Entropy-based analysis of single-qubit Otto and Carnot heat engines

TL;DR

The paper develops an entropy-based quantum thermodynamics framework in which heat corresponds to entropy changes and coherence work accounts for energy costs of non-unitary trajectories in a quantum regime. Using a single-qubit working medium, it analyzes Otto-like and Carnot-like cycles on the Bloch sphere, deriving explicit expressions for heat and coherence work and showing that the Carnot cycle achieves the classical efficiency $\eta_{Carnot}=1-\frac{T_L}{T_H}$ while the Otto cycle remains below this bound due to entropy production during isochoric steps. A geometric interpretation links the entropy production to an area in a $\tanh^{-1}(B)$ versus $B$ diagram, clarifying the irreversibility source and the efficiency gap. The work lays groundwork for future exploration of Stirling cycles and practical open-system control protocols to realize the dynamical strokes and assess experimental feasibility of quantum heat engines based on coherence work.

Abstract

From an entropy-based formulation of the first law of thermodynamics in the quantum regime, we investigate the performance of Otto-like and Carnot-like engines for a single-qubit working medium. Within this framework, the first law includes an additional contribution -- coherence work -- that quantifies the energetic cost of deviating the quantum trajectory from its natural unitary evolution. We focus on the efficiency of the heat-to-coherence work conversion and show that the Carnot cycle achieves the classical Carnot efficiency, while the performance of the Otto cycle is upper-bounded by the Carnot efficiency corresponding to the extreme temperatures of the cycle. We identify entropy generation during the isochoric stages as the key source of irreversibility limiting the Otto cycle's efficiency.

Figures (3)

• Figure 1: Representation of the Otto cycle on the Bloch sphere. The cycle is represented in the plane $B_x=0$ for convenience, though any vertical plane could have been chosen. The four strokes are: $1 \rightarrow 2$: isentropic process with $|\vec{B}| = B_1$, during which the Bloch vector rotates from angle $\theta_1$ to $\theta_2$; $2 \rightarrow 3$: isochoric heating at fixed angle $\theta_2$; $3 \rightarrow 4$: isentropic process with $|\vec{B}| = B_0$, where the vector rotates back from $\theta_2$ to $\theta_1$; and $4 \rightarrow 1$: isochoric cooling at fixed angle $\theta_1$.
• Figure 2: Diagrammatic representation of the Carnot cycle on the Bloch sphere. The four strokes are: $1\rightarrow2$: isentropic process with $|\vec{B}| = B_1$, where the Bloch vector rotates from $\theta_1$ to $\theta_2$; $2\rightarrow3$: isothermal heating at temperature $T_H$, from $\theta_2$ to $\theta_3$; $3\rightarrow4$: isentropic process with $|\vec{B}| = B_0$, from $\theta_3$ to $\theta_4$; and $4\rightarrow1$: isothermal cooling at temperature $T_L$, returning from $\theta_4$ to $\theta_1$.
• Figure 3: The entropy production in the quantum Otto cycle (in units of $k_{B}$) can be represented by the area of a rectangle in the $\tanh^{-1}(B)$ vs. $B$ diagram. For cycles with fixed width $\Delta B$, the rectangles have larger areas as their bases shift closer to 1, indicating greater entropy production.