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arXiv:2601.11480

Heat, work, and fluctuations in a driven quantum resonator

Riya Baruah, Pedro Portugal, Jun-Zhe Chen, Joachim Wabnig, Christian Flindt

quant-phcond-mat.mes-hallSubmitted Jan 16, 2026• Announced Jan 19, 2026• Revised Jun 26, 2026 (v2)
v1 · Jan 16, 2026v2 · Jun 26, 2026
arXivHTMLPDF

Abstract

A central building block of a heat engine is the working fluid, which mediates the conversion of heat into work. In nanoscale heat engines, the working fluid can be a quantum system whose behavior and dynamics are non-classical. A particularly versatile realization is a quantum resonator, which allows for precise control and coupling to thermal reservoirs, making it an ideal platform for exploring quantum thermodynamic processes. Here, we investigate the thermodynamic properties of a driven quantum resonator whose temperature is controlled by modulating its natural frequency. We evaluate the work performed by the external drive and the resulting heat flow between the resonator and its environment, both within linear response and beyond. To further elucidate these processes, we determine the full distribution of photon exchanges between the resonator and its environment, characterized by its first few cumulants. Our results provide quantitative insights into the interplay between heat, work, and fluctuations, and may help in designing future heat engines.

Journal reference
Phys. Rev. E 113, 064151 (2026)
DOI
10.1103/m7r2-vtdk
Comments
9 pages, 7 figures

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A central building block of a heat engine is the working fluid, which mediates the conversion of heat into work. In nanoscale heat engines, the working fluid can be a quantum system whose behavior and dynamics are non-classical. A particularly versatile realization is a quantum resonator, which allows for precise control and coupling to thermal reservoirs, making it an ideal platform for exploring quantum thermodynamic processes. Here, we investigate the thermodynamic properties of a driven quantum resonator whose temperature is controlled by modulating its natural frequency. We evaluate the work performed by the external drive and the resulting heat flow between the resonator and its environment, both within linear response and beyond. To further elucidate these processes, we determine the full distribution of photon exchanges between the resonator and its environment, characterized by its first few cumulants. Our results provide quantitative insights into the interplay between heat, work, and fluctuations, and may help in designing future heat engines.
Quantum thermodynamics sits at the intersection of quantum mechanics and thermodynamics, aiming to understand how key thermodynamic concepts such as heat, work, and entropy behave in quantum systems. A primary interest in this field is the study and development of quantum heat engines, devices that convert the flow of thermal energy into work using a quantum system as the working medium. Conventional quantum heat engines require consistent external control to drive their operating cycle. In contrast, autonomous quantum heat engines operate without such external control, instead, their cyclic behavior emerges entirely from internal dynamics and the coupling to thermal reservoirs. In this thesis, we study an autonomous quantum heat engine circuit based on superconducting quantum circuits, in which thermal noise from heat reservoirs drives cyclic energy exchange between two superconducting resonators coupled with a superconducting quantum interference device (SQUID). In this thesis, classical circuit simulations of the autonomous quantum heat engine are implemented in AWR Design Environment using lumped-element resonators, coupling capacitors, and Josephson junction models. By mapping quantum components to their classical analogues and varying the external magnetic flux threading the superconducting quantum interference device, the resonance frequencies, quality factors, and coupling dynamics of the circuit are analyzed across two parameter sets. The simulated quality factor of the second resonator exceeded the threshold required for negative dissipation in both parameter sets, consistent with conditions favorable for coherent microwave generation. Transient simulations further revealed that above a threshold drive power, energy builds up at the second resonator’s resonance frequency through SQUID-mediated coupling. This provides qualitative evidence that the classical model captures the essential inter-resonator coupling mechanism. However, systematic quantitative discrepancies in resonance frequencies and dissipation rates are identified, attributed primarily to the linearized SQUID representation used in the classical model. These results demonstrate that classical circuit simulations can qualitatively reproduce key dynamical features of the studied autonomous quantum heat engine, while also revealing the limitations of classical approximations in capturing its full quantum behavior.
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Quantum heat engines provide attractive means in quantum thermodynamics for studying the fundamentals of the flow of heat and work. Previous experimental implementations of heat engines operating at the level of a few excitation quanta have utilized external driving, which has made the observation of the produced work challenging. Conversely, autonomous quantum heat engines only require a flow of heat to operate and generate work. However, autonomous quantum heat engines have not yet been experimentally demonstrated in any system despite numerous theoretical investigations. Here, we experimentally realize an autonomous quantum heat engine based on superconducting circuits. We construct the engine circuit implementing an approximate Otto cycle by coupling two superconducting resonators with a superconducting quantum interference device, and coupling this system to spectrally filtered hot and cold reservoirs. By varying the experimental conditions, we observe coherent microwave power generation arising from the internal dynamics of the system driven only by the thermal reservoirs. Our results validate previous theoretical predictions for this circuit and pave the way for detailed studies of quantum effects in heat engines and for using heat-generated coherent microwaves in circuit quantum electrodynamics.
The fundamentals of a quantum heat engine are derived from first principles. The study is based on the equation of motion of a minimum set of operators, which is then used to define the state of the system. The relation between the quantum framework and the thermodynamical observables is examined. A four-stroke heat engine model with a coupled two-level system as a working fluid is used to explore the fundamental relations. In the model used, the internal Hamiltonian does not commute with the external control field, which defines the two adiabatic branches. Heat is transferred to the working fluid by coupling to hot and cold reservoirs under constant field values. Explicit quantum equations of motion for the relevant observables are derived on all branches. The dynamics on the heat transfer constant field branches is solved in closed form. On the adiabats, a general numerical solution is used and compared with a particular analytic solution. These solutions are combined to construct the cycle of operation. The engine is then analyzed in terms of the frequency-entropy and entropy-temperature graphs. The irreversible nature of the engine is the result of finite heat transfer rates and frictionlike behavior due to noncommutability of the internal and external Hamiltonians.
Conventional heat engines convert thermal energy into mechanical work by using a working medium such as steam or gas that operates between a hot and a cold reservoir. The medium absorbs heat from the hot reservoir, produces work, then releases some heat to the cold reservoir and returns to the initial state. With the advancement of quantum technologies, researchers strive to understand whether quantum media, such as atoms, spins, superconducting circuits and semiconductor quantum dots, can be used in quantum counterparts of the classical heat engines. After a recap of the basics of classical heat engines, this review explores the theoretical foundations and advancements of quantum heat engines, focusing on the proposed two- and multilevel systems used as the working medium for the quantum Otto cycle, which is the ideal thermodynamic cycle that describes the operation of a spark-ignition internal combustion engine. The review compares the efficiency and the work extraction of these systems, discussing their performance in the high- and low-temperature limits. Alternative cycles and engines are also presented to establish the adaptability of the working media when operating under different processes and non-equilibrium conditions. In conclusion, the review offers possible applications and future developments of quantum heat engines.
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