Abstract
The practical untenability of the quasistatic assumption makes any realistic engine intrinsically irreversible and its operating time finite, thus implying friction effects at short cycle times. An important technological goal is thus the design of maximally efficient engines working at the maximum possible power. We show that, by utilising shortcuts to adiabaticity in a quantum engine cycle, one can engineer a thermodynamic cycle working at finite power and zero friction. Our findings are illustrated using a harmonic oscillator undergoing a quantum Otto cycle.
Introduction
THermodynamics is the study of heat and its interconversion to mechanical work. It successfully describes the “equilibrium” properties of macroscopic systems ranging from refrigerators to black holes^{1}. The revolution potentially embodied by the implementation of quantum technologies is motivating the consideration of quantum devices going all the way down to the micro and nanoscale^{2}. This has forced us to revise our interpretation of thermodynamics to include ab initio both quantum and thermal fluctuations. The former, indeed, become quite prominent at such scales. In fact, far from equilibrium, quantum fluctuations become dominant and cannot be neglected. In turn, thermodynamic quantities such as work and heat become inherently stochastic and should be reformulated accordingly.
Recently discovered work fluctuation theorems (FTs) and the corresponding framework, which is known to hold both in quantum and classical systems, are extremely useful for the task of setting up a quantum apparatus for thermodynamics^{3,4,5,6,7}. FTs set fundamental constraints on the energy fluctuations of a general thermodynamic system and embody useful tools to understand the thermodynamic implications of finitetime transformations^{8,9}. Whether or not FTs can help us shedding light on the limitations or possible advantages of a quantum device operating in finitetime is a very important point to address, that goes beyond the scopes of this paper. However, it is sensible to expect that a convenient platform for the provision of quantitative answers in this sense could come from the study of quantum engines, for which the thermodynamic laws must be recast appropriately^{10,11,12,13}. In fact, although the working principles of a reversible engine might well be quantum mechanical, its efficiency would always be limited by the second law, thus making the quantum version of cycles similar to their classical counterparts.
The assumed reversibility (quasistationarity) of an engine cycle, which implies an infinitely long cycletime, determines its inevitable zeropower nature. This clearly crashes with the reality of any practical machine, either quantum or classical. However, the finitetime operation of a machine working in finitetime exposes it to the effects of frictioninduced losses. The key engineering goal in this context is thus to find the maximum efficiency allowed at the maximum possible power^{15,16}. Noticeably, in Ref. 17 an ultrafast cycle has been discussed that nevertheless attains Carnot efficiency.
Classical approaches to the maximization of the power output of thermal engines have been proposed and developed in the past^{18,19}. While Ref. 14 proposes the use of systematic noise to suppress friction in the expansion and compression stages of a quantum Otto cycle, here we devise an innovative way to run a finitetime, finitepower quantum cycle based on the use of quantum shortcuts to adiabaticity^{20,21,22,23,24}. Such techniques have been employed to show that the Hamiltonian of a quantum system can be manipulated in a way to mimic an adiabatic process via a nonadiabatic shortcut^{20,21,22,23,24}. In this context, the term “adiabatic” should be interpreted as “slow” and will be used, unless the context is evidently different (such as in the description of the cycle), as synonymous of “quasistatic”. A few experiments have demonstrated several of such proposals^{25,26,27}. In this paper we show that such Hamiltonianengineering techniques allow us to drive the expansion and compression stages of a cycle, which are prone to frictional effects, virtually without any loss affecting the performance of a quantum engine within the finitetime of its cycle. Inspired by a recent iontrap proposal^{13}, we will provide an example of a finitetime, fully frictionless quantum Otto cycle where the working medium is a quantum harmonic oscillator. Remarkable examples of recent works along the lines of our investigation have been reported recently^{28,29}, although they both address classical analogs of quantum shortcuts to adiabaticity with Deng et al. studying the performance of an Otto cycle from such perspective^{29}.
