# Dynamic Control of Spontaneous Emission Using Magnetized InSb Higher-Order-
Mode Antennas

Sina Aghili  School of Electrical Engineering and Computer Science, University
of Ottawa, Ottawa, Ontario, K1N 6N5, Canada  Rasoul Alaee  Department of
Physics, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada  Karlsruhe
Institute of Technology, Institute of Theoretical Solid State Physics, D-76131
Karlsruhe, Germany  Amirreza Ahmadnejad  Department of Electrical Engineering,
Sharif University of Technology, Tehran 11155-4365, Iran  Ehsan Mobini
Department of Physics, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada
Mohamadreza Mohamadpour  Department of Electrical Engineering, Sharif
University of Technology, Tehran 11155-4365, Iran  Carsten Rockstuhl
Karlsruhe Institute of Technology, Institute of Theoretical Solid State
Physics, D-76131 Karlsruhe, Germany  Institute of Nanotechnology, Karlsruhe
Institute of Technology, D-76344 Eggenstein-Leopoldshafen, Germany  Robert W.
Boyd  Department of Physics, University of Ottawa, Ottawa, Ontario K1N 6N5,
Canada  The Institute of Optics, University of Rochester, Rochester, New York
14627, United States  Ksenia Dolgaleva  School of Electrical Engineering and
Computer Science, University of Ottawa, Ottawa, Ontario, K1N 6N5, Canada
Department of Physics, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada

###### Abstract

We exploit InSb’s magnetic-induced optical properties to propose THz sub-
wavelength antenna designs that actively tune the radiative decay rates of
dipole emitters at their proximity. The proposed designs include a spherical
InSb antenna and a cylindrical Si-InSb hybrid antenna that demonstrate
distinct behaviors; the former dramatically enhances both radiative and non-
radiative decay rates in the epsilon-near-zero region due to the dominant
contribution of the Zeeman splitting electric octupole mode. The latter
realizes significant radiative decay rate enhancement via magnetic octupole
mode, mitigating the quenching process and accelerating the photon production
rate. A deep learning-based optimization of emitter positioning further
enhances the quantum efficiency of the proposed hybrid system. These novel
mechanisms are potentially promising for tunable THz single-photon sources in
integrated quantum networks.

_K_ eywords Active antenna, Indium Antimonide (InSb), Local density of states
(LDOS), Multipole moments, Radiative decay rate, Zeeman splitting effect,
III-V semiconductors

## 1 Introduction

It has been long appreciated that the local density of states (LDOS), as seen
by a quantum emitter, can be altered in a structured electromagnetic
environment [1, 2]. The spontaneous emission (SE) rate and transition dipole
strength, as the functional characteristics of a quantum emitter, greatly rely
on LDOS variations [3]. In simple terms, the electric or magnetic dipole
transition of quantum emitters, when considered as two-level systems, can
couple to photonic resonances. Then, the enhancement of the LDOS, thanks to
the structured photonic environment, accelerates the spontaneous transition
rates of the quantum emitter. The SE rate enhancement is of great interest to
diverse applications, including quantum source emission engineering [4, 5, 6],
ultra-brighter optoelectronic devices [7, 8, 9, 10], enhanced light-matter
interaction for fluorescence spectroscopy [11, 12, 13], and single-photon
sources (SPSs) [14, 15, 16].

Optical resonators are high-potential devices that can serve as the structured
electromagnetic media for quantum emitters by providing an enhanced LDOS [17,
18]. Radiative coupling channels emerge in a coupled emitter-resonator system,
allowing one to improve the SE rate of the quantum emitter. Microcavities
[19], photonic crystals, and hyperbolic metamaterials (HMMs) [20, 21, 22] are
closed-cavity platforms that have been intensively investigated for their
potential to increase the SE rate of quantum emitters. In recent decades,
optical antennas have been the subject of this extensive research. Optical
antennas are open resonators composed of sub-wavelength dielectric or metallic
elements that transfer the energy of their resonance modes to quantum emitters
via radiative and non-radiative channels [23, 24, 25]. Enhanced light-matter
interaction at the deep sub-wavelength scale is the distinguishing advantage
of antennas over cavities, leading to LDOS enhancement of quantum emitters in
the coupled emitter-antenna systems. In this direction, metallic antennas
enable one to modify the SE rate of quantum emitters due to the high near-
field enhancement attained by localized surface plasmon resonances (LSPRs)
[26, 27, 28, 29, 30]. Metallic nanorods [31] and ring resonators [32] are
highly effective plasmonic subwavelength antennas for enhancing the SE rate of
quantum emitters through the excitation of LSPRs, including transversal and
longitudinal plasmonic modes in nanorods, and concurrent electric and magnetic
resonant modes in ring resonators. In general, coupled emitter-plasmonic
antenna systems usually introduce dominant non-radiative channels due to high
Joule heating losses of metallic building blocks, resulting in low quantum
efficiency for the coupled systems.

Alternative designs have been investigated to mitigate dissipation losses of
plasmonic antennas. Interestingly, it was shown that plasmonic devices using a
dielectric spacer could avoid or limit non-radiative decay channels [33]. This
idea has flourished by employing dielectric optical antennas in a wide range
of applications such as surface-enhanced Raman spectroscopy [34], surface-
enhanced fluorescence [35], directional light emission [36], nonlinear
processes [37], and Huygens sources [38]. Furthermore, optically induced
electric and magnetic Mie resonances make high-index dielectric antennas
capable of efficiently engineering the SE rate of quantum emitters with
electric or magnetic transition dipole moment [39, 40, 41, 42]. Besides the
coexistence of electric and magnetic multipolar resonances at the sub-
wavelength scale, unique characteristics such as negligible inherent losses
and comparable near-field enhancement with metallic analogs enable high-index
dielectric structures to suppress non-radiative channels and robustify
radiative channels in the coupled emitter-dielectric antenna systems.
Concentric hollow structures [43], Yagi-Uda designs [44], dimers [45], and
oligomers [46] have been numerically and experimentally investigated to
exhibit the high potential of dielectric sub-wavelength antennas in
engineering the SE rate of quantum emitters.

Among the reported cases exploited to control the SE rate of quantum emitters
for specific applications, research has been mostly limited to passive
approaches such as design, material, and shape to improve the efficiency of
the coupled emitter-photonic systems. Here, we aim to employ active antennas
to dynamically control the SE or radiative decay rate of quantum emitters near
the proposed antennas. By applying an external static magnetic field, we
dynamically tune the scattering response of an optical antenna, leading to the
radiative decay rate variations of a quantum emitter. In this respect, we need
a material whose optical properties can be modulated upon magnetization [47].
Hence, Indium Antimonide (InSb), a III-V semiconductor with unique features of
nonreciprocity, magnetoplasmonics, and magnetic tunability in the THz region
[48], has been exploited to achieve an active sub-wavelength antenna. Without
a static magnetic field, the scattering response of a spherical InSb antenna
is divided into plasmonic, epsilon-near-zero (ENZ), and dielectric regions. By
placing an electric dipole (ED) source close to the antenna as the quantum
emitter, we observe different types of electromagnetic behaviors in the
coupled emitter-antenna system. Non-radiative channels dominate the plasmonic
region due to inherent losses of InSb below the plasma frequency, resulting in
the quenching process. A transparency window corresponds to the ENZ region;
hence, the quantum emitter does not experience the impact of the antenna. The
coexistence of electric and magnetic multipolar resonances is observed in the
dielectric region, leading to the radiative decay rate enhancement associated
with dominant radiative channels. In the presence of the static magnetic
field, the scattering characteristics of the InSb antenna are dramatically
changed. The quenching effect becomes much more intense in the plasmonic
region. Therefore, the radiative decay rate is negligible compared to the non-
radiative one. The ENZ region completely disappears due to the Zeeman-
splitting effect of plasmonic modes, and the quantum emitter experiences
considerable radiative decay rate variations compared to the unmagnetized
case. Despite dynamic control of the emission characteristics of the ED
emitter via the magnetized InSb antenna, dominant non-radiative processes
degrade the efficiency of the proposed antenna across the studied spectral
range. To solve this limitation, as the next step, a hybrid dielectric antenna
composed of Silicon (Si) and InSb layers has been proposed to serve as the
high-index dielectric antenna, leading to a high radiative decay rate
enhancement for the coupled magnetic dipole (MD) and ED emitter-antenna
systems. Dual-band SE rate enhancement, robust magnetic response, and high
quantum efficiency are important advantages realized by the magnetized hybrid
antenna, opening up a possibility for the emergence of tunable magnetic SPSs.

