pith. machine review for the scientific record. sign in

arxiv: 2604.26524 · v1 · submitted 2026-04-29 · 🪐 quant-ph · cond-mat.stat-mech

Recognition: unknown

Over forty years of research towards the understanding of Quantum Brownian Motion -- the contributions of A. O. Caldeira

Marcus V. S. Bonan\c{c}a , Sebastian Deffner , Gert-Ludwig Ingold
Authors on Pith no claims yet

Pith reviewed 2026-05-07 11:24 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.stat-mech
keywords Quantum Brownian motionCaldeira-Leggett modelQuantum tunnelingDissipationQuantum decoherenceQuantum thermodynamicsMetastable statesOpen quantum systems
0
0 comments X

The pith

Caldeira's models show that dissipation reduces the tunneling rate out of metastable quantum states.

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

The paper reviews Amir O. Caldeira's contributions to quantum Brownian motion across more than four decades. It first outlines the quantum regime of Brownian motion using his early work and emphasizes how dissipation affects escape from metastable states. Caldeira later developed alternative descriptions that extend beyond the standard model of a system coupled to a bath of oscillators. These ideas have shaped treatments of quantum decoherence and quantum thermodynamics. A sympathetic reader would care because the results explain how real environments alter quantum dynamics in systems ranging from microscopic particles to engineered devices.

Core claim

Caldeira established that in the quantum regime the coupling of a system to a dissipative environment, modeled as a thermal bath, quantitatively suppresses the rate at which a particle tunnels out of a metastable potential well, while also supplying consistent fluctuation-dissipation relations that connect to classical Brownian motion.

What carries the argument

The Caldeira-Leggett model, which represents the environment as a continuum of harmonic oscillators linearly coupled to the system coordinate and yields an exact path-integral treatment of dissipation.

If this is right

  • Dissipation can be tuned to control or suppress unwanted tunneling in quantum systems.
  • Decoherence times in open quantum systems follow from the same environment-coupling calculations.
  • Fluctuation-dissipation theorems derived in these models constrain quantum thermodynamic processes such as heat exchange at small scales.
  • Alternative formulations developed later allow treatment of non-Markovian dynamics and stronger couplings.

Where Pith is reading between the lines

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

  • Design of quantum devices could deliberately introduce controlled dissipation to stabilize states against tunneling leakage.
  • Experiments in superconducting circuits or trapped ions could directly test the predicted temperature and coupling dependence of the tunneling rate.
  • The same framework offers a route to derive quantum corrections for Brownian motion in complex media beyond the original harmonic-oscillator bath.

Load-bearing premise

The review assumes the selected papers and topics fully capture Caldeira's most important contributions without supplying a systematic selection method.

What would settle it

A controlled experiment that measures the tunneling rate from a metastable state while varying the system-environment coupling strength and finds the rate unchanged or increased would contradict the highlighted suppression effect.

read the original abstract

This article presents a brief account of Amir O. Caldeira's contributions to the theory of quantum Brownian motion. Motivated by its importance, we outline the description of Brownian motion in the quantum regime following Caldeira's first works. In this context, we particularly highlight the effect of dissipation on the tunneling rate out of a metastable state. We then journey along the alternative ways to approach quantum Brownian motion developed by Caldeira during his career, which go beyond the so-called Caldeira-Leggett model. We conclude by summarizing some of Caldeira's contributions to contemporary fields such as the theory of quantum decoherence and quantum thermodynamics, that were strongly inspired by his eponymous approach to quantum Brownian motion.

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

0 major / 1 minor

Summary. The manuscript provides a concise historical review of Amir O. Caldeira's contributions to the theory of quantum Brownian motion. It outlines the quantum description of Brownian motion from Caldeira's early works, highlights the effect of dissipation on the tunneling rate out of a metastable state, discusses alternative approaches developed by Caldeira that extend beyond the Caldeira-Leggett model, and summarizes influences on quantum decoherence and quantum thermodynamics.

