## Abstract

Ultra-thin metallic nanodisks, supporting localized plasmon (LP) modes, are used as a platform to facilitate high entanglement between distant quantum emitters (QEs). High Purcell factors, with values above 10^{3}, are probed for a QE placed near to an ultra-thin metallic nanodisk, composed of the noble metals Au, Ag, Al, and Cu. The disk supports two sets of localized plasmon modes, which can be excited by QEs with different transition dipole moment orientations. The two QEs are placed on opposite sides of the nanodisk, and their concurrence is used as a measure of the entanglement. We observe that the pair of QEs remains entangled for a duration that surpasses the relaxation time of the individual QE interacting with the metallic disk. Simultaneously, the QEs reach the entangled steady state faster than in the case where the QEs are in free space. Our results reveal a high concurrence value for a QES separation distance of 60 nm, and a transition energy of 0.8 eV (*λ* = 1550 nm). The robustness exhibited by this system under study paves the way for future quantum applications.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Controlling light-matter interactions at the nanoscale is important in a vast number of applications. The main obstacle is that these are weak interactions, and one way to enhance them is to take advantage of the surface plasmon (SP) modes to confine the light in nanometer dimensions, increasing not only the interaction strength, but also the interaction time between them. SP modes represent hybrid modes of the electromagnetic field and free electrons provided by conducting material [1,2]. Typically, noble metals are used for plasmonic applications, although they suffer from large material losses [3], and their optical response occurs mainly in the visible part of the spectrum.

Recently, a novel approach was introduced in noble metal plasmonics by fabricating ultra-thin metallic layers [4,5]. Considering the finite nanostructure of these ultra-thin layers, in particular nanoribbons, the authors were able to experimentally demonstrate the tuning of the plasmon modes in the near-infrared part of the spectrum by changing the width, spacing, and thickness of the nanoribbon pattern.

Various localized and extended surface modes have been used as mediators to enhance the interaction strength between quantum emitters and thus increase the interaction distance between them [6]. The emission properties and population state dynamics of the QEs placed in a nanostructured environment have been investigated with multiple materials and structures. QEs interacting with noble metal nanostructures [7–13], graphene nanostructures of different dimensionality (infinite planar structures, nanoribbon, nanodisk, nanotriangles, etc.) [14–18], carbon nanotubes [19–21], transition metal dichalchagogenides [22,23], and photonic crystals [24] have been investigated.

In this study, we investigate the entanglement dynamics between a pair of QEs in the presence of an ultra-thin noble metal nanodisk. The entanglement dynamics is typically investigated in the presence of Au and Ag plasmonic structures; the multilayer planar structures [25], 1D waveguides (cylindrical or channel) [11,12,26,27] and spherical geometries [28–32] support plasmon modes in the visible part of the spectrum. The ultra-thin nanodisk supports localized plasmon (LP) modes at the near-infrared part of the EM spectrum, at the telecommunication wavelength, thus increasing the functionality of the proposed scheme. Simultaneously, when infinite planar or 1D waveguides are considered, the SP mode travels along them in all available directions, and thus a large amount of the energy is lost without any use. The nanodisk supports localized modes, hence the nearby QEs can interact through them. In the case of metallic nanospheres, the separation between the QEs is small, causing a high direct interaction. The ultra-thin metal nanodisk allows efficient interaction between the QEs over larger distances. Finally, to promote the plasmonic response out of the visible part of the EM spectrum, patterned structures, such as metasurfaces, are considered. However, this requires an additional fabrication procedure, making the whole process more complicated compared to the case of nanodisk geometry.

To our knowledge, this is the first investigation of the entanglement dynamics between a pair of quantum emitters in the presence of an ultra-thin noble metal nanodisk. Our main findings are summarized as follows: a) the nanodisk supports entanglement between QEs that have a large separation distance, above $60\,$nm, and are placed on opposite sides of the nanodisk, b) the high entanglement between the two QEs is maintained over durations surpassing the relaxation rate of the individual QE in the presence of the metallic nanodisk, c) the entanglement of the QEs approaches the steady state faster than in the case of QEs in free space, and d) the ultra-thin metallic nanodisk provides localized plasmon (LP) resonances to the near-infrared part of the spectrum, at the telecommunication wavelength, such that we are not limited to the visible part of the spectrum.