## 2 Methodology

The spontaneous decay of a two-level system, the so-called quantum emitter, is
a radiating process at a quantum level that can be estimated by Fermi’s golden
rule [49]. When we consider the interaction of quantum emitters with optical
antennas as open resonators and lossy media, the normalized total decay rate
($\nicefrac{{\varGamma_{\mathrm{tot}}}}{{\varGamma_{0}}}$) must be defined by
LDOS in the form of [49]

$\frac{\varGamma_{\mathrm{tot}}}{\varGamma_{0}}=\frac{\rho_{n}(\boldsymbol{r}_{0},\omega)}{\rho_{0}(\boldsymbol{r}_{0},\omega)}=\frac{\boldsymbol{n}_{\boldsymbol{p}}^{\boldsymbol{T}}\;{\mathrm{Im}}\left[\overleftrightarrow{\boldsymbol{G}}_{\\!{s}}\left(\boldsymbol{r}_{0},\boldsymbol{r}_{0};\omega\right)+\overleftrightarrow{\boldsymbol{G}}_{\\!{0}}\left(\boldsymbol{r}_{0},\boldsymbol{r}_{0};\omega\right)\right]\boldsymbol{n}_{\boldsymbol{p}}}{\boldsymbol{n}_{\boldsymbol{p}}^{\boldsymbol{T}}\;{\mathrm{Im}}\left[\overleftrightarrow{\boldsymbol{G}}_{\\!{0}}\left(\boldsymbol{r}_{0},\boldsymbol{r}_{0};\omega\right)\right]\boldsymbol{n}_{\boldsymbol{p}}},$
(1)

where $\rho_{n}\left(\boldsymbol{r}_{0},\omega\right)$ and
$\rho_{0}\left(\boldsymbol{r}_{0},\omega\right)$ represent the LDOS at the
position $\boldsymbol{r}_{0}$ of the quantum emitter with a transition
frequency $\omega$ in the presence and absence of the optical sub-wavelength
antenna, respectively. The LDOS at the position of the quantum emitter is
computed using the imaginary part of the Green function [23], and
$\boldsymbol{n}_{\boldsymbol{p}}$ indicates the polarization’s unit vector.
Without the antenna, the LDOS only depends on the electromagnetic field
generated by the quantum emitter
$(\overleftrightarrow{\boldsymbol{G}_{0}}\left(\boldsymbol{r}_{0},\boldsymbol{r}_{0};\omega\right))$
at its position. Besides the self-generated field, in the presence of the
antenna, the LDOS has another contribution from the electromagnetic field
scattered by the antenna
$(\overleftrightarrow{\boldsymbol{G}_{s}}\left(\boldsymbol{r}_{0},\boldsymbol{r}_{0};\omega\right))$
at the emitter position, showing the impact of the local field provided by
optical sub-wavelength antennas on the total decay rate modification. The
contribution of the LDOS to engineering the decay rates of a quantum emitter
is a common point of classical and quantum electrodynamics [49]. From a
classical point of view, a quantum emitter can be modeled by a point-like
dipole as an oscillating current source located near an antenna. Both
electrodynamical and classical approaches agree when the intrinsic quantum
yield of the point-like quantum source is unity. The total decay rate is
defined as the ratio of the total emitted power of a dipole source in the
presence of the antenna to its total emitted power in the absence of the
antenna, in free space, in the form of

$\frac{\varGamma_{\mathrm{tot}}}{\varGamma_{0}}=\frac{\varGamma_{\mathrm{r}}+\varGamma_{\mathrm{nr}}}{\varGamma_{0}}=\frac{P_{\mathrm{r}}+P_{\mathrm{nr}}}{P_{0}},$
(2)

where radiative and non-radiative decay rates are denoted by
$\varGamma_{\mathrm{r}}$ and $\varGamma_{\mathrm{nr}}$, respectively. Here,
$P_{\mathrm{r}}$ and $P_{\mathrm{nr}}$ are also the scattered and absorbed
powers by the antenna, respectively. The part of the total power incident on
the antenna decays into the non-radiative channel due to the inherent losses
of the antenna [49, 23]. The remaining fraction of the radiation escapes from
the coupled emitter-antenna system to free space as far-field radiation. It is
thus necessary to calculate the corresponding powers represented in Eq. (2)
using the Poynting theorem [50]. The total emitted power is obtained by
integrating the Poynting vector flux over a surface enclosing the dipole
source without the antenna. The far-field radiation of the coupled emitter-
antenna system is computed by placing a Poynting vector surface integral
around the entire system. The dissipated power is proportional to the power
loss density, determined by a volume integral around the antenna. In this
study, we use COMSOL Multiphysics RF module package 6 to numerically compute
the coupled emitter-antenna system’s power rates corresponding to the dipole
source’s total decay rate [51, 52]. Owing to the inherent duality in classical
electrodynamics, particularly in Maxwell’s equations, the far-field radiation
patterns of a quantum emitter possessing a transition magnetic dipole (MD)
moment are indistinguishable from those with an electric dipole (ED) moment.
Consequently, the same numerical Poynting theorem applicable for EDs can be
employed within COMSOL’s RF module to compute the total decay rate of an MD
emitter. The distinction lies in the necessity to define a point source with a
transition MD moment within the simulation model.

## 3 Results

The primary purpose of this study is to investigate how a static magnetic
field applied to an optical antenna dynamically modifies the radiative decay
rate of a dipole emitter in the weak-coupling regime. We first consider a
spherical InSb antenna embedded in free space to serve as a sub-wavelength THz
antenna. The antenna’s radius is set to $30$ $\mu\mathrm{m}$, which
corresponds to the operation frequency range of 1.2 to 3 THz, matching the
range of InSb’s plasma frequency. The antenna’s geometry and frequency range
are chosen such that it can support multipolar electromagnetic resonance
modes. The dielectric permittivity of InSb becomes anisotropic upon magnetic
excitation perpendicular to the incidence plane, enabling the Zeeman-splitting
effects on the electronic energy levels of the material proportional to the
cyclotron frequency of $\omega_{c}=\nicefrac{{eB}}{{m^{*}}}$ depending on the
external magnetic field ($B$), electron charge ($e$) and effective mass of the
n-doped InSb ($m^{*}=0.0142m_{0}$).[53] (See supporting information for more
detail.)

To gain insight into the light interaction with InSb semiconductor, we
evaluate the scattering characteristics of the proposed antenna using
multipolar decomposition [54]. We consider a linearly $x$-polarized plane wave
with the wave vector $\mathbf{k}$ propagating along the $y$-direction as
illumination. By applying a static magnetic field along the $z$-direction, we
induce anisotropy in the dielectric permittivity of InSb, thereby dynamically
modifying the scattered-light characteristics (See Fig. S1 ).

Figure 1: (a, b) The comparison of SCS spectra of the spherical InSb antenna
in the presence (solid lines) and absence (dashed lines) of the static
magnetic field. The external magnetic field $(B_{z}=0.2\mathrm{T})$ is applied
along the $z$-direction to dynamically modify the scattering characteristics
of the antenna in the plasmonic (a) and ENZ (b) regions. (c, d) The spectral
features of the radiative decay rate of the $x$-oriented ED source, which is
placed at a distance of $d=3\>\mu\mathrm{m}$ from the antenna. (c) For
frequencies lower than $\omega_{p}$, the coupled system realizes an enhanced
multi-band radiative decay rate upon magnetization, while the unmagnetized
case demonstrates a larger enhancement factor in a single band. (d) Due to the
presence of Zeeman-splitting resonances, the radiative decay rate
significantly improves in the ENZ region, highlighted by the gray area. The
insets are the normalized electric-field distributions around the coupled
emitter-antenna system, displaying how the ED source excites electric resonant
modes of the antenna through the radiative coupling channels. Variations of
the radiative (e) and non-radiative (f) decay rates as a function of coupling
distance $(d)$ at the fixed frequency of $\omega/\omega_{p}=1.05$.

As shown in Fig. 1(a), the magnetically induced anisotropy lifts the mode
degeneracy, leading to separated plasmonic resonances in the magnetized InSb
antenna at frequencies lower than and around $\omega_{p}$. The Zeeman-
splitting effect results in new resonant modes whose spectral positions can be
predicted [53]. These atomic energy level splits are linearly proportional to
the Bohr magneton for weak magnetic fields. In the presence of the magnetic
field $B_{z}=0.2\mathrm{T}$, a narrow linewidth electric octupole (EO)
resonance at $\omega/\omega_{p}=0.951$ splits into multi-band EO resonances,
where a peak at $\omega/\omega_{p}=1.055$ considerably amplifies the antenna’s
scattering response. The magnitudes of the EO resonances have been multiplied
by ten to emphasize their critical role in the radiated power, as shown in
Fig. 1(a,b). A parent electric quadrupole (EQ) mode at
$\omega/\omega_{p}=0.94$ is split into two distinct EQ resonances at
$\omega_{-,EQ}/\omega_{p}=0.84$ and $\omega_{+,EQ}/\omega_{p}=1.035$.
Moreover, a broad electric dipole (ED) mode is also split into resonances at
$\omega_{-,ED}/\omega_{p}=0.8$ and $\omega_{+,ED}/\omega_{p}=,0.99$. Contrary
to the unmagnetized scenario, when an applied magnetic field couples with
localized surface plasmon resonances, it can excite magneto-plasmonic modes.
This results in the contribution of magnetic dipole (MD) resonances to the
antenna’s spectral response, as depicted in Fig. 1(a,b). The unmagnetized InSb
antenna also experiences a plasmonic-to-ENZ transition response at frequencies
slightly higher than $\omega_{p}$. This electromagnetic behavior provides
invisibility, highlighted by a gray box in Fig. 1(b). The inset depicts the
minimum contributions of the multipoles, leading to an almost negligible
scattering response, rendering the antenna nearly invisible. In the presence
of the magnetic field $B_{z}=0.2\mathrm{T}$, the electromagnetic response of
the antenna supports a strong splitting of the EQ mode at
$\omega/\omega_{p}=1.035$ and a high $Q$ EO mode at $\omega/\omega_{p}=1.055$,
leading to a transition from invisibility to visibility associated with a
60-fold enhancement of the total scattering cross section (SCS). Besides the
splitting of the EQ and EO modes, broadband ED and MD resonances also
contribute to the SCS of the magnetized InSb antenna.

