Entangled dark state mediated by a dielectric cavity within epsilon-near-zero materials

Two emitters can be entangled by manipulating them through optical fields within a photonic cavity. However, maintaining entanglement for a long time is challenging due to the decoherence of the entangled qubits, primarily caused by cavity loss and atomic decay. Here, we found the entangled dark state between two emitters mediated by a dielectric cavity within epsilon-near-zero (ENZ) materials, ensuring entanglement maintenance over an extended period. To obtain the entangled dark state, we derived an effective model with degenerate mode modulation. In the dielectric cavities within ENZ materials, the decay rate of emitters can be regarded as 0, which is the key to achieving the entangled dark state. Meanwhile, the dark state immune to cavity loss exists when two emitters are in symmetric positions in the dielectric cavity. Additionally, by adjusting the emitters to specific asymmetric positions, it is possible to achieve transient entanglement with higher concurrence. By overcoming the decoherence of the entangled qubits, this study demonstrates stable, long-term entanglement with ENZ materials, holding significant importance for applications such as nanodevice design for quantum communication and quantum information processing.


Introduction
Entanglement between qubits is a fundamental resource in the field of quantum information, offering remarkable potential for quantum computation and communication [1,2].Therefore, the control of entanglement is an essential task to realize the aforementioned quantum applications.Thanks to the significant optical field localization in nanostructures [3][4][5], the entanglement between two emitters can be efficiently manipulated by tailoring the light-emitter coupling [6].When the coupling strength between emitters and optical modes is weaker than the losses in the system, it is in the weak-coupling regime.In this situation, various nanocavities have the ability to amplify optical field intensity and improve quality factors, thus enhancing the entanglement and its stability [7][8][9].Surface plasmon Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.structures, such as metal nanoparticles [10][11][12], nanoantennas [13,14], and plasmonic waveguides [15][16][17][18], can enhance the entanglement between two emitters through near-field effects or the high local density of states.The combination of metal spheres and nanowires also provides a means to control entanglement [19].Utilizing metasurfaces to manipulate scattering fields enables the entanglement between qubits separated by a macroscopic distance [20].However, there are cavity losses due to absorption and scattering, and atomic decay due to emitters radiating to other channels in these systems, making it difficult for entanglement to maintain for a long time.While in the strong-coupling regime, due to the extremely strong emittermode interaction, the system can be considered robust to metal losses and atomic decay, so that long-lived entanglement can be achieved [21,22].However, achieving strong coupling is considerably more challenging.
To achieve long-lived entanglement between qubits, it is essential to reduce decoherence processes.Such long-time entanglement can benefit from the subradiant state, known as the dark state, which ideally does not decay into the ground state [23].In waveguide quantum electrodynamics, achieving a dark state that cannot emit photons into the waveguide is possible, but its lifetime is still limited by the non-radiative decoherence rates [24][25][26].In the weak-coupling regime, one possible approach to obtain a dark state is to ensure that the decay rate of the emitter is sufficiently small.The combination with ENZ materials can achieve this goal to some extent.As a kind of metamaterial, ENZ materials possess extraordinary electromagnetic manipulation capabilities, such as electromagnetic wave tunneling effects [27], directional radiation [28], nonlinear effects [29], etc., which can be utilized to enhance molecular fluorescence [30], enhance vortex harmonic radiation [31], and so on.Due to the large field enhancement and uniform phase distribution in ENZ materials, ENZ plasmonic waveguides have recently been used to control the decay rates of emitters and achieve long-time entanglement [32][33][34][35][36][37][38][39].However, in these systems, the plasmonic losses of the ENZ waveguide and the atomic decay make it difficult to achieve dark state entanglement.Interestingly, the decay rate of an emitter in the dielectric cavity within ENZ materials is nearly 0, which satisfies the requirement for dark state entanglement in the weak-coupling regime.
Here, we investigate the entangled dark state of two emitters located within spherical cavities embedded in ENZ materials.An effective model is derived to explain the generation of entangled dark states.According to the effective model, when the two emitters are symmetrically positioned in the cavity, the decay rate of an entangled eigenstate is equal to the atomic decay rate.In our system, as the emitters rarely emit into modes other than those interacting with them, the emitters' decay rate is nearly 0, which forms the basis of the dark state.Although our system is in the weak-coupling regime, the dark state is immune to cavity loss, allowing for long-lived entanglement.Moreover, transient entanglement with higher concurrence can be attained by strategically placing the emitters asymmetrically in the cavity.Our system achieves dark state entanglement between the two emitters in the weak-coupling regime, even though the emitter-cavity interaction strength is smaller than the cavity loss.These findings have various applications, such as the design of nanodevices for quantum communication and quantum information processing.

