Genuine pure decoherence: microscopic mechanisms, infrared singularities, and bath-structure dependence

In the strict pure-decoherence (energy-preserving) regime, the standard linear-coupling oscillator-bath models fail on both sides of their parameter range, as shown by Alicki (Open Syst. Inf. Dyn. 11, 53 (2004)): regular (superohmic) form factors produce only dressing plus coherent traveling waves, a 'false decoherence' with bounded suppression, while singular (ohmic/sub-ohmic) form factors reach finite rates only through infrared-divergent couplings whose limiting ground states are disjoint representations of the observable algebra, yielding a 'static' superselection whose physical status depends on the choice of algebra and limiting procedure. He further argued from level repulsion that chaotic baths can be less efficient pure decoherers than regular ones, and advocated elastic scattering as the proper microscopic mechanism. The open problem is the resulting program: construct mathematically controlled microscopic models of genuine pure decoherence with quantitative rates for macroscopic systems; determine how infrared-pathology resolutions depend on the observable algebra and limiting procedures and which are physically acceptable; establish under what conditions chaotic environments induce effective decoherence; and provide diagnostics separating true pure decoherence from dressing-induced false decoherence.

Background

Decoherence is the dynamical suppression of off-diagonal elements of the density matrix of a quantum system caused by entanglement with its environment; it is the standard mechanism invoked to explain why macroscopic superpositions ('Schrödinger cat' states) are not observed. Pure decoherence (dephasing) is the limiting regime in which this suppression occurs while energy exchange between system and environment remains negligible: either the interaction commutes with the system Hamiltonian, or the relevant level splitting $\epsilon$ is small enough that the decoherence time $\tau_{dec}$ satisfies $\hbar/\epsilon\gg\tau_{dec}$.

The workhorse microscopic models -- the Caldeira-Leggett model of quantum Brownian motion and the spin-boson model, in which the system is coupled linearly to a bath of harmonic oscillators -- are exactly solvable in this limit. For a two-level system coupled through $\sigma_3$ to a bosonic field with form factor $g(\omega)$, where the low-frequency scaling $|g(\omega)|^2\sim\omega^{\kappa-1}$ defines superohmic ($\kappa>0$), ohmic ($\kappa=0$), and sub-ohmic ($\kappa<0$) baths, the Hamiltonian in the limit $\epsilon\to 0$ splits into two commuting van Hove Hamiltonians $H_{\pm g}$: exactly solvable renormalization toy models built from Weyl displacement operators $W(f)=\exp\{a(f)-a^\dagger(f)\}$ on bosonic Fock space. The off-diagonal element of the reduced density matrix acquires the suppression factor $\exp(-\gamma_t)$ with $\gamma_t=2\|g-g_t\|^2$, $g_t(\omega)=e^{-i\omega t}g(\omega)$, and the decoherence rate is customarily identified with the zero-frequency value of the bath spectral function.

Alicki's analysis showed that both parameter regimes of this model class fail as models of genuine pure decoherence:

• Regular form factors ($\|g\|<\infty$; superohmic): the exact solution shows the suppression saturates at a finite value. The process describes the formation of dressed states -- eigenstates consisting of the bare system plus a coherent cloud of virtual bosons, $W(\pm g)\Omega$ -- accompanied by emission of coherent traveling waves, after which the dressed system is decoupled from the bath. Alicki called this 'false decoherence': the apparent loss of coherence is an artifact of describing the dressed, still-coherent system in the bare basis (a related observation is credited by him to Unruh).

• Singular form factors ($\|g\|=\infty$ with finite interaction energy; ohmic at finite temperature, sub-ohmic at zero temperature): the formal asymptotic rate is finite, but the infrared divergence destroys the model's physical interpretation. No ground states exist in Fock space; the limiting ground-state functionals are disjoint -- they define inequivalent representations of the observable algebra (the van Hove phenomenon) -- so their superpositions are operationally indistinguishable from mixtures, an induced superselection rule Alicki called 'static decoherence'. Following Spohn's rigorous study of spin-boson ground states, Alicki noted in a footnote that in the singular case the very definition of the ground states is not unique and depends on the limiting procedure and on the choice of algebra of physical observables; for the ohmic spin-boson model with finite tunneling, the thermal-limit ground states undergo spontaneous symmetry breaking at a critical coupling $\alpha_c\approx 1$ into two disjoint ground states. Whether such environment-induced superselection is physical or an unphysical artifact is a long-standing question of the einselection program -- Zurek's term for the environment-induced selection of preferred states and superselection rules (Zurek 1982; Olkiewicz 1999).

