Magnetic Control of Quantum Correlations in a Two-Qubit Spin System Under Dephasing

Abstract

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.

1. Introduction

The hydrogen atom, owing to its conceptual simplicity and analytical solvability, has profoundly influenced the foundations of quantum mechanics and remains a paradigmatic reference system in atomic physics, chemistry, and related disciplines [1,2,3,4]. In addition to its historical role, it constitutes an ideal platform for investigating elementary quantum-information concepts. Specifically, the coupled electron and proton spins realize a natural two-qubit system that enables precise examination of bipartite quantum correlations, including entanglement and state fidelity, through fundamental physical interactions. A key feature is the hyperfine structure arising from the magnetic dipole coupling between the spins. This interaction lifts the ground-state degeneracy, yielding singlet and triplet subspaces endowed with intrinsic quantum correlations. Application of an external magnetic field introduces Zeeman shifts that break rotational symmetry, modify the energy spectrum and eigenstates, and establish a versatile framework for studying coherent spin dynamics, level crossings, and field-controlled correlations. At low temperatures, the hyperfine states manifest nonclassical correlations governed by the competition between hyperfine splittings and thermal fluctuations; prior studies demonstrate that these correlations diminish with increasing temperature and vanish beyond a characteristic energy scale [5,6,7]. Complementary experiments on spin-polarized hydrogen in solid H₂ matrices reveal significant deviations from classical Boltzmann statistics at low temperature [5,8], emphasizing the pivotal role of quantum coherence in such regimes. These findings motivate a detailed theoretical investigation of how hyperfine and Zeeman interactions shape quantum correlations under realistic noisy conditions. In quantum information science, spin systems analogous to the electron–proton pair are recognized as promising quantum-information carriers. While electron spins in quantum dots [9,10] and nuclear spins in solid-state platforms [11,12,13] have been extensively explored as qubits, most prior works have addressed either spatial entanglement [14] or formal hyperfine treatments [15]. In contrast, the present study focuses on spin entanglement generated by the intrinsic hyperfine interaction, explicitly incorporating the Zeeman contribution from an external magnetic field. This consistent framework simultaneously addresses field-induced control and decoherence processes.

Recently, substantial interest has emerged in quantifying quantum correlations in open spin systems using well-established measures such as the entanglement of formation and Uhlmann fidelity. These tools provide complementary perspectives on the robustness of nonclassical correlations under environmental decoherence [16,17,18,19]. The entanglement of formation, derived from concurrence, offers a reliable and tractable quantifier of bipartite entanglement in two-qubit systems, including electron–proton spin pairs in hydrogen, and captures how hyperfine interactions generate correlations modulated by external magnetic fields. In parallel, the Uhlmann fidelity quantifies the stability of quantum states during dissipative evolution by measuring their overlap with a reference state, thereby characterizing the preservation of coherence under dephasing noise [17,20].

More generally, applications of these measures to atomic spin systems have revealed sharp transitions between classical and quantum regimes and highlighted the differing robustness of various quantifiers under decoherence. In particular, while entanglement typically decays rapidly, alternative indicators such as steering and related correlation measures may exhibit enhanced resilience, suggesting more robust pathways for maintaining quantum resources in noisy environments [18,21,22]. In this context, open-system analyses of hydrogenic hyperfine dynamics provide a useful platform for exploring the interplay between intrinsic atomic couplings, external magnetic fields, and environmental noise, with relevance to quantum information processing and precision sensing applications.

A faithful description of spin systems such as the hydrogen hyperfine manifold requires an open quantum system framework, since environmental noise inevitably induces decoherence and degrades quantum correlations. Open quantum systems theory provides the essential tools to describe the competition between coherent dynamics and dissipation [23]. In particular, the Lindblad master equation offers a standard framework for modeling Markovian dissipation in atomic and spin systems, ensuring complete positivity and enabling tractable analytical treatment of dephasing processes [23,24,25].

Within this framework, we consider the reduced dynamics of the hydrogen hyperfine spin system under Markovian dephasing, which may effectively arise from fluctuations of the external magnetic field. The evolution is governed by the interplay between intrinsic hyperfine coupling, Zeeman interaction, and environmental decoherence. To characterize the resulting quantum correlations, we employ three complementary measures: entanglement of formation (via concurrence), quantum steering (via the CJWR inequality), and Bell nonlocality (via the normalized CHSH violation). These quantifiers capture, respectively, progressively stronger forms of nonclassical correlations between the electron and proton spins.

By analyzing their time evolution, we identify regimes in which hyperfine-induced coherence exhibits partial robustness against decoherence and where the external magnetic field modifies the survival of quantum correlations. The present work thus provides a unified open-system description of the hydrogen hyperfine dynamics, highlighting how fundamental atomic interactions govern the generation and decay of quantum correlations within a single consistent framework.

Furthermore, quantum spin systems under external magnetic fields play a central role in spintronics and quantum information science, where precise control of spin degrees of freedom is essential for information processing and device applications. In particular, magnetic-field engineering provides a versatile mechanism to manipulate spin dynamics, enabling tunable coherence properties, controlled transitions, and robust operation of quantum states in both atomic and solid-state platforms [26,27,28,29,30]. Understanding the interplay between coherent control and environmental decoherence is therefore of fundamental importance for the design of reliable quantum technologies.

From a broader perspective, geometric and topological aspects of quantum evolution, such as the Berry phase, have been widely recognized as sensitive probes of system–environment interactions. In open quantum systems, geometric phases generally acquire corrections due to dissipation and noise, reflecting the competition between unitary dynamics and environmental coupling. While the present work does not explicitly address geometric-phase effects, it is consistent with this general framework in which external magnetic fields influence both dynamical evolution and the underlying phase structure of quantum states [31,32,33]. In this context, we investigate how hyperfine interaction, Zeeman splitting, and Markovian dephasing jointly govern the dynamics of quantum correlations in a two-spin system.

The rest of this paper is arranged as follows. We start in Section 2 by setting up the hyperfine Hamiltonian for a hydrogen atom exposed to an external magnetic field. Section 3 outlines the open-system dynamics captured by the Lindblad master equation. In Section 4, we introduce the quantifiers used to probe quantum correlations: entanglement, quantum steering, and nonlocality. Their evolution over time under dissipation is examined in Section 5. Finally, Section 6 concludes with a summary of our key results and a discussion of their physical importance.