The contribution of the LDOS to the total decay rate of a quantum emitter
implies that local electric field enhancement provided by the plasmonic
characteristic of the InSb antenna significantly improves the radiative decay
rate of a coupled emitter to the antenna. Such a unique feature motivates us
to study the dynamic engineering of a dipole emitter’s radiative decay rate in
microscale proximity to the spherical InSb antenna beyond a quasi-static
approximation.

Figures 1(c,d) depict the dependence of the radiative decay rate on frequency,
where the InSb antenna is fed by an $x$-oriented ED source with the separation
distance of $d=3\>\mu\mathrm{m}$. The excited resonant modes of the antenna
highly rely on the orientation and nature of the emitter [55]. Without the
magnetic field, as shown in Fig. 1(c), the radiative decay rate witnesses a
significant enhancement in the frequency range from
$\nicefrac{{\omega}}{{\omega_{p}}}=0.9$ to $0.95$, corresponding to the
spectral interval of the parent EQ and EO resonances. The radiative decay rate
experiences about 200 times enhancement due to the EO resonance with a high
$Q$ factor, while a broader EQ resonance leads to a lower enhancement. On the
other hand, the emitter coupled to the magnetized antenna realizes multi-band
radiative decay rate enhancement owing to the contribution of Zeeman-splitting
EO, EQ, and ED modes to the radiative coupling channels. The mechanism of the
radiative decay rate enhancement around frequencies of
$\omega/\omega_{p}=0.85$ and $0.925$ are similar to the unmagnetized case.
Lower Zeeman-splitting EO and EQ modes resonantly interact with the transition
moment of the ED source, realizing up to 50-fold radiative decay rate
enhancement at $\omega/\omega_{p}=0.85$. As shown in Fig. 1(c), another
enhancement arises from the EO and EQ modes at $\omega/\omega_{p}=0.925$,
where the latter broadens the spectral response of the radiative decay rate.
Other radiative decay rate maxima are observed at $\omega/\omega_{p}$= $0.98$,
resulting from the higher Zeeman-splitting EO resonance.

The interesting behavior of the magnetized InSb antenna is observed in Fig.
1(d), where the radiative decay rate enhances up to 60 times due to the
resonant interaction of the ED source’s transition moment with both the broad
EQ and higher Zeeman-splitting EO modes. However, without the magnetic field,
the InSb antenna is invisible, and thus the radiative decay rate is equal to
one and similar to that in free space across the spectral range highlighted by
the gray area. As mentioned earlier, the antenna can either radiatively or
non-radiatively dissipate the emitted energy of the ED source. In the coupled
emitter-antenna system, the radiative decay rate channel corresponds to the
antenna’s scattered electromagnetic field at the emitter position, while the
non-radiative decay rate channel is proportional to the absorption losses of
the antenna.

The separation distance between the antenna and the ED source called the
coupling distance, is another crucial parameter modifying the total decay
rate. Figure 1(e and f) indicates the distance dependence of the radiative and
non-radiative decay rates at the fixed frequency of $\omega/\omega_{p}=1.05$
associated with the transparent window of the InSb antenna. Under zero
magnetization, the radiative decay rate is unity, irrespective of the coupling
distance, since the antenna is invisible and the surrounding electromagnetic
environment evokes free space for the dipole source. By applying the static
magnetic field $B_{z}=0.2\>\mathrm{T}$, Zeeman-splitting modes realize a
60-fold enhancement for the radiative decay rate when the ED source and the
antenna are in close proximity. By increasing the coupling distance, the
resonant interaction of splitting modes with the ED source’s transition moment
weakens, and thus the enhancement factor decreases. For longer distances, the
enhancement factor is unity as the dipole source cannot “feel” the antenna
(see Fig. 1(e)). As shown in Fig. 1(f), the unmagnetized coupled emitter-
antenna experiences a zero non-radiative decay rate due to the invisibility of
the antenna. Under a biased magnetic field of $B_{z}=0.2\mathrm{T}$, a strong
near-field interaction between the ED source’s transition moment and the broad
ED mode of the antenna results in the Forster resonant energy transfer (FRET)
effect [56], and thus the non-radiative decay rate dramatically enhances for
shorter coupling distances. This non-radiative dipole-dipole interaction
becomes ineffective for longer coupling distances, so the non-radiative decay
rate falls to zero far from the antenna (see Fig. 1(f)). The magnetized InSb
antenna acting as an active photonic device allows us to realize dynamically
tunable multi-band SE rate enhancement. However, the total emitted power of
the ED source is mainly absorbed by the antenna in the near-field region due
to significant Joule heating losses of the InSb material, as shown in
Fig.1(f). Dominant non-radiative channels in the coupled system design give
rise to a low quantum efficiency for the antenna, impairing the high free-
space radiation rate required for SPSs. In the following, we propose an all-
dielectric antenna featuring a high quantum efficiency.

Let us consider a hybrid cylindrical antenna composed of silicon (Si) and InSb
layers with the same radii of $r=35\>\mu\mathrm{m}$ in the THz regime. The
silicone layer, as a high-index and low-loss material, comprises the antenna’s
upper part with a height of $h_{1}=80\>\mu\mathrm{m}$ to harness the Joule
heating losses of the design. The dielectric permittivity of Si is set to
$\varepsilon_{\mathrm{Si}}=10.6$ since the material shows a constant
permittivity with negligible inherent losses in the studied frequency range.
The lower part is made of InSb with a height of $h_{2}=8\>\mu\mathrm{m}$ to
supply the magneto-optical properties required for this active antenna. The
design principle resides on two aspects. First, the aspect ratio,
$\nicefrac{{(h_{1}+h_{2})}}{{r}}$, of the cylindrical antenna is optimized to
support scattering resonances with high $Q$ factors [57]. Second, a given
spectral range far from the main absorption band of InSb is chosen such that
the material shows moderate positive dielectric permittivity and low losses,
as shown in Fig. S1. When both constituent elements of the antenna exhibit
dielectric behaviors, an induced displacement current exceeds the conductive
one in the desired frequency range, and thus the all-dielectric antenna can
excite robust concurrent electric and magnetic resonances (See Fig. S3).

In Fig. 2, we investigate the radiative decay rate variations of an emitter
with ED transition, located near the hybrid antenna with the coupling distance
of $d=3\>\mu\mathrm{m}$. By applying an external static magnetic field, the
antenna’s scattering response is modified, enabling one to dynamically control
the total decay rate of the coupled ED source. As a comparison, we show the
SCS of the magnetized antenna under plane-wave illumination in Fig. 2(a). In
the presence of a magnetic field $B_{z}=0.2\mathrm{T}$, a dominant narrow
linewidth magnetic octupole (MO) mode is split into distinct resonances due to
the Zeeman effect, while the other broadband multipole resonances experience
slight blueshifts. Two MO resonances at frequencies of
$\omega_{-,MO}/\omega_{p}=1.202$ and $\omega_{+,MO}/\omega_{p}=1.206$ result
from a parent MO resonance at $\omega/\omega_{p}=1.204$. Positioning the
$x$-oriented ED source parallel to the hybrid antenna’s nearest surface,
specifically at the side, enhances the radiative decay rate by a factor of 20
around $\omega/\omega_{p}=1.24$, due to the strong contribution from the broad
EQ mode to the radiative coupling channels. In this configuration, the
high-$Q$ MO mode has only weak coupling with the ED transition moment of the
emitter, resulting in a more modest, 10-fold enhancement of the radiative
factor around the resonance frequency of the MO mode (see Fig. 2(b)).