Significance. If accurate, this targeted biographical review consolidates key technical developments in quantum open systems over four decades and provides useful context for Caldeira's foundational models and their documented influence on decoherence and thermodynamics. The narrower scope (one researcher's contributions rather than a comprehensive field survey) allows focused technical descriptions without requiring exhaustive justification of topic selection. No new derivations, machine-checked proofs, or reproducible code are presented, consistent with the review format.

minor comments (1)
  1. [Abstract] The abstract employs the phrasing 'we journey along the alternative ways' which is informal for a journal article; rephrasing to a direct statement such as 'we review' would improve the formal tone.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive assessment of our manuscript. The referee's summary accurately captures the scope and content of our review on A. O. Caldeira's contributions. Given the recommendation for minor revision but the absence of specific major comments, we have conducted a thorough review of the manuscript for any potential improvements and will incorporate minor editorial changes as appropriate.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper is a narrative historical review summarizing Amir Caldeira's published contributions to quantum Brownian motion, dissipation effects on tunneling, extensions beyond the Caldeira-Leggett model, and influences on decoherence and quantum thermodynamics. It contains no new derivations, equations, predictions, or fitted parameters. All technical content is attributed to prior external publications via citations, with no self-referential reduction of claims to the paper's own inputs or self-citations that bear the central load. The selection of topics is presented as motivated by importance rather than derived, and the reader's assessment of no derivations or predictions is confirmed by inspection of the full text.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The paper is a historical review and introduces no free parameters, axioms, or invented entities; all content refers to previously published models and results.

pith-pipeline@v0.9.0 · 5441 in / 960 out tokens · 34891 ms · 2026-05-07T11:24:31.825931+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

99 extracted references · 70 canonical work pages

  1. [1]

    Edinburgh New Philos

    Brown, R.: A brief account of microscopical observations made in the months of June, July, and August 1827, on the particles contained in the pollen of plants; and on the general existence of active molecules in organic and inorganic bodies. Edinburgh New Philos. J.5, 359–371 (1828)

  2. [2]

    In: Verm

    Ingen-Housz, J.: Bemerkungen ¨ uber den Gebrauch des Vergr¨ osserungsglases. In: Verm. Schriften Physisch-medicinischen Inhalts vol. 2, p. 123. C. F. Wappler, Wien (1784)

  3. [3]

    Gouy, L.G.: Note sur le mouvement brownien. J. Phys. Theor. Appl.7, 561–564 (1888) 14

  4. [4]

    Einstein, A.: ¨Uber die von der molekularkinetischen Theorie der W¨ arme geforderte Bewegung von in ruhenden Fl¨ ussigkeiten suspendierten Teilchen. Ann. Phys. (Leipzig)322, 549–560 (1905) https://doi.org/10.1002/andp. 19063261405

    work page doi:10.1002/andp 1905

  5. [5]

    Sutherland, W.: A dynamical theory of diffusion for non-electrolytes and the molecular mass of albumin. Phil. Mag. S. 69, 781–785 (1905)

    1905

  6. [6]

    Smoluchowski, M.: Zur kinetischen Theorie der Brownschen Molekularbewegung und der Suspensionen. Ann. Phys. (Leipzig)326, 756–780 (1906) https://doi. org/10.1002/andp.19063261405

    work page doi:10.1002/andp.19063261405 1906

  7. [7]

    Kubo, R.: The fluctuation-dissipation theorem. Rep. Prog. Phys.29, 255 (1966) https://doi.org/10.1088/0034-4885/29/1/306

    work page doi:10.1088/0034-4885/29/1/306 1966

  8. [8]

    Springer, Berlin (2004)

    Gardiner, C.W., Zoller, P.: Quantum Noise, 3rd edn. Springer, Berlin (2004)

    2004

  9. [9]

    H¨ anggi, P.: Escape from a metastable state. J. Stat. Phys.42, 105 (1986) https: //doi.org/10.1007/BF01010843

    work page doi:10.1007/bf01010843 1986

  10. [11]

    Wang, X.-Q., Zeng, R.-L., Zhang, Z.-Y., Tian, C., Zhang, S., Hemmerich, A., Xu, Z.-F.: Observation of Brownian motion of a Bose-Einstein condensate. Phys. Rev. Lett.134, 223402 (2025) https://doi.org/10.1103/tp71-51gv

    work page doi:10.1103/tp71-51gv 2025

  11. [12]

