Efficiency large deviation function of quantum heat engines

The efficiency of small thermal machines is typically a fluctuating quantity. We here study the efficiency large deviation function of two exemplary quantum heat engines, the harmonic oscillator and the two-level Otto cycles. While the efficiency statistics follows the 'universal' theory of Verley et al. [Nature Commun. 5, 4721 (2014)] for nonadiabatic driving, we find that the latter framework does not apply in the adiabatic regime. We relate this unusual property to the perfect anticorrelation between work output and heat input that generically occurs in the broad class of scale-invariant adiabatic quantum Otto heat engines and suppresses thermal as well as quantum fluctuations.

Fluctuations play a central role in the thermodynamics of small systems. Contrary to macroscopic thermodynamics that describes the average behavior of a vast number of particles, microscopic systems are characterized by stochastic variables, whose large fluctuations from mean values contain useful information on their dynamics [1]. At equilibrium, the probability distributions of thermal observables are conveniently obtained using the methods of equilibrium statistical physics [2]. However, their evaluation for nonequilibrium problems is often difficult. A powerful framework that allows the calculation of these distributions, both in equilibrium and nonequilibrium situations, is provided by large deviation theory [3][4][5][6]. From a physical point of view, the large deviation approach may be viewed as a generalization of the Einstein theory of fluctuations that relates the probability distribution to the entropy, P (x) ∼ exp[S(x)/k], where k is the Boltzmann constant. On the other hand, from a mathematical standpoint, it may be regarded as an extension of the law of large numbers and the central-limit theorem [3][4][5][6].
Large deviation techniques have found widespread application in many areas, ranging from Brownian motion and hydrodynamics to disordered and chaotic systems [3][4][5][6]. In the past few years, they have been successfully employed to investigate the efficiency statistics of small thermal machines [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. In microscopic systems, heat, work, and, consequently, efficiency are indeed random quantities owing to the presence of thermal [1] and, at low enough temperatures, quantum fluctuations [22,23]. Understanding their fluctuating properties is therefore essential. In particular, Refs. [7,8] have identified 'universal' features of the efficiency large deviation function, which exhibits a characteristic smooth form with two extrema, including a maximum at the Carnot efficiency. The latter value is thus remarkably the least likely in the long-time limit. These predictions have been experimentally verified for a stochastic harmonic heat engine based on an optically trapped colloidal particle [24].
In this paper, we compute the efficiency large deviation function of two paradigmatic quantum thermal machines, the harmonic oscillator quantum engine and the two-level system quantum motor [25][26][27][28][29][30][31][32][33][34][35][36]. Our study is motivated by the recent experimental implementation of a nanoscopic harmonic heat engine using a single trapped ion [37] and the realization of quantum spin-1/2 motors using NMR [38,39] and trapped ion [40] setups. We concretely consider the exemplary case of the quantum Otto cycle, a generalization of the ordinary four-stroke motor that has been extensively studied in the past thirty years [41]. We find that the efficiency large deviation functions follow the 'universal' form of Refs. [7,8] for nonadiabatic driving. However, in the adiabatic regime, which corresponds to maximum efficiency and may be reached exactly for a periodically driven two-level engine or using shortcut-to-adiabaticity techniques [42][43][44], we show that the large deviation functions take a markedly different shape, as the efficiency is deterministic and equal to the macroscopic Otto efficiency. This result holds generically for heat engines with scale-invariant Hamiltonians that describe a broad class of single-particle, many-body and nonlinear systems [45][46][47][48][49]. We trace this unusual behavior to the perfect anticorrelation between work output and heat input within the engine cycle that is established for adiabatic driving. This property completely suppresses the effects of fluctuations. As a consequence, microscopic adiabatic quantum Otto heat engines run at the nonfluctuating macroscopic efficiency.