2. Ground-State Hydrogen: Hyperfine Coupling in a Magnetic Field

The spin structure of the hydrogen atom in its ground state is governed by the magnetic interaction between the electron and proton, commonly referred to as the hyperfine coupling. This interaction originates from the magnetic dipole–dipole coupling of the two spin-$\frac{1}{2}$ constituents and can be written in the form

$${\widehat{H}}_{\mathrm{hf}}=\lambda ({\mathsf{\tau }}_e·{\mathsf{\tau }}_p),$$

(1)

where ${\mathsf{\tau }}_e=({\tau }_e^x,{\tau }_e^y,{\tau }_e^z)$ and ${\mathsf{\tau }}_p=({\tau }_p^x,{\tau }_p^y,{\tau }_p^z)$ denote the Pauli operator vectors acting on the electron and proton spin subspaces, respectively. The parameter $\lambda$ quantifies the strength of the contact interaction and is explicitly given by [34]

$$\lambda ={\displaystyle \frac{{\mu }_0{g}_e{g}_p{e}^2{\mathit{ћ}}^2}{12\pi m_e{m}_p{a}_0^3}},$$

(2)

with ${\mu }_0$ the vacuum permeability, $a_0$ the Bohr radius, and $g_e$, $g_p$ the corresponding Landé factors for the electron and proton. The quantities $m_e$ and $m_p$ denote their respective masses. This expression reflects the contact nature of the interaction through the electronic probability density at the origin.

When a static magnetic field $\mathbf{B}$ is applied, an additional contribution arises due to the Zeeman interaction of each spin with the field. This term is expressed as

$${\widehat{H}}_{\mathrm{Z}}=-{\mathsf{\mu }}_e·\mathbf{B}-{\mathsf{\mu }}_p·\mathbf{B}.$$

(3)

The magnetic moment operators are related to the spin operators via

$${\mathsf{\mu }}_e=-g_e{\mu }_B{\displaystyle \frac{{\mathsf{\tau }}_e}{2}},{\mathsf{\mu }}_p=g_p{\mu }_N{\displaystyle \frac{{\mathsf{\tau }}_p}{2}},$$

(4)

where ${\mu }_B$ and ${\mu }_N$ denote Bohr and nuclear magnetons, respectively. Substituting these relations leads to

$${\widehat{H}}_{\mathrm{Z}}={\gamma }_e({\mathsf{\tau }}_e·\mathbf{B})-{\gamma }_p({\mathsf{\tau }}_p·\mathbf{B}),$$

(5)

where we have introduced the constants ${\gamma }_e=\frac{g_e{\mu }_B}{2}$ and ${\gamma }_p=\frac{g_p{\mu }_N}{2}$ for compactness.

The complete Hamiltonian describing the coupled spin system in the presence of the external field then reads [35,36,37,38,39,40,41]

$$\widehat{H}=H_{hf}+H_Z=\lambda ({\mathsf{\tau }}_e·{\mathsf{\tau }}_p)+{\gamma }_e({\mathsf{\tau }}_e·\mathbf{B})-{\gamma }_p({\mathsf{\tau }}_p·\mathbf{B}).$$

(6)

This operator encapsulates the competition between hyperfine coupling and Zeeman splitting, which together determine the fine structure of the energy spectrum. The external magnetic field enters the Hamiltonian through the Zeeman term and therefore serves as a controllable parameter that modulates the spectral structure of the system, directly affecting the time evolution of quantum correlations under the Lindblad dynamics.

For completeness, we briefly comment on the structure of the full hyperfine Hamiltonian in the presence of an external magnetic field. The total Hamiltonian is given by (6), where ${\mathsf{\tau }}_e$ and ${\mathsf{\tau }}_p$ denote the electronic and nuclear angular momenta, respectively. In the absence of a magnetic field, the total angular momentum $\mathbf{F}={\mathsf{\tau }}_e+{\mathsf{\tau }}_p$ is conserved, whereas in the presence of a static field along the z-axis, only its projection $F_z$ remains a good quantum number.

Since $[H,F_z]=0$, the Hamiltonian does not couple states with different magnetic quantum number $M_F$, leading to a natural block-diagonal decomposition of the full matrix in the coupled basis. This symmetry reduction justifies the restriction to the relevant two-qubit subspace used in our effective model and ensures the consistency of the reduced Lindblad dynamics employed in this work.

Each spin degree of freedom is associated with a two-dimensional Hilbert space spanned by the eigenstates of the corresponding ${\tau }^z$ operator,

$${\mathcal{K}}_e={\left\{\right|\uparrow ⟩}_e{,|\downarrow ⟩}_e\},{\mathcal{K}}_p={\left\{\right|\uparrow ⟩}_p{,|\downarrow ⟩}_p\}.$$

(7)

The full state space is therefore the tensor product

$$\mathcal{K}={\mathcal{K}}_e\otimes {\mathcal{K}}_p,$$

(8)

which is spanned by the uncoupled basis

$${\left\{\right|\uparrow ⟩}_e\otimes {|\uparrow ⟩}_p{,|\uparrow ⟩}_e\otimes {|\downarrow ⟩}_p{,|\downarrow ⟩}_e\otimes {|\uparrow ⟩}_p{,|\downarrow ⟩}_e\otimes {|\downarrow ⟩}_p\}.$$

(9)

It is worth emphasizing that the relative signs in the Zeeman contributions play a crucial role in determining the spectral structure, leading to combinations proportional to $B({\gamma }_e\pm {\gamma }_p)$ that govern the dynamical evolution of the system. For a magnetic field aligned along the z-axis, the Hamiltonian admits an exact diagonalization. The corresponding eigenvalues and eigenvectors, expressed in the uncoupled basis, are derived explicitly in Appendix A to maintain the continuity of the main discussion.

3. Dephasing Effects in Open Quantum Systems

In practice, the coherent evolution of the electron–proton spin system is influenced by unavoidable interactions with the surrounding environment. Within the hyperfine manifold, a primary source of decoherence is pure dephasing, a mechanism that suppresses off-diagonal coherence terms while preserving population distributions.

Such behavior typically arises from temporal fluctuations of the external magnetic field, which couple asymmetrically to the two spin components through the Zeeman interaction. To describe this process, we employ the Markovian open quantum system formalism, where the reduced density operator $\varrho \left(t\right)$ evolves according to a Lindblad master equation. Working in natural units ($\mathit{ћ}=1$), the evolution equation takes the form [42,43,44]

$${\displaystyle \frac{d\varrho \left(t\right)}{dt}}=-i\left[\widehat{H},\varrho \left(t\right)\right]+\mathcal{D}\left[\varrho \left(t\right)\right],$$

(10)

where $\widehat{H}$ is the total Hamiltonian defined above, and $\mathcal{D}[·]$ represents the dissipative superoperator encoding the environmental effects.

