Quantum jump approach to microscopic heat engines

Modern technologies could soon make it possible to investigate the operation cycles of quantum heat engines by counting the photons that are emitted and absorbed by their working systems. Using the quantum jump approach to open-system dynamics, we show that such experiments would give access to a set of observables that determine the trade-off between power and efficiency in finite-time engine cycles. By analyzing the single-jump statistics of thermodynamic fluxes such as heat and entropy production, we obtain a family of general bounds on the power of microscopic heat engines. Our new bounds unify two earlier results and admit a transparent physical interpretation in terms of single-photon measurements. In addition, these bounds confirm that driving-induced coherence leads to an increase in dissipation that suppresses the efficiency of slowly driven quantum engines in the weak-coupling regime. A nanoscale heat engine based on a superconducting qubit serves as an experimentally relevant example and a guiding paradigm for the development of our theory.


I. INTRODUCTION
In classical thermodynamics, a heat engine is described as a machine that uses a periodic series of thermodynamic processes to convert thermal energy into mechanical work [1].Each process, or stroke, involves the transfer of work between the working medium and an external load, for example a movable piston, or the exchange of heat with either a hot or a cold reservoir.Output and input of the engine, that is, the net generated work and the heat uptake from the hot reservoir, depend on the applied protocol and the equations of state of the medium.Their ratio, however, the thermal efficiency, is subject to a universal upper bound, the Carnot limit, which follows directly from the first and the second law and is attained for optimal quasi-static, i.e., infinitely long, cycles [1].
Realistic machines have to operate at finite speed and therefore inevitably produce dissipative losses, which suppress their efficiency.But how close can a heat engine with fixed cycle time come to the Carnot limit?This question cannot be resolved within the framework of classical thermodynamics due to its lack of a fundamental time scale.Early on, this issue spurred the development of refined models for macroscopic heat engines that account for irreversible effects by phenomenological means, an approach that became known as finite-time thermodynamics [2][3][4][5].More recently, with the advent of stochastic and quantum thermodynamics, the focus has changed to microscopic heat engines, which, instead of a homogeneous medium, use a tiny object with few degrees of freedom to perform thermodynamic cycles [6][7][8][9][10].The input for such devices is provided by tunable heat sources, which control the temperature of the environment of the working system; work is extracted and injected by changing the internal energy of the system through external control parameters or by coupling it to a microscopic load.
In contrast to a macroscopic fluid, which contains a vast amount of particles, the working systems of a microscopic heat engine can be described on the level of trajectories or wave functions.Macroscopic equations of state are thereby replaced by stochastic equations of motion, which apply even far from equilibrium and create a direct link between micro-dynamics and thermodynamics.This approach has opened a wide range of possibilities to explore the basic principles that govern the performance of periodic heat engines.Recent investigations include the study of generalized cycles, which involve continuous temperature variations [11][12][13][14][15], the development of optimal control strategies [16][17][18][19][20][21][22][23][24], and the systematic investigation of the thermodynamic footprint of quantum effects, which become relevant at time and energy scales comparable to Planck's constant, see for instance [25][26][27][28][29][30][31][32][33][34].As a key result, this development led to the discovery of a family of trade-off relations between power, i.e., average work output per unit time, and efficiency, first in linear response [11,13,35] and then beyond [36][37][38].
