplete thermalization relations
Suppose the initial state of a qubit is given by the Bloch vector , where is the instantaneous Bloch components for . From the Refs. Breuerbook2003 (); Chakraborty2016 (), the master equation of a qubit interacting with a Markovian heat reservoir and whose Hamiltonian is fixed at is given by Eq. (2). The solution of the Bloch vector components at the end of the second stroke are given by
where , , and the remaining parameters have been defined in the main text. One of the coherence element at the end of the incomplete thermalization oscillates as
This oscillatory terms are the origin of the oscillations in Figs. 4(c) and .
Appendix B Decomposition of the divergence
In Refs. Santos2017 (); Francica2017 (), it was shown that , where is the dephasing map in the energy basis and is the relative entropy of coherence. This result can be generalized for any diagonal reference state as we now demonstrate.
Let be a diagonal state in the arbitrary basis . Thus, by definition
The first term is given by
Using this identity in the definition of the divergence and adding we obtain
after rearranging the terms.
Appendix C The thermal efficiency lag
In this appendix, we derive the expression for the thermal efficiency lag given by Eq. (4). The engine efficiency is given by
Using the first law of thermodynamics , we rewrite the efficiency as
where in the last equality we wrote the cold heat in terms of the internal energies.
Next, we use the following relation valid for the thermal divergence (see the Supplemental Material of Ref. Camati2016 () for a quick derivation). Let be an arbitrary state in some time with Hamiltonian and an associated Gibbs state with same Hamiltonian and some reference inverse temperature , then
where is the associated free energy and is the associated partition function.
We substitute the internal energies in the efficiency using the expressions
where we have used , and because the Hamiltonian is the same at times and (see Fig. A1). Hence,
where we already canceled the free energy terms and is the change in entropy during the fourth stroke. In the last equality we used the conservation of entropy , where is the change in entropy during the second stroke. The first and third strokes are unitary and hence do not contribute to the entropy change. Using the Eq. (C3), we substitute the von Neumann entropies
in order to obtain
where the Carnot efficiency and thermal efficiency lag were defined in the main text [see Eq. (4)].
Appendix D The quasistatic divergences
Before we demonstrate the expression for the quasistatic efficiency lag we need to obtain an expression for the quasistatic divergence similar to Eq. (C3) for the thermal divergence.
In the first stroke, the initial state is always the cold Gibbs state