Quasi-classical Limit of a Spin Coupled to a Reservoir

Michele Correggi; Marco Falconi; Michele Fantechi; Marco Merkli

doi:10.22331/q-2024-12-11-1561

Quasi-classical Limit of a Spin Coupled to a Reservoir

Michele Correggi¹, Marco Falconi¹, Michele Fantechi¹, and Marco Merkli²

¹Dipartimento di Matematica, Politecnico di Milano, P.zza Leonardo da Vinci, 32, 20133 Milano, Italy
²Department of Mathematics and Statistics, Memorial University of Newfoundland, NL A1C 5S7, St. John's, Canada

Published:	2024-12-11, volume 8, page 1561
Eprint:	arXiv:2408.02515v2
Doi:	https://doi.org/10.22331/q-2024-12-11-1561
Citation:	Quantum 8, 1561 (2024).

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Abstract

A spin (qubit) is in contact with a bosonic reservoir. The state of the reservoir contains a parameter $\varepsilon$ interpolating between quantum and classical reservoir features. We derive the explicit expression for the time-dependent reduced spin density matrix, valid for all values of $\varepsilon$ and for energy conserving interactions. We study decoherence and markovianity properties. Our main finding is that the spin decoherence is enhanced (full decoherence) when the spin is coupled to quantum reservoir states while it is dampened (partial decoherence) when coupled to classical reservoir states. The markovianity properties depend in a subtle way on the classicality parameter $\varepsilon$ and on the finer details of the spin-reservoir interaction. We further examine scattering and periodicity properties for energy exchange interactions.

Popular summary

We study the difference in the dynamics of a qubit when it is in contact with a `classical world' or a `quantum world'. By world we mean the surroundings the qubit is placed in. The degree of classicality of the world is parametrized by a number between 0 (classical) and 1 (fully quantum). Part of our work examines the environment without being coupled to the qubit. We examine explicit states of such environments, such as Bose-Einstein Condensate states, thermal states, coherent states and we analyze them for all values of the classicality parameter. When a qubit is placed into such an environment, it behaves differently depending on the degree of classicality of the environment. In general, we show that a higher degree of quantumness in the environment state leads to the qubit exhibiting stronger quantum features as well. Our methods involve both a mathematical treatment as well as numerical simulations.

► BibTeX data

@article{Correggi2024quasiclassicallimit,
  doi = {10.22331/q-2024-12-11-1561},
  url = {https://doi.org/10.22331/q-2024-12-11-1561},
  title = {Quasi-classical {L}imit of a {S}pin {C}oupled to a {R}eservoir},
  author = {Correggi, Michele and Falconi, Marco and Fantechi, Michele and Merkli, Marco},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {8},
  pages = {1561},
  month = dec,
  year = {2024}
}

► References

[1] M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions with Formulas, Graphs and Matematical Tables, volume 55. US Government printing office, 1964.

[2] Z. Ammari and F. Nier. Mean field limit for bosons and infinite dimensional phase-space analysis. Annales Henri Poincaré, 9 (8): 1503–1574, 2008. 10.1007/s00023-008-0393-5.
https://doi.org/10.1007/s00023-008-0393-5

[3] Z. Ammari and F. Nier. Mean field limit for bosons and propagation of Wigner measures. Journal of Mathematical Physics, 50 (4): 042107, 2009. 10.1063/1.3115046.
https://doi.org/10.1063/1.3115046

[4] Z. Ammari and F. Nier. Mean field propagation of Wigner measures and BBGKY hierarchies for general bosonic states. Journal de Mathématiques Pures et Appliquées, 95 (6): 585–626, 2011. 10.1016/j.matpur.2010.12.004.
https://doi.org/10.1016/j.matpur.2010.12.004

[5] Z. Ammari and F. Nier. Mean field propagation of infinite-dimensional Wigner measures with a singular two-body interaction potential. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, 14 (1): 155–220, 2015. 10.2422/2036-2145.201112_004.
https://doi.org/10.2422/2036-2145.201112_004

[6] L. Amour and J. Nourrigat. Semiclassical expansion of the ground state for a model of interacting spins in QED, 2016. arXiv:1603.08439.
https://doi.org/10.48550/arXiv.1603.08439
arXiv:1603.08439

[7] L. Amour, R. Lascar, and J. Nourrigat. Beals Characterization of Pseudodifferential Operators in Wiener Spaces. Applied Mathematics Research eXpress, 2017 (1): 242–270, 2016. 10.1093/amrx/abw001.
https://doi.org/10.1093/amrx/abw001

