Modification of the Emission Spectrum of a Quantum Emitter in the Vicinity of Bismuth Chalcogenide Microparticles

Abstract

We examine theoretically the effect of bismuth chalcogenide microparticles on the spontaneous emission of a double-V-type quantum emitter in free space. We have found, in particular, that the presence of a single microparticle causes a high degree of quantum interference in the way the quantum emitter releases energy in the process of spontaneous emission. This, in turn, leads to significant changes in the spectrum of the emitted energy. The quantum emitter’s initial state is crucial to how the energy is released in the presence of the microparticle. This observation has potential implications for quantum computing, particularly for reading the state of atomic qubits. When two microparticles are present, the effect is more pronounced, particularly when the quantum emitter is located in the gap between the particles.

1. Introduction

Quantum plasmonics is a rapidly growing field that combines quantum mechanics and plasmonics, the study of the collective oscillations of electrons in metals or metal-like structures [1,2,3,4]. In quantum plasmonics, the principles of quantum mechanics are used to manipulate and control light at the nanoscale. This opens up new avenues for applications in photonics and sensing, such as ultra-small optical devices, high-resolution imaging, and biosensing [5]. One of the key features of quantum plasmonics is the use of quantum entanglement, where the quantum states of two or more particles are correlated in such a way that the state of one particle affects the state of another. This provides new opportunities for applications in quantum communication, cryptography, and quantum computing. Overall, quantum plasmonics and quantum nanophotonics represent a promising direction for future technological advancements in the fields of nanophotonics and quantum technologies [6].

The implementation of quantum technologies that utilize plasmonic devices requires a significant interaction between a quantum emitter (QE) and light. This connection can be reinforced through the integration of the emitter with a plasmonic structure that intensifies the emission rates by elevating the availability of the local density of states (LDOS) to the emitter. The above phenomenon is known in the literature as the Purcell effect [7] and has since been acknowledged as a crucial aspect of quantum plasmonics. The so-called Purcell factor measures the change in the spontaneous emission (SE) rate of the emitter in the presence of a photonic/plasmonic environment in comparison to its rate in vacuum [8,9].

The mainstream materials supporting plasmon excitations are noble metals: silver, gold, and copper. However, due to their intrinsic losses, mainly in the optical regime, alternative materials have been suggested in order to mitigate losses. Such materials are high-index dielectrics and semiconductors, such as Si, GaAs, and AlGaAs [10]. The use of high-refractive-index materials in plasmonics offers several benefits, such as improved light–matter interactions, decreased losses, and increased compatibility with silicon-based electronics [11]. These benefits make plasmonics with high-refractive-index materials a rapidly growing field for applications in areas such as sensing, photonics, and quantum computing [12].

A recent advancement in materials science has given rise to chalcogenides, which showcase impressive dielectric properties due to their robust phonon resonances in the THz range. They are materials that contain elements from the chalcogen group on the periodic table, including sulfur (S), selenium (Se), and tellurium (Te). These materials have unique optical and electronic properties and have been of great interest for photonics and optoelectronics [13,14,15]. Bismuth chalcogenides are known for their high refractive index [16] and strong light confinement properties, making them attractive for use in nanophotonics and other optical technologies [17,18]. In recent years, chalcogenides have been particularly well studied for their potential use in polaritonic excitations and for enhancing light–matter interactions in quantum technologies [19].

The polaritonic excitations result in the strong confinement of light and high field enhancement at their surfaces. On the other hand, noble metal nanostructures may exhibit enhanced Purcell factors due to plasmon resonances; however, their large ohmic losses hinder efficient coupling with quantum emitters (QEs) [20]. Bismuth chalcogenides, on the other hand, exhibit lower ohmic losses in combination with high field enhancement, leading to improved coupling with QEs through polaritonic excitations [21,22,23]. The polaritonic excitations in these materials are eigenstates of the electromagnetic field and result in a high partially local density of states (PLDOS) around the bismuth chalcogenides particles [24], thus leading to substantial modifications and control over the spontaneous emission rate of a nearby QE.

In the present work, we will study the effect of the presence of bismuth chalcogenide microparticles on the spontaneous emission spectrum of a multi-level QE. Namely, we show that when the QE is placed near a single bismuth chalcogenide microparticle or in between two such particles, the spontaneous emission of the QE depends strongly on the initial state of the QE. This occurs as the presence of the microparticles boosts the phenomenon of quantum interference of spontaneous emission paths, which, in turn, generates the dependence of the emission spectrum on the initial quantum state of the QE.