These relations impose quantitative bounds on the efficiency of finite-time engine cycles, which go beyond the first and the second law and approach the Carnot limit only for infinite cycle times leading to vanishing power.This behavior is, in fact, generic for conventional systems and can be overcome only under exceptional conditions such as diverging fluctuations of thermodynamic quantities [38][39][40] or vanishing relaxation times enabled by fine-tuned dissipation mechanisms [41][42][43][44].
Over the last decade, microscopic heat engines have been realized on increasingly smaller length and energy scales with working systems such as a micrometer-sized silicon spring [45], colloidal particles [46][47][48][49][50], a single atom [51,52], nuclear [53] and electronic [54] spins or nitrogen-vacancy centers in diamond [55].In light of this rapid development, practical tests of trade-off relations between power and efficiency appear as a realistic challenge for near-future experiments.This endeavor will, besides measurements of produced work and absorbed arXiv:2005.12231v1[cond-mat.stat-mech]25 May 2020 heat, require the measurement of at least one additional parameter, which is necessary to match the physical dimensions of power and efficiency [38,56].Despite its practical importance, this problem has so far received only little attention and has not yet been addressed for the quantum regime.
In this article, we consider a promising solid-state platform for the realization of quantum thermal devices that is in reach of current technologies [57][58][59][60][61][62].This setup consists of an engineered working system and a mesoscopic reservoir, which acts as an effective environment.An engine cycle can be implemented by applying a periodic driving field to the system and modulating the base temperature of the reservoir.At the same time, the temperature of the reservoir is monitored with an ultrasensitive thermometer able to detect small variations due to the emission and absorption of single photons [63][64][65][66].Each detected event indicates the transfer of a specific amount of energy between the system and the reservoir and an abrupt change of the quantum state of the system, in other words, a quantum jump.Hence, the reservoir takes on a two-fold role; it functions as a source of thermal energy and as a small-scale calorimeter enabling the direct measurement of the exchanged heat and the quantitative observation of quantum jumps.As we show in the following, the statistics of these jumps encodes an operationally accessible trade-off relation between power and efficiency for quantum heat engines.
The quantum jump approach to dissipative dynamics was first conceived for applications in quantum optics [67][68][69][70] and has long been recognized as a powerful tool to extend the notions of stochastic thermodynamics into the quantum regime [10,71].The quantum jump record is thereby commonly treated as an analogue of a classical trajectory, along which fluctuating thermodynamic quantities can be consistently defined by invoking the two-point measurement scheme [72,73].Here, we pursue an alternative approach: instead of considering accumulated quantities over an entire record, we focus on the statistics of the quanta of thermodynamic fluxes that are exchanged between the system and its environment in individual quantum jumps.This strategy opens a new perspective on thermodynamic processes in systems with quantized energy levels and enables us to built a connection between the microscopic anatomy of the energy flow in a quantum engine cycle and its overall performance.
Our paper is organized as follows.In the next section, we set the stage by briefly reviewing the basics of the experimental setup proposed in Refs.[57][58][59][60][61][62] and discuss how it can be used to realize a microscopic heat engine with a superconducting qubit.We also introduce the concept of single-jump distributions and illustrate this idea with a numerical simulation.In Sec.III, we set up the theoretical framework for our analysis and proceed in several steps towards our main result: a family of new trade-off relations between power and efficiency for quantum heat engines, which hold for arbitrary multi-level systems and driving protocols.These relations involve only physically transparent parameters that can be determined through single-photon measurements and they unify several earlier results, which we recover as special cases.To demonstrate the quality of our bounds, we apply them to the qubit engine of Sec.II in Sec.IV.We discuss possible directions for future research and conclude in Sec.V.

