Bounds on fluctuations for ensembles of quantum thermal machines

Significant efforts in stochastic and quantum thermodynamics [1–5] are currently devoted to understanding trade-off relations in nanoscale thermal machines by weighting currents, their fluctuations, entropy production, and efficiency. As an example, the 'thermodynamic uncertainty relation' (TUR) describes a trade-off between precision (relative fluctuations) and cost (entropy production). References [6–19] constitute representative examples in this broad and active field. The TUR further constrains the performance of thermal engines, balancing output power, power fluctuations and the engine's efficiency [7].

A separate class of bounds, which are independent of the TUR, concerns ratios between fluctuations of different currents: the output power and input heat current [20–24]. For the process of refrigeration, for instance, one considers the ratio

$$\begin{equation}{\eta }^{(n)}\equiv \frac{\langle \langle {q}^{n}\rangle \rangle }{\langle \langle {w}^{n}\rangle \rangle },\end{equation} \tag{ 1 }$$

with q as the extracted stochastic cooling heat current and w the corresponding input heat current; ⟨⟨Aⁿ ⟩⟩, n = 1, 2, ... is the nth cumulant of a stochastic variable A. For n = 2 this ratio is lower and upper-bounded in linear response for continuous machines in steady state under time reversal symmetry [22],

$$\begin{equation}{\eta }^{2}\leqslant {\eta }^{(2)}\leqslant {\eta }_{\mathrm{C}}^{2}.\end{equation} \tag{ 2 }$$

η_C is the Carnot bound and η (our short notation for η⁽¹⁾) stands for the efficiency of the machine. Based on the upper bound on η⁽²⁾, a tighter-than-Carnot efficiency bound has been derived [20–23] and shown to be tighter than bounds received from the TUR [24].

In this paper, we focus on an ensemble of distinct, continuous, steady-state thermal machines, and inquire on universal aspects of their fluctuations—beyond linear response. The machines operate between the same affinities (temperatures, chemical potentials), but are different in their working parameters such as internal energies and system-bath coupling strength. We then ask the following basic question: assuming the relative fluctuations of an individual machine (operating beyond the linear response limit) are upper and lower bounded according to equation (2), do these bounds hold for an ensemble of distinct, uncorrelated machines?

This question is not trivial, as we show below. It is particularly difficult to address the lower bound: while all our machines are upper-bounded by the same Carnot limit, their efficiencies η (dictating the lower bound) are distinct. To make progress, we consider here individual machines operating in the so-called tight-coupling (TC) limit [25, 26]: in each machine, the stochastic input and output currents are proportional to each other, w ∝ q. Under the TC limit, one can readily prove, beyond linear response, the validity of the upper bound in equation (2), and show that the lower bound is saturated—for an individual machine [22]. However, TC does not necessarily hold when studying a collection of subsystems, even when the constituent elements follow it. Therefore, as we discuss here, proving in general the lower bound on η⁽²⁾ is a nontrivial task.

We now pose the problem to be addressed in this study. We consider individual TC machines (e.g. refrigerators) whose relative fluctuations ${\eta }_{k}^{(n)}$ [see equation (1)] for each member k are bounded according to

$$\begin{equation}{\eta }_{k}^{n}={\eta }_{k}^{(n)}\leqslant {\eta }_{\mathrm{C}}^{n},\quad n=1,2,\dots \end{equation} \tag{ 3 }$$

with the efficiency η_k possibly different for each member in the ensemble. Our objective is to find whether an ensemble of N > 1 distinct, independent machines satisfies the relations

$$\begin{equation}{\eta }_{N}^{n}\leqslant {\eta }_{N}^{(n)}\leqslant {\eta }_{\mathrm{C}}^{n},\quad n=2,3,\dots \end{equation} \tag{ 4 }$$

arbitrarily far from equilibrium; the validity of the upper and lower bounds in the linear response regime was proved in reference [22]. The currents considered in devising the ratios (1) are the total ones, $w={\sum }_{k=1}^{N}\enspace {w}_{k}$ and $q={\sum }_{k=1}^{N}\enspace {q}_{k}$. That is, e.g. ${\eta }_{N}^{(n)}$ is the efficiency (n = 1) or ratio of fluctuations (n = 2) of a system made of N constituents. Expressing equation (4) in words: considering ratios of cumulants of output to input currents in a collection of thermal machines, we would like to show that this ratio is upper bounded—by the nth power of the Carnot efficiency. Further, we interrogate whether a lower bound holds for the ensemble, given by the ensemble-averaged efficiency to the power n.

As we show in this paper, the upper bound in equation (4) can be readily proved for any order n. As for the lower bound, we mostly restrict ourselves to the behavior of fluctuations, n = 2, and we prove it in specific limits. Beyond those cases, we establish the lower bound more broadly for refrigerators based on numerical simulations of three-level quantum absorption refrigerators (3LQARs) and thermoelectric junctions. In contrast, we find examples of pairs (N = 2) of thermoelectric engines that violate the lower bound in equation (4) in the far from equilibrium regime.

The question posed here, on the validity of bounds for an ensemble of machines, is critical to our ability to experimentally confirm fundamental theoretical results. In some experiments, quantum machines are constructed from a collection of systems, such as trapped ions [27], quantum dots (see review [28]), and molecules [29]. Practically, even when aiming for homogeneity, these components cannot be made precisely identical in their energies and couplings to the environment. Moreover, even for machines made from an individual 'particle' [30–34], experiments inevitably suffer from a certain degree of uncertainty, requiring an ensemble average. Considering e.g. quantum absorption refrigerators based on superconducting circuits [35] or nitrogen vacancy centers in diamond [36], internal energies defining the refrigerator, as well as system-bath coupling parameters cannot be precisely-repeatedly realized, thus measurements necessarily rely on averaging.

We highlight that the ratio of cumulants as defined in equation (1) is unrelated to the concept of efficiency fluctuations explored e.g. in references [37–41]. In these studies, one defines the stochastic efficiency and studies its probability distribution function. In contrast, here we focus on the currents as the stochastic variables, we construct their cumulants, then look at their ratios to define η⁽ⁿ⁾.

As a final introductory comment, while our examples concern quantum thermal machines, their quantum nature only involves the discreteness of their energy levels and the quantum statistics of the baths (bosonic or fermionic). As such, bounds discussed in this work are directly applicable to classical systems.

The paper is organized as follows. In section 2 we briefly review preexisting results derived in the regime of linear response. The upper bound for an ensemble of TC machines is proved in section 3. In section 4, we discuss the lower bound and arrange it in alternative forms. We investigate the validity of the lower bound in different limits with two models for refrigerators: in section 5 we focus on 3LQARs, while in section 6 we examine thermoelectric refrigerators. In section 7 we demonstrate that the lower bound can be violated for thermoelectric engines operating far from equilibrium. Appendices A–C provide additional proofs and supporting simulations for the lower bound in different limits.

Continuous thermal machines operating close to equilibrium, such that the mean currents, ⟨q⟩ and ⟨w⟩, exhibit only linear dependence on thermodynamic affinities (differences in temperature, chemical potential), are said to operate in the regime of linear response. With two affinities, these currents are expressed in terms of the Onsager response matrix as $\langle {I}_{i}\rangle ={\sum }_{j=1}^{2}\enspace {L}_{ij}{A}_{j}$, where ⟨I_i ⟩ is either mean current (⟨q⟩ or ⟨w⟩), A_i is the corresponding affinity, and the coefficients ${L}_{ij}={\partial }_{{A}_{j}}\langle {I}_{i}\rangle {\vert }_{\left\{{A}_{k}\right\}=0}$ are the Onsager matrix elements. Under the time reversibility of the microscopic dynamics, a reciprocal relation holds with L_ij = L_ji . The fluctuation–dissipation relation gives the variance (fluctuations) in these currents as $\langle \langle {I}_{i}^{2}\rangle \rangle =2{L}_{ii}$, while the positivity of entropy production demands that det L ⩾ 0 [26].

The relative fluctuations for each current I_i may be defined as $\langle \langle {I}_{i}^{2}\rangle \rangle /{\langle {I}_{i}\rangle }^{2}$. Under microscopic reversibility, and provided that the system is in the appropriate operational regime (i.e. that it is indeed behaving as a refrigerator or engine, rather than undergoing only spontaneous processes), the properties of the Onsager matrix have been used to bound the ratio, $\mathcal{Q}$ of relative fluctuations of the output to input currents as $\mathcal{Q}\geqslant 1$. This inequality gives a result equivalent to the lower bound on η⁽²⁾ expressed in equation (2). The same assumptions have been used to prove the upper bound as well, giving the full hierarchy of equation (2) [22]. Since these linear response results are model-independent, they apply to the ensembles of TC machines that are the main topic of this paper.

There are, however, inherent limitations in the use of the Onsager formalism for deriving bounds on higher-order cumulants of currents in thermal machines. It cannot be applied to obtain any results for η⁽ⁿ⁾ if n ⩾ 3. Furthermore, it is not valid farther away from equilibrium, as the affinities grow in magnitude. Continuous thermal machines must operate far from equilibrium in order to generate a substantial power output, highlighting the importance of this regime for practical applications. While the level of model-independence achieved for results in the linear response regime cannot easily be obtained far from equilibrium, the following sections represent one approach to building an understanding of bounds on η⁽ⁿ⁾ under more general circumstances, albeit for a more restricted class of machines.

In this section, we prove that arbitrarily far from equilibrium, ratios of cumulants of order n of an ensemble with N noninteracting and uncorrelated distinct heat machines, operating under the same affinities, are bounded by

$$\begin{equation}{\eta }_{N}^{(n)}\leqslant {\eta }_{\mathrm{C}}^{n},\end{equation} \tag{ 5 }$$

so long as the inequality is satisfied at the level of the individual machine, ${\eta }_{k}^{(n)}\leqslant {\eta }_{\mathrm{C}}^{n}$. η_C is the Carnot bound dictated by the temperatures common to all machines.

