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Robust control and preservation of quantum steering, nonlocality, and coherence in open atomic systems

  • Kamal Berrada EMAIL logo and Abdelaziz Sabik
Published/Copyright: July 16, 2025
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Open Physics
From the journal Open Physics Volume 23 Issue 1

Abstract

We investigate the robustness of Einstein–Podolsky–Rosen (EPR) steering, nonlocality, and quantum coherence in a bipartite atomic system coupled to cavity fields under the influence of decoherence. The system consists of two non-interacting atoms, where each atom is confined within a cavity that interacts with another cavity field, which plays a crucial role in governing the dynamical evolution of two atoms. Through a combination of analytical and numerical investigations, we demonstrate that quantum steering, Bell nonlocality, and coherence can be not only preserved but also enhanced by appropriately tuning the cavity–cavity interaction strength, effectively mitigating environmental decoherence and extending the coherence lifetime of the system. Our results reveal that, under optimal conditions, steering, nonlocality, and coherence remain resilient against decoherence over extended timescales. These findings offer valuable insights into the controlled manipulation of quantum resources in open quantum systems and have significant implications for quantum information processing and secure communication technologies.

Keywords: open quantum systems; cavity–cavity interaction; photon loss; quantumness measures

1 Introduction

The concept of quantum steering was first introduced by Schrödinger in 1936 as a response to the Einstein–Podolsky–Rosen (EPR) paradox [1,2]. Decades later, Wiseman et al. established the fundamental connection between EPR steering, quantum entanglement, and Bell nonlocality, positioning steering as an intermediate quantum correlation [3]. Unlike entanglement, EPR steering exhibits an intrinsic asymmetry, allowing one party (Alice) to remotely influence the quantum state of another party (Bob), even when Bob lacks trust in Alice’s measurement apparatus [4–6]. Quantum steering is recognized as a pivotal quantum resource with diverse applications, including randomness certification [7], randomness generation [8], asymmetric quantum networks [9], subchannel discrimination [10], and quantum key distribution (QKD) [11]. Enhancing the robustness and practical utilization of EPR steering is crucial for quantum information transmission and foundational aspects of quantum communication protocols. Recent advances have focused on relaxing the no-signaling condition to optimize the efficiency of EPR steering resources [12]. Notably, it has been demonstrated that quantum steering can be sequentially distributed among multiple observers, either through standard projective measurements [13] or unsharp measurements [12]. This concept of steering sharing has been extensively studied in bipartite systems [14] and further extended to the realm of genuine multipartite steering reuse [15], shedding light on new possibilities for multi-user quantum networks and resource allocation in quantum technologies.

The phenomenon of quantum coherence, rooted in the superposition principle of quantum states, plays a fundamental role in quantum theory and technological applications. It serves as a key resource across various fields, including quantum information processing [16], quantum optics [17], solid-state physics [18], and even biological systems [19]. Over the years, substantial research efforts have been dedicated to developing a rigorous theoretical framework for quantum coherence as a physical resource [20,21], along with the establishment of quantitative measures for its characterization. A major breakthrough in this area was made by Baumgratz et al., who introduced a formal resource-theoretic framework for quantifying coherence [22]. This framework defines coherence through well-established measures such as the l 1 norm of coherence and the relative entropy of coherence, each with distinct physical interpretations. In multipath quantum interference experiments, as demonstrated in previous studies [23,24], the l 1 norm effectively quantifies coherence by capturing the wave-like nature of a quanton, making it an experimentally observable metric. On the contrary, in the asymptotic regime where the number of quantum copies approaches infinity [25], the relative entropy of coherence provides an optimal measure for the distillation of maximally coherent states via incoherent operations. In recent years, the resource theory of coherence has gained significant attention, with its applications extending beyond fundamental studies to practical implementations in various quantum technologies, including quantum metrology, quantum thermodynamics, and quantum computing [26,27]. These advancements underscore the pivotal role of coherence as a versatile quantum resource, further motivating the exploration of strategies for its protection and manipulation in open quantum systems.

In realistic quantum systems, decoherence and noise induced by interactions with the external environment pose fundamental challenges to the preservation of quantum resources. To mitigate these effects, various strategies have been proposed to protect and enhance quantum correlations and coherence [28–32]. Notably, extensive studies have demonstrated that non-Markovian environments, characterized by memory effects, can effectively preserve quantum coherence and correlations by enabling partial information backflow into the system [33–35]. Despite substantial advancements in improving the efficiency of EPR steering and coherence, an open question remains: How can these quantum resources be simultaneously protected against decoherence while being dynamically controlled? Given their crucial role in quantum technologies, understanding the interplay between quantum steering, nonlocality, and coherence is essential for optimizing quantum information processing and communication protocols. In this work, we investigate the preservation and manipulation of EPR steering, Bell nonlocality, and quantum coherence in a bipartite atomic system, where each atom is confined within a cavity that interacts with another cavity field. By systematically analyzing the system’s quantum dynamics, we demonstrate that these quantum resources can be sustained and even enhanced through strategic tuning of inter-cavity coupling strengths, enabling robust control over quantum correlations despite environmental decoherence. By comparing the time evolution of EPR steering, Bell nonlocality, and quantum coherence, we show that, in the ideal cavity limit, high levels of quantumness measures persist throughout the system’s evolution. Furthermore, we establish that an optimal selection of quantum model parameters allows for the long-term protection of quantum steering, nonlocality, and coherence, effectively mitigating decoherence effects.

The structure of this article is as follows. Section 2 presents the Hamiltonian formulation of the quantum system and describes its dynamical evolution, along with a concise review of quantum steering and coherence. Additionally, it provides a detailed numerical analysis, offering a comprehensive discussion of the obtained results. Finally, Section 3 summarizes the key findings and outlines potential directions for future investigations.

2 Model and quantum resources

In the quantum regime, physical systems must be treated as open systems due to their inevitable interactions with the surrounding environment. In this work, we consider a quantum model consisting of a single atom confined in a cavity, which is coupled to another cavity. The total Hamiltonian governing the atom–cavity system is given by

(1) H T = H A + H F + H A − F + H C − C .

