Coherent couplings between magnetic dipole transitions of quantum emitters and dielectric nanostructures

Qian Zhao; Zhong-Jian Yang; Jun He

doi:10.1364/PRJ.7.001142

1. INTRODUCTION

Dielectric nanostructures with high refractive index have recently drawn lots of interest as they exhibit strong magnetic and electric resonances while their material losses are low ^[1–3]. With these properties, they can find many nanophotonic applications, such as metamaterials ^[4], metasurfaces ^[5,6], structural colors ^[7], magnetic mirrors ^[8], and optical nanoantennas ^[9–11]. The strong optical responses of dielectric nanostructures are usually accompanied by considerable electric and/or magnetic near-field enhancements around the entire structure volume. These near-field enhancements could enable strong near-field couplings between different dielectric nanostructures ^[9,12–16] as well as couplings between the plasmonic and dielectric nanostructures ^[17–21]. The couplings of different dielectric nanostructures can induce the phenomena of hybridizations of electromagnetic modes ^[12,13] and Fano resonances ^[14–16]. The combinations of plasmonic and dielectric structures can also strongly modify their optical responses, where potential applications have been demonstrated including optical nanoantennas ^[20–22].

The hybrid photonic structures of dielectric nanoresonators and quantum emitters have begun to attract research interest recently. The strong near-field couplings between the electromagnetic modes of dielectric nanostructures and excitons have been reported ^[23–27], where the excitons can be excited in molecules or two-dimensional materials. This kind of coupling usually results in peak splitting on the scattering spectrum of a hybrid system. The emission properties of electric dipole emitters coupled to dielectric nanostructures have also been investigated ^[10,28–33]. Both decay rate and fluorescence enhancements have been realized experimentally ^[10,31,32]. The modifications of those photon–emitter interactions are based on the electric near-field enhancements in dielectric structures. Many of these studies are carried out by analogy to the plasmon–emitter hybrid systems, where strong plasmon–exciton couplings ^[34–43] and the modifications of the emission properties of emitters ^[44–46] have been extensively studied in plasmonic systems. It is well known that magnetic modes with magnetic near-field enhancements can be readily excited in a simple dielectric nanostructure. This feature makes dielectric structures attractive for enhancing the interactions between light and magnetic quantum emitters which exhibit magnetic dipole (MD) transitions. Strong MD transitions can be supported by many rare-earth ions ^[47–49]. Tailoring the emissions of MD emitters by dielectric nanostructures has begun to be studied ^{[29,30,50–53]}. The decay rate enhancement and directionality modification have been demonstrated. Note that plasmonic structures can also be used to modify the emission properties of MD emitters ^[54–57]. But this usually requires complex geometries because plasmonic nanostructures with simple shapes do not support efficient magnetic-mode resonances. Furthermore, plasmonic structures have higher material losses.

Here we study the optical responses of a hybrid system consisting of a dielectric nanostructure and MD emitters. The MD emitters are rare-earth ions with magnetic dipole resonances, which are taken as two-level emitters. The dielectric nanostructure is a silicon (Si) nanosphere. The modified magnetic dipole moments of the ions coupled with the dielectric sphere can be analytically obtained based on the master equation for the density matrix elements. With the above results, the extinction (or scattering) cross sections of the hybrid structure can be analytically calculated. Efficient magnetic near-field interaction can occur, which leads to giant modifications of the extinction spectra of the hybrid structures. For a given hybrid system, its extinction cross-section spectrum will show linear or nonlinear behaviors depending on the strength of the excitation field. For a weak excitation, the extinction of the ions is greatly enhanced. The hybrid structure (or the Si sphere) shows a pronounced dip on its extinction spectrum. For a strong excitation, the resonant extinction of the ions is only weakly enhanced, but the extinction spectrum becomes broader. The hybrid structure (or the Si sphere) shows a Fano-like line shape on its extinction spectrum. The effects from geometric and material parameters will also be considered.