Alicki further argued, contra a conjecture of Zurek's, that chaotic environments can be less efficient pure decoherers than regular ones: energy-conserving dephasing requires bath spectral weight at zero frequency, i.e. degenerate or nearly degenerate level pairs, while quantum-chaotic spectra exhibit level repulsion with Wigner-Dyson nearest-neighbor spacing $p(s)\sim s^\beta$. His mean-field model with a nearest-neighbor-spacing approximation yields zero pure-decoherence rate for chaotic baths and a finite one for Poisson (regular) spectra. As the constructive alternative, he advocated elastic scattering (in the tradition of Joos and Zeh's treatment of environmental scattering) as the simplest physically honest source of pure decoherence, since it changes the environment's state irreversibly while conserving energy.

Problem Statement

Establish, with mathematical control and physical acceptability, which microscopic energy-preserving mechanisms produce genuine pure decoherence -- irreversible and persistent suppression of coherences in macroscopic systems -- and characterize when and where it occurs. The linked components of this objective, faithful to the program implied by the source paper:

1. Construct and analyze explicit, mathematically controlled microscopic models of pure decoherence based on elastic scattering (or other energy-preserving processes), yielding quantitative decoherence rates applicable to realistic macroscopic systems, including dense or structured environments beyond dilute-gas and Markov approximations.

2. Clarify how the resolution of infrared pathologies -- existence and uniqueness of ground states, disjointness of limiting states, and the resulting induced superselection -- depends on the choice of observable algebra and on the limiting procedure for singular couplings, and determine physically acceptable criteria for model selection among inequivalent representations.

3. Extend the chaotic-versus-regular bath analysis beyond the mean-field nearest-neighbor-spacing approximation: determine under what conditions chaotic environments nevertheless induce effective pure decoherence (e.g. via degenerate subspaces, many-body effects, or nonperturbative couplings), reconciling the conflicting model-dependent results in the literature.

4. Provide diagnostics that distinguish true energy-preserving pure decoherence from dressing-induced false decoherence in systems where both pictures predict reduced coherences in the bare basis; the source's own criterion is theoretical -- the behavior of the bath spectral function at zero frequency, whose vanishing indicates absence of true decoherence, together with the dressed-state analysis of the exact solution.

Components 1-3 characterize the mechanisms and conditions of genuine pure decoherence; component 4 supplies its diagnostic test. Together they answer when environment-induced, energy-preserving suppression of quantum coherences is genuine and irreversible rather than an artifact of model choice, representation, or basis.

All four components are mandatory for a complete resolution of the objective, and each is independently attackable: a rigorous resolution of any single component is a self-standing contribution to the program, but the full objective is met only when all four are addressed.

