In quantum information science, entanglement [1] and coherence [2] have been considered as two important quantum resources for numerous quantum tasks. On the one hand, quantum entanglement is a fundamental aspect of quantum information science [3], playing a pivotal role in numerous phenomena, including quantum teleportation [4, 5], quantum cryptography [6], and quantum computation [7]. On the other hand, quantum coherence underlies most of the non-classical phenomena characteristic of quantum physics, such as entanglement, steering, and Bell nonlocality [8]. It plays a crucial role in quantum state merging [9], the Deutsch–Jozsa algorithm [10], Grover's search algorithm [11] and quantum metrology [12]. Over the past few years, coherence has received much attention, driven by the resource-theoretic formulation, see [2] for a comprehensive review. Quantum entanglement is nonlocal and exists in non-separable states while quantum coherence exists almost for all quantum states except diagonal (classically mixed) states. Quantum coherence can still play a significant role even without entanglement. For example, deterministic quantum computation with one qubit (DQC1) [13] and nonlocal deterministic quantum computation with two qubits (NDQC2) [14] can outperform those of the classical computation by quantum coherence even without entanglement. From this perspective, quantum coherence is a more general quantum information resource than quantum entanglement. Therefore, quantum coherence is highly worthy of investigation and is anticipated to play a pivotal role in the theory of quantum resources.

Although many coherence measures have been proposed [15], the l₁ norm of coherence stands out as one of the most important coherence monotone and easily computable measures [16]. It is particularly useful in studying speedup in quantum computation, as demonstrated by the Deutsch–Jozsa algorithm and Grover's algorithm, which is what we mentioned above. The l₁ norm of coherence also features prominently in alternative formulations of uncertainty relations, complementarity relations [17], and wave-particle duality in multipath interferometers [18–20]. Additionally, it plays a crucial role in quantifying the cohering and decohering powers of quantum operations. There is a significant relationship between coherence and entanglement in resource theory, as the l₁ norm of coherence is an upper bound for the negativity [8].

Decoherence is usually unavoidable due to the fact that any realistic quantum system is disturbed by its surrounding which may lead to loss of quantum coherence [21]. Artificially engineered structures, also termed as metamaterials, could be an effective way to address this issue [22, 23]. Metamaterials have played a significant role in the manipulation of electromagnetic fields. The unique properties of these materials have been recognized for their potential to enhance dipole-dipole interactions, facilitate energy harvesting, and enable long-range interactions of quantum emitters embedded in waveguide-like devices [24–27]. Plasmonic waveguide channels have been observed to facilitate extraordinary optical transmission upon stimulation and have been utilized in the subwavelength scale to facilitate long-distance interactions among quantum emitters [28]. Entanglement mediated by waveguides, such as nonreciprocal waveguides [29, 30] and topological materials [31] have been reported. Recently, the attention towards metamaterials exhibiting epsilon-near-zero (ENZ) permittivity response has grown significantly [32–35]. When ENZ waveguides operate near their cutoff wavelength, the effective ENZ mode can be induced, which possesses characteristics such as near-zero phase shift, high phase velocity, and low refractive index [36–38], finding potential applications such as the enhancement of optical nonlinearities [39, 40], the development of integrated photonic devices [41, 42], and the creation of efficient optical interconnects [43].

Numerous studies have linked ENZ metamaterials with quantum entanglement [44–47]. Research has indicated that the resonant energy transfer and entanglement performance of ENZ waveguides are significantly superior to other waveguide systems, including traditional V-groove and metal nanowire structures [48]. Compared to rectangular ENZ waveguides, rolled-up waveguides offer stronger resonant energy transfer and more entanglement in the ENZ mode [49–51]. Furthermore, the key advantage of the rolled-up ENZ waveguide over alternative plasmonic waveguide channels lies in its capability to seamlessly integrate diverse emitter technologies into the waveguide's core. For instance, it has been suggested to integrate quantum emitters into the core of the rolled ENZ waveguide by depositing the emitters onto the planar bilayer prior to beginning the rolling procedure [50]. Other types of ENZ waveguides have also been experimentally demonstrated to achieve effective coupling with qubits [52]. Another approach of microsyringe technology has been employed for the injection of colloidal nanoemitters. In this scenario, the propulsion of nanoemitters is facilitated by capillary forces [53].