The metallic nanodisk supports two sets of LP resonances that can be excited by the $x-$ and $z-$ transition dipole moment orientations of the QE. Moreover, the LP resonance energies can be tuned by changing the nanodisk radius or the noble metal material that they are composed of. The current fabrication capabilities for creating and patterning of ultra-thin nanostructures opens new possibilities for novel quantum applications.

In Sec. 2, we introduce the mathematical framework to describe the Purcell factor of an individual QE and the coupling terms between a pair of QEs. We employ the non-Hermitian description of the light matter interaction, using the Green’s tensor formalism [33,34]. We introduce the concurrence as the measure of entanglement between a pair of QEs. In Sec. 3, we present the Purcell factor of a single QE, the coupling terms between two QEs, and the influence they have on their values the two sets of LP modes supported by the noble metal nanodisk. By varying the nanodisk radius, we can tune the resonance energies of LP modes. We observe that when the transition energy matches the resonance modes supported by the nanodisk, high entanglement is observed. Finally, we compare the concurrence of QEs placed in free-space with an Au nanodisk between them. Sec. 4 presents the concluding remarks.

## 2. Theoretical model, noble metal quality factors, and plasmon resonances of the nanodisk

The QEs considered in this study are described as two energy level systems, where various natural (atoms, molecules) and artificial (quantum dots, color centers in diamond) systems can be approached with this way. We consider Au, Ag, Al, and Cu materials comprising the ultra-thin metallic nanodisk. Their optical response is given by the experimentally measured dielectric permittivity [35–37].

#### 2.1 Pair of quantum emitters interacting in a dissipative environment

The system we consider in this study consists of a pair of quantum emitters (QEs) interacting in the vicinity of a lossy environment. We consider a metallic nanodisk as the environment of the QEs in an arrangement shown in Fig. 1. The density matrix that describes the interaction between a pair of QEs is given by [11,12,25,26,30,38]:

We investigate the influence of the ultra-thin nanodisk to the entanglement of a pair of identical QEs in its vicinity. We consider that the QEs are placed on opposite sides of the center of the nanodisk at positions $\mathbf {r}_{1}=(0,0,z_{1})$ and $\mathbf {r}_{2}=(0,0,-z_{1})$, and the metallic disk is placed at $z=0$ at the $x-y$ plane, as shown in Fig. 1. This positioning simplifies the calculations of the density matrix, where we use the parameters $\Gamma (\mathbf {r},\omega )=\Gamma _{11}(\mathbf {r},\omega )=\Gamma _{22}(\mathbf {r},\omega )$, $\Gamma _{C}(\mathbf {r}_{1},\mathbf {r}_{2},\omega )=\Gamma _{12}(\mathbf {r}_{1},\mathbf {r}_{2},\omega )=\Gamma _{21}(\mathbf {r}_{1},\mathbf {r}_{2},\omega )$, and $\Omega _{C}(\mathbf {r}_{1},\mathbf {r}_{2},\omega )=\Omega _{12}(\mathbf {r}_{1},\mathbf {r}_{2},\omega )=\Omega _{21}(\mathbf {r}_{1},\mathbf {r}_{2},\omega )$; where the $\Gamma (\mathbf {r},\omega )$ is the relaxation of each QE in the presence of the nanodisk. $\Omega _{C}(\mathbf {r}_{1},\mathbf {r}_{2},\omega )$ and $\Gamma _{C}(\mathbf {r}_{1},\mathbf {r}_{2},\omega )$ are the real and imaginary parts of the coupling terms. $\Omega _{ii}$ denote the energy shifts of energy levels of the QEs, and they can be absorbed in the definition of the transition energy of each QE.