3.1. Independent Versus Correlated Dephasing

In this work, we primarily consider independent phase-noise channels acting separately on the electron and proton spins. Such a scenario is physically relevant when magnetic fluctuations vary over length scales comparable to or smaller than the electron–proton separation, or when each spin interacts with a distinct effective environment. Independent local dephasing models are widely used in the theory of open quantum systems and in the study of two-qubit spin dynamics within the Lindblad formalism [23,45,46].

Within this framework, the Lindblad operators are written as

$${\widehat{L}}_e={\tau }_e^z\otimes {\mathbb{I}}_p,{\widehat{L}}_p={\mathbb{I}}_e\otimes {\tau }_p^z,$$

(11)

with associated dephasing rates denoted by ${\kappa }_e$ and ${\kappa }_p$, respectively. The dissipative contribution then takes the form

$$\mathcal{D}\left[\varrho \right]={\kappa }_e\left({\widehat{L}}_e\varrho {\widehat{L}}_e-\varrho \right)+{\kappa }_p\left({\widehat{L}}_p\varrho {\widehat{L}}_p-\varrho \right),$$

(12)

where we used ${\widehat{L}}_e^2={\widehat{L}}_p^2=\mathbb{I}$.

In contrast, correlated (collective) dephasing arises when both spins couple to the same fluctuating magnetic field. In that case, the noise is described by a single operator

$${\widehat{L}}_{\mathrm{corr}}={\tau }_e^z\otimes {\mathbb{I}}_p+{\mathbb{I}}_e\otimes {\tau }_p^z,$$

(13)

which preserves coherence within specific symmetry-protected subspaces. Although such a regime can protect certain entangled states, the independent model adopted here provides a more general and experimentally relevant description of magnetic noise in hydrogen spin systems.

The validity of the Markovian approximation relies on the assumption that the environmental correlation time is much shorter than the intrinsic dynamical timescales set by the hyperfine interaction and Zeeman splitting. Under this separation of timescales, memory effects can be neglected and the reduced dynamics becomes effectively time-local, allowing for an analytically tractable description of decoherence processes.

3.2. Evolution of Density-Matrix Elements

For compactness, we introduce the effective parameters

$$\kappa ={\kappa }_e+{\kappa }_p,{\Omega }_{-}=B({\gamma }_e-{\gamma }_p),{\Omega }_{+}=B({\gamma }_e+{\gamma }_p).$$

(14)

In the product basis ${\left\{\right|\uparrow ⟩}_e{|\uparrow ⟩}_p{,|\uparrow ⟩}_e{|\downarrow ⟩}_p{,|\downarrow ⟩}_e{|\uparrow ⟩}_p{,|\downarrow ⟩}_e{|\downarrow ⟩}_p\}.$ The elements of the density matrix ${\varrho }_{ij}\left(t\right)$ obey the following equations under independent dephasing.

3.2.1. Population Subspace

$${\dot{\varrho }}_{11}\left(t\right)=0,$$

(15)

$${\dot{\varrho }}_{22}\left(t\right)=-2i\lambda \left[{\varrho }_{32}\left(t\right)-{\varrho }_{23}\left(t\right)\right],$$

(16)

$${\dot{\varrho }}_{33}\left(t\right)=+2i\lambda \left[{\varrho }_{32}\left(t\right)-{\varrho }_{23}\left(t\right)\right],$$

(17)

$${\dot{\varrho }}_{44}\left(t\right)=0.$$

(18)

Thus, the extremal populations remain invariant, while the intermediate populations are coupled coherently through the hyperfine interaction.

3.2.2. Coherence Subspace

Single-spin coherences

$${\dot{\varrho }}_{12}\left(t\right)=\left[-2i\lambda -i({\Omega }_{-}-{\Omega }_{+})-2{\kappa }_p\right]{\varrho }_{12}\left(t\right)+2i\lambda {\varrho }_{13}\left(t\right),$$

(19)

$${\dot{\varrho }}_{13}\left(t\right)=\left[-2i\lambda -i({\Omega }_{-}+{\Omega }_{+})-2{\kappa }_e\right]{\varrho }_{13}\left(t\right)+2i\lambda {\varrho }_{12}\left(t\right).$$

(20)

Two-spin coherence

$${\dot{\varrho }}_{14}\left(t\right)=-2\left(i{\Omega }_{-}+\kappa \right){\varrho }_{14}\left(t\right),$$

(21)

$${\dot{\varrho }}_{41}\left(t\right)=2\left(i{\Omega }_{-}-\kappa \right){\varrho }_{41}\left(t\right).$$

(22)

Single-excitation coherences

$${\dot{\varrho }}_{23}\left(t\right)=\left[-2i{\Omega }_{+}-2\kappa \right]{\varrho }_{23}\left(t\right)+2i\lambda \left[{\varrho }_{22}\left(t\right)-{\varrho }_{33}\left(t\right)\right],$$

(23)

$${\dot{\varrho }}_{32}\left(t\right)=\left[+2i{\Omega }_{+}-2\kappa \right]{\varrho }_{32}\left(t\right)-2i\lambda \left[{\varrho }_{22}\left(t\right)-{\varrho }_{33}\left(t\right)\right].$$

(24)

Remaining coherences

$${\dot{\varrho }}_{24}\left(t\right)=\left[2i\lambda -i({\Omega }_{-}+{\Omega }_{+})-2{\kappa }_e\right]{\varrho }_{24}\left(t\right)-2i\lambda {\varrho }_{34}\left(t\right),$$

(25)

$${\dot{\varrho }}_{34}\left(t\right)=\left[2i\lambda -i({\Omega }_{-}-{\Omega }_{+})-2{\kappa }_p\right]{\varrho }_{34}\left(t\right)-2i\lambda {\varrho }_{24}\left(t\right).$$

(26)

The above dynamical equations reveal a clear separation between coherent and incoherent processes. Independent phase noise acts solely on the off-diagonal elements, inducing exponential damping without directly affecting populations. Consequently, population transfer arises exclusively from the hyperfine coupling $\lambda$. The fully aligned configurations remain invariant under the combined dynamics and, therefore, form a decoherence-resistant subspace with respect to local phase noise. In contrast, the single-excitation states are coherently mixed and exhibit sensitivity to both hyperfine exchange and environmental fluctuations. The structure of the coherence equations shows that pairs such as $({\varrho }_{12},{\varrho }_{13})$ and $({\varrho }_{24},{\varrho }_{34})$ behave as coupled modes, undergoing oscillatory exchange governed by $\lambda$ and frequency shifts determined by ${\Omega }_{\pm }$, while decaying at rates set by the local noise strengths. The element ${\varrho }_{14}$ evolves independently, accumulates a relative phase and decays at the maximal rate $2\kappa$. In general, coherences that involve a single-spin decay at rates proportional to ${\kappa }_e$ or ${\kappa }_p$, while joint coherences are suppressed more rapidly at the combined rate $2\kappa$. This hierarchy highlights the role of noise locality in determining the robustness of quantum correlations.