    Troncoso, R.E., N´ u˜ nez, A.S.: Brownian motion of massive skyrmions in magnetic thin films. Ann. Phys.351, 850 (2014) https://doi.org/10.1016/j.aop.2014.10. 007

    work page doi:10.1016/j.aop.2014.10 2014

  12. [13]

    Nature Mater13, 241 (2014) https://doi.org/10.1038/nmat3862

    Mochisuzki, M., Yu, X.Z., Seki, S., Kanazawa, N., Koshibae, W., Zang, J., Mostovoy, M., Tokura, Y., Nagaosa, N.: Thermally driven ratchet motion of a skyrmion microcrystal and topological magnon Hall effect. Nature Mater13, 241 (2014) https://doi.org/10.1038/nmat3862

    work page doi:10.1038/nmat3862 2014

  13. [14]

    Petrovi´ c, A.P., Psaroudaki, C., Fischer, P., Garst, M., Panagopoulos, C.: Col- loquium: Quantum properties and functionalities of magnetic skyrmions. Rev. Mod. Phys.97, 031001 (2025) https://doi.org/10.1103/RevModPhys.97.031001

    work page doi:10.1103/revmodphys.97.031001 2025

  14. [15]

    Kosterlitz, J.M.: Nobel lecture: Topological defects and phase transitions. Rev. Mod. Phys.89, 040501 (2017) https://doi.org/10.1103/RevModPhys.89.040501

    work page doi:10.1103/revmodphys.89.040501 2017

  15. [16]

    Devoret, M.H., Martinis, J.M., Clarke, J.: Measurements of macroscopic quan- tum tunneling out of the zero-voltage state of a current-biased Josephson 15 junction. Phys. Rev. Lett.55, 1908–1911 (1985) https://doi.org/10.1103/ PhysRevLett.55.1908

    1908

  16. [18]

    Probabilities for quantum tunneling through a barrier with linear passive dissipation

    Caldeira, A.O., Leggett, A.J.: Comment on “Probabilities for quantum tunneling through a barrier with linear passive dissipation”. Phys. Rev. Lett.48, 1571 (1982) https://doi.org/10.1103/PhysRevLett.48.1571

    work page doi:10.1103/physrevlett.48.1571 1982

  17. [19]

    Caldeira, A.O., Leggett, A.J.: Quantum tunneling in a dissipative system. Ann. Phys. (N. Y.)149, 374–456 (1983) https://doi.org/10.1103/PhysRevLett.55. 1908

    work page doi:10.1103/physrevlett.55 1983

  18. [20]

    Physica121A, 587–616 (1983) https://doi.org/10.1016/0378-4371(83) 90013-4

    Caldeira, A.O., Leggett, A.J.: Path integral approach to quantum Brownian motion. Physica121A, 587–616 (1983) https://doi.org/10.1016/0378-4371(83) 90013-4

    work page doi:10.1016/0378-4371(83 1983

  19. [21]

    Chaos15, 026105 (2005) https://doi.org/10.1063/1.1853631

    H¨ anggi, P., Ingold, G.-L.: Fundamental aspects of quantum Brownian motion. Chaos15, 026105 (2005) https://doi.org/10.1063/1.1853631

    work page doi:10.1063/1.1853631 2005

  20. [22]

    The harmonic oscillator

    Senitzky, I.R.: Dissipation in quantum mechanics. The harmonic oscillator. Phys. Rev.119, 670–679 (1960) https://doi.org/10.1103/PhysRev.119.670

    work page doi:10.1103/physrev.119.670 1960

  21. [23]

    The harmonic oscillator

    Senitzky, I.R.: Dissipation in quantum mechanics. The harmonic oscillator. II. Phys. Rev.124, 642–648 (1961) https://doi.org/10.1103/PhysRev.124.642

    work page doi:10.1103/physrev.124.642 1961

  22. [24]

    Magalinski˘ ı, V.B.: Dynamical model in the theory of the Brownian motion. Sov. Phys. JETP9, 1381–1382 (1959)

    1959

  23. [25]

    Ford, G.W., Kac, M., Mazur, P.: Statistical mechanics of assemblies of coupled oscillators. J. Math. Phys.6, 504–515 (1965) https://doi.org/10.1063/1.1704304

    work page doi:10.1063/1.1704304 1965

  24. [26]