The proof holds for engines and refrigerators; we describe it in the language of refrigerators. We consider small, possibly quantum, refrigerators with $w={\sum }_{k=1}^{N}\enspace {w}_{k}$ the total stochastic heat current absorbed in the cooling process from the so-called work bath and $q={\sum }_{k=1}^{N}\enspace {q}_{k}$ the total extracted heat current from the cold bath. The components operate independently and are not correlated. We assume that each individual member of the ensemble satisfies

$$\begin{align} {\eta }_{k}& \equiv \frac{\langle {q}_{k}\rangle }{\langle {w}_{k}\rangle }\leqslant {\eta }_{\mathrm{C}}, \\ {\eta }_{k}^{(2)}& \equiv \frac{\langle \langle {q}_{k}^{2}\rangle \rangle }{\langle \langle {w}_{k}^{2}\rangle \rangle }\leqslant {\eta }_{\mathrm{C}}^{2}, \\ {\eta }_{k}^{(n)}& \equiv \frac{\langle \langle {q}_{k}^{n}\rangle \rangle }{\langle \langle {w}_{k}^{n}\rangle \rangle }\leqslant {\eta }_{\mathrm{C}}^{n},\quad n > 2. \end{align} \tag{ 6 }$$

To assist readers, we explicitly included the first two definitions. It can be shown that equation (6) holds in the TC limit even far from equilibrium—for individual machines [22]. Concrete systems operating in the TC limit are 3LQARs and resonant-level thermoelectric junctions [22].

For clarity, we begin with the case N = 2 and consider two refrigerators A and B. It is not difficult to prove that the ratio of their total fluctuations are bounded by the Carnot-efficiency squared,

$$\begin{align} {\eta }_{N=2}^{(n)}& \equiv \frac{\langle \langle {({q}_{A}+{q}_{B})}^{n}\rangle \rangle }{\langle \langle {({w}_{A}+{w}_{B})}^{n}\rangle \rangle }=\frac{\langle \langle {q}_{A}^{n}\rangle \rangle +\langle \langle {q}_{B}^{n}\rangle \rangle }{\langle \langle {w}_{A}^{n}\rangle \rangle +\langle \langle {w}_{B}^{n}\rangle \rangle } \\ & =\frac{\langle \langle {q}_{A}^{n}\rangle \rangle }{\langle \langle {w}_{A}^{n}\rangle \rangle }\left(\frac{\langle \langle {w}_{A}^{n}\rangle \rangle }{\langle \langle {w}_{A}^{n}\rangle \rangle +\langle \langle {w}_{B}^{n}\rangle \rangle }\right)+\frac{\langle \langle {q}_{B}^{n}\rangle \rangle }{\langle \langle {w}_{B}^{n}\rangle \rangle }\left(\frac{\langle \langle {w}_{B}^{n}\rangle \rangle }{\langle \langle {w}_{A}^{n}\rangle \rangle +\langle \langle {w}_{B}^{n}\rangle \rangle }\right) \\ & \leqslant {\eta }_{\mathrm{C}}^{n}\left(\frac{\langle \langle {w}_{A}^{n}\rangle \rangle }{\langle \langle {w}_{A}^{n}\rangle \rangle +\langle \langle {w}_{B}^{n}\rangle \rangle }\right)+{\eta }_{\mathrm{C}}^{n}\left(\frac{\langle \langle {w}_{B}^{n}\rangle \rangle }{\langle \langle {w}_{A}^{n}\rangle \rangle +\langle \langle {w}_{B}^{n}\rangle \rangle }\right)={\eta }_{\mathrm{C}}^{n}. \end{align} \tag{ 7 }$$

In the first line, we use the fact that the machines are uncorrelated. The last line is arrived at based on bounds for the individual machines.

Next, along the same principle we prove by induction the (N + 1)th inequality based on the validity of an upper bound for an N-member ensemble. We denote by q_k and w_k the stochastic current of the kth machine. q_N+1 and w_N+1 are the stochastic currents of the (N + 1)th member of the ensemble; η_N (and similarly for higher n) is the efficiency of an N-sized ensemble. We now write

$$\begin{align} {\eta }_{N+1}^{(n)}& \equiv \frac{{\sum }_{k=1}^{N+1}\langle \langle {q}_{k}^{n}\rangle \rangle }{{\sum }_{k=1}^{N+1}\langle \langle {w}_{k}^{n}\rangle \rangle } \\ & =\frac{{\sum }_{k=1}^{N}\langle \langle {q}_{k}^{n}\rangle \rangle +\langle \langle {q}_{N+1}^{n}\rangle \rangle }{{\sum }_{k=1}^{N}\langle \langle {w}_{k}^{n}\rangle \rangle +\langle \langle {w}_{N+1}^{n}\rangle \rangle } \\ & =\frac{{\sum }_{k=1}^{N}\langle \langle {q}_{k}^{n}\rangle \rangle }{{\sum }_{k=1}^{N}\langle \langle {w}_{k}^{n}\rangle \rangle }\left(\frac{{\sum }_{k=1}^{N}\langle \langle {w}_{k}^{n}\rangle \rangle }{{\sum }_{k=1}^{N}\langle \langle {w}_{k}^{n}\rangle \rangle +\langle \langle {w}_{N+1}^{n}\rangle \rangle }\right) \\ & \quad +\frac{\langle \langle {q}_{N+1}^{n}\rangle \rangle }{\langle \langle {w}_{N+1}^{n}\rangle \rangle }\left(\frac{\langle \langle {w}_{N+1}^{n}\rangle \rangle }{{\sum }_{k=1}^{N}\langle \langle {w}_{k}^{n}\rangle \rangle +\langle \langle {w}_{N+1}^{n}\rangle \rangle }\right) \\ & \leqslant {\eta }_{\mathrm{C}}^{n}. \end{align} \tag{ 8 }$$

In the last step we used $\frac{{\sum }_{k=1}^{N}\langle \langle {q}_{k}^{n}\rangle \rangle }{{\sum }_{k=1}^{N}\langle \langle {w}_{k}^{n}\rangle \rangle }\leqslant {\eta }_{\mathrm{C}}^{n}$ per our assumption of the validity of the upper bound for an ensemble with N elements. We also made use of $\frac{\langle \langle {q}_{N+1}^{n}\rangle \rangle }{\langle \langle {w}_{N+1}^{n}\rangle \rangle }\leqslant {\eta }_{\mathrm{C}}^{n}$, valid for every individual machine.

Summing up, we proved that if the inequality ${\eta }^{(n)}\leqslant {\eta }_{\mathrm{C}}^{n}$ holds for an individual machine, it also holds for a machine made of a collection of N > 1 distinct, uncorrelated systems—as long as they operate between the same temperatures, thus bounded by the same Carnot bound. The working elements of our machine are made distinct in their internal parameters and their coupling to the surroundings. For example, in figure 1 we depict a refrigerator with its working fluid including multiple three-level systems that are distinct in their energy spacings. In figure 4, we illustrate a thermoelectric device that comprises an array of independent junctions.

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Unlike the upper bound that we proved in section 3, we are not able to prove the lower bound in general, but only in specific cases (sections 5 and 6). Before discussing these cases, in this section, we limit ourselves to a pair of systems (N = 2) and organize the lower bound in a compatible, curious form.

We consider two machines, labelled A and B, both operating in the TC limit thus satisfying the relation ${\eta }_{k}^{2}={\eta }_{k}^{(2)}$. We focus on the relation

$$\begin{equation}{\eta }_{N=2}^{2}\leqslant {\eta }_{N=2}^{(2)}\end{equation} \tag{ 9 }$$

for the combined N = 2 system. The ratio of fluctuations for the pair is given by

$$\begin{align} {\eta }_{N=2}^{(2)}& =\frac{\langle \langle {({q}_{A}+{q}_{B})}^{2}\rangle \rangle }{\langle \langle {({w}_{A}+{w}_{B})}^{2}\rangle \rangle }=\frac{\langle \langle {q}_{A}^{2}\rangle \rangle +\langle \langle {q}_{B}^{2}\rangle \rangle }{\langle \langle {w}_{A}^{2}\rangle \rangle +\langle \langle {w}_{B}^{2}\rangle \rangle } \\ & ={\eta }_{A}^{(2)}\frac{\langle \langle {w}_{A}^{2}\rangle \rangle }{\langle \langle {w}^{2}\rangle \rangle }+{\eta }_{B}^{(2)}\frac{\langle \langle {w}_{B}^{2}\rangle \rangle }{\langle \langle {w}^{2}\rangle \rangle } \\ & =\lambda {\eta }_{A}^{(2)}+(1-\lambda ){\eta }_{B}^{(2)}, \end{align} \tag{ 10 }$$

where $\lambda \equiv \langle \langle {w}_{A}^{2}\rangle \rangle /\langle \langle {w}^{2}\rangle \rangle$. Crucially, 0 ⩽ λ ⩽ 1. If we assume without loss of generality that ${\eta }_{A}^{(2)}\geqslant {\eta }_{B}^{(2)}$, we have that ${\eta }_{B}^{(2)}\leqslant {\eta }_{N=2}^{(2)}\leqslant {\eta }_{A}^{(2)}$.

We may similarly expand the square of the efficiency itself,

$$\begin{align*} {\eta }_{N=2}^{2}& =\frac{{(\langle {q}_{A}\rangle +\langle {q}_{B}\rangle )}^{2}}{{\langle w\rangle }^{2}} \\ & ={\eta }_{A}^{2}\frac{{\langle {w}_{A}\rangle }^{2}}{{\langle w\rangle }^{2}}+{\eta }_{B}^{2}\frac{{\langle {w}_{B}\rangle }^{2}}{{\langle w\rangle }^{2}}+2{\eta }_{A}{\eta }_{B}\frac{\langle {w}_{A}\rangle \langle {w}_{B}\rangle }{{\langle w\rangle }^{2}} \\ & ={\eta }_{A}^{(2)}\frac{{\langle {w}_{A}\rangle }^{2}}{{\langle w\rangle }^{2}}+{\eta }_{B}^{(2)}\frac{{\langle {w}_{B}\rangle }^{2}}{{\langle w\rangle }^{2}}+2{\eta }_{A}{\eta }_{B}\frac{\langle {w}_{A}\rangle \langle {w}_{B}\rangle }{{\langle w\rangle }^{2}}, \end{align*}$$

where in the last line we have used the strict equality for individual machines due to the TC limit. Because of the TC limit, the relation ${\eta }_{A}^{(2)}\geqslant {\eta }_{B}^{(2)}$ further implies that η_A ⩾ η_B , and ${\eta }_{A}\geqslant \sqrt{{\eta }_{B}^{(2)}}$. With these relations, we get

$$\begin{align*} {\eta }_{N=2}^{2}& \leqslant {\eta }_{A}^{(2)}\left[\frac{{\langle {w}_{A}\rangle }^{2}}{{\langle w\rangle }^{2}}+\frac{{\langle {w}_{B}\rangle }^{2}}{{\langle w\rangle }^{2}}+2\frac{\langle {w}_{A}\rangle \langle {w}_{B}\rangle }{{\langle w\rangle }^{2}}\right]={\eta }_{A}^{(2)}, \\ {\eta }_{N=2}^{2}& \geqslant {\eta }_{B}^{(2)}\left[\frac{{\langle {w}_{A}\rangle }^{2}}{{\langle w\rangle }^{2}}+\frac{{\langle {w}_{B}\rangle }^{2}}{{\langle w\rangle }^{2}}+2\frac{\langle {w}_{A}\rangle \langle {w}_{B}\rangle }{{\langle w\rangle }^{2}}\right]={\eta }_{B}^{(2)}, \end{align*}$$

thus ${\eta }_{B}^{(2)}\leqslant {\eta }_{N=2}^{2}\leqslant {\eta }_{A}^{(2)}$. It is immediately clear that in the special case that ${\eta }_{A}^{(2)}={\eta }_{B}^{(2)}$, a strict equality, ${\eta }_{N=2}^{2}={\eta }_{N=2}^{(2)}$, is obtained. More generally, there exists some 0 ⩽ κ ⩽ 1 such that ${\eta }_{N=2}^{2}=\kappa {\eta }_{A}^{(2)}+(1-\kappa ){\eta }_{B}^{(2)}$. Solving for this coefficient gives

$$\begin{equation}\kappa =\frac{{\langle {w}_{A}\rangle }^{2}}{{\langle w\rangle }^{2}}+2\frac{{\eta }_{B}}{{\eta }_{A}+{\eta }_{B}}\frac{\langle {w}_{A}\rangle \langle {w}_{B}\rangle }{{\langle w\rangle }^{2}}.\end{equation} \tag{ 11 }$$

