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Quantum thermodynamics of the Caldeira-Leggett model with non-equilibrium Gaussian reservoirs

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Quantum thermodynamics of the Caldeira-Leggett model with non-equilibrium Gaussian reservoirs

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AbstractWe introduce a non-equilibrium version of the Caldeira-Leggett model in which a quantum particle is strongly coupled to a set of engineered reservoirs. The reservoirs are composed by collections of squeezed and displaced thermal modes, in contrast to the standard case in which the modes are assumed to be at equilibrium. The model proves to be very versatile. Strongly displaced/squeezed reservoirs can be used to generate an effective time dependence in the system Hamiltonian and can be identified as sources of pure work. In the case of squeezing, the time dependence is stochastic and breaks the fluctuation-dissipation relation, this can be reconciled with the second law of thermodynamics by correctly accounting for the energy used to generate the initial non-equilibrium conditions. To go beyond the average description and compute the full heat statistics, we treat squeezing and displacement as generalized Hamiltonians on a modified Keldysh contour. As an application of this technique, we show the quantum-classical correspondence between the heat statistics in the non-equilibrium Caldeira-Leggett model and the statistics of a classical Langevin particle under the action of squeezed and displaced colored noises. Finally, we discuss thermodynamic symmetries of the heat generating function, proving a fluctuation theorem for the energy balance and showing that the conservation of energy at the trajectory level emerges in the classical limit.Featured image: Pictorial representation of the different reservoirs considered in the paper. The two reservoirs on the left are standard equilibrium reservoirs, whereas the two on the right are out of equilibrium through squeezing and displacement, respectively. In this graphical representation we depict thermal, squeezed, and displaced modes in a way that is reminiscent of their phase-space distributions. Non-equilibrium reservoirs are non-stationary, and contact with them can induce a time-dependent control of the system coupled to them.Popular summaryQuantum thermodynamics studies thermodynamic processes at the microscopic scale, where quantum effects become important. Besides average quantities such as work and heat, the field also investigates their fluctuations. One approach to the description of work fluctuations is to model work sources explicitly as auxiliary quantum systems, often called work reservoirs or batteries, whose coupling to the system induces an effective time dependence of its Hamiltonian. In this paper we investigate a non-equilibrium version of the Caldeira–Leggett model in which squeezed and displaced reservoirs act as work sources. The well-understood structure of the Caldeira–Leggett model allows us to interpolate between the standard heat limit, where a system is in contact with a thermal environment, and a work-source limit that is achieved when the squeezing or displacement is sufficiently strong, i.e. when the environment is driven far from equilibrium. This paper presents a paradigmatic model in which the autonomous work-source limit emerges from a microscopic description. This perspective also ensures that the thermodynamic description is fully consistent, as demonstrated by the validity of the fluctuation theorem.► BibTeX data@article{Cavina2026quantum, doi = {10.22331/q-2026-06-15-2136}, url = {https://doi.org/10.22331/q-2026-06-15-2136}, title = {Quantum thermodynamics of the {C}aldeira-{L}eggett model with non-equilibrium {G}aussian reservoirs}, author = {Cavina, Vasco and Esposito, Massimiliano}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2136}, month = jun, year = {2026} }► References [1] Amir O. Caldeira and Anthony J. Leggett. ``Path integral approach to quantum brownian motion''. Physica A: Statistical mechanics and its Applications 121, 587–616 (1983). https://doi.org/10.1016/0378-4371(83)90013-4 [2] Anthony J. Leggett, Sudit Chakravarty, Alan T. Dorsey, Matthew P.A. Fisher, Anupam Garg, and Wilhelm Zwerger. ``Dynamics of the dissipative two-state system''. Reviews of Modern Physics 59, 1 (1987). https://doi.org/10.1103/RevModPhys.59.1 [3] Milena Grifoni and Peter Hänggi. ``Driven quantum tunneling''. Physics Reports 304, 229–354 (1998). https://doi.org/10.1016/S0370-1573(98)00022-2 [4] Hermann Grabert and Ulrich Weiss. ``Quantum tunneling rates for asymmetric double-well systems with ohmic dissipation''. Physical review letters 54, 1605 (1985). https://doi.org/10.1103/PhysRevLett.54.1605 [5] Matthew P.A. Fisher and Alan T. Dorsey. ``Dissipative quantum tunneling in a biased double-well system at finite temperatures''. Physical review letters 54, 1609 (1985). https://doi.org/10.1103/PhysRevLett.54.2647.2 [6] Alex Kamenev. ``Field theory of non-equilibrium systems''.