    Ullersma, P.: An exactly solvable model for Brownian motion. Physica32, 27–96 (1965) https://doi.org/10.1016/0031-8914(66)90102-9,https://doi.org/10.1016/ 0031-8914(66)90103-0,https://doi.org/10.1016/0031-8914(66)90104-2,https: //doi.org/10.1016/0031-8914(66)90105-4

    work page doi:10.1016/0031-8914(66)90102-9 1965

  25. [27]

    Zwanzig, R.: Nonlinear generalized Langevin equations. J. Stat. Phys.9, 215–220 (1973) https://doi.org/10.1007/BF01008729

    work page doi:10.1007/bf01008729 1973

  26. [28]

    Widom, A., Clark, T.D.: Probabilities for quantum tunneling through a barrier with linear passive dissipation. Phys. Rev. Lett.48, 63–65 (1982) https://doi. org/10.1103/PhysRevLett.48.63

    work page doi:10.1103/physrevlett.48.63 1982

  27. [29]

    Grabert, H., Schramm, P., Ingold, G.-L.: Quantum Brownian motion: The func- tional integral approach. Phys. Rep.168, 115 (1988) https://doi.org/10.1016/ 16 0370-1573(88)90023-3

    1988

  28. [30]

    Leggett, A.J.: Macroscopic quantum systems and the quantum theory of mea- surement. Suppl. Progr. Theor. Phys.69, 80–100 (1980) https://doi.org/10. 1143/PTP.69.80

    1980

  29. [31]

    Leggett, A.J.: Schr¨ odinger’s cat and her laboratory cousins. Contemp. Phys.25, 583–598 (1984) https://doi.org/10.1080/00107518408210731

    work page doi:10.1080/00107518408210731 1984

  30. [32]

    Caldeira, A.O., Leggett, A.J.: Influence of dissipation on quantum tunneling in macroscopic systems. Phys. Rev. Lett.46, 211–214 (1981) https://doi.org/10. 1103/PhysRevLett.46.211

    1981

  31. [33]

    Langer, J.S.: Theory of the condensation point. Ann. Phys. (N.Y.)41, 108–157 (1967) https://doi.org/10.1016/0003-4916(67)90200-X

    work page doi:10.1016/0003-4916(67)90200-x 1967

  32. [34]

    Callan Jr., C.G., Coleman, S.: Fate of the false vacuum. II. First quantum correc- tions. Phys. Rev. D16, 1762–1768 (1977) https://doi.org/10.1103/PhysRevD. 16.1762

    work page doi:10.1103/physrevd 1977

  33. [35]

    Feynman, R.P., Vernon Jr., F.L.: The theory of a general quantum system inter- acting with a linear dissipative system. Ann. Phys. (N.Y.)24, 118–173 (1963) https://doi.org/10.1016/0003-4916(63)90068-X

    work page doi:10.1016/0003-4916(63)90068-x 1963

  34. [36]

    Hakim, V., Ambegaokar, V.: Quantum theory of a free particle interacting with a linear dissipative environment. Phys. Rev. A32, 423 (1985) https://doi.org/ 10.1103/PhysRevA.32.423

    work page doi:10.1103/physreva.32.423 1985

  35. [37]

    Morais Smith, C., Caldeira, A.O.: Generalized Feynman-Vernon approach to quantum dissipative systems. Phys. Rev. A36, 3509 (1987) https://doi.org/10. 1103/PhysRevA.36.3509

    1987

  36. [38]

    Morais Smith, C., Caldeira, A.O.: Application of the generalized Feynman- Vernon approach to a simple system: The damped harmonic oscillator. Phys. Rev. A41, 3103–3115 (1990) https://doi.org/10.1103/PhysRevA.41.3103

    work page doi:10.1103/physreva.41.3103 1990

  37. [39]

    Oxford Univ

    Breuer, H.-P., Petruccione, F.: The Theory of Open Quantum Systems. Oxford Univ. Press, Oxford (2007)

    2007

  38. [40]

    World Scientific, Singapore (2021)

    Weiss, U.: Quantum Dissipative Systems, 5th edn. World Scientific, Singapore (2021)

    2021

  39. [41]

