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Intrinsic Hamiltonian of Mean Force and Strong-Coupling Quantum Thermodynamics

Ignacio González1,*, Sagnik Chakraborty1,†, and Ángel Rivas1,2,‡

  • *Contact author: ignago10@ucm.es
  • †Contact author: sagnikch@ucm.es
  • ‡Contact author: anrivas@ucm.es
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PRX Quantum 7, 043014 – Published 21 October, 2026

DOI: https://doi.org/10.1103/pwws-21cx

Abstract

We present a universal thermodynamic framework for quantum systems strongly coupled to thermal reservoirs, identifying the class of initial conditions required for consistent thermodynamic behavior. Unlike other approaches, our framework provides a unified description of both thermodynamic processes and thermostatic properties, while preserving the same gauge freedoms as in the standard weak-coupling regime and retaining the von Neumann expression for the thermodynamic entropy. Furthermore, the formulation is entirely in terms of the system’s reduced state, relying only on variables accessible through local microscopic control. It is therefore directly connected to experimentally accessible quantities, making it possible to address operational questions concerning the controlled system—such as determining the minimum work required to drive it between two prescribed states in the presence of a strongly coupled reservoir. We validate our approach using a paradigmatic model of a structured bosonic reservoir.

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Popular Summary

In the quantum realm, small systems like atoms or superconducting circuits interact with their environments in complex ways. When these interactions are strong, standard thermodynamic concepts like heat or internal energy become ambiguous. This poses challenges for designing, controlling, or measuring energy flow in quantum devices. Our work addresses this by developing a new theoretical framework that extends thermodynamics to strongly interacting quantum systems, while preserving its foundational links with information theory. Existing approaches to strong-coupling thermodynamics often rely on assumptions that make experimental implementation difficult for example, requiring detailed knowledge of the environment. Others sacrifice consistency with well-known thermodynamic relations or lose the intuitive connection to quantum information. Our framework overcomes these limitations by introducing an “intrinsic Hamiltonian of mean force,” which depends solely on the system’s reduced state. This ensures that thermodynamic quantities—such as internal energy, entropy, and free energy—can be defined and measured using only system-level information, regardless of how strongly it interacts with its surroundings. We validate our approach using a model where a quantum oscillator interacts strongly with a structured environment. The results show that our method accurately captures equilibrium and nonequilibrium thermodynamic behavior, including heat flow and entropy production, while preserving standard relations even in regimes of strong coupling. Looking ahead, this framework could serve as a foundation for understanding energy and information flow in next-generation quantum technologies, such as quantum batteries, heat engines, or quantum simulators operating beyond the weak-coupling limit.

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PRX Quantum 7, 043014– Published 21 October, 2026
  • Received 18 June 2025
  • Revised 15 April 2026
  • Accepted 14 September 2026
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