Results
Quantum Otto Cycle. In an Otto engine, a working medium (coupled alternatively to two baths at different temperatures T_{i}, i = 1, 2) undergoes a fourstroke cycle. In its quantum version, the state of the working medium is described by a density operator ρ(λ(t)) that is changed by the Hamiltonian . Here, λ(t) is a work parameter, typical of the specific setting used to implement , whose value determines the equilibrium configuration of the system. As illustrated in Fig. 1, the cycle steps are as follows:

1
An adiabatic expansion performed by the change λ_{0} ≡ λ(0) → λ_{1} ≡ λ(τ_{1}), where τ_{1} is the time at which this step ends. As a result of this transformation, work is extracted from the medium due to the change in its internal energy.

2
A cold isochore where heat is transferred from the working medium to the cold bath. This is associated with a heat flow from the medium to the cold reservoir.

3
An adiabatic compression performed by the reverse change of the work parameter λ_{1} → λ_{0} and during which work is done on the medium.

4
A hot isochore during which heat is taken from the hot reservoir by the working medium.
If the engine is run in a finitetime, i.e. we abandon the usual quasistatic assumption, friction is generated along the expansion and compression steps. We will elucidate the nature of this friction later on. In addition, one may (realistically) assume imperfect heat conduction during the isochores. Under such conditions, the work W done by/on the engine and the heat Q exchanged by the medium with the baths, become stochastic quantities. The efficiency of the engine is then defined as the ratio between the average total work per cycle and the average heat received from the hot bath, that is
where 〈K〉_{j} (for K = Q, W) is evaluated during step j = 1, …, 4. The power of the engine is then
where τ_{j} is the time needed for step j. Here we consider the case where friction only occurs along the adiabatic transformations and neglect fluctuations in the heat flow. On the other hand, we should not forget about the thermalisation process inherent in the isochores, which are associated with the production of entropy. We thus assume to have identified a regime such that the entropy produced by such thermalisation steps is negligible and associated with finite values of τ_{2,4} (see Supplementary Information for the determination of such a working point and an estimate of both the order of magnitude of such time intervals and of the corresponding production of entropy). Needless to say, the entropy produced during the isochores is the same regardless of the way we perform the adiabats. Moreover, as we will describe in the second part of this paper, the strategy based on shortcuts to adiabaticity that we propose makes 〈W〉_{i} independent of τ_{i}. In these conditions, the maximisation of Eq. (2) is achieved by minimizing τ_{1,3}. We now describe our protocol to reduce the time needed for the adiabats and keep the associated friction at bay.
Finitetime thermodynamics. Before we quantify the efficiency of the engine, we need to define the probability distribution of work of which 〈W〉 is the first moment. We consider a Hamiltonian applied to a system prepared into the Gibbs state . with inverse temperature β and λ(t ≤ 0) = λ_{0}. Here, is the partition function. At t = 0, the systemreservoir coupling is removed and λ is changed from its initial value λ_{0} to λ_{1} at t = τ. Such process could be the expansion/compression step of the Otto cycle, represented by a change of λ in , where n(λ)〉 is the n^{th} eigenstate with eigenvalue ε_{n}(λ). In this context, work should be reformulated in a way to account for both the statistics of the initial state of the system and the nondeterministic nature of quantum measurements^{30}. Under the assumption , the corresponding work distribution P(W;t) reads
Here we use the notation shortcut n(t)〉 = n(λ(t))〉 and call the probability that, under the action of the evolution operator associated with , the system goes from the initial state n(0)〉 to the final one k(t)〉. Finally, is the occupation probability of the initial state n(0)〉, which for a Gibbs ensemble reads . For , this expression needs to be modified^{31}. However, regardless of the value of such commutator, the first two moments of the work distribution read .