Considering the above conditions, the density matrix equations reduce to the form:

for the states $\left |e\right \rangle =\left |e_{1}e_{2}\right \rangle$, $\left |g\right \rangle =\left |g_{1}g_{2}\right \rangle$, $\left |s\right \rangle =1/\sqrt {2}\left (\left |e_{1}g_{2}\right \rangle +\left |g_{1}e_{2}\right \rangle \right )$ and $\left |a\right \rangle\! =\! 1/\sqrt {2}\left (\left |e_{1}g_{2}\right \rangle \!-\!\left |g_{1}e_{2}\right \rangle \right )$, where the last two states are the symmetric and antisymmetric states [25,30]. The symmetric state $\left |s\right \rangle$, is the state of maximum entanglement. The solution of these equations is given byTo evaluate the Purcell factor $\Gamma$ of the individual QEs, and the coupling terms $\Gamma _{C}$ and $\Omega _{C}$, between a pair of QEs, we must know the value of the Green’s tensor. Green’s tensor has a simple classical analogue, it gives the response of the geometry under consideration to a point like dipole excitation. The total Green’s tensor is written as $\mathfrak {G}(\mathbf {r}_{1},\mathbf {r}_{2},\omega )=\mathfrak {G}^{0}(\mathbf {r}_{1},\mathbf {r}_{2},\omega )+\mathfrak {G}^{\textrm {ind}}(\mathbf {r}_{1},\mathbf {r}_{2},\omega )$; where $\mathfrak {G}^{0}(\mathbf {r}_{1},\mathbf {r}_{2},\omega )$ represents the homogeneous part of Green’s tensor, accounting for the direct interaction between the two QEs, whereas $\mathfrak {G}^{\textrm {ind}}(\mathbf {r}_{1},\mathbf {r}_{2},\omega )$ is the induced part of the Green’s tensor and accounts for the interaction between the two QE induced by the presence of the metallic nanodisk. The emission energies of the QE considered in this study are below $1.5\,$eV (above wavelengths of $800\,$nm), while their separation is below 100 nm. Thus, we use the electrostatic approximation to approach the homogeneous and the induced part by the nanodisk. We also verify that the electrostatic approximation is valid by comparing its results with those obtained from a full numerical solution using the boundary element method (BEM) via the general public license software MNPBEM.

The homogeneous part of the Green’s tensor for the case the QEs are placed at $\mathbf {r}_{1}=(0,0,z_{1})$ and $\mathbf {r}_{2}=(0,0,z_{2})$ is given by:

#### 2.2 Quality factor of ultra-thin noble metal layers

The optical response of the noble metals considered is described through experimentally measured dielectric permittivity values; for Au JC and Ag JC Ref. [35], for Au CRC Ref. [37], and for Cu Palik and Ag Palik ref. [36] tabulated data are used. We focus on the two-dimensional nature of the problem considered and transform the dielectric permittivity of the noble metals $\varepsilon _{i}$ to the surface conductivity $\sigma _{i}$ through the relation $\sigma _{i}(\omega )=\frac {\omega d}{4\pi i}\left (\varepsilon _{i}(\omega )-1\right )$, where $i=$Au, Ag, Al, and Cu. $d$ denotes the thickness of the ultra-thin nanostructure. In this subsection, we discuss the case where we have an infinite planar layer of ultra-thin noble metal to present the influence of the material parameters to the quality of the plasmon mode.