Theory
In photonic nanocavities, the entanglement between two emitters will be damaged by the cavity losses, including scattering, absorption, and atomic decay.This hinders the maintenance of entanglement for a long time [20,34,40].The use of dielectric cavities within ENZ materials partially solves this problem.Because, except for the resonant modes, the emitter rarely radiates into other modes, resulting in a nearly zero decay rate.Two emitters will be entangled when placed in a spherical dielectric cavity embedded in an ENZ material, as shown in figure 1(a).This system is in the weak-coupling regime because the cavity loss is much larger than the modeemitter interaction.The primary source of cavity losses is the absorption of ENZ materials [41], with radiation and scattering being minimal.The effective model derived below suggests that the symmetrical placement of the emitters results in an entangled dark state that is immune to cavity loss.In the weak-coupling regime, our system achieves sustained dark state entanglement for a considerable time.
In figure 1(a), a dielectric cavity with a dielectric constant of ε 1 = 1 and a magnetic permeability of μ 1 = 1 is embedded in the ENZ materials with ε 2 = 0.01i and μ 2 = 1.The application of ENZ materials results in an infinite refractive index contrast between the inside and outside of the cavity, leading to the phenomenon of mode degeneracy.Specifically, 2 l -TM and 2 l+1 -TE modes resonate at the same wavelength in the same cavity [41].Without loss of generality, we chose the dielectric cavity that supports degenerate 2-TM and 4-TE resonances at frequencies ω 1 and ω 2 , respectively, where ω 1 ≈ ω 2 .These resonances are driven by a classical field E(ω) = E 0 e i ω t + c. c.Two two-level emitters with emission frequencies ω a and ω b are placed at distances d a and d b from the center of the sphere along the x-axis.As shown in figure 1(b), there are interactions between the emitters and cavity modes.While there is no interaction between the two cavity modes because they are orthogonal (details in supplementary material).Under the rotating wave approximation and dipole approximation, the Hamiltonian of this system can be written as: ) † is the free Hamiltonian for emitters and cavity modes, .
is the interaction between the emitters and modes, and is the driving term.Here, ω is the frequency of the driving field, σ a (s a † ) and σ b (s b † ) denote the lowering (raising) operators for the emitters, and a 1 (a 1 † ), a 2 (a 2 † ) are the bosonic annihilation (creation) operators of the 2-TM and 4-TE modes, respectively.g j1 (g j2 ) ( j = a, b) is the coupling strength between the emitter and 2-TM (4-TE) cavity mode.Ω 1 and Ω 2 are the Rabi frequencies of the driving field on 2-TM and 4-TE modes, respectively.
The dynamic process of the system is governed by the master equation where ρ is the density matrix of the system, κ i is the cavity loss of the i-th mode, with i = 1 corresponding to the 2-TM mode and i = 2 corresponding to the 4-TE mode.Cavity losses mainly come from the absorption of ENZ materials, unlike plasmonic cavity structures, where the cavity loss mainly originates from the radiation and Ohmic losses [19,35].In our system, κ i corresponds to the half-height width of the cavity mode radiation power spectra.The decay rate of 2-TM mode is κ 1 = 37.51 meV, and for 4-TE mode κ 2 = 12.94 meV [41].Since κ i is about tens of meV, much larger than g ji (<0.5 meV), the system is in the weak-coupling regime.Besides, γ j is the decay rate of the jth emitter.In our system, it is mainly due to the emitter's radiation into modes other than 2-TM/4-TE.Due to the application of ENZ materials, the radiation losses of the cavity modes are extremely low, resulting in narrow linewidths in the radiation power curve Radiation power spectra of the dielectric sphere in ENZ materials.There is minimal overlap between the 2-TM/4-TE mode and other modes.
The modulation of cavity modes on the interactions between emitters is reflected in coefficients such as g eff , γ eff , and Ω eff , allowing for a more straightforward illustration of how the modes affect entanglement.The effective transition frequencies w m eff , the effective coupling strength g mn eff , the effective Rabi frequencies W m eff , and the effective decay rates g g , , , , where m, n = a, b and m ≠ n.The interactions between two emitters, reflected in the effective parameters, are modulated by coupling strength g ji and cavity loss κ i .Due to the presence of degenerate modes within the dielectric cavity in ENZ materials, the effective parameters include two different terms with κ 1 and κ 2 , respectively.For the simplest case, when ω 1 = ω 2 = ω a = ω b = ω 0 , and two emitters are placed symmetrically d a = d b , we can obtain that g a1 = g b1 , g a2 = g b2 , so .The cavity mode regulates entanglement by affecting the effective spontaneous emission rate g mm eff and mutual interactions g mn eff between emitters.Here g g = mm mn eff eff , which is the basis of the dark state entanglement.Due to the interactions between emitters, the initially separate decay channels of the two emitters will be coupled.For simplicity, we consider the case where g eff mn is greater than 0. In the absence of the driving field, by diagonalizing the effective Hamiltonian, the eigenstates of the effective system are (shown in figure 3  , γ +− = γ −+ = 0.It can be seen that the decay of | − 〉 is immune to the cavity loss κ i .The state | − 〉 will not decay into |gg〉 or transit to | + 〉 so that it can be maintained, which means itʼs a dark state.It is worth noting that our approach to obtaining the long-time entanglement is in the weak-coupling regime.Different from the situation in the strong-coupling regime where the system can be considered robust to cavity loss and atomic decay [21], the dark state in our system is based on near-zero atomic decay in the cavity within ENZ material and the symmetry of the emitter position.With the existence of this entangled dark state, entanglement in our system can be maintained for a very long time.
We use concurrence to measure entanglement, which is defined as [44]  There is also transient entanglement in our system.When the emitters are placed asymmetrically, the decay rate γ − is greater than 0 due to the symmetry breaking, so the entangled state | − 〉 will decay with time.However, if ρ |+〉〈+| and ρ |−〉〈−| differ greatly, it is possible to achieve transient entanglement with high concurrence.Although it will decay faster over time, concurrence up to 0.6 can be achieved.