The atomic Hamiltonian is expressed as

(2) H A = ℏ ω 0 2 σ z ,

where ω 0 denotes the atomic transition frequency, and σ z is the Pauli matrix. The Hamiltonian describing the cavity fields is

(3) H F = ℏ ∑ i = 1,2 ω i c i † c i ,

where ω = ω 1 = ω 2 represents the frequency of the cavity modes, and c i ( c i † ) are the annihilation (creation) operators. The interaction Hamiltonians governing the atom-field coupling ( H A − F ) and the inter-cavity coupling ( H C − C ) are given by

(4) H A − F = ℏ α ( c 1 σ + + c 1 † σ − ) ,

(5) H C − C = ℏ β ( c 1 c 2 † + c 1 † c 2 ) ,

where α and β represent the coupling strengths of the atom-field and cavity–cavity interactions, respectively. The operators σ − and σ + denote the atomic lowering and raising operators. The Hamiltonian (1) considered in this work describes an atom trapped within a cavity that is linked to a second cavity, a configuration that holds significant importance in the field of cavity quantum electrodynamics (QED). This model effectively encapsulates the fundamental dynamics of open quantum systems, where interactions with the environment play a crucial role. By introducing coupling between the two cavities, this Hamiltonian goes beyond the traditional Jaynes–Cummings model, enabling the exploration of quantum correlations across multiple parties, specifically EPR steering and Bell nonlocality, within a system of two atoms. The model features adjustable parameters, including the strength of the atom’s interaction with the field ( α ) and the strength of the connection between the cavities ( β ). These parameters provide a means to manipulate the quantum properties of the system, which is particularly useful for examining how these properties can be maintained in the presence of decoherence [28,36]. One of the strengths of this model is its ability to realistically represent physical systems while remaining mathematically controllable.

Considering the dissipative effects in both cavities, the time evolution of the density operator ρ , which describes the combined system of the atom and the cavities, is governed by the following master equation:

(6) ∂ ρ ∂ t = i ℏ [ ρ , H T ] − ∑ i = 1 2 μ i 2 [ c i † c i ρ + ρ c i † c i − 2 c i ρ c i † ] ,

where μ 1 and μ 2 denote the photon decay rates of the respective cavities. The distinction between non-Markovian and Markovian dynamics is determined by conditions α > μ 1 ∕ 4 and α ≤ μ 1 ∕ 4 [28,37,38], respectively.

The atomic density operator evolves as

(7) ρ A ( t ) = ρ A 00 ( 0 ) ∣ κ ( t ) ∣ 2 ρ A 01 ( 0 ) κ ( t ) ρ A 10 ( 0 ) κ * ( t ) 1 − ρ A 11 ( 0 ) ∣ κ ( t ) ∣ 2 .

The function κ ( t ) , which encapsulates the system’s temporal evolution, is given by [28]

(8) κ ( t ) = L − 1 F ( s ) G ( s ) ,

where

(9) F ( s ) = − 4 β 2 − ( 2 s + 2 i ω + μ 1 ) ( 2 s + 2 i ω + μ 2 ) , G ( s ) = 2 α 2 ( 2 s + 2 i ω + μ 2 ) + ( s + i ( ω + δ ) ) [ 4 ( β 2 + ( s + i ω ) 2 ) + 2 μ 2 ( s + i ω ) + μ 1 ( 2 s + 2 i ω + μ 2 ) ] .

Here, L − 1 represents the inverse Laplace transform and δ = ω 0 − ω represents the atom-field detuning.

It is well-established that for non-interacting subsystems, the full dynamics of a two-qubit system can be determined from the evolution of each individual qubit coupled to its respective environment [37]. Using the dynamics of a single qubit, we can derive the time-evolved density matrix for the two-atom system, with its elements expressed as follows:

(10) ρ A − A 11 ( t ) = ρ A − A 11 ( 0 ) ∣ κ ( t ) ∣ 4 , ρ A − A 22 ( t ) = ρ A − A 11 ( 0 ) ∣ κ ( t ) ∣ 2 ( 1 − ∣ κ ( t ) ∣ 2 ) + ρ A − A 22 ( 0 ) ∣ κ ( t ) ∣ 2 , ρ A − A 33 ( t ) = ρ A − A 11 ( 0 ) ∣ κ ( t ) ∣ 2 ( 1 − ∣ κ ( t ) ∣ 2 ) + ρ A − A 33 ( 0 ) ∣ κ ( t ) ∣ 2 , ρ A − A 44 ( t ) = 1 − ( ρ A − A 11 ( t ) + ρ A − A 22 ( t ) + ρ A − A 33 ( t ) ) .

The non-diagonal density matrix elements evolve as

(11) ρ A − A 12 ( t ) = ρ A − A 12 ( 0 ) κ ( t ) ∣ κ ( t ) ∣ 2 , ρ A − A 13 ( t ) = ρ A − A 13 ( 0 ) κ ( t ) ∣ κ ( t ) ∣ 2 , ρ A − A 14 ( t ) = ρ A − A 14 ( 0 ) κ ( t ) 2 , ρ A − A 23 ( t ) = ρ A − A 23 ( 0 ) ∣ κ ( t ) ∣ 2 , ρ A − A 24 ( t ) = ρ A − A 13 ( 0 ) κ ( t ) ( 1 − ∣ κ ( t ) ∣ 2 ) + ρ A − A 24 ( 0 ) κ ( t ) , ρ A − A 34 ( t ) = ρ A − A 12 ( 0 ) κ ( t ) ( 1 − ∣ κ ( t ) ∣ 2 ) + ρ A − A 34 ( 0 ) κ ( t ) .

The condition ρ A − A i j ( t ) = ρ A − A j i ( t ) ensures the Hermiticity of the density matrix. Using equations (10) and (11), the two-atom dynamics for any given initial state can be fully determined. This allows for the investigation of quantum resources, such as quantum steering, nonlocality, and coherence, as a function of the model parameters.