For finite systems, the statistical nature of work requires the second law of thermodynamics to be revised to 〈W〉 ≥ ΔF, with ΔF the change in free energy and the equality holding for a quasistatic isothermal process (the inequality holding strictly for all quasistatic processes performed without the coupling to a thermal reservoir). For nonideal processes, the deficit between 〈W〉 and ΔF can be accounted for by the introduction of the average irreversible work 〈W_{irr}〉 as 〈W〉 = 〈W_{irr}〉 + ΔF. The behavior of 〈W〉 will be later compared with the average work 〈W_{ad}〉 performed onto (or made by) the system in an adiabatic process. For such quantity, in the absence of a heat bath, we have 〈W_{ad}〉 > ΔF. For a closed quantum system, the incoming heat flow is null and the irreversible entropy is ΔS_{irr} = β(〈W〉 − ΔF) = β〈W_{irr}〉, which can be recast as ^{32} with S(ρ_{A}ρ_{B}) = Tr(ρ_{A}lnρ_{A} − ρ_{A}lnρ_{B}) the relative entropy between two density matrices ρ_{A} and ρ_{B}^{33}, ρ_{t} the timeevoluted state and the corresponding equilibrium state at the temperature 1/β. Here, 〈W_{irr}〉 quantifies the degree of friction caused by the finitetime protocol on the expansion or compression stage of the engine cycle. When a bath is reconnected, such friction results in dissipation and hence the decrease in the overall efficiency of the motor. For the point of demonstration we allow only this form of irreversibility in our cycle, although in principle the same analysis can be done for fluctuating heat flows^{34,35}.
Frictionfree finitetime engine. Recently, substantial work has been devoted to the design of superadiabatic protocols, i.e. shortcuts to states which are usually reached by slow adiabatic processes^{20,21,22,24}. A typical approach for shortcuts to adiabaticity is to use ad hoc dynamical invariants to engineer a Hamiltonian model that connects a specific eigenstate of a model from an initial to a final configuration determined by a dynamical process. Here we will rely on an approach based on engineered nonadiabatic dynamics achieved using selfsimilar transformations^{23,36}.
Let us consider a quantum harmonic oscillator with timedependent frequency ω(t) as the working medium of the engine cycle^{23}. The Hamiltonian model that we consider is thus , where and are the position and momentum operators of an oscillator of mass m. Inspired by the scheme in Ref. 13, we will use the tuneable harmonic frequency to implement the compression and expansion steps of the Otto cycle. In line with such proposal, the frequency of the harmonic trap embodies the volume of the chamber into which the working medium is placed, while the corresponding pressure is defined in terms of the change of energy per unit frequency.
Clearly, in the compression or expansion stage of the Otto cycle, the frequency of the trap will have to be varied, so that ω(t) takes here the role of a work parameter. We now suppose to subject the working medium to a change in the work parameter occurring in a time τ and corresponding to one of the frictionprone steps of the Otto cycle. Our goal is to design an appropriate shortcut to adiabaticity to arrange for a fast, frictionless evolution between the configurations of the working medium at t = 0 and that at t = τ. In order to do this, we remind that the wavefunction ϕ_{n}(x, t = 0) = 〈xn(0)〉 of an initial eigenstate n(0)〉 of is known to follow the selfsimilar evolution^{23}
where , ε_{n}(0) is the energy of the eigenstate being considered at t = 0 and the scaling factor b is the solution of the Ermakov equation
with the initial conditions b(0) = 1 and . Needless to say, while the physically relevant parameter is the timedependent frequency ω(t), the determination of the exact scaling parameter b(t) is key for the engineering of the correct shortcut to adiabaticity. This is found by inverting the Ermakov equation and complementing the previous set of boundary conditions with and with ω_{0} = ω(0) and ω_{f} = ω(τ). An instance of the solution to this problem can be found in the Methods, where we give the explicit form of b(t) such that the finitetime dynamics taking the initial state ϕ_{n}(x, t = 0) = 〈xn(0)〉 to the final one mimics the wanted adiabatic evolution (albeit for any t ∈ (0, τ), ϕ_{n}(x, t) is in general different from the eigenstate n(t)〉 of ). The choice of a harmonic oscillator is not a unique example as analogous selfsimilar dynamics can be induced in a large family of manybody systems^{36} and other trapping potentials, such as a quantum piston^{37}. The resilience of the shortcuts to adiabaticity approach to imperfections in the engineering of the exact functional form of the timedependent protocol embodied by ω(t) is an important point to address. Overall, shortcuts to adiabaticity are known to be robust against perturbations, as discussed in Ref. 23 for the case of an approximately harmonic trap and in Ref. 36 for other trapping potentials.