Results
In this section, we first discussed the crucial role of | − 〉 in the initial state for achieving dark state entanglement.Then, the implementation of transient entanglement was studied, which can achieve a concurrence greater than 0.5.Finally, we found that the entanglement between the two emitters exhibits a certain degree of robustness to their positions.Consequently, large entanglement can be achieved over a larger spacial range in the cavity.
In our system, the radius of the dielectric cavity in ENZ materials is 450.5 nm and its dielectric constant is ε = 1.2-TM and 4-TE modes are degenerate in this cavity, with resonance frequencies ω 1 = ω 2 = 1968 meV, corresponding to the wavelength around 630 nm.The transition frequencies of the emitters are also ω a = ω b = 1968meV, and the dipole moment of emitters is μ = 0.5 enm.The coupling strengths between the emitters and modes are obtained from the electric field distributions of 2-TM and 4-TE modes, as described in the supplementary material.As shown in figure 4, the coupling strengths are below 0.5 meV.The dielectric constant of ENZ material is ε 1 = 0.01i, so the cavity losses of 2-TM and 4-TE modes are κ 1 = 37.51 meV and κ 2 = 12.94 meV, respectively.Since κ 1 , κ 2 are tens of times larger than g a1 , g a2 , g b1 , and g a2 (all smaller than 0.5 meV), our system is in the weak-coupling regime.Besides, as there is almost no decay from emitters to other channels, the decay rate γ a , γ b of emitters can be considered as 0, as discussed earlier.
First, we discuss the influence of the initial state |j 0 〉 on the realization of the entangled dark state.The results show that the presence of | − 〉 in the initial state is crucial for achieving dark state entanglement.In this condition, we position the emitters symmetrically with d a = d b = 225.3nm (R/2), so g a1 = g b1 , g a2 = g b2 , making them identical.As mentioned earlier, the decay rate of , while the decay rate of | − 〉 is γ − = 0. Therefore, as shown in figure 5(a), when the initial state is |j 0 〉 = | + 〉, the concurrence decays rapidly from 1 to 0, while when the initial state is |j 0 〉 = | − 〉, the concurrence remains at 1. | − 〉 is regarded as the dark state, which almost stops evolving.Moreover, when the initial state is |j 0 〉 = |e a , g b 〉 or |j 0 〉 = |g a , e b 〉, the evolution process of the concurrence is exactly the same, gradually increasing over time and eventually approaching 0.5.Neither |e a , g b 〉 nor |g a , e b 〉 is an entangled state, so the concurrence is initially 0. At first, For |gg〉, C = 0, and for | − 〉, C = 1, so the concurrence remains at 0.5.However, when the initial state is |j 0 〉 = |ee〉, the system decays almost exclusively through the channel |ee〉 → | + 〉 → |gg〉, which has a greater decay rate, so there is no entanglement maintenance.Since the realization of dark state entanglement in our system is based on γ − = 0, the existence of | − 〉 in the initial state is crucial.While in the case of strong coupling, steady state entanglement is related to the redistribution of the energy of the initially excited quantum dots, so the initial state is a single quantum dot excitation [21].
