Optimal cycles for low-dissipation heat engines

Paolo Abiuso∗

ICFO – Institut de Ciencies Fotoniques, The Barcelona Instituteof Science and Technology,08860 Castelldefels (Barcelona), Spain

Martı Perarnau-LlobetMax-Planck-Institut fur Quantenoptik, D-85748 Garching, Germany

(Dated: March 9, 2020)

We consider the optimization of a finite-time Carnot engine characterized by small dissipations.We bound the power with a simple inequality and show that the optimal strategy is to performsmall cycles around a given working point, which can be thus chosen optimally. Remarkably, thisoptimal point is independent of the figure of merit combining power and efficiency that is beingmaximized. Furthermore, for a general class of dissipative dynamics the maximal power outputbecomes proportional to the heat capacity of the working substance. Since the heat capacity canscale supra-extensively with the number of constituents of the engine, this enables us to designoptimal many-body Carnot engines reaching maximum efficiency at finite power per constituent inthe thermodynamic limit.

The Carnot engine has a pivotal role in thermo-dynamics, both from a fundamental and applied per-spective, being the reference point for other enginesin terms of efficiency [1, 2]. It is thus of paramountimportance to understand its limits and strategies forits best utilization. In this article, we consider theoptimization of a finite-time Carnot cycle within theso called low-dissipation (LD) regime [3–14], wherethe driving of the control parameters is slow but fi-nite. Previous studies of Carnot engines in the LDregime have considered bounds on the reachable effi-ciencies [3], tradeoffs between efficiency and power [7–9, 15], the coefficient of performance of refrigera-tors [12, 13], the impact of the spectral density of thethermal baths [14], and other thermodynamic figuresof merit [10, 11]. Despite this remarkable progress, thefollowing crucial question has remained unaddressed:given a certain level of control on the working sub-stance (e.g. some parameters of the Hamiltonian, orsome macroscopic variables such as volume or pres-sure), what is the optimal cyclic modulation of thecontrol parameters to maximise the power output (or,more generally, any figure of merit involving powerand efficiency [7–9, 15]) of a finite-time Carnot en-gine? Such an optimal cycle has been designed for asingle-qubit engine in [16, 17], but a general under-standing is lacking.

Using recent insights on a geometrical approach tothermodynamics [18–29], we show that, given any rea-sonable figure of merit involving power and efficiency,the optimal control strategy is to perform infinitesi-mal cycles around a fixed point. Furthermore, whenthe thermalization of the relevant quantities can bedescribed by a single time-scale τeq (see details be-low), the optimal power output becomes proportionalto C/τeq, where C is the heat capacity of the workingsubstance (WS). Hence, the optimization of the heatengine cycle becomes intimately related to the maxi-mization of C of the WS (interestingly, maximizing Cis also crucial in thermometry [30–33]).

∗ paolo.abiuso@icfo.eu

We then use these insights to design many-bodyheat engines that can operate at Carnot efficiencywith finite power per constituent of the WS througha supraextensive scaling of C/τeq (e.g. in a phasetransition), in the spirit of [34, 35] (see also [35–38]).Despite differences w.r.t. previous proposals, whichwere based on Otto engines [34, 35], we find the sameasymptotic scalings for performance, while provingby construction their optimality in the slow-drivingregime. Other recent proposals towards the possibil-ity of reaching Carnot efficiency at finite power havebeen developed in [34–36, 39–45] (see also [15, 46, 47]for no-go results [48]).

Finite-time Carnot cycle. We consider a quan-tum working substance (WS) with a driven Hamil-tonian H(t), which interacts alternatively with acold (Bc) and a hot (Bh) heat bath at tempera-ture Tc and Th, respectively (the results presentedin this article can be naturally extended to classi-cal systems). The Carnot cycle consists of four steps:(i) While being coupled to Bc, H(t) is modified con-tinuously from H(0) = H(X) to H(τ−c ) = H(Y )

for a time τc. (ii) With the system isolated fromthe reservoirs, an adiabatic process is performed tak-ing H(Y ) → H(Y )Th/Tc. During this process H(t)satisfies [H(t), H(t′)] = 0 ∀t, t′ and commutes withthe initial state (a Gibbs state w.r.t H(τ−c )), so itis possible to perform it arbitrarily quickly withoutaffecting the state (hence corresponding to Hamil-tonian quenches). (iii) While being coupled to Bh,H(t) is modified back from H(τ+

c ) = H(Y )Th/Tc toH(τc + τh) = H(X)Th/Tc in a time τh. (iv) Finally asecond quench is performed to restore H(X)Th/Tc →H(X), closing the cycle.

It is convenient to introduce the adimensionalHamiltonian G(t) := βH(t), where β = 1/kBT isthe inverse temperature of the bath that the WS iscoupled to (we shall set the Boltzmann constant kBequal to kB = 1). Note that the Carnot cycle becomessmooth with respect to G(t), and in what follows, wetake the driving protocol to be time-reversal symmet-ric, more precisely that G(sτc) = G(τc + τh(1 − s))with s ∈ [0, 1]. This property is always satisfied byoptimal heat engines in the LD regime as long as the

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two baths have the same spectral density [14], whichis the subject of interest of this work. By expressingG(t) as

G(sτc) =∑j

λj(s)Xj , s ∈ [0, 1] (1)

where λj(s) are the control parameters and Xj theconjugated forces, the cycle control can be charac-

terised by τc, τh and its shape ~λ(s) (notice that τc 6= τhin general; the time-reversal symmetry is intended inthe adimensional unit s). We will not write explicitlythe dependence on s, which will be clear from the con-text, but will indicate with a dot the time derivativew.r.t. s, i.e. G ≡ ∂

∂sG ≡ τxddtG, x = (h, c).

We now assume the slow driving or low dissipationapproximation, which is crucial in this work. That is,ddtG is finite but small, so that we can expand the rele-vant quantities and keep only leading terms (formally,the small parameter of the expansion is τeq/τx, whereτeq is the relaxation time of the dynamics). In thisregime, the state of the WS is always close to ther-mal equilibrium, and the heat exchanged during theisotherms (steps (i) and (iii)) can be divided as

Qx = Tx∆Sx −W dissx , x = (h, c) (2)

where ∆Sx is the reversible contribution obtained inthe quasistatic limit τx → ∞, which is given by∆S ≡ ∆Sh = −∆Sc = S(ω(τc)) − S(ω(0)) withω(t) = e−G(t)/Tr(e−G(t)) and where S(ρ) is the VonNeumann entropy. The irreversible term W diss

x canbe described in this regime by the so called thermody-namic length [20–25, 28]

W dissx =

Txτx

∫ 1

0

∑ij

λimij λj ds ≡ TxΣxτx, (3)

where mij is given, when the driven observables 〈Xj〉relax to equilibrium with the same time-scale τeq,by [20–25, 28]:

mij = τeq∂2 lnZ∂λi∂λj

. (4)

where Z = Tr(e−G) is the partition function. Giventhe time-reversal symmetry of the driving protocol,it follows that Σh = Σc (symmetric low-dissipationregime SLD). Importantly, while the results presentedin the main text are based upon the standard ther-modynamic metric (4), generalizations (including τeq

depending on the trajectory [24, 25], the possibilityof having several relaxation time-scales, general Lind-bladian dynamics [24, 28], and protocols in whichΣh 6= Σc) are developed in the Supp. Mat. [49].

Optimisation of the cycle. We now optimise thepower (and efficiency) of the Carnot engine over τc, τhand its shape λj(s), which are all the possible degreesof freedom. This enables us to obtain a fundamentalupper bound on the power in the slow-driving regime,as well as the corresponding optimal control.

The work extracted during a cycle is given by W =Qh+Qc, and the total time is τ = τc+ τh. The powerhence reads P = (Qh +Qc)/τ , and the efficiency η =

(Qh+Qc)/Qh. By substituting the expressions (2)-(3)and appropriately setting τc and τh, one can maximizethe power of the engine (∂P/∂τj = 0) obtaining [3, 50]

P (max) =(∆S)2

4Σ(√Th −

√Tc)

2 (5)

and the corresponding efficiency at maximum power(EMP) is given by the Curzon-Ahlborn EMP, ηCA =

1−√Tc/Th [51]. In the most general case one might

seek, in order to not sacrifice completely the efficiencyoptimization over the power, to maximize a hybridfigure of merit [7–9, 15]. The maximum efficiency forany given power output of the engine has been derivedin [9] (see also [7]). Analogously, we can express thebest power for a given efficiency, fixed to be a fractionof the maximum one:

η = γηC , γ ≤ 1, (6)

where ηC = 1 − Tc/Th is the Carnot efficiency. Inthe SLD regime this leads to a maximum power (cf.Supp.Mat. [49]),

P (max)γ =

(∆S)2

4Σ

(Th − Tc)2γ(1− γ)

γTc + (1− γ)Th, (7)

obtained by setting τc = 2ΣTc/(∆S (Th − Tc) (1− γ))and τh = τc(ThT

−1c (1−γ)+γ). Essentially, by tuning

γ in τc and τh, one can interpolate between a max-imally powerful engine with power (5) at efficiencyηCA, and a Carnot engine with maximal efficiency andnull power.

