1. Introduction

Light can be confined to sub-diffraction lengths by taking advantage of the surface modes that are confined in the interface between insulator media and materials with a negative real part of the frequency-depended dielectric permittivity. Among various such modes, the most studied one is the plasmon-polariton mode, a hybrid mode of the electromagnetic (EM) field and the free electrons provided by a noble metal [1]. The plasmon mode is usually resonant to the visible part of the EM spectrum and its functionality is hindered due to the high material losses [2]. The optical response of the noble metals can be engineered by nanostructuring to support plasmon resonances to the infrared part of the EM spectrum, where the material losses of the noble metals are reduced, with the cost, however, of additional fabrication steps.

Hybrid modes of nanostructured materials and metamaterials with EM fields are of great interest in the infra-red part of the EM spectrum [3–5]. A simple way to make a metamaterial is to consider a multilayer nanostructure of metal and dielectric layers; this type of multilayer geometry mimics the form of a photonic crystal [6,7]. Hexagonal boron nitride (h-BN) is a natural hyperbolic material that has its optical response in the mid-infrared part of the EM spectrum, an area sparse of optically active Van der Waals materials. This material supports phonon-polariton modes and is a platform for promoting strong light-matter interaction [4,5,8–12].

The electromagnetic interaction between two-level or multi-level quantum emitters (QEs) with their environment is important for quantum applications, where information can be stored to the QE and through the interaction with the environment can be transferred back and forth. The light-matter interaction between QEs placed in a nanostructured environment has been studied extensively, both experimentally and theoretically, by considering different materials and nanostructures as the environment of the QE [5,13–15]. For example, the QE spontaneous emission spectrum and the population dynamics have been investigated close to various nanostructures, including metallic nanostructures [16–30], metal-dielectric nanostructures [31–34], graphene [35–37], transition metal dichalcogenide materials [38–43], and photonic crystals [44–48], where reversible population dynamics due to strong light-matter coupling has been predicted in various studies.

Here, we focus on the population dynamics of a QE interacting with a h-BN layer, Fig. 1(a). As the QE/h-BN layer separation distance is reduced, the coupling strength between them is increased, resulting into an increasingly reversible population dynamics behavior, which depends on the transition energy and free-space decay rate of the QE, as well as the QE/h-BN separation distance and h-BN thickness. The reversible population dynamics means that the components of the QE/h-BN layer system interaction is within the strong coupling regime. Under specific conditions, the initially excited QE does not fully relax, at the $t\to \infty$ limit [20,26,28,32], a clear indication that a bound hybrid-state between the QE and the EM electromagnetic mode continuum is formed [26,28,32]. This type of exotic light-matter state is important for quantum applications [49,50]. The conditions for the formation of the bound state are also analyzed.

 

Fig. 1. (a) The geometry under consideration, a quantum emitter interacting with a h-BN layer, the contour plot presents the scattering part of the EM field distribution. (b) Real parts of the parallel, $\varepsilon _{x}(\omega )$, and perpedincular, $\varepsilon _{z}(\omega )$, dielectric permittivities of hBN layer.

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The manuscript starts by presenting the theory of the macroscopic quantum electrodynamics (QED) description of the QE population dynamics due to the light-matter interaction between the QE and the h-BN layer, as well as calculate the Purcell factor of the QE in proximity to the h-BN layer using the EM Green’s tensor method. In the results part, we calculate the QE excited state population dynamics for various system parameters, probe the conditions under which the necessary strong coupling regime for the QE/h-BN layer interaction is achieved, and explore the behavior of the population as $t\to \infty$, especially under conditions which allow for the formation of bound states in the continuum. We also compare semi-analytical results of the bound state population at long times with results obtained by solving numerically the time-dependent Schrödinger equation. Finally, the concluding results are presented.

2. Theory and material parameters

2.1 Purcell factor calculation for the h-BN layer

In this work we consider two-level quantum systems with transition frequencies in the mid-infrared part of the EM spectrum, such as quantum wells with inter-subband transitions [51]. The ground and excited states of the QE are denoted as $|g\rangle$ and $|e\rangle$, respectively; moreover, $\omega _{0}$ stands for the transition frequency and $\boldsymbol {\mu }$ denotes the transition dipole of the QE. An excited QE near an h-BN layer interacts with the EM modes continuum, thus, relaxing to the ground state by spontaneously emitting a photon, or forming a hybrid state with the EM mode continuum as modified by the h-BN layer in case the coupling strength between the QE and the EM continuum is strong enough for this purpose. The EM continuum states in proximity to the h-BN layer are the phonon-polariton modes supported by the h-BN layer.

