Optimising finite-time quantum information engines using Pareto bounds

TL;DR

The paper addresses how finite-time measurement impacts information engines that convert information into work. It introduces a concrete model with a TLS measured by a quantum harmonic oscillator, analyzes three key metrics (information efficiency, power, and thermodynamic efficiency), and demonstrates that there exist Pareto-optimal trade-offs among these metrics. Using Pareto fronts and NSGA-II optimization, it identifies parameter regimes where positive power and high efficiency are achievable, and shows how measurement time, coupling strength, and energy scales govern performance. The results offer actionable design principles for experimental implementations in nano-mechanical and circuit-QED systems, highlighting the necessity of balancing information acquisition costs against work extraction. Overall, the work provides a finite-time, information-centric framework to optimize quantum information engines under realistic constraints, with broad relevance to nanoscale energy transduction and quantum thermodynamics.

Abstract

Information engines harness measurement and feedback to convert energy into useful work. In this study, we investigate the fundamental trade-offs between ergotropic output power, thermodynamic efficiency and information-to-work conversion efficiency in such engines, explicitly accounting for the finite time required for measurement. As a model engine, we consider a two-level quantum system from which work is extracted via a temporarily coupled quantum harmonic oscillator that serves as the measurement device. This quantum device is subsequently read out by a classical apparatus. We compute trade-offs for the performance of the information engine using Pareto optimisation, which has recently been successfully used to optimise performance in engineering and biological physics. Our results offer design principles for future experimental implementations of information engines, such as in nano-mechanical systems and circuit QED platforms.

Figures (15)

• Figure 1: Schematic of the information engine, showing a concrete model in which the system is a two-level system () and the meter is a quantum harmonic oscillator . Coupling them via the time-dependent interaction $\hat{V}_{\rm I}$ costs the measurement work $W_\mathrm{meas}$ and goes along with information transfer $I$. In the $j$-th cycle, the is projectively measured in its energy eigenstate $n_j$ with a classical measurement device with a corresponding entropy flow $\mathcal{S}$. The cycle is used to extract energy $W_{\text{ext}}$. After each cycle the and the are coupled to thermal baths at temperatures $T_{\text{S}}$ and $T_{\text{M}}$.
• Figure 2: Energy flow diagram for the information engine operating between two heat baths at temperatures $T_{\text{S}},$ and $T_{\text{M}}$ respectively. The engine is bipartite and consists of the system S from which energy $W_{\text{ext}}$ can be extracted upon measurement and information acquisition by a meter M with energetic cost $W_{\text{meas}}$. The extracted work is compensated by the heat $\mathcal{Q}_{\rm S}$ from the system's thermal bath while all energy invested for the measurement is released as heat $\mathcal{Q}_{\rm M}$ into the meter bath.
• Figure 3: Conditional probabilities of the two-level system states $P(i|n,t_{\text{m}}=0)$ for $i=0,1$ given a measurement outcome $n$. The solid lines show the conditional probabilities $P(i|n,t_{\text{m}}=0)$ at time $t_{\text{m}}=0$, i.e., when the system and meter are uncorrelated. The dotted lines show $P(i|n,t_{\text{m}})$ at finite measurement time $t_{\text{m}}$ which depend on the measurement outcome $n$. The dashed vertical line marks the first measurement outcome $n'$ at which $P(1|n,t_{\text{m}}) > P(0|n,t_{\text{m}})$. The chosen parameters are $\omega t_{\text{m}} = \pi/2$, $\Delta E =k_{\text{B}}T_{\text{S}} = g^2_{\text{eff}}$, $\hbar\omega = 0.1\Delta E$, $T_{\text{M}}/T_{\text{S}} = 0.3$.
• Figure 4: Information efficiency $\eta_\text{info}$ (solid black line), \ref{['eq:eta_info']}, on the left vertical axis, and the lowest lying meter level $n'$ (solid blue line) at which the condition $P(1|n,t_{\text{m}}) > P(0|n,t_{\text{m}})$ is first satisfied on the right vertical axis. Both are plotted as functions of the relative temperature $T_{\text{M}}/T_{\text{S}}$. The chosen parameters are $g_\text{eff}^2/\Delta E= 0.1$, $\Delta E = 4k_{\text{B}}T_{\text{S}}$, $\omega t_{\text{m}} = \pi/2$, and $\hbar\omega= 1.5k_{\text{B}}T_{\text{S}}$. The $\times$ denotes a specific operating point, serving as reference in \ref{['fig:P_eta_both']}(a) and \ref{['fig:P_eta_info_both']}.
• Figure 5: Information efficiency, \ref{['eq:eta_info']}, as a function of time for $g_\text{eff}^2/\Delta E=0.01,\,0.1,\, 1.0$. The other parameters are $T_{\text{M}}/T_{\text{S}} = 0.2$, $\Delta E = 4k_{\text{B}}T_{\text{S}}$, $\hbar\omega=1.5k_{\text{B}}T_{\text{S}}$. The $\times$ denotes a specific operating point, serving as reference in \ref{['fig:P_eta_both']}(a) and \ref{['fig:P_eta_info_both']}.