However, research on quantum coherence between qubits when mediated by ENZ waveguides is exceedingly rare. Given its remarkable potential in quantum computing, we have investigated quantum coherence mediated by ENZ waveguides, with the expectation that our findings may assist other researchers in this field. In addition, silicon nitride (Si₃N₄) is a commonly used material for fabricating waveguides due to its high refractive index, low optical loss, high temperature resistance, and corrosion resistance [54–56]. Given that metals have a negative dielectric constant and silicon nitride has a positive dielectric constant, combining them in a specific manner allows the design of a rolled-up ENZ waveguide that can excite specific ENZ modes. We innovatively introduce Si₃N₄ to replace SiO₂ in the waveguide structure [50] and establish a new rolled-up waveguide that serves as the medium for our discussion on the dynamics of quantum coherence (and, for comparison, the quantum entanglement) between two qubits mediated by the ENZ waveguide. We find that replacing SiO₂ with Si₃N₄ can achieve higher quantum coherence and entanglement over certain distances. This is because substituting SiO₂ for Si₃N₄ can result in a higher maximum coherent coupling and a slower decay of incoherent coupling within the ENZ region [50]. We derive analytical expressions for both quantum coherence and quantum entanglement, allowing for direct comparison within a unified framework. To achieve stable quantum coherence, classical field driving is introduced. We find that stable coherence is much larger and easier mediated than that of stable entanglement.

To begin with, we artificially design an ENZ waveguide and perform mode analysis of the waveguide to ensure it operates in the ENZ mode. Here, we utilize two finite element analysis software programs: Ansys Lumerical and COMSOL Multiphysics. With numerical finite element eigen mode (FEEM) simulations, the desired ENZ modes are verified with consistent results from both software packages. Figure 1 shows the waveguide structure and the electric-field distribution. As depicted in figure 1(a), the structure consists of 12 alternating layers of Au and Si₃N₄, where each Au layer is 10 nm thick, and each Si₃N₄ layer is 5 nm thick. The inner diameter and the length of the ENZ waveguide is D = 700 nm and L = 25 μm. The operating wavelength is set as λ = 1580 nm. A coordinate system is established with the circular base as the x − y plane and the radial direction of the rolled-up waveguide as the z direction. For more modeling details, please refer to the appendix.

Figure 1(b) shows the different modes excited in the rolled-up waveguide at different effective refractive indices (n_eff). Due to the excitation of the plasmonic modes, the electric-field modes with effective refractive index (n_eff ≠ 0) are mostly confined within the metal-dielectric cladding region. By contrast, the TE₁₁ mode with a nearly zero index (n_eff ≃ 0) corresponds to the ENZ mode, with the mode distribution predominantly confined within the central region of the rolled-up waveguide. In what follows, we are interested in the ENZ case which is present at the wavelength λ = 1580 nm in this work. Since the ENZ phenomenon displays a stronger electric field with a longer wavelength, it is expected to mediate the longer-range coupling between qubits (quantum emitters) embedded within the waveguide.