The relation on the pair of machines, ${\eta }_{N=2}^{2}\leqslant {\eta }_{N=2}^{(2)}$, then, is satisfied exactly when κ ⩽ λ, or,

$$\begin{equation}\frac{{\langle {w}_{A}\rangle }^{2}}{{\langle w\rangle }^{2}}+2\frac{{\eta }_{B}}{{\eta }_{A}+{\eta }_{B}}\frac{\langle {w}_{A}\rangle \langle {w}_{B}\rangle }{{\langle w\rangle }^{2}}\leqslant \frac{\langle \langle {w}_{A}^{2}\rangle \rangle }{\langle \langle {w}^{2}\rangle \rangle }.\end{equation} \tag{ 12 }$$

Since 2η_B /(η_A + η_B ) ⩽ 1, this is always the case for a pair of machines meeting the stronger condition, $\langle \langle {w}_{A}^{2}\rangle \rangle /\langle \langle {w}^{2}\rangle \rangle \geqslant \langle {w}_{A}\rangle /\langle w\rangle$, or, equivalently,

$$\begin{equation}\frac{\langle \langle {w}_{A}^{2}\rangle \rangle }{\langle \langle {w}_{B}^{2}\rangle \rangle }\geqslant \frac{\langle {w}_{A}\rangle }{\langle {w}_{B}\rangle },\quad \mathrm{f}\mathrm{o}\mathrm{r}\enspace {\eta }_{A}\geqslant {\eta }_{B}.\end{equation} \tag{ 13 }$$

Equation (12) is equivalent to the lower bound (9). In contrast, equation (13) provides a stronger condition: satisfying equation (13) necessarily means obeying the original relation, (9). However, we may violate the inequality (13) yet still satisfy the lower bound (9).

Fundamental results on autonomous-continuous thermal machines are often illustrated and examined within simple models for quantum absorption refrigerators [42–44]; recent experiments realized a QAR using trapped ions [27, 32]. In a QAR, heat is extracted from a cold (q) bath and released into a hot (h) bath by absorbing heat from the so-called work (w) reservoir. The reversed operation realizes a heat engine. We identify three temperatures, T_w > T_h > T_q in a QAR, and three stochastic heat currents, w, h, q, defined positive when flowing towards the system.

The performance of QARs has been investigated in different models for elucidating principles in quantum thermodynamics. For example, QARs have been analyzed from weak to strong couplings to the baths [45–48], in models supporting multiple competing cycles [49, 50], and when quantum coherences between eigenstates survive in the steady-state limit [51, 52]; these are illustrative examples out of a rich literature. In this paper, we utilize the three-level model of Scovil and Schulz-DuBois [53] to illustrate bounds on relative fluctuations of currents for an ensemble of QARs. We limit our discussion to the weak system-bath coupling limit, which can be handled with a perturbative quantum Master equation, providing the full counting statistics of the model [52, 54].

A schematic diagram of our model is displayed in figure 1(a). An individual 3LQAR, illustrated in figure 1(b), has particularly served to elucidate concepts in quantum thermodynamics, since at weak system-bath coupling it can be solved analytically. In this model, the three baths are coupled selectively to the different transitions: the cold (q) bath allows the transitions across θ_q , from the ground state to the intermediate level. The work (w) bath is coupled to a transition of energy gap θ_w , from the intermediate level to the top one. In a cooling process, the hot bath (h) extracts the heat, θ_q + θ_w .

5.1. Single refrigerator, N = 1

Consider an individual 3LQAR of spacings θ_q and θ_w . The cooling efficiency of the engine is defined as $\eta =\frac{\langle q\rangle }{\langle w\rangle }$ with q (w) the stochastic cooling (work) heat currents. It can be shown that in the weak system-bath coupling limit, an individual 3LQAR operates in the TC limit: for every quanta θ_q absorbed from T_q , a quanta θ_w must be absorbed from the work bath. The efficiency and η⁽ⁿ⁾ therefore obey the following relations [54]:

$$\begin{align} \eta & \equiv \frac{\langle q\rangle }{\langle w\rangle }=\frac{{\theta }_{q}}{{\theta }_{w}},\enspace \enspace \enspace \enspace \enspace \enspace \\ {\eta }^{(n)}& \equiv \frac{\langle \langle {q}^{n}\rangle \rangle }{\langle \langle {w}^{n}\rangle \rangle }={\left(\frac{{\theta }_{q}}{{\theta }_{w}}\right)}^{n}={\eta }^{n}\quad n=2,3,\dots \end{align} \tag{ 14 }$$

Furthermore, the cooling condition is [54]

$$\begin{equation}\langle q\rangle \propto {n}_{q}({\theta }_{q}){n}_{w}({\theta }_{w})[{n}_{h}({\theta }_{h})+1]-[{n}_{q}({\theta }_{q})+1][{n}_{w}({\theta }_{w})+1]{n}_{h}({\theta }_{h})\geqslant 0,\end{equation} \tag{ 15 }$$

with ${n}_{i}({\theta }_{i})={[{\text{e}}^{{\beta }_{i}{\theta }_{i}}-1]}^{-1}$ the Bose Einstein distribution function of the bath i = h, w, q with transition θ_i . The cooling condition can be equivalently written as

$$\begin{equation}\frac{{\theta }_{q}}{{\theta }_{w}}\leqslant \frac{{\beta }_{h}-{\beta }_{w}}{{\beta }_{q}-{\beta }_{h}},\end{equation} \tag{ 16 }$$

with β_i = 1/T_i the inverse temperature. We identify the left-hand side by the efficiency, η, and the right-hand side by the Carnot efficiency, thus

$$\begin{equation}\eta \leqslant {\eta }_{\mathrm{C}}=\frac{{\beta }_{h}-{\beta }_{w}}{{\beta }_{q}-{\beta }_{h}}.\end{equation} \tag{ 17 }$$

Altogether, an individual 3LQAR operates in the TC limit and it satisfies

$$\begin{equation}{\eta }^{n}={\eta }^{(n)}\leqslant {\eta }_{\mathrm{C}}^{n}.\end{equation} \tag{ 18 }$$

5.2. Ensemble of N > 1 distinct refrigerators

An ensemble of distinct, uncorrelated refrigerators, with possibly different spacings θ_q and θ_w and different system-bath coupling energies, provides a nontrivial setting for exemplifying the lower bound on ratios of fluctuations. We represent such an ensemble in figure 1(c). We assume that all our systems are operating between the same heat baths at temperatures T_i , i = w, h, q, and we fix θ_h .

First, given the validity of the upper bound, equation (18), for each individual machine, we conclude that a similar upper bound holds for the ensemble, as we proved in section 3. In what follows we therefore focus on establishing a lower bound on ${\eta }_{N}^{(n)}$.

5.2.1. QARs with different system-bath couplings

We consider an ensemble of three-level systems, each with different coupling strength to the baths but with the same gaps θ_q and θ_w . Members of this ensemble sit vertically (at the same θ_q ) along different curves, as exemplified in figure 2(a) with asterisks. While the heat currents depend on both the coupling parameters and the energy spacings, notably for any individual refrigerator the ratio η⁽ⁿ⁾ depends only on the latter. Next we prove the saturation of the lower bound,

$$\begin{equation}{\eta }_{N}^{n}={\eta }_{N}^{(n)}.\end{equation} \tag{ 19 }$$

For each 3LQAR, ⟨w_k ⟩/⟨q_k ⟩ = θ_w /θ_q ≡ α, see equation (14). Therefore, the efficiency of the ensemble of refrigerators to the power n is

$$\begin{align} {\eta }_{N}^{n}& ={\left(\frac{{\sum }_{k=1}^{N}\langle {q}_{k}\rangle }{{\sum }_{k=1}^{N}\langle {w}_{k}\rangle }\right)}^{n} \\ & ={\left(\sum\limits _{j}\frac{\langle {q}_{j}\rangle }{\langle {w}_{j}\rangle }\frac{\langle {w}_{j}\rangle }{{\sum }_{k}\langle {w}_{k}\rangle }\right)}^{n}=\frac{1}{{\alpha }^{n}}. \end{align} \tag{ 20 }$$

Similarly, based on equation (14),

$$\begin{equation}{\eta }_{N}^{(n)}={\left(\frac{{\sum }_{k=1}^{N}\langle \langle {q}_{k}^{n}\rangle \rangle }{{\sum }_{k=1}^{N}\langle \langle {w}_{k}^{n}\rangle \rangle }\right)}^{n}=\frac{1}{{\alpha }^{n}},\end{equation} \tag{ 21 }$$

and we confirm equation (19).

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5.2.2. QARs with distinct gaps

We now prove the lower bound for an ensemble of 3LQARs characterized by distinct energy gaps, but coupled in the same manner to the different baths; we highlight that each three-level system may couple asymmetrically to the three different baths, but all the three-level systems follow the same coupling parameters. The proof described in this section holds in the low-cooling regime identified as region I in figure 2(a); points marked by circles exemplify members of this ensemble. In appendix A we describe a complementary proof (for N = 2 and n = 2) that holds in region II, at the edge of the cooling window marked in figure 2(a).

