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Entropy

Quantum Coherence and Purity in Dissipative Hydrogen Atoms: Insights from the Lindblad Master Equation

K. Berrada, Smail B. Bougouffa

Published Aug 10, 2025• 4 citations
Publisher

Abstract

In this work, we investigate the quantum coherence and purity in hydrogen atoms under dissipative dynamics, with a focus on the hyperfine structure states arising from the electron-proton spin interaction. Using the Lindblad master equation, we model the time evolution of the density matrix of the system, incorporating both the unitary dynamics driven by the hyperfine Hamiltonian and the dissipative effects due to environmental interactions. Quantum coherence is quantified using the L1 norm and relative entropy measures, while purity is assessed via von Neumann entropy, for initial states, including a maximally entangled Bell state and a separable state. Our results reveal distinct dynamics: for the Bell states, both coherence and purity decay exponentially with a rate proportional to the dissipation parameter, whereas for a kind of separable state, coherence exhibits oscillatory behavior modulated via the hyperfine coupling constant, superimposed on an exponential decay, and accompanied by a steady increase in entropy. Higher dissipation rates accelerate the loss of coherence and the growth of von Neumann entropy, underscoring the environment's role in suppressing quantum superposition and driving the system towards mixed states. These findings enhance our understanding of coherence and purity preservation in atomic systems and offer insights for quantum information applications where robustness against dissipation is critical.

DOI
10.3390/e27080848

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In this work, we investigate the quantum coherence and purity in hydrogen atoms under dissipative dynamics, with a focus on the hyperfine structure states arising from the electron-proton spin interaction. Using the Lindblad master equation, we model the time evolution of the density matrix of the system, incorporating both the unitary dynamics driven by the hyperfine Hamiltonian and the dissipative effects due to environmental interactions. Quantum coherence is quantified using the L1 norm and relative entropy measures, while purity is assessed via von Neumann entropy, for initial states, including a maximally entangled Bell state and a separable state. Our results reveal distinct dynamics: for the Bell states, both coherence and purity decay exponentially with a rate proportional to the dissipation parameter, whereas for a kind of separable state, coherence exhibits oscillatory behavior modulated via the hyperfine coupling constant, superimposed on an exponential decay, and accompanied by a steady increase in entropy. Higher dissipation rates accelerate the loss of coherence and the growth of von Neumann entropy, underscoring the environment's role in suppressing quantum superposition and driving the system towards mixed states. These findings enhance our understanding of coherence and purity preservation in atomic systems and offer insights for quantum information applications where robustness against dissipation is critical.
We investigate the quantum dynamics of coherence in the hyperfine structure of hydrogen atoms subjected to dephasing noise, modeled using the Lindblad master equation. The effective Hamiltonian describes the spin–spin interaction between the electron and proton, with dephasing introduced via Lindblad operators. Analytical solutions for the time-dependent density matrix are derived for various initial states, including separable, partially entangled, and maximally entangled configurations. Quantum coherence is quantified through the l1-norm measures, while purity is evaluated to assess mixedness. Results demonstrate that coherence exhibits oscillatory decay modulated by the dephasing rate, with antiparallel spin states showing greater resilience against noise compared to parallel configurations. These findings highlight the interplay between coherent hyperfine dynamics and environmental dephasing, offering insights into preserving quantum resources in atomic systems for applications in quantum information science.
We investigate the quantum dynamics of entanglement and fidelity in the hyperfine structure of hydrogen atoms under dephasing noise, modeled via the Lindblad master equation. The effective Hamiltonian captures the spin–spin interaction between the electron and proton, with dephasing incorporated through local Lindblad operators. Analytical solutions for the time-dependent density matrix are derived for various initial states, including separable, partially entangled, and maximally entangled configurations. Entanglement is quantified using the concurrence, while fidelity measures the similarity between the evolving state and the initial state. Numerical results demonstrate that entanglement exhibits oscillatory decay modulated by the dephasing rate, with anti-parallel spin states displaying greater robustness compared to parallel configurations, often leading to entanglement sudden death. Fidelity dynamics reveal similar damped oscillations, underscoring the interplay between coherent hyperfine evolution and environmental dephasing. These insights elucidate strategies for preserving quantum correlations in atomic systems, with implications for quantum information processing and metrology.
The hyperfine structure of the hydrogen atom provides a clean, experimentally relevant two-qubit platform in which the coupled electron and proton spins exhibit rich quantum behavior. We investigate the open-system dynamics of this system under the simultaneous influence of the intrinsic hyperfine coupling, an external static magnetic field (via the Zeeman interaction), and local Markovian dephasing noise. Employing the Lindblad master equation, we derive the exact time evolution of the density matrix for general X-shaped initial states and focus on two complementary measures of quantum coherence—the L1-norm coherence CL(t) and the relative entropy of coherence CR(t)—together with the state purity P(t). Numerical results reveal that all three quantities display characteristic damped oscillatory evolution. For a vanishing magnetic field, the decay is relatively rapid and smooth, whereas increasing the proton magnetic parameter markedly raises the oscillation frequency and slows the overall envelope of both coherence and purity. Even under stronger dephasing rates, a suitably chosen external field can substantially postpone the loss of quantum features, acting effectively as a control knob that reshapes the coherent unitary dynamics to counteract dissipative effects. These findings underscore the delicate competition between intrinsic atomic interactions and environmental noise, while offering a practical route for protecting quantum resources in spin-based systems. Our work bridges fundamental atomic physics with resource-theoretic concepts and highlights promising strategies for coherence preservation in realistic, controllable quantum platforms.
In the present work, we investigate the detection and control of quantum correlations in the hyperfine structure of hydrogen atoms under dissipative dynamics. The hydrogen atom serves as a fundamental platform for studying bipartite quantum correlations between the electron and proton spins, where entanglement is quantified by concurrence and EPR steering is witnessed through violation of the entropic steering inequality. By modeling the open-system evolution via the Lindblad master equation, we analyze the time-dependent behavior of concurrence, steering, and fidelity under varying dissipation rates. Numerical illustrations highlight the resilience of entanglement relative to steering in noisy environments, providing insights into the control of non-classical phenomena.
In this work, we investigate quantum correlations, including entanglement and quantum discord, within the hyperfine structure of the hydrogen atom using the Lindblad master equation to model its dynamics as an open quantum system interacting with an environment. By incorporating realistic environmental influences, we examine the time evolution of two key measures of quantum correlations: concurrence, which quantifies entanglement, and local quantum uncertainty (LQU), a broader indicator of quantumness. Our analysis spans various initial states, including coherent superpositions of hyperfine states, to capture a wide range of possible configurations and demonstrate how these measures capture distinct aspects of quantum behavior. The results reveal the robustness of LQU in regimes where entanglement may vanish. This resilience of LQU underscores its utility as a robust measure of quantum correlations beyond entanglement alone in the hydrogen atom. By elucidating the dynamics of quantum correlations in the hydrogen atom under realistic conditions, this work not only deepens our fundamental understanding of atomic systems but also highlights their potential relevance to quantum information science and the development of quantum technologies.
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.