Mixed eigenstates in spin-boson systems with one-photon and two-photon interactions

arXiv:2604.05037 · quant-ph, cond-mat.stat-mech, nlin.CD · Submitted 2026-04-06 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: Today's paper: "Mixed eigenstates in spin-boson systems with one-photon and two-photon interactions".

Mira: The gist: This work investigates mixed eigenstates in spin-boson systems by comparing one-photon and two-photon interactions, proposing a generalized phase-space overlap index to identify genuine mixed eigenstates,

Kai: First, who's behind it and why it matters.

Title and authors: Mira: The paper focuses on how they analyze the dynamics of these spin-boson systems, specifically looking at the transition from predictable, regular motion to chaotic motion in both one-photon and two-photon models. They are trying to map out that messy middle ground where neither behavior is strictly dominant.

Kai: And they do this by defining chaos in a few ways—first classically through things like Lyapunov exponents, and then quantum mechanically through spectral statistics, specifically looking at level repulsion in the Gaussian orthogonal ensemble distribution from random matrix theory.

Lev: That level repulsion stuff is what tells us about the underlying quantum correlations when we look at chaos in the system. The paper uses a normalized spectral ratio called r c to quantify that transition, where r c = one means fully chaotic and r c = zero means regular <ref:2604.05037#pg2>.

Mira: And they don't just stop there; they introduce this novel index, the generalized phase-space overlap index, M nu, which is based on the nu-moment of the Poincaré-Husimi function. This is their main proposed way to identify those mixed eigenstates clearly.

The paper's summary: Kai: So, what they actually found is that this new index helps them distinguish between regular, chaotic, and mixed states by setting boundaries based on values like M nu in

-one M-) for regular and M nu in (M+, one: for chaotic <ref:2604.05037#pg3>.

Mira: They specifically identify eigenstates where the overlap index is in the interval M = M+ - M- as being genuine mixed eigenstates. And they show that these states behave in a specific way: when you increase the moment nu, say to M four an eigenstate that started as mixed tends to transition into a regular state where M four is around-one.

Lev: That transition suggests that higher moments of the Husimi function are actually helping them filter out the noise, which helps in identifying those mixed states. It’s like using a sharper lens to see what’s happening in the phase space.

Kai: And they're testing this on both models. For the one-photon Dicke model, they found that an eigenstate with a significant localization and M one in that mixed interval is a genuine mixed eigenstate <ref:2604.05037#pg1>.

The paper's improvements: Mira: One of the key things they’re pushing here is their use of higher moments, nu > one <ref:2604.05037#pg1>. They argue that these moments effectively remove regions of minor contribution in phase space. This is meant to prevent those spurious overlaps where regular and chaotic regions might look like they are touching when you only look at the first moment.

Lev: So, if you think the first moment might be misleading, using higher moments helps clean up the picture by focusing on what’s more significant in terms of phase space contribution. That makes for a much clearer identification of those mixed eigenstates we talked about earlier.

Kai: And they use this filtering to see how the Husimi function of an initially mixed eigenstate actually changes as you move to higher moments, showing that it tends to regularize towards the boundary where M four about-one.

Mira: They also use this framework to explore the principle of uniform semiclassical condensation, or PUSC. They conjecture that in the semiclassical limit, all eigenstates become strictly regular or chaotic.

Conclusion: Kai: So to wrap up, they confirmed that both the one-photon and two-photon Dicke models follow this PUSC principle. They define a mixed eigenstate fraction eta nu and show it decays as a power law of the form eta nu proportional to j - xi nu as the system approaches the semiclassical limit, which is consistent in both systems.

Lev: That power-law decay is really something specific to systems that have mixed phase space, suggesting it’s a universal feature across different interaction strengths.

Mira: They also noted that even though the one-photon system has a higher fraction of mixed eigenstates compared to the two-photon system, both models share a similar chaotic component when you look at mu c, which is about zero point eight seven for one photon and zero point eight four for two photons.

Kai: So they’re showing that the mechanism for how these quantum states mix, even when you change the photon interaction from one to two, follows a consistent scaling law in the semiclassical limit. That’s what this paper on mixed eigenstates in spin-boson systems with one-photon and two-photon interactions tells us.