Henceforth, for simplicity and without loss of generality we set the total gap at θ_h = 1. Assuming that θ_q ≪ 1, we Taylor-expand the cooling current (note that the expansion coefficients, β and γ, in the following are not to be confused with inverse temperature and coupling strength, respectively),

$$\begin{equation}\langle q\rangle \approx \alpha {\theta }_{q}+\beta {\theta }_{q}^{2}+\cdots \end{equation} \tag{ 22 }$$

Plugging this expansion into $\eta =\frac{\langle q\rangle }{\langle w\rangle }=\frac{{\theta }_{q}}{{\theta }_{w}}$ provides a consistent expansion for the input work,

$$\begin{equation}\langle w\rangle \approx \alpha -(\alpha -\beta ){\theta }_{q}+\cdots \end{equation} \tag{ 23 }$$

Similarly, we write a Taylor expansion for the fluctuations of the cooling current,

$$\begin{equation}\langle \langle {q}^{2}\rangle \rangle \approx \gamma {\theta }_{q}^{2}+\delta {\theta }_{q}^{3}+\cdots \end{equation} \tag{ 24 }$$

and using ${\eta }^{(2)}=\frac{{\theta }_{q}^{2}}{{\theta }_{w}^{2}}=\frac{\langle \langle {q}^{2}\rangle \rangle }{\langle \langle {w}^{2}\rangle \rangle }$ put together the consistent expansion for the fluctuations of the work current,

$$\begin{equation}\langle \langle {w}^{2}\rangle \rangle \approx \gamma +{\theta }_{q}(\delta -2\gamma )+\cdots \end{equation} \tag{ 25 }$$

Since we assume small cooling current and correspondingly small fluctuations (see figure 2), θ_q ≪ 1, βθ_q ≪ α, and δθ_q ≪ γ, the lowest order expansions for the currents and their fluctuations are,

$$\begin{align} \langle q\rangle & \approx \alpha {\theta }_{q},\qquad \langle w\rangle \approx \alpha . \\ \langle \langle {q}^{2}\rangle \rangle & \approx \gamma {\theta }_{q}^{2},\qquad \langle \langle {w}^{2}\rangle \rangle \approx \gamma . \end{align} \tag{ 26 }$$

These expressions are consistent with equation (14); recall that we set here the total gap at θ_h = 1. We are now ready to test the lower bound. We begin with two refrigerators A and B of spacings θ_qA and θ_qB that are coupled in the same manner to the baths (for example, A and B are marked by circles in figure 2(a)). Since these systems lie on the same curve, ⟨q_A ⟩ ≈ αθ_qA , ⟨q_B ⟩ ≈ αθ_qB , ⟨w_A ⟩ ≈ ⟨w_A ⟩ ≈ α. Similarly, $\langle \langle {q}_{A}^{2}\rangle \rangle \approx \gamma {\theta }_{qA}^{2}$, $\langle \langle {q}_{B}^{2}\rangle \rangle \approx \gamma {\theta }_{qB}^{2}$, $\langle \langle {w}_{A}^{2}\rangle \rangle \approx \langle \langle {w}_{B}^{2}\rangle \rangle \approx \gamma$. We now test the inequality

$$\begin{align} {\eta }_{N=2}^{2}={\left(\frac{\langle {q}_{A}\rangle +\langle {q}_{B}\rangle }{\langle {w}_{A}\rangle +\langle {w}_{B}\rangle }\right)}^{2}\leqslant \frac{\langle \langle {q}_{A}^{2}\rangle \rangle +\langle \langle {q}_{B}^{2}\rangle \rangle }{\langle \langle {w}_{A}^{2}\rangle \rangle +\langle \langle {w}_{B}^{2}\rangle \rangle }={\eta }_{N=2}^{(2)},\end{align} \tag{ 27 }$$

by substituting the currents and fluctuations. It reduces to

$$\begin{equation}\frac{{({\theta }_{qA}+{\theta }_{qB})}^{2}}{4}\leqslant \frac{{\theta }_{qA}^{2}+{\theta }_{qB}^{2}}{2},\end{equation} \tag{ 28 }$$

which is true since ${({\theta }_{qA}-{\theta }_{qB})}^{2}\geqslant 0$.

This proof can be generalized to the bound on η⁽ⁿ⁾ for an ensemble of N refrigerators. Given the small-θ_q expansions of the nth cumulant [54], $\langle \langle {q}_{k}^{n}\rangle \rangle \approx {\gamma }_{n}{\theta }_{qk}^{n}$, $\langle \langle {w}_{k}^{n}\rangle \rangle \approx {\gamma }_{n}$, the lower-bound inequality reads

$$\begin{equation}{\eta }_{N}^{n}={\left(\frac{{\sum }_{k}{\theta }_{qk}}{N}\right)}^{n}\leqslant \frac{{\sum }_{k}{\theta }_{qk}^{n}}{N}={\eta }_{N}^{(n)}.\end{equation} \tag{ 29 }$$

Taking the nth root gives the desired result by the power mean inequality [55], with power 1 on the left-hand side and n on the right,

$$\begin{align} {\eta }_{N}=\left(\frac{1}{N}\sum\limits _{k}{\theta }_{qk}\right)\leqslant {\left(\frac{1}{N}\sum\limits _{k}{\theta }_{qk}^{n}\right)}^{1/n}={\left[{\eta }_{N}^{(n)}\right]}^{1/n}.\end{align} \tag{ 30 }$$

The power mean inequality also implies that n = 2 provides the tightest bound on η_N .

We note that in general, the assumption of the cooling current being linear in the gap, ⟨q⟩ = αθ_q , does not necessarily correspond to a linear response limit, ⟨q⟩ ∝ (T_w − T_q ). However, for the 3LQARs, this is in fact the case [54]: a linear dependence of ⟨q⟩ with θ_q develops hand in hand with the current becoming linear in the temperature difference, though the temperature difference could be large, ΔT/T_i ≫ 1.

In appendix A, we prove the lower bound in region II as marked in figure 2(a), albeit limited to n = 2 and N = 2.

5.2.3. Simulations

We exemplify in figure 2 the behavior of individual 3LQARs. The population dynamics, heat currents and their fluctuations are calculated as described in reference [54] using kinetic-like quantum master equations. The three-level system is coupled to the heat baths with an Ohmic spectral density function, with the excitation rate constant, e.g. between the lowest state and the intermediate one, k_q = Γ_q (θ_q )n_q (θ_q ) where we use an Ohmic model, Γ_q (θ_q ) = γ_q θ_q ; n_i (θ_i ) is the Bose–Einstein distribution function. The relaxation rate constant follows from local detailed balance. γ_i are dimensionless coefficients that control the coupling strength. Simulations in figure 2 agree with theoretical results for the efficiency η = θ_q /θ_w and the ratio of fluctuations, ${\eta }^{(2)}={({\theta }_{q}/{\theta }_{w})}^{2}$.

In figure 3(a), we present simulation results for ${\eta }_{N=2}^{(2)}$. We generate a randomized set of 300 refrigerators by sampling uniformly over different values of θ_q within the cooling domain. As for the coupling strength, they are sampled from a uniform distribution 0 ⩽ γ_i ⩽ 1, and they can be distinct and unequal at each realization (γ_h ≠ γ_q ≠ γ_w ). We then consider every possible pair of refrigerators and calculate for this pair the total cooling (⟨q_A ⟩ + ⟨q_B ⟩) and work currents (⟨w_A ⟩ + ⟨w_B ⟩), as well as their fluctuations, $\langle \langle {q}_{A}^{2}\rangle \rangle +\langle \langle {q}_{B}^{2}\rangle \rangle$ and $\langle \langle {w}_{A}^{2}\rangle \rangle +\langle \langle {w}_{B}^{2}\rangle \rangle$. This allows us to calculate the efficiency for each pair, η_N=2 = (⟨q_A ⟩ + ⟨q_B ⟩)/(⟨w_A ⟩ + ⟨w_B ⟩), and the ratio of their fluctuations, ${\eta }_{N=2}^{(2)}$. Though most of our refrigerators lie outside regions I and II, we do not observe violations to the lower bound, ${\eta }_{N=2}^{2}\leqslant {\eta }_{N=2}^{(2)}$; note that parameters are outside the linear response regime [22].

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In figure 3(b), we select 90 000 samples of triplet 3LQARs and calculate their total current and fluctuations; the total cooling current is given by ⟨q_A ⟩ + ⟨q_B ⟩ + ⟨q_C⟩ and the associated fluctuations by $\langle \langle {q}_{A}^{2}\rangle \rangle +\langle \langle {q}_{B}^{2}\rangle \rangle +\langle \langle {q}_{C}^{2}\rangle \rangle$, with analogous expressions giving the work current and its fluctuations. We then obtain the efficiency and the ratio of fluctuations for these triplet 3LQARs. Again we confirm based on simulations that the lower bound is satisfied.

5.2.4. Discussion

We organize our observations up to this point on the validity of the bounds (4) for an ensemble of TC refrigerators: (i) the upper bound holds without additional assumptions (section 3). As for a lower bound on η⁽ⁿ⁾, for the 3LQARs as discussed in this section, we proved that: (ii) the lower bound saturates for an ensemble of systems with identical spacings, θ_q , but distinct couplings to the baths. (iii) The lower bound holds for an ensemble of systems operating with equal coupling schemes. This proof holds in the limit of small cooling currents, in region I, characterized by a vanishing θ_q . We were also able to prove (appendix A) the lower bound in region II, characterized by a vanishing input work current, albeit limited to n = 2 and N = 2. (iv) Extensive simulations confirmed the validity of the lower bound more broadly beyond linear response. Even more so, our simulations showed that an inequality more general than the lower bound holds, namely equation (13). (v) In appendix B we discuss the validity of the lower bound for a three-level model operating as an engine.

A thermoelectric junction comprises a left (L) and right (R) metal lead, between which charge and energy transport may be mediated through an embedded quantum system. The leads serve as thermal baths, differing in temperature and chemical potential such that the associated affinities oppose one another [26]; we suppose that T_L < T_R and μ_L > μ_R. In what follows, we assume resonant electron transport through a discrete level [31], and discuss the existence of the lower bound on ${\eta }_{N}^{(n)}$ for an ensemble with N uncorrelated thermoelectric junctions as depicted in figure 4.