In theory, as long as the emitters are placed symmetrically, they can be considered identical, so the entangled dark state can be achieved.However, when emitters are placed at different symmetrical positions and the initial state is |j 0 〉 = |e a , g b 〉, the concurrences ultimately peak at 0.5 at that when the coupling strengths are different, the decay rate γ + of | + 〉 will be different, so the rate of evolution to the dark state entanglement will be influenced by the positions of emitters.In our system, the realization rate of steady state entanglement can be manipulated by regulating the emitters at different symmetrical positions.In contrast, in the strong-coupling regime, the emitters usually need to be very close to the metallic structure, typically within a few nanometers, to achieve strong coupling [21].
Transient entanglement can also be realized in our system.When emitters are placed asymmetrically, the coupling strengths g ai and g bi are different, rendering the two emitters incapable of being considered identical anymore.As a result of the lack of symmetry, dark state entanglement cannot be achieved, implying that concurrence will decay to 0 over time.However, when the initial state is |j 0 〉 = |e a , g b 〉, the concurrence is no longer limited to a maximum of 0.5.As shown in figure 7(a), when emitters are placed in certain asymmetric positions, concurrence can be greater than 0.5, even reaching a maximum of 0.6.As the symmetry is broken, γ − is no longer 0, so the entangled state | − 〉 will decay over time.But at the same time, γ +− , γ −+ are also no longer 0, so there will be a transition between | + 〉 and | − 〉.When    greater than 200 nm, the concurrence almost always remains greater than 0.4 at κ 2 t = 5000.This is because the entanglement between emitters in our system [41] is mediated by degenerate 2-TM and 4-TE modes, based on the characteristics of the dielectric cavity in ENZ materials.So the entanglement is simultaneously regulated by the coupling strengths g a1 , g b1 , g a2 , g b1 , and cavity losses κ 1 , κ 2 .Compared to traditional single-mode resonant structures [10,46,47], our system provides more degrees of freedom to modulate entanglement, giving the system greater robustness to the positions of emitters.
In practice, ENZ materials can be realized by doped semiconductors [48], topological insulators [49], dielectric photonic crystals at the Dirac cone [50], etc.In the above discussion, we consider the dielectric cavity to be an air cavity, which may be prepared in ENZ materials using laser pulses by generating a material modification in the ENZ material [51].DNA origami [52] can be employed to connect single-molecule emitters [53,54] with ENZ materials.Additionally, the key to the entangled dark state lies in the high refractive index contrast between the inside and outside of the cavity.Conventional dielectric materials such as Si, SiO 2 , etc, are viable options for the cavity.Materials like quantum dots [39,55,56] can be appropriately positioned as emitters.However, interactions between the emitter and phonons induced by the lattice vibrations can lead to pure dephasing of the entangled qubits, causing the loss of quantum information and affecting the preservation of entangled dark states [57].This leads to an increase in the decay rate γ j of the emitter, which may no longer be considered negligible, thereby affecting the implementation of entangled dark states.Nevertheless, we can mitigate the decoherence process by cooling the system.For instance, at low temperatures, the decay rate of InGaAs quantum dots is only a few μeV [58], much smaller than the coupling strength g ji ( ̃0.3 meV) and the cavity losses κ i (tens of meV).Additionally, decoherence can be reduced by selecting suitable ligands to lower the phonon density [59].Furthermore, for the air cavity employed in the main text, the low phonon density in the environment has a negligible impact on the emitter.