II. QUBIT ENGINE: SETUP
We consider a solid-state realization of a microscopic heat engine with a superconducting qubit.This working system can be described by the Hamiltonian where Ω sets the overall energy scale and the dimensionless parameters ∆ and V correspond to the characteristic tunneling energy of the qubit and the flux-tunable energy bias [74,75]; σ x and σ z are the usual Pauli matrices and denotes Planck's constant.The qubit is coupled to the electronic degrees of freedom of a metallic island, which acts as a mesoscopic reservoir, see Fig. 1(a).An engine cycle is realized by periodically changing the energy bias V and the base temperature T of the island.For simplicity, we here focus on harmonic driving protocols given by (2) so that the energy bias V t and the normalized temperature T t /T 0 oscillate between 1 and 2, where T 0 is the base temperature of the island and T denotes the cycle time.
In the low-temperature regime, the electron gas inside the island features a low heat capacity and a short internal relaxation time of the order of nanoseconds.After absorbing or emitting a photon, the electronic subsystem therefore first settles to an internal equilibrium state before returning to the base temperature of the island via electron-phonon mediated heat flow to the substrate.This equilibration process takes place on a much longer time scale, on the order of 100 ns [62,76,77].This mechanism leads to spikes and dips in the temperature trace Tt of the electron gas, which can be detected with an ultra-sensitive electron thermometer, see Fig. 1(b).It thus becomes possible to detect the exchange of single photons and to measure a quantum jump record, for every operation cycle of the engine.Here, t k is the time at which the event k occurs and the binary variable d k indicates whether a photon was emitted (d k = +) or absorbed (d k = −) by the reservoir, that is, whether the qubit jumped to its excited or ground state.Each detected event indicates the transfer of discrete amounts of heat and effective thermal energy, Q and Û , The left box represents the qubit, whose two energy levels are sketched as functions of the energy bias V .The two boxes on the right are the calorimeter, which consists of the electron gas inside a metallic island, and an ultra-sensitive thermometer monitoring its temperature.The small panels on the bottom show the driving protocols for the energy bias of the qubit and the base temperature of the island, cf.Eq. ( 2).Each photon exchange between the qubit and the island is accompanied by a quantum jump of the qubit and a short-lived variation of the electron temperature.(b) Temperature fluctuations of the electron gas, δ Tt ≡ Tt − Tt, as detected by the thermometer for five different operation cycles.(c) Single-jump distribution of the exchanged heat between qubit and reservoir, cf.Eq. (4a).(d) Single-jump distribution of the effective thermal input from the heat source, cf.Eq. (4b).(e) Power vs efficiency for ∆ = 0, 0.5, 0.95 from top to bottom, cf.Eq. ( 5).The plots in panels (b)-(d) were obtained from a stochastic wave-function simulation with the jump operators (54) for T = 1/Ω [70].For the plots (c) and (d), averages were taken over 5000 cycles.The plot (e) was prepared by numerically calculating the time-dependent density matrix of the qubit for increasing cycle times running from T = 0.1/Ω to T = 250/Ω.For all numerical calculations we have set κ = 10 and T0 = Ω, see Sec.IV for details.
between the qubit and the reservoir.The statistics of these thermal quanta is described by the single-jump distributions which can be determined by running the experiment over a large number of cycles, see Figs. 1(c) and 1(d).Here, the symbol E indicates the average over all jump records, ε t ≡ ΩV t is the time-dependent level splitting of the qubit, η t ≡ 1 − T 0 /T t is the instantaneous Carnot factor and the normalization factor A corresponds to the mean number of jumps per cycle.Note that, throughout this article, we use hats to indicate thermodynamic quantities associated with single quantum jumps.
The mean value Q ≡ Q/A of the heat flux Q determines the average output of the engine since the first law requires Q = W , where Q is net heat uptake of the working system per cycle and W the produced work.In analogy, the mean value Û ≡ U/A of the effective thermal energy flux Û can be regarded as the input provided by the external heat source [11,13].This identification leads to a consistent generalization of the standard thermal efficiency for heat engines that are driven by continuous temperature variations, ( The upper bound 1 of this figure of merit follows directly from the second law and corresponds to the Carnot limit, see Sec.III A for details.To further motivate this definition, it is instructive to consider the special case of Carnot-type engines, which operate between two fixed temperature levels T 0 and T 1 > T 0 .For such cycles, the effective input becomes U = η C Q 1 , where η C ≡ 1−T 0 /T 1 denotes the Carnot factor and Q 1 is the heat uptake during the hot phase of the cycle, which is considered the input of a heat engine in classical thermodynamics.The generalized efficiency (5) thus reduces to the normalized thermal efficiency and we have η = η th /η C with η th ≡ W/Q 1 .Hence, our approach is consistent with the standard model of classical thermodynamics.
In the plot in Fig. 1(e), we observe that the efficiency η approaches the ideal value 1 only in the quasi-static limit T → ∞, where the power P ≡ W/T goes to zero.At the same time, the power at fixed efficiency decreases with the tunneling energy ∆.In the following section, we will show that this behavior can be understood from general trade-off relations, which are determined by the singlejump statistics of the thermodynamic fluxes between the working system and its environment.