After determining the photonic properties of the ENZ waveguide, we proceed to employ it as a medium for quantum information processing. We design a two-qubit quantum system composed of two independent quantum emitters (with the ground state $\left|g\right\rangle $ and the excited state $\left|e\right\rangle $) but with the same transition frequency ω₀, each placed along the central axis (the z-axis) of the rolled-up waveguide shown in figure 1(a). The distance between the two qubits is r (in this context, r represents the actual distance between two qubits. However, in subsequent discussions, we will primarily use the dimensionless parameter d = r/λ to characterize the relative distance between the two qubits). The dynamics of the quantum system coupled to the ENZ waveguide can be described by the quantum master equation with relevant parameters supported by the dyadic Green's function [28]. For clarity, the two qubits are denoted as qubit A and qubit B and their quantum state is the density matrix ρ_AB(t). Assuming the Born–Markov and rotating wave approximations, the quantum master equation is given by [57]:where ℏ = h/(2π) is the reduced Planck constant and the Hamiltonian H_eff contributing the coherent part of the dynamics is expressed asHere, ${\sigma }_{i}^{+}=| e\rangle \,\langle g| $ and ${\sigma }_{i}^{-}=| g\rangle \,\langle e| $ are the raising and lowering operators for qubit i = A, B. At optical frequencies, the Lamb shift g_ii becomes very small for a qubit-waveguide distance exceeding approximately 10 nm [47], which will be neglected in our setup. In addition, the parameter g_ij with i ≠ j represents the effective (coherent) coupling between two qubits while the parameters γ_ij with i = j and i ≠ j characterize the self-decay and cross-decay rates (the decay rate between the two qubits mediated by the ENZ waveguide channel), which are related to the (incoherent) dissipation. Coherent and incoherent couplings represent the two fundamental mechanisms employed for mediating two emitters. Specifically, coherent coupling, also referred to as dipole-dipole interaction between the two emitters. On the other hand, incoherent coupling governs the dissipative term in quantum master equation (1). Here, we employ the dyadic Green's function $\overleftrightarrow{G}\left({\vec{r}}_{i},{\vec{r}}_{j},{\omega }_{0}\right)$, which satisfies the classical Maxwell equations for a minute dipole source positioned at the spatial coordinate ${\vec{r}}_{j}$. Physically, the dyadic Green's function $\overleftrightarrow{G}\left({\vec{r}}_{i},{\vec{r}}_{j},{\omega }_{0}\right)$ encapsulates the electromagnetic interaction that propagates from the spatial point ${\vec{r}}_{j}$ to another point ${\vec{r}}_{i}$ [28]. By integrating the dyadic Green's function of the electromagnetic field in the frequency domain, one may obtain the coherent-coupling parameter g_ij and the dissipation parameter γ_ij as follows [58]:where ${\vec{\mu }}_{i}$ represents the dipole moment of qubit i, ϵ₀ is the free space permittivity, c is the speed of light. The result is that the coherent and incoherent contributions to the coupling are proportional to the real and imaginary parts of the Green's function, respectively, which is achieved by applying the Kramers–Kronig relation between the real and imaginary components of the Green's function [28]. The dyadic Green's function $\overleftrightarrow{G}\left({\vec{r}}_{i},{\vec{r}}_{j},{\omega }_{0}\right)$ can be numerically obtained via Ansys Lumerical FDTD simulation software (more detailed information will be presented in the appendix) and then the coherent-coupling parameter g_ij and the dissipation parameter γ_ij are also numerically determined by equation (3). For clarity, we denote γ_AA = γ_BB = γ₀ and γ_AB = γ_BA = γ₁ respectively to characterize the rates of spontaneous decay and cross-decay. Here, we would like to employ three dimensionless parameters γ = γ₁/γ₀, g = g_AB/γ₀ and d = r/λ. Figure 2 shows the dimensionless coherent and incoherent coupling parameters g and γ as functions of the dimensionless distance d for different wavelengths λ. At λ = 1450 nm, 1500 nm, the incoherent coupling γ decays rapidly to a minimum value below zero and damping oscillatory behaviors for the two wavelengths are observed. By contrast, at the cutoff wavelength λ = 1580 nm, the incoherent coupling γ decays monotonically and asymptotically in a much slower manner to the vanishing value. For the coherent interaction g, it also displays an oscillatory behavior after reaching its local maximum value at λ = 1450 nm, 1500 nm while it is greatly enhanced at the cutoff wavelength λ = 1580 nm. These phenomena are consistent with the results reported in [50]. However, we shall point out that the maximum value can even exceed the incoherent coupling γ, which has not been reported anywhere. This also indicates that our setup by replacing SiO₂ by Si₃N₄ actually may achieve higher quantum information since the enhancement of γ will be beneficial for inducing entanglement [50].