Conclusion
We have demonstrated the dark state entanglement between two two-level emitters embedded in a spherical dielectric cavity within an ENZ material.An effective model has been derived to explain the mechanism of the dark state entanglement.We found that the decay rate of emitters in the dielectric cavity within ENZ materials is nearly 0. When the emitters are in symmetric positions, dark state entanglement is achieved, allowing the entanglement to be maintained for a long time.At the same time, when the emitters are in certain asymmetric positions, transient entanglement with high concurrence can also be achieved.Moreover, the relatively high entanglement also exhibits considerable robustness to variations in the position of the emitters.These findings provide potential applications such as quantum communication and quantum information processing in nanodevices, and are expected to promote the development of quantum technology.

Figure 1 .
Figure 1.(a) Schematic diagram of the system.A spherical dielectric cavity (white) is placed inside an ENZ material (blue).Two emitters are positioned at distances d a and d b from the center of the sphere along the x-axis.The long-lived entangled dark state is achieved.(b)Schematic diagram of the interaction in the system, which includes: coupling strength g ji between cavity modes and emitters, cavity losses κ i , and emitter decay rates γ i .
[41].According to the normalized radiation power spectra of several modes shown in figure2, there is almost no overlap between 4-TM/8-TE and 2-TM/4-TE modes, or 2-TE and 2-TM/4-TE modes.At λ = 630 nm, the ratio of the radiation power of 2-TM&4-TE modes and other modes (here we mainly consider 2-TE, 4-TM, and 8-TE modes) is approximately 0.9992: 0.0008.The radiation power of the emitters with dipole moments μ a = μ b = 0.5 enm in the cavity is around 1.42 × 10 −9 J so that γ a = γ b = 2.39 × 10 −6 meV.γ j is 10 5 times smaller than the coupling strength g ji , so we can treat γ j as 0 in the following, and this is crucial for achieving the dark state.By eliminating the field operators a i from Hamiltonian[42], an effective model involving only the equivalent emitter-emitter interactions can be derived (details are shown in supplementary material).Using the property that κ i ?γ i , we eliminated the rapidly dissipative terms associated with κ i and incorporated the modulation of κ i into various parameters of the effective model.With the above considerations, the effective Hamiltonian and the effective master equation can be written ): the excited state |ee〉 = |e a , e b 〉, the symmetric state | + 〉 = α + |e a , g b 〉 + β + |g a , e b 〉, the antisymmetric state | − 〉 = α − |e a , g b 〉 + β − |g a , e b 〉, the ground state |gg〉 = |g a , g b 〉, and the corresponding eigenvalues are

