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arxiv: 2405.00215 · v3 · pith:WM5BFMHAnew · submitted 2024-04-30 · 🪐 quant-ph · cond-mat.stat-mech

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

Vasco Cavina , Massimiliano Esposito This is my paper

Pith reviewed 2026-05-24 01:10 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mech
keywords quantum thermodynamicsCaldeira-Leggett modelsqueezed reservoirsdisplaced reservoirsfluctuation theoremheat statisticsKeldysh contournon-equilibrium baths
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The pith

Non-equilibrium squeezed and displaced reservoirs act as pure work sources and generate effective time-dependent Hamiltonians in the Caldeira-Leggett model.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a non-equilibrium extension of the Caldeira-Leggett model in which a quantum particle couples strongly to collections of squeezed and displaced thermal modes rather than equilibrium ones. These reservoirs generate an effective time dependence in the system Hamiltonian and function as sources of pure work. For the squeezed case the resulting stochastic time dependence breaks the fluctuation-dissipation relation, yet remains consistent with the second law once the energy cost of preparing the initial non-equilibrium state is included. Full heat statistics are computed by representing squeezing and displacement as generalized Hamiltonians on a modified Keldysh contour; this yields a quantum-classical correspondence with the statistics of a classical Langevin particle under squeezed and displaced colored noise, together with a fluctuation theorem for the energy balance.

Core claim

Strongly displaced and squeezed reservoirs in the non-equilibrium Caldeira-Leggett model generate effective time dependence in the system Hamiltonian and act as pure work sources. Treating squeezing and displacement as generalized Hamiltonians on a modified Keldysh contour allows computation of the full heat statistics, which correspond to those of a classical Langevin particle driven by squeezed and displaced colored noises, while a fluctuation theorem for the energy balance holds and energy conservation at the trajectory level appears in the classical limit.

What carries the argument

Modified Keldysh contour in which squeezing and displacement are treated as generalized Hamiltonians, allowing exact extraction of heat statistics for Gaussian non-equilibrium reservoirs.

If this is right

  • Displaced reservoirs can engineer effective time-dependent Hamiltonians without external driving of the system.
  • Squeezed reservoirs produce stochastic time dependence that violates fluctuation-dissipation yet is reconciled by counting preparation energy.
  • The heat generating function obeys thermodynamic symmetries that imply a fluctuation theorem for the energy balance.
  • Heat statistics of the quantum model reduce exactly to those of a classical Langevin equation driven by colored noises.
  • Conservation of energy holds at the trajectory level only after the classical limit is taken.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same contour technique may extend to other open quantum systems where non-equilibrium Gaussian baths are engineered.
  • Work extraction protocols could be designed that rely solely on reservoir preparation rather than explicit time-dependent controls.
  • Trapped-ion or circuit-QED platforms could test the predicted classical-limit emergence of trajectory-wise energy conservation.

Load-bearing premise

Treating squeezing and displacement as generalized Hamiltonians on a modified Keldysh contour yields the full heat statistics without additional approximations beyond the Gaussian reservoir structure.

What would settle it

A direct computation or measurement of heat statistics for a specific squeezing strength that deviates from the predicted quantum-to-classical correspondence would falsify the central claim.

Figures

Figures reproduced from arXiv: 2405.00215 by Massimiliano Esposito, Vasco Cavina.

Figure 1
Figure 1. Figure 1: FIG. 1. Pictorial representation of the example considered at [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. In the limit in which the coupling is small ( [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Schematization of the TPEM for the four reservoirs example introduced at the end of sec [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The contour of integration for the MGF in Eq. ( [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Schematic representation of the application of the [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. A summary of the results of Secs. [PITH_FULL_IMAGE:figures/full_fig_p014_6.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript introduces a non-equilibrium Caldeira-Leggett model in which a quantum particle is coupled to collections of squeezed and displaced thermal modes. It argues that strongly displaced or squeezed reservoirs can generate effective time dependence in the system Hamiltonian and serve as sources of pure work. For squeezing, this leads to stochastic time dependence that breaks the fluctuation-dissipation relation, which is reconciled with the second law by accounting for the energy to prepare the initial state. The full heat statistics are computed by treating squeezing and displacement as generalized Hamiltonians on a modified Keldysh contour. This is used to demonstrate quantum-classical correspondence between the heat statistics and those of a classical Langevin particle driven by squeezed and displaced colored noises. Thermodynamic symmetries are discussed, including a fluctuation theorem for the energy balance and emergence of trajectory-level energy conservation in the classical limit.