2. Quantum Interference in a Double-V-Type Quantum Emitter

Quantum interference (QI) is a result of the simultaneous emission of two nearby excited states of a quantum emitter into a shared ground state, creating the possibility for interference between the two emission paths. The typical quantum system for this to happen is the V-type system. The spectral pattern of the emission is shaped by the energy structure and quantum coherence; however, for interference to happen, the transition dipole moments of the two emission pathways in the V-type system must not be perpendicular. Achieving this in a pure quantum system can be challenging. However, Agarwal demonstrated in 2000 [25] that the creation of an anisotropic quantum vacuum can result in QI between the decay channels of perpendicular nearby states [26]. This occurs when there is a substantial difference in decay rates between orthogonal dipole moments, resulting in a noticeable degree of QI [27]. This phenomenon in a quantum emitter could result in exciting phenomena in quantum optics and coherent nonlinear optics, including coherent population trapping [28,29,30,31,32], a large degree of entanglement [33,34,35], amplification without inversion [36,37], controllable spontaneous emission or resonance fluorescence spectra [26,38,39,40], optical transparency accompanied with slow light [41,42,43], strengthened Kerr nonlinearity [44,45,46,47,48], and distinct non-Markovian dynamics [49].

Figure 1 shows the quantum system we study in the present work. We consider a double-V-type quantum system as having two closely lying upper states, $|2\rangle$ and $|3\rangle$, and two lower-energy states, $|0\rangle$ and $|1\rangle$. We take states $|2\rangle$ and $|3\rangle$ to characterize two Zeeman sublevels. Then, the dipole moment operator is taken as $\overrightarrow{\mu }={\mu }^{\prime }\left(\right|2\rangle \langle 0|{\widehat{\epsilon }}_{-}+|3\rangle \langle 0\left|{\widehat{\epsilon }}_{+}\right)+\mu \left(\right|2\rangle \langle 1|{\widehat{\epsilon }}_{-}+|3\rangle \langle 1\left|{\widehat{\epsilon }}_{+}\right)+H.c.$, where ${\widehat{\epsilon }}_{\pm }=({\mathbf{e}}_z\pm i{\mathbf{e}}_x)/\sqrt{2}$, is used for the description of right-rotating (${\widehat{\epsilon }}_{+}$) or left-rotating (${\widehat{\epsilon }}_{-}$) unit vectors. Furthermore, the dipole matrix elements $\mu$, ${\mu }^{\prime }$ are taken to be real.

Spontaneous emission occurs from states $|2\rangle$ and $|3\rangle$ to the lower state $|0\rangle$ with decay rates $2{\gamma }_2^{\prime }$ and $2{\gamma }_3^{\prime }$, respectively. Furthermore, spontaneous decay occurs from states $|2\rangle$ and $|3\rangle$ to the lower state $|1\rangle$ with decay rates $2{\gamma }_2$ and $2{\gamma }_3$, respectively. We assume that the transitions $|2\rangle$ and $|3\rangle$ to $|1\rangle$ are in energies that are influenced by the presence of nearby macroscopic objects (bismuth chalcogenide microparticles in our case). The transitions $|2\rangle$ and $|3\rangle$ to $|0\rangle$ are in quite different energies from the transitions $|2\rangle$ and $|3\rangle$ to $|1\rangle$, and we assume that they are not influenced by the presence of the microparticles. So, these transitions occur from the interaction of the quantum system with free-space vacuum, and we will address this decay below as the free-space decay. In the following part of the paper, we will calculate the spontaneous decay rate of this process.

The method of calculation of the spontaneous emission spectrum is based on the probability amplitude approach [50]. Under the application of the rotating-wave approximation and using the Weisskopf–Wigner theory of spontaneous emission, the probability amplitudes of states $|2\rangle$, $|3\rangle$, and $|0\rangle$ are given by [26,42,51] (we assume $\hslash =1$ throughout this work):

$$\begin{array}{ccc}\hfill {\dot{b}}_2\left(t\right)& =& \frac{1}{2}(i{\omega }_{32}-2{\gamma }_2-2{\gamma }_2^{\prime })b_2\left(t\right)-{\kappa }_3{b}_3\left(t\right),\hfill \end{array}$$

(1)

$$\begin{array}{ccc}\hfill {\dot{b}}_3\left(t\right)& =& -{\kappa }_2{b}_2\left(t\right)-\frac{1}{2}(i{\omega }_{32}+2{\gamma }_3+2{\gamma }_3^{\prime })b_3\left(t\right),\hfill \end{array}$$