The initial state of the system is denoted as $|i\rangle =|e\rangle \otimes |0\rangle$, where the QE is in the excited state and the EM continuum, as modified by the h-BN layer, is in its vacuum state. After the decay of the QE, the EM field will be in a $|1(\mathbf {k},p)\rangle =\hat {f}_{i}^{\dagger }(\mathbf {r},\omega )|0\rangle$ state, with $p$ and $\mathbf {k}$ being the polarization and wave-vector, respectively. $\hat {f}_{i}^{\dagger }$ is the bosonic creation operator which accounts for the far-field photon emission or the medium-dressed states, the phonon polariton modes discussed here. The final state of the entire system therefore has the form $|f\rangle =|g\rangle \otimes \hat {f}_{i}^{\dagger }(\mathbf {r},\omega )|0\rangle$. By applying Fermi’s golden rule and summing over all final states, the expression for the relaxation rate $\Gamma$ is given by $\Gamma (\mathbf {r},\omega )=\frac {2\omega ^{2}\mu ^{2}}{\hbar \varepsilon _{0}c^{2}}\hat {\mathbf {n}}\cdot \textrm {Im}\,G(\mathbf {r},\mathbf {r},\omega )\cdot \hat {\mathbf {n}}$, where $\mathbf {\hat {n}}$ is a unit vector along the direction of the transition dipole moment $\boldsymbol {\mu }$, and $G(\mathbf {s},\mathbf {r},\omega )$ is the Green’s tensor representing the response of the geometry under consideration to a point-like excitation located at $\mathbf {r}$, where $\mathbf {s}$ is the position at which its response is calculated [52–54].

The influence of the h-BN layer on the QE emission is related to the Purcell factor, $\tilde {\Gamma }_i(\mathbf {r},\omega )=\Gamma _i(\mathbf {r},\omega )/\Gamma _{0}$, of the QE, where $\Gamma _{0}=\omega ^{3}\mu ^{2}/3\pi c^{3}\hbar \varepsilon _{0}$. The optical response of the h-BN layer is given by its dielectric permittivity

h-BN is a natural uniaxial material, the two dielectric permittivities related to the two spatial dimensions lead to the existence of two Reststrahlen bands, (a) the lower band, when real $\left (\varepsilon _{z}\right )<0$ and real $\left (\varepsilon _{x}\right )>0$ corresponding to Type 1 hyperbolicity, and (b) the upper band, when real $\left (\varepsilon _{z}\right )>0$ and real $\left (\varepsilon _{x}\right )<0$ corresponding to Type 2 hyperbolicity [55–58]. The Reststrahlen bands are the energy intervals between the longitudinal (LO) and transverse (TO) optical phonon frequencies. In Fig. 1 we present the two Reststrahlen bands and we also show the areas where the Type 1 and 2 hyperbolicities exist.

The Purcell factor of a QE, with a transition dipole moment along $z$, that is placed at $z_{\textrm {QE}}$ above a h-BN layer is given by

2.2 Population dynamics

The general state of a two-level system placed at distance $z$ above a h-BN layer is given by [20,31]:

The population dynamics of the excited state of the two-level system is obtained by calculating the coefficient $c_{1}(t)$. For this purpose, we use the macroscopic QED Hamiltonian, acting to the system state Eq. (5) is used,

The population of the QE is found by numerically solve the following integro-differential equation:

3. Results and discussion

3.1 Purcell factor

The spontaneous emission relaxation process of a QE is determined by its interaction with the EM field. The Purcell factor is related to the interaction strength between a QE and the EM mode continuum as modified by the h-BN layer. In Figs. 2(a,c) and Figs. 2(b,d) we present the Purcell factor of a QE placed above a h-BN layer of thickness $d$, as function of the QE transition energy, $\hbar \omega _0$, with a $z$-oriented and $x$-oriented, respectively, transition dipole moment. We consider different h-BN layer thickness values, $d=10$ nm and $d=50$ nm, and different distances between the QE and the layer, $z_{\textrm {QE}}=10,\,20,\,50$ and $100\,$nm.