In this section, we would like to explore the dynamics of quantum information (including quantum coherence and quantum entanglement) induced by the ENZ waveguide. First, we need to obtain the dynamics of the quantum state for the two-qubit system via solving the master equation (1). We consider the initial state ρ_AB(0) of the two-qubit system has the form ofin the basis {∣ee⟩, ∣eg⟩, ∣ge⟩, ∣gg⟩}, which is an X-state [59], an important family of two-qubit states commonly used in quantum information science. Analytical solutions have been derived for the state density matrix ρ_AB(t) which has the same form as equation (4) with all the non-vanishing elements [57] ${\alpha }_{\pm }(t)=\left({\rho }_{22}(t)\pm {\rho }_{33}(t)\right)$/2, ${\beta }_{\pm }(t)=\left({\rho }_{23}(t)\pm {\rho }_{23}(t)* \right)$/2, ${\rho }_{11}(t)={\rho }_{11}(0){{\rm{e}}}^{-2{\gamma }_{0}t}$, ${\rho }_{14}(t)={\rho }_{14}(0){{\rm{e}}}^{-\left({\gamma }_{0}+2{\rm{i}}{\omega }_{0}\right)t}$, ρ₂₂(t) = α₊(t) + α₋(t), ρ₃₃(t) = α₊(t) − α₋(t), ρ₂₃(t) = β₊(t) + β₋(t) and ρ₄₄ = 1 − ρ₁₁(t) − ρ₂₂(t) − ρ₃₃(t), whereSubsequently, appropriate measures of quantum information are employed for discussion of the ENZ waveguide induced quantum information. In the field of quantum information science, entanglement and coherence are considered two important quantum resources for numerous quantum tasks. Quantum coherence lies at the root of many quantum phenomena, such as entanglement, steering, and Bell nonlocality and it is a fundamental resource for demonstrating quantum advantages in various information processing tasks, including quantum metrology, quantum cryptography, and quantum computation [8]. Although many coherence measures have been proposed, the l₁ norm of coherence stands out as one of the most important and easily computable coherence monotones. Considering a quantum state ρ = ∑_jkρ_jk∣j⟩⟨k∣ in a finite-dimensional orthonormal basis {∣j⟩}, the definition of l₁ norm of coherence is given by [16]:To compare the coherence and entanglement in the same frame, we employ negativity rather than concurrence as the entanglement measure, which is the lower bound for the l₁ norm of coherence [8]. The definition of negativity is given by:where ${\rho }^{{{\rm{T}}}_{{\rm{A}}}}$ denotes the partial transpose of ρ with respect to subsystem A, and ∥ · ∥₁ is the Schatten 1-norm (trace norm). It has been proven that negativity is exactly the lower bound of the l₁ norm of coherence [8], satisfying ${ \mathcal N }(\rho )\leqslant { \mathcal C }(\rho )$, which means they are directly comparable and quantum coherence characterizes more quantum information than entanglement does. With the density matrix ρ_AB(t) of the two-qubit system mediated by the rolled-up waveguide, the quantum coherence and entanglement can be exactly calculated via equations (6) and (7) respectively. To explore the mediation effects of the ENZ waveguide, the two-qubit system is assumed to be initially in a product state ∣eg⟩ without quantum information, in which case the only non-vanishing element in the density matrix is ρ₂₂(0) = 1. By substituting, the analytical expressions of ${ \mathcal C }(\rho )$ and ${ \mathcal N }(\rho )$ in equations (6) and (7) can be derived, which provide a highly convenient and efficient means for calculating quantum coherence and quantum entanglement. They verify that the negativity is indeed the lower bound for the l₁ norm of coherence [8]. If we set ${\sin }^{2}(2{gt})+{\sinh }^{2}\left({\gamma }_{1}t\right)=a$ and $\left({{\rm{e}}}^{{\gamma }_{0}t}-\cosh \left({\gamma }_{1}t\right)\right)=b$, then $\sqrt{{\sin }^{2}(2{gt})+{\sinh }^{2}\left({\gamma }_{1}t\right)+{\left({{\rm{e}}}^{{\gamma }_{0}t}-\cosh \left({\gamma }_{1}t\right)\right)}^{2}}-\left({{\rm{e}}}^{{\gamma }_{0}t}-\cosh \left({\gamma }_{1}t\right)\right)\,=\sqrt{{a}^{2}+{b}^{2}}-b$, and it is evident that $a\geqslant \sqrt{{a}^{2}+{b}^{2}}-b$. This is also the reason why we intentionally simplify quantum coherence and entanglement into those forms.