At this point, we note a crucial observation: af-ter the optimisation of P over τc and τh, the re-maining figure of merit is always (∆S)2/Σ, indepen-dently of the value of γ. In fact, this is a propertythat can be argued to be general given any figure ofmerit combining power and efficiency [49]. We nowshow how to maximize it by an opportune use of aCauchy-Schwarz inequality. First, using the formulafor the derivative of an exponential [52]: ∂e−G/∂λj =

−∫ 1

0e−GXje

−(1−s)G, we can express again Σ in (3) inthe more compact form

Σ = τeq

∫ 1

0

ds covω(G, G) (8)

where covω(A,B) is the Kubo-Mori-Boguliobov in-

ner product: covω(A,B) =∫ 1

0ds Tr(Aω1−s(B −

Tr(ωB))ωs). Next, we note that the Von Neumann

entropy S = −Tr[ω lnω] satisfies: S = −Tr(ω lnω) =

−covω(lnω, G) = covω(G, G) where we used again the

formula ω = −∫ 1

0dx ω1−x(G−Tr(ωG))ωx, as well as

Tr(ω) = 0 and covω(A,1) = 0. Hence, we can writethe total change in entropy ∆S as:

∆S = −∫ 1

0

ds covω(G, G). (9)

Crucially, both ∆S and Σ can be expressed throughan infinite-dimensional scalar product given by:

〈A,B〉ω =∫ 1

0ds covω(A,B), that depends on the

3

path {λj(s)}. Using the Cauchy-Schwarz (C-S) in-equality |〈A,B〉|2 ≤ 〈A,A〉〈B,B〉, the ratio (∆S)2/Σcan then be bounded as:

(∆S)2

Σ≤ 1

τeq

∫ 1

0

ds covω(G,G) ≡ 1

τeq

∫ 1

0

ds C (10)

where C is the heat capacity of the WS,

C = −β2 ∂2 lnZ∂β2

= Tr(ωG2)− Tr(ωG)2 (11)

i.e. the variance of the adimensional Hamiltonian Gfor its thermal state. The C-S inequality is saturatedfor G ∝ G, which means that creating quantum coher-ence cannot favour the power output in the slow driv-ing regime, in agreement with Refs. [53, 54]. More im-portantly, it shows that optimal thermodynamic pro-tocols take the simple form G(t) = λ(t)G(0).

We can further maximize (63) by noting that∫ 1

0ds C ≤ maxs C. To saturate this inequality in prac-

tice, one needs to consider cycles where the modula-tion of G is small

G(τc) = G(0)(1 + ε) ε� 1,

G(τc + τh) = G(0), (12)

around an optimal point G(0) where C/τeq is max-imised (as long as ε is small enough so that G(sτc)with s ∈ [0, 1] does not change substantially along thecycle, the precise form of G(sτc) is not important; seethe SM for examples of explicit cycles). In this case,in the limit ε → 0 the maximal power (7) for a givenefficiency γηC is given by

P (max)γ =

C4τeq

(Th − Tc)2γ(1− γ)

γTc + (1− γ)Th. (13)

where C is the heat capacity at G(0). We stress that(13) has been obtained after maximising P (for a fixedη) over all degrees of freedom: τc, τh and the pro-tocol {λj(s)}. This result shows a fundamental linkbetween maximal power of a finite-time Carnot cycleand the heat capacity of the WS.

The simplicity of (13) can be contrasted to exact op-timisations of heat engines [45, 55–58], where the fullsolution easily becomes too complex or not even com-putable with the size of the WS; and with other ge-ometric optimisations which require solving geodesicequations to design optimal paths in the parameterspace [24, 26, 28, 29, 59–61]. In our approach, fromthe point of view of optimization all that is left todo in (13) is to maximise C over the control parame-ters {λj} to identify the optimal working point G(0)in (12). In Fig. 6 explicit results are reported for thevalue of maximum C, for different paradigmatic levelsof control on the same system of N qubits.

Crucially, this approach can be generalized to anymetric mij in (3) describing dissipation: in the SM[49] we show that the optimal control problem is al-ways reduced to infinitesimal cycles, and the optimalworking point can be found by a scalar maximisationproblem. We work out as well examples of standardmicroscopical dynamics in open quantum systems: a

Full control

Ising chain

Qubits array

1 2 5 10 20 50

0.5

1

5

10

50

100

Figure 1. Maximum adimensional C for a thermal sys-tem of N qubits with different degrees of control [49].1) Cmax ' 0.44N for N independent 2-level systems

with gap control. 2) For an Ising chain H(N) =

−λ1(t)∑Ni=1 σ

zi σ

zi+1 − λ2(t)

∑Ni=1 σ

zi , we obtain Cmax '

0.59N . 3) Given complete control over the spectrum with2N levels, Cmax ' N2/4. Details can be found in [49].

qubit, a qutrit, or an harmonic oscillator as a WSin contact with bosonic thermal baths with differentspectral densities.

It is important to point out that (13) should beunderstood as a theoretical ultimate upper bound onpower, obtained by taking ε → 0 in (12). In prac-tice, any experimental or realistic protocol will havefinite ε. In this case, as long as C is sufficiently smoothalong the thermodynamic cycle (12), the power outputP (given by the integrated C in (63)) will not changeconsiderably, so that realistic cycles will provide a sim-ilar P than the theoretically maximal one. In practice,keeping ε finite is also important to ensure the con-sistency of the slow-driving approximation τeq/τ � 1,given that for the optimized protocols that lead to thepower (7), one has τ ∝ ε, more precisely

τeq/τ ∼ ε−1(Th/Tc − 1)(1− γ). (14)

Note that this can always be guaranteed for high ef-ficiencies γ → 1. From the same equation we noteincidentally that engines whose maximum efficiencyis constrained to be low (Tc/Th . 1), i.e. arguablythose engines that mostly need optimization in thehigh efficiency regime, show better convergence to theabsolute bound.

Reaching Carnot efficiency at finite power. As anexample of application of the previous results, we nowuse the designed optimal finite-time Carnot cycles(Eqs. (12) where G(0) will depend on each model ofinterest) to explore the possibility of reaching Carnotefficiency ηC at finite power in the macroscopic limit.We follow the approach put forward in Refs. [34, 35]:considering a N -particle WS, we aim at approachingCarnot efficiency in the macroscopic limit N → ∞without giving up power per constituent.

To reach Carnot efficiency, we need γ = 1 in (6), andhence we take 1−γ = N−ξ, where ξ > 0 can be chosen

at will. On the other hand, the maximal power P(max)γ

in (13) depends only on C and τeq; we then assumeC = c0N

1+a and τeq = τ0Nb, where the meaning of

4

the different constants will later be described for eachmodel of interest. Expanding the relevant quantitiesfor N � 1, we obtain at leading order in N :

P (max)γ =

c0(Tc − Th)2

4τ0TcN1+a−b−ξ

τc =2εTc

Th − Tcτ0N

b+ξ , τh = τc ,

W = (Th − Tc)c0εN1+a, σ2w = 2(Th − Tc)

W

ε. (15)

where σ2w = 〈w2〉 − (〈w〉)2 is the variance of the work

distribution, which measures the work fluctuations percycle of the engine (see Supp.Mat. [49] for detailson the calculation). Let us now discuss two separatecases, inspired by [34] and [35], respectively.(a) Control on the engine and the engine-

bath interaction. We first assume full control overthe engine Hamiltonian with 2N levels: that is, alllevels can be modified at will by the experimental-ist. While this is extremelly challenging in practice,it is useful to obtain fundamental upper bounds onthe maximal power. The optimal Hamiltonian max-imising C then consists in a ground level and a 2N − 1degenerate level (see [30, 62]) which, as shown in Fig.6,leads to C ∝ N2, i.e., a = 1. This supralinear scalingis obtained in an increasingly smaller region of the pa-rameter space, which requires taking ε ∝ 1/N in (12)(see details in Supp. Mat. [49]), and this constraintsfrom Eq. (14) also 1−γ to scale accordingly, i.e. ξ = 1.Furthermore, it is possible to reach in realistic colli-sional scenarios τeq ∝

√N (i.e. b = 1/2), or constant

τ (b = 0) if one is allowed to fine-tune the interactionbetween the WS and the baths [34][63], in agreementwith Ref. [34]. In the Supp.Mat. [49], we solve exactlythis proposal for a feasible driving protocol close to theoptimal one.

(b) Engine working on a phase transitionpoint. A promising platform to obtain supralinearscaling of power with realistic control is by choos-ing the engine to work in a phase transition point ofthe many-body WS. For a finite WS operating closeto the critical point, finite size scaling theory tellsus that C develops a peak of height C ∝ N1+α/(νd)

and width δ ∝ N−1/(dν), while τeq ∝ Nz/d (here α,ν and z correspond to the specific heat, correlationlength and dynamical critical exponents, while d isthe spatial dimension of the engine [64, 65]). In orderto exploit the critical scaling of the WS, we need toperform the cycle (12) where C becomes peaked, andhence ε ∝ δ ∝ N−1/(dν), which implies ξ = 1/(dν)from Eq. (14). Then, from (15) with a = α/(νd) and

b = z/d, supralinear scaling of P(max)γ is possible if

α− zν − 1 > 0 . (16)

This condition is the same found for the Otto cycleproposed in [35]. Examples of physical systems where(16) is satisfied are also provided in [35], particularlyin the presence of critical speed-ups of thermalisation(z < 0 [66–68]).

Besides efficiency and power, another crucial aspectof a heat engine is its reliability, i.e. the fluctuationsin the output power. In fact, it has been recently

pointed out in [36] that the Otto-cycle of [35] suf-fers from macroscopic fluctuations in the thermody-namic limit. For the Carnot-cycle considered here,from (15) the relative work fluctuations read fw =

σw/W =√

2(Th − Tc)/εW. First of all, in the case(a) where a = 1 and ε ∝ N−1 in (15), one hasfw ∼ O(1) in the macroscopic limit. A similar sit-uation takes place for the critical heat engine (b) asone simultaneously has ε ∝ N−1/(dν) and a = α/(dν),and hence fw ∝ N (−2+dν+α)/(dν). Using the relationdν = 2 − α [69], we hence obtain fw ∼ O(1). There-fore, for both proposals fw ∼ O(1) in the thermody-namic limit, hence hindering their reliability, which isthe same result found in [36] for the Otto cycle. De-spite fw ∼ O(1), these fluctuations can be suitablyavoided when the number M of cycles is large (the ar-gument below applies to the Otto and Carnot cycle).

