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arXiv:2405.00215 · Quantum

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

Vasco Cavina, Massimiliano Esposito

Published Jun 15, 2026• 2 citations

Abstract

We 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.

DOI: 10.22331/q-2026-06-15-2136

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We investigate the heat statistics in a relaxation process of quantum Brownian motion described by the Caldeira-Leggett model. By employing the normal mode transformation and the phase-space formulation approach, we can analyze the quantum heat distribution within an exactly dynamical framework beyond the traditional paradigm of Born-Markovian and weak-coupling approximations. It is revealed that the exchange fluctuation theorem for quantum heat generally breaks down in the strongly non-Markovian regime. Our results may improve the understanding about the nonequilibrium thermodynamics of open quantum systems when the usual Markovian treatment is no longer appropriate.

We study the Caldeira-Leggett model where a quantum Brownian particle interacts with an environment or a bath consisting of a collection of harmonic oscillators in the path-integral formalism. Compared to the contours that the paths take in the conventional Schwinger-Keldysh formalism, the paths in our study are deformed in the complex time plane as suggested by the recent study by C. Aron, G. Biroli, and L. F. Cugliandolo [SciPost Phys. 4, 008 (2018)10.21468/SciPostPhys.4.1.008]. This is done to investigate the connection between the symmetry properties in the Schwinger-Keldysh action and the equilibrium or nonequilibrium nature of the dynamics in an open quantum system. We derive the influence functional explicitly in this setting, which captures the effect of the coupling to the bath. We show that in equilibrium the action and the influence functional are invariant under a set of transformations of path-integral variables. The fluctuation-dissipation relation is obtained as a consequence of this symmetry. When the system is driven by an external time-dependent protocol, the symmetry is broken. From the terms that break the symmetry, we derive a quantum Jarzynski-like equality for a quantum mechanical worklike quantity given as a function of fluctuating quantum trajectory. In the classical limit, the transformations becomes those used in the functional integral formalism of the classical stochastic thermodynamics to derive the classical fluctuation theorem.

We calculate the reduced density matrix for a system of coupled harmonic oscillators in a bosonic heat bath using the Born-Markov approximation and show that the expectation values of position and momentum evolve like classical quantities. We consider the cases where every oscillator is coupled to its own reservoir and where all oscillators are coupled to the same heat bath, pointing out the differences between these two models. In particular, for systems coupled to a common reservoir, we present an uncommon dissipation mechanism, which disappears, when the dynamics of the system are gouverned by the low frequency modes of the environment (Caldeira-Leggett limit). We also address the interesting phenomenon of reservoir-induced interactions.

We derive the quantum thermodynamics of quantum Brownian motion from the exact solution of its reduced density matrix. We start from the total equilibrium thermal state between the Brownian particle and its reservoir, and solve analytically and exactly the reduced density matrix of the system by taking the partial trace over all the reservoir states. We find that the reduced Hamiltonian and the reduced partition function of the Brownian motion must be renormalized significantly, as shown in the general nonperturbative renormalization theory of quantum thermodynamics for open quantum systems we developed recently [Phys. Rev. Res. 4, 023141 (2022)]. The reduced Hamiltonian contains not only a frequency shift but also a squeezing pairing interaction, where a momentum-dependent potential is generated naturally from the strong coupling between the Brownian particle and the reservoir, after traced over all the reservoir states. The resulting exact reduced density matrix of the Brownian motion is given by a squeezing thermal state. Moreover, beyond the weak coupling limit, in order to obtain correctly the reduced partition function of the Brownian motion, one must take into account the non-negligible changes of the reservoir state induced by the system-reservoir coupling. Using the exact solutions of the reduced density matrix, the reduced Hamiltonian as well as the reduced partition function of the Brownian motion, we show that the controversial results obtained from the different definitions of internal energy and the issue of the negative heat capacity in the previous studies of strong-coupling quantum thermodynamics are resolved.