A. Thermodynamics
For a general model of a quantum heat engine, we now consider an arbitrary multi-level system, whose Hamiltonian H t can be modulated through external fields to extract mechanical work.The temperature T t of the reservoir that forms the environment is controlled by a heat source that provides thermal energy.The thermodynamics of the system is governed by the first and the second law, In the weak-coupling regime, which we focus on in this article, the internal energy and entropy of the system can be expressed in terms of its density matrix ρ t as [78] Furthermore, the rate of heat uptake from the environment and the extracted mechanical power are given by Φ t = tr ρt H t and P t = − tr ρ t Ḣt , the symbol Σ t denotes the total rate of entropy production and dots indicate time derivatives.Note that we set Boltzmann's constant to 1 throughout.Under continuous periodic driving, the system settles to a limit cycle state and its internal energy and entropy become periodic functions of time.Integrating the first law over a full period T thus gives the identity being the mean heat uptake and work output per cycle.Analogously, the second law leads to the relation Here, ∆S tot is the average total entropy production per cycle.We further recall that η t ≡ 1 − T 0 /T t is the instantaneous Carnot factor with respect to the base temperature T 0 ≤ T t of the environment and that U corresponds to the effective input of thermal energy from the heat source [11,13].
The inequality (10) shows that the efficiency of a general engine cycle can be consistently defined as This figure attains its upper bound 1 in the reversible limit, for which ∆S tot = 0.This condition, however, can be met in generic systems only under quasi-static driving leading to vanishing power.A quantitative description of this trade-off between power and efficiency cannot be derived from the elementary laws of thermodynamics and requires a microscopic model for the dynamics of the working system, which we introduce in the next section.

B. Dynamical Model
The density matrix of the working system evolves according to a linear master equation with the form [70] The structure of the generator L t thereby depends on the coupling mechanism between the system and its environment and on the hierarchy of the involved time scales.Here, we focus on the adiabatic weak-coupling regime, where the system-environment interactions can be treated perturbatively and the applied driving is slow compared to the both the unperturbed evolution of the working system and the relaxation dynamics of the surrounding reservoir.The fluctuations Tt of the reservoir temperature displayed in Fig. 1(b) therefore do not have to be taken into account in the discussion of the system dynamics.The generator L t can then be divided into a unitary part, which describes the evolution of the bare working system, and a dissipation superoperator, which accounts for the influence of the environment, that is, Here, [A, B] ≡ AB − BA denotes the commutator.Owing to micro-reversibility, the dissipation superoperator can be further decomposed into independent Markovian dissipation channels.Specifically, we have The jump operators J α+ t and J α− t , which respectively describe the emission and absorption of a photon with energy ε α t > 0 by the reservoir, fulfill the relation and the detailed-balance condition For a detailed discussion of the microscopic basis and the range of validity of the adiabatic weak-coupling approach, see for example Refs.[78][79][80][81].To ensure that the working system settles to a unique limit cycle state, we further require that the jump operators connect all energy levels of the working system during a finite fraction of the cycle [82].
Using the master equation ( 12) and the structure of the generator (13), the rate of heat uptake and the total rate of entropy production can be expressed as denotes the instantaneous Gibbs state of the working system.Upon observing that D t [R t ] = 0 as a consequence of the conditions ( 15) and ( 16), it follows from Spohn's theorem that Σ t ≥ 0 for any ρ t [84].This result shows that the adiabatic weak-coupling approach is inherently consistent with the second law, for details, see Ref. [13].