[8] L. Amour, L. Jager, and J. Nourrigat. Infinite dimensional semiclassical analysis and applications to a model in nuclear magnetic resonance. Journal of Mathematical Physics, 60: 071503, 2019a. 10.1063/1.5094396.
https://doi.org/10.1063/1.5094396

[9] L. Amour, R. Lascar, and J. Nourrigat. Weyl calculus in Wiener spaces and in QED. Journal of Pseudo-Differential Operators and Applications, 10: 1–47, 2019b. 10.1007/s11868-018-0269-5.
https://doi.org/10.1007/s11868-018-0269-5

[10] L. Amour, R. Lascar, and J. Nourrigat. Weyl calculus in QED I. The unitary group. Journal of Mathematical Physics, 58, 2019c. 10.1063/1.4973742.
https://doi.org/10.1063/1.4973742

[11] H. Araki and E. Woods. Representations of the canonical commutation relations describing a non-relativistic infinite free bose gas. Journal of Mathematical Physics, 4 (5): 637–662, 1963. 10.1063/1.1704002.
https://doi.org/10.1063/1.1704002

[12] V. Beaud, W. Dybalski, and G. M. Graf. Infraparticle States in the Massless Nelson Model: Revisited. Annales Henri Poincaré, 25: 173–212, 2024. 10.1007/s00023-022-01261-2.
https://doi.org/10.1007/s00023-022-01261-2

[13] V. I. Bogachev. Gaussian Measures. Mathematical Surveys and Monographs, 6, 1998. 10.1090/surv/062.
https://doi.org/10.1090/surv/062

[14] S. Breteaux, M. Correggi, M. Falconi, and J. Faupin. Quantum Point Charges Interacting with Quasi-classical Electromagnetic Fields, 2024. arXiv:2407.18600.
https://doi.org/10.48550/arXiv.2407.18600
arXiv:2407.18600

[15] H. P. Breuer and F. Petruccione. The Theory of Open Quantum Systems. Oxford University Press, Oxford, 2002. URL https://doi.org/10.1093/acprof:oso/9780199213900.001.0001.
https://doi.org/10.1093/acprof:oso/9780199213900.001.0001

[16] R. Carlone, M. Correggi, M. Falconi, and M. Olivieri. Emergence of Time-Dependent Point Interactions in Polaron Models. SIAM Journal on Mathematical Analysis, 53 (4): 4657–4691, 2021. 10.1137/20M1381344.
https://doi.org/10.1137/20M1381344

[17] M. Correggi and M. Falconi. Effective Potentials Generated by Field Interaction in the Quasi–Classical Limit. Annales Henri Poincaré, 19 (1): 189–235, 2018. 10.1007/s00023-017-0612-z.
https://doi.org/10.1007/s00023-017-0612-z

[18] M. Correggi, M. Falconi, and M. Olivieri. Magnetic Schrödinger Operators as the Quasi–Classical Limit of Pauli–Fierz–type Models. Journal of Spectral Theory, 9 (4): 1287–1325, 2019. 10.4171/JST/277.
https://doi.org/10.4171/JST/277

[19] M. Correggi, M. Falconi, and M. Olivieri. Quasi-classical dynamics. Journal of the European Mathematical Society, 25 (2): 731–783, 2021. 10.4171/JEMS/1197.
https://doi.org/10.4171/JEMS/1197

[20] M. Correggi, M. Falconi, and M. Olivieri. Ground State Properties in the Quasiclassical Regime. Analysis & PDE, 16 (8): 1745–1798, 2023a. 10.2140/apde.2023.16.1745.
https://doi.org/10.2140/apde.2023.16.1745

[21] Michele Correggi, Marco Falconi, and Marco Merkli. Quasi-Classical Spin Boson Models. In Michele Correggi and Marco Falconi, editors, Quantum Mathematics I, pages 107–127. Springer Nature Singapore, 2023b. 10.1007/978-981-99-5894-8_3.
https://doi.org/10.1007/978-981-99-5894-8_3

[22] J. D. Dollard. Quantum-Mechanical scattering theory for short-range and Coulomb interactions. Rocky Mountain Journal of Mathematics, 1 (1): 5–88, 1971. 10.1216/RMJ-1971-1-1-5.
https://doi.org/10.1216/RMJ-1971-1-1-5

[23] M. Falconi. Cylindrical Wigner Measures. Documenta Mathematica, 23: 1677–1756, 2018. 10.4171/DM/658.
https://doi.org/10.4171/DM/658

