Unlocking inaccessible performance of the quantum refrigerator with catalysts

TL;DR

This work addresses the challenge of boosting quantum refrigerator performance beyond conventional limits by introducing a finite-dimensional catalyst into a two-stroke refrigerator built from two TLSs. Using catalytic majorization and permutation protocols, it derives closed-form expressions for the COP and cooling window, showing that the catalyst can exceed the Otto bound and expand operational regimes while preserving cyclicity. A key finding is that two distinct permutation types are required to simultaneously enhance COP and the accessible operating range for refrigerators (unlike heat engines, which may be improved with a single permutation). These results provide a concrete framework for high-performance quantum cooling devices and offer a route to experimentally test catalytic quantum thermodynamics in nanoscale platforms.

Abstract

Quantum thermal machines offer promising platforms for exploring the fundamental limits of thermodynamics at the microscopic scale. The previous study demonstrated that the incorporation of a catalyst can significantly enhance the performance of a heat engine by broadening its operational regime and achieving a more favorable trade-off between work output and efficiency. Building on this powerful framework and innovative idea, here we further extend the concept to a two-stroke quantum refrigerator that extracts heat from a cold reservoir via discrete strokes powered by external work. The working medium consists of two two-level systems (TLSs) and two heat reservoirs at different temperatures and is assisted by an auxiliary system acting as a catalyst. Remarkably, the catalyst remains unchanged after each cycle, ensuring that heat extraction is driven entirely by the work input. We show that the presence of the catalyst leads to two significant enhancements: it enables the coefficient of performance (COP) and cooling capacity to exceed the Otto bound and allows the refrigerator to operate in frequency and temperature regimes that are inaccessible without a catalyst. Furthermore, through a comparison with catalytic heat engines, our analysis reveals that two distinct permutation types are necessary to simultaneously enhance the COP and operational range of refrigerators, in contrast to heat engines for which a single permutation suffices. These results highlight the potential of catalytic mechanisms to broaden the operational capabilities of quantum thermal devices and to surpass conventional thermodynamic performance limits.

Figures (9)

• Figure 1: Two-stroke quantum refrigerator assisted by a catalyst. The device is operated to cool a target cold reservoir characterized by inverse temperature $\beta_c$, transferring the extracted energy to a hot environment at inverse temperature $\beta_h$ with the assistance of an external work source $W$ and a catalytic auxiliary system.
• Figure 2: The schematic illustrates a permutation that exchanges the populations between the second and third excited states, resulting in an optimal coefficient of performance (COP) for the noncatalytic two-stroke refrigerator.
• Figure 3: The permutation scheme employs a catalyst to enhance the COP of the refrigerator. The figure illustrates all energy levels of the composite system, where $|i,j,k\rangle \equiv |i\rangle_s |j\rangle_h |k\rangle_c$. The catalyst expands the original four energy levels (two TLSs) into $d$-node groups, where $d$ represents the dimension of the catalyst. Each column corresponds to the $i$ th catalyst state ($i \in [1,d]$) acts on original TLSs). The permutation of energy levels is illustrated with red arrows, which indicate the exchange of populations between the corresponding levels. The initial populations corresponding to the energy levels $|i,0,0\rangle_{s,h,c}$, $|i,0,1\rangle_{s,h,c}$, $|i,1,0\rangle_{s,h,c}$, $|i,1,1\rangle_{s,h,c}$ are $\frac{p_i}{(1+a_c)(1+a_h)}$, $\frac{p_ia_c}{(1+a_c)(1+a_h)}$, $\frac{p_ia_h}{(1+a_c)(1+a_h)}$, $\frac{p_ia_ca_h}{(1+a_c)(1+a_h)}$, respectively. The permutation between energy levels in the $i$ and $i+1$ node-groups lead to a net population flow form the $i$th node to the $(i + 1)$th, denoted by $\delta P_i$ and indicated with blue arrows. The region enclosed by the yellow dashed rectangle represents the hot subspace. Summing the populations within this area yields the total population of the excited state of the hot qubit. Similarly, the region enclosed by the green dashed line represents the cold subspace, and the sum of the populations within it corresponds to the excited-state population of the cold qubit.
• Figure 4: The energy level permutation scheme that utilizes a catalyst to expand the operation regime of the refrigerator.
• Figure 5: The entropy production $\sigma$ (the left axis) and $\delta P$ (the right axis) varying with the catalyst dimension $d/n'$. The blue and pink dash-dotted curve plots $\sigma$ from Eq. \ref{['2nd_prove_2']} and $\delta P$ from Eq. \ref{['delta_P']}, respectively. The vertical green solid line indicates the theoretical critical point $d/n^{\prime} = \beta_h \omega_h / (\beta_c \omega_c)$, and the horizontal yellow solid line marks the condition $\sigma=0$.