We take into consideration the EPR steering inequality [39–41] in order to study the dynamics of quantum steering for the two-atom state in the present model. If the steering inequality is violated, the atom state is steerable. For a quantum state in X -form with the Bloch decomposition in terms of vectors r → = ( 0, 0 , r ) and s → = ( 0, 0 , s ) , the quantum steering inequality for the two atoms via performing the Pauli measurements is formulated as

(12) ∑ l = 1,2 [ ( 1 − b l ) log ( 1 − b l ) + ( 1 + b l ) log ( 1 + b l ) ] − [ ( 1 + r ) log ( 1 + r ) + ( 1 − r ) log ( 1 − r ) ] + 1 2 ( 1 + b 3 − r − s ) log ( 1 + b 3 − r − s ) + 1 2 ( 1 + b 3 + r + s ) log ( 1 + b 3 + r + s ) + 1 2 ( 1 − b 3 + r − s ) log ( 1 − b 3 + r − s ) + 1 2 ( 1 − b 3 − r + s ) log ( 1 − b 3 − r + s ) ≤ 2 ,

where

(13) b 1 = 2 ( ρ A − A 23 ρ A − A 14 ) b 2 = 2 ( ρ A − A 23 − ρ A − A 14 ) b 3 = ρ A − A 11 − ρ A − A 22 − ρ A − A 33 + ρ A − A 44 r = ρ A − A 11 + ρ A − A 22 − ρ A − A 33 − ρ A − A 44 s = ρ A − A 11 − ρ A − A 22 + ρ A − A 33 − ρ A − A 44 .

When the inequality is violated, the two-atom steering is obtained.

Bell nonlocality is a fundamental manifestation of quantum mechanics, providing a means to test quantum correlations that cannot be reconciled with classical explanations. The Bell–Clauser–Horne–Shimony–Holt (BCHSH) inequality serves as a standard criterion for quantifying nonlocality in quantum systems. A violation of this inequality signifies the presence of nonlocal correlations and can be formulated as follows:

(14) B l = 2 max { B l1 , B l2 } ,

where the quantities B l1 and B l2 are given by

(15) B l1 = b 1 + b 2 , B l2 = b 1 + b 3 .

Here, b 1 , b 2 , and b 3 are parameters derived from the elements of the system’s density matrix, defined as

(16) b 1 = 4 ( ∣ ρ A − A 14 ∣ + ∣ ρ A − A 23 ∣ ) 2 , b 2 = 4 ( ∣ ρ A − A 14 ∣ − ∣ ρ A − A 23 ∣ ) 2 , b 3 = ( ρ A − A 11 − ρ A − A 22 − ρ A − A 33 + ρ A − A 44 ) 2 .

These expressions highlight the pivotal role of off-diagonal density matrix elements in governing the degree of nonlocality exhibited by the system. Notably, the relationship b 1 ≥ b 2 ensures that B l accurately captures the maximal violation of the BCHSH inequality. This formalism is particularly relevant to two-qubit X states, where the density matrix structure simplifies the analysis, offering deeper insights into the emergence and characterization of nonlocal quantum correlations.

The dependence of two measures on the parameters β and δ during the dynamics is shown in Figures 1, 2, 3, 4. For a generic case, the time evolution of quantum resources is affected by the coupling strength and the detuning parameter. As shown in the figures, we have displayed the measures of steering and Bell nonlocality as a function of the time μ 1 t for various β values. We find that the quantum measures decreases, from a maximally steerable state with Bell nonlocality with Tsirelson’s bound, as the time increases. In the presence cavity loss μ 2 ≠ 0 , the increase in the parameter β leads to delay the measures loss during the dynamics and the atoms state will be steerable with Bell nonlocality for long periods of time. This indicates that the increase in the interaction between the cavities will enhance the quantum steerability and Bell nonlocality. In the absence of cavity loss with μ 2 ≠ 0 (perfect cavity), we can observe the trapping phenomenon of quantum measures and the atoms state is always steerable with Bell nonlocality only in the large values of parameter β . In this context, the physical parameters act in a similar way on the quantum measures and when the Bell nonlocality’s is satisfied, the quantum seteering inequality is violated for the atoms state. On the contrary, the presence of detuning effects can enhance the quantum measures and the atoms state can be steerable with Bell nonlocality more periods of time. From the obtained results, the EPR steering and nonlocality for the atoms state can be controlled during the time evolution through a proper choice of β and δ values.

Figure 1 Dynamics of the Q S {Q}_{S} (EPR steering measure) for the two atoms, governed by the EPR steering inequality defined in equation (12), plotted as a function of time μ 1 t {\mu }_{1}t for various β \beta values with δ = 0 \delta =0 when the atoms start from a Bell state ρ A − A ( 0 ) {\rho }_{A-A}\left(0) = 1 ∕ 2 ( ∣ 01 ⟩ ⟨ 01 ∣ 1/2(| 01\rangle \langle 01| + ∣ 01 ⟩ ⟨ 10 ∣ | 01\rangle \langle 10| + ∣ 10 ⟩ ⟨ 01 ∣ + ∣ 10 ⟩ ⟨ 01 ∣ ) | 10\rangle \langle 01| +| 10\rangle \langle 01| ) . Labels (a), (b), (c), and (d) are for α = 0.24 ∕ 0.5 μ 2 = 0.24 μ 1 \alpha =0.24/0.5{\mu }_{2}=0.24{\mu }_{1} , α = 0.4 ∕ 0.5 μ 2 = 0.4 μ 1 \alpha =0.4/0.5{\mu }_{2}=0.4{\mu }_{1} , α = 0.24 μ 1 \alpha =0.24{\mu }_{1} with μ 2 = 0 {\mu }_{2}=0 , and α = 0.4 μ 1 \alpha =0.4{\mu }_{1} with μ 2 = 0 {\mu }_{2}=0 , respectively. For the case of (a) and (b): black line, green line, blue dotted line, blue dot-dashed line, and red dashed line are for β = 2 μ 1 \beta =2{\mu }_{1} , β = 1.5 μ 1 \beta =1.5{\mu }_{1} , β = 1 μ 1 \beta =1{\mu }_{1} , β = 0.5 μ 1 \beta =0.5{\mu }_{1} , and β = 0 \beta =0 , respectively. For the case of (c) and (d): black line, green line, blue dotted line, blue dot-dashed line, and red dashed line are for β = 1.5 μ 1 \beta =1.5{\mu }_{1} , β = 0.5 μ 1 \beta =0.5{\mu }_{1} , β = 0.3 μ 1 \beta =0.3{\mu }_{1} , β = 0.2 μ 1 \beta =0.2{\mu }_{1} , and β = 0.1 μ 1 \beta =0.1{\mu }_{1} , respectively.
Figure 1