C. Quantum Jump Statistics
To develop a quantum jump description of microscopic heat engines, we now assume that the reservoir can be continuously monitored such that the external observer obtains a channel-resolved quantum jump record R for every operation cycle of the device.Extending the notation introduced in Sec.II, we write where t k is the time at which the event k in the dissipation channel α k is detected and d k = ± indicates whether a photon was emitted (+) or absorbed (−) by the reservoir.After collecting sufficiently many records, the single-jump distribution can be determined for every thermodynamic flux X that is exchanged between the system and the reservoir.Recall that E denotes the average over all possible records and A is the mean number of jumps per cycle; by X α t , we denote the amount of the quantity X that is carried by a photon in the channel α at the time t.For example, the fluxes of heat and of effective thermal energy are characterized by Q α t = ε α t and U α t = ε α t η α t , respectively.To derive an explicit expression for the distribution (19), we have to analyze the dynamics of the engine under continuous monitoring.To this end, we use the stochastic wave function method, which unravels the master equation (12) into measurement-conditioned quantum trajectories with piecewise deterministic evolution of the pure state |ψ t of the system [67][68][69][70][71].In this approach, every detected event (d, α) corresponds to a quantum jump, which is described by the transformation Here, |ψ 2 ≡ ψ | ψ denotes the norm of the state |ψ .Between two consecutive jumps at the times t and t > t, the state changes continuously according to the transformation where the non-unitary time evolution operator is given by the anti-chronologically ordered exponential with the effective Hamiltonian Hence, if the record R is observed over the period T , the initial state |ψ 0 undergoes the transformation where the record-conditioned time evolution operator is found by successively applying the transformation rules ( 21) and (22).The arrow in Eq. ( 25) indicates the the product is ordered anti-chronologically, M is the total number of events in the record R and we set t 0 ≡ 0. The probability density to observe a given record R for the initial state |ψ 0 can now be expressed as Consequently, if the system is initially in the mixed state ρ 0 = j r j 0 |ψ j 0 ψ j 0 |, the cycle average of any recorddependent observable X can be expressed as where we have introduced the notation for the sum over all records between 0 and T .As we show in App.B, this formula leads to the compact expression for the moments of the single-jump distribution P[ X], which we will use in the next section.The variables correspond to the mean flux of photons that is absorbed (+) or emitted (−) by the system through the channel α at the time t and the activity A is given by

D. Bounds on Entropy Production
The total entropy production ∆S tot provides a measure for the thermodynamic cost of running a cyclic heat engine in finite time.In the following, we first show that this cost can be divided into two non-negative contributions, one arising from quantum jumps and one stemming from the decay of coherences.We then derive a lower bound on the jump entropy production, which depends only on the activity A and the dimensionless parameter which we refer to as the homogeneity of the flux X.These results will provide the basis for the derivation of our new trade-off relation between power and efficiency.

Decomposition of entropy production
We begin our analysis by observing that, upon inserting the spectral decomposition of the density matrix, ρ t = j r j t |ψ j t ψ j t |, the expressions ( 17) and ( 29) for the total rate of entropy production and the average photon fluxes can be rewritten as  [85], which yields [86] After integrating both sides of this relation over a full cycle, we end up with the result (34) This bound admits a transparent physical interpretation, which derives from the observation that the quantity ∆S j can be expressed as where the flux Σj corresponds to the entropy production of single quantum jumps.The quantity ∆S j thus provides a measure for the average thermodynamic cost of all jumps in one cycle.The remainder of the total entropy production, ∆S c ≡ ∆S tot − ∆S j ≥ 0, (36) stems from the non-unitary evolution of the system between the jumps, that is, from the decay of superpositions between different energy levels [87].It can therefore be interpreted as a measure for the thermodynamic cost of coherence.As we show in App.C, the contribution ∆S c indeed vanishes in the quasi-classical regime, where the density matrix of the system commutes with its Hamiltonian throughout the cycle and every jump operator can be identified with a single transition between two energy levels.Under these conditions, equality is attained in Eq. ( 33) and the bound (34) becomes trivial.