Efficiency large deviation function. We consider a generic quantum system with a time-dependent Hamiltonian H t as the working medium of a quantum Otto engine. The engine is alternatingly coupled to two heat baths at inverse temperatures β i = 1/(kT i ), (i = c, h), where k is the Boltzmann constant. The quantum Otto cycle consists of the following four consecutive steps [41]: (1) Unitary expansion: the Hamiltonian is changed from H 0 to H τ1 in a time τ 1 , consuming an amount of work W 1 , (2) Hot isochore: the system is weakly coupled to the hot bath at inverse temperature β h to absorb heat Q 2 in a time τ 2 , (3) Unitary compression: the isolated system is driven from H τ1 back to H 0 in a time τ 3 , producing an amount of work W 3 , and (4) Cold isochore: the cycle is closed by connecting the system to the cold bath at inverse temperature β c , releasing heat Q 4 in a time τ 4 . Work and heat are positive, when added to the system. We further assume that heating and cooling times, τ 2,4 , are longer than the relaxation time, so that the system can fully thermalize after each isochore, as in the experimental quantum Otto engines of Refs. [38,39]. Without loss of generality, we additionally set τ 1 = τ 3 = τ .
The stochastic efficiency of the microscopic quantum arXiv:2008.00778v2 [quant-ph] 4 Aug 2020 heat engine is defined as the ratio of work output and heat input, η = −W/Q 2 , where W = W 1 + W 3 denotes the total work [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21]. It should not be confused with the thermodynamic efficiency of macroscopic engines, η th = − W / Q 2 , which is given by the ratio of the mean work output and the mean heat input, and is thus a deterministic quantity. We investigate the efficiency statistics of the quantum engine in the long-time limit using large deviation theory [3][4][5][6]. Following Refs. [7,8], we write the joint distribution of work and heat, P s (Q 2 , W ), as well as the efficiency distribution P s (η), for a large number of cycles (s 1), in the asymptotic form, The two large deviation functions I(Q 2 , W ) and J(η) describe the exponentially unlikely deviations of the variables Q 2 , W and η from their typical values. The rate function J(η) follows from I(Q 2 , W ) by contraction [6], An alternative, more practical, expression may be obtained by introducing the bivariate scaled cumulant generating function of the mean heat and mean work per cycle, q Using the Legendre-Fenchel transform, one then finds [8], The efficiency large deviation function J(η) may thus be determined from the scaled cumulant generating function φ(γ 1 , γ 2 ). In the following, we evaluate φ(γ 1 , γ 2 ) by taking the logarithm of the moment generating function, that is, the Wick transformed characteristic function G(γ 1 , γ 2 ) = exp(−iγ 1 Q 2 − iγ 2 W ) [51]. Work-heat correlations. Work output and heat input are usually correlated in a closed quantum heat engine cycle. Despite their fundamental importance, their correlations have received little attention so far [52]. We next derive their joint probability distribution using the standard two-projective-measurement approach [53]. In this method, energy changes of a quantum system during single realizations of a process are identified with the difference of energy eigenvalues obtained though projective measurements at the beginning and at the end of the process. In the quantum Otto cycle, work is performed during the unitary expansion and compression stages, while heat is exchanged during the nonunitary heating and cooling steps. We obtain the distributions of work and heat by applying the two-projective-measurement scheme to the respective expansion, hot isochore and compression branches. The corresponding joint distribution for work output and heat input reads accordingly [54], are the respective energy eigenvalues at the beginning and at the end of the expansion (compression) step, with corresponding unitary operator U exp (U com ). The thermal distribution at the beginning of the expansion (compression) stage is given by . The occupation probabilities P 0 n (β c ) and P τ k (β h ) account for thermal fluctuations, while the transition probabilities | n| U exp (τ ) |m | 2 and | k| U com (τ ) |l | 2 for both quantum fluctuations and quantum dynamics [50].
