Optimizing energy conversion with nonthermal resources in steady-state quantum devices

TL;DR

The paper addresses energy conversion in two-terminal coherent quantum devices driven by nonthermal resources, where distributions $f(\varepsilon)$ and $g(\varepsilon)$ differ and standard thermodynamic notions are not directly applicable. It develops a scattering-theory framework combined with a Lagrange-multiplier constraint to optimize performance (output current, efficiency, precision, or TUR-like trade-offs) at a fixed $I^x$, showing that the optimal energy filtering consists of a series of boxcar transmissions with window locations set by crossing points of spectral currents and occupation differences. When applied to cooling tasks, the nonthermal resources (from competing environments or irradiation) yield higher cooling power, improved efficiency, and reduced noise compared to equivalent thermal resources, and the boxcar structure provides practical design guidelines for nanoelectronic energy converters. These findings offer fundamental bounds and actionable guidance for exploiting nonthermal resources in future quantum devices, illustrating how carefully engineered energy filters can significantly enhance energy-conversion performance.

Abstract

We provide a framework for optimizing energy conversion processes in coherent quantum conductors fed by nonthermal resources. Such nonthermal resources, which cannot be characterized by temperatures or electrochemical potentials, occur in small-scale systems that are smaller than their thermalization length. Using scattering theory in combination with a Lagrange multiplier method, we optimize the device's performance based on the efficiency, precision, or a trade-off between the two at a given output current. The transmission properties leading to this optimal performance are identified. We showcase our findings with the example of a refrigerator exploiting experimentally relevant nonthermal resources, which could result from competing environments or from light irradiation. We show that the performance is improved compared to a device exploiting a thermal resource. Our results can serve as guidelines for the design of energy-conversion processes in future nanoelectronic devices.

Figures (12)

• Figure 1: Sketch of the energy landscape of a two-terminal conductor with a central coherent scattering region. The goal is to extract heat from the left contact thereby cooling it. This is done by exploiting energy filtering adapted to the distribution of the right contact acting as a resource, which is either (a) hotter than the left one or (b) given by a nonthermal distribution.
• Figure 2: Occupation probabilities of the thermal contact of temperature $T$ (blue) and of the nonthermal contact (purple) as function of the energy $\mu\equiv0$ in the thermal contact. Energy windows in which the transmission should be set to zero in order to achieve optimal cooling power $I^x=I^{Q,\mathrm{L}}$ are shaded in gray.
• Figure 3: Illustration of how (a) currents change with transmission probability $d_\gamma$ (with $\gamma=i,j$) of two example energy slices $d_i$,$d_{j}$ and (b) how the noise changes with transmission probability of two example energy slices $d_i$,$d_{j}$. The minimum is in both cases always located at the boundary of the functions (indicated by filled circles).
• Figure 4: Finding the transmission function with crossing points. Purple lines are related to quantities of the nonthermal contact, blue lines to quantities of the cold one. (a) Two distribution functions, where red indicates regions where the second factor in the argument of Eq. \ref{['eq:opt_eff_transmission']} is positive. (b) The two current coefficients entering the first factor of Eq. \ref{['eq:opt_eff_transmission']}; $y(\varepsilon)$ gives the entropy current into the nonthermal bath and $\lambda x(\varepsilon)$ the entropy current out of the cold bath, weighted by the Lagrange multiplier. Panel (c) illustrates where in the energy spectrum the two factors are negative. In the fainter red areas, the step-function in Eq. \ref{['eq:opt_eff_transmission']} yields zero. In the white and in the darker red area, the factors in the argument of Eq. \ref{['eq:opt_eff_transmission']} are both positive or both negative, such that the transmission function is set to 1.
• Figure 5: Example for the characteristic efficiency as function of energy (purple) compared to the minimum characteristic energy (black) fixed by the given output current. When the spectral currents $i^x,i^y$ have opposite signs, no cooling takes place (gray areas). When $\eta^\mathrm{char}<\eta^\mathrm{char}_\mathrm{min}=1/\lambda$, cooling is relatively inefficient and hence excluded by the optimization. The white areas are the ones where $\mathcal{D}(\varepsilon)=1$ leads to optimal efficiency at a given output current.