(2)

$$\begin{array}{ccc}\hfill {\dot{b}}_{\mathbf{k}}\left(t\right)& =& -i{\delta }_{\mathbf{k}}{b}_{\mathbf{k}}\left(t\right)-i{g}_{\mathbf{k}2}{b}_2\left(t\right)-i{g}_{\mathbf{k}3}{b}_3\left(t\right),\hfill \end{array}$$

(3)

here ${\omega }_{32}={\omega }_3-{\omega }_2$, where ${\omega }_n$ is the angular frequency of level n (with $n=2,3$). Furthermore, the coupling constant of transition $|n\rangle \leftrightarrow |0\rangle$ with the $\mathbf{k}$-th vacuum mode is $g_{\mathbf{k}n}=-i\sqrt{\frac{2\pi {\omega }_{\mathbf{k}}}{V}}{\overrightarrow{{\mu }^{\prime }}}_{n0}·{\widehat{\epsilon }}_{\mathbf{k}}$. In addition, ${\widehat{\epsilon }}_{\mathbf{k}}$ denotes the unit polarization vector of the $\mathbf{k}$-th vacuum mode with angular frequency ${\omega }_{\mathbf{k}}$, and V is the quantization volume. Furthermore, ${\delta }_{\mathbf{k}}={\omega }_{\mathbf{k}}-({\omega }_{20}+{\omega }_{30})/2$, where ${\omega }_{n0}={\omega }_n-{\omega }_0$. Finally, important quantities are the parameters ${\kappa }_2$ and ${\kappa }_3$, which describe the coupling coefficients between states $|2\rangle$ and $|3\rangle$ due to spontaneous emission in the altered photonic environment due to the presence of the microparticles [25,51,52,53,54,55,56]. We will call these terms below the QI terms. The values of ${\gamma }_2$ and ${\gamma }_3$, and ${\kappa }_2$ and ${\kappa }_3$, are given by [51,53]

$$\begin{array}{ccc}\hfill {\gamma }_2& =& \frac{1}{2}\left({\Gamma }_{\perp }^{{\omega }_{21}}+{\Gamma }_{\Vert }^{{\omega }_{21}}\right),\hfill \end{array}$$

(4)

$$\begin{array}{ccc}\hfill {\gamma }_3& =& \frac{1}{2}\left({\Gamma }_{\perp }^{{\omega }_{31}}+{\Gamma }_{\Vert }^{{\omega }_{31}}\right),\hfill \end{array}$$

(5)

$$\begin{array}{ccc}\hfill {\kappa }_2& =& \frac{1}{2}\left({\Gamma }_{\perp }^{{\omega }_{21}}-{\Gamma }_{\Vert }^{{\omega }_{21}}\right),\hfill \end{array}$$

(6)

$$\begin{array}{ccc}\hfill {\kappa }_3& =& \frac{1}{2}\left({\Gamma }_{\perp }^{{\omega }_{31}}-{\Gamma }_{\Vert }^{{\omega }_{31}}\right).\hfill \end{array}$$

(7)

In the above equations, $\mathbf{r}$ refers to the position of the quantum emitter and ${\omega }_{n1}={\omega }_n-{\omega }_1$. Furthermore, it is convenient to define the spontaneous emission rates normal and parallel to the surface of the microstructure of interest, as ${\Gamma }_{\perp }^{{\omega }_{n1}}={\mu }^2{\omega }_{n1}^2\mathrm{Im}\left[G_{\perp }(\mathbf{r},\mathbf{r};{\omega }_{n1})\right]$ and ${\Gamma }_{\Vert }^{{\omega }_{n1}}={\mu }^2{\omega }_{n1}^2\mathrm{Im}\left[G_{\Vert }(\mathbf{r},\mathbf{r};{\omega }_{n1})\right]$, respectively.