Dynamics of the Q S (EPR steering measure) for the two atoms, governed by the EPR steering inequality defined in equation (12), plotted as a function of time μ 1 t for various β values with δ = 0 when the atoms start from a Bell state ρ A − A ( 0 ) = 1 ∕ 2 ( ∣ 01 ⟩ ⟨ 01 ∣ + ∣ 01 ⟩ ⟨ 10 ∣ + ∣ 10 ⟩ ⟨ 01 ∣ + ∣ 10 ⟩ ⟨ 01 ∣ ) . Labels (a), (b), (c), and (d) are for α = 0.24 ∕ 0.5 μ 2 = 0.24 μ 1 , α = 0.4 ∕ 0.5 μ 2 = 0.4 μ 1 , α = 0.24 μ 1 with μ 2 = 0 , and α = 0.4 μ 1 with μ 2 = 0 , respectively. For the case of (a) and (b): black line, green line, blue dotted line, blue dot-dashed line, and red dashed line are for β = 2 μ 1 , β = 1.5 μ 1 , β = 1 μ 1 , β = 0.5 μ 1 , and β = 0 , respectively. For the case of (c) and (d): black line, green line, blue dotted line, blue dot-dashed line, and red dashed line are for β = 1.5 μ 1 , β = 0.5 μ 1 , β = 0.3 μ 1 , β = 0.2 μ 1 , and β = 0.1 μ 1 , respectively.

Figure 2 Dynamics of the Q S {Q}_{S} (EPR steering measure) for the two atoms, governed by the EPR steering inequality defined in equation (12), plotted as a function of time μ 1 t {\mu }_{1}t for various δ \delta values with β = μ 1 \beta ={\mu }_{1} when the atoms start from a Bell state ρ A − A ( 0 ) = 1 ∕ 2 ( ∣ 01 ⟩ ⟨ 01 ∣ + ∣ 01 ⟩ ⟨ 10 ∣ + ∣ 10 ⟩ ⟨ 01 ∣ + ∣ 10 ⟩ ⟨ 01 ∣ ) {\rho }_{A-A}\left(0)=1/2(| 01\rangle \langle 01| +| 01\rangle \langle 10| +| 10\rangle \langle 01| +| 10\rangle \langle 01| ) . Labels (a), (b), (c), and (d) are for α = 0.24 ∕ 0.5 μ 2 = 0.24 μ 1 \alpha =0.24/0.5{\mu }_{2}=0.24{\mu }_{1} , α = 0.4 ∕ 0.5 μ 2 = 0.4 μ 1 \alpha =0.4/0.5{\mu }_{2}=0.4{\mu }_{1} , α = 0.24 μ 1 \alpha =0.24{\mu }_{1} with μ 2 = 0 {\mu }_{2}=0 , and α = 0.4 μ 1 \alpha =0.4{\mu }_{1} with μ 2 = 0 {\mu }_{2}=0 , respectively. Black line, green line, blue dotted line, blue dot-dashed line, and red dashed line are for δ = 0 \delta =0 , δ = 0.5 μ 1 \delta =0.5{\mu }_{1} , δ = 2 μ 1 \delta =2{\mu }_{1} , δ = 3 μ 1 \delta =3{\mu }_{1} , and δ = 5 μ 1 \delta =5{\mu }_{1} , respectively.
Figure 2

Dynamics of the Q S (EPR steering measure) for the two atoms, governed by the EPR steering inequality defined in equation (12), plotted as a function of time μ 1 t for various δ values with β = μ 1 when the atoms start from a Bell state ρ A − A ( 0 ) = 1 ∕ 2 ( ∣ 01 ⟩ ⟨ 01 ∣ + ∣ 01 ⟩ ⟨ 10 ∣ + ∣ 10 ⟩ ⟨ 01 ∣ + ∣ 10 ⟩ ⟨ 01 ∣ ) . Labels (a), (b), (c), and (d) are for α = 0.24 ∕ 0.5 μ 2 = 0.24 μ 1 , α = 0.4 ∕ 0.5 μ 2 = 0.4 μ 1 , α = 0.24 μ 1 with μ 2 = 0 , and α = 0.4 μ 1 with μ 2 = 0 , respectively. Black line, green line, blue dotted line, blue dot-dashed line, and red dashed line are for δ = 0 , δ = 0.5 μ 1 , δ = 2 μ 1 , δ = 3 μ 1 , and δ = 5 μ 1 , respectively.