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6.1. Single junction, N = 1

We identify a stochastic charge current j_k , for a junction labelled by k, focusing here on the TC limit [25, 26], wherein the mean energy current is _k ⟨j_k ⟩-proportional to the charge current, where _k is the resonant level of a single quantum dot characterizing the system through which transport is mediated. In this limit, the charge current and fluctuations are given by [56, 57]

$$\begin{align} \langle {j}_{k}\rangle & ={\mathcal{T}}_{1}\left[{f}_{\mathrm{L}}({{\epsilon}}_{k})-{f}_{\mathrm{R}}({{\epsilon}}_{k})\right], \\ \langle \langle {j}_{k}^{2}\rangle \rangle & ={\mathcal{T}}_{1}\left[{f}_{\mathrm{L}}({{\epsilon}}_{k})(1-{f}_{\mathrm{R}}({{\epsilon}}_{k}))+{f}_{\mathrm{R}}({{\epsilon}}_{k})(1-{f}_{\mathrm{L}}({{\epsilon}}_{k}))\right]-{\mathcal{T}}_{2}{\left[{f}_{\mathrm{L}}({{\epsilon}}_{k})-{f}_{\mathrm{R}}({{\epsilon}}_{k})\right]}^{2}. \end{align} \tag{ 31 }$$

The coefficients ${\mathcal{T}}_{1}$ and ${\mathcal{T}}_{2}$ are determined by the coupling strengths Γ_L and Γ_R between the system and the two leads. For the resonant level model, ${\mathcal{T}}_{1}=\frac{{{\Gamma}}_{\mathrm{L}}{{\Gamma}}_{\mathrm{R}}}{{{\Gamma}}_{\mathrm{L}}+{{\Gamma}}_{\mathrm{R}}}$ and ${\mathcal{T}}_{2}=\frac{2{{\Gamma}}_{\mathrm{L}}^{2}{{\Gamma}}_{\mathrm{R}}^{2}}{{({{\Gamma}}_{\mathrm{L}}+{{\Gamma}}_{\mathrm{R}})}^{3}}$; see references [17, 18]. f_L() and f_R() are the Fermi–Dirac distributions for the two leads. A thermoelectric device may act as a refrigerator or an engine in various operational regimes, as determined by the directions of power and heat currents, which are proportional to the charge current in the TC limit. The mean currents thus obey ⟨w_k ⟩ = Δμ⟨j_k ⟩, ⟨q_k ⟩ = (_k − μ_ν )⟨j_k ⟩ (ν = L for a refrigerator, R for an engine) with Δμ = μ_L − μ_R. The fluctuations in these quantities are similarly determined by those for the charge current: $\langle \langle {w}_{k}^{2}\rangle \rangle ={\Delta}{\mu }^{2}\langle \langle {j}_{k}^{2}\rangle \rangle$, $\langle \langle {q}_{k}^{2}\rangle \rangle ={({{\epsilon}}_{k}-{\mu }_{\nu })}^{2}\langle \langle {j}_{k}^{2}\rangle \rangle$.

An individual TC thermoelectric junction has been shown to satisfy the equality ${\eta }_{k}^{n}={\eta }_{k}^{(n)}$ for any n = 1, 2, 3, ... [22], where, with _k the dot energy for the machine k,

$$\begin{align} {\eta }_{k}& =\frac{{\Delta}\mu }{{{\epsilon}}_{k}-{\mu }_{\mathrm{R}}}\enspace \enspace (\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{i}\mathrm{n}\mathrm{e}) \\ {\eta }_{k}& =\frac{{{\epsilon}}_{k}-{\mu }_{\mathrm{L}}}{{\Delta}\mu }\enspace \enspace (\mathrm{r}\mathrm{e}\mathrm{f}\mathrm{r}\mathrm{i}\mathrm{g}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r}). \end{align} \tag{ 32 }$$

Immediately, the upper bound, ${\eta }_{k}^{(n)}\leqslant {\eta }_{\mathrm{C}}^{n}$, also holds [22], where η_C represents the operational Carnot bound.

6.2. Ensemble of N > 1 distinct junctions

We now consider an ensemble of distinct TC thermoelectric junctions, labelled by indices k, whose internal parameters such as _k and system–lead coupling strengths may differ, but which operate between leads with the same set of bath parameters. For example, we envision a thermoelectric device with parallel quantum dots each conducting resonantly with their own energy and coupling strengths to the metal electrodes; see figure 4.

The validity of the upper bound for a single junction, along with the general result expressed in equation (8), leads to the analogous upper bound for an ensemble of such junctions as discussed in section 3. The following discussion will therefore focus on when the lower bound, ηⁿ ⩽ η⁽ⁿ⁾, holds for such an ensemble, in the n = 2 case.

6.2.1. Thermoelectric junctions with different system-bath couplings

Consider an ensemble of N thermoelectric junctions, labelled by k, with the same resonant level, , characterizing the quantum system, as represented in figure 5(a) by the pair of asterisks. The cooling efficiency, ⟨q_k ⟩/⟨w_k ⟩ = ( − μ_L)/Δμ is equal for all such refrigerators since it does not depend on the system-bath coupling. As such, one can readily prove, via an argument mirroring that in section 5.2.1, the strict equality ${\eta }_{N}^{n}={\eta }_{N}^{(n)}$ for the ensemble.

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6.2.2. Thermoelectric junctions with distinct energy levels

Now, we will focus specifically on the case of a pair (N = 2) of TC thermoelectric junctions, A and B, operating as refrigerators, with cooling currents flowing from the left (cold) lead, ⟨q_k ⟩ = (_k − μ_L)⟨j_k ⟩, k = A, B. The cooling and work currents are taken, by convention, to be positive. We will show that for this pair taken as a single device, the lower bound, ${\eta }_{N=2}^{2}\leqslant {\eta }_{N=2}^{(2)}$ holds, provided we restrict ourselves to a specific range of values for _A and _B , namely, region II, near the edge of the cooling window such that ⟨j_k ⟩ is small. For instance, such an ensemble may be represented by the pair of circles in figure 5(a). Furthermore, we will suppose that one refrigerator, say B, has a significantly larger charge current than the other (⟨j_A ⟩/⟨j_B ⟩ ≪ 1). This requires that _A > _B , so η_A > η_B . The complementary proof for region I is given in appendix C.

We express the dot energy for each thermoelectric refrigerator as _k = μ_R + Δμβ_L/Δβ − u_k with Δβ = β_L − β_R, Δμ = μ_L − μ_R. Note that the charge current vanishes when u_k = 0 since f_L(_k ) = f_R(_k ) at this point [58], and initially grows linearly with u_k : ⟨j_k ⟩ ≈ αu_k , where α is some constant coefficient. Then we have that u_A /u_B ≪ 1, and we may write out the full efficiency of the pair, η_N=2 = [(_A − μ_L)u_A + (_B − μ_L)u_B ]/[(u_A + u_B )Δμ]. Squaring and truncating after first order in u_A /u_B , we get

$$\begin{equation}{\eta }_{N=2}^{2}\approx {\eta }_{B}^{(2)}\left[1+2\left(\frac{{{\epsilon}}_{A}-{\mu }_{\mathrm{L}}}{{{\epsilon}}_{B}-{\mu }_{\mathrm{L}}}-1\right)\frac{{u}_{A}}{{u}_{B}}\right].\end{equation} \tag{ 33 }$$

Choosing sufficiently large _k leads to a small value for f_L(_k ) − f_R(_k ). We therefore ignore the contribution to the fluctuations, as given by equation (31), proportional to ${({f}_{\mathrm{L}}({{\epsilon}}_{k})-{f}_{\mathrm{R}}({{\epsilon}}_{k}))}^{2}$. This corresponds to the assumption that cotunneling processes may be neglected. We write

$$\begin{equation}\langle \langle {j}_{k}^{2}\rangle \rangle ={\mathcal{T}}_{1}\left[{f}_{\mathrm{L}}({{\epsilon}}_{k})(1-{f}_{\mathrm{R}}({{\epsilon}}_{k}))+{f}_{\mathrm{R}}({{\epsilon}}_{k})(1-{f}_{\mathrm{L}}({{\epsilon}}_{k}))\right].\end{equation} \tag{ 34 }$$

This can be shown to be proportional to the charge current,

$$\begin{equation}\langle \langle {j}_{k}^{2}\rangle \rangle =\langle {j}_{k}\rangle \mathrm{coth}\left(\frac{{\Delta}\beta {u}_{k}}{2}\right)\equiv {c}_{k}\langle {j}_{k}\rangle ,\end{equation} \tag{ 35 }$$

with c_k defined from this relation. The ratio of fluctuations of the cooling current to work current is given in terms of u_A and u_B as

$$\begin{equation}{\eta }_{N=2}^{(2)}={\eta }_{B}^{(2)}\frac{1+\frac{{({{\epsilon}}_{A}-{\mu }_{\mathrm{L}})}^{2}}{{({{\epsilon}}_{B}-{\mu }_{\mathrm{L}})}^{2}}\frac{{c}_{A}{u}_{A}}{{c}_{B}{u}_{B}}}{1+\frac{{c}_{A}{u}_{A}}{{c}_{B}{u}_{B}}}.\end{equation} \tag{ 36 }$$

As u_k approaches zero, c_k goes to infinity, so we cannot assume c_A u_A /c_B u_B is small. However, we can still compare ${\eta }_{N=2}^{2}$ and ${\eta }_{N=2}^{(2)}$, finding, after keeping only terms up to first order in u_A /u_B , that the lower bound is equivalent to the inequality

$$\begin{equation}2\left(\frac{{{\epsilon}}_{A}-{\mu }_{\mathrm{L}}}{{{\epsilon}}_{B}-{\mu }_{\mathrm{L}}}-1\right)\leqslant \frac{{c}_{A}}{{c}_{B}}\left[\frac{{({{\epsilon}}_{A}-{\mu }_{\mathrm{L}})}^{2}}{{({{\epsilon}}_{B}-{\mu }_{\mathrm{L}})}^{2}}-1\right].\end{equation} \tag{ 37 }$$

Since _A > _B , this holds as long as c_A /c_B ⩾ 2. We note that this is certainly the case since, by assumption, u_A and u_B are small, so c_A /c_B ≈ u_B /u_A ≫ 1. We may conclude that in this regime, ${\eta }_{N=2}^{2}\leqslant {\eta }_{N=2}^{(2)}$.

The proof outlined in this section is quite limited. It holds for a pair of refrigerators, N = 2, and only for the second cumulant, n = 2. Complementing this proof, in appendix C, we prove the lower bound in region I, now for an ensemble of arbitrary size N and to the order n. Furthermore, figure 6 demonstrates, via the results of simulations, that the upper and lower bounds (for n = 2) appear to hold for pairs and triplets of thermoelectric refrigerators outside regions I and II.