    Hedeg˚ ard, P., Caldeira, A.O.: Quantum dynamics of a particle in a fermionic environment. Phys. Scr.35, 609 (1987) https://doi.org/10.1088/0031-8949/35/ 5/001

    work page doi:10.1088/0031-8949/35/ 1987

  40. [42]

    Hedeg˚ ard, P., Caldeira, A.O.: Static properties of a particle coupled to a fermionic environment. Phys. Rev. B36, 2770 (1987) https://doi.org/10.1103/ 17 PhysRevB.36.2770

    1987

  41. [43]

    Barone, P.M.V.B., Caldeira, A.O.: Quantum mechanics of radiation damping. Phys. Rev. A43, 57 (1991) https://doi.org/10.1103/PhysRevA.43.57

    work page doi:10.1103/physreva.43.57 1991

  42. [44]

    Castro Neto, A.H., Caldeira, A.O.: New model of dissipation in quan- tum mechanics. Phys. Rev. Lett.67, 1960 (1991) https://doi.org/10.1103/ PhysRevLett.67.1960

    1960

  43. [45]

    Aslangul, C., Pottier, N., Saint-James, D.: Time behavior of the correlation functions in a simple dissipative quantum model. J. Stat. Phys.40, 167–189 (1985) https://doi.org/10.1007/BF01010531

    work page doi:10.1007/bf01010531 1985

  44. [46]

    Schramm, P., Grabert, H.: Low-temperature and long-time anomalies of a damped quantum particle. J. Stat. Phys.49, 767–810 (1987) https://doi.org/ 10.1007/BF01009356

    work page doi:10.1007/bf01009356 1987

  45. [47]

    Castro Neto, A.H., Caldeira, A.O.: Alternative approach to the dynamics of polarons in one dimension. Phys. Rev. B46, 8858 (1992) https://doi.org/10. 1103/PhysRevB.46.8858

    1992

  46. [48]

    Castro Neto, A.H., Caldeira, A.O.: Transport properties of solitons. Phys. Rev. E48, 4037 (1993) https://doi.org/10.1103/PhysRevE.48.4037

    work page doi:10.1103/physreve.48.4037 1993

  47. [49]

    North-Holland, Amsterdam (1982)

    Rajaraman, R.: Solitons and Instantons: an Introduction to Solitons and Instantons in Quantum Field Theory. North-Holland, Amsterdam (1982)

    1982

  48. [50]

    Castro Neto, A.H., Caldeira, A.O.: Mobility and diffusion of a particle cou- pled to Luttinger liquid. Phys. Rev. B50, 4863 (1994) https://doi.org/10.1103/ PhysRevB.50.4863

    1994

  49. [51]

    Caldeira, A.O., Castro Neto, A.H.: Motion of heavy particles coupled to fermionic and bosonic environments in one dimension. Phys. Rev. B52, 4198 (1995) https://doi.org/10.1103/PhysRevB.52.4198

    work page doi:10.1103/physrevb.52.4198 1995

  50. [52]

    Morais Smith, C., Caldeira, A.O., Blatter, G.: Creep of vortices from columnar defects. Phys. Rev. B54, 784 (1996) https://doi.org/10.1103/PhysRevB.54.784

    work page doi:10.1103/physrevb.54.784 1996

  51. [53]

    Ferrer, A.V., Caldeira, A.O.: Skyrmion dynamics in quantum Hall ferromagnets. Phys. Rev. B61, 2755 (2000) https://doi.org/10.1103/PhysRevB.61.2755

    work page doi:10.1103/physrevb.61.2755 2000

  52. [54]

    Desposito, M.A., Ferrer, A.V., Caldeira, A.O., Castro Neto, A.H.: Mobility of Bloch walls via the collective coordinate method. Phys. Rev. B62, 919 (2000) https://doi.org/10.1103/PhysRevB.62.919

    work page doi:10.1103/physrevb.62.919 2000

  53. [55]

    Ferrer, A.V., Caldeira, A.O.: Dynamical properties of a gas of solitons in one- dimensional quantum antiferromagnets. Phys. Rev. B64, 104425 (2001) https: //doi.org/10.1103/PhysRevB.64.104425 18

    work page doi:10.1103/physrevb.64.104425 2001

  54. [56]