When the two upper levels $|2\rangle$ and $|3\rangle$ are degenerate, then ${\omega }_{21}={\omega }_{31}=\overline{\omega }$$({\omega }_{32}=0)$. Then, ${\gamma }_2={\gamma }_3=\overline{\gamma }=({\Gamma }_{\perp }+{\Gamma }_{\Vert })/2$ and ${\kappa }_2={\kappa }_3=\overline{\kappa }=({\Gamma }_{\perp }-{\Gamma }_{\Vert })/2$. We can now define the spontaneous decay rates ${\Gamma }_{\perp ,\Vert }={\mu }^2{\overline{\omega }}^2\mathrm{Im}\left[G_{\perp ,\Vert }(\mathbf{r},\mathbf{r};\overline{\omega })\right]$, and the degree of QI is given by

$$p=\frac{\overline{\kappa }}{\overline{\gamma }}=\frac{{\Gamma }_{\perp }-{\Gamma }_{\Vert }}{{\Gamma }_{\perp }+{\Gamma }_{\Vert }}.$$

(8)

In the case that $p=1$, we have maximum QI. This can be achieved by placing the emitter close to a structure that completely inhibits ${\Gamma }_{\Vert }$. Obviously, when the emitter is placed in vacuum, ${\Gamma }_{\perp }={\Gamma }_{\Vert }$ and $\overline{\kappa }=0$, so no QI occurs in the system, as should be expected.

The spontaneous emission spectrum in the long-time limit is given by [50]

$$S\left({\delta }_{\mathbf{k}}\right)\propto {\left|b_{\mathbf{k}}(t\to \infty )\right|}^2.$$

(9)

In order to calculate the spectrum, we apply the Laplace transform in Equations (1)–(3), and use the final value theorem [50] to obtain [42]

$$\begin{array}{ccc}\hfill S\left({\delta }_{\mathbf{k}}\right)& \propto & {\gamma }_2^{\prime }\frac{{\left|(-i{\delta }_{\mathbf{k}}+i{\omega }_{32}/2+{\gamma }_3+{\gamma }_3^{\prime })b_2\left(0\right)-{\kappa }_3{b}_3\left(0\right)\right|}^2}{{\left|D\right({\delta }_{\mathbf{k}}\left)\right|}^2}\hfill \\ & +& {\gamma }_3^{\prime }\frac{{\left|(-i{\delta }_{\mathbf{k}}-i{\omega }_{32}/2+{\gamma }_2+{\gamma }_2^{\prime })b_3\left(0\right)-{\kappa }_2{b}_2\left(0\right)\right|}^2}{{\left|D\right({\delta }_{\mathbf{k}}\left)\right|}^2},\hfill \end{array}$$

(10)

where

$$\begin{array}{ccc}\hfill D\left({\delta }_{\mathbf{k}}\right)& =& (-i{\delta }_{\mathbf{k}}+i{\omega }_{32}/2+{\gamma }_3+{\gamma }_3^{\prime })(-i{\delta }_{\mathbf{k}}-i{\omega }_{32}/2\hfill \\ & +& {\gamma }_2+{\gamma }_2^{\prime })-{\kappa }_2{\kappa }_3.\hfill \end{array}$$

(11)

3. Results and Discussion

The computational setup of the present work is shown in Figure 2. However, we first assume that the QE of Figure 1 is in vacuum. The corresponding spontaneous emission spectrum is presented in Figure 3. It is calculated using Equation (10) and presuming that the emitter is in a vacuum environment. The results show that when ${\gamma }_2^{\prime }$ = ${\gamma }_3^{\prime }$ = ${\gamma }_2$ = ${\gamma }_3$ = $\gamma$, and ${\omega }_{32}=0$, the spectrum is a Lorentzian with the following form:

$$S\left({\delta }_{\mathbf{k}}\right)\propto \frac{\gamma }{{\delta }_{\mathbf{k}}^2+4{\gamma }^2}.$$

(12)

This means that the shape of the spectrum is independent of the specific initial conditions. However, when ${\omega }_{32}\ne 0$, the spectrum is given by the sum of two Lorenzians located at $\pm {\omega }_{32}/2$ and weighted by ${\left|b_2\left(0\right)\right|}^2$ and ${\left|b_3\left(0\right)\right|}^2$, respectively. The spectrum can be strongly influenced by the initial conditions in this case, as ${\left|b_2\left(0\right)\right|}^2$ and therefore the two Lorenzians will have different heights. However, for the particular parameters chosen in Figure 3b, the initial conditions have no impact on the shape of the spectrum, since ${\left|b_2\left(0\right)\right|}^2={\left|b_3\left(0\right)\right|}^2=1/2$.