Figure 2: (a) The scattering spectra of the hybrid antenna under plane-wave
illumination in the presence (solid lines) and absence (dashed lines) of the
static magnetic field. By applying the static magnetic field
$(B_{z}=0.2\>\mathrm{T})$ along the $z$-direction, multipole moments
experience a slight redshift due to the Zeeman effect, while the high-$Q$
parent MO mode splits into two resonant modes in the spectral interval from
$\omega/\omega_{p}=1.2$ to $1.22$. (b) The radiative decay rate of the
$x$-oriented ED source located near the hybrid antenna with the coupling
distance of $d=3\>\mu\mathrm{m}$. The spectral response of the radiative decay
rate demonstrates a strong dual-band enhancement factor upon magnetization,
whereas the maximum radiative decay rate enhancement is recorded under zero
magnetic bias. The insets display the normalized electric-field distribution
of the MO resonance peak excited by the ED emitter. (c) The enhancement of the
radiative decay rate for the coupled system is driven by an $z$-oriented ED
emitter as a function of frequency. In this orientation, every multipole
moment contributes radiatively to the coupling channels. The insets, showing
the normalized electric-field distributions for each resonance peak, highlight
the relatively weak coupling between the multipole moments and the ED source.
Spectral variations of (d) non-radiative decay rates of the magnetized system
upon the $z$-oriented ED emitter excitation at the fixed frequency of
$\omega/\omega_{p}=1.205$ corresponding to the resonance frequency of the MO
mode. It clearly shows that the non-radiative decay rate of the emitter
coupled to the dielectric antenna is significantly lower than that in the
coupled emitter-plasmonic antenna system. In this figure, the coupling
distance is set to be $3\>\mu\mathrm{m}$.

Under a biased magnetic field $B_{z}=0.2\>\mathrm{T}$, the coupled system
retains multi-band radiative decay rate enhancement, akin to the unmagnetized
case. However, the contribution of the MO splitting modes to the radiative
coupling channels undergoes further attenuation due to the Zeeman effect.

Considering the significant impact of the emitter’s orientation on decay-rate
enhancement, we examine a configuration in which a $z$-oriented ED emitter is
aligned parallel to the incident polarization and situated at the side
position relative to the antenna, as illustrated in Fig. 2(c). Under zero
magnetization, the MO mode strongly contributes to the radiative coupling
channels, and the radiative decay rate experiences up to a 140-fold
enhancement. As shown in Fig. 2(c), the coupled system can enhance the dual-
band radiative decay rate when the magnetized antenna supports the MO
splitting resonances. Insets display the normalized electric field
distributions of the coupled system, allowing us to perceive the contribution
of multipole moments in the coupling channels. The electric-field profiles
verify that the magnetic field application only affects the spectral resonance
positions, irrespective of their inherent properties.

Figure 2(d) depicts the non-radiative decay rate of the $z$-oriented ED source
as a function of the coupling distance ($d$), highlighting that the power
absorbed by the antenna at the MO mode’s resonance frequency, corresponding to
the maximum radiative decay rate, is negligible. This stresses the proficiency
of the proposed all-dielectric antenna in suppressing dissipative losses
compared to its plasmonic counterpart, as evidenced by the considerably lower
non-radiative decay rate relative to the radiative decay rate in the near-
field region. Therefore, it is expected to have a high quantum efficiency for
the magnetized hybrid antenna that can be exploited for various applications
demanding high multi-band radiation.

As demonstrated in Fig. 2, a magnetic hotspot is achieved due to the MO
resonance with a high $Q$ factor, promising highly efficient control of the
radiative decay rate of an MD emitter. An $x$-oriented MD emitter located
perpendicular to the closest surface of the antenna enables the strong
radiative coupling channels with the MO resonant mode in the spectral range
from $\nicefrac{{\omega}}{{\omega_{p}=}}$ $1.2$ to $1.21$. In the coupled MD
emitter-antenna system, the radiative decay rate demonstrates a 600-fold
enhancement due to a narrow-linewidth MO resonance under zero-bias magnetic
field that can be engineered in the presence of the applied magnetic field
(See Fig. 3.(a)). Hence, selective emission is another promising advantage of
the magnetized hybrid antenna in which spectral bands of the enhanced
radiative decay highly depend on the applied magnetic field’s strength.
Increasing the applied magnetic field makes resonance peaks well-separated
across a broader spectral range. The Zeeman-splitting effect is linearly
proportional to the cyclotron frequency for magnetic fields
$B_{z}<0.3\>\mathrm{T}$, and the split lines are symmetrical with respect to
the original resonance peak observed at $B_{z}=0$. For larger magnetic field
values, we can not estimate the spectral position of splitting resonances
based on Bohr magneton approximation, and new resonance peaks are
asymmetrically separated compared to the original resonance. As demonstrated
in Fig.3(b), higher-frequency splitting resonances experience considerable
blue-shifts while lower ones are slightly red-shifted under a biased magnetic
field larger than $B_{z}=0.3\>\mathrm{T}$.

Figure 3: (a) Frequency-dependent radiative decay rate of the $x$-oriented MD
emitter positioned near the hybrid antenna. The high-$Q$ MO resonance
generates substantial radiative coupling with the emitter’s transition MD
moment, leading to a 600-fold enhancement in the photon production rate. (b).
Dynamically tunable dual-band radiative decay rate enhancement of the
$x$-oriented MD source employing the applied magnetic field increment in the
frequency range corresponding to the high-$Q$ MO mode.

As highlighted previously, the LDOS enhancement factor and the normalized
radiative and non-radiative decay rates depend on the position, orientation,
and type of the dipole source or quantum emitter. Evaluating decay rates based
on varied antenna parameters presents complexity and labor intensiveness such
that numerical and semi-analytical methods often ignore mutual interactions,
leading to suboptimal results. To address this challenge, we employ advanced
deep-learning methods recognized for navigating complex and multi-dimensional
datasets. We propose a convolutional neural network (CNN) architecture [58,
59] to discern spatial intricacies within the coupled emitter-antenna system.
This CNN design adopts several convolutional layers, employing Rectified
Linear Unit (ReLU) activation to determine hierarchical data representations.
CNN output undergoes flattening and feeds into dense layers for predictive
analysis.

As illustrated in Fig. 4(a), the CNN inputs encompass the spatial coordinates
of the quantum emitter, the operational frequency, the value of the applied
static magnetic field, and the type of quantum emitter. The initial
computational step involves the convolution layer of
$C_{i}=\mathrm{ReLU}(\mathrm{Conv}(X,W_{i})+b_{i})$, where $C_{i}$ determines
each layer output. Here, $X$ represents the input data, $W_{i}$ is the weight
matrix for the $i^{th}$ filter, and $b_{i}$ is its corresponding bias. The
ReLU function introduces non-linearity, allowing the model to adjust its
prediction. This operation is repeated for $i=1$ to $N$, where $N$ is the
total number of filters in the convolution layer.

The output is passed through fully connected layers, and the network undergoes
a flattening process, converting the 2D data structure from the convolution
layer into a 1D vector. This process of transformation is represented as
$FC=\mathrm{ReLU}(W_{FC}\times\mathrm{Flatten}(C_{N})+b_{FC})$, where $W_{FC}$
represents the weight matrix and $b_{FC}$ is the bias for the fully connected
layer. The network concludes with the output layer, which uses the expression
of $\hat{Y}=W_{\text{out}}\times FC+b_{\text{out}}$, where the output
$\hat{Y}$ demonstrates the estimated radiative or non-radiative decay rate.

Utilizing our approach, we can determine both the type and optimal positioning
of a quantum emitter near an antenna to optimize radiative decay rates. Figure
4(b) illustrates the radiative decay rate for a dipole source positioned
variably around the antenna’s external surface. Notably, an $x$-oriented MD
source placed adjacent to the hybrid antenna at
($37\>\mu\mathrm{m},0\>\mu\mathrm{m},40\>\mu\mathrm{m}$) reaches an
enhancement factor of 700 at the frequency range corresponding to the MO
mode’s resonance frequency under zero magnetization. To verify this
prediction, Fig. 4(c) displays the simulation result of both radiative and
non-radiative decay rates for $m_{x}$ dipole source at its prime location,
mirroring CNN’s predictive output. The inset presents the far-field radiation
pattern of the maximum value corresponding to the dominant MO mode in a non-
magnetized state.

Figure 4:  (a) The CNN architecture tailored to identify the ideal position of
a variably oriented dipole source that yields the maximum radiative decay rate
near the hybrid antenna. This approach holds the constant coupling distance
($d=3\>\mu\mathrm{m}$). (b) The radiative decay rate of the $m_{x}$ dipole
source at varied positions around the hybrid antenna’s outer surface. The CNN
prediction indicates a seven-fold enhancement factor when $m_{x}$ is situated
at ($37\>\mu\mathrm{m},0\>\mu\mathrm{m},40\>\mu\mathrm{m}$) within the
frequency range corresponding to the MO mode in a non-magnetized state. (c)
The radiative and non-radiative decay rates of the $m_{x}$ dipole source at
its optimal position, aligning with the MO mode’s resonant frequency in a non-
magnetized environment under numerical simulation. The maximum value of the
radiative decay rate validates the CNN prediction. The inset highlights the
far-field radiation pattern, emphasizing the dominant radiation efficiency of
the MO mode into free space. (d) The radiative decay rate variations for three
different values of the applied magnetic field. The Zeeman-splitting effect
becomes broader with respect to the increment in the applied magnetic field,
while the enhancement factor decreases compared to the absence of the magnetic
field. Insets exhibit the spatial variations of the far-field electric-field
patterns corresponding to radiative decay rate maxima at the resonant
frequencies of $\omega/\omega_{p}=1.207,(B_{z}=1.5T)$ and
$\omega/\omega_{p}=1.264\>(B_{z}=1T)$.