2. THEORETICAL APPROACH

We consider a hybrid system composed of a dielectric nanosphere and an ion cluster, as is shown in Fig. 1. We take the commonly used Si as the high-refractive-index dielectric material. The radius of the sphere is $R$ . The ion cluster is placed at the center of the Si nanosphere. The whole system is excited by a plane wave light. We use the classical Mie scattering theory to describe the optical responses of a Si sphere. For the optical responses of the ion cluster, we use the density matrix equation based on quantum optics. The working frequency is near the magnetic dipole resonance of the Si sphere, and the ion cluster only exhibits MD transitions at this frequency. The response spectrum of the ion cluster is much smaller than that of the Si sphere. We take a single magnetic ion as a simple two-level system. The energies of the ground and excited states are $E_{1}$ and $E_{2}$ , respectively. The frequency and matrix element of the MD transition of an ion are $ω_{0} = (E_{2} - E_{1}) / ℏ$ and $μ_{MD}$ , respectively.

Figure 1.(a) Schematic of a hybrid system under study. (b) Quantum transition of a MD emitter.

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The Hamiltonian for an ion in the hybrid system can be written as ^[58,59] ${\hat{a}}^{†}$ and $\hat{a}$ are the creation and annihilation operators of the excited state, respectively. $B_{MD}$ is the total magnetic field felt by the ion. In the coupled system, the total magnetic field $B_{MD}$ consists of the magnetic field $B_{0}$ of the excitation plane wave and the magnetic field of the Si sphere, namely $B_{MD} = B_{0} + {X m}_{Si}$ , where $X m_{Si}$ is the contribution of the Si sphere, $m_{Si}$ is the magnetic dipole moment of the Si sphere, and $X$ represents the proportional coefficient between the magnetic field produced by the Si sphere and its magnetic dipole moment $m_{Si}$ . The magnetic dipole moment can be written as $m_{Si} = μ_{0}^{- 1} α_{M} B_{Si}$ , where $B_{Si}$ is the total magnetic field felt by the Si sphere, $α_{M}$ is the magnetic polarizability of the Si sphere, and $μ_{0}$ is the permeability of vacuum. The total magnetic field felt by the Si sphere consists of the magnetic field $B_{0}$ of the plane wave and that from the magnetic dipole of the ion. Thus, we have $B_{Si} = B_{0} + {Y m}_{MD}$ , where ${Y m}_{MD}$ is the contribution of the magnetic ion and $m_{MD}$ is the magnetic dipole moment of the ion. Note that the Si sphere is much larger than the ion. Thus, the magnetic field felt by the Si sphere, which is produced by the ion, is not inhomogeneous. Here, for the sake of simplicity, we assume that there is an equivalent uniform magnetic field contributed from the ion, and it is written as ${Y m}_{MD}$ , which is proportional to the magnetic dipole of moment $m_{MD}$ of the ion, where $Y$ is the proportional coefficient. We calculate the magnetic dipole moment of a magnetic ion using the density matrix $μ_{MD} (ρ_{12} + ρ_{21})$ . For an ion cluster, one can take it as an equivalent ion with a larger dipole moment ^[58,59]. If the number of the ions is $N$ , the magnetic dipole moment of the ion cluster can be written as $m_{MD} (t) = N μ_{MD} ({\bar{ρ}}_{12} e^{i ω t} + {\bar{ρ}}_{21} e^{- i ω t}) = (m_{MD} / 2) e^{- i ω t} + (m_{MD}^{*} / 2) e^{i ω t}$ . ${\bar{ρ}}_{12}$ and ${\bar{ρ}}_{21}$ are used to separate the high-frequency part. The magnetic field of incident light as a function of time can be written as $B (t) = (B_{0} / 2) e^{i ω t} + (B_{0} / 2) e^{- i ω t}$ . The magnetic field felt by the Si sphere is $B_{Si} (t) = (B_{0} / 2 + N Y μ_{MD} {\bar{ρ}}_{21}) e^{- i ω t} + (B_{0} / 2 + N Y μ_{MD} {\bar{ρ}}_{12}) e^{i ω t}$ . The magnetic field felt by ion cluster can be written as where $Ω = \frac{μ_{MD}}{ℏ} (1 + \frac{X α_{M}}{μ_{0}}) \frac{B_{0}}{2}$ , $G = \frac{μ_{MD}^{2} X Y N α_{M}}{ℏ μ_{0}}$ .