Given M cycles, we have that fw ∝ 1/√M as the

average work W ∝ M whereas the work fluctuationsσw ∝

√M (think of a biased random walk). There-

fore, the ratio between the fluctuations per unit timeand the power goes to zero as M grows even when thefluctuations per single cycle are large. Since we havethat M ∝ τtot/τeq, for a total time τtot of observation,fluctuations can be suitably avoided e.g. for criticalspeed-ups where τeq ∝ N b with b < 0.

In actual implementations, the technical require-ments to realise such optimised Carnot cycles are:global control of the WS, H(t) = λ(t)H(0), andenough precision to engineer small cycles in the regionwhere C/τeq has supralinear scaling with N . Since thewidth ε of this region shrinks with N , in a realisticimplementation the supralinear scaling will be even-tually lost as the control precision is limited [70]. Weremark that even when the experimental control maybe limited, our considerations provide upper boundson the maximal power of finite-time Carnot engines.

Conclusions. We have characterized the opti-mal cycle of a finite-time Carnot engine in the low-dissipation regime. The dissipation has been char-acterized by the thermodynamic metric (4), whichis justified when the thermalization of the workingsubstance (WS) is well described by a single time-scale τeq. In this case, the optimal cycle turns out tobe remarkably simple: it consists of modulations inthe form λ(t)H(0), where H(0) is the Hamiltonian ofthe WS. The power output is then proportional to theheat capacity C of the WS, linking the optimal per-formance to the nature of the WS: as an applicationwe showed how the critical scaling of C can enablethe design of optimal engines with extensive powerreaching Carnot efficiency. These results have beengeneralised to general metrics in the Supp. Mat. [49],which we have used to derive the optimal cycle andcorresponding power output of different WS (qubit, 3-level system, or harmonic oscillator) interacting witha bosonic thermal bath. Putting everything together,our reults provide a general framework to efficientlyoptimise the control of slowly driven Carnot engines.

We hope this work stimulates further investigationsin the interplay between many-body physics and heatengines [35–38, 71–77], as well as connections betweenperformance, fluctuations, and degree of control [78].

5

In particular our results hint at the answer for twoopen problems: 1) small cycles are optimal for en-gine performance in all regimes [17, 45, 57], and 2)the performance of the proposals [34, 35] cannot beimproved.

Acknowledgments. We thank Harry J. D. Millerand Karen Hovhannisyan for insightful discussions.We also thank the anonymous referees for construc-

tive criticisms and useful feedback. P.A. is sup-ported by “la Caixa” Foundation (ID 100010434, fel-lowship code LCF/BQ/DI19/11730023), the Span-ish MINECO (QIBEQI FIS2016-80773-P, and SeveroOchoa SEV-2015-0522), Generalitat de Catalunya(SGR1381 and CERCA Programme), Fundacio Pri-vada Cellex.

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the present analysis, as we are concerned with reach-ing Carnot efficiency asymptotically.

[49] P. Abiuso and M. Perarnau-Llobet, Supplemental ma-terial for ”Optimal cycles for low-dissipation heat en-gines”, which contains Refs. [79–90].

[50] T. Schmiedl and U. Seifert, EPL (Europhysics Let-ters) 81, 20003 (2007).

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[52] F. Hiai and D. Petz, in Introduction to Matrix Analy-sis and Applications (Springer International Publish-ing, 2014) pp. 101–135.

[53] K. Brandner, M. Bauer, and U. Seifert, Phys. Rev.Lett. 119, 170602 (2017).

6

[54] Y.-H. Ma, D. Xu, H. Dong, and C.-P. Sun, Phys.Rev. E 98, 022133 (2018).

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[63] We note that because of the degeneracies in theHamiltonian of the WS the steady state might not beunique (as the population in each degenerate energystate might depend on the initial state). However, this

is is irrelevant as we are considering a driving G ∝ G,where the power output depends only on the totalpopulation in each degenerate energy level.

[64] M. E. Fisher and M. N. Barber, Phys. Rev. Lett. 28,1516 (1972).

[65] M. Suzuki, Progress of Theoretical Physics 58, 1142(1977).

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[69] K. Huang, Introduction to statistical physics (Chap-man and Hall/CRC, 2009).

[70] Other possible implementation-dependent limitationsmay be the cost of turning on/off the interactionwith the baths [72, 91], and a non-zero time for thequenches.

[71] E. Mascarenhas, H. Braganca, R. Dorner, M. Fran caSantos, V. Vedral, K. Modi, and J. Goold, Phys. Rev.E 89, 062103 (2014).

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[76] J. Jaramillo, M. Beau, and A. del Campo, New Jour-nal of Physics 18, 075019 (2016).

[77] W. Niedenzu and G. Kurizki, New Journal of Physics20, 113038 (2018).

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[79] R. Alicki, Journal of Physics A: Mathematical andGeneral 12, L103 (1979).

[80] J. Anders and V. Giovannetti, New Journal of Physics15, 033022 (2013).

[81] T. D. Kieu, Phys. Rev. Lett. 93, 140403 (2004).[82] S. Vinjanampathy and J. Anders, Contemporary

Physics 57, 545 (2016).[83] M. Campisi, S. Denisov, and P. Hanggi, Phys. Rev.

A 86, 032114 (2012).

[84] T. V. Acconcia, M. V. Bonanca, and S. Deffner, Phys-ical Review E 92, 042148 (2015).

[85] M. F. Ludovico, F. Battista, F. von Oppen, andL. Arrachea, Phys. Rev. B 93, 075136 (2016).

[86] H.-P. Breuer, F. Petruccione, et al., The theory ofopen quantum systems (Oxford University Press onDemand, 2002).

[87] T. Albash, S. Boixo, and D. A. Lidar, N. J. Phys 14,123016 (2012).

[88] R. Dann, A. Levy, and R. Kosloff, Phys. Rev. A 98,052129 (2018).

[89] M. Yamaguchi, T. Yuge, and T. Ogawa, Phys. Rev.E 95, 012136 (2017).

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95, 032139 (2017).[92] G. E. Crooks, (2018).

7

SUPPLEMENTAL MATERIAL

This Supplemental Material (SM) contains a generalisation of the results presented in the main text to openquantum systems, as well as technical details. We start in Sec. I with a review the quantum Carnot cyclein the quasistatic limit. In Sec. II we introduce finite-time corrections for open quantum systems dynamicsand show to characterise them geometrically. In Sec. III we generalise the main result (the optimal cycleand corresponding maximal power) for general open quantum system dynamics by giving the solution as themaximisation of a scalar function. In Sec. IV, we present the maximisation of the heat capacity discussed inthe main text. In Sec. IV we discuss the technical details of the asymptotic expansions of power and efficiencypresented in the main text. Finally, in Sec. VI we solve analytically an illustrative solvable case. To help thereader find information in this Supplemental Material we introduced Table I.

Section I Description of a quasistatic, reversible Carnot cycle for a general quantum system.Section II Introduction of finite-time Carnot cycle and thermodynamic length.IIA Thermodynamic metric for exponential relaxations of the observables.IIB Thermodynamic metric from microscopical models.IIB1 Example: Qubit with bosonic baths.IIB2 Example: Harmonic oscillator with bosonic baths.IIB3 Example: 3-level system with detailed balance evolution.Section III Generalizations of cycle optimization to general metrics and protocols.IIIA Generalization for asymmetric dissipation and symmetric protocols.IIIA1 Optimization for systems with point-dependent thermalization timescale.IIIA2 Optimization for general metrics.IIIA3 Solution of the optimization for the models presented in Section IIB1-2-3.IIIB Bounds on completely asymmetric protocols.Section IV Maximization of the heat capacity for systems with different degree of control.Section V Time-tuning optimization of a low-dissipation Carnot engine.VA Critical scalings of power, efficiency and fluctuations of optimal protocols.VB Comparison with Otto cycle of Ref.[36]Section VI Explicit analytical control protocol for a D − 1 degenerate model.

Table I. Guide table for the Supplemental Material.

I. QUASISTATIC QUANTUM CARNOT CYCLE

For completeness, in this first section we review the quantum Carnot cycle in the quasistatic limit (i.e. withoutincluding finite-time corrections).

The internal energy of a system with Hamiltonian H in the state ρ is defined as

U = Tr[ρH] . (17)

Considering the variation dU , it is possible to identify [79–82] the work and heat contributions

dW = Tr[ρ dH] , (18)

dQ = Tr[dρ H] . (19)

To simulate sensible restraints on the system, external control is assumed on dynamical parameters of the local

Hamiltonian of the system Ht = H(~l(t)). When in contact with a reservoir at temperature T = 1/β (we useunits in which the Boltzmann’s constant is kB = 1), such a system relaxes to the the Gibbs state

ωβ(~l) = e−βH/Zβ(H) (20)

(here Zβ(H) = Tr[e−βH ] is the partition function of the system in the canonical ensemble).A Carnot Cycle [14, 17, 57, 82] is identified with a 4 steps process, that is two isothermal strokes alternated

with two isoentropic (adiabatic) strokes (cf. Fig. 2). Consider a system with a controlled Hamiltonian Ht whichcan be coupled independently to two reservoirs with temperature Th > Tc. In the ideal quasistatic limit the

operations are performed slowly enough to allow the system to be in thermal equilibrium ρ(t) ≡ ωβ(~l(t)) atevery instant. The 4 steps are:

1) while being coupled to the cold reservoir, the Hamiltonian is modified continuously from H(X) to H(Y )

such that Tr[ωβHt] is negative, in order for heat to be released to the cold source.

