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Equivalence of Discrete and Continuous Otto-Like Engines assisted by Catalysts: Mapping Catalytic Advantages from the Discrete to the Continuous Framework

This paper establishes a theoretical equivalence between discrete and continuous Otto-like engines by mapping their respective dynamics, thereby demonstrating that the catalytic advantages observed in discrete two-stroke systems—such as enhanced power and efficiency—can be successfully translated into and realized within experimentally feasible continuous frameworks.

Original authors: Marcin Łobejko, Tanmoy Biswas, Michał Horodecki

Published 2026-09-09

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CC BY 4.0

Original authors: Marcin Łobejko, Tanmoy Biswas, Michał Horodecki

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Heat engines are the workhorses of our modern world, from the pistons in a car to the turbines in a power plant. At their core, they are simple machines that convert heat into motion. They operate by taking energy from a hot source, letting some of it escape to a cold sink, and using the difference to do work. For over a century, physicists have studied the theoretical limits of how efficient these machines can be. In the microscopic realm, where the rules of quantum mechanics take over, scientists have been designing tiny engines that operate on single atoms or particles. These microscopic engines often follow a specific rhythm known as the Otto cycle, named after the inventor of the gasoline engine. In this cycle, the engine works in distinct steps: it first absorbs heat, then changes its internal energy to extract work, then releases heat, and finally resets. While this step-by-step approach is easy to calculate on paper, building a real machine that switches between these steps with perfect timing is incredibly difficult. It requires precise external control to coordinate every single move, which is often too complex for current laboratory technology.

A newer idea in this field involves using a "catalyst" to help the engine run better. In chemistry, a catalyst is a substance that speeds up a reaction without being used up. In the world of quantum engines, a catalyst is an extra system that interacts with the engine to boost its performance—making it more powerful or more efficient—but returns to its exact starting state at the end of the cycle, leaving no trace of its involvement. Previous theoretical studies showed that adding such a catalyst to a discrete, step-by-step engine could push its efficiency far beyond what was previously thought possible, even approaching the absolute maximum limit allowed by the laws of physics. However, because these studies relied on the difficult-to-build step-by-step model, it remained unclear whether these impressive gains could actually be achieved in a real, continuous machine that runs smoothly without stopping and starting.

This is the central puzzle that Marcin Łobejko, Tanmoy Biswas, and Michał Horodecki set out to solve. They wanted to know if the theoretical advantages of these catalytic, step-by-step engines could be translated into the language of continuous engines, which are much more practical to build. The researchers developed a mathematical bridge that connects the two worlds. Instead of trying to force a continuous machine to mimic the jerky steps of a discrete one, they showed that the two are fundamentally the same thing, just described in different ways. They demonstrated that the "flow" of probability in a step-by-step engine is mathematically identical to the "current" of probability in a continuous engine. By replacing the idea of a discrete jump with a steady flow, they proved that the performance benefits seen in the theoretical step-by-step models hold true for the continuous ones as well.

The team focused on a specific type of engine that uses a simple two-level system as a catalyst. In the discrete version, this catalyst helps the engine shuffle energy in a way that extracts more work than a standard engine could. The researchers showed that when you map this setup onto a continuous engine—one that is constantly connected to both a hot and a cold bath while being driven by an external field—the results are identical. The continuous engine achieves the same boost in efficiency and power output. Crucially, they found that the continuous engine does not need to be controlled with the same frantic precision as the discrete one. It runs naturally, driven by the interaction between the engine, the catalyst, and the heat baths. The catalyst still plays its role, helping to guide the energy flow, but it does so while remaining thermally isolated, ensuring it returns to its original state after every cycle.

One of the most significant findings is that the efficiency of the engine remains unchanged during this translation from the discrete to the continuous world. If a discrete engine with a catalyst can reach a certain level of efficiency, its continuous counterpart will reach that exact same level. Furthermore, the researchers showed that the continuous engine can actually outperform the standard non-catalytic version in terms of power output across a wide range of efficiencies. In their simulations, the catalytic continuous engine produced more power than the standard engine for every possible efficiency setting, proving that the advantage is not just a theoretical curiosity but a robust feature of the system. They also identified a specific relationship between the time it takes for a discrete engine to complete a cycle and the rate at which a continuous engine produces power, linking the two concepts through a single characteristic time constant.

The study provides a clear path forward for experimentalists. Because continuous engines are easier to build and control than their step-by-step counterparts, this mapping suggests that the impressive gains predicted by theory can be realized in the lab. The researchers point to existing platforms, such as trapped ions and superconducting qubits, as potential testbeds where these catalytic advantages could be demonstrated. By showing that the complex, step-by-step models are equivalent to smoother, continuous ones, the work removes a major barrier between theory and practice. It confirms that the laws of thermodynamics, when aided by quantum catalysts, allow for machines that are not only more efficient but also more powerful, and that these machines can be built in a way that is compatible with current technology. The results suggest that the future of microscopic energy conversion may lie not in perfecting the timing of discrete steps, but in harnessing the steady, continuous flow of energy with the help of a catalyst that never gets tired.