The dispersion relation $\hbar \omega \left (k_{\textrm {SP}}\right )$ connects the energy of the applied EM wave and the in-plane wave vector, and presents the available modes supported from an infinite metallic layer. The noble metal film supports a symmetric and an anti-symmetric plasmon mode, and as the thickness of the film is decreased, the anti-symmetric mode dispersion relation approaches the light-line, such that the anti-symmetric SP mode is not a confined mode, while the symmetric mode breaks further away from the light-line. The surface plasmon wavevector of the symmetric mode is $k_{\textrm {SP}}=\frac {\omega }{c}\sqrt {\varepsilon _{1}-\varepsilon _{1}^{2}c^{2}/\left (4\pi ^{2}\sigma _{i}^{2}\right )}$, considering a constant dielectric permittivity host medium $\varepsilon _{1}$, and $c$ is the speed of light. The quality of the surface plasmon (SP) modes, supported by an ultra-thin noble metal layer, is described by their propagation length $L_{\textrm {SP}}$ and the penetration depth $\delta _{\textrm {SP}}$. The propagation length depicts the distance that the SP mode propagates along the metallic layer before is damped to $1/e$ over its initial value, and it is given by the relation $L_{\textrm {SP}}=1/(2\textrm{Imag}{(k_{\textrm {SP}})})$. The penetration depth represents the extent of the SP mode over the perpendicular direction to the thin metallic film, for our case along the $z$ axis; for a given energy, it defines the length within which a nearby QE can efficiently excite the SP modes through its near field. The penetration depth is defined as $\delta _{\textrm {SP}}=1/\left (2\textrm {Imag}\left (k_{z}^{\textrm {SP}}\right )\right )$, where $k_{z}^{\textrm {SP}}=\sqrt {k_{1}^{2}-k_{\textrm {SP}}^{2}}\sim ik_{\textrm {SP}}$ and thus $\delta _{\textrm {SP}}=1/\left (2\textrm {Real}\left (k_{\textrm {SP}}\right )\right )$. To evaluate the quality of the modes supported by the metallic layer, we define the quality factor $F$,

which indicates the trade off between the SP mode propagation over its perpendicular extension, before damping comes into play, where the maximum of this value represents the better quality of the SP mode [2,43]. This analysis indicates the influence of the noble metal material parameters on the SP modes.In Fig. 2, we present the quality factor $F=L_{\textrm {SP}}/\delta _{\textrm {SP}}$ of the SP modes of the ultra-thin metal layer. We observe that Ag supports SP modes with a high quality factor, $F>40$, over a broad part of the EM spectrum. Typically, in the field of plasmonics, Au and Ag materials are considered, with their optical response when infinite or finite structures are considered is in the visible part of the spectrum. Figure 2 shows that the Au and Cu layers support good quality modes at the near infrared part of the spectrum, while the Al layer supports poor quality modes in this region. The observed behavior is connected with material losses of noble metals, where Au, Ag, and Cu have lower material losses in the near-infrared part of the spectrum, making them excellent candidates for photonic applications close to the energy range of the telecommunication wavelength $1550\,$nm ($\hbar \omega =0.8\,$eV). Thus, it is detrimental to have nanostructures where they can support plasmon modes in the infrared part of the spectrum. When a noble metal nanosphere is considered, the dipole LP mode is supported when Real$\left (\varepsilon _{metal}\right )=-2\varepsilon _{host}$; considering the host medium to be air and an Au sphere, this condition is satisfied at the visible part of the spectrum at $2.5\,$eV. This limits the usefulness of noble metals in the telecommunication wavelength. In the following section, we introduce the ultra-thin nanodisk geometry and observe that its dipole LP resonance has its highest response in the near-infrared part of the spectrum.

For completeness, in Fig. 2 we consider tabulated data from CRC, Ref. [37], and JC, Ref. [35], for Au, and we observe that they present a slightly different optical response for the surface plasmon modes of the metallic layer. This shows the importance of the fabrication method. The use of pristine materials in the fabrication makes it possible to achieve even higher values of the quality factor of the SP mode.