Figure 3 Dynamics of Bell nonlocality for the two atoms, quantified by the Bell-CHSH measure B l {B}_{l} from equation (14), plotted as a function of time μ 1 t {\mu }_{1}t for various β \beta values with δ = 0 \delta =0 when the atoms start from a Bell state ρ A − A ( 0 ) = 1 ∕ 2 ( ∣ 01 ⟩ ⟨ 01 ∣ + ∣ 01 ⟩ ⟨ 10 ∣ + ∣ 10 ⟩ ⟨ 01 ∣ + ∣ 10 ⟩ ⟨ 01 ∣ ) {\rho }_{A-A}\left(0)=1/2(| 01\rangle \langle 01| +| 01\rangle \langle 10| +| 10\rangle \langle 01| +| 10\rangle \langle 01| ) . Labels (a), (b), (c), and (d) are for α = 0.24 ∕ 0.5 μ 2 = 0.24 μ 1 \alpha =0.24/0.5{\mu }_{2}=0.24{\mu }_{1} , α = 0.4 ∕ 0.5 μ 2 = 0.4 μ 1 \alpha =0.4/0.5{\mu }_{2}=0.4{\mu }_{1} , α = 0.24 μ 1 \alpha =0.24{\mu }_{1} with μ 2 = 0 {\mu }_{2}=0 , and α = 0.4 μ 1 \alpha =0.4{\mu }_{1} with μ 2 = 0 {\mu }_{2}=0 , respectively. For the case of (a) and (b): black line, green line, blue dotted line, blue dot-dashed line, and red dashed line are for β = 2 μ 1 \beta =2{\mu }_{1} , β = 1.5 μ 1 \beta =1.5{\mu }_{1} , β = 1 μ 1 \beta =1{\mu }_{1} , β = 0.5 μ 1 \beta =0.5{\mu }_{1} , and β = 0 \beta =0 , respectively. For the case of (c) and (d): black line, green line, blue dotted line, blue dot-dashed line, and red dashed line are for β = 1.5 μ 1 \beta =1.5{\mu }_{1} , β = 0.5 μ 1 \beta =0.5{\mu }_{1} , β = 0.3 μ 1 \beta =0.3{\mu }_{1} , β = 0.2 μ 1 \beta =0.2{\mu }_{1} , and β = 0.1 μ 1 \beta =0.1{\mu }_{1} , respectively.
Figure 3

Dynamics of Bell nonlocality for the two atoms, quantified by the Bell-CHSH measure B l from equation (14), plotted as a function of time μ 1 t for various β values with δ = 0 when the atoms start from a Bell state ρ A − A ( 0 ) = 1 ∕ 2 ( ∣ 01 ⟩ ⟨ 01 ∣ + ∣ 01 ⟩ ⟨ 10 ∣ + ∣ 10 ⟩ ⟨ 01 ∣ + ∣ 10 ⟩ ⟨ 01 ∣ ) . Labels (a), (b), (c), and (d) are for α = 0.24 ∕ 0.5 μ 2 = 0.24 μ 1 , α = 0.4 ∕ 0.5 μ 2 = 0.4 μ 1 , α = 0.24 μ 1 with μ 2 = 0 , and α = 0.4 μ 1 with μ 2 = 0 , respectively. For the case of (a) and (b): black line, green line, blue dotted line, blue dot-dashed line, and red dashed line are for β = 2 μ 1 , β = 1.5 μ 1 , β = 1 μ 1 , β = 0.5 μ 1 , and β = 0 , respectively. For the case of (c) and (d): black line, green line, blue dotted line, blue dot-dashed line, and red dashed line are for β = 1.5 μ 1 , β = 0.5 μ 1 , β = 0.3 μ 1 , β = 0.2 μ 1 , and β = 0.1 μ 1 , respectively.

Figure 4 Dynamics of Bell nonlocality for the two atoms, quantified by the Bell-CHSH measure B l {B}_{l} from equation (14), plotted as a function of time μ 1 t {\mu }_{1}t for various δ \delta values with β = μ 1 \beta ={\mu }_{1} when the atoms start from a Bell state ρ A − A ( 0 ) = 1 ∕ 2 ( ∣ 01 ⟩ ⟨ 01 ∣ + ∣ 01 ⟩ ⟨ 10 ∣ + ∣ 10 ⟩ ⟨ 01 ∣ + ∣ 10 ⟩ ⟨ 01 ∣ ) {\rho }_{A-A}\left(0)=1/2(| 01\rangle \langle 01| +| 01\rangle \langle 10| +| 10\rangle \langle 01| +| 10\rangle \langle 01| ) . Labels (a), (b), (c), and (d) are for α = 0.24 ∕ 0.5 μ 2 = 0.24 μ 1 \alpha =0.24/0.5{\mu }_{2}=0.24{\mu }_{1} , α = 0.4 ∕ 0.5 μ 2 = 0.4 μ 1 \alpha =0.4/0.5{\mu }_{2}=0.4{\mu }_{1} , α = 0.24 μ 1 \alpha =0.24{\mu }_{1} with μ 2 = 0 {\mu }_{2}=0 , and α = 0.4 μ 1 \alpha =0.4{\mu }_{1} with μ 2 = 0 {\mu }_{2}=0 , respectively. Black line, green line, blue dotted line, blue dot-dashed line, and red dashed line are for δ = 0 \delta =0 , δ = 0.5 μ 1 \delta =0.5{\mu }_{1} , δ = 2 μ 1 \delta =2{\mu }_{1} , δ = 3 μ 1 \delta =3{\mu }_{1} , and δ = 5 μ 1 \delta =5{\mu }_{1} , respectively.
Figure 4

Dynamics of Bell nonlocality for the two atoms, quantified by the Bell-CHSH measure B l from equation (14), plotted as a function of time μ 1 t for various δ values with β = μ 1 when the atoms start from a Bell state ρ A − A ( 0 ) = 1 ∕ 2 ( ∣ 01 ⟩ ⟨ 01 ∣ + ∣ 01 ⟩ ⟨ 10 ∣ + ∣ 10 ⟩ ⟨ 01 ∣ + ∣ 10 ⟩ ⟨ 01 ∣ ) . Labels (a), (b), (c), and (d) are for α = 0.24 ∕ 0.5 μ 2 = 0.24 μ 1 , α = 0.4 ∕ 0.5 μ 2 = 0.4 μ 1 , α = 0.24 μ 1 with μ 2 = 0 , and α = 0.4 μ 1 with μ 2 = 0 , respectively. Black line, green line, blue dotted line, blue dot-dashed line, and red dashed line are for δ = 0 , δ = 0.5 μ 1 , δ = 2 μ 1 , δ = 3 μ 1 , and δ = 5 μ 1 , respectively.