We first remark that work output and heat input are perfectly anticorrelated in this case, with a Pearson coefficient [55], ρ = cov(Q 2 , W )/σ Q2 σ W = −1 (Fig. 1) [54]. On the other hand, the minimization in Eq. (4) leads to a rate function that plateaus at infinity, except at the macroscopic efficiency η th where it vanishes (Fig. 2). As a consequence, the microscopic stochastic efficiency is deterministic and equal to the macroscopic value η th . Adiabatic quantum Otto engines hence lie outside the universality class of Refs. [7,8]. This may be understood by noting that quantum work fluctuations are suppressed in the adiabatic regime. Thermal fluctuations are additionally canceled by the perfect work-heat anticorrelation.
Quantum heat engines. Let us now examine the workheat correlations and the efficiency large deviations, both in the adiabatic and nonadiabatic regimes, for two exactly solvable quantum Otto engines. We first evaluate the characteristic function G(γ 1 , γ 2 ) for a solvable twolevel quantum motor. Inspired by the recent NMR ex- periments [38,39], we consider the expansion Hamiltonian, H exp t = ωσ z /2 + λ(t) (cos ωt σ x + sin ωt σ y ), that describes a spin-1/2 driven by a constant magnetic field with strength ω/2 along the z-axis and a rotating magnetic field with varying strength λ(t) in the (x-y)-plane, where σ i , i = (x, y, z), are the standard Pauli operators (with = 1). The rotation frequency is chosen to be ω = π/2τ to ensure a complete rotation from the x-axis to the y-axis during time τ . The amplitude of the rotating field, λ(t) = λ 1 (1 − t/τ ) + λ 2 (t/τ ), is increased from λ 1 at time zero to λ 2 at time τ . This driving leads to a widening of the energy spacing of the two-level system from 2ν 0 = 4λ(0) 2 + ω 2 to 2ν τ = 4λ(τ ) 2 + ω 2 . The compression Hamiltonian is obtained from the time reversed process, H com The characteristic function G(γ 1 , γ 2 ) may be determined by solving the time evolution of the engine. It is explicitly given by [54], where u = 1 − v denotes the probability of no-level transition (0 ≤ u ≤ 1), x = β c ν 0 and y = β h ν τ [54]. The twolevel engine operates adiabatically, when the adiabaticity parameter, defined as the ratio of the nonadiabatic and adiabatic mean energies, Q * TL = 2u − 1 = 1 (or u = 1). We emphasize that, since the driving is periodic, the adiabatic regime is here reached exactly for and not just asymptotically for large driving times [54]. Equation (7) contains all the information needed to investigate the work-heat correlations and the efficiency large deviation function of the quantum two-level heat engine.
We next consider a (unit mass) harmonic oscillator engine with expansion Hamiltonian H exp τ t/τ . The reversed compression protocol is again obtained with the replacement t = τ − t. This quantum Otto engine model is analytically solvable [34] and we find the work-heat characteristic function [54], where we have again introduced the adiabaticity parameter Q * HO , which is equal to 1 for adiabatic driving Results. We begin by analyzing the work-heat correlations within the quantum Otto cycle using the Pearson coefficient. Figure 1 shows the correlation coefficient ρ for the qubit (orange solid) and the harmonic (red dotted-dashed) quantum heat engines as a function of the respective adiabaticity parameters (we have set their frequencies equal, ω j = ν j , in order to compare the two cases). We observe that work output and heat input are generally negatively correlated in both examples. However, contrary to the harmonic engine, the two-level motor displays a nonmonotonous dependence on Q * due to the finite dimension of its Hilbert space. We, moreover, see that the amount of correlations increases with decreasing nonadiabaticity. In particular, work output and heat input are perfectly anticorrelated for adiabatic cycles, in agreement with the result obtained for scaleinvariant engines. Figure 2 exhibits the large deviation function J(η) for both working media. For nonadiabatic driving (Fig. 2a), we recognize the characteristic form obtained in Refs. [7,8], with a maximum at the Carnot efficiency η ca (the least likely value) and a minimum located at the macroscopic Otto efficiency η th (the most likely value). The harmonic rate function is, furthermore, strictly above that of the qubit (with the exception of the root at η th ), indicating that the harmonic heat engine converges faster towards the macroscopic efficiency η th than the two-level engine. By contrast, for adiabatic driving (Fig. 2b), when work output and heat input are perfectly anticorrelated, the rate function of both systems noticeably departs from that general form: it is zero at the thermodynamic efficiency η th and infinite everywhere else, confirming that the efficiency behaves deterministically. It is important to stress that these findings are not restricted to the strict adiabatic limit [54]. They are also valid in the linear response regime, which is often used to examine the finitetime dynamics of quantum heat engines [58][59][60].