2) with the system isolated from the reservoirs, a quench is performed taking H(Y ) → H(Y ) ThTc

.

8

Figure 2. Pictorial, bidimensional representation of a Carnot cycle, in the ”phase space” defined by its state andHamiltonian: two isothermal strokes, where the Hamiltonian is modified and the system approximately follows theGibbs state, alternated with two rapid quenches where the system keeps the same state.

3) while being coupled to the hot reservoir, the Hamiltonian is modified continuously fromH(Y ) ThTc

toH(X) ThTc

.

4) again isolating the system a quench is performed to restore H(X) ThTc→ H(X) Th

TcTcTh

= H(X).

Note the factors ThTc

and TcTh

are chosen in order for the state to be continuous during the quenches. In fact

the thermal state uniquely depends on βHT (cf. (20)); i.e. for example during the quench 2) the relationβcH

(Y ) = βhH(Y ) Th

Tcguarantees ωβc(H

(Y )) = ωβh(H(Y ) ThTc

). We shall thus define then the adimensional

Hamiltonian at temperature 1/β (note that the temperature takes only two values, that depend on the respectiveisotherms)

Gt′ := βHt′τ (21)

so that the thermal state is ω′t = e−Gt′/Tr[e−Gt′ ] on both the cold and hot isotherm. The time reparametrizationis conceived in order to isolate the shape of the control Gt′ , with 0 ≤ t′ ≤ 1. Note that G (and hence ω) iscontinuous also on the quenches; that is, we can consider the cold isotherm consisting in a transformation(ω(X), G(X)) → (ω(Y ), G(Y )) with duration τc and the hot isotherm the opposite (ω(Y ), G(Y )) → (ω(X), G(X)),with duration τh. In a Carnot cycle heat is exchanged only during the 1), 3) steps (absorbed from the hotsource, released to the cold one), hence we can compute the efficiency using the observation that over a cycle

∆Q+ ∆W = 0 and the heat is absorbed from the hot bath Q(0)abs = Q

(0)h (we use the superscript(0) to indicate

quantities in the quasistatic regime)

ηCarnot =−∆W (0)

Q(0)abs

= 1 +Q

(0)c

Q(0)h

= 1− TcTh

, (22)

where we have used in the last step that in the quasistatic limit the above mentioned considerations togetherwith Eq. (19) give

Q(0)c

Tc=

∫ Y

X

Tr[dω G] = −∫ X

Y

Tr[dω G] = −Q

(0)h

Th. (23)

II. FINITE-TIME CARNOT CYCLE AND THERMODYNAMIC LENGTH

We now consider finite-time corrections on the above quasistatic Carnot cycle through a quantum open systemapproach, where the dissipation in linear response can be described in a geometric form by following [28]. Weconsider an isothermal process where the Hamiltonian H(t) of the WS is driven for t ∈ [0, τ ] in contact witha thermal bath at temperature T = 1/β. In order to characterise the process beyond the quasistatic limit, weneed to assume some structure on the thermalization processes of the working substance (WS) induced by the

9

reservoirs. In a rather generic scenario we consider that the relaxation of the WS can be described by a masterequation with the thermal state as a unique fixed point:

d

dtρ = L[ρ] s.t. L[ωβ(H(t))] = 0. (24)

where we note that L ≡ L(t) is also time-dependent. Following [14], the solution of this master equation canbe found perturbatively around 1/τ . Using the renormalised time s ∈ [0, 1] with the convection Hs ≡ H(sτ)and ρs ≡ ρ(sτ), the solution is given by (indicating with a dot ˙ the time derivative w.r.t. s):

ρs = ωs +1

τL−1s [ωs] + ... (25)

with ωs = e−βHs/Tr(e−βHs), and where L−1s is the inverse of Ls within the traceless subspace of density

matrices, the so called Drazin inverse (see e.g. [14, 17, 28, 92]). Plugging this expression into Q =∫ 1

0ds Tr(ρH)

and using integration by parts we obtain:

Q = T∆S − T

τΣ (26)

with the first order correction to the quasistatic limit given by:

Σ = β

∫ 1

0

ds Tr(L−1s [ωs]Hs) (27)

In terms of the adimensional Hamiltonian Gs ≡ βHs, and using the formula for the derivative of an exponential,

ω = −∫ 1

0

dx ω1−x(G− Tr(ωG))ωx ≡ Jω[G], (28)

we can write the variation of entropy as

∆S = −∫ 1

0

dsTr[Jωs [Gs] lnωs] =

∫ 1

0

dsTr[GsJωs [Gs]] , (29)

while Σ can be reexpressed in the convenient quadratic form:

Σ = −∫ 1

0

dsTr[GsL−1s [Jωs [Gs]]] . (30)

Now, expanding Gs as

Gs =∑j

λj(s)Xj , (31)

we can conveniently write ∆S and Σ as

∆S =∑i

∫ 1

0

ds siλi , Σ =∑ij

∫ 1

0

ds λimij λj (32)

with

si = Tr[GsJωs [Xi]] and mij = −1

2

(Tr(XiL−1

s Jωs [Xj ] +XjL−1s Jωs [Xi])

). (33)

The matrix mij is symmetric, positive-definite due to the second law dΣ ≥ 0, and it depends smoothly on thebase point ω; hence it defines a metric.

Finally, we note that the linear expansion (32) can also be obtained in exact treatments of the system-bathHamiltonian dynamics through linear-response theory [25, 83–85]. As a rule, the expansion (32) can be arguedto be general for any system with dissipations that are linear (at the lowest order) in the speed of the driving.Suppose indeed that the dissipation along an infinitesimal segment of the trajectory depends only on the localpoint and the local driving. As a consequence it must be in the form dΣ = dλifi(λ, λ, λ, ...), but the 1/τ scalingimplies that the first derivative terms enter linearly in the product while higher orders are suppressed, whichimplies dΣ = dλimij(λ)λj , which is equivalent to (32).

10

A. Thermodynamic metric for single or multiple time scales

We first show how to obtain the Kubo-Mori-Boguliobov metric used in the main text (Eqs. (4) and (8)). Thiscan be easily done by taking an exponential relaxation of ρ to equilibrium with a single time-scale, as describedby the Lindbladian:

Ls[ρs] = τ−1eq (ωs − ρs) (34)

which has the Drazin inverse L−1s [.] = τeq(ωsTr(.)− I). In this case, using that Tr(Jω[G]) = 0 one finds that Σ

in (30) is given by:

Σ = τeq

∫ 1

0

dsTr[Gs[Jωs [Gs]]] . (35)

that is Eq. (8) from the main text, as the generalised covariance is given by cov(A,B) = Tr[A[Jωs [B]]]. Wehence see that the standard thermodynamic length (Eq. (4) in the main text) can be obtained by an heuristicmodel of thermalisation with a single time-scale.

It is important to keep in mind that these considerations are only relevant close to equilibrium, where themetric (4) in the main text becomes a good thermodynamic description. Furthermore, although we have assumedthat the whole state ρ converges to equilibrium as in (34), strictly speaking it is only necessary that the driven

observables (the Xj in (31)) converge to equilibrium with a single time-scale, i.e., 〈Xi〉ρ = τ−1eq (〈Xi〉ω − 〈Xi〉ρ),

with 〈X〉ρ = Tr(Xρ). This is especially relevant in complex systems (e.g. many-body systems), where the fulldissipative dynamics can be extremely complex but the equilibration of some macroscopic observables can bewell described by an exponential relaxation with a suitable time-scale. In this sense, it is also worth pointingout that if each generalised observable Xi decays with a different time-scale τi

〈Xi〉ρ = τ−1i (〈Xi〉ω − 〈Xi〉ρ) , (36)

then the metric Eq. (4) of the main text can be easily generalised as [28]:

mij =τi + τj

2

∂2

∂λi∂λjlnZ, (37)

where we have absorbed the dependence on τi in the metric and where τi can in principle depend on the pointof the trajectory.

As a final remark, we note that while we have derived the metrics (35) and (37) with an heuristic approachbased on exponential relaxation near equilibrium, one can derive them from a microscopic derivation basedupon linear-response [24, 25], in which case τi will in general depend on the point of the trajectory, i.e., G. Thiscase will be treated in Sec. III A 2.

B. Thermodynamic metric on standard microscopical models

In this section, we apply our general considerations to derive thermodynamic metrics for systems describedby quantum master equations. Before, let us discuss here some generic properties. Following [14, 86], we havein standard scenarios where a quantum system is coupled to a bosonic bath:

Ls[ρs] =∑ν>0

γ0να(

(N(βν) + 1)DAν [ρs] +N(βν)DA†ν[ρs])

(38)

where γ0να is the spectral density of the bath (α defines the Ohmicity),

DX [ρ] = XρX† − 1

2(X†Xρ+ ρX†X) (39)

and

N(βν) =1

eβν − 1(40)

is the Bose-Einstein distribution (one can consider ferimonic baths by replacing N(βν) by the Fermi distribu-tion). The general form (38) of L depends on Hs, β and the spectral density J(ν) = γ0ν

α. Consider nowa Carnot cycle with two baths at temperature βc and βh, and the action of each bath being described by aL(HS , β, J(ν)). Assuming that the spectral density of both baths is the same, then both Lindbladians are

11

related by the transformation β → λ−1β and HS → λHS , with β ≡ βc and λ ≡ βc/βh. In terms of theLindbladian, note that [14]:

L(λHS , λ−1β) = λαL(HS , β), (41)

which shows how the generators of the dynamics are related between the cold and hot isotherm, i.e.