 

Fig. 2. Purcell factor of a quantum emitter, interacting with a h-BN layer of thickness $d$. The position of the QE is $\mathbf {r}=(0,0,z_{\textrm {QE}}+d)$. Two h-BN layer thickness values are considered: (a,c) $d=10\,$nm and (b,d) $d=50\,$nm. Also, two different orientations of the transition dipole moment of the QE are considered: (a,b) $z$-oriented and (c,d) $x$-oriented. The h-BN layer is placed at the $x-y$ plane with the lower edge at $z=0$.

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We observe that when the QE/h-BN separation distance is small, e.g. $z_{\textrm {QE}}=10\,$nm, then the Purcell enhancement is larger than six orders of magnitude, due to the coupling of the near field of the QE with the phonon-polariton modes. As $z_{\textrm {QE}}$ increases, the enhancement decreases, but still its value remains over 100 at very large distance, $z_{\textrm {QE}}=100\,$nm. As the thickness $d$ of the h-BN layer is increased, the Purcell enhancement does not practically change, attaining a similar value like we have a semi-infinite h-BN substrate. Furthermore, for the thinnest h-BN layer, $d=10\,$nm, we observe that for the higher energy Reststrahlen band, the peak of the Purcell factor is red-shifted as the distance of the QE/h-BN layer is increased, while in case of a $d=50\,$nm h-BN layer thickness, the position of the peak value remains unchanged as $z_{\textrm {QE}}$ increases. The reason for this lies in the fact that for the thinner h-BN layer, the near field of the QE decouples from the layer easier than in case of a thicker one, since the phonon-polariton modes are more tightly confined on the h-BN layer in comparison to the thicker layer. Moreover, we further observe that a QE with $z$-oriented transition dipole moment features a larger Purcell enhancement, than in the case of a QE with a $x$- oriented one; we find that roughly $\Gamma _{z}/\Gamma _{x}=2$, which is a common property of a QE placed in proximity to infinite planar nanostructures [43,60]. More details for the connection of the phonon polariton modes and the Purcell factor are given in Ref. [11].

3.2 Excited state population dynamics

Here, we present the population dynamics of a QE in proximity to a h-BN layer. The h-BN layer thickness is $d=10$nm and we consider various values for the QE/h-BN layer separation, $z_{\textrm {QE}}=20,\,30,\,40,\,50$ nm. In Figs. 3(a,b) we focus on a QE with transition energy $\hbar \omega _{0}=0.1024\,$eV and free-space lifetime $\tau =0.5\,$ns; in Figs. 3(c,d) we focus on a QE with transition energy $\hbar \omega _{0}=0.1902\,$eV and free-space lifetime $\tau =0.3\,$ns. Moreover, two different transition dipole moment orientations are considered, along $z$ in Figs. 3(a) and (c), and along $x$ in Figs. 3(b) and (d). The values of the transition energy of the QE is chosen so as to match the energies at which the directional Purcell factor has its peak value, as shown in Figs. 2(a) and (c). We choose the specific values of $\tau$ in order to probe dynamics in the strong coupling regime. Such values are commonly met in molecular systems at transition frequencies lying in the infrared regime. We remind to the reader that $\tau =1/ \Gamma _0$ connecting the free-space lifetime to the free-space relaxation rate.

 

Fig. 3. Population of the excited state of a QE, placed at $z_{\textrm {QE}}$, with a free-space lifetime $\tau$, next to a h-BN layer of thickness $d=10\,$nm. Several QE/h-BN layer separation distances are considered: $z_{\textrm {QE}}=20,\,30,\,40$ and $50\,$nm. Panels (a,b): The QE transition energy is $\hbar \omega _{0}=0.1024\,$eV and $\tau =0.5\,$ns. Panels (c,d): The QE transition energy is $\hbar \omega _{0}=0.1902\,$eV and $\tau =0.3\,$ns. Two different transition dipole moment orientations of the QE are considered: (a,c) $z-$oriented and (b,d) $x-$oriented.

