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arXiv:2106.06870 · Physical Review A

Natural and magnetically induced entanglement of hyperfine-structure states in atomic hydrogen

Yusef Maleki, Sergei Sheludiakov, Vladimir V. Khmelenko, Marlan O. Scully, David M. Lee, Aleksei M. Zheltikov

Published May 6, 2021• 12 citations
arXivHTMLPDF

Abstract

The spectrum of atomic hydrogen has long been viewed as a Rosetta stone that bears the key to decode the writings of quantum mechanics in a vast variety of physical, chemical, and biological systems. Here, we show that, in addition to its role as a basic model of quantum mechanics, the hydrogen atom provides a fundamental building block of quantum information. Through its electron- and nuclear-spin degrees of freedom, the hydrogen atom is shown to lend a physically meaningful frame and a suitable Hilbert space for bipartite entanglement, the two-qubit concurrence and quantum coherence of which can be expressed in terms of the fundamental physical constants---the Planck and Boltzmann constants, electron and proton masses, the fine-structure constant, as well as the Bohr radius and the Bohr magneton. The intrinsic, natural entanglement that the hyperfine-structure (HFS) states of the H atom store at low temperatures rapidly decreases with a growth in temperature, vanishing above a ${\ensuremath{\tau}}_{c}\phantom{\rule{4pt}{0ex}}\ensuremath{\approx}5.35\phantom{\rule{4pt}{0ex}}\ensuremath{\mu}\mathrm{eV}$ threshold. An external magnetic field, however, can overcome this thermal loss of HFS entanglement. As one of the central findings of this paper, we show that an external magnetic field can induce and sustain an HFS entanglement, against all the odds of thermal effects, at temperatures well above the ${\ensuremath{\tau}}_{c}$ threshold, thus enabling magnetic-field-assisted entanglement engineering in low-temperature gases and solids.