Homogeneity bound
In order to derive a lower bound on the jump entropy production, we first introduce the weighting factors and the rescaled photon fluxes , which fulfill the relations We thereby defined the auxiliary variable The expression (34) can now be cast into the form Finally, because the right-hand side of the inequality ( 41) is monotonically decreasing in |Ξ|, this variable can be eliminated by replacing it with its upper bound which again follows from Jensen's inequality.Recalling the definition (31) thus leaves us with the compact result which shows that the jump entropy production is bounded from below by a monotonically increasing function of the homogeneity of any thermodynamic flux X.This figure attains its upper limit 1, for which the right-hand side of Eq. ( 43) diverges, if the corresponding single-jump distribution P[ X] has zero width, indicating that either emissions or absorptions are fully suppressed and every photon carries the same amount of the quantity X.Any deviation of λ X from 1 signifies fluctuations in the single-jump units of X with the lower limit 0 being attained if no net exchange of the quantity X takes place between the system and the reservoir, i.e., if X = 0.The relation (43) then reduces to the trivial bound ∆S j ≥ 0.

E. Performance Bounds for Quantum Heat Engines
Our bounds on entropy production ( 34) and ( 43) imply a whole family of trade-off relations between power and efficiency, which we derive in two steps.In the first one, we obtain a simple relation, which depends on the second single-jump moment of the effective thermal input and allows us to recover two earlier results.We then derive an optimal trade-off relation, which is stronger than the simple one but involves more parameters.

Simple trade-off relation
We first consider the effective thermal input Û and note that the bounds (34) and (43) together imply (44) Upon recalling Eqs. ( 5), (10) and (31), the first of these bounds can be rewritten as a trade-off relation between the power and the efficiency, which is given by with γ ≡ A/T denoting the average jump rate.This result shows that, for generic systems with finite γ, the power output of any cyclic engine must go to zero as its efficiency approaches the ideal value 1.The linear slope of this decay is determined by the second singlejump moment of thermal input Û .Moreover, the bound (45) becomes successively stronger as the parameter Ψ decreases, that is, as the coherence-induced entropy production ∆S c = ∆S tot − ∆S j increases.In line with previous results [7,13,15,29], this behavior indicates that coherence is generally detrimental to the performance of microscopic heat engines, at least under weak-coupling and slow-driving conditions.The trade-off relation (45) includes two earlier results as special cases.First, for small driving amplitudes, it reduces to the bound that was obtained in Ref. [13] as we show in App.D. Second, for Carnot-type cycles with two heat baths at different temperatures, Eq. ( 45) becomes denoting the second single-jump moment of the heat uptake during the hot phase of the cycle T 1 ; recall that η C and η th denote the Carnot factor and the thermal efficiency.Upon noting that tanh[a] ≤ a for a ≥ 0, this bound can be reduced to the weaker trade-off relation which was derived in Refs.[36,37] by Shiraishi and coworkers.
Before moving on, it is worth noting that applying the bound (43) to the heat uptake Q instead of the thermal input Û yields the alternative trade-off relation which is, however, weaker than the one in Eq. ( 45), since Û 2 ≤ Q2 ≤ Q2 /η 2 and the hyperbolic tangent is a monotonically increasing function.

Optimal trade-off relation
We now consider the flux whose first and second moment are given by with R α t ≡ ε α t √ η t and ϕ being an arbitrary real number.
Upon applying the bound (43), this ansatz yields the general trade-off relation which includes the two results (45) and (48) as limiting cases for ϕ → 0 and ϕ → ∞, respectively.Its strongest form is obtained by choosing ϕ such that the right-hand side of the inequality (51) becomes minimal.This value, which can be found by inspection, also maximizes the homogeneity λ Ŷϕ of the flux Ŷϕ and is given by Inserting this result into Eq.( 51) gives the optimal tradeoff relation As we will show in the following section, this bound can be significantly stronger than the simple one in Eq. ( 45).We stress that, despite its complex structure, the tradeoff relation ( 53) could be tested in experiments since it involves only parameters that would be accessible through single-photon measurements.