Let us consider the fluctuations induced in the expansion and compression stages of the Otto cycle when the above shortcut to adiabaticity is implemented. Let us consider a driving Hamiltonian with instantaneous eigenstates n(t)〉 and eigenvalues ε_{n}(t). In the adiabatic limit, the corresponding transition probabilities tend to 〈n(t)k(t)〉^{2} = δ_{k}_{,n}(t) for all t ∈ [0, τ]. The average work is then . On the other hand, in a shortcut to adiabaticity, only the weaker condition holds. For the timedependent harmonic oscillator, it follows that
In the adiabatic limit and .
Fig. 2(a) shows 〈W〉 along a shortcut to an adiabatic expansion in comparison with the corresponding adiabatic process 〈W_{ad}(t)〉 (the behavior observed during a shortcut to a compression is mirrored in time). We stress that 〈W〉 is the work done on either adiabat until the reconnection with the bath, i.e. just prior to the isochoric heating/cooling stage. Fig. 2(b) displays the standard deviation ΔW = [〈W^{2}〉 − 〈W〉^{2}]^{1/2}, which provides a further characterisation of the work fluctuations along the shortcut through the width of P(W;t). Interestingly, upon completion of the stroke, the nonequilibrium deviation of both the average work and the standard deviation from the adiabatic trajectory disappear.
We shall now analyse the nonequilibrium deviation δW = 〈W〉 − 〈W_{ad}(t)〉 with respect to 〈W_{ad}(t)〉. This is equivalent to the deviation of the mean energy of the motor along the superadiabats from its (instantaneous) adiabatic expression. For a reversible isothermal process 〈W_{ad}〉 = ΔF and δW = 〈W_{irr}〉. Differently, for the adiabatic dynamics of stages 1 and 3 of the Otto cycle, conservation of the population in n(t)〉 is satisfied provided that β_{t} = βε_{n}(0)/ε_{n}(t), as it is the case for a largeclass of selfsimilar processes, as discussed in Refs. 23,36,37 and remarked in the Supplementary Information. Here, β_{t} is introduced by noticing that the physical adiabatic state at time t is characterised by the occupation probabilities . Therefore, the reference state is not the physical instantaneous equilibrium state resulting from the adiabatic dynamics and we find
Therefore, in general δW ≠ 0. However, one can check that at the end of the process we have , which implies δW = 0 and thus the frictionless nature of the process [cf. Fig. 2(c)]. The timeevolution of the different contribution to δW, i.e. and , are displayed in Fig. 2(d). This result is remarkable in the context of the quantum Otto cycle: If the baths are reconnected at time τ after both the compression and expansion stages, then the efficiency of an ideal reversible engine can be reached in finitetime, thus implementing a frictionless finitetime cycle. As friction is the only source of irreversibility in our scheme, the superadiabatic engine reaches the maximum efficiency of an ideal quasistatic engine in a finitetime only.