Technical Summary: Equivalence of Discrete and Continuous Otto-Like Engines Assisted by Catalysts

Problem Statement
Theoretical models of quantum thermal machines, particularly Otto engines, have traditionally relied on discrete frameworks where work extraction and thermalization occur in distinct, sequential strokes. While these discrete models are analytically tractable and have demonstrated that ancillary systems (catalysts) can enhance efficiency and power beyond standard non-catalytic limits, they pose significant challenges for experimental implementation. Realizing discrete strokes requires precise external control to coordinate the sequence of operations, which is technically demanding. Conversely, continuous engines—systems simultaneously coupled to heat baths and a driving field—are more experimentally feasible as they operate without distinct strokes. However, a rigorous theoretical bridge connecting the performance metrics of discrete catalytic engines to their continuous counterparts has been lacking. Specifically, it remains unclear whether the catalytic advantages observed in discrete Otto-like engines (such as efficiencies between the Otto and Carnot limits) translate directly to continuous regimes.

Methodology
The authors establish a direct mathematical equivalence between discrete and continuous Otto-like engines, both with and without catalysts. The methodology involves:

Discrete Framework Formulation: Defining a discrete engine using a density matrix $\rho$ and a Hamiltonian $H_0$ split into hot and cold degrees of freedom. The work stroke is modeled as a unitary permutation $S$ that swaps specific states ( $|u_i\rangle \leftrightarrow |d_i\rangle$ ), while the heat stroke involves thermalization. Heat and work are defined via probability flows ( $\Delta p_i$ ) and energy differences ( $\Delta \varepsilon^k_i$ ).
Continuous Framework Formulation: Constructing a continuous analog using a total Hamiltonian $H_0 + V_t$ , where $V_t$ represents a resonant driving field. The system evolves under a local master equation (Lindblad formalism) incorporating dissipators for the hot and cold baths.
Mapping Strategy: The core of the approach is mapping the discrete "probability flow" ( $\Delta p_i$ ) to the continuous "probability current" ( $\langle \dot{p}_i \rangle$ ). By introducing a characteristic time constant $\tau_i = \Delta p_i / \langle \dot{p}_i \rangle$ , the authors relate the work per cycle in the discrete model to the power (work per unit time) in the continuous model.
Catalytic Conditions: The study extends this mapping to catalytic engines. For discrete engines, the catalyst condition requires the marginal state of the catalyst to remain invariant after the work stroke. For continuous engines, this is translated into a condition on the probability currents where the net flow associated with the catalyst's projectors vanishes.

Key Contributions and Results

Establishment of Equivalence: The paper proves that discrete and continuous Otto-like engines are equivalent under the mapping $\Delta p_i \leftrightarrow \langle \dot{p}_i \rangle$ . This equivalence preserves the thermodynamic structure, including the First Law ( $W = Q_h + Q_c$ vs. $P = J_h + J_c$ ) and the Second Law (Clausius inequality).
Preservation of Catalytic Advantage: A central finding is that the efficiency formula for catalytic engines, $\eta = 1 + \frac{\sum \Delta \varepsilon^c_i \Delta p_i}{\sum \Delta \varepsilon^h_i \Delta p_i}$ , remains invariant under the mapping to the continuous regime. Consequently, the catalytic enhancement in efficiency—allowing values between the Otto efficiency ( $\eta_{Otto}$ ) and the Carnot efficiency ( $\eta_{Carnot}$ )—is fully retained in the continuous model.
Power-Efficiency Trade-off: The authors demonstrate that for a specific class of "simple permutations" (where probability flows are equal), the continuous engine reproduces the work output advantages of the discrete catalytic engine. Specifically, if a discrete catalytic engine outperforms a non-catalytic one in work output ( $W_{Catalytic} > W_{Otto}$ ), the corresponding continuous engine exhibits higher power ( $P_{Catalytic} > P_{Otto}$ ) for the same efficiency, provided the characteristic cycle time of the catalytic engine is shorter or equal to that of the standard engine.
Specific Model Demonstration: The mapping is explicitly illustrated for the simplest catalytic extension of the Otto engine using a two-dimensional catalyst. The results confirm that the continuous model achieves the same efficiency bounds and power enhancements as the discrete model, visualized through power-efficiency trade-off curves.

Significance
The paper claims that this work provides a unified theoretical picture for quantum thermal machines, validating the use of discrete models as a tool for understanding and optimizing continuous, experimentally realizable engines. By showing that catalytic advantages are not artifacts of the discrete stroke approximation but are robust features that persist in continuous dynamics, the study bridges the gap between theoretical quantum thermodynamics and experimental feasibility. The authors emphasize that their results open the door for experimental demonstrations of catalytic advantages in platforms such as trapped ions and superconducting qubits, where continuous coupling is more natural than discrete stroke control. The work suggests that the insights gained from analytically simpler discrete models can be directly applied to the design of more practical continuous thermal machines.

Technical Summary: Equivalence of Discrete and Continuous Otto-Like Engines Assisted by Catalysts

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