3.3. Initial States

To analyze the dynamical behavior of the hyperfine–Zeeman spin system in a systematic manner, we consider a general initial two-qubit state of the X form with maximally mixed marginals. This choice is sufficiently general to encompass a wide range of physically relevant states, including pure entangled states and mixed states of practical interest.

The initial density matrix is taken as

$$\varrho \left(0\right)={\displaystyle \frac{1}{4}}\left(\begin{array}{cccc}1+c_3& 0& 0& c_1-c_2\\ 0& 1-c_3& c_1+c_2& 0\\ 0& c_1+c_2& 1-c_3& 0\\ c_1-c_2& 0& 0& 1+c_3\end{array}\right),$$

(27)

where the real parameters $c_1,c_2,c_3\in [-1,1]$ fully characterize the initial correlations of the system.

This parametrization provides a versatile framework for studying the evolution of quantum correlations. In particular, it includes as special cases maximally entangled Bell states, Werner states, and pure states confined to specific excitation subspaces. Therefore, it allows for a systematic and consistent investigation of coherence and entanglement dynamics without restricting the analysis to particular initial configurations.

The explicit time evolution of the density matrix elements corresponding to this general initial condition is presented in Appendix B.

4. Quantifiers of Quantum Correlations: Entanglement of Formation, Quantum Steering via the CJWR Inequality, and Bell Nonlocality

In order to gain a comprehensive understanding of the quantum correlations generated in the system, we analyze several complementary quantifiers that capture different aspects of non-classical behavior. In particular, we consider the entanglement of formation as a measure of bipartite entanglement, quantum steering based on the CJWR inequality [47] as an indicator of directional quantum correlations beyond entanglement, and Bell nonlocality as a stringent test of the incompatibility with local hidden-variable theories. These three quantifiers form a hierarchy of quantum correlations, allowing us to explore how environmental effects and system parameters influence the transition between different regimes of quantum correlations.

4.1. Entanglement of Formation

With the analytical solution available, we proceed to examine the quantum correlations present in the system. In this work, we focus exclusively on entanglement, which represents a fundamental signature of nonclassical correlations in composite quantum states.

To quantify bipartite entanglement, we employ the entanglement of formation, a widely used measure that is particularly well suited for two-qubit systems. This quantity provides a direct way to characterize the degree of entanglement, ranging from fully separable states to maximally entangled ones. It is conveniently expressed in terms of the concurrence [16].

For a given density matrix $\rho$, the entanglement of formation is defined as

$$E\left(\rho \right)=H\left[{\displaystyle \frac{1}{2}}\left(1+\sqrt{1-C^2\left(\rho \right)}\right)\right],$$

(28)

where the function $H\left(f\right)$ denotes the binary entropy [16,48,49,50],

$$H\left(f\right)=-f{log}_{2}f-(1-f){log}_2(1-f).$$

(29)

The concurrence $C\left(\rho \right)$ is given by [16,51]

$$C\left(\rho \right)=max\left\{0,\sqrt{{\lambda }_1}-\sqrt{{\lambda }_2}-\sqrt{{\lambda }_3}-\sqrt{{\lambda }_4}\right\},$$

(30)

where ${\lambda }_i$ are the eigenvalues, in decreasing order, of the operator

$$R=\sqrt{\rho }\tilde{\rho }\sqrt{\rho }.$$

(31)

Here, the matrix

$$\tilde{\rho }=({\sigma }_y\otimes {\sigma }_y){\rho }^{*}({\sigma }_y\otimes {\sigma }_y)$$

(32)

represents the spin-flipped density operator, with ${\rho }^{*}$ denoting complex conjugation.

The concurrence satisfies $0\le C\le 1$, where $C=0$ corresponds to separable states and $C=1$ to maximally entangled states. Intermediate values indicate partial entanglement. This measure is particularly useful for analyzing the dynamical behavior of entanglement in open quantum systems, as it provides a clear and quantitative description of how environmental effects degrade quantum correlations.

4.2. Quantum Steering Based on the CJWR Inequality

To quantify quantum steering, we employ the inequality formulated by Cavalcanti, Jones, Wiseman, and Reid (CJWR) [47]. Later, Angelo et al. [52] introduced a practical, normalized steering measure that extracts the degree of steerability directly from the largest violation of these CJWR inequalities. For an arbitrary bipartite state $\rho$ the CJWR expression is given by

$$F_n(\rho ,\mu )={\displaystyle \frac{1}{\sqrt{n}}}\left|{\displaystyle \sum _{i=1}^n}⟨A_i\otimes B_i⟩\right|⩽1,$$

where $A_i={\overrightarrow{u}}_i·\overrightarrow{\sigma }$ and $B_i={\overrightarrow{v}}_i·\overrightarrow{\sigma }$ are local dichotomic observables constructed from the Pauli vector $\overrightarrow{\sigma }=({\sigma }_1,{\sigma }_2,{\sigma }_3)$. Here the unit vectors ${\overrightarrow{u}}_i\in {\mathbb{R}}^3$ specify Alice’s measurement directions, while the ${\overrightarrow{v}}_i$ form an orthonormal set for Bob’s measurements. The symbol $\mu$ simply denotes the complete set of chosen directions. In the present work we restrict attention to the two-observable case ($n=2$), which is both experimentally accessible and theoretically convenient. Alice and Rob each choose between a pair of dichotomic measurements, $\{A_1,A_2\}$ and $\{B_1,B_2\}$, with binary outcomes labelled $\pm 1$. For this setting the steering quantifier takes the compact form [52]

$$S_2:=max\left\{0,{\displaystyle \frac{F_2\left(\rho \right)-1}{F_2^{max}-1}}\right\},$$

where $F_2\left(\rho \right)={max}_{\mu }{F}_2(\rho ,\mu )$ is the largest value of the CJWR function over all admissible measurement pairs $\mu$, and $F_2^{max}=\sqrt{2}$ is the ultimate quantum bound attainable by any two-qubit state. The function itself admits the simple geometric expression

$$F_2\left(\rho \right)=\sqrt{c^2-c_{min}^2},$$

with $c=\left|\overrightarrow{c}\right|$ and $c_{min}$ the smallest absolute component of the three-dimensional correlation vector $\overrightarrow{c}=(c_1,c_2,c_3)$ appearing in the Bloch decomposition of $\rho$. This CJWR-based measure offers a clear geometric picture in terms of the correlation matrix and is especially useful when comparing steering with Bell nonlocality, since the two notions become indistinguishable in the two-measurement scenario [52]. It therefore complements the entropic approach and provides a second independent witness of steerability for the accelerated detector system studied below.