The above matrix elements should satisfy the master equation ^[58], where the relaxation elements of diagonal and nondiagonal elements correspond to $Γ_{22} = - Γ_{11} = 1 / T_{1}$ and $Γ_{12} = Γ_{21} = 1 / T_{2}$ , respectively, where $T_{1}$ is relaxation time of ions at equilibrium (the longitudinal homogeneous lifetime) and $T_{2}$ is the time about loss of phase coherence (the transverse homogeneous lifetime) ^[58,60]. From Eq. (3), we have where ${(ρ_{11} - ρ_{22})}_{0}$ is the initial population difference, which is taken to be ${(ρ_{11} - ρ_{22})}_{0} = 1$ . We write $Δ = ρ_{11} - ρ_{22}$ , ${\bar{ρ}}_{12} = C + i D$ , ${\bar{ρ}}_{21} = C - i D$ . With the rotating wave approximation, we find where $Ω_{R}$ and $Ω_{I}$ are the real and imaginary parts of $Ω$ , respectively. $G_{R}$ and $G_{I}$ are the real and imaginary parts of $G$ , respectively. In the steady-state situation, the left-hand side of Eq. (5) satisfies $d C / d t = d D / d t = d Δ / d t = 0$ . We obtain The extinction cross sections of the system of the ions and Si sphere can be calculated based on the above results. As the size of the ion cluster is much smaller than the wavelength of light, the extinction and scattering cross section can be expressed as ${(σ_{ext})}_{MD} = μ_{0} k Im (m_{MD} B_{MD}^{*}) / B_{0}^{2} = k Im (α_{MD}) | B_{MD}^{2} | / B_{0}^{2}$ , ${(σ_{scat})}_{MD} = k^{4} | m_{MD}^{2} | / 6 π μ_{0}^{2} B_{0}^{2} = k^{4} | α_{M}^{2} | | B_{MD}^{2} | / 6 π B_{0}^{2}$ , respectively, where $k = 2 π / λ$ is a wave vector and $λ$ is the wavelength of light ^[61]. The magnetic dipole moment of ion cluster $m_{MD}$ can be expressed as $m_{MD} = N μ_{MD} [(C + i D) e^{i ω t} + (C - i D) e^{- i ω t}]$ . $m_{MD}$ can also be written as $m_{MD} = μ_{0}^{- 1} B_{MD} α_{MD}$ , $α_{MD}$ is the magnetic dipole polarizability of an ion cluster. Thus, according to Eq. (4), the polarizability of an ion cluster can be written as We obtain the analytical formula of the extinction cross section of ion cluster as The full width at half-maximum (FWHM) of the extinction spectrum is where $B_{MD 1}$ is the magnetic field at the frequency $ω = ω_{0} \pm Δ ω / 2$ (the magnitudes of the two magnetic fields at positions of the full width at half-maximum are close to each other, so they are approximately taken as the same value), and $B_{MD 0}$ is the magnetic field at the frequency of the MD transition $ω_{0}$ . For an individual magnetic ion cluster which corresponds to $X = Y = 0$ , its extinction cross section becomes which agrees with the reported results ^[58]. The corresponding FWHM is now $Δ ω = 2 \sqrt{1 + μ_{MD}^{2} B_{0}^{2} T_{1} T_{2} / ℏ^{2}} / T_{2}$ .