Based on the study of Baumgratz et al. [22], the concept of L 1 norm is introduced for detecting the amount of coherence with respect to the off-diagonal elements of the density operator ρ 12 . Mathematically, this coherence measure is defined as

(17) C L 1 = ∑ i , j ∣ ρ A − A i j ∣ with i ≠ j .

The measure of coherence corresponding to the atom state is displayed in Figures 5 and 6 versus the time μ 1 t for various β and δ values in the presence and absence of the decay rate effect. Generally, the L 1 norm of quantum coherence is first decreased from its maximal value as the time μ 1 t increases. In the existence of cavity loss, the increase in the interaction between the cavities results in an enhancement in the amount of coherence. Additionally, we obtain that as the parameter β increases, coherence decays more slowly. In the absence of cavity loss, we can observe that coherence saturates at different maximum values for various β values. On the contrary, the increase of the parameter δ accompanied by an enhancement in the amount of coherence. As a result, the possibility for the control and preservation of L 1 norm of quantum coherence may occur through a proper choice of parameters β and δ with respect to the decay rate effect.

Figure 5 Time evolution of the quantum coherence C L 1 {C}_{{L}_{1}} (equation (17)) for the two atoms as a function of time μ 1 t {\mu }_{1}t for various β \beta values with δ = 0 \delta =0 when the atoms start from a Bell state ρ A − A ( 0 ) = 1 ∕ 2 ( ∣ 01 ⟩ ⟨ 01 ∣ + ∣ 01 ⟩ ⟨ 10 ∣ + ∣ 10 ⟩ ⟨ 01 ∣ + ∣ 10 ⟩ ⟨ 01 ∣ ) {\rho }_{A-A}\left(0)=1/2(| 01\rangle \langle 01| +| 01\rangle \langle 10| +| 10\rangle \langle 01| +| 10\rangle \langle 01| ) . Labels (a), (b), (c), and (d) are for α = 0.24 ∕ 0.5 μ 2 = 0.24 μ 1 \alpha =0.24/0.5{\mu }_{2}=0.24{\mu }_{1} , α = 0.4 ∕ 0.5 μ 2 = 0.4 μ 1 \alpha =0.4/0.5{\mu }_{2}=0.4{\mu }_{1} , α = 0.24 μ 1 \alpha =0.24{\mu }_{1} with μ 2 = 0 {\mu }_{2}=0 , and α = 0.4 μ 1 \alpha =0.4{\mu }_{1} with μ 2 = 0 {\mu }_{2}=0 , respectively. For the case of (a) and (b): black line, green line, blue dotted line, blue dot-dashed line, and red dashed line are for β = 2 μ 1 \beta =2{\mu }_{1} , β = 1.5 μ 1 \beta =1.5{\mu }_{1} , β = 1 μ 1 \beta =1{\mu }_{1} , β = 0.5 μ 1 \beta =0.5{\mu }_{1} , and β = 0 \beta =0 , respectively. For the case of (c) and (d): black line, green line, blue dotted line, blue dot-dashed line, and red dashed line are for β = 1.5 μ 1 \beta =1.5{\mu }_{1} , β = 0.5 μ 1 \beta =0.5{\mu }_{1} , β = 0.3 μ 1 \beta =0.3{\mu }_{1} , β = 0.2 μ 1 \beta =0.2{\mu }_{1} , and β = 0.1 μ 1 \beta =0.1{\mu }_{1} , respectively.
Figure 5

Time evolution of the quantum coherence C L 1 (equation (17)) for the two atoms as a function of time μ 1 t for various β values with δ = 0 when the atoms start from a Bell state ρ A − A ( 0 ) = 1 ∕ 2 ( ∣ 01 ⟩ ⟨ 01 ∣ + ∣ 01 ⟩ ⟨ 10 ∣ + ∣ 10 ⟩ ⟨ 01 ∣ + ∣ 10 ⟩ ⟨ 01 ∣ ) . Labels (a), (b), (c), and (d) are for α = 0.24 ∕ 0.5 μ 2 = 0.24 μ 1 , α = 0.4 ∕ 0.5 μ 2 = 0.4 μ 1 , α = 0.24 μ 1 with μ 2 = 0 , and α = 0.4 μ 1 with μ 2 = 0 , respectively. For the case of (a) and (b): black line, green line, blue dotted line, blue dot-dashed line, and red dashed line are for β = 2 μ 1 , β = 1.5 μ 1 , β = 1 μ 1 , β = 0.5 μ 1 , and β = 0 , respectively. For the case of (c) and (d): black line, green line, blue dotted line, blue dot-dashed line, and red dashed line are for β = 1.5 μ 1 , β = 0.5 μ 1 , β = 0.3 μ 1 , β = 0.2 μ 1 , and β = 0.1 μ 1 , respectively.

Figure 6 Time evolution of the quantum coherence C L 1 {C}_{{L}_{1}} (equation (17)) for the two atoms as a function of time μ 1 t {\mu }_{1}t for various δ \delta values with β = μ 1 \beta ={\mu }_{1} when the atoms start from a Bell state ρ A − A ( 0 ) = 1 ∕ 2 ( ∣ 01 ⟩ ⟨ 01 ∣ + ∣ 01 ⟩ ⟨ 10 ∣ + ∣ 10 ⟩ ⟨ 01 ∣ + ∣ 10 ⟩ ⟨ 01 ∣ ) {\rho }_{A-A}\left(0)=1/2(| 01\rangle \langle 01| +| 01\rangle \langle 10| +| 10\rangle \langle 01| +| 10\rangle \langle 01| ) . Labels (a), (b), (c), and (d) are for α = 0.24 ∕ 0.5 μ 2 = 0.24 μ 1 \alpha =0.24/0.5{\mu }_{2}=0.24{\mu }_{1} , α = 0.4 ∕ 0.5 μ 2 = 0.4 μ 1 \alpha =0.4/0.5{\mu }_{2}=0.4{\mu }_{1} , α = 0.24 μ 1 \alpha =0.24{\mu }_{1} with μ 2 = 0 {\mu }_{2}=0 , and α = 0.4 μ 1 \alpha =0.4{\mu }_{1} with μ 2 = 0 {\mu }_{2}=0 , respectively. Black line, green line, blue dotted line, blue dot-dashed line, and red dashed line are for δ = 0 \delta =0 , δ = 0.5 μ 1 \delta =0.5{\mu }_{1} , δ = 2 μ 1 \delta =2{\mu }_{1} , δ = 3 μ 1 \delta =3{\mu }_{1} , and δ = 5 μ 1 \delta =5{\mu }_{1} , respectively.
Figure 6