4.3. Bell Nonlocality

Bell nonlocality is the most compelling signature of quantum correlations, as it rules out any description based on local hidden variables. For the electron–proton spin pair in the hydrogen ground state, this feature is probed through the violation of the Clauser–Horne–Shimony–Holt (CHSH) inequality [53,54,55,56,57,58], which serves as the standard test for two-qubit systems. The CHSH correlator takes the form

$$\mathcal{B}\left(\rho \right)=\underset{\{A_i,B_j\}}{max}\left|⟨A_1\otimes B_1⟩+⟨A_1\otimes B_2⟩+⟨A_2\otimes B_1⟩-⟨A_2\otimes B_2⟩\right|,$$

where $A_i={\overrightarrow{a}}_i·\overrightarrow{\sigma }$ and $B_j={\overrightarrow{b}}_j·\overrightarrow{\sigma }$ ($i,j=1,2$) are local dichotomic observables with unit vectors ${\overrightarrow{a}}_i,{\overrightarrow{b}}_j\in {\mathbb{R}}^3$. Any local realistic theory requires $\mathcal{B}\left(\rho \right)⩽2$, while quantum mechanics permits values up to the Tsirelson bound $2\sqrt{2}$. Following the normalization used for the steering quantifier, we define the amount of Bell nonlocality as

$$\mathcal{N}\left(\rho \right):=max\left\{0,{\displaystyle \frac{\mathcal{B}\left(\rho \right)-2}{2\sqrt{2}-2}}\right\}.$$

This measure vanishes for states that satisfy the classical CHSH bound and reaches unity when the maximal quantum violation is attained. For two-qubit states the maximum $\mathcal{B}\left(\rho \right)$ admits a closed-form expression $2\sqrt{m_1+m_2}$ in terms of the two largest eigenvalues $m_1\ge m_2$ of the correlation matrix $U^{T}U$, where $U_{ij}=Tr(\rho {\sigma }_i\otimes {\sigma }_j)$. In the present context, comparing $\mathcal{N}\left(\rho \right)$ directly with the CJWR-based steering $S_2$ reveals how the hyperfine coupling, Zeeman splitting induced by the external magnetic field, and local dephasing noise affect the hierarchy of quantum correlations between the electron and proton spins.

5. Numerical Results and Analysis of Quantum Correlation Quantifiers

In this section we present the numerical results for the time evolution of the three main quantum correlation quantifiers—entanglement of formation $E_F\left(t\right)$, EPR steering parameter $S_{EPR}\left(t\right)$, and maximum CHSH Bell nonlocality $B_{max}\left(t\right)$—under the combined influence of hyperfine coupling, Zeeman splitting, and independent Markovian dephasing. The results are displayed in Figure 1, Figure 2, Figure 3, Figure 4, Figure 5 and Figure 6 as functions of the scaled time t (in units of ${\lambda }^{-1}$), allowing a direct comparison of how the external magnetic field and environmental noise jointly shape the different layers of quantum correlations in the hydrogen hyperfine system.

Figure 1. Entanglement of formation $E_F\left(t\right)$ of the proton–electron hyperfine X-state as a function of scaled time t (in units of ${\lambda }^{-1}$) for initial state parameters $c_1=1$, $c_2=1$, $c_3=-1$, and different proton magnetic parameters $M_p=0$ (red), $0.5\lambda$ (blue), $\lambda$ (green), $2\lambda$ (black), with $\Gamma =0.05\lambda$ and $M_n=0$.

Figure 2. Entanglement of formation $E_F\left(t\right)$ of the proton–electron hyperfine X-state as a function of scaled time t (in units of ${\lambda }^{-1}$) for initial state parameters $c_1=1$, $c_2=1$, $c_3=-1$, and different proton magnetic parameters $M_p=0$ (red), $0.5\lambda$ (blue), $\lambda$ (green), $2\lambda$ (black), with $\Gamma =0.1\lambda$ and $M_n=0$.

Figure 3. EPR steering parameter $S_{\mathrm{EPR}}\left(t\right)$ of the proton–electron hyperfine X-state as a function of scaled time t (in units of ${\lambda }^{-1}$) for initial state parameters $c_1=1$, $c_2=1$, $c_3=-1$, and different proton magnetic parameters $M_p=0$ (red), $0.5\lambda$ (blue), $\lambda$ (green), $2\lambda$ (black), with $\Gamma =0.05\lambda$ and $M_n=0$.

Figure 4. EPR steering parameter $S_{\mathrm{EPR}}\left(t\right)$ of the proton–electron hyperfine X-state as a function of scaled time t (in units of ${\lambda }^{-1}$) for initial state parameters $c_1=1$, $c_2=1$, $c_3=-1$, and different proton magnetic parameters $M_p=0$ (red), $0.5\lambda$ (blue), $\lambda$ (green), $2\lambda$ (black), with $\Gamma =0.1\lambda$ and $M_n=0$.

Figure 5. Maximum CHSH Bell nonlocality $B_{\max }\left(t\right)$ of the proton–electron hyperfine X-state as a function of scaled time t (in units of ${\lambda }^{-1}$) for initial state parameters $c_1=1$, $c_2=1$, $c_3=-1$, and different proton magnetic parameters $M_p=0$ (red), $0.5\lambda$ (blue), $\lambda$ (green), $2\lambda$ (black), with $\Gamma =0.05\lambda$ and $M_n=0$.

Figure 6. Maximum CHSH Bell nonlocality $B_{\max }\left(t\right)$ of the proton–electron hyperfine X-state as a function of scaled time t (in units of ${\lambda }^{-1}$) for initial state parameters $c_1=1$, $c_2=1$, $c_3=-1$, and different proton magnetic parameters $M_p=0$ (red), $0.5\lambda$ (blue), $\lambda$ (green), $2\lambda$ (black), with $\Gamma =0.1\lambda$ and $M_n=0$.