    Juricic, V., Benfatto, L., Caldeira, A.O., Morais Smith, C.: Dynamics of topo- logical defects in a spiral: A scenario for the spin-glass phase of cuprates. Phys. Rev. Lett.92, 137202 (2004) https://doi.org/10.1103/PhysRevLett.92.137202

    work page doi:10.1103/physrevlett.92.137202 2004

  55. [57]

    Juricic, V., Benfatto, L., Caldeira, A.O., Morais Smith, C.: Dissipative dynamics of topological defects in frustrated Heisenberg spin systems. Phys. Rev. B71, 064421 (2005)

    2005

  56. [58]

    Caldeira, A.O., Castro Neto, A.H., Carvalho, T.O.: Dissipative quantum systems modeled by two-level-reservoir coupling. Phys. Rev. B48, 13974 (1993) https: //doi.org/10.1103/PhysRevB.48.13974

    work page doi:10.1103/physrevb.48.13974 1993

  57. [59]

    Duarte, O.S., Caldeira, A.O.: Effective coupling between two Brownian particles. Phys. Rev. Lett.97, 250601 (2006) https://doi.org/10.1103/PhysRevLett.97. 250601

    work page doi:10.1103/physrevlett.97 2006

  58. [60]

    Physica A89, 373 (1977) https://doi.org/10.1016/0378-4371(77)90111-X

    Felderhof, B.U.: Hydrodynamic interaction between two spheres. Physica A89, 373 (1977) https://doi.org/10.1016/0378-4371(77)90111-X

    work page doi:10.1016/0378-4371(77)90111-x 1977

  59. [61]

    Felderhof, B.U.: Diffusion of interacting Brownian particles. J. Phys. A11, 929 (1978) https://doi.org/10.1088/0305-4470/11/5/022

    work page doi:10.1088/0305-4470/11/5/022 1978

  60. [62]

    Duarte, O.S., Caldeira, A.O.: Effective quantum dynamics of two Brownian par- ticles. Phys. Rev. A80, 032110 (2009) https://doi.org/10.1103/PhysRevA.80. 032110

    work page doi:10.1103/physreva.80 2009

  61. [63]

    Valente, D.M., Caldeira, A.O.: Thermal equilibrium of two quantum Brownian particles. Phys. Rev. A81, 012117 (2010) https://doi.org/10.1103/PhysRevA. 81.012117

    work page doi:10.1103/physreva 2010

  62. [64]

    Leggett, A.J., Chakravarty, S., Dorsey, A.T., Fisher, M.P.A., Garg, A., Zwerger, W.: Dynamics of the dissipative two-state system. Rev. Mod. Phys.59, 1 (1987) https://doi.org/10.1103/RevModPhys.59.1

    work page doi:10.1103/revmodphys.59.1 1987

  63. [65]

    Brito, F., Caldeira, A.O.: Dissipative dynamics of a two-level system resonantly coupled to a harmonic mode. New J. Phys.10(11), 115014 (2008) https://doi. org/10.1088/1367-2630/10/11/115014

    work page doi:10.1088/1367-2630/10/11/115014 2008

  64. [66]

    Science299, 1869 (2003) https://doi

    Chiorescu, I., Nakamura, Y., Harmans, C.J.P.M., Mooij, J.E.: Coherent quantum dynamics of a superconducting flux qubit. Science299, 1869 (2003) https://doi. org/10.1126/science.1081045

    work page doi:10.1126/science.1081045 2003

  65. [67]

    Quantum Computing in the NISQ era and beyond.Quantum, 2:79, August 2018

    Preskill, J.: Quantum Computing in the NISQ era and beyond. Quantum2, 79 (2018) https://doi.org/10.22331/q-2018-08-06-79

    work page Pith review doi:10.22331/q-2018-08-06-79 2018

  66. [68]

    Zurek, W.H.: Decoherence, einselection, and the quantum origins of the classical. Rev. Mod. Phys.75, 715–775 (2003) https://doi.org/10.1103/RevModPhys.75. 715 19

    work page doi:10.1103/revmodphys.75 2003

  67. [69]