#### 2.3 Modes supported by ultra-thin metallic nanodisk

Plasmons in ultra-thin infinite films exhibit confined excitation with lateral wave vectors $k_{\textrm {SP}}$, exceeding the free-space light wave vector $2\pi /\lambda _{0}$ by far. Thus, a direct light-plasmon coupling is not possible due to momentum mismatch. An ultra-thin metallic nanodisk supports localized plasmon (LP) modes at specific resonance frequencies, which can be found by expanding the charge density over an appropriate set of functions, $\rho (r)=\sum _{n=0}^{\infty }r^{l}c_{n}^{l}P_{n}^{(l,0)}\left (1-2r^{2}\right )$, where $l$ is the angular and $n$ is the radial eigenmode, and $P_{n}^{(l,0)}$ are the Jacobi polynomials. Solving the Poisson equation for the nanodisk specific charge density resonances emerge at eigenfrequencies, which are solutions of the equation [18,42]

where $\zeta _{n}^{l}$ are the geometric eigenmodes that can be calculated through an eigenmode equation [42], forming a unique set for the disk geometry independent of its radius, and $\omega _{n}^{l}$ are the relevant eigenfrequencies for each radius $R$. The LP modes with different angular eigenvalue $l$ can be excited by a QE with different transition dipole moment orientations, when they are placed at the center of the nanodisk, namely, $l=0$ the $z-$oriented and for $l=1$ the $x-$oriented dipole moments.We can consider the surface conductivity $\sigma _{i}$, for $i=$Au, Ag, Cu, and Al, fitted by a Drude surface conductivity, $\sigma _{i}(\omega )=\frac {\omega d}{4\pi i}\left (\varepsilon _{i}(\omega )-1\right )=\frac {\omega _{pi}^{2}d}{4\pi }\frac {i}{\omega +i\gamma _{i}}$ [44], where $\omega _{pi}$ is the plasma frequency and $\gamma _{i}$ are material losses, which are frequency dependent quantities. Then, by ignoring the losses $\gamma _{i}$, from Eq. (21) we extract an $\textit {approximate}$ expression in the electrostatic limit for the eigenfrequencies,

which reveals various trends for varying the different geometrical parameters. In particular, we observe that for increasing the value of the radius the resonance frequency is redshifted following a $1/\sqrt {R}$ dependence. Moreover, we observe that by increasing the thickness of the disk, $d$, the resonance is blueshifted. The electrostatic nature of Eq. (22) is observed by the scaling of LP frequency from the $\sqrt {d/R}$ term, although the 2D nature of our analysis dictates the thickness of the nanodisk to be small. In the results presented, we consider the disk radii, $R$, larger than $10\,$nm to avoid quantum effects in the description of the optical response of the nanodisk [45].We provide further details on the dipole and breathing LP modes supported by the metallic nanodisk. The metallic nanodisk confines the light in all three dimensions, and only specific surface charge distributions $\rho _{n}^{l}(r)$ are allowed, associated with the angular $l$ and radial $n$ eigenmodes . The $l=1$ and $n=0$ is the dipole LP mode and can be excited by a QE with a $x$-oriented transition dipole moment [46]. The dipole mode is a radiative mode that can be excited by a far field excitation. In Fig. 3(a), we present the charge distribution of the dipole LP mode. We observe that this mode is highly confined at the edge of the nanodisk. The SP wave number of this dipolar mode is $k_{\textrm {LP}}=2\pi /\lambda _{\textrm {LP}}$, where the LP wavelength of the dipole mode is given by $\lambda _{\textrm {LP}}=2\pi R$, due to the confinement of the charge density around the periphery of the nanodisk.

In addition to the dipole mode, we also have the $l=0$ and $n=1$ breathing mode, which is a special mode, as it is a dark mode and cannot be excited by direct light illumination [18,47]. A QE with a $z-$orientation transition dipole moment can directly excite the breathing mode. This mode has been experimentally probed for the case of a thick noble metal disk and a graphene disk [48,49]. We present that the breathing mode is sustained down to the 2D limit for the thickness of the noble metal nanodisk. Figure 3(b) shows that the breathing mode is an LP mode that can be described as a standing plasmon wave that is confined to the metallic nanodisk, and its plasmon wavelength is related to the radius of the nanodisk via $\lambda _{\textrm {LP}}=2R$, with $k_{\textrm {LP}}=2\pi /\lambda _{\textrm {LP}}=\pi /R$.