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In all panels of Fig. 3, we observe that as the separation distance between the QE and the h-BN layer is decreased, the population dynamics shows a highly oscillatory behavior. At the higher distances for the transition energy of $\hbar \omega _0=0.1024\,$eV we observe that the population dynamics follows sinusoidal-type oscillation, the so-call Rabi oscillations. Reducing the QE/h-BN layer the oscillations have a smaller period and break away from the simple Rabi-oscillation model, due to the enhancement of the light-matter interaction coupling between the QE and the h-BN layer, as indicated by the corresponding enhancement of the Purcell factor shown in Fig. 2. For the lower transition energy of the QE, $\hbar \omega _0 = 0.1024\,$eV, we observe that the amplitude of the oscillations of the excited state population of the QE, for both transition dipole moment orientations, decrease as the time evolve. Also, there is a fine structure in the excited state dynamics for 20 nm distance and $\hbar \omega _{0} = 0.1024$ eV. This fine structure at these conditions is an indication of the highly non-Markovian character of the dynamics due to the very strong light-matter coupling. In such a case, the dynamics can be affected not only by the phonon polariton mode resonant to the QE transition frequency but also by other energetically close phonon polariton modes. For a QE with transition energy $\hbar \omega _0=0.1902\,$eV we observe that by decreasing the QE/h-BN layer distance, the population remains only transiently trapped. Meaning that at later times the population of the excited state of the QE drops, not shown here due to the high oscillations frequency.

In Fig. 4 we investigate the QE/h-BN layer interaction when $z_{\textrm {QE}}=10\,$nm, and the QE parameters are the same as Fig. 3. For the $z$-oriented transition dipole moment in both cases of QE transition energies, we find that the population of the QE is fully trapped, since it attains a constant value, about 0.25 for $\hbar \omega _0=0.1024$ eV and 0.15 for $\hbar \omega _0=0.1902$ eV, respectively, as time grows, shown in Figs. 4(a) and (c). This is a sign of the formation of a bound state. The bound states in relation with the Figs. 4(a) and (c) are defined as hybrid states between the two-level system QE and the EM continuum, into which population cannot fully decay out at the $t\to \infty$ limit. We note that the infinite time limit is understood as the time that is larger than any other time scale involved in the system, which in our case, is given by the QE lifetime $\tau$. When the transition dipole moment of the QE is along $x$, the QE/h-BN layer interaction drops to half, thus in Figs. 4(b) and (d) we observe that the excited state population drops versus time and no population trapping is observed.

 

Fig. 4. Population of the excited state of a QE, placed at $z_{\textrm {QE}}$, with a free-space lifetime $\tau$, next to a h-BN layer of thickness $d=10\,$nm; the QE/h-BN layer separation is $10\;$nm. Panels (a,b): The QE transition energy is $\hbar \omega _{0}=0.1024\,$eV and $\tau =0.5\,$ns. Panels (c,d): The QE transition energy is $\hbar \omega _{0}=0.1902\,$eV and $\tau =0.3\,$ns. Two different transition dipole moment orientations of the QE are considered: (a,c) $z-$oriented and (b,d) $x-$oriented.

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The QE/environment bound states are usually found when nanostructures presenting a band gap in the available mode are considered [6,7,44,46,50,61], while among such structures the photonic crystals are the most well studied. The physical reason is that, if a QE is placed in a photonic crystal and has transition energy in the bang gap of the available EM modes, it cannot relax to the ground state and a bound state is formed. Thus, the bound state results in a localization of the EM field at the QE location [6,7,44,46].

As we have already described, the h-BN layer supports phonon-polariton modes at specific energies. When the transition energy of the QE is resonant with them and the QE/h-BN layer separation is small, the Purcell factor enhancement can reach such high values that the light-matter interaction is in the strong coupling regime. Then, the population dynamics of the QE features strongly non-Markovian behavior, where population trapping is present, as it is shown in Figs. 4(a) and (c), for longer times as the QE/h-BN distance is decreased. Further increasing the QE/h-BN interaction strength, by reducing the QE/h-BN separation, the population trapping will sustain for long times. In the following part we derive a semi-analytical estimate for value of the trapped population when a bound state is formed.