    Schlosshauer, M.: Decoherence, the measurement problem, and interpretations of quantum mechanics. Rev. Mod. Phys.76, 1267–1305 (2005) https://doi.org/ 10.1103/RevModPhys.76.1267

    work page Pith review doi:10.1103/revmodphys.76.1267 2005

  68. [70]

    Gamble, J.K., Lindner, J.F.: Demystifying decoherence and the master equation of quantum Brownian motion. Am. J. Phys.77(3), 244–252 (2009) https://doi. org/10.1119/1.3043847

    work page doi:10.1119/1.3043847 2009

  69. [71]

    Schlosshauer, M.: Quantum decoherence. Phys. Rep.831, 1–57 (2019) https: //doi.org/10.1016/j.physrep.2019.10.001

    work page doi:10.1016/j.physrep.2019.10.001 2019

  70. [72]

    Physica A199(3), 517–526 (1993) https://doi.org/10.1016/0378-4371(93)90065-C

    Di´ osi, L.: Calderia-Leggett master equation and medium temperatures. Physica A199(3), 517–526 (1993) https://doi.org/10.1016/0378-4371(93)90065-C

    work page doi:10.1016/0378-4371(93)90065-c 1993

  71. [73]

    Nicacio, F., Koide, T.: Complete positivity and thermal relaxation in quadratic quantum master equations. Phys. Rev. E110, 054116 (2024) https://doi.org/ 10.1103/PhysRevE.110.054116

    work page doi:10.1103/physreve.110.054116 2024

  72. [74]

    Ferialdi, L.: Dissipation in the Caldeira-Leggett model. Phys. Rev. A95, 052109 (2017) https://doi.org/10.1103/PhysRevA.95.052109

    work page doi:10.1103/physreva.95.052109 2017

  73. [75]

    Hu, B.L., Paz, J.P., Zhang, Y.: Quantum Brownian motion in a general environ- ment: Exact master equation with nonlocal dissipation and colored noise. Phys. Rev. D45, 2843–2861 (1992) https://doi.org/10.1103/PhysRevD.45.2843

    work page doi:10.1103/physrevd.45.2843 1992

  74. [76]

    Halliwell, J.J., Yu, T.: Alternative derivation of the Hu-Paz-Zhang master equation of quantum Brownian motion. Phys. Rev. D53, 2012–2019 (1996) https://doi.org/10.1103/PhysRevD.53.2012

    work page doi:10.1103/physrevd.53.2012 2012

  75. [77]

    Castro Neto, A.H., Caldeira, A.O.: Quantum dynamics of an electromagnetic mode in a cavity. Phys. Rev. A42, 6884–6893 (1990) https://doi.org/10.1103/ PhysRevA.42.6884

    1990

  76. [78]

    Moussa, M.H.Y., Mizrahi, S.S., Caldeira, A.O.: Quantum coherence in a dissipative-driven system and the optical Stern-Gerlach experiment. Phys. Lett. A221(3), 145–152 (1996) https://doi.org/10.1016/0375-9601(96)00577-4

    work page doi:10.1016/0375-9601(96)00577-4 1996

  77. [79]

    Oliveira, T.R., Caldeira, A.O.: Dissipative Stern-Gerlach recombination exper- iment. Phys. Rev. A73, 042502 (2006) https://doi.org/10.1103/PhysRevA.73. 042502

    work page doi:10.1103/physreva.73 2006

  78. [80]

    Westfahl, H., Caldeira, A.O., Medeiros-Ribeiro, G., Cerro, M.: Dissipative dynamics of spins in quantum dots. Phys. Rev. B70, 195320 (2004) https: //doi.org/10.1103/PhysRevB.70.195320

    work page doi:10.1103/physrevb.70.195320 2004

  79. [81]

    D´ avila Romero, L., Pablo Paz, J.: Decoherence and initial correlations in quan- tum Brownian motion. Phys. Rev. A55, 4070–4083 (1997) https://doi.org/10. 20 1103/PhysRevA.55.4070

    1997

  80. [82]

    Halliwell, J., Zoupas, A.: Post-decoherence density matrix propagator for quan- tum Brownian motion. Phys. Rev. D55, 4697–4704 (1997) https://doi.org/10. 1103/PhysRevD.55.4697

    1997

Showing first 80 references.