The Au, Ag, Cu, and Al noble metal nanodisks support the dipole and breathing modes with the charge distributions presented in Fig. 3, at different eigenenergies $\hbar \omega _{n}^{l}$ and maximum/minimum values, although these values do not change the physics discussed.

## 3. Results

#### 3.1 Purcell factor and coupling terms

In Fig. 4, we present the Purcell factor of a QE interacting with a metallic disk, varying its emission energy $\hbar \omega$. The QE is placed above the center of the nanodisk, $\mathbf {r}=(0,0,30\,\textrm {nm})$, and two different transition dipole moment orientations are considered, (a) along the $x-$axis and (b) along the $z-$axis. The nanodisk radius is $R=30\,$nm, and the Au, Ag, Al, and Cu noble metal materials are considered. We observe that for both orientations the Purcell factor of the QE, when its emission energy matches the LP eigenenergies, reaches values above of $10^{2}$, which means that the relaxation rate is enhanced above two orders of magnitude compared to the free space value. The two different transition dipole moment orientations excite the two different sets of LP modes supported by the noble metal nanodisk, when the transition dipole moment of the QE is along $x$ the dipole resonance is excited; a QE with $z$ transition dipole moment can excite the breathing mode. We observe that the dipole LP resonance provides an efficient relaxation channel for the QE at lower energies. The breathing mode, excited for a $z$ oriented QE, is consistently at higher energies for a given disk radius, per material. In Fig. 4(a), we observe that the higher order LP modes of the metallic nanodisk, which emerge at higher energies, exhibit lower Purcell factor values. This effect is due to the fact that the higher order LP modes are tightly confined on the disk, thus they decouple faster from the near field of the QE, as its separation distance with the nanodidk is increased. Interestingly, for QE-metallic disk separations of $30\,$nm we observe high Purcell factors of the QE close to the telecommunication wavelength of $0.8\,$eV ($\lambda =1550\,$nm), for a QE oriented along $x$. We see that Au and Ag have the highest response in terms of enhancement of the relaxation rate of a QE, compared to the free space value; Cu has also similar behavior to the Au, while Al in not of interest in the near infrared part of the spectrum.

To fully understand the interaction between a pair of quantum emitters (QEs), we must also investigate the coupling terms $\Gamma _{C}(\mathbf {r}_{1},\mathbf {r}_{2},\omega )$, and $\Omega _{C}(\mathbf {r}_{1},\mathbf {r}_{2},\omega )$ besides the relaxation rate of the individual QEs $\Gamma (\mathbf {r}_{i},\omega )$, with $i=1,2$. In the following sections, we concentrate on the Au and Ag noble metals, as they are the most experimentally investigated materials, and because they exhibit highly intriguing behavior in the telecommunication part of the spectrum (Fig. 4). As we have already observed, Al behaves poorly in the near-infrared part of the spectrum, and the Cu disk has almost the same response in the near infrared part of the spectrum as Au, such that all results for the Au disk apply for Cu. In Fig. 5, we present the coupling terms $\Gamma _{C}$ and $\Omega _{C}$ between two QEs placed on opposite sides of a metallic disk, $\mathbf {r}_{1}=(0,0,30\,\textrm {nm})$ and $\mathbf {r}_{2}=(0,0,-30\,\textrm {nm})$, for varying the emission energy of the QEs. The same figures plot the Purcell factor $\Gamma (\mathbf {r}_{1},\omega )$ of the individual QE. We consider two different orientations for the transition dipole moment of the QE, and we observe that for each of the orientations, a different set of modes is excited. All values are normalized to the free-space relaxation rate of the individual QEs $\Gamma _{0}$. We observe that when the LP modes are excited, there is a large enhancement of the coupling terms compared to the free space relaxation rate.

In the insets of Figs. 5(a) and 5(b), we compare the mode expansion method used in this study, Eqs. 18 and 19, and the full numerical solution using boundary element method via the general public license software MNPBEM [50,51]. We observe a good agreement between our method and the full numerical solution, where we focus on the Au disk, indicating that the electrostatic method introduced is valid.