L((βc/βh)HS , βh) =

(βcβh

)αL(HS , βc) . (42)

After noticing that the dissipation is related to L through Eq. (30), we can write a simple proportionalityrelation for the metric describing the dissipation on the hot and cold isotherm,

m(h)ij =

(TcTh

)αm

(c)ij , (43)

which implies, for time-reversal symmetric protocols described in the main text,

Σh =

(TcTh

)αΣc . (44)

While from this relation we see that the considerations of the main text (for symmetric protocols Σc = Σh) inprinciple only apply for flat spectral densities α = 0, we will show in Sec. III A that the same figure of merit∆S2/Σc (or equivalently ∆S2/Σh) needs to be maximised to obtain maximal power.

An important comment is now in order. Whereas the dissipator (38) is usually derived for time-independentHamiltonians [86], here we are interested in slowly driven Hamiltonians. Nevertheless, the same form forthe dissipator is justified as long as the bath dynamics are fast compared to the driving rate of the systemHamiltonian, which leads to the well-known adiabatic master equation [87–89].

1. Two-level system with a bosonic bath

The well-known optical master equation [86] can describe a two-level system with HamiltonianG ≡ βH = w2 σ

z

relaxing in a bosonic thermal bath at temperature 1/β, and from (38) and working in the interaction picture ittakes the form

ρ = Γ(1 +N(w))(σ−ρσ+ − 1

2{σ+σ−, ρ}

)+ ΓN(w)

(σ+ρσ− − 1

2{σ−σ+, ρ}

), (45)

where the rate Γ ∝ wα depends on the ohmicity of the bath (for a standard atomic-optical field interactionα = 3 [86]) and σ± = (σx± iσy)/2. It is easy to translate this equation on the single Bloch-vector components

d

dt〈σx〉 = −Γ(2N + 1)

2〈σx〉 , (46a)

d

dt〈σy〉 = −Γ(2N + 1)

2〈σy〉 , (46b)

d

dt〈σz〉 = −Γ(2N + 1)〈σz〉 − Γ . (46c)

This equations being in the form (36), imply that the thermodynamic metric is indeed in the form (37).Moreover the covariances of different cartesian components decouple as cov(σi, σj) ∝ δij (with i, j,= x, y, z)hence implying

mij =1

Γ(2N(w) + 1)

2cov(σx, σx) 0 00 2cov(σy, σy) 00 0 cov(σz, σz)

(47)

Consider now the external control on the qubit, i.e. (n(t) is a unit vector)

G(t) ≡ βH(t)w(t)

2n(t) · ~σ. (48)

Assuming that the driving is much slower than the internal dynamics of the bath (i.e. the adiabatic masterequation [87–89]), the same master equation (45) and metric (47) instantaneously holds in the rotated basiswhere σz(t) = n(t) · ~σ and with ω → ω(t) becoming time dependent. The same metric is expressed in polarcoordinates in [28].

12

2. Harmonic oscillator with a bosonic bath

As in the previous example, the optical master equation can be written similarly for an harmonic oscillatorwith gap control βH(t) = w(t)a†a (again working on the interaction picture and assuming an adiabatic masterequation)

ρ = Γ(1 +N(w))(aρa† − 1

2{a†a, ρ}

)+ ΓN(w)

(a†ρa− 1

2{aa†, ρ}

). (49)

Considering a modulation of the level splitting w, we have for the average occupation number

d

dt〈a†a〉 = Γ(N(w)− 〈a†a〉) (50)

Thus the metric for the single control parameter w(t) is in this case simply

mww =1

Γ. (51)

Note that a frequency-controlled harmonic oscillator can be realized in a different set up, i.e.

βH(t) =mw(t)2X2

2+P 2

2m(52)

(to see how this control is not equivalent to (49) it is sufficient to notice [H,H] 6= 0 in general). The metricarising with this control is being worked out in [90].

3. Three-level system with a bosonic bath

For a final example we consider here a three-level system satisfying a detailed balance master equation. Weassume control on each of the energy levels, while keeping the basis fixed (this is motivated by the observationthat creating coherence does not increase power in the linear-response regime [78]). Without loss of generalitywe can assume the ground state energy to be zero (βE0 ≡ 0) and the two excited states with energies β−1E1(t) ≤β−1E2(t), hence the thermal state being characterized by a probability vector

~ω(t) =1

1 + e−E1 + e−E2

1e−E1

e−E2

(53)

The evolution of the probability vector is described by the Markovian master equation [86]

pi =∑j

Γijpj . (54)

The rate matrix Γij can be obtained from (38), yielding

Γi<j = Γ(N(Ej − Ei) + 1)(Ej − Ei)α

Γi>j = ΓN(Ej − Ei)(Ej − Ei)α

Γii = −∑j 6=i

Γji (55)

where Γ is a rate, N(w) = 1/(ew−1) for bosonic baths, and α defines the spectral density of the bath. Note thatthese dynamics cannot be described by the simple exponential decay (36), so that we need to use the more general

approach (33) to find the metric. Indeed, we use Eq.(33), with a simplified version of Jω[G] due to commuting

operators inside it (i.e. in this case Eq.(28) corresponds to −∫ 1

0dx ω1−x(G − Tr(ωG))ωx = −Gω + ωTr[Gω]),

obtaining

Σ =∑ij

EimijEj with mij = −∑k

Γ−1ik ∂jωk (56)

where Γ−1ij is the Drazin inverse of Γij (see e.g. [14, 17, 28, 92] for details on the Drazin inverse) while ∂jωk is

the variation of ωk due to Ej , i.e. ∂jωk = −ωkδjk +ωjωk. The analytic form of mij can be computed from thisexpression even for larger systems with more than 3 levels using symbolic computation software.

13

III. GENERALISATION OF THE MAIN RESULT: OPTIMAL CYCLES FOR GENERALMETRICS AND ASYMMETRIC DISSIPATIONS

We now have the necessary tools to generalise the optimisation to more generic metrics and thermodynamicprocesses. In the most general case, we can consider the Carnot cycle where the cold isotherm is characterised

by the control parameters {λ(c)j (s)} and the hot one by {λ(h)

j (s)} in adimensional time units (recall s ∈ [0, 1]),

as well as the durations τc, τh. For time-reversal cycles considered in the main text, {λ(c)j (s)} = {λ(h)

j (1 − s)}(but remind that in general τc 6= τh).

We first note that under optimization of a very general class of processes where m(h)ij and m

(c)ij are simply

proportional to each other (this class includes e.g. all the standard microscopic scenarios with same spectraldensity of the baths, as showed in II B), the time-reversal property is not an assumption but simply followsfrom the optimization process [14]. This implies, as we show in the following III A, that the relevant figure ofmerit to be maximized is still ∆S2/Σ. Therefore, we show how to accomplish this task for different structures

of the metric mh,cij , up to the most general case where the dependence on the temperature, spectral density, etc.

is encoded in L−1 in the expression (33).

Finally in Section III B we will consider generalizations to cases where m(h)ij and m

(c)ij differ significantly, so

that {λ(c)j (s)} 6= {λ(h)

j (1− s)}.

A. Simple asymmetric dissipations

We consider here cases in which the dissipations along the hot isotherm are proportional to the dissipationsalong the cold isotherm via a constant factor that does not depend on the specific control protocol:

Σc = σΣh ≡ σΣ , (57)

where σ is a number. Importantly this class of cases includes the scenario where the two baths have the same

(non-flat) spectral density, as explained in Sec. II B and in [14], where we obtained that m(c)ij = (Th/Tc)

αm(h)ij

(here m(c)ij (m

(h)ij ) is the metric associated to the dissipation of the cold (hot) bath), which implies Σc =

(Th/Tc)αΣh.

Starting from (57), the power to maximize is given by

P =(Th − Tc)∆S − Σ(Thτh + σTc

τc)

τh + τc(58)

under the efficiency constraint

1−Tc(1 + σΣ

∆Sτc)

Th(1− Σ∆Sτh

)= γ

(1− Tc

Th

)(59)

As the only time unit is Σ/∆S ≡ τ∗ and the temperature ratio r = TcTh

the problem can be rephrased as

Th∆S

τ∗maxxc,xh

[(1− r)− ( 1

xh+ rσ

xc)

xh + xc

]with 1− r1 + σ/xc

1− 1/xh= γ(1− r) (60)

where xj = τj/τ∗. Here the maximization with its constraint is expressed in full adimensional terms, meaning

that after solving it the resulting power will be in the form Th(∆S)2

Σ f(γ, r, σ) for some scalar function f . Thegeneral form of the function f can be obtained analytically but it is in general non-trivial and very lengthy (sowe do not show it here), but in the limit of high efficiency it simplifies to

P(max)γ≈1 =

Th∆S2

Σ

(1− r)2

r(1 +√σ)2

(1− γ) +O((1− γ)2) . (61)

At this point the optimisation of the power boils down to the maximisation of (∆S)2/Σ, as in the main text,. Inthe following we show how to maximise ∆S2/Σ in general scenarios, and prove that asymptotically infinitesimalcycles are optimal.

