Correlated quantum machines beyond the standard second law

TL;DR

The paper extends thermodynamics to correlated quantum machines by deriving exact generalized laws for cyclic processes with arbitrary system–bath correlations. It introduces a generalized first law $W = \sum_j Q_j + \Delta U$ and a generalized second law $\sum_i \Delta S(\rho_i) + \sum_j Q_j/(kT_j) = \Delta \Sigma$, where $\Delta \Sigma$ captures correlations and nonequilibrium effects; notably, a negative $\Delta \Sigma$ indicates entropic resources that can drive work. A universal efficiency formula $\eta = \gamma\left(\eta_{th} - \frac{k T_{\min} \Delta \sigma}{Q^{in}} \right)$ is derived, enabling work to be extracted from entropic resources and potentially exceed the Carnot bound in the athermal regime. These results are illustrated with a two-oscillator engine, where initial correlations can dominate early cycles before entropy production returns the system to thermal operation. The work provides a unified framework to quantify and optimize correlated quantum machines and links to prior Clausius-type bounds while highlighting novel entropic resources as a resource for work.

Abstract

The laws of thermodynamics strongly restrict the performance of thermal machines. Standard thermodynamics, initially developed for uncorrelated macroscopic systems, does not hold for microscopic systems correlated with their environments. We here derive an exact formula for the efficiency of any cyclically driven quantum engine by using generalized laws of quantum thermodynamics that account for all possible correlations between all involved parties, including initial correlations. Furthermore, we demonstrate the existence of two basic modes of engine operation: the usual thermal case, where heat is converted into work, and a novel athermal regime, where work is extracted from entropic resources, such as system-bath correlations. In the latter regime, the efficiency is not bounded by the usual Carnot formula. Our results provide a unified formalism to determine the efficiency of correlated microscopic quantum machines.

Figures (4)

• Figure 1: Operation regimes of correlated engines. Depending on the value of the ratio, Eq. \ref{['6']}, of thermal and athermal contributions, correlated engines may produce work by converting heat (thermal regime) or entropic resources, such as correlations (athermal regime). The plot shows the logarithm of the upper bound of Eq. \ref{['6']} (normalized to $[-1,1]$), as a function of the thermal efficiency $\eta_\text{th}$ and the coefficient $\gamma$. Engines are guaranteed to run in the thermal regime when $1 / \gamma \eta_\text{th}>2$ (solid line). The dashed rectangle indicates the region explored by the two-oscillator-engine example of Fig. 3.
• Figure 2: Two-oscillator quantum engine. The working substance $\mathcal{S}$ is composed of two harmonic oscillators, with respective frequencies $\omega_\text{c}$ and $\omega_\text{h}$, each coupled to its own reservoir, $\mathcal{R}_\text{c}$ and $\mathcal{R}_\text{h}$, with temperatures $T_\text{c}$ and $T_\text{h}$, and coupling constants $\lambda_\text{c}$ and $\lambda_\text{h}$. Work is produced by periodically switching the oscillator interaction $\lambda f(t)$ on and off. The oscillators thermalize with their respective baths during the off phases, leading to the creation of correlations between them.
• Figure 3: Operation regimes of the two-oscillator engine. The plot shows the (normalized) logarithm of Eq. \ref{['6']}, after the first cycle. The black circles indicate the exact boundary between thermal and athermal regimes given by $\eta / \gamma \eta_\text{th}=2$, whereas the solid line corresponds to the device-independent upper bound of Fig. 1. Parameters are $\omega_\text{c} = 1$, $\omega_\text{h} = 2$, $\lambda = 0.08$, $\lambda / 15 \leq \lambda_\text{c} = \lambda_\text{h} \leq \lambda / 2$, $T_\text{c} = 0.8$ and $1.7 \leq T_\text{h} \leq 8$.
• Figure 4: Performance of the two-oscillator engine. abc) In the thermal regime, the engine produces work from heat, while initial correlation are not exploited ($T_\text{c} \Delta \sigma >0$ after a transient). The efficiency $\eta$ is always smaller than the Carnot efficiency $\eta_\text{C}$ and quickly converges to the Otto efficiency $\eta_\text{O}$. Parameters are $\lambda = 0.08$, $\lambda_\text{c} = \lambda_\text{h} = \lambda / 2$, $T_\text{c} = 0.8$ and $T_\text{h} = 8$ (corresponding to the upper right corner in Fig. \ref{['fig:quotientSimulation']}). def) In the athermal regime, the engine predominantly produces work from entropic resources such as system-bath correlations that are created during the thermalization with the baths: $T_\text{c} \Delta \sigma$ initially grows more negative, work is larger than the absorbed heat and the efficiency exceeds the Carnot efficiency. As the number of cycles increases, nonequilibrium entropy produces leads to $T_\text{c} \Delta \sigma>0$, pushing the engine to thermal operation. The 'classical' entropic term $T_\text{c} \Delta \tilde{\sigma}$ deviates from the fully quantum expression $T_\text{c} \Delta \sigma$. Parameters are $\lambda = 0.08$, $\lambda_\text{c} = \lambda_\text{h} = \lambda / 15$, $T_\text{c} = 0.8$ and $T_\text{h} = 1.7$ (corresponding to the lower left corner in Fig. \ref{['fig:quotientSimulation']}).