In Fig. 6, we present a contour plot of the coupling terms between a pair of QEs, where we vary their transition energy and the radius of the nanodisk. The QEs are placed in the same positions as in Fig. 5, $\mathbf {r}_{1}=(0,0,30\,\textrm {nm})$ and $\mathbf {r}_{2}=(0,0,-30\,\textrm {nm})$. We now concentrate on a pair of QEs with their transition dipole moment along $x$, where the dipole LP mode is involved in the calculation. The dipole mode has the largest extension in the perpendicular dimension away from the $x-y$ plane that the nanodisk lies in, and the dipole mode is the one that is the most redshifted, thus allowing us to excite the LP modes within the telecommunication wavelength. We focus on Au disks, and similar results hold for the Ag, Cu, and Al materials, where the main difference is the eigenenergy values of the dipole resonance. Figure 5(a) presents the real, and Fig. 5(b) the imaginary parts of coupling terms, $\Omega _{Cx}$ and $\Gamma _{Cx}$, respectively. These terms achieve their highest/lowest values when they excite the LP modes. We observe that as the radius of the nanodisk increases, the resonance modes are redshifted, following the relation given by Eq. (22). A higher value of the coupling terms indicates a stronger interaction between the two QEs.

#### 3.2 Entanglement dynamics between a pair of quantum emitters

To study the effect the metallic disk has on the entanglement between the two QEs, we calculate the concurrence $C(t)$, Eq. (15). When the value of $C(t)$ is one, it means that the entanglement between the QEs is at the highest value, while for zero, the entanglement is at its lowest. We observe in Figs. 4 and 5 that a QE with a $z-$axis transition dipole moment excites the breathing LP mode which is always at higher energies compared to the dipole mode, for a given disk radius. Thus, we focus on investigating the entanglement between QEs with a $x-$axis transition dipole moment orientation.

A contour plot of the concurrence $C(t)$ is presented in Fig. 7, where we vary the transition energy of the QEs and the time evolution of the system. We consider the position of the QEs to be fixed at $\mathbf {r}_{1}=(0,0,30\,\textrm {nm})$ and $\mathbf {r}_{2}=(0,0,-30\,\textrm {nm})$, and the disk radius is $R=40\,$nm; the ultra-thin disk radius is chosen to have a dipole LP mode close to the telecommunication wavelength. In Fig. 7 the transition dipole moment of the QEs is along the $x-$axis, such the QE can excite the dipole mode supported by the metallic nanodisk at the transition energy of $0.85\,$eV. Two noble metal materials are considered, (a) Au CRC and (b) Ag JC. For both cases, we observe that when the emission energy of the QEs matches the energy of the LP resonances of the nanodisk, the concurrence obtains high values for long times. The time is scaled to the Purcell factor of a single QE $\Gamma (\mathbf {r}_{1,}\omega )$, where we see that when the transition energy of the QEs matches the LP resonances, then the entanglement between the two QEs is longer than the relaxation rate $\Gamma (\mathbf {r}_{1},\omega )$ of the individual QE. This is an important feature of the proposed scheme. We also observe that between the resonance modes there are fast oscillations of the concurrence $C(t)$, where after a few $\Gamma _{0}$ cycles $C$ approaches zero. Moreover, we observe that the dipole LP resonance, at $\hbar \omega =0.85$eV for Fig. 7(a) and $\hbar \omega =0.90$eV for Fig. 7(b), facilitates higher entanglement between the two QEs compared with the higher energy LP resonances.