Before, let us note that given any reasonable figure of merit between efficiency and power, the relevantfigure of merit will always be (∆S)2/Σ. Given as objective any figure of merit f(η, P ) expressed in terms ofthe efficiency and the power, after optimization on τh, τc the resulting value f := f(η, P ) (we indicate with ¯

quantities after time optimization) will be given, by η = η(Th, TC , σ) and P = P (Th, Tc, σ)(∆S)2/Σ for some

14

adimensional function η and homogeneous function P of the temperatures, by dimensional analysis. Moreover,any reasonable f will be monotonously increasing in both its parameters singularly (i.e. for a given efficiencywe wish to enhance the power and vice-versa), therefore after speed optimization it will still be possible toimprove the engine performance by increasing the ratio (∆S)2/Σ. The role of the above mentioned ratio as acharacteristic scale defining the performance of both engines and refrigerators was already noticed in [6] withoutfurther analysis.

1. Dependence of the time scale in G

As a simple first extension of our results, we consider the case of the standard thermodynamic metric witha time-scale that can depend on the point of the parameter space G = βH. In other words, we considerthe heuristic model of exponential thermalisation (34) with a G-dependent τeq (see also [25] for a microscopicderivation). Following the reasoning of the main text, we can define an infinite-dimensional scalar product givenby

〈A,B〉ω ≡∫ 1

0

ds covωs(A,B) (62)

with ωs = e−Gs/Tr(e−Gs), and where we note that the scalar product is defined for a fixed trajectory Gs withs ∈ [0, 1]. We then have,

(∆S)2

Σ=

(−∫ 1

0ds covω(G, G)

)2

∫ 1

0ds covω(G, G)τeq

=

(〈τ−1/2G, τ1/2G〉ω

)2

〈τ1/2G, τ1/2G〉ω≤ 〈τ−1/2G, τ−1/2G〉ω =

∫ 1

0

dsCτeq

, (63)

so we conclude that the results of the main text can be naturally extended to time-scales that depend onβH, meaning that the performance is upper-bounded by small cycles with proportional modulation of theHamiltonian H ∝ H performed on the point where the ratio between heat capacity and relaxation time of thesystem is maximum.

2. General metrics

Let us consider the maximisation of ∆S2/Σ for more general protocols where the thermodynamic metric isgiven by the general expression (33) which depends on the particular Lindbladian describing the thermalisationdynamics. Following the considerations of the main text, to maximise (∆S)2/Σ, we can expand them as

Gs =∑j

λj(s)Xj ⇒ ∆S =∑i

∫ 1

0

siλi (64)

with si = Tr[(Tr[Gω]ω −Gω)Xi], and

Σ =∑ij

∫ 1

0

dt λimij λj . (65)

In contrast to the main text, where mij is given by Eq. (4) in the main text, here we a general metric of theform (33). The Cauchy-Schwarz inequality can be applied by considering the following:

Inequality. Consider two vectors ~a,~b and a quadratic invertible form g > 0 defined on their vector space.

Then the standard C-S inequality applied to g1/2~a and g−1/2~b tells

(~a ·~b)2 =((g1/2~a) · (g−1/2~b)

)2 ≤ |g1/2~a|2|g−1/2~b|2 = (~aT g~a)(~bT g−1~b) . (66)

If we now consider this inequality applied to two vectors

sj(t), λj(t) j = 1, . . . , k 0 ≤ t ≤ 1 , (67)

and the metric

git,jt′ = mij(t)δ(t− t′) , (68)

where m(t) is a positive time-dependent quadratic form in Rk , we have

g−1 = m−1ij (t)δ(t− t′) , (69)

15

and thus it is possible to write the C-S inequality as(∫ 1

0

sj(t)λj(t)

)2

≤(∫ 1

0

si(t)m−1ij (t)sj(t)

)(∫ 1

0

λi(t)mij(t)λj(t)

). (70)

This inequality allows us to bound the figure of merit for the maximum power as

(∆S)2

Σ≤ max

λ

∑ij

sim−1ij sj . (71)

where we recall that si = Tr[(Tr[Gω]ω − Gω)Xi], and the Xi are given by the expansion (64). Hence, theinitial optimisation over all possible trajectories {λj(s)} boils down to maximising a single real function giventhe control parameters. In other words, our result shows that in order to design optimal finite-time Carnotcycles one does not need to optimise over all trajectories {λj(s)} with s ∈ [0, 1], but it is enough to search asuitable point in the thermodynamic space {λj}. To saturate it in practice, one needs to maximise ~sm−1~s over

the control parameters ~λ,

λ∗ = argmax

∑ij

sim−1ij sj

, (72)

and consider infinitesimal variations around this optimal point:

~λ = ~λ∗ + εt~µ, ε� 1, ~µ = m−1~s , (73)

where we have used the more compact vector notation: ~s = {s1, s2, ...}, etc. The direction of the modulationsis defined, from the Cauchy-Schwarz saturation condition, by the vector m−1~s, while the normalization of themodulation is in principle to be taken infinitesimal. The abstract expression for ~µ corresponds, in the case of

the analysed in the main text, to ~λ∗ itself, i.e. the modulations of the control parameters are proportional tothe control themselves in that case.

3. Examples

a. Qubit with optical master equation. We first consider here a slowly driven qubit with full hamilto-nian control (here G is the adimensional Hamiltonian in temperature energy units)

G(t) =w(t)

2n(t) · ~σ (74)

in contact with a bosonic bath, so that the evolution is described by the adiabatic master equation (in theinteraction picture)

ρ = Γ(1 +N(w))(σ−ρσ+ − 1

2{σ+σ−, ρ}

)+ ΓN(w)

(σ+ρσ− − 1

2{σ−σ+, ρ}

)(75)

Here, σ± ≡ σ±(t) = σx(t)± σy(t) in the basis where σz(t) = n(t) · ~σ, and ω ≡ ω(t), as in Section II B 1. Giventhe metric (47) and considering that the variation of entropy is non-zero only along the instantaneous z directionn(t) it is evident how, to maximize ∆S/Σ2, changing the direction of n is useless (as noted in [28], generationof coherences is detrimental, and the eigenvectors of mij (47) are larger along the tangential directions x andy), hence the optimal trajectories can be recognized to be in the form of simple gap modulation w(t), reducingthe metric to

mww =1

Γ(2N(w) + 1)cov(σz, σz) , implying Σ =

∫cov(G, G)

Γ(2N(w) + 1). (76)

Therefore we fall into the category of systems analysed in Section III A 1, for which the optimal control consistsin modulations on the point where the ratio between the heat capacity Varω(G) and the relaxation timescaleis maximum. In this case τeq ≡ Γ−1(2N(w) + 1)−1 and CQ = w2e−w/(1 + e−w) − (we−w/(1 + e−w))2. InFig. 3 we show the maximization of CQΓ(2N(w) + 1), corresponding to the maximum value of (∆S)2/Σ in thiscase, choosing Γ constant (i.e. flat spectral density, α = 0). Allowing α 6= 0 would not change significantly thedifficulty of the maximization problem, here α = 0 has been chosen to allow a comparison with the case studiedin the main text, where the thermalization timescale does not depend on the value of the Hamiltonian. To sumup, by making use of equation (61), the full maximization of power for a qubit in contact with bosonic thermalbaths, leads to

P(max)γ≈1 = max

w{CQwα(2N(w) + 1)} Γ(Th − Tc)2

Tc(1 +√

(Th/Tc)α)2(1− γ) +O((1− γ)2) (77)

where the maximization result depends on α.

16

0 2 4 6 8 10

0.0

0.1

0.2

0.3

0.4

0.5

Figure 3. Maximization, as function of the gap w of the (adimensional) heat capacity of a Qubit divided by thethermalization scale τeq ≡ Γ−1(2N(w) + 1)−1, in units of Γ. A comparison can be made with the maximization of theheat capacity alone.

0 2 4 6 8 10

0.0

0.5

1.0

1.5

Figure 4. Similar maximization for an harmonic oscillator attached to a bath of spectral density α = 3.

b. Harmonic oscillator The above analysis for the qubit can be replicated thoroughly for the controlledharmonic oscillator described in II B 2, and the final result

P(max)γ≈1 = max

w{CHOwα}

Γ(Th − Tc)2

Tc(1 +√

(Th/Tc)α)2(1− γ) +O((1− γ)2) . (78)

where heat capacity of the harmonic oscillator is equal to CHO = (2 sinh(w/2))−2. In Fig. 4 an example is shownfor α = 3.

c. 3-level system with master equation Finally we consider a 3-level system as described in II B 3,controlled by the energy levels of the two excited states E1 and E2, and whose evolution is given by a standarddetailed balanced master equation (54)-(55). By following the general approach described in III A 2, we bound∆S2/Σ by

∑ij sim

−1ij sj , see (71). We symbolically compute the metric (56) and the vector {si} (see details

in Sec. II B 3) in order to express∑ij sim

−1ij sj as a function of {E1, E2}, and then maximise it over {E1, E2}

numerically. An example is shown in Fig. 5, where we plot∑ij sim

−1ij sj for a linear spectral density α = 1.

Note that the maximum is obtained for E1 = E2. We also find that in this case the vector ~µ (73) is proportional

to

(11

)on the bisector, meaning the modulation on the optimal working point are symmetric on E1 and E2.