A deeper understanding of the stark differences between adiabatic and nonadiabatic driving in the quantum Otto cycle may be gained by applying the geometric approach of Ref. [20] to the present instance of quantum heat engines. According to Eq. (4), the rate function J(η) is obtained for fixed η by minimizing the cumulant generating function φ(γ 1 , γ 2 ) along the line γ 1 = ηγ 2 . The theory of Refs. [7,8] then only applies when there is a unique minimum. This is the case for nonadiabatic driving, as can be seen from the contour plot of φ(γ 1 , γ 2 ) for the two-level quantum motor (Fig. 3a). By contrast, for adiabatic driving, the isocontours of φ(γ 1 , γ 2 ) are parallel lines with slope η th (Fig. 3b). As a result, the minimum is degenerate, leading to the plateau of the large deviation function at infinity (except at the macroscopic efficiency η th ) and the breakdown of the formalism of Refs. [7,8]. A similar behavior is observed for the example of the harmonic quantum heat engine [54].
Conclusions. We have investigated the work-heat correlations and the efficiency statistics of the quantum Otto cycle with a working medium consisting of a two-level system or a harmonic oscillator. We have found that work output and heat input are in general negatively correlated, with perfect anticorrelation achieved for adiabatic driving. As a consequence, the microscopic quantum efficiency is equal to the deterministic macroscopic Otto efficiency and the efficiency large deviation function strongly deviates from the characteristic form obtained in Refs. [7,8]. These results not only hold for quantum heat engines that operate in the adiabatic limit, such as shortcut-to-adiabaticity engines, but also in the linear response regime. Our findings are thus important for the study of the performance of small quantum thermal machines that run close to the adiabatic regime.
We acknowledge financial support from the German Science Foundation (DFG) under project FOR 2724.

SUPPLEMENTAL MATERIAL I. WORK-HEAT CORRELATIONS
We here derive the joint probability distribution of work output and heat input P (Q 2 , W ), Eq. (5) in the main text, using the two-projective-measurement scheme [53]. Performing a projective energy measurement at the beginning and at end of the expansion step, we obtain the expansion work distribution P (W 1 ), where E 0 n and E τ m are the respective energy eigenvalues, P 0 n (β c ) = exp(−β c E 0 n )/Z 0 is the initial thermal occupation probability and P τ n→m = | n| U exp |m | 2 the transition probability between the instantaneous eigenstates |n and |m . The corresponding unitary is denoted by U exp . Similarly, the probability density of the heat Q 2 during the following hot isochore, given the expansion work W 1 , is equal to the conditional distribution [61], where the occupation probability at time τ is P τ k = δ km when the system is in eigenstate |m after the second projective energy measurement during the expansion step. Noting that the state of the system is thermal with inverse temperature β h at the end of the isochore, we further have P τ2 k→l = P τ2 l (β h ) = exp(−β h E τ l )/Z τ . The quantum work distribution for compression, given the expansion work W 1 and the heat Q 2 , is moreover, with the occupation probability P τ +τ2 i = δ il when the system is in eigenstate |l after the third projective energy measurement. The transition probability P τ i→j = | i| U com |j | 2 is fully specified by the unitary time evolution operator for compression U com .