Figure 3 .
Figure 3.The effective energy levels of two emitters.| + 〉 and | − 〉 are entangled states, and the two decay channels have different decay rates.
ρ is the density matrix of the emitters (with 4 × 4 elements), λ i (i = 1, 2, 3, 4) are eigenvalues of rr ˜, and can be measured by quantum state tomography as mentioned in reference[45].Moreover, in the case of the entangled dark state with symmetrically positioned emitters, the Hanbury-Brown-Twiss experiments measuring the photon-photon correlations between the photons emitted by both emitters can also indicate the degree of entanglement[16].The value of concurrence ranges from 0 (separable) to 1 (maximal entanglement).When two emitters are identical, | − 〉 is a maximally entangled state with C = 1.Furthermore, the decay rate of | − 〉 is considered to be 0. Therefore, when the initial state contains | − 〉, the concurrence will persist at a value greater than 0, indicating the presence of dark state entanglement.

Figure 4 .
Figure 4.The coupling strength between emitters and 2-TM, 4-TE modes along x-axis, respectively.The 2-TM mode is represented by the black curve, while the 4-TE mode is represented by the red curve.

Figure 5 .
Figure 5. (a) The evolution of concurrence over time corresponding to different initial states.When the initial states are |e a , g b 〉 and |g a , e b 〉, the evolution of concurrence is exactly the same.(b) Expectation values of the diagonal element of the density matrix.Here, emitters are placed symmetrically at R/2, and κ 2 = 12.94 meV.

Figure 6 .
Figure 6.(a) The evolution of concurrence when emitters are placed at different symmetrical positions, with |j 0 〉 = |e a , g b 〉.When d a = d b = 261 nm and d a = d b = 423 nm, the rates of concurrence evolution are almost the same.(b) The decay rate γ + of | + 〉 when emitters are placed at distinct marked symmetrical locations.

Figure 7 .
Figure 7.The evolution of concurrence when emitters are placed at asymmetric positions, with (a) |j 0 〉 = |e a , g b 〉 and (b) |j 0 〉 = |ee〉.The maximum concurrence of transient entanglement can reach 0.6.

ρ
|+〉〈+| and ρ |−〉〈−| differ greatly, it is possible to achieve transient entanglement with higher concurrence.In particular, when d a = R/2 and d b = 370 nm, γ − = 6.8 × 10 −4 meV is relatively small, allowing for the long-time maintenance of a substantial entanglement with a concurrence exceeding 0.6.Similarly, as illustrated in figure 7(b), when the initial state is |j 0 〉 = |ee〉 and the emitters are placed in asymmetric positions, due to the transition from | + 〉 to | − 〉, the concurrence can break through 0 and even be maintained for a long time when γ − is close to 0. When we simultaneously change the positions of two emitters, we observe that the relatively high entanglement exhibits a certain robustness to the location of emitters.When the initial state is |j 0 〉 = |e a , g b 〉, the concurrence with different emitter positions d a and d b at κ 2 t = 5000 is obtained.κ 2 t = 5000 is chosen because this time is long enough and can reflect the characteristics of transient entanglement.As shown in figure 8, in the area around d a = d b and d a , d b

Figure 8 .
Figure 8.The concurrence when two emitters are in different positions with κ 2 t = 5000.The relatively high entanglement exhibits a certain robustness to the locations of emitters.