Next, we assume that the QE of Figure 1 is placed in the vicinity of a single ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ microsphere with a 2 $\mathsf{\mu }$m radius (see Figure 2a). The dielectric function of ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ is provided by a triple sum of Lorentz-oscillator-type terms [18]

$$ϵ\left(\omega \right)=\sum _{j=\alpha ,\beta ,f}\frac{{\omega }_{pj}^2}{{\omega }_{0j}^2-{\omega }^2-i{\gamma }_j\omega }.$$

(13)

The subscripts in Equation (13) represent contributions from $\alpha$ and $\beta$ phonons, as well as bulk free-charge carriers f. These parameters can be found in [18] and are derived from fitting to experimental data on bulk ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ found in [57]. ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ is a well-known topological insulator, and its dielectric function has an extra component due to surface states, which decrease as the sphere radius increases [17]. This surface-state term only has a substantial impact on nanospheres, and is negligible in our case as we are using microspheres. We note that the dielectric function of Equation (13) assumes colossal values in the THz regime due to the existence of phonon–polariton modes [17].

In order to calculate the parameters of Equations (7) and (8), we need the spontaneous emission rates for a radially and tangentially (relative to a sphere) oriented dipole (QE) for the case of a single ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ microsphere. These are calculated via the EM Green’s tensor [58,59] based on Mie scattering theory. Figure 4 shows the degree of QI, p Equation (8), as a function of frequency and distance from the surface of a single ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ microsphere [22]. It is evident that there is a wide region of both parameters (frequency and distance), where the degree of QI assumes very high values. This region spans roughly, in frequency range, from 10 to 14 THz and, in distance range, from 0.8 to 1.7 $\mathsf{\mu }$m. We note that, when conducting high-precision quantum-optics experiments to investigate QI-related phenomena, using the photonic reservoir of modes supported by a ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ sphere is a more reliable method. This is because the QI values are much higher across a broad range of frequencies and distances from the surface, and a precise positioning of the QE is not critical. Similar effects have also been reported for another type of bismuth chalcogenide microparticle, namely, ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ microparticles [23], whose dielectric function is similar to that of Equation (13). We want to emphasize that for bismuth chalcogenide spheres with smaller sizes, i.e., nanospheres, the degree of interference is reduced, and a significant enhancement of the Purcell factors is obtained; therefore, the observed phenomena reported in our manuscript cannot be seen.

Based on Figure 4, the highest value of p (QI) for a $2\mathsf{\mu }$m ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ microsphere is $p=0.95$ and occurs at frequency 12.1 THz, when a QE (dipole) is placed at $2\mathsf{\mu }$m from the surface of the microsphere. In this case, the corresponding spontaneous emission rates are ${\gamma }_{\perp }=16.34\gamma$ (radially oriented dipole) and ${\gamma }_{\Vert }=0.41\gamma$ (tangentially oriented dipole). This remarkable difference in the spontaneous emission rates between the two different dipole orientations is the source of the high value $p=0.95$ of the degree of QI. By substituting the above values of ${\gamma }_{\perp }$ and ${\gamma }_{\Vert }$ into Equations (7), we obtain the corresponding values for ${\gamma }_2,{\gamma }_3,{\kappa }_2,{\kappa }_3$ needed for the calculation of the spectrum of Equation (10). The results are depicted in Figure 5. Namely, in Figure 5, we depict the free-space spontaneous emission spectrum Equation (10) for the same parameters as in Figure 3, except from the fact that the QE of Figure 6 is not assumed to be in free space but close to a ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ microsphere. We observe that the degeneracy of the spectrum for the two different initial states of the QE no longer exists. Even more, the presence of the ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ microparticle has opposite effects on the emission spectrum for the two initial quantum states. Roughly speaking, for the anti-symmetric state ($b_2\left(0\right)=\frac{1}{\sqrt{2}}$, $b_3\left(0\right)=-\frac{1}{\sqrt{2}}$) there is a 20-fold increase in the peak value, whereas for the symmetric state ($b_2\left(0\right)=b_3\left(0\right)=\frac{1}{\sqrt{2}}$) there is a 10-fold decrease in the same quantity. This is a strong manifestation of the symmetry break introduced by the presence of the microsphere, which creates an artificial ’anisotropic vacuum’ [25], which, in turn, promotes the presence of QI in the spontaneous emission paths of the QE of Figure 1. At the same time, we observe that the double-peak structure, in the absence of the microparticle and for ${\omega }_{32}=8\gamma$ (see Figure 3), becomes single-peaked when it is present. This is due to the higher values of the rates ${\kappa }_{2,3},{\gamma }_{2,3}$, making the two Lorentzian peaks of the vacuum case merge into a single peak [42] when the bismuth chalcogenide microparticle is present. Contrary to the case of ${\omega }_{32}=0$, when ${\omega }_{32}\ne 0$, there are no simple formulae for the emission spectrum $S\left({\delta }_{\mathbf{k}}\right)$, such as those of Equations (14) and (15).