Figure 3 gives a plot of the maximal coherence ${{ \mathcal C }}_{\max }$ and entanglement ${{ \mathcal N }}_{\max }$ induced by the ENZ waveguide as functions of interatomic distance d. It is easily observed that one always has ${{ \mathcal C }}_{\max }\geqslant {{ \mathcal N }}_{\max }$, and both of them are significantly enhanced when d → 0 . The maximal quantum information does not monotonically decay versus the interatomic distance, since it is rather dependent on $\left|g\right|$, which is not a monotonic function and a revival to its local maximum is present at d ≈ 0.2. Figure 3 also shows that replacing SiO₂ with Si₃N₄ can achieve higher quantum coherence and entanglement over certain distances. This is because substituting SiO₂ for Si₃N₄ can result in a higher maximum coherent coupling and a slower decay of incoherent coupling within the ENZ region [50]. The choice of a too small interatomic distance is meaningless since we would like to explore quantum information induced at a distance of order λ. In this sense, we will choose the same interatomic distance d = 0.5 as [49, 50] in the following discussions, in which case our setup by replacing SiO₂ with Si₃N₄ gives rise to a higher maximal coherence ${{ \mathcal C }}_{\max }$ and entanglement ${{ \mathcal N }}_{\max }$.

Generally speaking the induced quantum information will finally vanish due to the loss. Next, we consider the scheme to achieve stable quantum coherence and stable quantum entanglement by locally applying classical pumpings. When the two qubits are under continuous drivings, new terms $\displaystyle \sum _{i=A,B}\,{\hslash }\ {\rm{\Omega }}{\ }_{i}\left({\sigma }_{i}^{+}{{\rm{e}}}^{-{\rm{i}}{\omega }_{{ct}}}+{\sigma }_{i}^{-}{{\rm{e}}}^{{\rm{i}}{\omega }_{{ct}}}\right)$ need to be added to the effective Hamiltonian H_eff in equation (2), which can be expressed aswhere Ω_i represents the coupling strength between qubit i and and its classical field of frequency ω_c. As demonstrated in [28], stable entanglement can be achieved even for specific-separated qubits when the classical field is resonant (ω_c = ω₀). Due to the introduction of the driving concept, which disrupts the X-state in equation (4), an analytical expression for quantum coherence and quantum entanglement cannot be obtained. Consequently, numerical methods are employed in subsequent calculations for equation (10), using the Wolfram Mathematica software. Of course, the analytical expressions in equations (8) and (9) can still be used to calculate quantum coherence and quantum entanglement without driving ( Ω₁ = Ω₂ = 0), making the calculations highly convenient and efficient.