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The validity of the lower bound has been proven for three-level refrigerators and thermoelectric refrigerators in corresponding cases, in regions I and II. We now consider an analogous set of assumptions as discussed in section 6.2, but applied to ensembles of thermoelectric junctions operating instead as engines. We find that there exist circumstances under which the lower bound must be violated, identifying a limitation to the applicability of the lower bound on η⁽²⁾ for ensembles of thermal machines, and highlighting an interesting distinction between the behaviours of thermoelectric engines and refrigerators.

Specifically, we consider a pair of thermoelectric engines in the TC limit, far from equilibrium, operating in the regime of high _k , for both k = A, B, and thus small charge currents. Note that, due to the assumption of TC, this operational regime has no upper boundary with respect to _k . For simplicity, we take the two junctions to have identical coupling strengths to the leads. Furthermore, we suppose that the magnitude of the charge current is considerably greater for engine A than for B. Since, in this regime, charge current decreases with _k , _B > _A , so, referring to equation (32), η_A > η_B . We will show that, under these assumptions, the lower bound on the ratio of fluctuations, ${\eta }_{N=2}^{2}\leqslant {\eta }_{N=2}^{(2)}$, is violated.

In contrast to the refrigerator case, we now define the heat current with respect to the right (hot) lead-⟨q_k ⟩ = (_k − μ_R)⟨j_k ⟩. We first write out the efficiency of the pair of engines,

$$\begin{equation} {\eta }_{N=2}=\frac{\langle {w}_{A}\rangle +\langle {w}_{B}\rangle }{\langle {q}_{A}\rangle +\langle {q}_{B}\rangle }=\frac{{\Delta}\mu (\langle {j}_{A}\rangle +\langle {j}_{B}\rangle )}{({{\epsilon}}_{A}-{\mu }_{\mathrm{R}})\langle {j}_{A}\rangle +({{\epsilon}}_{B}-{\mu }_{\mathrm{R}})\langle {j}_{B}\rangle }={\eta }_{A}\frac{1+\frac{\langle {j}_{B}\rangle }{\langle {j}_{A}\rangle }}{1+\frac{{{\epsilon}}_{B}-{\mu }_{\mathrm{R}}}{{{\epsilon}}_{A}-{\mu }_{\mathrm{R}}}\frac{\langle {j}_{B}\rangle }{\langle {j}_{A}\rangle }}.\end{equation} \tag{ 38 }$$

Noting that ⟨j_B ⟩/⟨j_A ⟩ ≪ 1, we expand to first order in this ratio. After squaring and using that ${\eta }_{A}^{2}={\eta }_{A}^{(2)}$, we have

$$\begin{equation}{\eta }_{N=2}^{2}\approx {\eta }_{A}^{(2)}\left[1+2\left(1-\frac{{{\epsilon}}_{B}-{\mu }_{\mathrm{R}}}{{{\epsilon}}_{A}-{\mu }_{\mathrm{R}}}\right)\frac{\langle {j}_{B}\rangle }{\langle {j}_{A}\rangle }\right].\end{equation} \tag{ 39 }$$

Next, we consider the fluctuations in the work and heat currents. The assumption of small currents allows us to neglect cotunneling processes—that is, contributions proportional to ${({f}_{\mathrm{L}}({{\epsilon}}_{k})-{f}_{\mathrm{R}}({{\epsilon}}_{k}))}^{2}$. Furthermore, we use the identity ${f}_{\mathrm{L}}({{\epsilon}}_{k})\left[1-{f}_{\mathrm{R}}({{\epsilon}}_{k})\right]+{f}_{\mathrm{R}}({{\epsilon}}_{k})\left[1-{f}_{\mathrm{L}}({{\epsilon}}_{k})\right]=[{f}_{\mathrm{L}}({{\epsilon}}_{k})-{f}_{\mathrm{R}}({{\epsilon}}_{k})]\mathrm{coth}[(-{\Delta}\beta {{\epsilon}}_{k}+\bar{\beta }{\Delta}\mu )/2]\equiv {c}_{k}[{f}_{\mathrm{L}}({{\epsilon}}_{k})-{f}_{\mathrm{R}}({{\epsilon}}_{k})]$ (note that Δβ = β_L − β_R > 0, $\bar{\beta }=({\beta }_{\mathrm{L}}+{\beta }_{\mathrm{R}})/2$) to express the charge current fluctuations for each engine in terms of the charge current itself:

$$\begin{equation}\langle \langle {j}_{k}^{2}\rangle \rangle ={c}_{k}\langle {j}_{k}\rangle .\end{equation} \tag{ 40 }$$

Then, arguments similar to those used to derive equation (39) give

$$\begin{equation}{\eta }_{N=2}^{(2)}\approx {\eta }_{A}^{(2)}\left[1+\left(1-{\left(\frac{{{\epsilon}}_{B}-{\mu }_{\mathrm{R}}}{{{\epsilon}}_{A}-{\mu }_{\mathrm{R}}}\right)}^{2}\right)\frac{{c}_{B}\langle {j}_{B}\rangle }{{c}_{A}\langle {j}_{A}\rangle }\right].\end{equation} \tag{ 41 }$$

Now, comparing equations (39) and (41), we find that the lower bound ${\eta }_{N=2}^{2}\leqslant {\eta }_{N=2}^{(2)}$ is equivalent to the inequality

$$\begin{equation}2\left(1-\frac{{{\epsilon}}_{B}-{\mu }_{\mathrm{R}}}{{{\epsilon}}_{A}-{\mu }_{\mathrm{R}}}\right)\leqslant \frac{{c}_{B}}{{c}_{A}}\left[1-{\left(\frac{{{\epsilon}}_{B}-{\mu }_{\mathrm{R}}}{{{\epsilon}}_{A}-{\mu }_{\mathrm{R}}}\right)}^{2}\right].\end{equation} \tag{ 42 }$$

Recall that ${c}_{k}=\mathrm{coth}[(-{\Delta}\beta {{\epsilon}}_{k}+\bar{\beta }{\Delta}\mu )/2]$; working far from equilibrium, we suppose that Δβ is sufficiently large that _A and _B meeting our above assumptions may be chosen such that both c_A ≈ −1 and c_B ≈ −1. Thus, c_B /c_A ≈ 1. This simplifies the necessary and sufficient condition for the lower bound to

$$\begin{equation}2\left(1-\frac{{{\epsilon}}_{B}-{\mu }_{\mathrm{R}}}{{{\epsilon}}_{A}-{\mu }_{\mathrm{R}}}\right)\leqslant \left[1-{\left(\frac{{{\epsilon}}_{B}-{\mu }_{\mathrm{R}}}{{{\epsilon}}_{A}-{\mu }_{\mathrm{R}}}\right)}^{2}\right],\end{equation} \tag{ 43 }$$

or, equivalently,

$$\begin{equation}{\left(\frac{{{\epsilon}}_{B}-{\mu }_{\mathrm{R}}}{{{\epsilon}}_{A}-{\mu }_{\mathrm{R}}}-1\right)}^{2}\leqslant 0.\end{equation} \tag{ 44 }$$

Our assumption that _B > _A renders equation (44) a clear contradiction. Therefore, the set of assumptions considered here describes a pair of thermoelectric engines that must violate the lower bound on the ratio of work to heat current fluctuations, ${\eta }_{N=2}^{(2)}$. We emphasize that this derivation relies on the thermoelectric engine operating far from equilibrium. In linear response, a small value for Δβ would conflict with the ability to achieve c_B /c_A ≈ 1 for any choice of _A and _B satisfying the rest of the assumptions. Thus, there is no contradiction with reference [22].

In figure 7 we provide supporting numerical evidence for the violation of the lower bound for thermoelectric engines operating in region II. While we use approximate expressions for the fluctuations in our analysis, the numerical simulations use the full expressions of equation (31), demonstrating that these violations are not merely a consequence of the assumptions made.

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We studied universal bounds on ratios of fluctuations, as well as higher order cumulants, in an ensemble of distinct, uncorrelated thermal machines operating in steady state and arbitrarily far from equilibrium. We built our bounds for the ensemble of machines from bounds on the individual machine, ${\eta }_{k}^{n}={\eta }_{k}^{(n)}\leqslant {\eta }_{\mathrm{C}}^{n}$. These relations hold, e.g. for systems obeying the TC limit, such as three-level absorption refrigerators and resonant-level thermoelectric junctions. η_C is the Carnot efficiency and η⁽ⁿ⁾ is ratio of the nth cumulants of the output current to input current. We proved that:

• (a)  
  An upper bound holds for an ensemble of N distinct machines, engines or refrigerators, arbitrarily far from equilibrium, ${\eta }_{N}^{(n)}\leqslant {\eta }_{\mathrm{C}}^{n}$.
• (b)  
  The lower bound on the ensemble, ${\eta }_{N}^{n}\leqslant {\eta }_{N}^{(n)}$, is more limited in scope. While it seems to hold for refrigerators, based on analytic and numerical work, we demonstrated that thermoelectric engines may violate the lower bound for N ⩾ 2.
• (c)  
  Focusing on 3LQARs and thermoelectric junctions, we proved the validity of the lower bound, ${\eta }_{N}^{n}\leqslant {\eta }_{N}^{(n)}$, in different cases: when the cooling current was small, and for an ensemble of distinct machines with identical efficiencies.
• (d)  
  While analytic proofs of the lower bound for refrigerators were limited in scope (e.g. to n = 2 and N = 2 in region II near the edge of the cooling window), simulations beyond these strict regimes support its validity more broadly.

The significance of our study is in developing bounds for ensembles of machines. While bounds on the individual system may be trivial, as in the TC limit, a machine that collects input and output from several tightly-coupled components (recall figures 1 and 4) may behave nontrivially in this respect, even under classical laws and in the absence of interactions between components. Adding interactions to the working fluid, e.g. by using an interacting atomic gas or introducing Coulomb interactions between quantum dots in a thermoelectric device, opens up the door to new effects, such as many-particle enhancement of performance [59–64]. Furthermore, quantum statistical effects combined with interactions or collective unitary operations on the components can enhance performance, as predicted in reference [65].

An important result of our work is in tightening bounds on efficiency: as an outcome of the lower bound, equation (30), one immediately finds that $\eta \leqslant {\left[{\eta }^{(n)}\right]}^{1/n}\leqslant {\eta }_{\mathrm{C}}$, and that the case with n = 2 provides the tightest upper bound on the efficiency. Whether this is a general result, or only valid for refrigerators in the small-θ small-cooling domain, remains an open question.