To further understand the influence of the ultra-thin metallic nanodisk to the entanglement of two QEs, with fixed transition energy $\hbar \omega =0.8\,$eV, we study the concurrence $C$ when we vary the radius $R$ of the nanodisk and time $t$. We observe that for the larger radii of the nanodisk, higher order resonances are excited; these resonances are more confined to the nanodisk and thus at a distance of $30\,$nm away from the Au disk, $\mathbf {r}_{1}=(0,0,30\,\textrm {nm})$ and $\mathbf {r}_{2}=(0,0,-30\,\textrm {nm})$, lead to a short lived entanglement, due to the reduced interaction. The time scale of the plot of Fig. 8(a) is in units of the relaxation rate $\Gamma (\mathbf {r}_{1,}\omega )$ of the individual QE in the presence of the Au nanodisk, we observe that for a disk radius of $R=45\,$nm the entanglement of the pair of QEs is larger than the relaxation rates of the QE alone. At radius $R=45\,$nm we also observe that the entanglement approaches fast the steady state, while for radius lower or greater some oscillations are observed. An oscillatory behavior of the entanglement is unwanted for practical quantum applications.

The fact that the time scale of Fig. 8(a) is normalized to the relaxation rate of the QE in the presence of the Au nanodisk does not unveil the actual time evolution of the entanglement, as it is orders of magnitude faster than the free space relaxation of the QE. Moreover, the transition energy of the QEs is $\hbar \omega =0.8\,$eV ($\lambda =1550\,$nm), hence the QEs with a separation of $60\,$nm interact through their near field, meaning that the homogeneous part of the Green’s tensor, accounting for direct interaction between the two QEs, it is not negligible. Thus, in Fig. 8(b) we compare the entanglement between two QEs that are placed $60\,$nm away, at positions $\mathbf {r}_{1}=(0,0,30\,\textrm {nm})$ and $\mathbf {r}_{2}=(0,0,-30\,\textrm {nm})$, when they are in free-space and when a Au nanodisk is placed between them. We observe that for the case where the QEs are in free space, the value of the concurrence $C(t)$ oscillates very fast between $C=1$ and $C=0$. This is problematic when using the QEs for quantum applications in free-space or in a homogeneous dielectric environment. In contrast, when an ultra-thin nanodisk of radius $R=45\,$nm is placed between the pair of QEs, the entanglement rapidly approaches the steady state, and high entanglement, $C(t)\sim 0.5$, is persistent over long durations, making the full structure QE and Au nanodisk a good candidate for quantum applications. Furthermore, for longer durations, we observe that the entanglement between QEs in the free-space strongly oscillates with smaller amplitude, as shown in the inset of Fig. 8(b), whereas when the Au nanodisk is included between them, a constant entanglement is observed.

## 4. Conclusions

We theoretically study the entanglement dynamics between a pair of quantum emitters (QEs) in the presence of an ultra-thin metallic nanodisk. The metallic nanodisk is composed of Au, Ag, Al, and Cu noble metal materials and supports localized plasmon (LP) modes. The LP modes are categorized according to the angle, $l$, eigenvalues and are excited by the near field of the QEs, depending on the orientation of their transition dipole moment. The QEs are placed on opposite sides of the nanodisk at its center, the main quantities of interest are the Purcell factor of the individual QE and the real and imaginary parts of the coupling terms. These quantities are calculated by the Green’s tensor approach; we calculate the Green’s tensor in the electrostatic regime using an expansion over an orthogonal set of polynomials, whose method has excellent agreement with the one of the completely numerical BEM.

When the emission energy of the QEs matches the energies of the LP modes supported by the metallic nanodisk, the Purcell factor and the coupling terms are enhanced by several orders of magnitude, compared to the free space relaxation rate of the QE. Once the LP modes are excited, the nearby pair of QEs are strongly entangled; this entanglement reach the steady state rapidly, faster than the free space case and also overcomes the relaxation rate of the individual QE in the presence of the nanodisk. Our results focus on the energies of $\hbar \omega =0.8\,$eV, the telecommunication wavelength of $\lambda =1550\,$nm, a region of the EM spectrum that is very important for quantum technology applications, where the proposed scheme is also compatible with the silicon photonics applications currently used.

## Disclosures

The authors declare no conflicts of interest.

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