[24] H. Fröhlich. Theory of electrical breakdown in ionic crystals. Proc. Roy. Soc. London Ser. A Math. Phys. Engrg. Sci., 160: 230–241, 1937. 10.1098/rspa.1937.0106.
https://doi.org/10.1098/rspa.1937.0106

[25] J. Ginibre, F. Nironi, and G. Velo. Partially classical limit of the Nelson model. Annales Henri Poincaré, 7 (1): 21–43, 2006. 10.1007/s00023-005-0240-x.
https://doi.org/10.1007/s00023-005-0240-x

[26] M. Grifoni and P. Hänggi. Driven quantum tunneling. Physics Reports, 304 (5): 229–354, 1998. 10.1016/S0370-1573(98)00022-2.
https://doi.org/10.1016/S0370-1573(98)00022-2

[27] D. Hasler, B. Hinrichs, and O. Siebert. Non-Fock ground states in the translation-invariant Nelson model revisited non-perturbatively. Journal of Functional Analysis, 286 (7), 2024. 10.1016/j.jfa.2024.110319.
https://doi.org/10.1016/j.jfa.2024.110319

[28] T. Hida. Brownian Motion. Applications of Mathematics 11, Springer Verlag, 1980. URL https://doi.org/10.1007/978-1-4612-6030-1.
https://doi.org/10.1007/978-1-4612-6030-1

[29] J. S. Howland. Stationary Scattering Theory for Time-dependent Hamiltonians. Mathematische Annalen, 207 (4): 315–335, 1974. 10.1007/BF01351346.
https://doi.org/10.1007/BF01351346

[30] M. Hübner and H. Spohn. Spectral properties of the spin-boson Hamiltonian. Ann. Inst. Henri Poincaré, 62 (3): 289–323, 1995. URL http://www.numdam.org/item/AIHPA_1995__62_3_289_0/.
http://www.numdam.org/item/AIHPA_1995__62_3_289_0/

[31] V. Jaksic and C.-A. Pillet. On a model for quantum friction. III. Ergodic properties of the spin-boson system. Communications in Mathematical Physics, 178 (3): 627–651, 1996. 10.1007/BF02108818.
https://doi.org/10.1007/BF02108818

[32] T. Kato and S. T. Kuroda. The abstract theory of scattering. Rocky Mountain Journal of Mathematics, 1 (1): 127–171, 1971. 10.1216/RMJ-1971-1-1-127.
https://doi.org/10.1216/RMJ-1971-1-1-127

[33] M. Könenberg, M. Merkli, and H. Song. Ergodicity of the spin-boson model for arbitrary coupling strength. Communications in Mathematical Physics, 336 (1): 261–285, 2014. 10.1007/s00220-014-2242-3.
https://doi.org/10.1007/s00220-014-2242-3

[34] E.-M. Laine, J. Piilo, and H.-P. Breuer. Measure for the non-Markovianity of quantum processes. Physical Review A, 81 (6): 062115, 2010. 10.1103/PhysRevA.81.062115.
https://doi.org/10.1103/PhysRevA.81.062115

[35] A. J. Leggett, S. Chakravarty, A. T. Dorsey, M. P. A. Fisher, A. Garg, and W. Zwerger. Dynamics of the dissipative two-state system. Reviews of Modern Physics, 59 (1): 1–85, 1987. 10.1103/RevModPhys.59.1.
https://doi.org/10.1103/RevModPhys.59.1

[36] D. Lonigro. Generalized spin-boson models with non-normalizable form factors. Journal of Mathematical Physics, 63: 072105, 2022. 10.1063/5.0085576.
https://doi.org/10.1063/5.0085576

[37] D. Lonigro. Self-Adjointness of a Class of Multi-Spin-Boson Models with Ultraviolet Divergences. Mathematical Physics, Analysis and Geometry, 26 (2): 15, 2023. 10.1007/s11040-023-09457-6.
https://doi.org/10.1007/s11040-023-09457-6

[38] M. Merkli. The Ideal Quantum Gas. In Lecture Notes in Mathematics, pages 183–233. Springer, 2006. URL https://doi.org/10.1007/3-540-33922-1_5.
https://doi.org/10.1007/3-540-33922-1_5

[39] M. Merkli. Resonance Dynamics and Decoherence. Analysis and Mathematical Physics, Trends in Mathematics, pages 405–422, 2009. 10.1007/978-3-7643-9906-1_20.
https://doi.org/10.1007/978-3-7643-9906-1_20

[40] M. Merkli. Dynamics of Open Quantum Systems I, Oscillation and Decay. Quantum, 6: 615, 2022a. 10.22331/q-2022-01-03-615.
https://doi.org/10.22331/q-2022-01-03-615