In future work we will focus on understanding the validity of the lower bound on η⁽ⁿ⁾ in general settings, and on exploring the fundamental differences between engines and refrigerators as they pertain to bounds on fluctuations.

DS acknowledges support from an NSERC Discovery Grant and the Canada Research Chair program. The work of NK was supported by the Ontario Graduate Scholarship (OGS). The research of MG was supported by the NSERC Canada Graduate Scholarship-Master's and the OGS. Na'im Kalantar and Matthew Gerry contributed equally to this work and are joint 'first authors'.

The data that support the findings of this study are available upon reasonable request from the authors.

Here, we present a proof of the validity of a lower bound for two 3LQARs, applicable in the limit of small currents. This proof holds at the boundary of the cooling domain (see region II in figure 2). The proof presented here is limited to two QARs, and its extensions to N refrigerators is not obvious.

For simplicity, we define the scaled currents q/θ_q → q and w/θ_w → w. We also assume that T_w → ∞. The cooling current and its noise are given in terms of the bath-induced transition rate constants Γ_q,h,w , see text in section 5.2.3 and reference [54]. We also use the short notation ${\tilde{{\Gamma}}}_{i}={{\Gamma}}_{i}({\theta }_{i}){n}_{i}({\theta }_{i})$. The cooling current and its fluctuations are given by [54]

$$\begin{equation}\langle q\rangle =\frac{{\tilde{{\Gamma}}}_{h}{\tilde{{\Gamma}}}_{q}{\tilde{{\Gamma}}}_{w}\left({\mathrm{e}}^{{\theta }_{h}/{T}_{h}}-{\mathrm{e}}^{{\theta }_{q}/{T}_{q}}\right)}{M},\end{equation} \tag{ A1 }$$

$$\begin{align*} \langle \langle {q}^{2}\rangle \rangle & =\frac{2\left({\tilde{{\Gamma}}}_{h}+{\tilde{{\Gamma}}}_{q}+{\tilde{{\Gamma}}}_{w}+{\tilde{{\Gamma}}}_{q}{\mathrm{e}}^{{\theta }_{q}/{T}_{q}}+{\tilde{{\Gamma}}}_{w}+{\tilde{{\Gamma}}}_{h}{\mathrm{e}}^{{\theta }_{h}/{T}_{h}}\right)\langle {q\rangle }^{2}}{M} \\ & \quad +\frac{{\tilde{{\Gamma}}}_{q}{\tilde{{\Gamma}}}_{h}{\tilde{{\Gamma}}}_{w}\left({\mathrm{e}}^{{\theta }_{h}/{T}_{h}}+{\mathrm{e}}^{{\theta }_{q}/{T}_{q}}\right)}{M} \end{align*}$$

$$\begin{align} \text{with}\;\;M& =\left({\tilde{{\Gamma}}}_{c}{\mathrm{e}}^{{\theta }_{q}/{T}_{q}}+{\tilde{{\Gamma}}}_{w}\right)\left({\tilde{{\Gamma}}}_{w}+{\tilde{{\Gamma}}}_{q}{\mathrm{e}}^{{\theta }_{h}/{T}_{h}}\right)+\left({\tilde{{\Gamma}}}_{q}+{\tilde{{\Gamma}}}_{h}\right)\left({\tilde{{\Gamma}}}_{w}+{\tilde{{\Gamma}}}_{h}{\mathrm{e}}^{{\theta }_{q}/{T}_{q}}\right) \\ & \quad +\left({\tilde{{\Gamma}}}_{q}+{\tilde{{\Gamma}}}_{h}\right)\left({\tilde{{\Gamma}}}_{q}{\mathrm{e}}^{{\theta }_{q}/{T}_{q}}+{\tilde{{\Gamma}}}_{w}\right)-{\tilde{{\Gamma}}}_{q}^{2}{\mathrm{e}}^{{\theta }_{q}/{T}_{q}}-{\tilde{{\Gamma}}}_{w}^{2}-{\tilde{{\Gamma}}}_{h}^{2}{\mathrm{e}}^{{\theta }_{h}/{T}_{h}}. \end{align} \tag{ A2 }$$

The cooling window is defined by the condition $0< {\theta }_{q}< {\theta }_{h}\frac{{T}_{q}}{{T}_{h}}$. When $\langle q\rangle \ll {{\Pi}}_{i=h,q,w}{\tilde{{\Gamma}}}_{i}$, the second term in the noise dominates, thus it is given by

$$\begin{equation}\langle \langle {q}^{2}\rangle \rangle \approx \left\vert \frac{{\mathrm{e}}^{{\theta }_{h}/{T}_{h}}+{\mathrm{e}}^{{\theta }_{q}/{T}_{q}}}{{\mathrm{e}}^{{\theta }_{h}/{T}_{h}}-{\mathrm{e}}^{{\theta }_{q}/{T}_{q}}}\right\vert \langle q\rangle =\left\vert \mathrm{coth}\left(\frac{{\theta }_{h}}{2{T}_{h}}-\frac{{\theta }_{q}}{2{T}_{q}}\right)\right\vert \langle q\rangle \equiv F({\theta }_{q})\langle q\rangle .\end{equation} \tag{ A3 }$$

We now consider two refrigerators, A and B, with identical total gaps θ_h , but different internal energies, ${\theta }_{{q}_{A}}\ne {\theta }_{{q}_{B}}$. Correspondingly, each QAR supports the cooling current ⟨q_k ⟩ with the associated noise $\langle \langle {q}_{k}^{2}\rangle \rangle$. As for the lower bound, we would like to show that (recovering the energy gaps in the expressions for the currents and noises):

$$\begin{equation}{\eta }^{2}=\frac{{({\theta }_{{q}_{A}}\langle {q}_{A}\rangle +{\theta }_{{q}_{B}}\langle {q}_{B}\rangle )}^{2}}{{({\theta }_{{w}_{A}}\langle {q}_{A}\rangle +{\theta }_{{w}_{B}}\langle {q}_{B}\rangle )}^{2}}\leqslant \frac{{\theta }_{{q}_{A}}^{2}\langle \langle {q}_{A}^{2}\rangle \rangle +{\theta }_{{q}_{B}}^{2}\langle \langle {q}_{B}^{2}\rangle \rangle }{{\theta }_{{w}_{A}}^{2}\langle \langle {q}_{A}^{2}\rangle \rangle +{\theta }_{{w}_{B}}^{2}\langle \langle {q}_{B}^{2}\rangle \rangle }={\eta }^{(2)}.\end{equation} \tag{ A4 }$$

Using equation (A3) and that θ_w = θ_h − θ_q , we get

$$\begin{align} & \frac{{[{\theta }_{{q}_{A}}\langle {q}_{A}\rangle +{\theta }_{{q}_{B}}\langle {q}_{B}\rangle ]}^{2}}{{[({\theta }_{h}-{\theta }_{{q}_{A}})\langle {q}_{A}\rangle +({\theta }_{h}-{\theta }_{{q}_{B}})\langle {q}_{B}\rangle ]}^{2}} \\ & \qquad \leqslant \frac{{\theta }_{{q}_{A}}^{2}F({\theta }_{{q}_{A}})\langle {q}_{A}\rangle +{\theta }_{{q}_{B}}^{2}F({\theta }_{{q}_{B}})\langle {q}_{B}\rangle }{{({\theta }_{h}-{\theta }_{{q}_{A}})}^{2}F({\theta }_{{q}_{A}})\langle {q}_{A}\rangle +{({\theta }_{h}-{\theta }_{{q}_{B}})}^{2}F({\theta }_{{q}_{B}})\langle {q}_{B}\rangle }. \end{align} \tag{ A5 }$$

After cross-multiplying, all the ⟨q⟩² terms vanish and we are left with

$$\begin{align} & \sum\limits _{i\ne k}\left({\theta }_{{q}_{i}}^{2}{({\theta }_{h}-{\theta }_{{q}_{k}})}^{2}F({\theta }_{{q}_{k}})+2{\theta }_{{q}_{i}}{\theta }_{{q}_{k}}{({\theta }_{h}-{\theta }_{{q}_{i}})}^{2}F({\theta }_{{q}_{i}})\right)\langle {q}_{i}\rangle \\ & \qquad \leqslant \sum\limits _{i\ne k}\left({\theta }_{{q}_{k}}^{2}{({\theta }_{h}-{\theta }_{{q}_{i}})}^{2}F({\theta }_{{q}_{k}})+2{\theta }_{{q}_{i}}^{2}({\theta }_{h}-{\theta }_{{q}_{i}})({\theta }_{h}-{\theta }_{{q}_{k}})F({\theta }_{{q}_{i}})\right)\langle {q}_{i}\rangle . \end{align} \tag{ A6 }$$

Since each current ⟨q⟩ can be set freely by setting the transition rates, the inequality is true if and only if each ⟨q⟩ part obeys the inequality:

$$\begin{align} & {\theta }_{{q}_{A}}^{2}{({\theta }_{h}-{\theta }_{{q}_{B}})}^{2}F({\theta }_{{q}_{B}})+2{\theta }_{{q}_{A}}{\theta }_{{q}_{B}}{({\theta }_{h}-{\theta }_{{q}_{A}})}^{2}F({\theta }_{{q}_{A}}) \\ & \qquad \leqslant {\theta }_{{q}_{B}}^{2}{({\theta }_{h}-{\theta }_{{q}_{A}})}^{2}F({\theta }_{{q}_{B}})+2{\theta }_{{q}_{A}}^{2}({\theta }_{h}-{\theta }_{{q}_{A}})({\theta }_{h}-{\theta }_{{q}_{B}})F({\theta }_{{q}_{A}}). \end{align} \tag{ A7 }$$

After additional manipulations we get

$$\begin{align} & \left[{\theta }_{h}^{2}\left({\theta }_{{q}_{A}}^{2}-{\theta }_{{q}_{B}}^{2}\right)-2{\theta }_{h}{\theta }_{{q}_{A}}{\theta }_{{q}_{B}}({\theta }_{{q}_{A}}-{\theta }_{{q}_{B}})\right]F({\theta }_{{q}_{B}}) \\ & \qquad +\left[2{\theta }_{{q}_{A}}{\theta }_{h}({\theta }_{h}-{\theta }_{{q}_{A}})({\theta }_{{q}_{B}}-{\theta }_{{q}_{A}})\right]F({\theta }_{{q}_{A}})\leqslant 0. \end{align} \tag{ A8 }$$

The function F(θ) is positive, and it grows exponentially as θ approaches the asymptote at the edge of the cooling window, ${\theta }_{q}={\theta }_{h}\frac{{T}_{q}}{{T}_{h}}$, from below. Therefore, if ${\theta }_{{q}_{A}}< {\theta }_{{q}_{B}}< {\theta }_{h}\frac{{T}_{q}}{{T}_{h}}$, the first term in equation (A8) dominates. It is given by

$$\begin{equation}{\theta }_{h}({\theta }_{{q}_{A}}+{\theta }_{{q}_{B}})({\theta }_{{q}_{A}}-{\theta }_{{q}_{B}})-2{\theta }_{{q}_{A}}{\theta }_{{q}_{B}}({\theta }_{{q}_{A}}-{\theta }_{{q}_{B}})\leqslant 0,\end{equation} \tag{ A9 }$$

which is reduced to

$$\begin{equation}{\theta }_{h}({\theta }_{{q}_{A}}+{\theta }_{{q}_{B}})-2{\theta }_{{q}_{A}}{\theta }_{{q}_{B}}\geqslant 0.\end{equation} \tag{ A10 }$$

This is true since θ_h > θ_qA and θ_h > θ_qB .