Lets assume that the system under consideration has a stationary bound state, of the form:

Using the above two equations and the identity $\textrm {Imag}[G(\mathbf {r},\mathbf {r},\omega )]=\int {dr}^\prime \frac {\omega ^2}{c^2}\varepsilon _I(\mathbf {r}^\prime ,\omega )\mathbf {n}^*\cdot G^*(\mathbf {r}^\prime ,\mathbf {r},\omega ) G(\mathbf {r}^\prime ,\mathbf {r},\omega )\cdot \mathbf {n}$, where $\mathbf {n}$ is the transition dipole moment orientation of the QE and $\mathbf {r}_0$ is its position, we extract the relation

Lets now consider the time evolution of a state describing the QE being initially excited and the dressed EM field in the ground state, $\left |\Psi (0)\right \rangle =\left |e,\left \{ 0_{\mathbf {r},\omega }\right \} \right \rangle$; the time evolution of this bound state follows $\left |\Psi (t)\right \rangle =c_{1B}e^{-i\omega _{B}t}\left |\Phi _{B}\right \rangle$, with $c_{1B}=\left \langle \Phi _{B}|\Psi (0)\right \rangle$ at $t=0$. The overlap with the system initial state is given by $\left \langle \Psi (0)|\Psi (t)\right \rangle =\left |c_{1B}\right |^{2}e^{-i\omega _{B}t}$.

From Eq. (10), and the normalization condition $\left \langle \psi _{B}|\psi _{B}\right \rangle =1$, the population coefficients satisfy the relation,

Thus the QE population for $t\to \infty$ when a bound state is formed, is given by the expression:

The existence of a bound state in the QE/h-BN layer system depends on various parameters of the QE; internal factors, such as the QE transition energy, its free-space lifetime and its transition dipole moment orientation, and external factors, such as the position of the QE above the h-BN layer and the thickness of the h-BN layer. We further know that the higher the Purcell enhancement factor, the stronger the coupling of the QE with the EM modes continuum. Also, the value of the transition dipole strength is very important for approaching the strong coupling limit, since the higher its value the less enhancement of the Purcell factor of the QE is needed [13]. In the following, we investigate in detail the influence of the above parameters on the QE/h-BN layer system dynamics.

In Fig. 5 we present the dependence of the excited bound state probability on the free-space lifetime $\tau$ of the QE. We consider a QE that is placed at $z_{\textrm {QE}}=10\,$nm above the h-BN layer. Two thicknesses of the h-BN layer are considered $d=10$ and $50\,$nm. In Figs. 5(a) and (b), we consider a QE with transition energy $\hbar \omega _{0}=0.1024\,$eV and in Figs. 5(c) and (d) a QE with $\hbar \omega _{0}=0.1902\,$eV; these energies match the Purcell factor peak values at the lower and higher Reststrahlen band, respectively, as shown in Fig. 2(a).

 

Fig. 5. Population of the excited state of the QE placed at $z_{\textrm {QE}}=10\,$nm, when a bound state of the QE/h-BN layer composite system exists, as function of the QE free-space lifetime $\tau$. Two values for the thickness of the h-BN layer are considered $d=10$ and $50\,$nm. Panels (a,b): The QE transition energy is $\hbar \omega _{0}=0.1024\,$eV and the transition dipole moment is $z$-oriented (a) and $x$-oriented (b). Panels (c,d): The QE transition energy is $\hbar \omega _{0}=0.1902\,$eV the transition dipole moment is $z$-oriented (c) and $x$-oriented (d). The insets present the bound state energy, $E_B$, as a function of $\tau$, for the different set of parameters.

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In Fig. 5(a) we observe that the QE/h-BN layer system can form a bound state as long as the QE lifetime is $\tau \le 0.8\,$ns, for $d=10\,$nm, and $\tau \le 0.9\,$ns, for $d=50\,$nm, and its transition dipole moment is $z$-oriented; when a QE with a $x$-oriented transition dipole moment is considered, as shown in Fig. 5(b), the critical free-space lifetime for forming a bound is decreased to $0.4\,$ns, for $d=10\,$nm, and $0.45\,$ns, for $d=50\,$nm. This effect can be explained by comparing the value of the QE Purcell enhancement for $z-$ and $x-$orientations of its transition dipole moment, where $\Gamma _{z}/\Gamma _{x}\sim 2$, which implies that the QE/h-BN layer interaction is weaker when the transition dipole moment is along the $x$ axis. Also, we observe for Fig. 2 that for the thicker h-BN layer the enhancement of the Purcell factor spectrum of the QE is higher, for a given transition dipole moment orientation.