[41] M. Merkli. Dynamics of Open Quantum Systems II, Markovian Approximation. Quantum, 6: 616, 2022b. 10.22331/q-2022-01-03-616.
https://doi.org/10.22331/q-2022-01-03-616

[42] M. Merkli. Correlation decay and markovianity in open systems. Annales Henri Poincaré, 24: 751–782, 2023. 10.1007/s00023-022-01226-5.
https://doi.org/10.1007/s00023-022-01226-5

[43] M. Merkli, G. P. Berman, R. T. Sayre, S. Gnanakaran, M. Könenberg, A. I. Nesterov, and H. Song. Dynamics of a Chlorophyll Dimer in Collective and Local Thermal Environments. Journal of Mathematical Chemistry, 54 (4): 866–917, 2016. 10.1007/s10910-016-0593-z.
https://doi.org/10.1007/s10910-016-0593-z

[44] P. M. Morse and H. Feshbach. Methods of Theoretical Physics. McGraw Hill, 1953.

[45] M. Palma, K.-A. Suominen, and A. Ekert. Quantum Computers and Dissipation. Proc. R. Soc. Lond. A, 452: 567–584, 1996. 10.1098/rspa.1996.0029.
https://doi.org/10.1098/rspa.1996.0029

[46] D. Petz. An Invitation to the Algebra of Canonical Commutation Relations. Leuven Notes in Mathematical and Theoretical Physics, Vol. 2. Leuven Univ. Pr., Leuven, 1989.

[47] A. Rivas and S. F. Huelga. Open Quantum Systems. An Introduction, Springer Verlag, 2011. URL https://doi.org/10.1007/978-3-642-23354-8.
https://doi.org/10.1007/978-3-642-23354-8

[48] A. V. Skorohod. Integration in Hilbert space. Ergebnisse der Mathematik und ihrer Grenzgebiete, 1974. URL https://doi.org/10.1007/978-3-642-65632-3.
https://doi.org/10.1007/978-3-642-65632-3

[49] T. Takaesu. On generalized spin-boson models with singular perturbations. Hokkaido Mathematical Journal, 39: 317–349, 2010. 10.14492/hokmj/1288357972.
https://doi.org/10.14492/hokmj/1288357972

[50] A. Trushechkin. Long-Term Behaviour in an Exactly Solvable Model of Pure Decoherence and the Problem of Markovian Embedding. Mathematics, 12 (1), 2024. 10.3390/math12010001.
https://doi.org/10.3390/math12010001

[51] A. S. Trushechkin, M. Merkli, J. D. Cresser, and J. Anders. Open quantum system dynamics and the mean force gibbs state. AVS Quantum Science, 4 (1): 012301, 2022. 10.1116/5.0073853.
https://doi.org/10.1116/5.0073853

[52] N. N. Vakhania, V. I. Tarieladze, and S. A. Chobanyan. Probability Distibutions on Banach Spaces. Mathematics and Its Applications (Soviet Series), D. Reidel Publishing Company, Kluwer Academic Publishers, 1987. URL https://doi.org/10.1007/978-94-009-3873-1.
https://doi.org/10.1007/978-94-009-3873-1

Cited by

[1] Dong Hao Ou Yang, "Approach to equilibrium in open quantum statistical mechanics under detailed balance condition", Journal of Mathematical Physics 66 10, 103303 (2025).

[2] Michele Fantechi and Marco Merkli, "Bosonization of Noise Effects in Nonlocal Quantum Dynamics", Physical Review Letters 136 6, 060402 (2026).

[3] Michele Fantechi and Marco Merkli, "Quantum systems coupled to environments via mean field interactions", Annals of Physics 476, 169981 (2025).

[4] Marco Falconi, Benjamin Hinrichs, and Javier Valentín Martín, "Non-Trivial Renormalization of Spin-Boson Models with Supercritical Form Factors", arXiv:2508.00805, (2025).

[5] Zied Ammari, Michele Correggi, Marco Falconi, and Raphaël Gautier, "Semiclassical limit of entropies and free energies", arXiv:2510.15777, (2025).

[6] Marco Falconi, Benjamin Hinrichs, and Javier Valentín Martín, "Wave Function Renormalization for Particle-Field Interactions", arXiv:2603.07045, (2026).

The above citations are from Crossref's cited-by service (last updated successfully 2026-08-08 19:09:12) and SAO/NASA ADS (last updated successfully 2026-08-08 19:09:13). The list may be incomplete as not all publishers provide suitable and complete citation data.

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