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 four-stroke 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.

Figure 1 Pressure-volume diagram of a (quantum) Otto cycle. The numbers relate the processes to the description of each step given in the main text. We identify the steps where heat enters (exits) the working medium and those where work is performed by (done onto) it as a result of a corresponding change in the work parameter λ (t). Full size image

If the engine is run in a finite-time, i.e. we abandon the usual quasi-static 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.

Finite-time 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 system-reservoir 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 nth 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 non-deterministic nature of quantum measurements30. 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 modified31. 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 quasi-static isothermal process (the inequality holding strictly for all quasi-static processes performed without the coupling to a thermal reservoir). For non-ideal 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 time-evoluted state and the corresponding equilibrium state at the temperature 1/β. Here, 〈W irr 〉 quantifies the degree of friction caused by the finite-time 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 flows34,35.

Friction-free finite-time engine. Recently, substantial work has been devoted to the design of super-adiabatic protocols, i.e. shortcuts to states which are usually reached by slow adiabatic processes20,21,22,24. 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 non-adiabatic dynamics achieved using self-similar transformations23,36.

Let us consider a quantum harmonic oscillator with time-dependent frequency ω(t) as the working medium of the engine cycle23. 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 friction-prone 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) = 〈x|n(0)〉 of an initial eigenstate |n(0)〉 of is known to follow the self-similar evolution23

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 time-dependent 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 finite-time dynamics taking the initial state ϕ n (x, t = 0) = 〈x|n(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 self-similar dynamics can be induced in a large family of many-body systems36 and other trapping potentials, such as a quantum piston37. The resilience of the shortcuts to adiabaticity approach to imperfections in the engineering of the exact functional form of the time-dependent 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 time-dependent 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 = [〈W2〉 − 〈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 non-equilibrium deviation of both the average work and the standard deviation from the adiabatic trajectory disappear.

Figure 2 Work fluctuations along a shortcuts to an adiabaticity expansion. (a) Average work; (b) Standard deviation of the work; (c) Non-equilibrium deviations from the adiabatic average mean work; (d) We show ( and ) and ( and ) [cf. Eq. (7)] for the same processes shown in the other panels. All quantities are plotted in units of ħω 0 (β = 1). Full size image

We shall now analyse the non-equilibrium 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 super-adiabats 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 large-class of self-similar 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 time-evolution 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 finite-time, thus implementing a frictionless finite-time cycle. As friction is the only source of irreversibility in our scheme, the super-adiabatic engine reaches the maximum efficiency of an ideal quasi-static engine in a finite-time only.

Let us address a final point: The efficiency in Eq. (1) diminishes with the breakdown of adiabaticity13. In contrast, our super-adiabatic engine achieves the maximum possible value . Clearly, if unlimited resources are available, there is no fundamental lower-bound on the running time of the adiabats. However, we take a pragmatic approach and quantify the energy cost associated with the implementation of our super-adiabatic engine, which would provide a significant cost function for such part of the cycle. We have thus considered the time-averaged dissipated work , ensuring ω2(t) > 0 for t ∈ [0, τ]. The cut-off time τ c was taken as the maximum running time along the shortcut of each super-adiabat before the trap is inverted (cf. Methods section). When this occurs, the adiabatic eigen-energies 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 trap36. 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 super-adiabatic engine exhibits a 〈δW〉 ~ 1/τ behavior for a wide range of parameters, as shown in Fig. 3. This demonstrates the existence of a trade-off between the running time of the super-adiabatic transformations and the corresponding amount of time-averaged 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.