The maximum power results in such a case

P(max)γ≈1 ≈ 2.8

Γ(Th − Tc)2

Tc(1 +√

(Th/Tc))2(1− γ) +O((1− γ)2) . (79)

Furthermore, in Fig. 4 (right), we also compare∑ij sim

−1ij sj with the heat capacity C divided by the

characteristic timescale τeq = Γ−1w−1 (but remember the thermalization process is not simply exponential inthis case), and it can be seen how this ratio provides a rather good approximation of the maximal power. Thatis, while C/τeq is not exactly the correct figure of merit in this case, it can be used as a relevant figure of merit

17

0 2 4 6 8 10

0.0

0.5

1.0

1.5

2.0

2.5

Figure 5. (Left) The value of∑ij sim

−1ij sj (which bounds (∆S)2/Σ, as shown in (71)), for the 3-level system described

in II B 3, with α = 1. (Right) Plot of the bound compared to the maximization of the heat capacity multiplied by therate Γw (in units of Γ), on the bisector E1 = E2.

to determine where the optimal power is. This example also illustrates that our approach can be used to dealwith more complex dissipative dynamics than the standard exponential decays discussed in the main text.

B. General asymmetric protocols

Finally, we consider the most general setting (time-reversal asymmetric protocol, different baths, genericdependence on the temperature of L): Σc 6= Σh. Then, the maximum power expression after isotherms-durationtuning (with no efficiency constraints) is given by [9]

P (max) =(∆S)2(√

TcΣc +√ThΣh

)2 (Th − Tc)2

4(80)

which can be thus written as from Eq.s (30)-(33)

(∑j

∫ 1

0dt sj λj)

2(√Tc∑ij

∫ 1

0dt λim

(c)ij λj +

√Th∫ 1

0dt λim

(h)ij λj

)2

(Th − Tc)2

4, (81)

where now two different metrics m(j) appear in the denominator. In this case we lose the simple structure(scalar product)2/(quadratic form), hence it is not possible to apply directly the C-S inequality; nevertheless itis possible to upper-bound Eq. (81) using simple inequalities e.g.

(81) ≤ maxx=c;h

[(∑j

∫ 1

0dt sj λj)

2

4Tx∑ij

∫ 1

0dt λim

(x)ij λj

](Th − Tc)2

4using

√a+√b ≥ 2 min(

√a,√b) ; (82)

(81) ≤(∑j

∫ 1

0dt sj λj)

2∑ij

∫ 1

0dt λi[Tcm(c) + Thm(h)]ij λj

(Th − Tc)2

4using

√a+√b ≥√a+ b . (83)

These can be now optimized using C-S as we did in the previous section, and depending on the relative size (in theinterval 1

3 ≤ab ≤ 3 the former is tighter, otherwise the latter) they will give a bound (not tight in general). Note

that while inequalities (82-83) are useful to give upperbounds to the maximum power theoretically obtainable,in practical terms it is also possible to give lower bounds, which are useful to certify that it is possible to reachat least a given value of the power, and maximizing it by the same methods we used in the main work. Namelythe bounds can be written

(81) ≥ minx=c;h

[(∑j

∫ 1

0dt sj λj)

2

4Tx∑ij

∫ 1

0dt λim

(x)ij λj

](Th − Tc)2

4using

√a+√b ≤ 2 max(

√a,√b) , (84)

(81) ≥(∑j

∫ 1

0dt sj λj)

2

√2∑ij

∫ 1

0dt λi[Tcm(c) + Thm(h)]ij λj

(Th − Tc)2

4using

√a+√b ≤

√2(a+ b) . (85)

18

As a final observation, we note how all of the above can be applied also to the power in the high efficiencyregime, which is (cf. Eq. (61))

P(max)γ≈1 =

∆S2(Th − Tc)2

Tc(√

Σh +√

Σc)2(1− γ) +O((1− γ)2) . (86)

IV. MAXIMIZING THE HEAT CAPACITY VS. DEGREE OF CONTROL

The optimal cycles described in the main text (induced by the standard thermodynamic metric (4)) make themaximal power proportional to the heat capacity (13) of the chosen working point of the infinitesimal Carnotcycle. The task of maximizing the power is then translated in finding the point in the control parameter spacethat maximizes the heat capacity, i.e. the variance of the adimensional Hamiltonian G for a thermal stateω = e−G/Tr[e−G]. We show in the following 3 paradigmatic cases that are summed up in Fig.6 showing howdifferent degrees of control offer different possible performance on the same system, which is made of N qubits.(a) Full control over the spectrum. We first assume total control on the Hamiltonian of the N qubits,

which is made up of D = 2N energy levels. That is, any desired (possibly long-range) interaction can beengineered, so that the D-level spectrum can be controlled at will. While this might be extremely challengingto realise in practice, it is useful to consider this situation to obtain a fundamental upper bound on the maximalpower. Indeed, the maximization of heat capacity C of a general D-dimensional system at thermal equilibriumhas been carried out in [30, 62]. The optimal Hamiltonian consists of a ground level and a D − 1 degeneratelevel, with an optimal gap x in adimensional units (i.e. rescaled by the temperature) defined by ex = (D −1)(x + 2)/(x − 2) and the corresponding C is Cmax = x2ex(D − 1)/(D − 1 + ex)2. This expression gives in theasymptotic regime (D →∞) x ' lnD, hence Cmax ' (lnD)2/4; which in terms of the particle number N ∝ lnDmeans Cmax ' N2/4 for N � 1, i.e., a quadratic scaling.

On the robustness of result (a). Suppose the control of the optimal energy gap x has some imprecision (orsimply needs to be modulated to perform the cycle). We show here that as far as the modulation/imprecisiondoesn’t scale with the dimension, the heat capacity behaves smoothly. Suppose indeed that x = (1+ε) ln(D−1).Then the adimensional variance of the flat Hamiltonian with d ≡ D − 1 degenerate excited states and gap x is

de−xx2

1 + de−x− d2e−2xx2

(1 + de−x)2= (ln d)2(1 + ε)2

( d−ε

1 + d−ε− d−2ε

(1 + d−ε)2

)= (ln d)2 (1 + ε)2

4 cosh2( ε2 ln d)(87)

which, to keep the scaling as (ln d)2 ∼ N2 needs the rest to stay finite, i.e. ε to scale as 1/ ln d = 1/N .(b) N independent qubits. As another extreme case, corresponding to almost no control on the spectrum,

we can consider N independent qubits, that is an Hamiltonian H(N) =∑Ni=1 λi(t)σ

zi . Note that given the

qubits do not interact the thermal heat capacity will result additive, and the optimal gap will be the same forall the qubits λi = λj . This means it is enough to solve the previous case for D = 2 (i.e. N = 1), to findCmax ≈ 0.439N (with an optimal gap λ∗i ≈ 2.40), in agreement with [17].

(c) Ising chain. Finally, we consider an Ising chain (H(N) = −λ1(t)∑Ni=1 σ

zi σ

zi+1 − λ2(t)

∑Ni=1 σ

zi ), and

assume control over λ1,2(t). Numerical results in Fig. 6 show how the interactions allow for substantiallyimproving on (b), although asymptotically we obtain a linear scaling in N , Cmax ' 0.59N for N � 1. Thislinear scaling is in fact expected for systems away of a phase transition point (e.g., a rigorous proof that C isextensive for translationally invariant gapped systems in the low-temperature regime is provided in AppendixA of [32]). The optimal controls are found to be λ∗2 ≈ 5, λ∗1 ≈ 8.

V. ASYMPTOTIC EXPANSIONS AND CRITICAL SCALING

Inspired by Ref. [9] we can find the best power for a given fixed efficiency, i.e. η = γηC = γ(1− Tc

Th

), which

in the symmetric low-dissipation regime means to fix (we identify ∆S ≡ ∆hS)

Qh +QcQh

=(Th − Tc)∆S − Σ(Thτh + Tc

τc)

Th(∆S − Σ/τh)= γ

(1− Tc

Th

), (88)

which relates τc and τh, after which it is possible to maximize the power

P =Qh +Qcτc + τh

=(Th − Tc)∆S − Σ(Thτh + Tc

τc)

τc + τh(89)

by enforcing ∂τxP = 0 (here τx can be either τc(τh) or τh(τc) via Eq. (88)), to obtain

P (max)γ =

∆S2

4Σ

(Tc − Th)2γ(1− γ)

Th(1− γ) + Tcγ(90)

19

Full control

Ising chain

Qubits array

1 2 5 10 20 50

0.5

1

5

10

50

100

Figure 6. Maximum heat capacity Cmax (corresponding to the maximum power P(max)γ in adimensional units), for a

system of N qubits with different degrees of control. We obtain asymptotically: Cmax ' 0.44N for N independent qubitswith a single control parameter, Cmax ' 0.59N for an Ising chain with two control parameters, and Cmax ' N2/4 for fullcontrol on the spectrum.

by choosing

τc =2Σ

∆S(ThTc− 1)

(1− γ), τh =

2Σ(ThTc

(1− γ) + γ)

∆S(ThTc− 1)

(1− γ). (91)

The correspondent work is

Wγ =∆S(Th − Tc)γ(Tc(1 + γ) + Th(1− γ))

2(Th + γ(Tc − Th)). (92)

A. Critical scaling

Now we consider the optimal engine described in the main text whose cycle consists of G(sτc) = (1+εg(s))G(0)with s ∈ (0, 1), ε � 1 but finite, and G(τc + sτh) = G((1 − s)τc). G is chosen appropriately to maximize theheat capacity given the allowed control. In the limit ε � 1 the shape g(s) becomes irrelevant as far as it is asmooth function with g(0) = 0 and g(1) = 1, which implies 〈g〉 = 1. This leads to

∆S ' ε covω(G,G) (93)

Σ ' ε2τeq covω(G,G) . (94)

We also consider that we scale up the engine with N while approaching γ → 1, by setting (using the notationfor critical exponents of a second order phase transition as in [35, 36])