Likewise, the second term in equation (A8) dominates in the opposite case, if ${\theta }_{{q}_{B}}< {\theta }_{{q}_{A}}< {\theta }_{h}\frac{{T}_{q}}{{T}_{h}}$. We then check whether

$$\begin{equation}2{\theta }_{{q}_{A}}{\theta }_{h}({\theta }_{h}-{\theta }_{{q}_{A}})({\theta }_{{q}_{B}}-{\theta }_{{q}_{A}})\leqslant 0,\end{equation} \tag{ A11 }$$

which is true, since the last term is negative and the rest are positive.

We consider here the performance of the three-level model as an engine, rather than a refrigerator. We show that in the limit of small currents, at the edge of the engine's window, the lower bound for η⁽²⁾ holds. This is to be contrasted with behavior observed for a thermoelectric engine, section 7, which shows violations to the lower bound in a corresponding domain.

The system acts as an engine when ${\theta }_{h}\frac{{T}_{q}}{{T}_{h}}< {\theta }_{q}< {\theta }_{h}$. Heat is absorbed from the hot (h) bath and emitted at the w bath as useful work, with leftover heat released at the q bath. In this engine's regime, assuming the thermal noise dominates (since the current is small), we write

$$\begin{equation}\langle \langle {q}^{2}\rangle \rangle =-F({\theta }_{q})\langle q\rangle ,\end{equation} \tag{ B1 }$$

where was defined in equation (A3). Recall that by our conventions, ⟨q⟩ is positive when flowing into the system, and thus it is negative when the system acts as an engine. Furthermore, when the system operates as an engine, F is positive and decreasing with θ_q .

For an engine, the efficiency is the work current over the heat input from the hot bath. As usual, we set the total gap size θ_h the same for both systems. Rewriting equation (A4) for the two engines gives

$$\begin{equation}{\eta }^{2}=\frac{{({\theta }_{{w}_{A}}\langle {q}_{A}\rangle +{\theta }_{{w}_{B}}\langle {q}_{B}\rangle )}^{2}}{{({\theta }_{{h}_{A}}\langle {q}_{A}\rangle +{\theta }_{{h}_{B}}\langle {q}_{B}\rangle )}^{2}}\leqslant \frac{{\theta }_{{w}_{A}}^{2}\langle \langle {q}_{A}^{2}\rangle \rangle +{\theta }_{{w}_{B}}^{2}\langle \langle {q}_{B}^{2}\rangle \rangle }{{\theta }_{{h}_{A}}^{2}\langle \langle {q}_{A}^{2}\rangle \rangle +{\theta }_{{h}_{B}}^{2}\langle \langle {q}_{B}^{2}\rangle \rangle }={\eta }^{(2)}.\end{equation} \tag{ B2 }$$

Recall that, according to our convention used in appendices A and B ⟨q⟩ is a scaled measure, i.e. it counts the number of quanta exchanged, rather than the heat current itself. Following the same steps as in appendix A, we end with equation (A7), but with θ_q → θ_w , except in the F functions, and θ_w → θ_h . As before, we chose without loss of generality only the ⟨q_A ⟩ terms. The lower bound is true if

$$\begin{equation}2({\theta }_{{q}_{A}}-{\theta }_{{q}_{B}})({\theta }_{h}-{\theta }_{{q}_{A}})F({\theta }_{{q}_{A}})\leqslant ({\theta }_{{q}_{A}}-{\theta }_{{q}_{B}})(2{\theta }_{h}-{\theta }_{{q}_{A}}-{\theta }_{{q}_{B}})F({\theta }_{{q}_{B}}).\end{equation} \tag{ B3 }$$

When ${\theta }_{{q}_{A}} > {\theta }_{{q}_{B}}$,

$$\begin{equation}2({\theta }_{h}-{\theta }_{{q}_{A}})F({\theta }_{{q}_{A}})\leqslant (2{\theta }_{h}-{\theta }_{{q}_{A}}-{\theta }_{{q}_{B}})F({\theta }_{{q}_{B}})\end{equation} \tag{ B4 }$$

which is true, using that F is a decreasing function of θ_q . When ${\theta }_{{q}_{A}}< {\theta }_{{q}_{B}}$,

$$\begin{equation}2({\theta }_{h}-{\theta }_{{q}_{A}})F({\theta }_{{q}_{A}})\geqslant (2{\theta }_{h}-{\theta }_{{q}_{A}}-{\theta }_{{q}_{B}})F({\theta }_{{q}_{B}}),\end{equation} \tag{ B5 }$$

which also holds.

In figure 8, we search numerically for violations of the lower bound for an ensemble of three-level absorption engines in a broad parameter space. As we are not able to identify such violations, we hypothesize that the three-level system acting as either a refrigerator or an engine satisfies the lower bound on ${\eta }_{N}^{(2)}$.

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We now turn our attention to a pair of TC thermoelectric refrigerators, A and B, with both _A and _B at the opposite end of the cooling window from those discussed in section 6. We refer to this as 'region I', as depicted in figure 5. In this region, ⟨q_k ⟩, k = A, B, is small, vanishing when _k = μ_L, however, the charge current itself ⟨j_k ⟩ remains finite and nonzero. We suppose that $\langle {j}_{A}\rangle \approx \langle {j}_{B}\rangle \approx \langle {j}_{m}\rangle {\vert }_{{{\epsilon}}_{m}={\mu }_{\mathrm{L}}}\equiv J$. Defining u_k ≡ _k − μ_L, we have that ⟨q_k ⟩ = ⟨j_k ⟩u_k ≈ Ju_k . The cooling efficiency of a single refrigerator is η_k = u_k /Δμ, and the TC limit gives ${\eta }_{k}^{(2)}={({u}_{k}/{\Delta}\mu )}^{2}$.

Writing out the efficiency of the pair, we have

$$\begin{equation}{\eta }_{N=2}\approx \frac{\langle {q}_{A}\rangle +\langle {q}_{B}\rangle }{2{\Delta}\mu J}\approx \frac{{u}_{A}+{u}_{B}}{2{\Delta}\mu }=\frac{{\eta }_{A}}{2}\left(1+\frac{{u}_{B}}{{u}_{A}}\right).\end{equation} \tag{ C1 }$$

Since region I is not characterized by a small charge current, we cannot assume that the fluctuations in the charge or heat currents are given predominantly by thermal noise. However, we may echo our assumption above by supposing that $\langle \langle {j}_{A}^{2}\rangle \rangle \approx \langle \langle {j}_{B}^{2}\rangle \rangle \approx \langle \langle {j}_{m}^{2}\rangle \rangle {\vert }_{{{\epsilon}}_{m}={\mu }_{\mathrm{L}}}\equiv S$. Then, fluctuations in the cooling currents are given by $\langle \langle {q}_{k}^{2}\rangle \rangle \approx S{u}_{k}^{2}$, and the ratio of cooling to work current fluctuations is given by

$$\begin{equation}{\eta }_{N=2}^{(2)}\approx \frac{{u}_{A}^{2}+{u}_{B}^{2}}{2{\Delta}{\mu }^{2}}=\frac{{\eta }_{A}^{(2)}}{2}\left[1+{\left(\frac{{u}_{B}}{{u}_{A}}\right)}^{2}\right].\end{equation} \tag{ C2 }$$

In figure 9 we demonstrate that equations (C1) and (C2) indeed provide a good approximation for the efficiency and the ratio of fluctuations for refrigerators in region I.

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Squaring equation (C1), utilizing that ${\eta }_{A}^{2}={\eta }_{A}^{(2)}$, and comparing it to equation (C2), we see that the lower bound, ${\eta }_{N=2}^{2}\leqslant {\eta }_{N=2}^{(2)}$, is equivalent to the inequality

$$\begin{equation}\frac{1}{4}{\left(1+\frac{{u}_{B}}{{u}_{A}}\right)}^{2}\leqslant \frac{1}{2}\left[1+{\left(\frac{{u}_{B}}{{u}_{A}}\right)}^{2}\right],\end{equation} \tag{ C3 }$$

or, rearranging,

$$\begin{equation}{\left(\frac{{u}_{B}}{{u}_{A}}-1\right)}^{2}\geqslant 0.\end{equation} \tag{ C4 }$$

This is clearly satisfied for any choice of u_A and u_B , so we conclude that the lower bound ${\eta }_{N=2}^{2}\leqslant {\eta }_{N=2}^{(2)}$ is satisfied for a pair of refrigerators in this regime.

We now extend this argument more generally to n > 2 and N > 2, by getting expressions for η_N and ${\eta }_{N}^{(n)}$ of the ensemble analogous to equations (C1) and (C2),

$$\begin{align} {\eta }_{N}& =\frac{{\sum }_{k=1}^{N}\enspace {u}_{k}}{N{\Delta}\mu }, \\ {\eta }_{N}^{(n)}& =\frac{{\sum }_{k=1}^{N}\enspace {u}_{k}^{n}}{N{\Delta}{\mu }^{n}}. \end{align} \tag{ C5 }$$

Taking the nth root of ${\eta }_{N}^{(n)}$, we see that the lower bound, ${\eta }_{N}^{n}\leqslant {\eta }_{N}^{(n)}$, is equivalent to

$$\begin{equation}\frac{1}{N}\sum\limits _{k=1}^{N}\enspace {u}_{k}\leqslant {\left(\frac{1}{N}\sum\limits _{k=1}^{N}\enspace {u}_{k}^{n}\right)}^{\frac{1}{n}},\end{equation} \tag{ C6 }$$

which is always true as a result of the power mean inequality [55].