Geometric Bound for Trade-off Relation in Quantum Tricycle

Abstract

We establish a finite-time quantum tricycle driven by an external field and investigate its thermodynamic performance in the slow-driving regime. By developing a perturbative expansion of heat with respect to operation time, we capture the dynamics of heat exchange processes beyond the quasistatic limit. Within a geometric framework, we derive fundamental bounds on trade-offs between the cooling rate, coefficient of performance, and dissipation, governed by the thermodynamic length and trajectory geometry in control space. Our findings unveil intrinsic limits to the performance of quantum thermal machines and highlight the role of geometry in shaping finite-time thermodynamics. This work advances the fundamental understanding of quantum thermodynamic processes and offers guiding principles for the design of next-generation quantum technologies.

Figures (2)

• Figure 1: (a) Schematic representation of a quantum tricycle for refrigeration and (b)Temperature--entropy ($T{\b@ld{bold} \text{\bfseries–}}–S$) diagram illustrating the thermodynamic cycle.
• Figure 2: Numerical verification of the two-level system. We plot the reduced thermodynamic length squared $\bar{\mathcal{\mathscr{L}}^{2}}$, the left-hand side (LH) defined as $R\left(\varepsilon_{r}/\varepsilon-1\right)$, and the right-hand side (RH) defined as $\bar{\mathcal{\mathscr{L}}^{2}}/\tau$, as functions of the contact time $\tau_{c}$ with the cold reservoir. Figs (a) and (b) correspond to spectral parameters $\alpha=0.8$ (sub-Ohmic) and $\alpha=1.2$ (super-Ohmic), respectively, with insets showing the optimized durations $\tau_{h}$ and $\tau_{p}$ as $\tau_{c}$ varies. Fig. (c) displays the dependence of $\bar{\mathcal{\mathscr{L}}^{2}}$, LH, and RH on $\alpha$ at fixed $\tau_{c}=20$, with the inset illustrating the optimized values of $\tau_{h}$ and $\tau_{p}$ as $\alpha$ changes. The remaining parameters are fixed as $\gamma_{0}=k_{B}=\hbar=1$, $\delta_{c}=k_{B}T_{c}$, $\zeta_{c}=\zeta_{h}=2$, $T_{h}=6$, $T_{p}=2.4$, and $T_{c}=2$.