IV. QUBIT ENGINE: NEW BOUNDS
To probe the quality of our new trade-off relations, we now return to the qubit engine discussed in Sec.II.The dissipative dynamics of this system can be described with the two jump operators [76] where the dimensionless parameter κ determines the average jump frequency and |E + and |E − are the eigenvectors of the Hamiltonian (1) with corresponding eigenvalues E ± = ± ΩV .Upon inserting the protocols (2) for the energy bias V and the base temperature of the reservoir T , the periodic density matrix of the qubit can be determined by numerically solving the master equation (12).The work output W , the thermal input U and the total entropy production ∆S tot of the engine can then be evaluated using Eqs.( 8), ( 9) and ( 10) [15].Furthermore, the second single-jump moments of the fluxes Q, Û , R and the first moment of Σj , which enter the trade-off relations (45) and (53), can be evaluated with the help of Eqs. ( 29) and (28).
The results of this analysis are plotted in Fig. 2. They show that the simple trade-off relation (45) overestimates the power of the qubit engine by a factor between 4 and 9.By contrast, the optimal bound (53) closely follows the exact power-efficiency curve and practically saturates for small η.Exact saturation is, in fact, achieved for small driving amplitudes and optimal protocols as we show in App.D. As a second key observation, we find that the power at fixed efficiency is uniformly suppressed along with its upper bounds in the tunneling energy ∆.This behavior can be understood by noting the engine is quasiclassical in the limit ∆ → 0, where the eigenstates of the Hamiltonian (1) become independent of the energy bias V .As ∆ deviates from 0, the driving generates superpositions between the two energy levels of the system.This effect leads to coherence-induced dissipation and thus reduces the performance of the engine.