Time evolution of the quantum coherence C L 1 (equation (17)) for the two atoms as a function of time μ 1 t for various δ values with β = μ 1 when the atoms start from a Bell state ρ A − A ( 0 ) = 1 ∕ 2 ( ∣ 01 ⟩ ⟨ 01 ∣ + ∣ 01 ⟩ ⟨ 10 ∣ + ∣ 10 ⟩ ⟨ 01 ∣ + ∣ 10 ⟩ ⟨ 01 ∣ ) . Labels (a), (b), (c), and (d) are for α = 0.24 ∕ 0.5 μ 2 = 0.24 μ 1 , α = 0.4 ∕ 0.5 μ 2 = 0.4 μ 1 , α = 0.24 μ 1 with μ 2 = 0 , and α = 0.4 μ 1 with μ 2 = 0 , respectively. Black line, green line, blue dotted line, blue dot-dashed line, and red dashed line are for δ = 0 , δ = 0.5 μ 1 , δ = 2 μ 1 , δ = 3 μ 1 , and δ = 5 μ 1 , respectively.

3 Conclusion

In this work, we have investigated the robustness and control of quantum steering, Bell nonlocality, and coherence in a bipartite atomic system where each atom is confined within a cavity that interacts with another cavity field. By systematically analyzing the quantum dynamics under decoherence effects, we demonstrated how key system parameters – particularly inter-cavity coupling strength and detuning – govern the resilience and evolution of quantumness measures. Our results reveal that quantum steering, nonlocality, and coherence can be preserved and even enhanced through optimal tuning of system parameters, allowing significant levels of quantumness measures to persist over extended timescales. Specifically, increasing the inter-cavity interaction strength was found to delay the degradation of quantum resources, while an appropriate choice of detuning parameters further improved coherence, nonlocality, and steering, effectively mitigating the impact of decoherence. A significant outcome of our study is that, in the ideal cavity limit, where cavity losses are minimized, the system retains robust quantum correlations, reinforcing the feasibility of coherence protection during the quantum dynamics. Furthermore, our findings demonstrate that EPR steering and Bell nonlocality exhibit similar dynamical behaviors under certain parameter conditions, indicating their interdependence in non-Markovian environments. Additionally, we showed that quantum coherence plays a crucial role in sustaining nonlocal correlations, and its controlled enhancement via cavity coupling provides a means for long-term coherence preservation. These findings offer valuable insights into the fundamental mechanisms governing quantum resource dynamics in open quantum systems, with direct implications for quantum information processing and secure quantum communication technologies. The ability to suppress decoherence and sustain quantum correlations is essential for the implementation of high-fidelity quantum operations, quantum state transfer, and distributed quantum networks. Future research should focus on extending these results to multipartite and high-dimensional quantum systems, exploring experimental realizations in cavity QED, superconducting circuits, or trapped-ion platforms, and developing advanced control techniques, including feedback optimization and adaptive tuning strategies, to further enhance quantum coherence, nonlocality and steering in realistic environments. These directions will contribute to the development of scalable, noise-resilient quantum technologies for future applications in quantum computing and quantum communication networks.

  1. Funding information: This work was supported and funded by the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) (grant number IMSIU-DDRSP2503).

  2. Author contributions: Kamal Berrada: writing – original draft. Abdelaziz Sabik: visualization and supervision. All authors have accepted responsibility for the entire content of this manuscript and approved its submission.

  3. Conflict of interest: The authors state no conflict of interest.

  4. Data availability statement: All data generated or analyzed during this study are included in this published article.

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Received: 2025-02-06
Revised: 2025-05-19
Accepted: 2025-06-16
Published Online: 2025-07-16

© 2025 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/phys-2025-0178
Keywords for this article
open quantum systems; cavity–cavity interaction; photon loss; quantumness measures
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BY 4.0