Figure 1 and Figure 2 display the time evolution of the entanglement of formation $E_F\left(t\right)$ of the proton–electron hyperfine X-state (initial parameters $c_1=c_2=1$, $c_3=-1$) as a function of scaled time t (in units of ${\lambda }^{-1}$) under independent Markovian dephasing with $M_n=0$. Figure 1 corresponds to a weak dephasing rate $\Gamma =0.05\lambda$, while Figure 2 shows the dynamics at the doubled rate $\Gamma =0.1\lambda$; in both panels the four curves correspond to different values of the effective proton magnetic parameter $M_p=0$ (red), $0.5\lambda$ (blue), $\lambda$ (green), and $2\lambda$ (black). Starting from its maximum value of unity (corresponding to a maximally entangled Bell-like initial state), the entanglement exhibits pronounced damped oscillations in both cases, whose frequency and decay envelope are strongly modulated by $M_p$. This behavior arises from the coherent competition between the hyperfine coupling $\lambda$ (which mixes the single-excitation subspace) and the Zeeman splitting ${\Omega }_{\pm }=B({\gamma }_e\pm {\gamma }_p)$ induced by the external magnetic field, as explicitly diagonalized in Appendix A, while the local dephasing suppresses off-diagonal coherences (at combined rate $\kappa ={\kappa }_e+{\kappa }_p$) without affecting the invariant aligned-spin populations. Compared with the slower decay in Figure 1, the stronger noise intensity in Figure 2 visibly accelerates the suppression of quantum correlations, leading to a faster approach to the incoherent stationary state. Together, the two figures illustrate how the external magnetic field provides a tunable control parameter over the robustness of hyperfine entanglement against environmental noise, highlighting the sensitivity of bipartite entanglement to both dephasing strength and Zeeman tuning.

Figure 3 and Figure 4 show the dynamical behavior of the EPR steering parameter $S_{EPR}\left(t\right)$ for the proton–electron hyperfine X-state (initial parameters $c_1=c_2=1$, $c_3=-1$) versus scaled time t (in units of ${\lambda }^{-1}$), again with the four curves representing proton magnetic parameters $M_p=0$ (red), $0.5\lambda$ (blue), $\lambda$ (green), and $2\lambda$ (black) at $M_n=0$. Figure 3 corresponds to the weaker dephasing rate $\Gamma =0.05\lambda$, while Figure 4 uses the stronger rate $\Gamma =0.1\lambda$. In both panels the steering starts close to its upper bound of 1 and displays clear damped oscillations whose frequency and damping profile are clearly shaped by the value of $M_p$. Physically, these oscillations arise from the coherent interplay of the hyperfine spin-exchange interaction $\lambda$ (which couples the single-excitation states) and the Zeeman energy shifts induced by the external magnetic field, while the independent dephasing channels damp the off-diagonal elements at the combined rate $\kappa ={\kappa }_e+{\kappa }_p$ without altering the fixed populations in the aligned-spin subspace. Notably, the steering signal persists significantly longer than the entanglement of formation seen in Figure 1 and Figure 2—especially under the milder noise of Figure 3—demonstrating the well-known hierarchy in which steering is more resilient to decoherence than full bipartite entanglement. This robustness carries important physical meaning: even after entanglement has largely decayed, one party can still remotely “steer” the statistics of the other’s measurements, a resource directly relevant for one-sided quantum tasks such as secure quantum communication or one-way quantum computation in noisy atomic platforms. Overall, the pair of figures highlights how the external magnetic field serves as a practical tuning mechanism that can be used to prolong steerability in realistic hydrogen-atom environments, reinforcing the idea that different layers of quantum correlations respond differently to the same noise and magnetic conditions.

Figure 5 and Figure 6 present the time evolution of the maximum CHSH Bell nonlocality $B_{max}\left(t\right)$ for the proton–electron hyperfine X-state with the same initial parameters ($c_1=c_2=1$, $c_3=-1$) as a function of scaled time t (in units of ${\lambda }^{-1}$). The four curves in each figure again correspond to proton magnetic parameters $M_p=0$ (red), $0.5\lambda$ (blue), $\lambda$ (green), and $2\lambda$ (black) at $M_n=0$, with Figure 5 using $\Gamma =0.05\lambda$ and Figure 6 using the stronger dephasing $\Gamma =0.1\lambda$. The Bell nonlocality starts near its quantum upper limit of roughly 2.828 and shows damped oscillatory decay, with the oscillation frequency and lifetime clearly influenced by the choice of $M_p$. These oscillations reflect the coherent dynamics driven by the hyperfine interaction $\lambda$ mixing the relevant spin states and the magnetic-field-induced Zeeman splittings, while independent dephasing progressively damps the coherences responsible for the CHSH violation. Interestingly, Bell nonlocality fades away more rapidly than both entanglement and steering seen in the earlier figures—a clear demonstration of the standard hierarchy of quantum correlations, where nonlocality is the most fragile against noise. From a physical perspective, this quantity is crucial because it certifies genuine quantum behavior incompatible with any local hidden-variable model, which is essential for device-independent quantum cryptography and fundamental tests of quantum mechanics in atomic systems. The results nicely show that even though the external magnetic field can be used to adjust the persistence of Bell nonlocality, the effect remains quite sensitive to dephasing, completing the picture of how different layers of quantum correlations behave under realistic conditions in the hydrogen hyperfine system.

Figures corresponding to the time evolution of entanglement of formation, quantum steering, and Bell nonlocality exhibit a damped oscillatory behavior for different values of the external magnetic field. As expected, the magnetic field modifies the effective Zeeman splitting, which in turn affects the oscillation frequency of the quantum correlations. However, the full dissipative dynamics resulting from the interplay between hyperfine coupling and Markovian dephasing is more intricate than a simple frequency shift.

Although the role of the magnetic field may qualitatively be associated with this Zeeman-induced modification, its impact on the decay profiles is not straightforward in the presence of environmental noise. Our numerical analysis therefore provides quantitative insight into how the magnetic field simultaneously reshapes the oscillation frequency, decay rates, and revival structures of the three correlation measures. In particular, it allows us to identify distinct robustness regimes for entanglement, steering, and Bell nonlocality, which cannot be inferred from qualitative arguments alone.

Moreover, a clear hierarchy is observed across all parameter regimes, where entanglement of formation decays most rapidly, followed by quantum steering, while Bell nonlocality is the most fragile under local dephasing. This behavior becomes increasingly evident as the magnetic field strength is varied, highlighting a nontrivial interplay between coherent control and environmental decoherence.

