Analytical Lindblad Dynamics of Field-Controlled Entanglement and State Fidelity in the Hydrogen Electron-Proton Spins: Interplay of Hyperfine Coupling, Zeeman Effects, and Pure Dephasing

In this paper, we investigate the dynamics of quantum correlations in the ground-state hyperfine manifold of the hydrogen atom subjected to a static external magnetic field and local pure dephasing. The electron–proton spin pair is modeled as a bipartite two-qubit system evolving under the combined effects of hyperfine coupling, Zeeman splitting, and a Lindblad master equation that describes Markovian dissipative processes. Employing exact analytical solutions for the time-dependent density matrix elements (derived in the Markovian open-system framework), we quantify entanglement persistence via concurrence and state stability via Uhlmann fidelity with respect to the initial preparation. For an initial Werner state, numerical results reveal that the external magnetic field substantially modifies the system dynamics: Both concurrence and fidelity exhibit pronounced dependence on the Zeeman parameter, producing field-controlled oscillations, delayed entanglement sudden death, and altered decoherence rates. This behavior originates from Zeeman-induced lifting of hyperfine degeneracies, symmetry breaking of the isotropic Werner state, and redistribution of populations and coherences. Unlike previous studies that treat hyperfine interactions, Zeeman splitting, or decoherence in isolation, the present work provides a unified analytical treatment that simultaneously incorporates all three mechanisms. The findings underscore the competition between coherent hyperfine coupling and environmental noise and open new pathways for precision spectroscopy and robust quantum information protocols based on atomic spin degrees of freedom.