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  25. Numerical examination of the chemically reactive MHD flow of hybrid nanofluids over a two-dimensional stretching surface with the Cattaneo–Christov model and slip conditions
  26. Impacts of sinusoidal heat flux and embraced heated rectangular cavity on natural convection within a square enclosure partially filled with porous medium and Casson-hybrid nanofluid
  27. Stability analysis of unsteady ternary nanofluid flow past a stretching/shrinking wedge
  28. Solitonic wave solutions of a Hamiltonian nonlinear atom chain model through the Hirota bilinear transformation method
  29. Bilinear form and soltion solutions for (3+1)-dimensional negative-order KdV-CBS equation
  30. Solitary chirp pulses and soliton control for variable coefficients cubic–quintic nonlinear Schrödinger equation in nonuniform management system
  31. Influence of decaying heat source and temperature-dependent thermal conductivity on photo-hydro-elasto semiconductor media
  32. Dissipative disorder optimization in the radiative thin film flow of partially ionized non-Newtonian hybrid nanofluid with second-order slip condition
  33. Bifurcation, chaotic behavior, and traveling wave solutions for the fractional (4+1)-dimensional Davey–Stewartson–Kadomtsev–Petviashvili model
  34. New investigation on soliton solutions of two nonlinear PDEs in mathematical physics with a dynamical property: Bifurcation analysis
  35. Mathematical analysis of nanoparticle type and volume fraction on heat transfer efficiency of nanofluids
  36. Creation of single-wing Lorenz-like attractors via a ten-ninths-degree term
  37. Optical soliton solutions, bifurcation analysis, chaotic behaviors of nonlinear Schrödinger equation and modulation instability in optical fiber
  38. Chaotic dynamics and some solutions for the (n + 1)-dimensional modified Zakharov–Kuznetsov equation in plasma physics
  39. Fractal formation and chaotic soliton phenomena in nonlinear conformable Heisenberg ferromagnetic spin chain equation
  40. Single-step fabrication of Mn(iv) oxide-Mn(ii) sulfide/poly-2-mercaptoaniline porous network nanocomposite for pseudo-supercapacitors and charge storage
  41. Novel constructed dynamical analytical solutions and conserved quantities of the new (2+1)-dimensional KdV model describing acoustic wave propagation
  42. Tavis–Cummings model in the presence of a deformed field and time-dependent coupling
  43. Spinning dynamics of stress-dependent viscosity of generalized Cross-nonlinear materials affected by gravitationally swirling disk
  44. Design and prediction of high optical density photovoltaic polymers using machine learning-DFT studies
  45. Robust control and preservation of quantum steering, nonlocality, and coherence in open atomic systems
  46. Coating thickness and process efficiency of reverse roll coating using a magnetized hybrid nanomaterial flow
  47. Dynamic analysis, circuit realization, and its synchronization of a new chaotic hyperjerk system
  48. Decoherence of steerability and coherence dynamics induced by nonlinear qubit–cavity interactions
  49. Finite element analysis of turbulent thermal enhancement in grooved channels with flat- and plus-shaped fins
  50. Modulational instability and associated ion-acoustic modulated envelope solitons in a quantum plasma having ion beams
  51. Statistical inference of constant-stress partially accelerated life tests under type II generalized hybrid censored data from Burr III distribution
  52. On solutions of the Dirac equation for 1D hydrogenic atoms or ions
  53. Entropy optimization for chemically reactive magnetized unsteady thin film hybrid nanofluid flow on inclined surface subject to nonlinear mixed convection and variable temperature
  54. Stability analysis, circuit simulation, and color image encryption of a novel four-dimensional hyperchaotic model with hidden and self-excited attractors
  55. A high-accuracy exponential time integration scheme for the Darcy–Forchheimer Williamson fluid flow with temperature-dependent conductivity
  56. Novel analysis of fractional regularized long-wave equation in plasma dynamics
  57. Development of a photoelectrode based on a bismuth(iii) oxyiodide/intercalated iodide-poly(1H-pyrrole) rough spherical nanocomposite for green hydrogen generation
  58. Investigation of solar radiation effects on the energy performance of the (Al2O3–CuO–Cu)/H2O ternary nanofluidic system through a convectively heated cylinder
  59. Quantum resources for a system of two atoms interacting with a deformed field in the presence of intensity-dependent coupling
  60. Studying bifurcations and chaotic dynamics in the generalized hyperelastic-rod wave equation through Hamiltonian mechanics
  61. A new numerical technique for the solution of time-fractional nonlinear Klein–Gordon equation involving Atangana–Baleanu derivative using cubic B-spline functions
  62. Interaction solutions of high-order breathers and lumps for a (3+1)-dimensional conformable fractional potential-YTSF-like model
  63. Hydraulic fracturing radioactive source tracing technology based on hydraulic fracturing tracing mechanics model
  64. Numerical solution and stability analysis of non-Newtonian hybrid nanofluid flow subject to exponential heat source/sink over a Riga sheet
  65. Numerical investigation of mixed convection and viscous dissipation in couple stress nanofluid flow: A merged Adomian decomposition method and Mohand transform
  66. Effectual quintic B-spline functions for solving the time fractional coupled Boussinesq–Burgers equation arising in shallow water waves
  67. Analysis of MHD hybrid nanofluid flow over cone and wedge with exponential and thermal heat source and activation energy
  68. Solitons and travelling waves structure for M-fractional Kairat-II equation using three explicit methods
  69. Impact of nanoparticle shapes on the heat transfer properties of Cu and CuO nanofluids flowing over a stretching surface with slip effects: A computational study
  70. Computational simulation of heat transfer and nanofluid flow for two-sided lid-driven square cavity under the influence of magnetic field
  71. Irreversibility analysis of a bioconvective two-phase nanofluid in a Maxwell (non-Newtonian) flow induced by a rotating disk with thermal radiation
  72. Hydrodynamic and sensitivity analysis of a polymeric calendering process for non-Newtonian fluids with temperature-dependent viscosity
  73. Exploring the peakon solitons molecules and solitary wave structure to the nonlinear damped Kortewege–de Vries equation through efficient technique
  74. Modeling and heat transfer analysis of magnetized hybrid micropolar blood-based nanofluid flow in Darcy–Forchheimer porous stenosis narrow arteries
  75. Activation energy and cross-diffusion effects on 3D rotating nanofluid flow in a Darcy–Forchheimer porous medium with radiation and convective heating
  76. Insights into chemical reactions occurring in generalized nanomaterials due to spinning surface with melting constraints
  77. Review Article
  78. Examination of the gamma radiation shielding properties of different clay and sand materials in the Adrar region
  79. Special Issue on Fundamental Physics from Atoms to Cosmos - Part II
  80. Possible explanation for the neutron lifetime puzzle
  81. Special Issue on Nanomaterial utilization and structural optimization - Part III
  82. Numerical investigation on fluid-thermal-electric performance of a thermoelectric-integrated helically coiled tube heat exchanger for coal mine air cooling
  83. Special Issue on Nonlinear Dynamics and Chaos in Physical Systems
  84. Analysis of the fractional relativistic isothermal gas sphere with application to neutron stars
  85. Abundant wave symmetries in the (3+1)-dimensional Chafee–Infante equation through the Hirota bilinear transformation technique
  86. Successive midpoint method for fractional differential equations with nonlocal kernels: Error analysis, stability, and applications
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