The joint probability of having certain values of W 3 , Q 2 and W 1 during the cycle follows from the chain rule for conditional probabilities, P (W 3 , Q 2 , W 1 ) = P (W 3 |Q 2 , W 1 )P (Q 2 |W 1 )P (W 1 ) [51]. We find [62], The joint distribution P (Q 2 , W ) then follows by integrating over the work contributions W 1 and W 2 , We next compute the characteristic function, Eq. (6) of the main text, and the Pearson coefficient for adiabatic scale invariant quantum Otto heat engines with In the adiabatic regime, | m| U exp |n | 2 = δ nm and | k| U com |l | 2 = δ kl , we have, The characteristic function G(γ 1 , γ 2 ) = exp(−iγ 1 Q 2 − iγ 2 W ) , Eq. (6) of the main text, is readily obtained after Fourier transformation. The Pearson coefficient in this case follows as, We observe that work-heat correlations are always maximal in the adiabatic regime. By further considering the heat engine conditions, we find that 1 − ε −2 τ ≤ 0. As a result, work output and heat input are perfectly anticorrelated for an adiabatic quantum Otto engine, ρ = −1. We can thus conclude that, even though the engine is still subjected to nonvanishing heat and work fluctuations, they fluctuate in unison such that its efficiency is deterministic.

II. CHARACTERISTIC FUNCTIONS
We next evaluate the characteristic function G TL (γ 1 , γ 2 ), Eq. (7) of the main text, for the exactly solvable two-level quantum Otto engine. The time evolution operator U exp for the expansion branch may be calculated using the methods of Refs. [62][63][64], U exp = e −iωt/2 cos I ie −iωt/2 sin I ie iωt/2 sin I e iωt/2 cos I , where I = − t 0 dt λ(t ) is the integral over the increasing strength of the rotating magnetic field. The operator U com follows from U exp by the replacement t with τ − t. The probability of no level transition during expansion or compression are identical for τ 1 = τ 3 = τ and reads, The probability of a (nonadiabatic) level transition during either driving phases is accordingly v = 1 − u. The adiabaticity parameter is defined as the ratio Q * TL = H τ nad / H τ ad = 2u − 1 [49] and is equal to 1 for adiabatic driving, u = 1. Inserting the above expressions for the transition probabilities into P (Q 2 , W ) and performing the Fourier transform, dW dQ 2 exp(−iγ 1 Q 2 − iγ 2 W )P (Q 2 , W ), then yields the characteristic function G TL (γ 1 , γ 2 ).
The characteristic function G HO (γ 1 , γ 2 ), Eq. (8) of the main text, for the exactly solvable harmonic quantum Otto engine may be directly evaluated using a result of Ref. [57]. The generating function of the transition probabilities for expansion is indeed given by, and a similar expression for the compression step. We then determine the characteristic function G HO (γ 1 , γ 2 ) by comparing the terms of different powers in (n, m, k, l) of the Fourier transform of Eq. (5) of the main text with the ones in Eq. (18).
where the parameters x, y, u, v are unchanged. On the other hand, a Taylor expansion in first-order around Q * HO = 1 yields the work-heat characteristic function for the harmonic quantum Otto heat engine, where the parameters x 0 , y 0 , u 0 , v 0 are also unchanged. The corresponding approximate and exact large deviation functions J(η) are shown in Fig. 4a) for the two-level engine and in Fig. 4b) for the harmonic motor. We observe in both cases that the maximum at the Carnot efficiency η ca has effectively disappeared and that the narrow peak at the minimum located at the macroscopic efficiency η th has instead broadened.

IV. HARMONIC SCALED CUMULANT GENERATING FUNCTION
We finally show the contour plots of the scaled cumulant generating function φ(γ 1 , γ 2 ) of the harmonic quantum Otto engine in the nonadiabatic (Fig. 5a) and adiabatic (Fig. 5b) regimes. They are qualitatively similar to those of the two-level quantum motor represented in Fig. 3 of the main text. In the nonadiabatic case, we find regions in the (γ 1 , γ 2 )-plane for which the cumulant generating function is undefined (dark blue), contrary to what happens for the two-level Otto engine. This might lead to additional deviations from the 'universal' theory of Refs. [7,8] as those already pointed out in Ref. [20]. In the adiabatic case, we again observe parallel lines with slope η th , leading to a degenerate minimum in the minimization procedure of the rate function J(η).