V. CONCLUDING PERSPECTIVES
Power and efficiency are arguably the two most important benchmarks for the performance of a heat engine.Quantitative bounds that make it possible to assess the trade-off between these two figures are key results of the theory of microscopic heat engines that has emerged over the last years.This paper contributes to these ongoing developments in two ways.On the conceptual side, our analysis shows that, within the adiabatic weak-coupling regime, a whole family of trade-off relations between power and efficiency can be derived in a technically simple and transparent manner.These relations, which unify and extend previous results, were obtained only through the repeated application of Jensen's inequality.As we show in App.A, it is straightforward to generalize this technique for setups involving multiple reservoirs and other types of thermal devices such as microscopic refrigerators.From a practical perspective, our approach delivers a clear physical interpretation of the additional parameters that determine the relation-ship between the power and the efficiency of microscopic heat engines.Inspired by current developments in the area of superconducting circuits, our theory provides a promising avenue towards practical tests of thermodynamic trade-off relations in future experiments, which could shed new light on the working mechanisms of microscopic thermal devices.
Further theoretical research will be necessary to expand our approach into the regimes of fast driving and strong coupling, where the performance of heat engines can be enhanced through coherence [22,[90][91][92].In addition, a fully realistic model of small-scale calorimetric measurements must account for the finite size of the electronic reservoir and the back action of its temperature fluctuations on the working system.Investigating how our bounds will be altered by these effects is an important subject for future work.In paving the way for such studies, our paper contributes to the general goal of a unified and experimentally confirmed theory of thermodynamic trade-off relations for microscopic thermal devices.
All three of the bounds (A20), (A21) and (A23) show that, first, the Carnot limit ω C can, for finite jump rates γ ν , be attained only at the price of vanishing cooling power.Second, the maximum cooling power at given efficiency decreases uniformly with the coherence factor Ψ. Hence, like microscopic heat engines, micro-coolers can be expected to perform best in the quasi-classical limit, as has been observed before for qubit-based devices [21,94].Which of the bounds (A20), (A21) and (A23) is strongest, in general, depends on the specific setting.which corresponds to Eq. ( 45) for Ψ = 1, since we focus on the quasi-classical limit here.This result was derived earlier in Ref. [13].For two-temperature cycles, it becomes the linear-response version of the bound (47), which goes back to Refs.[36,37].
The bound (D9) becomes strongest for as can be easily verified by inspection.For this choice, we obtain the optimal trade-off relation which corresponds to Eq. ( 53).Note that the scaling factor between power and efficiency in the bounds (D12) and (D14) is independent of the mechanical protocol f w t .Furthermore, Eq. (D14) implies the efficiency-independent bound on power, which was derived in Ref. [29].Finally, it is instructive to note that the optimal tradeoff relation (D14) can be saturated if the variable G w is proportional to the unperturbed Hamiltonian H 0 and the equilibrium energy correlation function decays mono-exponentially, that is, if G w = H 0 /ζ and (D16) where ζ and µ > 0 are real constants.These conditions are met, for example, for the qubit engine discussed in the main text.In this case, the mechanical protocol f w * t that generates the maximal work for a given temperature protocol f u t and fixed efficiency η is given by This result can be derived by expanding the protocols f w t and f u t into Fourier series and maximizing the work W with respect to the Fourier coefficients of f w t under the constraint W − ηU = 0, for details see Refs.[13,19].Evaluating the kinetic coefficients (D3) for the protocols (D17) and using the conditions (D16) shows that the bound (D14) is indeed saturated with being proportional to the equilibrium energy fluctuations of the working system.Hence, we can conclude that our optimal trade-off relation between power and efficiency, Eq. ( 53), can be saturated in linear response.The analogous relation holds for any non-negative function ϕ[x] on D with φ ≡

Figure 1 .
Figure1.Quantum jump thermodynamics of a single-qubit engine.(a) Setup.The left box represents the qubit, whose two energy levels are sketched as functions of the energy bias V .The two boxes on the right are the calorimeter, which consists of the electron gas inside a metallic island, and an ultra-sensitive thermometer monitoring its temperature.The small panels on the bottom show the driving protocols for the energy bias of the qubit and the base temperature of the island, cf.Eq. (2).Each photon exchange between the qubit and the island is accompanied by a quantum jump of the qubit and a short-lived variation of the electron temperature.(b) Temperature fluctuations of the electron gas, δ Tt ≡ Tt − Tt, as detected by the thermometer for five different operation cycles.(c) Single-jump distribution of the exchanged heat between qubit and reservoir, cf.Eq. (4a).(d) Single-jump distribution of the effective thermal input from the heat source, cf.Eq. (4b).(e) Power vs efficiency for ∆ = 0, 0.5, 0.95 from top to bottom, cf.Eq. (5).The plots in panels (b)-(d) were obtained from a stochastic wave-function simulation with the jump operators (54) for T = 1/Ω[70].For the plots (c) and (d), averages were taken over 5000 cycles.The plot (e) was prepared by numerically calculating the time-dependent density matrix of the qubit for increasing cycle times running from T = 0.1/Ω to T = 250/Ω.For all numerical calculations we have set κ = 10 and T0 = Ω, see Sec.IV for details.

Figure 2 .
Figure 2. Power and efficiency of the qubit engine for three different tunneling energies.In each panel, the three curves, from top to bottom, show the simple bound (45), the optimal bound (53) and the actual power output of the engine as a function of its efficiency.The shaded areas under the two upper curves indicate the admissible regions of the power-efficiency plane for the corresponding bounds.All plots were prepared by varying the cycle time from T = 0.1/Ω to T = 250/Ω for κ = 10 and T0 = Ω.