Vol. 33 No. 2 (2026) in progress:

Open Access

Article

Article ID: 3929

Quantum purity exchange dynamics in a qubit–resonator system subject to squeezed-vacuum driving

by Leila Abdelgader, Chafaa Hamrouni

Advances in Differential Equations and Control Processes, Vol.33, No.2, 2026;

This work presents a theoretical framework to study the non-Markovian dynamics of a two-level quantum emitter interacting with a broadband squeezed electromagnetic reservoir, and both one- and two-photon interaction processes are incorporated. Mathematical modeling uses a time-convolution less projection operator technique. This yields a time-local master equation. The coefficients of this equation are derived from integrals over the reservoir's squeezed correlation functions:  and . The model is validated through rigorous numerical simulation of the resulting dynamical equations. Testing involves computing key physical observables: the transient emission spectrum  and the field linear entropy . These predictions are systematically analyzed against variations in squeezing parameters , coupling strengths , and detector bandwidth . The results confirm that the model successfully captures phase-dependent decoherence, spectral modulation, and purity oscillations. Notably, two-photon processes suppress decoherence under strong squeezing. The consistency between analytical derivations and numerical outcomes validates the framework. It is established as a predictive tool for quantum optics in engineered nonclassical environments. This study directly connects engineered reservoir properties specifically its nonclassical photon statistics to observable, time-dependent quantum phenomena. The findings offer fundamental insights and a predictive tool for quantum control, sensing, and information processing in tailored electromagnetic environments.

Open Access

Perspective

Article ID: 4007

From linear w-γ correlation to redshift-dependent dynamics: A complete phenomenological framework and testing roadmap for dark energy

by Tongfeng Zhao

Advances in Differential Equations and Control Processes, Vol.33, No.2, 2026;

Based on an adaptive universe model, this perspective article presents a phenomenological framework that correlates the dark energy equation of state w with the cosmic growth index γ via the linear relation w(a) = −1 + η(γ(a) − 0.55). Recognizing that the coupling between dark energy and structure formation may evolve with cosmic time, the framework is extended to a redshift-dependent formulation: w(z) = −1 + η(z)[γ(z) − 0.55] + Δw_bg(z), where η(z) captures the structure-dependent coupling and Δw_bg(z) accounts for intrinsic background evolution. Several physically motivated parameterizations of η(z) are proposed, including continuous forms (smooth transition and oscillatory) and a phenomenological piecewise model aligned with distinct phases of structure formation history. Built upon an interacting dark sector model that strictly conserves energy and momentum, the framework maintains the spacetime geometry of General Relativity while naturally addressing both the Hubble tension (via enhanced late-time expansion) and the S₈ tension (via suppressed structure growth). A hierarchical Bayesian testing roadmap with Fisher forecasts demonstrates that upcoming surveys (DESI, Euclid, Roman) can decisively detect couplings of magnitude |η| ≳ 0.05 at high significance. The framework yields distinctive, testable predictions—including correlated w(z) and fσ₈(z) evolution, a gravitational slip parameter η_slip = 1 that distinguishes it from modified gravity theories, and scale-dependent signatures in the nonlinear regime—providing a comprehensive path to either validate or falsify the hypothesized dark energy–structure growth connection.