DOI
10.1103/physreva.103.052804

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The spectrum of atomic hydrogen has long been viewed as a Rosetta stone that bears the key to decode the writings of quantum mechanics in a vast variety of physical, chemical, and biological systems. Here, we show that, in addition to its role as a basic model of quantum mechanics, the hydrogen atom provides a fundamental building block of quantum information. Through its electron- and nuclear-spin degrees of freedom, the hydrogen atom is shown to lend a physically meaningful frame and a suitable Hilbert space for bipartite entanglement, the two-qubit concurrence and quantum coherence of which can be expressed in terms of the fundamental physical constants---the Planck and Boltzmann constants, electron and proton masses, the fine-structure constant, as well as the Bohr radius and the Bohr magneton. The intrinsic, natural entanglement that the hyperfine-structure (HFS) states of the H atom store at low temperatures rapidly decreases with a growth in temperature, vanishing above a \ensuremathτc\ensuremath≈5.35\ensuremathμeV{\ensuremath{\tau}}_{c}\phantom{\rule{4pt}{0ex}}\ensuremath{\approx}5.35\phantom{\rule{4pt}{0ex}}\ensuremath{\mu}\mathrm{eV}\ensuremathτc​\ensuremath≈5.35\ensuremathμeV threshold. An external magnetic field, however, can overcome this thermal loss of HFS entanglement. As one of the central findings of this paper, we show that an external magnetic field can induce and sustain an HFS entanglement, against all the odds of thermal effects, at temperatures well above the \ensuremathτc{\ensuremath{\tau}}_{c}\ensuremathτc​ threshold, thus enabling magnetic-field-assisted entanglement engineering in low-temperature gases and solids.
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.
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.
The hydrogen atom is the Rosetta stone of the early twentieth century atomic physics. The attempt to decipher its structure and properties led to the development of quantum mechanics and the unraveling of many of the mysteries of atomic, molecular, and solid state physics, and a good deal of chemistry and modern biology. Unlike the various one-dimensional model problems that we have been studying in the previous chapters, the hydrogen atom is a real physical system in three dimensions. It consists of an electron moving in a spherically symmetric potential well due to the Coulomb attraction of the positively charged nucleus. In three dimensions, the electron is not constrained to move linearly. It can execute orbital motions and, thus, has angular momentum. Not only is the total energy of the electron in the atom quantized, its angular momentum also has interesting and unexpected quantized properties that cannot possibly be understood on the basis of classical mechanics and electrodynamics. They are, however, the natural and necessary consequences of the basic postulates of quantum mechanics, as will be shown in this chapter. According to classical mechanics and electrodynamics, it is not possible to have a stable structure consisting of a small positively charged nucleus at the center of an electrically neutral atom with an electron sitting in its vicinity. For the electron not to be attracted into the positive charge, it must be orbiting around the nucleus so that the centrifugal force will counter the Coulomb attraction of the nucleus and maintain a constant electron orbit.
We begin our study of atomic physics by relying on the phenomenological notions summarized in chapter 1 and by fully exploiting the quantum mechanical formalism outlined in chapter 2. A hierarchy of approximations will be introduced in the first instance to describe the hydrogen atom, which allows one to focus on the main features of its energy spectrum and quantum properties, including the key ‘atomic orbital’ concept. Next, these results will be extended to the more general case of hydrogenic atoms. The physical picture will be made more rigorous by introducing the magnetic interactions, included those linked to the intrinsic spin degree of freedom, leading to the full description of the fine and hyperfine structure of one-electron atoms. The effects of a static external magnetic or electric field will be eventually discussed by calculating the corresponding energy shifts and splittings of the energy levels.
We investigate the time evolution of bipartite quantum correlations in the ground-state hyperfine manifold of the hydrogen atom subjected to an external magnetic field and independent Markovian dephasing. Treating the electron–proton spin pair as an effective two-qubit system, we derive the exact solution of the Lindblad master equation for an X-shaped initial state and quantify the dynamics using three complementary measures: entanglement of formation (through concurrence), quantum steering (through the CJWR inequality) and Bell nonlocality (through normalized CHSH violation). The dynamics are obtained within a unified open-system framework that combines hyperfine interaction, Zeeman splitting, and Markovian dissipation in a single analytically solvable Lindblad model, allowing a complete operator-level characterization of the correlation decay. This exact treatment provides a transparent link between the underlying spectral structure of the Hamiltonian and the observed hierarchy in the robustness of quantum correlations. Our results reveal that all three quantities exhibit damped oscillations whose frequency and decay rate are strongly tuned by the proton magnetic parameter through the Zeeman splitting. While entanglement decays relatively quickly, steering persists noticeably longer and Bell nonlocality proves to be the most fragile, confirming the expected hierarchy of quantum correlations under local dephasing. The external magnetic field emerges as a practical control knob that can extend the lifetime of these resources even in the presence of noise. These findings provide a clear physical picture of how hyperfine coupling, Zeeman effects, and environmental fluctuations jointly govern quantum coherence in atomic spin systems, with direct implications for spin-based quantum technologies and fundamental tests of nonlocality in realistic laboratory settings.
Hydrogen is the most dominant atom in the universe and is considered the main component of baryonic matter. Thus far, the quantum features of the unbounded hydrogen atoms in the background of the universe and the possibility of emerging unique quantum effects, such as entanglement on the cosmological scale, have not been considered. In this work, we demonstrate that the dynamical expansion of the universe leads to the emergence of natural entanglement in the hyperfine structure of atomic hydrogen. Our findings reveal that there exists a critical age for the universe where hydrogen atoms naturally build up entanglement, resulting from the expansion of the universe. More precisely, when the universe reaches the age of about 2.5 × 1018 seconds (about 80 billion years old), the hyperfine structure entanglement in hydrogen atoms naturally takes off, demonstrating a peculiar quantum phenomenon known as entanglement sudden birth. This expansion-induced entanglement becomes maximum at about 3.6 × 1018 seconds (about 115 billion years), after the Big Bang. By analyzing the fate of seed atoms formed in the early universe, this study underscores the significance of unique quantum mechanical features, such as entanglement, on cosmological scales. The authors investigate quantum entanglement in the hyperfine structure of the neutral hydrogen atom in thermal equilibrium with the cosmological microwave background radiation. They demonstrate that when the universe is around 80 billion years old, neutral hydrogen atoms begin to form entangled states, displaying a phenomenon known as entanglement sudden birth.