## 4 Conclusion

In this study, we numerically investigated the impact of an active antenna on
the total decay rate of an emitter. The active antenna enables one to
manipulate its scattering response by external agents dynamically. InSb
material, a III-V semiconductor with strong magneto-optical properties in the
THz region has been employed to realize the active antenna that can modify
light-matter interaction via an applied static magnetic field. We used the
multipole decomposition method to obtain the SCS of the antenna under plane-
wave illumination. The scattering response of the antenna allows us to
determine which type of emitters efficiently interact with the antenna. In the
first section, we proposed a spherical InSb antenna to modify the radiative
decay rate of an ED source upon magnetization. We have shown that the
radiative decay rate experiences a multi-band enhancement when the magnetized
antenna has plasmonic features, whereas the maximum enhancement factor of
about 200-fold is observed for the unmagnetized case. Moreover, a transition
from invisibility to visibility has been shown in the ENZ region such that the
transparent antenna turns into a strong scatterer under a biased magnetic
field, and thus the radiative decay rate dramatically enhances up to 60 times.
However, the proposed active antenna suffers from high Joule heating losses,
impairing its quantum efficiency associated with a low photon production rate.
To mitigate this limitation, we proposed a cylindrical hybrid antenna composed
of Si and InSb layers to act as an all-dielectric active antenna. Due to its
robust magnetic resonance, we have found that the hybrid antenna is very
promising for dynamic control of the radiative decay rate of an MD emitter.
Results show that the magnetized hybrid antenna achieves an enhanced tunable
dual-band radiative decay rate for the MD source, so each band represents
several hundred-fold enhancement factors. Moreover, we have found that the
hybrid antenna harnesses the non-radiative processes, leading to high quantum
efficiency. Such unique features open up a possibility for tunable SPSs
required in quantum information processing. Finally, we employed a modified
deep-learning CNN to determine the emitter’s position within the coupled
system precisely. This position maximizes the radiative decay rate when an
$x$-oriented MD source, located at coordinates
($37\>\mu\mathrm{m},0\>\mu\mathrm{m},40\>\mu\mathrm{m}$), couples with the
hybrid antenna. Remarkably, this configuration boosts the radiative decay rate
up to 720 times under zero magnetization. This finding was further validated
through numerical simulations using Comsol, confirming the prediction of our
modified deep-learning network. Offering dynamic splitting bandwidth via the
proposed coupled system holds significant potential for designing tunable THz
single-photon sources suitable for practical quantum applications.

Supplemental Material

##  Appendix A Dielectric Permittivity of InSb

We use the Drude model to express the permittivity tensor of InSb at THz
frequencies, denoted by

$\varepsilon\left(\omega\right)=\left[\begin{array}[]{ccc}\varepsilon_{xx}&\varepsilon_{xy}&0\\\
\varepsilon_{yx}&\varepsilon_{yy}&0\\\
0&0&\varepsilon_{zz}\end{array}\right],$ (3)

where the tensor elements of Eq. (3) are defined as

$\varepsilon_{zz}\left(\omega\right)=\varepsilon_{\infty}-\frac{\omega_{p}^{2}}{\omega^{2}+i\gamma_{p}\omega}$
(4)
$\varepsilon_{xx}\left(\omega\right)=\varepsilon_{yy}\left(\omega\right)=\varepsilon_{\infty}-\frac{\omega_{p}^{2}\left(\omega^{2}+i\gamma_{p}\omega\right)}{\left(\omega^{2}+i\gamma_{p}\omega\right)^{2}-\omega_{c}^{2}\omega^{2}}$
(5)
$\varepsilon_{xy}\left(\omega\right)=-\varepsilon_{yx}\left(\omega\right)=-i\frac{\omega_{p}^{2}\omega_{c}\omega}{\left(\omega^{2}+i\gamma_{p}\omega\right)^{2}-\omega_{c}^{2}\omega^{2}}.$
(6)

We consider the parameters of an n-doped InSb semiconductor [60, 47], where
$\varepsilon_{\infty}=15.6$ is the background permittivity,
$\omega_{p}=12.56\times 10^{12}\nicefrac{\mathrm{rad}}{\mathrm{s}}$ is the
plasma frequency, $\gamma_{p}=0.01\omega_{p}$ as the damping constant
represents the Ohmic losses of InSb, and
$\omega_{c}=\nicefrac{{eB_{z}}}{{m^{*}}}$ is the cyclotron frequency depending
on the external magnetic field ($B_{z}$), electron charge ($e$) and effective
mass ($m^{*}=0.0142m_{0}$). The low effective mass of InSb results in a strong
anisotropy under a weak external magnetic field [48, 47, 53]. Figure S1
compares the spectral dependency of InSb’s complex permittivities in the
presence and absence of the applied magnetic field.

Figure S1: (a, c) Real and imaginary parts of InSb’s dielectric permittivity
versus frequency under zero magnetization $(B_{z}=0)$. Diagonal elements are
identical, while off-diagonal ones vanish due to cyclotron frequency
dependence. (a) The real part of the diagonal elements is negative at
frequencies $\omega/\omega_{p}<1$, representing the plasmonic region of InSb.
For frequencies $\omega/\omega_{p}>1$, InSb is a dielectric material due to
the positive real part of its permittivity. (c) The positive imaginary part of
diagonal elements is larger in the plasmonic region, denoting the Joule
heating losses of InSb as the plasmonic material. (b, d) Real and imaginary
parts of InSb’s permittivity as a function of frequency upon magnetization
$(B_{z}=0.2\mathrm{T})$. (b) The real part of diagonal elements increases
compared to the unmagnetized case, and $\varepsilon_{xx}(\omega)$ dissociates
from $\varepsilon_{zz}(\omega)$ at lower frequencies. On the other hand, off-
diagonal elements remain almost unchanged, similar to the previous case. (d)
In the presence of the applied magnetic field, the imaginary part of diagonal
elements is negligible, while off-diagonal ones demonstrate considerable
variations. $\varepsilon_{xy}(\omega)$ shows negative values across the
studied spectral range so that it improves the plasmonic behavior of InSb at
frequencies $\omega/\omega_{p}<1$. On the other hand,
$\varepsilon_{yx}(\omega)$ is positive throughout the broad spectral range,
increasing absorption losses of the material.

Without the magnetic field $(B_{z}=0)$, off-diagonal elements of the InSb
permittivity disappear due to the cyclotron frequency dependence. On the other
hand, diagonal elements are identical, showing negative real parts at lower
frequencies than $\omega_{p}$, and positive values for the imaginary parts
across the whole spectral range (Fig. S1a and c). Under zero magnetization,
the optical response of InSb only depends on the real part of
$\varepsilon_{zz}(\omega)$ so that its negative values result in a plasmonic
behavior [61], and the positive ones turn the material into a relatively high-
index dielectric [47]. Positive values of its imaginary part demonstrate the
Ohmic losses of InSb, specifically in the plasmonic region. In the presence of
the magnetic field ($B_{z}=0.2\mathrm{T}$), the real parts of the components
remain almost unchanged, and only $\varepsilon_{xx}(\omega)$ increases at
lower frequencies compared to the previous case (Fig. S1b). On the other hand,
the imaginary parts of the magnetized components experience main variations,
in which off-diagonal components demonstrate the same values with different
signs. As shown in Fig. S1(d), $\varepsilon_{xy}(\omega)$ indicates negative
values that can robustify the plasmonic behavior of InSb in agreement with
negative real parts of the diagonal components, whereas
$\varepsilon_{yx}(\omega)$ increases the Ohmic losses of the material in the
plasmonic region due to its considerable positive values.

##  Appendix B Scattering Cross Section of the Single Resonant Antenna

We first need to evaluate the scattering characteristics of the proposed
antenna using multipolar decomposition [55]. The incident light can induce an
electric conductive or displacement current on the antenna such that the
scattered field originates from the induced oscillating current carrying on
multipole moments mentioned in Eq. (11). The respective induced current is
described as
$J(r,\omega)=i\omega\varepsilon_{0}(\varepsilon(\omega)-1)E(r,\omega)$, where
$\varepsilon(\omega)$ is the dielectric permittivity of the antenna and
$E(r,\omega)$ describes the electric field distribution around and inside the
antenna that can be numerically calculated by the RF module of COMSOL version
6 software. By having the electric-induced current, we can obtain the proposed
antenna’s total scattering cross-section (SCS) based on the contribution of
multipole moments as follows

$p_{\alpha}=-\frac{1}{i\omega}\left[\int
J_{\alpha}\>dv\>+\frac{k^{2}}{10}\int\left(\mathrm{\mathbf{J.r}}\right)r_{\alpha}-2r^{2}J_{\alpha}\>dv\>+\frac{k^{4}}{280}\int
3r^{4}J_{\alpha}-2r^{2}\left(\mathrm{\mathbf{r.J}}\right)r_{\alpha}\>dv\right],$
(7)

$m_{\alpha}=\frac{1}{2}\left[\int\left(\mathrm{\mathbf{r\times
J}}\right)_{\alpha}\>dv\>-\frac{k^{2}}{10}\int
r^{2}\left(\mathrm{\mathbf{r\times J}}\right)_{\alpha}\>dv\right],$ (8)