Significance. If the modified Keldysh contour technique is shown to be exact for the heat generating function based on the Gaussian structure, the work would offer a useful extension of open quantum system methods to non-equilibrium reservoirs, enabling studies of heat fluctuations and fluctuation theorems in driven systems while highlighting the role of reservoir engineering in thermodynamics.

major comments (1)
  1. [Abstract (paragraph on heat statistics)] The central claim that displaced/squeezed reservoirs act as pure work sources and that a fluctuation theorem follows requires that treating them as generalized Hamiltonians on a modified Keldysh contour produces the exact heat generating function from the Gaussian reservoir structure alone. No explicit justification is given for why the contour deformation preserves the correct operator ordering or counting-field insertion for heat without further approximations (e.g., in the definition of the system-bath interaction or the initial-state factorization). This is the load-bearing step for both the quantum statistics and the claimed quantum-classical correspondence.
minor comments (1)
  1. The abstract is dense; consider breaking the description of the model and results into clearer paragraphs for readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the central role of the modified Keldysh contour construction. We address the single major comment below and will strengthen the presentation accordingly.

read point-by-point responses

  1. Referee: [Abstract (paragraph on heat statistics)] The central claim that displaced/squeezed reservoirs act as pure work sources and that a fluctuation theorem follows requires that treating them as generalized Hamiltonians on a modified Keldysh contour produces the exact heat generating function from the Gaussian reservoir structure alone. No explicit justification is given for why the contour deformation preserves the correct operator ordering or counting-field insertion for heat without further approximations (e.g., in the definition of the system-bath interaction or the initial-state factorization). This is the load-bearing step for both the quantum statistics and the claimed quantum-classical correspondence.

    Authors: We agree that an explicit derivation of the exactness of the heat generating function is essential. The Gaussian character of the reservoirs permits an exact influence functional on the Keldysh contour; the counting field for heat is introduced by a contour shift that corresponds to a displacement in the bath operators, preserving the time-ordering because the bath correlation functions remain Gaussian. The initial-state factorization is the standard product-state assumption used throughout the Caldeira-Leggett literature and is not an additional approximation. We will insert a dedicated subsection (new Sec. III C) that derives the generating function from the Gaussian path integral, showing that no further approximation is required beyond the model definition. This will also clarify the quantum-classical correspondence at the level of the characteristic function. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained on standard techniques

full rationale

The paper applies the standard Caldeira-Leggett model to non-equilibrium Gaussian reservoirs (squeezed/displaced modes) and introduces a modified Keldysh contour to compute heat statistics. No load-bearing steps reduce by construction to fitted parameters, self-definitions, or self-citation chains; the contour technique is presented as a direct extension without the result being presupposed in the inputs. The fluctuation theorem and quantum-classical limit follow from the model equations without renaming known results or importing uniqueness from prior author work. The framework remains externally falsifiable via standard open-system benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claims rest on the standard quantum open-system assumptions plus the new reservoir preparation; no new particles or forces are postulated.

free parameters (1)
  • squeezing and displacement amplitudes
    Parameters that define the non-equilibrium state of each reservoir mode; their values control the effective drive and work input.
axioms (2)
  • domain assumption Reservoirs remain Gaussian after squeezing and displacement
    The model treats the baths as collections of Gaussian modes whose statistics are fully captured by displacement and squeezing parameters.
  • domain assumption Modified Keldysh contour correctly encodes the non-equilibrium initial conditions
    The technique is invoked to compute heat statistics without further justification in the abstract.

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

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Reference graph

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