1− γ = N−ξ, ξ > 0,

covω(G,G) = c0N1+α/(dν), α ≥ 0,

τeq = τ0Nz/d. (95)

Then, expanding the relevant quantities for N � 1, we obtain at leading order in N :

P =c0(Tc − Th)2

4τ0TcN1+α/(dν)−z/d−ξ +O(N1+α/(dν)−z/d−2ξ)

τc =2ε

ThTc− 1

τ0Nξ+z/d

τh = τc +O(1)

W = εc0N1+α/(dν)(Th − Tc) +O(N1+α/(dν)−ξ)

= (Th − Tc)∆S +O(N1+α/(dν)−ξ) (96)

20

Now we look at the fluctuations of work. The cycle consists of four processes: Two (quasi)-isothermalprocesses and two quenches. We note that as N →∞, we have that W → (Th − Tc)∆S, so that the isothermalprocesses become exact at leading order in N which implies that they become fluctuationless [29]; note that thisis expected as τc →∞ with N →∞, which makes the low-dissipation assumption also more and more exact aswe increase N . Secondly, regarding the two quenches: H → HTh/Tc and the inverse one H ← HTh/Tc; it iseasy to see that the work fluctuations are given by

σ2W = 2 (Th − Tc)2 [

Tr(ωG2)− (Tr(ωG)2]

= 2 (Th − Tc)2covω(G,G) , (97)

and thus

σ2W = 2 (Th − Tc)

W

ε. (98)

Hence we have that

fw =σWW∝ 1√

εW∝ 1

ε√N1+α/(dν)

. (99)

As explained in the main text, if we want to exploit the critical scaling in the phase transition of the system εmust satisfy ε ≤ N−1/(2−α). Then, using dν = 2− α [69], we obtain

fw ∝1

ε√N1+α/dν

=1

εN (1+α/(2−α))/2≥ N1/(2−α)

N1/(2−α)= 1 , (100)

which is the same result found in [36] for the Otto engine proposal. More details on the necessary criticalexponents, and on the fluctuations of the engine, are provided in the main text.

B. Comparison with Otto cycle

Let us check in more detail how our results compare to Ref. [35], where an Otto cycle is considered. Following[35], let us define the internal energy

UH(β) = Tr

(H

e−βH

Tr(e−βH)

)(101)

and the heat capacity

CH(β) = −β2 ∂UH(β)

∂β. (102)

The work output of a Otto cycle working between Hamiltonians λhH ← λcH for a fixed efficiency η = γηC ,∆η = ηC(1− γ) is given by

W = −λhηCγ [UH(βhλh + βcλh∆η)− UH(βhλh)] . (103)

Expanding for low ∆η which corresponds to λcHc ≈ λhHh, using (102) and keeping only leading terms in ∆ηwe obtain

W ≈ η2C

βcβ2h

(1− γ)γC , (104)

where we defined

C ≡ CλhHh(βh) ≈ CλcHc(βc) . (105)

It is important to note that the linear expansion (104) of (103) is only justified close to the phase transitionpoint (105), whose width scales as δ ∝ N−1/(dν). This sets an extra requirement on ∆η:

∆η = ηC(1− γ) ∝ N−1/(dν) (106)

The power is simply W/τ where τ is the time of the cycle, which involves two thermalization processes [35].Let us take τ = 2κτeq, where κ measures how exact the thermalisation process is (the error being exponentiallysmall with κ if one assumes a standard exponential relaxation). Then the power reads

POtto ≈1

2κτeqη2C

βhβ2c

(1− γ)γC . (107)

21

Let us now consider the Carnot cycle. After maximisation over (∆S)2/Σ we have shown in the main text that

P (max)γ =

C4τeq

(Th − Tc)2γ(1− γ)

γTc + (1− γ)Th. (108)

Expanding around γ ≈ 1 , we have

Pmaxγ≈1 ≈

κPotto

2, (109)

which is correct for a fixed γ close to 1.

VI. EXPLICIT PROTOCOL FOR A 2N − 1 DEGENERATE SYSTEM

For the sake of completeness, we show here how the results of the paper apply explicitly for a feasible drivingprotocol close to the optimal one, in the case of a D-level system with a ground state and D − 1 degenerateexcited states. As we explained in Sec. IV case (a), this Hamiltonian is motivated as it maximises the heatcapacity given a D-level system. For comparison purposes, we will sometimes use N satisfying D = 2N , so thatthe Hamiltonian can be thought of N suitably interacting particles. Note that the case N = 1 corresponds tothe driving of a qubit [17, 57].

Consider a D-level system in a thermal state at temperature 1/β with an engineered Hamiltonian with aground state and a d ≡ D − 1 degenerate excited states with gap E/β, i.e.

H(t) =E(t)

βΠ(d), Π(d) =

d∑i=1

|i〉〈i|, d ≡ 2N − 1, (110)

such that the thermal state is

ω(t) =|0〉〈0|+ Π(d)e−E(t)

1 + de−E(t)≡ q(t)|0〉〈0|+ (1− q(t))Π(d), (111)

where we defined the driving ground state population

q(t) =1

1 + de−E(t). (112)

Consider the dynamics given by the equation

d

dtρ(t) = Γ(ω(t)− ρ(t)), Γ ≡ 1

τeq(113)

(which induces the metric (4) of the main text in the context of continuous time dynamics, as explained inSection II A or Ref. [28]). We can then consider a solution in the form

ρ(t) = p(t)|0〉〈0|+ (1− p(t))Π(d) (114)

characterized in terms of the ground state probability, which by the slow-driving approximation [14] can besolved to be

p(t) = q(t)− 1

Γ

d

dtq(t) +

1

Γ2

d2

dt2q(t) + . . . (115)

that is, expressing quantities in terms of the adimensional time unit ρ(t) = ρ(sτ) (here τ is the duration of the

driving), and using the notation A = ∂sA = τ ddtA,

ρ =

∞∑i=0

ρ(j), ρ(j) = (−1)j1

τ jΓj∂(j)s ω (116)

The heat exchange can be computed at all the orders

βQ(j) =

∫ 1

0

Tr[βHρ(j)]ds =

∫ 1

0

d∑i=1

E(−τΓ)−j∂(j+1)s

(1− qd

)ds = −

∫ 1

0

E

(−τΓ)j∂(j+1)s q ds . (117)

22

Quasistatic

1st order

2nd order

0.0 0.2 0.4 0.6 0.8 1.0

0.46

0.48

0.50

0.52

0.54

Ideal max power

Approximated protocol

Full series protocol

10 20 30 40 50 60 70

0.5

1

2

Figure 7. (Left) Ground state probability evolution in adimensional units p(s), for the driving given in Eq. (118) forε = 0.1, Tc/Th = 0.9, γ = 0.9, N = 10. Different orders of approximation are compared.(Right) Extensive Power scaling of the heat engine as a function of its size N (the number of qubits it is composed of),obtained by the above explicit feasible driving of Eq. (118), with ε = 0.1, Tc/Th = 0.9, in natural units. The efficiencyapproximates the Carnot’s one as N grows γ = 1 − 1

N. The blue circles represent the maximal ideal bound, the yellow

triangles represent the performance achieved by the protocol using the slow-driving approximation, while the green ringsshow the correction to the full analytical series.

The point of maximum heat capacity that optimizes the power output of this system, as shown in Sec. IV case(a), is asymptotically E ∼ ln d which implies q ∼ 1

2 . We consider thus a modulation near this point, i.e. a

driving protocol in the form q(t) = 12

(1 + ε cos(πt/τ)

), with a small abuse of notation

q(s) =1

2

(1 + ε cos(πs)

), (118)

or equivalently E(s) = ln( dq(s)1−q(s) ). Notice here that we chose to use the letter ε instead of ε used in the main

text and in Eq. (87), because the latter multiplies the Hamiltonian, which scales linearly in N on the optimalpoint, while here the modulation of E(s) is of order ε with no pre-factor. Hence for scaling comparisons it isimportant to remember ε ∼ ε/N .

The total heat can be explicitly computed from (117)

βQ =

∫ 1

0

−E(q − q

τΓ+

...q

τ2Γ2+ . . .

)=

∫ 1

0

−E ε2

(− π sin(πs) +

π2 cos(πs)

τΓ+π3 sin(πs)

τ2Γ2+ . . .

)∫ 1

0

−E−πε2

(sin(πs)− π

τΓcos(πs)

)(1− π2

τ2Γ2+

π4

τ4Γ4− . . .

)=πε

2

1

1 + π2

τ2Γ2

∫ 1

0

E(

sin(πs)− π

τΓcos(πs)

).

(119)

The low-dissipation we considered in the text consists in neglecting all the terms of order O( 1τ2Γ2 ) on (which in

this specific case would consist only in a renormalization of the total result), hence identifying βQ = ∆S−Σ/τ :

∆SLD =πε

2

∫ 1

0

E(s) sin(πs)ds ΣLD =π2ε

2Γ

∫ 1

0

E(s) cos(πs)). (120)

In the high efficiency regime we can then write from the asymptotic expansions of Eq. (90) and (91)

P (max)γ ' ∆S2

Σ

(Tc − Th)2(1− γ)

4Tcγ(121)

τh ' τc '2Σ

∆S(ThTc− 1)

(1− γ). (122)

In Figure 7 we show the evolution of the ground state probability p(s) (which characterizes the whole statesolution) according to different approximations, as well as the actual performance of the engine compared tothe maximal one (which is obtained in the limit ε→ 0), for feasible choices of the parameters. The slow-drivingapproximation gives results that almost coincide with the full analytical series for this particular protocol, andthe performance is close to the optimal one.