CAMTP - Center for Applied Mathematics and Theoretical Physics, University of Maribor

quant-ph, cond-mat.stat-mech, nlin.CD

Submitted: 2026-04-06

Updated: 2026-10-09

Comments: 22 pages, 13 figures

Journal ref: Phys. Rev. E 114, 044206 (2026)

DOI: 10.1103/4sc6-546j

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 81/100

The gist: The gist: This work investigates mixed eigenstates in spin-boson systems by comparing one-photon and two-photon interactions, proposing a generalized phase-space overlap index to identify genuine

Key concepts

Spin-Boson Systems: These are physical models used to study how quantum systems interact with a single bosonic mode, often representing light or phonons. They are important for exploring quantum technologies and understanding transitions between regular and chaotic behavior when control parameters change.
Generalized Phase-Space Overlap Index (Mν): This is a novel tool based on the $ u$-moment of the Poincaré-Husimi function. It helps identify genuine mixed eigenstates by using higher moments ($ u > 1$) to enhance regions where certain states are localized, effectively separating regular and chaotic phase space contributions.
Uniform Semiclassical Condensation (PUSC): This principle suggests that in the semiclassical limit, quantum eigenstates of a mixed system must become strictly regular or chaotic. The paper verifies this by showing that the fraction of mixed eigenstates decays according to a power law in both one-photon and two-photon systems.

Terminology

Phase Transition — 6× in this paper · explained in 2 episodes
Chaotic Dynamics — 2× in this paper · explained in 2 episodes
Entanglement — 2× in this paper · explained in 4 episodes
Quantization — 2× in this paper · explained in 13 episodes
Quantum Physics — 2× in this paper · explained in 12 episodes
Quantum sensing — 2× in this paper · explained in 2 episodes
Superradiance — 2× in this paper · explained in 3 episodes
Clustering — 1× in this paper · explained in 2 episodes
Hilbert Space — 1× in this paper · explained in 2 episodes
Phase Transitions — 1× in this paper · explained in 2 episodes
Quantum fluctuations — 1× in this paper · explained in 2 episodes

Summary

The gist: This work investigates mixed eigenstates in spin-boson systems by comparing one-photon and two-photon interactions, proposing a generalized phase-space overlap index to identify genuine mixed eigenstates, and providing complementary evidence supporting the principle of uniform semiclassical condensation (PUSC) of quasiprobability functions.

Model Description

The study focuses on the Dicke model with one-photon and two-photon interactions, where the generalized spin-boson Hamiltonian is given by Eq. (1) The value f = 1 describes one-photon interactions in the system, while the value f = 2 describes two-photon interactions In the first process, one bosonic excitation is created (or annihilated) for each collective atomic excitation However, in the second process, two bosonic excitations are created (or annihilated) for each collective atomic excitation Three main parameters appear in the above Hamiltonian: the radiation bosonic frequency is ω and the energy splitting between two atomic levels is represented by ω0 The interaction strength between atoms and photons is modulated by the coupling parameter γ For the two-photon system (f = 2), the spin-boson coupling is limited to a critical value γsc = ω/2, where the spectral collapse occurs

Classical and Quantum Chaos Signatures

The analysis explores the classical transition from integrability to chaotic motion in both systems by examining phase space For the one-photon Dicke model, a zone of mixed dynamics is identified in the energy range ϵ ∈ (−1, 1] For the two-photon Dicke model, mixed dynamics are observed in the energy interval ϵ ∈ [0, 4], along with fluctuating zones in the range ϵ ∈ [4, 8) The percentage of chaotic motion is quantified by µc = (1/Vϵ) Σ Mϵ dx λ(Q, P) In the quantum realm, the presence of chaotic motion corresponds to quantum correlations within the system These manifest as level repulsion in quantum spectra described by the Gaussian orthogonal ensemble (GOE) distribution from random matrix theory A normalized spectral ratio rc = (⟨r⟩ - ⟨r⟩P) / (⟨r⟩GOE - ⟨r⟩P) is used, where rc = 1 identifies a fully chaotic quantum system and rc = 0 represents a regular quantum system

Generalized Phase-Space Overlap Index

A novel tool introduced is the generalized phase-space overlap index Mν, which is based on the ν-moment of the Poincar´e-Husimi function The definition is Mkν = (1/Bkν) Σ dQdP Qeνk(Q, P)χ(Q, P), where Bkν = R dQdP Qeνk(Q, P) is the normalization constant and χ is a classical characteristic function that takes two values: χ = 1 for classical regions with positive Lyapunov exponent (chaos) and χ = −1 for regions with zero Lyapunov exponent (integrability) The moment ν = 1 recovers the standard definition of the phase-space overlap index Mk1 = (1/Bk1) R dQdP Qek(Q, P)χ(Q, P) Higher moments of the Husimi function can enhance regions where certain eigenstates are localized, which helps to prevent the mistaken overlap of regular and chaotic regions