References

1. R. Alicki, Pure decoherence in quantum systems, Open Systems & Information Dynamics 11, 53 (2004). DOI: 10.1023/b:opsy.0000024755.58888.ac, arXiv:0801.0936. URL: https://arxiv.org/abs/0801.0936
2. H. Spohn, Ground state(s) of the spin-boson hamiltonian, Communications in Mathematical Physics 123, 277 (1989). DOI: 10.1007/BF01238859. URL: https://doi.org/10.1007/BF01238859
3. A. Arai, M. Hirokawa, On the existence and uniqueness of ground states of a generalized spin-boson model, Journal of Functional Analysis 151, 455 (1997). DOI: 10.1006/jfan.1997.3140. URL: https://doi.org/10.1006/jfan.1997.3140
4. D. Hasler, I. Herbst, Ground states in the spin boson model, Annales Henri Poincaré 12, 621 (2011). DOI: 10.1007/s00023-011-0091-6. URL: https://doi.org/10.1007/s00023-011-0091-6
5. V. Bach, M. Ballesteros, M. Könenberg, L. Menrath, Existence of ground state eigenvalues for the spin-boson model with critical infrared divergence and multiscale analysis, Journal of Mathematical Analysis and Applications 453, 773 (2017). DOI: 10.1016/j.jmaa.2017.03.075, arXiv:1605.08348. URL: https://arxiv.org/abs/1605.08348
6. D. Hasler, B. Hinrichs, O. Siebert, On existence of ground states in the spin boson model, Communications in Mathematical Physics 388, 419 (2021). DOI: 10.1007/s00220-021-04185-w, arXiv:2102.13373. URL: https://arxiv.org/abs/2102.13373
7. B. Hinrichs, S. Polzer, Wiener-type theorems for the Laplace transform. With applications to ground state problems (2025). arXiv:2511.02867. URL: https://arxiv.org/abs/2511.02867
8. W. G. Unruh, False loss of coherence (2011). arXiv:1110.2199. URL: https://arxiv.org/abs/1110.2199
9. B. Bellomo, G. Compagno, F. Petruccione, Loss of coherence and dressing in QED, Physical Review A 74, 052112 (2006). DOI: 10.1103/PhysRevA.74.052112, arXiv:quant-ph/0612192. URL: https://arxiv.org/abs/quant-ph/0612192
10. K. Hornberger, Monitoring approach to open quantum dynamics using scattering theory, Europhysics Letters 77, 50007 (2007). DOI: 10.1209/0295-5075/77/50007, arXiv:quant-ph/0612078. URL: https://arxiv.org/abs/quant-ph/0612078
11. K. Hornberger, B. Vacchini, Monitoring derivation of the quantum linear Boltzmann equation, Physical Review A 77, 022112 (2008). DOI: 10.1103/PhysRevA.77.022112. URL: https://doi.org/10.1103/PhysRevA.77.022112
12. C. J. Hemming, R. V. Krems, Collisional decoherence of internal state superpositions in a trapped ultracold gas, Physical Review A 81, 052701 (2010). DOI: 10.1103/PhysRevA.81.052701, arXiv:0908.3204. URL: https://arxiv.org/abs/0908.3204
13. H. Uys, M. J. Biercuk, A. P. VanDevender, C. Ospelkaus, D. Meiser, R. Ozeri, J. J. Bollinger, Decoherence due to elastic Rayleigh scattering, Physical Review Letters 105, 200401 (2010). DOI: 10.1103/PhysRevLett.105.200401, arXiv:1007.2661. URL: https://arxiv.org/abs/1007.2661
14. M. Schut, P. Andriolo, M. Toroš, S. Bose, A. Mazumdar, Decoherence rate expression due to air molecule scattering in spatial qubits (2024). arXiv:2410.20910. URL: https://arxiv.org/abs/2410.20910
15. J. N. Bandyopadhyay, Quantum chaotic system as a model of decohering environment, Europhysics Letters 85, 50006 (2009). DOI: 10.1209/0295-5075/85/50006, arXiv:0806.4517. URL: https://arxiv.org/abs/0806.4517
16. N. Burić, Decoherence and classical behavior with a chaotic macroscopic environment, Physics Letters A 374, 4090 (2010). DOI: 10.1016/j.physleta.2010.05.009. URL: https://doi.org/10.1016/j.physleta.2010.05.009
17. H. Brox, J. Bergli, Y. M. Galperin, Importance of level statistics in the decoherence of a central spin due to a spin environment, Physical Review A 85, 052117 (2012). DOI: 10.1103/PhysRevA.85.052117, arXiv:1202.0676. URL: https://arxiv.org/abs/1202.0676
18. A. S. Sanz, Y. Elran, P. Brumer, Temperature crossover of decoherence rates in chaotic and regular bath dynamics, Physical Review E 85, 036218 (2012). DOI: 10.1103/PhysRevE.85.036218, arXiv:1203.4629. URL: https://arxiv.org/abs/1203.4629
19. Z. Xu, L. P. García-Pintos, A. Chenu, A. del Campo, Extreme decoherence and quantum chaos, Physical Review Letters 122, 014103 (2019). DOI: 10.1103/PhysRevLett.122.014103, arXiv:1810.02319. URL: https://arxiv.org/abs/1810.02319
20. P. Machnikowski, Change of decoherence scenario and appearance of localization due to reservoir anharmonicity, Physical Review Letters 96, 140405 (2006). DOI: 10.1103/PhysRevLett.96.140405, arXiv:quant-ph/0504035. URL: https://arxiv.org/abs/quant-ph/0504035
21. A. Trushechkin, Long-term behaviour in an exactly solvable model of pure decoherence and the problem of Markovian embedding, Mathematics 12, article 1 (2024). DOI: 10.3390/math12010001. URL: https://doi.org/10.3390/math12010001
22. J. Nishimura, H. Watanabe, Quantum decoherence in the Caldeira-Leggett model by the real-time path integral on a computer (2025). arXiv:2503.20699. URL: https://arxiv.org/abs/2503.20699
23. A. Albrecht, R. Baunach, A. Arrasmith, Adapted Caldeira-Leggett model, Physical Review Research 5, 023187 (2023). DOI: 10.1103/PhysRevResearch.5.023187, arXiv:2105.14040. URL: https://arxiv.org/abs/2105.14040
24. W. H. Zurek, Environment-induced superselection rules, Physical Review D 26, 1862 (1982). DOI: 10.1103/PhysRevD.26.1862. URL: https://doi.org/10.1103/PhysRevD.26.1862
25. W. H. Zurek, Sub-Planck spots of Schroedinger cats and quantum decoherence, Nature 412, 712 (2001). DOI: 10.1038/35089017, arXiv:quant-ph/0201118. URL: https://arxiv.org/abs/quant-ph/0201118
26. E. Joos, H. D. Zeh, The emergence of classical properties through interaction with the environment, Zeitschrift für Physik B 59, 223 (1985). DOI: 10.1007/BF01725541. URL: https://doi.org/10.1007/BF01725541
27. R. Olkiewicz, Environment-induced superselection rules in Markovian regime, Communications in Mathematical Physics 204, 245 (1999). DOI: 10.1007/s002200050756. URL: https://doi.org/10.1007/s002200050756