First, we consider the impact of the classical resonant fields (ω_c = ω₀) and plot the dynamics of coherence and negativity under different driving conditions in figure 4. As shown by the black solid curves, the two types of quantum information can indeed be induced without drivings but they will monotonically decay after the one-time induction. In figure 4(a), for quantum coherence, the peak value is low without driving ( Ω₁ = Ω₂ = 0) and gradually decays to zero. When an asymmetric driving ( Ω₁ = 0.4 , Ω₂ = 0) or anti-symmetric driving ( Ω₁ =0.2 , Ω₂ = − 0.2) is applied, coherence reaches a maximum value and maintains a relatively high stable value (t → ∞ ). The asymmetric driving is slightly more effective in this regard. With symmetric driving ( Ω₁ = Ω₂ = 0.2), although the stable value achieved is lower than the asymmetric and anti-symmetric driving cases, but it is still much higher than in the non-driving case. The continuous drivings are effectively to enhance the induced coherence as well as stabilizing the coherence. By contrast, the effects of these drivings varies for negativity. Firstly, the induced maximal values ${{ \mathcal N }}_{\max }$ of negativity are almost the same as the non-driving case and the stable values are always less than the induced maximal values ${{ \mathcal N }}_{\max }$ under the drivings. Secondly, among the three driving conditions, the anti-symmetric driving ( Ω₁ = 0.2 , Ω₂ = − 0.2) works most efficiently, in which case the stable value is close to the maximal value. Overall, considering both coherence and negativity, applying anti-symmetric driving yields the best performance.

After discussing the dynamics of driven quantum coherence and entanglement, we next examine the effects of the inter-qubit distance d on the stable values. Since very close distances between the two qubits are not meaningful, we start our discussion from d = 0.2. Under the three driving conditions, the variations of stable quantum coherence and negativity as functions of the inter-qubit distance d = 0.2 are shown in figure 5.

As shown in figure 5, the variation of quantum coherence and negativity versus d under the three driving conditions is distinct, demonstrating that different values of d significantly impact the steady-state coherence ${{ \mathcal C }}_{\mathrm{st}}$ and negativity ${{ \mathcal N }}_{\mathrm{st}}$ under t → ∞ . We find that, similar to the situation depicted in figure 4, within most regions of the inter-qubit distance d, the anti-symmetric driving still remains the most effective approach, both in terms of coherence and negativity. Under the anti-symmetric driving, the steady-state coherence ${{ \mathcal C }}_{\mathrm{st}}$ is maximized around one wavelength d ≈ 1, while the steady-state negativity ${{ \mathcal N }}_{\mathrm{st}}$ is maximized around half a wavelength d ≈ 1/2. Under different drivings, the ${{ \mathcal C }}_{\mathrm{st}}$ remains considerably large within the full range of d considered while the ${{ \mathcal N }}_{\mathrm{st}}$ is almost vanishing for d > 1.5. In this regard, classical drivings are highly effective at stabilizing coherence, and notably, coherence serves as a more universal quantum resource compared to entanglement. This can be explained by the fact that coherence can be created through local unitary operations (local continuous driving), even though non-classical correlations like entanglement do not change in this scenario. This also explains why the stable coherence value can exceed the maximum negativity value without driving, as shown in figure 4.

From the above discussion, we can conclude that negativity and coherence depend on the configuration of the driving parameters Ω₁ and Ω₂ in equation (10) with ω_c = ω₀. To gain a more detailed understanding of the effects of different driving schemes on negativity and coherence, we will conduct a more in-depth analysis. To ensure a fair contribution from Ω₁ and Ω₂, we assume each qubit is initially in its ground state, i.e. the initial state of the two-qubit system is ∣gg⟩. Selecting d = 0.5 in figure 6, we plot ${{ \mathcal C }}_{\mathrm{st}}$ and ${{ \mathcal N }}_{\mathrm{st}}$ as functions of Ω₁ and Ω₂. The results indicate that the values of coherence and negativity do not necessarily increase with enhancing Ω₁ and Ω₂. As depicted in figure 6(a), the region of maximum coherence is found near the symmetric or anti-symmetric double-driving schemes, such as around regions Ω₁ = Ω₂ = 0.5 or Ω₁ = − Ω₂ = 0.5. The situation for negativity is entirely different. As shown in figure 6(b), the maximum value of negativity is relatively high under anti-symmetric driving ( Ω₁ = 0.2, Ω₂ = − 0.2), which is consistent with our discussion in figure 5. Moreover, we can see that negativity is quite sensitive to the driving conditions, with many driving scenarios leading to negligible negativity while coherence can be well maintained under a wide range of driving conditions.