Let us address a final point: The efficiency in Eq. (1) diminishes with the breakdown of adiabaticity^{13}. In contrast, our superadiabatic engine achieves the maximum possible value . Clearly, if unlimited resources are available, there is no fundamental lowerbound on the running time of the adiabats. However, we take a pragmatic approach and quantify the energy cost associated with the implementation of our superadiabatic engine, which would provide a significant cost function for such part of the cycle. We have thus considered the timeaveraged dissipated work , ensuring ω^{2}(t) > 0 for t ∈ [0, τ]. The cutoff time τ_{c} was taken as the maximum running time along the shortcut of each superadiabat before the trap is inverted (cf. Methods section). When this occurs, the adiabatic eigenenergies are not well defined and our formalism breaks down. For the shortcut to adiabaticity discussed here, such inversion occurs when the expansion time is smaller than the inverse of the initial frequency of the trap^{36}. The exact value of such critical running time, which depends on the expansion factor and can be found numerically, is different for expansion and compression stages, being larger in the former case. While the steps necessary for the calculation of 〈δW〉 are reported in the Supplementary Information, here is enough to mention that the cost of running the superadiabatic engine exhibits a 〈δW〉 ~ 1/τ behavior for a wide range of parameters, as shown in Fig. 3. This demonstrates the existence of a tradeoff between the running time of the superadiabatic transformations and the corresponding amount of timeaveraged dissipated work, in line with the analogous compromise between the irreversible entropy produced along the isochores and the running time of the transformations. An upper bound for the power of an engine run can be calculated using the fundamental limitations set by quantum speed limit. The key steps of such calculations are discussed in the Supplementary Information.
Discussion
We have demonstrated the possibility to perform a fully frictionless quantum cycle in a finitetime. Our proposal exploits the idea of shortcuts to adiabaticity, which allowed us to bypass the effects of friction on the compression and expansion stages in an important cycle such as Otto's. Our study embodies one example of the potential brought about by the combination of shortcuts to adiabaticity and the framework for outofequilibrium dynamics of a quantum system. The possibility to achieve maximum efficiency of a quantum engine at finite time with virtually no friction is tantalizing in the perspective of designing micro and nanoscale motors operating at the verge of quantum mechanics.
Methods
Driving protocol of the superadiabats
Here we illustrate the formal procedure for the determination of the scaling factor b(t) used in the superadiabatic steps of our proposal. The simplest interpolation of the actual solution of the Ermakov equation in the main body of the paper with the boundary conditions stated in the main text is found to be the polynomial
with s = t/τ. This solution guarantees that, in the adiabatic limit, b·(t) → 0 and b(t) → b_{ad}, while more generally the eigenstates of the initial oscillator will evolve according to the scaling law in Eq. (4). The modulation of ω(t) is the responsible for the speedup of the transformations performed along the superadiabats. In turn, the implementation of such modulation is the price to pay for the achievement of such advantage. The explicit form ω(t) can be extracted from the Ermakov equation, , using Eq. (8) for b(t), see Fig. (4). For sufficiently small values of ω_{0}τ, ω(t) can become purely imaginary (requiring the inversion of the trap into an expelling barrier). The condition ω^{2}(t) > 0 provides the cutoff time τ_{c} used in Fig. 3.
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Acknowledgements
We are grateful to G. De Chiara, B. Damski, S. Deffner, R. Dorner, R. Fazio, C. Jarzynski, E. Passemar and N. Sinitsyn for discussions and comments on this work. AdC is supported by the U.S. Department of Energy through the LANL/LDRD Program and a LANL J. Robert Oppenheimer fellowship. JG acknowledges funding from IRCSET through a Marie Curie International Mobility fellowship. MP thanks the UK EPSRC for a Career Acceleration Fellowship and a grant under the “New Directions for EPSRC Research Leaders” initiative (EP/G004579/1) the John Templeton Foundation (grant 43467), the EU Collaborative project TherMiQ (Grant Agreement 618074) and the COST Action MP1209.
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A.d.C. and J.G. developed the calculation for the concept proposed by M.P. All authors interpreted the results and wrote the manuscript.
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Campo, A., Goold, J. & Paternostro, M. More bang for your buck: Superadiabatic quantum engines. Sci Rep 4, 6208 (2014). https://doi.org/10.1038/srep06208
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DOI: https://doi.org/10.1038/srep06208
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