6. Conclusions

In conclusion, we have presented a comprehensive analysis of the dynamics of quantum correlations in the ground-state hyperfine structure of the hydrogen atom subjected to an external magnetic field and independent Markovian dephasing. By deriving the exact analytical solution of the Lindblad master equation for an X-shaped initial state, we have characterized the time evolution of three fundamental quantifiers of nonclassical correlations—entanglement of formation, quantum steering via the CJWR inequality, and Bell nonlocality via the normalized CHSH violation—across different regimes of the proton magnetic parameter and dephasing strengths. The results reveal a robust damped oscillatory behavior in all three measures, where both the oscillation frequency and the decay envelope are effectively controlled by the Zeeman interaction. A clear hierarchy is consistently observed: entanglement of formation decays most rapidly, while quantum steering and Bell nonlocality exhibit progressively higher resilience against local dephasing noise. This hierarchy highlights the differing robustness of quantum resources under environmental decoherence in a minimal two-qubit atomic system. Beyond these specific findings, our work provides a unified open-system framework that combines hyperfine coupling, magnetic-field-induced level splitting, and Markovian dissipation within a single analytically solvable model. These results emphasize the role of the external magnetic field as a practical control parameter for tailoring the lifetime of quantum correlations, with potential relevance to atomic spin control and spin-based quantum technologies. Finally, extensions of this study to correlated or non-Markovian environments, finite-temperature effects, and geometric or topological aspects of quantum evolution may provide further insight into the protection and control of quantum resources in realistic settings.

Appendix A.1. Matrix Representation

For a field $\mathbf{B}=B\widehat{z}$, the total Hamiltonian introduced in Section 2 reduces to

$$\widehat{H}=\lambda ({\mathsf{\tau }}_e·{\mathsf{\tau }}_p)+{\gamma }_{e}B{\tau }_e^z-{\gamma }_{p}B{\tau }_p^z.$$

(A1)

In the product basis ${\left\{\right|\uparrow ⟩}_e{|\uparrow ⟩}_p{,|\uparrow ⟩}_e{|\downarrow ⟩}_p{,|\downarrow ⟩}_e{|\uparrow ⟩}_p{,|\downarrow ⟩}_e{|\downarrow ⟩}_p\},$ the Hamiltonian takes a block-diagonal form.

The Zeeman contribution is diagonal and can be written as

$${\gamma }_{e}B{\tau }_e^z-{\gamma }_{p}B{\tau }_p^z=\mathrm{diag}({\Omega }_{-},{\Omega }_{+},-{\Omega }_{+},-{\Omega }_{-}),$$

(A2)

where we recall the definitions ${\Omega }_{\pm }=B({\gamma }_e\pm {\gamma }_p).$

The hyperfine interaction is represented by

$${\mathsf{\tau }}_e·{\mathsf{\tau }}_p=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& -1& 2& 0\\ 0& 2& -1& 0\\ 0& 0& 0& 1\end{array}\right).$$

(A3)

Combining both contributions, the total Hamiltonian reads

$$\widehat{H}=\left(\begin{array}{cccc}\lambda +{\Omega }_{-}& 0& 0& 0\\ 0& -\lambda +{\Omega }_{+}& 2\lambda & 0\\ 0& 2\lambda & -\lambda -{\Omega }_{+}& 0\\ 0& 0& 0& \lambda -{\Omega }_{-}\end{array}\right).$$

(A4)

Appendix A.2. Eigenvalues and Eigenstates

The states with parallel spin alignment, ${|\uparrow ⟩}_e{|\uparrow ⟩}_p$ and ${|\downarrow ⟩}_e{|\downarrow ⟩}_p,$ are already eigenstates of $\widehat{H}$, with eigenvalues

$$\begin{array}{cc}\hfill E_1& =\lambda +{\Omega }_{-},\hfill \end{array}$$

(A5)

$$\begin{array}{cc}\hfill E_2& =\lambda -{\Omega }_{-}.\hfill \end{array}$$

(A6)

The remaining subspace spanned by ${\left\{\right|\uparrow ⟩}_e{|\downarrow ⟩}_p{,|\downarrow ⟩}_e{|\uparrow ⟩}_p\}$ is described by the effective $2\times 2$ block

$$\left(\begin{array}{cc}-\lambda +{\Omega }_{+}& 2\lambda \\ 2\lambda & -\lambda -{\Omega }_{+}\end{array}\right).$$

(A7)

Its eigenvalues are obtained as

$$E_{\pm }=-\lambda \pm \sqrt{4{\lambda }^2+{\Omega }_{+}^2}\equiv -\lambda \pm \Delta ,$$

(A8)

where we introduced the generalized splitting $\Delta =\sqrt{4{\lambda }^2+{\Omega }_{+}^2}.$

The corresponding normalized eigenstates can be written as

$$|{\Psi }_{+}⟩=cos\theta |{\uparrow ⟩}_e{|\downarrow ⟩}_p+sin\theta |{\downarrow ⟩}_e{|\uparrow ⟩}_p,$$

(A9)

$$|{\Psi }_{-}⟩=sin\theta |{\uparrow ⟩}_e{|\downarrow ⟩}_p-cos\theta |{\downarrow ⟩}_e{|\uparrow ⟩}_p,$$

(A10)

where the mixing angle $\theta$ is defined through

$$tan\left(2\theta \right)={\displaystyle \frac{2\lambda }{{\Omega }_{+}}}.$$

(A11)

Equivalently, the coefficient ratio satisfies

$${\displaystyle \frac{tan\theta }{1}}={\displaystyle \frac{\Delta -{\Omega }_{+}}{2\lambda }}.$$

(A12)

Appendix A.3. Limiting Regimes

In the absence of an external magnetic field ($B=0$), one has ${\Omega }_{+}=0$, and the eigenstates reduce to the well-known symmetric and antisymmetric combinations

$$|{\Psi }_{+}⟩ ={\displaystyle \frac{1}{\sqrt{2}}}\left({|\uparrow ⟩}_e{|\downarrow ⟩}_p+{|\downarrow ⟩}_e{|\uparrow ⟩}_p\right),$$

(A13)

$$|{\Psi }_{-}⟩ ={\displaystyle \frac{1}{\sqrt{2}}}\left({|\uparrow ⟩}_e{|\downarrow ⟩}_p-{|\downarrow ⟩}_e{|\uparrow ⟩}_p\right),$$

(A14)

with energies $E_{\pm }=-\lambda \pm 2\lambda$, corresponding to the triplet and singlet manifolds [34,59].

In the opposite regime of strong magnetic field, $|{\Omega }_{+}|\gg \lambda$, the mixing angle vanishes, and the eigenstates approach the uncoupled basis states,

$$|{\Psi }_{+}{⟩\approx |\uparrow ⟩}_e{|\downarrow ⟩}_p,|{\Psi }_{-}{⟩\approx |\downarrow ⟩}_e{|\uparrow ⟩}_p,$$

(A15)

with corrections of order $\lambda /{\Omega }_{+}$. This reflects the suppression of hyperfine-induced mixing by the dominant Zeeman splitting.