The observed behavior of Figure 5a (${\omega }_{32}=0$) can be explained by looking at Equation (10) and considering different initial conditions. When $b_2\left(0\right)=b_3\left(0\right)=1/\sqrt{2}$, the resulting spectrum takes the form

$$S\left({\delta }_{\mathbf{k}}\right)\propto \frac{\gamma }{{\delta }_{\mathbf{k}}^2+{(\gamma +{\gamma }_{\perp })}^2}.$$

(14)

In this case, ${\gamma }_{\perp }$ is significantly higher than the decay rate in vacuum, which results in a significant increase in the denominator of the spectrum, leading to an overall suppression of the spectrum. On the other hand, when $b_2\left(0\right)=1/\sqrt{2}$ and $b_3\left(0\right)=-1/\sqrt{2}$, the spectrum takes the form

$$S\left({\delta }_{\mathbf{k}}\right)\propto \frac{\gamma }{{\delta }_{\mathbf{k}}^2+{(\gamma +{\gamma }_{\Vert })}^2}.$$

(15)

In this case, ${\gamma }_{\Vert }$ is significantly lower than the corresponding vacuum value, which leads to an enhancement of the spectrum and a narrowing of the line.

Next, we consider the case of a dimer of ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ microspheres with a 2 $\mathsf{\mu }$m radius, where the QE of Figure 1 is placed between the microspheres at equal distance from each sphere (see Figure 2b). With this particular setup, the degree of QI, p, is boosted even further due to the strong confinement of the photonic modes in the gap between the microspheres [23]. Figure 6 shows the spontaneous emission rates for the radial (a) and tangential (b) orientation of the QE relative to the sphere surfaces for three different sphere separations (gaps): $D=2000,200,100$ nm. First of all, it is evident that, overall, the spontaneous emission rates significantly increase with decreasing gap size for both types of QE orientations. This is due to the enhancement of the PLDOS within the gap, which acts as a cavity resonator. Secondly, the increase is not the same for both orientations, leading to very high values for the degree of QI, p. Evidently, from Figure 6c, we can see that p reaches values very close to unity. As an example, for a gap size of 2 $\mathsf{\mu }$m and frequency of 11.6 THz, the spontaneous emission rates become ${\gamma }_{\perp }=69.67\gamma$ (radially oriented dipole) and ${\gamma }_{\Vert }=0.35\gamma$ (radially oriented dipole), which provide a value of $p=0.99$ for the degree of QI. By substituting the above values of ${\gamma }_{\perp }$ and ${\gamma }_{\Vert }$ into Equations (7), we obtain the corresponding values for ${\gamma }_2,{\gamma }_3,{\kappa }_2,{\kappa }_3$ needed for the calculation of the spectrum of Equation (10). The results are depicted in Figure 7. It is evident that the findings of Figure 5 are even more pronounced for the case of the microsphere dimer of Figure 7. Especially for ${\omega }_{32}=0$, the spectrum for the symmetric case (top panel of Figure 7) is almost flattened out. We note that the spectral differences in Figure 7 can, again, be explained in terms of Equations (14) and (15). It is worth noting that more complicated setups involving additional microspheres are not required, given that the dimer already produces a value of $p=0.99$, which is almost as high as the theoretical maximum of unity.

4. Conclusions

We have investigated the impact of bismuth chalcogenide microparticles on the spontaneous emission of a double-V-type quantum emitter in free space. Specifically, we analyzed a system where one V-type transition is influenced by the interaction with the microparticles, while the other V-type transition interacts solely with a free-space vacuum. We demonstrated numerically that the presence of a single bismuth chalcogenide microparticle evokes a high degree of quantum interference in the spontaneous emission paths of the quantum emitter, which, in turn, leads to dramatic modifications to the spontaneous emission spectrum relative to the case where the microparticle is absent. Moreover, in the presence of the bismuth chalcogenide microparticle, the spontaneous emission spectrum depends critically on the initial state of the quantum. This effect would have potential application in quantum computing for reading out the state of atomic qubits. These phenomena are more magnified for the case of a dimer of bismuth chalcogenide microparticles, in which case the quantum emitter resides in the gap space between the microparticles.