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[4] Vasco Cavina, Antonio D'Abbruzzo, and Vittorio Giovannetti, "Unifying quantum stochastic methods using Wick's theorem on the Keldysh contour", Physical Review Research 7 4, 043262 (2025). [5] H. F. A. Coleman, R. A. Morrison, A. D. Armour, and E. K. Twyeffort, "Convergence to semiclassicality in the quantum Rabi model", arXiv:2604.09314, (2026). The above citations are from SAO/NASA ADS (last updated successfully 2026-06-15 13:19:55). The list may be incomplete as not all publishers provide suitable and complete citation data.Could not fetch Crossref cited-by data during last attempt 2026-06-15 13:19:53: Could not fetch cited-by data for 10.22331/q-2026-06-15-2136 from Crossref. This is normal if the DOI was registered recently.This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions. AbstractWe introduce a non-equilibrium version of the Caldeira-Leggett model in which a quantum particle is strongly coupled to a set of engineered reservoirs. The reservoirs are composed by collections of squeezed and displaced thermal modes, in contrast to the standard case in which the modes are assumed to be at equilibrium. The model proves to be very versatile. Strongly displaced/squeezed reservoirs can be used to generate an effective time dependence in the system Hamiltonian and can be identified as sources of pure work. In the case of squeezing, the time dependence is stochastic and breaks the fluctuation-dissipation relation, this can be reconciled with the second law of thermodynamics by correctly accounting for the energy used to generate the initial non-equilibrium conditions. To go beyond the average description and compute the full heat statistics, we treat squeezing and displacement as generalized Hamiltonians on a modified Keldysh contour. As an application of this technique, we show the quantum-classical correspondence between the heat statistics in the non-equilibrium Caldeira-Leggett model and the statistics of a classical Langevin particle under the action of squeezed and displaced colored noises. Finally, we discuss thermodynamic symmetries of the heat generating function, proving a fluctuation theorem for the energy balance and showing that the conservation of energy at the trajectory level emerges in the classical limit.Featured image: Pictorial representation of the different reservoirs considered in the paper. The two reservoirs on the left are standard equilibrium reservoirs, whereas the two on the right are out of equilibrium through squeezing and displacement, respectively. In this graphical representation we depict thermal, squeezed, and displaced modes in a way that is reminiscent of their phase-space distributions. Non-equilibrium reservoirs are non-stationary, and contact with them can induce a time-dependent control of the system coupled to them.Popular summaryQuantum thermodynamics studies thermodynamic processes at the microscopic scale, where quantum effects become important. Besides average quantities such as work and heat, the field also investigates their fluctuations. One approach to the description of work fluctuations is to model work sources explicitly as auxiliary quantum systems, often called work reservoirs or batteries, whose coupling to the system induces an effective time dependence of its Hamiltonian. In this paper we investigate a non-equilibrium version of the Caldeira–Leggett model in which squeezed and displaced reservoirs act as work sources. The well-understood structure of the Caldeira–Leggett model allows us to interpolate between the standard heat limit, where a system is in contact with a thermal environment, and a work-source limit that is achieved when the squeezing or displacement is sufficiently strong, i.e. when the environment is driven far from equilibrium. This paper presents a paradigmatic model in which the autonomous work-source limit emerges from a microscopic description. This perspective also ensures that the thermodynamic description is fully consistent, as demonstrated by the validity of the fluctuation theorem.► BibTeX data@article{Cavina2026quantum, doi = {10.22331/q-2026-06-15-2136}, url = {https://doi.org/10.22331/q-2026-06-15-2136}, title = {Quantum thermodynamics of the {C}aldeira-{L}eggett model with non-equilibrium {G}aussian reservoirs}, author = {Cavina, Vasco and Esposito, Massimiliano}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {10}, pages = {2136}, month = jun, year = {2026} }► References [1] Amir O. Caldeira and Anthony J. Leggett. ``Path integral approach to quantum brownian motion''. Physica A: Statistical mechanics and its Applications 121, 587–616 (1983). https://doi.org/10.1016/0378-4371(83)90013-4 [2] Anthony J. Leggett, Sudit Chakravarty, Alan T. Dorsey, Matthew P.A. Fisher, Anupam Garg, and Wilhelm Zwerger. ``Dynamics of the dissipative two-state system''. Reviews of Modern Physics 59, 1 (1987). https://doi.org/10.1103/RevModPhys.59.1 [3] Milena Grifoni and Peter Hänggi. ``Driven quantum tunneling''. Physics Reports 304, 229–354 (1998). https://doi.org/10.1016/S0370-1573(98)00022-2 [4] Hermann Grabert and Ulrich Weiss. ``Quantum tunneling rates for asymmetric double-well systems with ohmic dissipation''. Physical review letters 54, 1605 (1985). https://doi.org/10.1103/PhysRevLett.54.1605 [5] Matthew P.A. Fisher and Alan T. Dorsey. ``Dissipative quantum tunneling in a biased double-well system at finite temperatures''. Physical review letters 54, 1609 (1985). https://doi.org/10.1103/PhysRevLett.54.2647.2 [6] Alex Kamenev. ``Field theory of non-equilibrium systems''.

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Quantum thermodynamics of the Caldeira-Leggett model with non-equilibrium Gaussian reservoirs

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