In the preceding discussion, we have consistently assumed ω_c = ω₀. In equation (10), we define the detuning δ = ω_c − ω₀. Next, by selecting Ω₂ = 0, we continue to explore the effect of detuning of the driving frequency ω_c from the qubit frequency ω₀. Figure 7 plots ${{ \mathcal C }}_{\mathrm{st}}$ and ${{ \mathcal N }}_{\mathrm{st}}$ as functions of Ω₁ and δ. There exists some regions (such as the region close to δ = ω_c − ω₀ = 0.5), where ${{ \mathcal C }}_{\mathrm{st}}$ is quasi-optimized and ${{ \mathcal N }}_{\mathrm{st}}$ is optimized by variation of Ω₁. We also discovered a surprising phenomenon: selecting specific distances d (the primary impact is on g and γ, as each d corresponds to a specific g and γ) can break the symmetry of negativity and coherence, as shown in figure 7. We find that the symmetry of coherence and negativity is more sensitive to g but is less sensitive to γ. After selecting a broader range of g and γ values for our calculations, we find the closer g is to zero, the better the symmetry. With more extreme values, as illustrated in figure 8, when d = 1.37 and $\left|g\right|\simeq 0$ (as can be seen from figure 2), the symmetric distribution of ${{ \mathcal C }}_{\mathrm{st}}$ and ${{ \mathcal N }}_{\mathrm{st}}$ along Ω₁ = 0 and δ = 0 is recovered, in which case the resonant drivings are the optimal choice to maximal ${{ \mathcal C }}_{\mathrm{st}}$ and ${{ \mathcal N }}_{\mathrm{st}}$. Therefore, non-resonant drivings may be a preferable option for qubits that are in close proximity.

In summary, we have investigated the quantum coherence between two qubits mediated by an ENZ waveguides. By replacing SiO₂ (investigated by other researchers) with Si₃N₄, we have proposed a new ENZ waveguide composed of alternating layers of Au and Si₃N₄, and perform the mode analysis to ensure it operates in the ENZ mode. We describe its coupling with quantum qubits using coherent coupling and incoherent coupling (via the electromagnetic dyadic Green's function). To compare the quantum coherence and quantum entanglement in the same frame, we employ the l₁ norm of coherence and negativity (rather than concurrence) as measures of quantum coherence and entanglement, respectively. Their analytical expressions are derived and l₁ norm of quantum coherence is always no less than negativity, which confirms that quantum coherence is a more general and robust resource than entanglement.

Upon selecting an appropriate inter-qubit distance, the two types of quantum resources can be induced. Substituting Si₃N₄ for SiO₂ results in a higher maximum coherent coupling and a slower decay of incoherent coupling within the ENZ region, which are benifical to induce higher quantum resources. However, they will monotonically decay to vanishing after a one-time induction.

To achieve stable coherence and negativity, classical fields are applied locally. We discuss the effects of different driving schemes on coherence and negativity, finding that the continuous drivings are effectively to enhance the induced coherence, as well as stabilizing the coherence and anti-symmetric driving is generally the best choice among the four driving schemes. Both in terms of coherence and negativity within most region of the inter-qubit distance, we find that classical drivings are highly effective at stabilizing coherence.

