Microscopic thermal machines promise to play an important role in future quantum technologies. Making such devices widely applicable will require effective strategies to channel their output into easily accessible storage systems like classical degrees of freedom. Here, we develop a self-consistent theoretical framework that makes it possible to model such quantum-classical hybrid devices in a thermodynamically consistent manner. Our approach is based on the assumption that the quantum part of the device is subject to strong decoherence and dissipation induced by a thermal reservoir. Due to the ensuing separation of time scales between slowly evolving classical and fast relaxing quantum degrees of freedom, the dynamics of the hybrid system can be described by means of adiabatic-response theory. We show that, upon including fluctuations in a minimally consistent way, the resulting equations of motion can be equipped with a first and second law, both on the ensemble level and on the level of individual trajectories of the classical part of the system, where thermodynamic quantities like heat and work become stochastic variables. As an application of our theory, we work out a physically transparent model of a quantum-classical hybrid engine, whose working system consists of a chain of Rydberg atoms, which is confined in an optical cavity and driven by periodic temperature variations. We demonstrate through numerical simulations that the engine can sustain periodic oscillations of a movable mirror, which acts as a classical load, against external friction and extract the full distributions of input heat and output work. By making the statistics of thermodynamic processes in quantum-classical hybrid systems accessible without the need to further specify a measurement protocol, our work contributes to bridging the long-standing gap between classical and quantum stochastic thermodynamics.

中文翻译：

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量子经典边界的随机热力学：基于绝热响应理论的自洽框架


微型热机有望在未来的量子技术中发挥重要作用。要使此类设备广泛适用，需要有效的策略将其输出引导到易于访问的存储系统（例如经典自由度）中。在这里，我们开发了一个自洽的理论框架，使得以热力学一致的方式模拟这种量子经典混合器件成为可能。我们的方法基于这样的假设：设备的量子部分会受到热库引起的强烈退相干和耗散的影响。由于缓慢演化的经典自由度和快速弛豫量子自由度之间的时间尺度随之分离，混合系统的动力学可以通过绝热响应理论来描述。我们证明，在以最小一致的方式包含波动时，所得到的运动方程可以在系综层面和系统经典部分的个体轨迹层面上配备第一和第二定律，其中热力学热量和功等量成为随机变量。作为我们理论的应用，我们提出了量子经典混合发动机的物理透明模型，其工作系统由里德伯原子链组成，该原子链被限制在光学腔中并由周期性温度变化驱动。我们通过数值模拟证明，发动机可以承受可移动镜子（充当经典负载）的周期性振荡，抵抗外部摩擦，并提取输入热量和输出功的完整分布。 通过使量子经典混合系统中的热力学过程的统计数据无需进一步指定测量协议，我们的工作有助于弥合经典和量子随机热力学之间长期存在的差距。