$Q_{\alpha\beta}^{e}=-\frac{1}{i\omega}\left[\int
r_{\alpha}J_{\beta}+r_{\beta}J_{\alpha}-\frac{2}{3}\left(\mathrm{\mathbf{r.J}}\right)\delta_{\alpha\beta}\>dv\>+\frac{k^{2}}{42}\int
4\left(\mathrm{\mathbf{r.J}}\right)r_{\alpha}r_{\beta}+2\delta\alpha\beta\left(\mathrm{\mathbf{J.r}}\right)r^{2}-5r^{2}\left(r_{\alpha}J_{\beta}+r_{\beta}J_{\alpha}\right)\>dv\right],$
(9)

$Q_{\alpha\beta}^{m}=\frac{1}{3}\left[\int
r_{\alpha}\left(\mathrm{\mathbf{r\times
J}}\right)_{\beta}+r_{\beta}\left(\mathrm{\mathbf{r\times
J}}\right)_{\alpha}\>dv\>-\frac{k^{2}}{4}\int
r^{2}\left(r_{\alpha}\left(\mathrm{\mathbf{r\times
J}}\right)_{\beta}+r_{\beta}\left(\mathrm{\mathbf{r\times
J}}\right)_{\alpha}\>dv\right)\right],$ (10)

$\begin{split}O_{\alpha\beta\gamma}^{e}=-\frac{1}{i\omega}\big{[}\int
r_{\alpha}r_{\beta}J_{\gamma}+r_{\beta}r_{\gamma}J_{\alpha}+r_{\gamma}r_{\alpha}J_{\beta}-\frac{1}{5}\delta_{\alpha\beta}\left(r^{2}J_{\gamma}+2\left(\mathrm{\mathbf{r.J}}\right)r_{\gamma}\right)\\\
-\frac{1}{5}\delta_{\beta\gamma}\left(r^{2}J_{\alpha}+2\left(\mathrm{\mathbf{r.J}}\right)r_{\alpha}\right)-\frac{1}{5}\delta_{\gamma\alpha}\left(r^{2}J_{\beta}+2\left(\mathrm{\mathbf{r.J}}\right)r_{\beta}\right)\>dv\big{]},\end{split}$
(11)

$\begin{split}O_{\alpha\beta\gamma}^{m}=\frac{1}{24}\big{[}\int\left(r_{\alpha}r_{\beta}\left(\mathrm{\mathbf{r\times
J}}\right)_{\gamma}+r_{\beta}r_{\gamma}\left(\mathrm{\mathbf{r\times
J}}\right)_{\alpha}+r_{\gamma}r_{\alpha}\left(\mathrm{\mathbf{r\times
J}}\right)_{\beta}\right)\\\
-\frac{1}{5}\delta_{\alpha\beta}r^{2}\left(\mathrm{\mathbf{r\times
J}}\right)_{\gamma}-\frac{1}{5}\delta_{\beta\gamma}r^{2}\left(\mathrm{\mathbf{r\times
J}}\right)_{\alpha}-\frac{1}{5}\delta_{\gamma\alpha}r^{2}\left(\mathrm{\mathbf{r\times
J}}\right)_{\beta}\>dv\big{]},\end{split}$ (12)

and the scattering cross-section from the antenna can be expressed in terms of
multipole moments provided by Eqns. (5) to (10) as

$\begin{split}C_{\mathrm{sca}}^{\,\mathrm{tot}}=\frac{k^{4}}{6\pi\varepsilon_{0}^{2}\left|E_{\mathrm{inc}}\right|^{2}}\left|p_{\alpha}\right|^{2}+\frac{k^{4}}{6\pi
c^{2}\varepsilon_{0}^{2}\left|E_{\mathrm{inc}}\right|^{2}}\left|m_{\alpha}\right|^{2}+\frac{k^{6}}{160\pi\varepsilon_{0}^{2}\left|E_{\mathrm{inc}}\right|^{2}}\sum_{\alpha\beta}\left|Q_{\alpha\beta}^{e}\right|^{2}\\\
+\frac{k^{6}}{80\pi
c^{2}\varepsilon_{0}^{2}\left|E_{\mathrm{inc}}\right|^{2}}\sum_{\alpha\beta}\left|Q_{\alpha\beta}^{m}\right|^{2}+\frac{k^{8}}{1890\pi\varepsilon_{0}^{2}\left|E_{\mathrm{inc}}\right|^{2}}\sum_{\alpha\beta\gamma}\left|O_{\alpha\beta\gamma}^{e}\right|^{2}+\frac{k^{8}}{1890\pi
c^{2}\varepsilon_{0}^{2}\left|E_{\mathrm{inc}}\right|^{2}}\sum_{\alpha\beta\gamma}\left|O_{\alpha\beta\gamma}^{m}\right|^{2}.\end{split}$
(13)

Here $p_{\alpha}$/$m_{\alpha}$, $Q_{\alpha\beta}^{e}$/$Q_{\alpha\beta}^{m}$,
and $O_{\alpha\beta\gamma}^{e}$/$O_{\alpha\beta\gamma}^{m}$ are
electric/magnetic dipole moments, electric/magnetic quadrupole moments, and
electric/magnetic octupole moments, respectively. $c$ is the speed of light,
$E_{0}$ represents the electric field magnitude of the incident light, and
$\alpha,\beta,\gamma=x,y,z$.

###  B.1 Spherical InSb Antenna

We consider a linearly $x$-polarized plane wave propagating with the
wavevector $k$ along the $y$-direction as illumination. By applying a static
magnetic field, we dynamically modify the scattered light characteristics of
the antenna owing to the anisotropic behavior observed in InSb dielectric
permittivity (See Fig. 1).

Figure S2: The scattering characteristics of the spherical InSb antenna with
the radius $r=30\mu\mathrm{m}$ under $x$-polarized plane wave illumination
with the wavevector $k_{y}$, (a) in the absence and (b) presence of the
applied magnetic field. (a) The total SCS of the antenna indicates five
resonance peaks arising from the contribution of dominant ED, MD, EQ, MQ, and
EO modes. The insets demonstrate the antenna’s instantaneous normalized
electric field distributions corresponding to the total SCS maxima, further
clarifying the nature of electromagnetic resonant modes. (b) The scattering
characteristics of the antenna are modified upon magnetization where the
Zeeman splitting effect makes visibility from Invisibility around
$\nicefrac{{\omega}}{{\omega_{p}}}=1$. The insets specifically focus on the
SCS of the magnetized antenna in the ENZ region. Here, the high-$Q$ multiband
EO resonant modes significantly influence the antenna’s scattering response,
thereby enhancing light manipulation.

Figure S2 shows the SCS of the InSb antenna, allowing us to identify the
electromagnetic resonant modes excited in the system. The spectral response of
the SCS is divided into three specific regions corresponding to InSb’s
dielectric permittivity shown in Fig. S1(a). The induced electric conductive
current excites a broad ED, a narrow EQ, and an ultra-narrow linewidth EO with
a very small magnitude corresponding to negative values of InSb’permittivity
to realize a plasmonic antenna in the studied frequency range of
$\nicefrac{{\omega}}{{\omega_{p}}}<1$. The total SCS of the antenna at
frequencies slightly higher than $\omega_{p}$ nearly vanishes to provide a
transparent region associated with near-zero values of InSb permittivity. An
induced circular displacement current is excited inside the antenna at higher
frequencies, so the total SCS also has contributions from magnetic dipole (MD)
and magnetic quadrupole (MQ) modes. The concurrent support of multipolar
electric and magnetic resonances realizes a dielectric antenna associated with
positive real parts of InSb permittivity. The scattering response of the
antenna is dramatically enhanced in the transparent region under a biased
magnetic field due to the contribution of the splitting EQ and EO resonant
modes, as shown in Fig. S2(b).

###  B.2  Hybrid Antenna

In this section, we scrutinize the scattering response of the hybrid antenna
excited by a linearly $z$-polarized plane wave impinging along the $x$-axis,
as shown in Fig. S3. The antenna is made of a lower InSb layer and an upper Si
layer with a radius of $35\mu\mathrm{m}$ and a height of $8\mu\mathrm{m}$ and
$80\mu\mathrm{m}$ respectively. Under zero magnetization, the multipolar
decomposition of SCS reveals a resonance with a high $Q$ factor in the
spectral interval around $\omega/\omega_{p}=1.22$, corresponding to
contributions of the dominant MO mode, which is splitted into two resonant
modes with high-$Q$ factors upon weak magnetization. Insets demonstrate the
normalized instantaneous electric field distributions of the total SCS maxima,
allowing us to elucidate how the incident wave resonantly interacts with the
antenna. The anti-parallel electric field’polarizability at opposite sides of
the dielectric cylinder induces a circular displacement current, leading to a
narrow linewidth MO resonance inside the particle.