The above diagonalization provides a complete description of the hyperfine–Zeeman spectrum of hydrogen for an arbitrary magnetic field aligned along the z-axis. The competition between $\lambda$ and ${\Omega }_{+}$ governs the transition from maximally entangled eigenstates at weak field to nearly separable states in the strong-field regime, a feature that plays a central role in the dynamical behavior discussed in the main text.

Appendix B.1. Block Structure of the Dynamics

Due to the symmetry of the Hamiltonian and the local dephasing operators, the dynamics naturally decomposes into two independent subspaces:

(i)

The aligned-spin subspace ${\left\{\right|\uparrow ⟩}_e{|\uparrow ⟩}_p{,|\downarrow ⟩}_e{|\downarrow ⟩}_p\}$,

(ii)

The single-excitation subspace ${\left\{\right|\uparrow ⟩}_e{|\downarrow ⟩}_p{,|\downarrow ⟩}_e{|\uparrow ⟩}_p\}$.

This separation allows for a simplified analytical treatment.

Appendix B.2. Aligned-Spin Subspace

The populations of the fully aligned states are constants of motion:

$${\varrho }_{11}\left(t\right)={\varrho }_{11}\left(0\right)={\displaystyle \frac{1}{4}}(1+c_3),{\varrho }_{44}\left(t\right)={\varrho }_{44}\left(0\right)={\displaystyle \frac{1}{4}}(1+c_3).$$

(A16)

The corresponding coherence evolves independently as

$${\displaystyle \frac{d}{dt}}{\varrho }_{14}\left(t\right)=-2\left(i{\Omega }_{-}+\kappa \right){\varrho }_{14}\left(t\right),$$

(A17)

with solution

$${\varrho }_{14}\left(t\right)={\displaystyle \frac{1}{4}}(c_1-c_2)e^{-2\kappa t}{e}^{-2i{\Omega }_{-}t},{\varrho }_{41}\left(t\right)={\varrho }_{14}^{*}\left(t\right).$$

(A18)

All coherences connecting this subspace to the rest of the Hilbert space remain identically zero due to the initial X-structure and the symmetry of the dynamics.

Appendix B.3. Single-Excitation Subspace

The nontrivial dynamics is confined to the subspace spanned by ${\varrho }_{22}\left(t\right)$, ${\varrho }_{33}\left(t\right)$, and ${\varrho }_{23}\left(t\right)$.

We define

$$S\left(t\right)={\varrho }_{22}\left(t\right)+{\varrho }_{33}\left(t\right),\Delta \left(t\right)={\varrho }_{22}\left(t\right)-{\varrho }_{33}\left(t\right),$$

(A19)

and decompose the coherence as

$${\varrho }_{23}\left(t\right)=X\left(t\right)+iY\left(t\right).$$

(A20)

From the evolution equations, the total population is conserved:

$$S\left(t\right)=S\left(0\right)={\displaystyle \frac{1}{2}}(1-c_3).$$

(A21)

The remaining variables satisfy the closed linear system

$$\left\{\begin{array}{c}\dot{X}=-2\kappa X+2{\Omega }_{+}Y,\hfill \\ \dot{Y}=-2\kappa Y-2{\Omega }_{+}X+2\lambda \Delta ,\hfill \\ \dot{\Delta }=-8\lambda Y,\hfill \end{array}\right.$$

(A22)

with initial conditions

$$X\left(0\right)={\displaystyle \frac{1}{4}}(c_1+c_2),Y\left(0\right)=0,\Delta \left(0\right)=0.$$

(A23)

This system can be written compactly as

$$\dot{\mathbf{V}}\left(t\right)=\mathbb{M}\mathbf{V}\left(t\right),\mathbf{V}\left(t\right)={(X,Y,\Delta )}^T,$$

(A24)

where

$$\mathbb{M}=\left(\begin{array}{ccc}-2\kappa & 2{\Omega }_{+}& 0\\ -2{\Omega }_{+}& -2\kappa & 2\lambda \\ 0& -8\lambda & 0\end{array}\right).$$

(A25)

The solution is obtained by diagonalizing $\mathbb{M}$. Its characteristic polynomial reads

$$\chi \left(\mu \right)={\mu }^3+4\kappa {\mu }^2+\left(4{\kappa }^2+4{\Omega }_{+}^2+16{\lambda }^2\right)\mu +32\kappa {\lambda }^2.$$

(A26)

Denoting the three eigenvalues by ${\mu }_k$ ($k=1,2,3$), the general solution takes the form

$$\mathbf{V}\left(t\right)={\displaystyle \sum _{k=1}^3}{d}_k{\mathbf{w}}_k{e}^{{\mu }_{k}t},$$

(A27)

where ${\mathbf{w}}_k$ are the corresponding eigenvectors and the coefficients $d_k$ are fixed by the initial condition.

Once $(X,Y,\Delta )$ are determined, the density-matrix elements follow as

$${\varrho }_{22}\left(t\right)={\displaystyle \frac{S\left(t\right)+\Delta \left(t\right)}{2}},$$

(A28)

$${\varrho }_{33}\left(t\right)={\displaystyle \frac{S\left(t\right)-\Delta \left(t\right)}{2}},$$

(A29)

$${\varrho }_{23}\left(t\right)=X\left(t\right)+iY\left(t\right),$$

(A30)

$${\varrho }_{32}\left(t\right)=X\left(t\right)-iY\left(t\right).$$

(A31)

Appendix B.4. Remaining Components

All other matrix elements vanish identically for all times:

$${\varrho }_{ij}\left(t\right)=0,(i,j)\notin \{(1,1),(4,4),(1,4),(4,1),(2,2),(3,3),(2,3),(3,2)\}.$$

(A32)

The analytical solution highlights a clear separation of dynamical roles. The aligned-spin subspace forms a decoherence-resistant subspace at the level of populations, while its coherence decays exponentially with rate $2\kappa$ and accumulates a Zeeman phase determined by ${\Omega }_{-}$. In contrast, the single-excitation subspace exhibits nontrivial coupled dynamics driven by the competition between hyperfine interaction ($\lambda$) and Zeeman splitting (${\Omega }_{+}$), with dephasing introducing an overall damping. The interplay of these mechanisms leads to damped oscillations of both population imbalance and coherence. In the long-time limit, all coherences vanish, and the system approaches an incoherent stationary state determined solely by the conserved populations, providing a natural framework for analyzing the decay of quantum correlations in the hyperfine–Zeeman system.