When the QE is placed at the same position but its transition energy is resonant to the higher Reststrahlen band, $\hbar \omega _{0}=0.1902\,$eV, as shown in at Figs. 5(c) and (d), we observe that the QE/h-BN layer system can form a bound state for smaller QE lifetimes. For the z-oriented transition dipole moment when $\tau >0.4\,$ns, for $d=10\,$nm, and $\tau >0.46\,$ns, for $d=50\,$nm and for x-oriented transition dipole moment when $\tau >0.21\,$ns, for $d=10\,$nm, and when $\tau >0.23\,$ns, for $d=50\,$nm, there are no bound states. Therefore, when a QE emits spontaneously at the lower Reststrahlen band frequency, the QE/h-BN layer light-matter interaction is stronger and the system forms a bound state, even for a QE with relatively long free-space lifetimes. This effect is again clearly attributable to the corresponding Purcell enhancement spectrum that features a sharper peak at the lower energy band, compared to a wider one at the higher energy band.

The insets of Fig. 5 present the bound state energy, $E_B$, for the different set of parameters, varying the free-space lifetime, $\tau$. We observe that the higher the QE/h-BN layer coupling the bound state energy, $E_B$, is getting more negative; the more negative values translates into the bound state being deeper in the band-gap. We clearly observe that the bound state formation, translates to $E_B<0$.

To further investigate the influence of the lifetime and the transition energy of the QE, interacting with a h-BN layer of thickness $d=10\,$nm, to the bound state formation, in Fig. 6 we present contour plots of the bound state population, when we vary the above two intrinsic properties of the QE. More specifically, in panels (a) and (b) of this figure, we consider a QE with $z$- and $x$- transition dipole moment orientations, respectively. Similar results can be drawn for the bound state probability of the QE/h-BN layer system for thicker layers, although the particular population time evolution will not follow the same dynamics profile.

 

Fig. 6. Contour plot of the bound state population as function of the QE transition energy, $\hbar \omega _{0}$, and lifetime, $\tau$. The QE is placed at $z_{\textrm {QE}}=10\,$nm above a h-BN layer of thickness $d=10\,$nm. Two QE transition dipole moment orientations are considered: (a) along the $z$ axis and (b) along the $x$ axis. The absence of color defines the region that there are no bound states in the continuum (BIC).

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As expected, when the transition dipole moment of the QE is along the $z$-axis, a bound state can be formed for larger values of the free-space lifetime, compared to the case of a QE with $x$-oriented transition dipole moment. Moreover, we observe that at lower QE transition energies, bound states can be formed for a wider set of parameters, even when the transition energy of the QE is away from the lower and higher energy Reststrahlen bands, where the phonon-polariton modes exist. Interestingly, this comes in contrast to the common understanding that for achieving the strong coupling regime for a QE/nanostructure composite system, the transition energy of the QE should significantly overlap with the resonances supported by the nanostructure. We should also keep in mind that the state dynamics calculated through Eq. (7) takes in account the whole energy range of the Purcell enhancement spectrum, which obviously includes the phonon-polariton resonances frequencies, as shown in Fig. 2. Furthermore, as the QE transition energy decreases close to zero, the transition wavelength of the QE increases to significant values; so a QE/h-BN separation distance of $10\,$nm, implies that the QE is within the extreme near field, thus the Purcell enhancement reaches high values due to the excitation of the bulk non-propagating modes that are connected with losses, which further implies that a QE emitting at these energy range can easily approach the strong light-matter coupling regime. We believe that this can be a general characteristic, which requires further investigation though.

4. Conclusions

In this work we investigated the population dynamics of a two-level QE when interacts with a h-BN layer. We considered this interaction in the weak and strong coupling regimes. We showed that the composite system QE/h-BN layer can form bound states, under certain conditions. The formation of the bound state due to the QE/h-BN layer interaction depends on the intrinsic properties of the QE, such as its transition energy and its free-space lifetime, and extrinsic properties of the composite system, such as the separation between QE/h-BN layer and the h-BN layer thickness.

Firstly, we considered the population dynamics of a QE with fixed intrinsic properties, by decreasing the QE/h-BN separation distance. We found that by tuning the separation distance, we tune the interaction from the weak to strong coupling regime, which even results to the formation of the bound state of the composite system. Secondly, we presented a semi-analytical way to calculate the bound state population and further investigate the range of the intrinsic parameters of the QE in order the total system QE/h-BN can form a bound state.

In conclusion, here, we showed that the h-BN layer can be a platform for investigating the light-matter interaction up to the strong coupling limit, where Rabi oscillations in the population dynamics are present. The material parameters used here to describe the optical response of the h-BN layer and the two-level QE are experimentally accessible making our finding directly accessible to experimental testing.