Figure S3: The SCS of the hybrid cylindrical antenna under $z$-polarized plane
wave illumination with the wavevector $k_{x}$ (a) in the absence and (b)
presence of the applied static magnetic field. The scattering response of the
antenna supports all multipole moments across the studied spectral range in
which the contribution of the MO mode is significant and experiences the
Zeeman splitting effect upon magnetization, leading to the splitted MO
resonant modes, as shown in panel (b). The upper part of the antenna is made
of a Si layer with the height and radius of $80$ and $35\mu\mathrm{m}$,
respectively. InSb layer comprises the lower part of the antenna with the
height and radius of $8$ and $35\mu\mathrm{m}$, respectively. Insets show the
instantaneous normalized electric field distributions around the antenna
corresponding to the total SCS maxima.

##  Appendix C Optimization Method Algorithm

Our methodology employs stochastic gradient descent (SGD) to refine our
model’s parameters iteratively. SGD fine-tunes the CNN’s weights and biases by
minimizing the MSE loss. Given the intricate high-dimensional nature of the
coupled system features, we use a genetic algorithm to pinpoint the most
relevant feature subset. This algorithm evolves a population of feature sets
over multiple iterations, leveraging crossover and mutation techniques. The
effectiveness of each feature set is assessed through the accuracy of CNN’s
predictions.

In our validation, we conduct rigorous tests on a contemporary dataset
encompassing diverse antenna systems. Our deep learning model outperforms
conventional regression techniques. Furthermore, the genetic algorithm adaptly
discerns the paramount features crucial for precise decay rate prediction.
Overall, integrating deep learning with genetic algorithm-driven feature
selection presents a significant advancement in addressing challenges in
coupled emitter-antenna systems (See. Algorithm 1).

Algorithm 1 Enhanced Deep Learning and Genetic Algorithms for Radiative Decay
Rate Prediction

1:Data Preprocessing:

2:Normalize spatial coordinates ($x$, $y$, $z$)

3:Apply one-hot encoding for frequency and applied magnetic field

4:Neural Network Initialization:

5:Set up neural network architecture with random weights and biases

6:Genetic Algorithm Parameters:

7:Set population size $P$, number of generations $G$, and mutation rate $M$

8:Initialize population of feature sets $F$ with $P$ individuals

9:Deep Learning Training:

10:Define the number of epochs and learning rate

11:for each epoch do

12: Compute predicted radiative decay rate $\hat{Y}$

13: Compute MSE loss: $MSE=\frac{1}{n}\sum_{i=1}^{n}(Y_{i}-\hat{Y}_{i})^{2}$

14: Update model parameters via optimizer

15:end for

16:Genetic Optimization:

17:for generation $g=1$ to $G$ do

18: Evaluate fitness of each feature set $f\in F$ using the neural network

19: Select best-performing individuals for mating pool $MatingPool$

20: Populate $NewF$ by applying genetic operations on $MatingPool$

21: for each feature set $f\in NewF$ do

22: if random value $<$ $M$ then

23: Introduce mutation to $f$

24: end if

25: end for

26: Update population: $F\leftarrow NewF$

27:end for

28:Optimal Features:

29:Select feature set with the topmost fitness value from the final population

30:Final Prediction:

31:Predict radiative decay rate using the selected feature set

## References

  * [1] KH Drexhage.  Influence of a dielectric interface on fluorescence decay time.  Journal of luminescence, 1:693–701, 1970.
  * [2] Edward M Purcell, Henry Cutler Torrey, and Robert V Pound.  Resonance absorption by nuclear magnetic moments in a solid.  Physical review, 69(1-2):37, 1946.
  * [3] Roy J Glauber and M Lewenstein.  Quantum optics of dielectric media.  Physical Review A, 43(1):467, 1991.
  * [4] Stefano Paesani, Massimo Borghi, Stefano Signorini, Alexandre Maïnos, Lorenzo Pavesi, and Anthony Laing.  Near-ideal spontaneous photon sources in silicon quantum photonics.  Nature communications, 11(1):1–6, 2020.
  * [5] Sumin Choi, Toan Trong Tran, Christopher Elbadawi, Charlene Lobo, Xuewen Wang, Saulius Juodkazis, Gediminas Seniutinas, Milos Toth, and Igor Aharonovich.  Engineering and localization of quantum emitters in large hexagonal boron nitride layers.  ACS applied materials & interfaces, 8(43):29642–29648, 2016.
  * [6] Ruoxi Wang, Michio Ikezawa, Yoshiki Sakuma, Hiroyuki Takeda, Naoki Ikeda, Yoshimasa Sugimoto, Kazuaki Sakoda, Yuuta Yamada, and Yasuaki Masumoto.  Enhanced spontaneous emission rates for single isoelectronic luminescence centers in photonic crystal cavities.  ACS Photonics, 7(2):321–326, 2020.
  * [7] David Arbel, Nikolai Berkovitch, Amir Nevet, Andrea Peer, Shimon Cohen, Dan Ritter, and Meir Orenstein.  Light emission rate enhancement from inp mqw by plasmon nano-antenna arrays.  Optics Express, 19(10):9807–9813, 2011.
  * [8] Bruno Romeira and Andrea Fiore.  Purcell effect in the stimulated and spontaneous emission rates of nanoscale semiconductor lasers.  IEEE Journal of Quantum Electronics, 54(2):1–12, 2018.
  * [9] Fuzhi Wang, Shengli Jin, Wenda Sun, Jingting Lin, Baogui You, Yang Li, Bing Zhang, Tasawar Hayat, Ahmed Alsaedi, and Zhan’ao Tan.  Enhancing the performance of blue quantum dots light-emitting diodes through interface engineering with deoxyribonucleic acid.  Advanced Optical Materials, 6(21):1800578, 2018.
  * [10] Peng Yu, Ziyuan Li, Tongwei Wu, Yi-Tao Wang, Xin Tong, Chuan-Feng Li, Zhongchang Wang, Su-Huai Wei, Yunyan Zhang, Huiyun Liu, et al.  Nanowire quantum dot surface engineering for high temperature single photon emission.  ACS nano, 13(11):13492–13500, 2019.
  * [11] Kasey J Russell, Tsung-Li Liu, Shanying Cui, and Evelyn L Hu.  Large spontaneous emission enhancement in plasmonic nanocavities.  Nature Photonics, 6(7):459–462, 2012.
  * [12] Felicia Tam, Glenn P Goodrich, Bruce R Johnson, and Naomi J Halas.  Plasmonic enhancement of molecular fluorescence.  Nano letters, 7(2):496–501, 2007.
  * [13] Javier Cambiasso, Gustavo Grinblat, Yi Li, Aliaksandra Rakovich, Emiliano Cortés, and Stefan A Maier.  Bridging the gap between dielectric nanophotonics and the visible regime with effectively lossless gallium phosphide antennas.  Nano letters, 17(2):1219–1225, 2017.
  * [14] Igor Aharonovich, Dirk Englund, and Milos Toth.  Solid-state single-photon emitters.  Nature Photonics, 10(10):631–641, 2016.
  * [15] Sonia Buckley, Kelley Rivoire, and Jelena Vučković.  Engineered quantum dot single-photon sources.  Reports on Progress in Physics, 75(12):126503, 2012.
  * [16] Alexander Lohrmann, BC Johnson, JC McCallum, and Stefania Castelletto.  A review on single photon sources in silicon carbide.  Reports on Progress in Physics, 80(3):034502, 2017.
  * [17] Edward Mills Purcell.  Spontaneous emission probabilities at radio frequencies.  In Confined Electrons and Photons, pages 839–839. Springer, 1995.
  * [18] Serge Haroche, Jean-Michel Raimond, and Jonathan P Dowling.  Exploring the quantum: Atoms, cavities, and photons.  American Journal of Physics, 82(1):86–87, 2014.
  * [19] Zhiyuan Qian, Lingxiao Shan, Xinchen Zhang, Qi Liu, Yun Ma, Qihuang Gong, and Ying Gu.  Spontaneous emission in micro-or nanophotonic structures.  PhotoniX, 2(1):1–35, 2021.
  * [20] Faraz A Inam, Nadeem Ahmed, Michael J Steel, and Stefania Castelletto.  Hyperbolic metamaterial resonator–antenna scheme for large, broadband emission enhancement and single-photon collection.  JOSA B, 35(9):2153–2162, 2018.
  * [21] Nadeem Ahmed, Saba Akhtar, and Faraz A Inam.  Hyperbolic metamaterial-based metal–dielectric resonator-antenna designs for ghz photon collection rates from wide-range solid-state single-photon sources.  JOSA B, 37(11):3469–3477, 2020.
  * [22] Caner Guclu, Ting Shan Luk, George T Wang, and Filippo Capolino.  Radiative emission enhancement using nano-antennas made of hyperbolic metamaterial resonators.  Applied Physics Letters, 105(12):123101, 2014.
  * [23] Nicolas Bonod.  Controlling spontaneous emission with dielectric optical antennas.  In Dielectric Metamaterials, pages 109–144. Elsevier, 2020.
  * [24] Mario Agio.  Optical antennas as nanoscale resonators.  Nanoscale, 4(3):692–706, 2012.
  * [25] TH Taminiau, FD Stefani, Franciscus B Segerink, and NF Van Hulst.  Optical antennas direct single-molecule emission.  Nature photonics, 2(4):234–237, 2008.
  * [26] Sergei Kühn, Ulf Håkanson, Lavinia Rogobete, and Vahid Sandoghdar.  Enhancement of single-molecule fluorescence using a gold nanoparticle as an optical nanoantenna.  Physical review letters, 97(1):017402, 2006.
  * [27] Pascal Anger, Palash Bharadwaj, and Lukas Novotny.  Enhancement and quenching of single-molecule fluorescence.  Physical review letters, 96(11):113002, 2006.
  * [28] Hua Mi, Le Wang, Yanpei Zhang, Guangtao Zhao, and Ruibin Jiang.  Control of the emission from electric and magnetic dipoles by gold nanocup antennas.  Optics Express, 27(10):14221–14230, 2019.
  * [29] Quanbo Jiang, Prithu Roy, Jean-Benoît Claude, and Jérôme Wenger.  Single photon source from a nanoantenna-trapped single quantum dot.  Nano Letters, 21(16):7030–7036, 2021.
  * [30] Sina Aghili, Saeed Golmohammadi, and Aydin Amini.  Proposing an on/off optical router in telecom wavelength using plasmonic tweezer.  Optics Communications, 427:95–100, 2018.
  * [31] Lavinia Rogobete, Franziska Kaminski, Mario Agio, and Vahid Sandoghdar.  Design of plasmonic nanoantennae for enhancing spontaneous emission.  Optics letters, 32(12):1623–1625, 2007.
  * [32] Sven M Hein and Harald Giessen.  Tailoring magnetic dipole emission with plasmonic split-ring resonators.  Physical review letters, 111(2):026803, 2013.
  * [33] Amit Raj Dhawan, Chérif Belacel, Juan Uriel Esparza-Villa, Michel Nasilowski, Zhiming Wang, Catherine Schwob, Jean-Paul Hugonin, Laurent Coolen, Benoît Dubertret, Pascale Senellart, et al.  Extreme multiexciton emission from deterministically assembled single-emitter subwavelength plasmonic patch antennas.  Light: Science & Applications, 9(1):1–9, 2020.
  * [34] Pavel A Dmitriev, Denis G Baranov, Valentin A Milichko, Sergey V Makarov, Ivan S Mukhin, Anton K Samusev, Alexander E Krasnok, Pavel A Belov, and Yuri S Kivshar.  Resonant raman scattering from silicon nanoparticles enhanced by magnetic response.  Nanoscale, 8(18):9721–9726, 2016.
  * [35] Masanobu Iwanaga.  All-dielectric metasurfaces with high-fluorescence-enhancing capability.  Applied Sciences, 8(8):1328, 2018.
  * [36] Lin Cheng, Rasoul Alaee, Akbar Safari, Mohammad Karimi, Lei Zhang, and Robert W Boyd.  Superscattering, superabsorption, and nonreciprocity in nonlinear antennas.  ACS Photonics, 8(2):585–591, 2021.
  * [37] Ehsan Mobini, Rasoul Alaee, Robert W Boyd, and Ksenia Dolgaleva.  Giant asymmetric second-harmonic generation in bianisotropic metasurfaces based on bound states in the continuum.  ACS Photonics, 8(11):3234–3240, 2021.
  * [38] Sheng Liu, Aleksandr Vaskin, Salvatore Campione, Omri Wolf, Michael B Sinclair, John Reno, Gordon A Keeler, Isabelle Staude, and Igal Brener.  Huygens metasurfaces enabled by magnetic dipole resonance tuning in split dielectric nanoresonators.  Nano letters, 17(7):4297–4303, 2017.
  * [39] Yang Yang, Bofeng Zhu, and Haitao Dai.  Strong magnetic field enhancement and magnetic purcell effect in a dielectric disk-ring composite nanocavity.  JOSA B, 37(3):702–708, 2020.
  * [40] Brice Rolly, Betina Bebey, Sebastien Bidault, Brian Stout, and Nicolas Bonod.  Promoting magnetic dipolar transition in trivalent lanthanide ions with lossless mie resonances.  Physical Review B, 85(24):245432, 2012.
  * [41] P Elli Stamatopoulou and Christos Tserkezis.  Role of emitter position and orientation on silicon nanoparticle-enhanced fluorescence.  OSA Continuum, 4(3):918–932, 2021.
  * [42] H Chew.  Transition rates of atoms near spherical surfaces.  The Journal of chemical physics, 87(2):1355–1360, 1987.
  * [43] Erdem Aslan.  Engineering of dual-band magnetic dipole decay rate enhancement with concentric hollow nanodisk resonators.  Optical Materials, 113:110871, 2021.
  * [44] Dmitry S Filonov, Alexander E Krasnok, Alexey P Slobozhanyuk, Polina V Kapitanova, Elizaveta A Nenasheva, Yuri S Kivshar, and Pavel A Belov.  Experimental verification of the concept of all-dielectric nanoantennas.  Applied Physics Letters, 100(20):201113, 2012.
  * [45] Alexander Krasnok, Stanislav Glybovski, Mihail Petrov, Sergey Makarov, Roman Savelev, Pavel Belov, Constantin Simovski, and Yuri Kivshar.  Demonstration of the enhanced purcell factor in all-dielectric structures.  Applied Physics Letters, 108(21):211105, 2016.
  * [46] Davide Rocco, Aristeidis Lamprianidis, Andrey E Miroshnichenko, and Costantino De Angelis.  Giant electric and magnetic purcell factor in dielectric oligomers.  JOSA B, 37(9):2738–2744, 2020.
  * [47] Grigorios P Zouros, Georgios D Kolezas, Evangelos Almpanis, and Kosmas L Tsakmakidis.  Three-dimensional giant invisibility to superscattering enhancement induced by zeeman-split modes.  ACS Photonics, 8(5):1407–1412, 2021.
  * [48] Jan Chochol, Kamil Postava, Michael Čada, Mathias Vanwolleghem, Lukáš Halagačka, Jean-François Lampin, and Jaromír Pištora.  Magneto-optical properties of insb for terahertz applications.  AIP Advances, 6(11):115021, 2016.
  * [49] Lukas Novotny and Bert Hecht.  Principles of nano-optics.  Cambridge university press, 2012.
  * [50] Hiba Alhalaby, Haitham Zaraket, and Maria Principe.  Enhanced photoluminescence with dielectric nanostructures: A review.  Results in Optics, 3:100073, 2021.
  * [51] Comsol Software.  www.comsol.com.
  * [52] Yuntian Chen, Torben Roland Nielsen, Niels Gregersen, Peter Lodahl, and Jesper Mørk.  Finite-element modeling of spontaneous emission of a quantum emitter at nanoscale proximity to plasmonic waveguides.  Physical Review B, 81(12):125431, 2010.
  * [53] A Márquez and R Esquivel-Sirvent.  Terahertz response of plasmonic nanoparticles: Plasmonic zeeman effect.  Optics Express, 28(26):39005–39016, 2020.
  * [54] Rasoul Alaee, Carsten Rockstuhl, and Ivan Fernandez-Corbaton.  An electromagnetic multipole expansion beyond the long-wavelength approximation.  Optics Communications, 407:17–21, 2018.
  * [55] Mikolaj K Schmidt, Ruben Esteban, JJl Sáenz, I Suárez-Lacalle, S Mackowski, and Javier Aizpurua.  Dielectric antennas-a suitable platform for controlling magnetic dipolar emission.  Optics express, 20(13):13636–13650, 2012.
  * [56] JR Lackowicz.  Principles of fluorescence spectroscopy, 1999 kluwer academic.
  * [57] Mikhail V Rybin, Kirill L Koshelev, Zarina F Sadrieva, Kirill B Samusev, Andrey A Bogdanov, Mikhail F Limonov, and Yuri S Kivshar.  High-q supercavity modes in subwavelength dielectric resonators.  Physical review letters, 119(24):243901, 2017.
  * [58] Serkan Kiranyaz, Onur Avci, Osama Abdeljaber, Turker Ince, Moncef Gabbouj, and Daniel J Inman.  1d convolutional neural networks and applications: A survey.  Mechanical systems and signal processing, 151:107398, 2021.
  * [59] Amirreza Ahmadnejad, Ahmad Mahmmodian Darviishani, Mohmmad Mehrdad Asadi, Sajjad Saffariyeh, Pedram Yousef, and Emad Fatemizadeh.  Tacnet: Temporal audio source counting network.  arXiv preprint arXiv:2311.02369, 2023.
  * [60] Grigorios P Zouros, Georgios D Kolezas, Evangelos Almpanis, Konstantinos Baskourelos, Tomasz P Stefański, and Kosmas L Tsakmakidis.  Magnetic switching of kerker scattering in spherical microresonators.  Nanophotonics, 9(12):4033–4041, 2020.
  * [61] Sina Aghili, Aydin Amini, Leyla Shirafkan Dizaj, and Ksenia Dolgaleva.  Thz plasmonic metasurface based on a periodic array of insb metamolecules with narrow resonances.  Optics Communications, 508:127805, 2022.