We have selected a variety of driving parameters to analyze their influence on the enhancement of quantum coherence and negativity. The values of coherence and negativity do not necessarily increase with the enhancement of driving parameters. Negativity is particularly sensitive to driving conditions, with many driving scenarios resulting in little negativity, while coherence can be well maintained under a wide range of driving conditions. This also indicates that quantum coherence is a more general form of quantum resource compared to entanglement. We also investigate the effects of detuning on the dissipation-driving stability of coherence and negativity, demonstrating that they can still be effectively maintained under certain detuning conditions. Selecting small inter-qubit distance d (large $\left|g\right|$) can break the symmetry of coherence and negativity, and the maximum values of coherence and negativity may not necessarily occur at the resonance case, i.e., non-resonant drivings may be a preferable option for qubits that are in close proximity. Our work contributes to the creation of a new stable quantum resource via an ENZ waveguide.

The Green's function is a mathematical tool widely used to solve partial differential equations, particularly in fields such as electromagnetism, quantum mechanics, and acoustics. It primarily describes the system's response to a point source or impulse excitation. In the context of electromagnetism, the dyadic Green's function $\overleftrightarrow{G}\left({\vec{r}}_{i},{\vec{r}}_{j},{\omega }_{0}\right)$ is used, which satisfies the classical Maxwell equations for an infinitesimal dipole source positioned at the spatial coordinate ${\vec{r}}_{j}$. Physically, the dyadic Green's function $\overleftrightarrow{G}\left({\vec{r}}_{i},{\vec{r}}_{j},{\omega }_{0}\right)$ responds to the electromagnetic interaction that propagates from the spatial point ${\vec{r}}_{j}$ to another point ${\vec{r}}_{i}$, which can be expressed as [28]where μ₀ is the vacuum permeability, $\vec{\mu }$ is the dipole moment, $\overleftrightarrow{G}\left({\vec{r}}_{i},{\vec{r}}_{j},{\omega }_{0}\right)$ is a 3 × 3 symmetric matrix given byFor example, G_yy can be calculated from a dipole oriented along the y direction:Due to the cylindrical symmetry of the waveguide system, for simplicity, it can be assumed that the dipole moment is along a certain coordinate axis, such as the y-axis. Then, the coherent coupling parameter g_ij and the incoherent coupling parameter γ_ij in equation (3) can be simplified asAfter obtaining the coherent coupling parameter g_ij and the incoherent coupling parameter γ_ij, they can be substituted into equations (1) and (2) to solve for the density matrix ρ_AB(t). By inserting ρ_AB(t) into equations (6) and (7) the quantum coherence and negativity can be calculated.

The electrical field E_y in the waveguide can be numerically implemented utilizing the Ansys Lumerical software. After obtaining the coherent coupling parameter g_ij and the incoherent coupling parameter γ_ij, we use the Wolfram Mathematica software to calculate the subsequent quantum coherence and negativity. In the Ansys Lumerical software setting, a 24-layer hollow waveguide composed of alternating layers of Au and Si₃N₄ is modeled. The interior of this waveguide is occupied by an air column, with an inner diameter D = 700 nm and a length of L = 25 μm. The waveguide consists of 12 layers of Au and 12 layers of Si₃N₄, each having a thickness of 10 nanometers for Au and 5 nanometers for Si₃N₄, respectively, with the innermost layer being Au. The complex refractive index of the Au and Si₃N₄ used in the simulation is from the material data of Johnson and Christy, Kischkat, respectively. A dipole positioned at the center of the waveguide and oriented along the y-axis, is added as the excitation source to simulate the quantum emitter. The dipole moment is set to unity. The mesh refinement is set to conformal variant 0 with a minimum mesh set of 0.25 nm. An additional mesh is used to increase the step size of the simulation and enhance resolution. The number of simulation boundary layers and simulation time is increased to ensure that there is sufficient time for the radiated field to decay completely.