Mixed Eigenstate Characterization

Mixed eigenstates are identified by defining an overlap index interval ∆M = M+ − M−, where M+ −1 Eigenstates with Mν ∈ [−1, M−) are classified as regular; eigenstates with Mν ∈ (M+, 1] are considered chaotic; and eigenstates with Mν ∈ ∆M are identified as mixed In the analysis of representative eigenstates, an eigenstate characterized by significant localization and an overlap index M1 ∈ ∆M is identified as a genuine mixed eigenstate The Husimi function of an initially mixed eigenstate with M1 ∈ ∆M transitions to a regular eigenstate with M4 ≈ −1 as the moment increases

Universal Semiclassical Condensation (PUSC)

The principle of uniform semiclassical condensation (PUSC) states that in the semiclassical limit, the eigenstates of a mixed quantum system become strictly regular or chaotic This decay pattern is verified in various systems, including the onephoton Dicke model and the two-photon Dicke model The generalized mixed-eigenstate fraction is defined by ην = N∆M / N∆ϵ, where N∆M represents the number of mixed eigenstates within the interval ∆M, and N∆ϵ denotes the total number of eigenstates within the energy interval ∆ϵ The PUSC conjectures that this fraction decays as a power law of the form ην ∝ j − ξν as the system approaches the semiclassical limit This power-law decay is observed in both one-photon and two-photon systems, supporting its validity The one-photon system has a chaos fraction of µc = 0.87, while the two-photon system has a chaos fraction of µc = 0.84 The proportion of mixed eigenstates is higher in the one-photon system compared to the two-photon system

Conclusion

The study confirms that two-photon processes also adhere to the PUSC, suggesting that a power-law decay of the mixed-eigenstate fraction is a universal characteristic of quantum systems with mixed phase space The introduction of the generalized phase-space overlap index Mν demonstrates that higher moments, ν > 1, effectively remove regions of minor contribution in phase space, leading to clearer identification of genuine mixed eigenstates While the one-photon system exhibits a higher fraction of mixed eigenstates compared to the two-photon system, both systems display a similar fraction of the classical chaotic component.

Appendix A Details

The Fock basis provides a matrix representation of the Dicke Hamiltonian with elements defined as Eq. (A2) The efficient basis is written in terms of a displaced bosonic annihilation operator Aˆ = ˆa + GJˆx with G = 2γ/(ω√N) The convergence criterion ensures that the wavefunction remains unchanged within the truncated Hilbert space and is considered well-converged In all panels (c1)-(d3), we show the Anderson-Darling parameter A2 The vertical green dashed line in Figs. 3(a1)-3(c1) indicates a classical energy characterized by mixed statistics in the one-photon Dicke model, ϵ = 0 In Fig. 5(b1), we illustrate the Poincar´e section projected onto the atomic phase space for the classical one-photon Dicke Hamiltonian h1(x) and the classical energy ϵ = 0 The vertical green dashed line in Figs. 3(a2)-3(c2) represents an energy with mixed statistics for the two-photon Dicke model, ϵ = 1 In both systems, we observe a linear-like correlation between the localization measures However, in Fig.

Improvements for AI systems

Improved classification of quantum states based on phase-space overlap index: The generalized phase-space overlap index Mν provides a robust tool to identify genuine mixed eigenstates, which can be used to distinguish between regular, chaotic, and mixed states by defining boundaries as eigenstates with Mν ∈ [−1, M−) are classified as regular; eigenstates with Mν ∈ (M+, 1] are considered chaotic; and eigenstates with Mν ∈ ∆M are identified as mixed.
Quantification of semiclassical decay rates: The study allows for the exploration of the validity of the PUSC, confirming a power-law decay of the mixed-eigenstate fraction as we approached the semiclassical limit in both one-photon and two-photon systems, providing a universal characteristic for quantum systems with mixed phase space.
Moment-dependent filtering for localization analysis: By utilizing higher moments of the Husimi function, the system can effectively remove regions of minor contribution in phase space, which helps in clearer identification of mixed eigenstates by eliminating spurious overlaps between simultaneous regular and chaotic regions caused by broad Husimi functions.
Distinction between one-photon and two-photon chaos thresholds: The analysis reveals a difference in the fraction of mixed eigenstates, showing that the one-photon system exhibits a higher fraction of mixed eigenstates compared to the two-photon system, even though both systems display a similar fraction of the classical chaotic component.
System size dependence assessment for semiclassical convergence: The research demonstrates that the semiclassical limit is more readily achieved in systems with two-photon interactions compared to those with one-photon interactions, as evidenced by the observation that finite system size effects diminish as the system size increases in the two-photon model but are present in the one-photon model.