For a Si nanosphere with the relative refractive index $m = n_{1} + i n_{2}$ , $k R ≪ 1$ is approximately satisfied, but $y = m k R ≪ 1$ is not satisfied. Its extinction and scattering cross section can be written as ${(σ_{ext})}_{Si} = μ_{0} k Im (m_{Si} B_{Si}^{*}) / B_{0}^{2} + k Im (P_{Si} E_{Si}^{*}) / ε_{0} E_{0}^{2}$ and ${(σ_{sca})}_{Si} = (k^{4} / 6 π) (| α_{M}^{2} | | B_{Si}^{2} | / B_{0}^{2} + | α_{E}^{2} | | E_{Si}^{2} | / E_{0}^{2})$ , respectively, based on the Mie theory ^[22,61,62]. Here the electric dipole mode also contributes to the optical responses of the Si sphere ^[62]. The electric and magnetic dipole moments can be written as $P_{Si} = ε_{0} α_{E} E_{Si}$ and $m_{Si} = μ_{0}^{- 1} α_{M} B_{Si}$ , respectively. $ε_{0}$ is the vacuum dielectric constant, and $α_{E}$ and $α_{M}$ are the electric dipole and magnetic dipole polarizabilities of the Si sphere, respectively. According to the Mie theory, $α_{E}$ and $α_{M}$ can be written as where where $j_{n} (x)$ and $y_{n} (x)$ are the spherical Bessel and Neumann functions, respectively. As $B_{Si} = B_{0} + {Y m}_{MD}$ , the analytical formula of extinction cross section of a coupled Si sphere can be written as The scattering cross section can also be written as ${(σ_{scat})}_{Si} = (k^{4} / 6 π) | α_{M}^{2} | {{[B_{0} + Re ({Y m}_{MD})]}^{2} + {[Im ({Y m}_{MD})]}^{2}} / B_{0}^{2} + (k^{4} / 6 π) | α_{E}^{2} |$ . Although the electric dipole mode contributes the responses of a Si sphere, this mode does not interact with the ion cluster. In the spherical system we considered, $X$ can be calculated analytically according to the Mie theory. $Y$ can be equivalently obtained by using the theory of magnetic dipole radiation enhancement (see Appendix A for the relevant discussion).

3. RESULTS AND DISCUSSION

For numerical calculations, the radius of the Si sphere is chosen to be $R = 60 nm$ , and its refractive index is taken from the Palik’s book ^[63]. The resonance of its magnetic dipole mode is near $λ = 520 nm$ . The MD transition of the rare-earth ions is chosen to be at $λ = 525 nm$ , which is spectrally near the resonant position of Si sphere. The transverse homogeneous lifetime $T_{2}$ and the longitudinal homogeneous lifetime $T_{1}$ are both taken to be 0.2 ns. The matrix element for the MD transition is $0.3 μ_{B}$ ( $μ_{B}$ is the Bohr magneton). The number of ions is $N = 300,000$ . From Eq. (6), the magnetic polarizability of the ionic cluster varies with $B_{MD}$ and $B_{MD}$ varies with the incident $B_{0}$ , which can cause the system to show nonlinear optical responses. When the intensity of excitation light is weak, $μ_{MD}^{2} {| B_{MD} |}^{2} T_{1} T_{2} / ℏ^{2} ≪ 1$ . The magnetic polarizability of the ion cluster is approximately written as $α_{MD} = (N μ_{0} μ_{MD}^{2} T_{2}) / ℏ [(ω_{0} - ω) T_{2} - i]$ . The magnetic ion cluster exhibits approximately linear optical response, namely the extinction spectrum [Eq. (7)] does not vary with the intensity of the light ( $B_{MD} / B_{0}$ can be represented by $X$ , $Y$ , $α_{MD}$ , and $α_{M}$ ). When the light intensity increases so that $μ_{MD}^{2} {| B_{MD} |}^{2} T_{1} T_{2} / ℏ^{2} ≪ 1$ cannot be satisfied, the ion exhibits nonlinear optical response. In order to quantify the linear and nonlinear response regimes of the system, we define $μ_{MD}^{2} {| B_{MD} |}^{2} T_{1} T_{2} / ℏ^{2} = 0.1$ as the threshold from the linear to nonlinear region. Based on this threshold, the magnetic field of the excitation light for the above system is calculated to be $B_{0} = 3.1 \times 10^{- 3} T$ , which corresponds to the light intensity of $I_{0} = 1.1554 \times 10^{5} W / {cm}^{2}$ .

We first consider a case with a relatively low intensity of the excitation light $I_{0} = 10^{4} W / {cm}^{2}$ ( $B_{0} = 9.1453 \times 10^{- 4} T$ ). Figure 2(a) shows extinction spectra of the individual and coupled ion cluster. The inset of Fig. 2(a) illustrates the response spectrum of $Δ = ρ_{11} - ρ_{22}$ . The absorption of the ions is greatly enhanced (more than 200 times) with the Si sphere. This can be understood with the behavior of $Δ$ and the magnetic field enhancement of the Si sphere. With a weak excitation, $Δ$ is very close to 1. This means that an ion is probably at the ground state and it can easily absorb energy. At the same time, the Si sphere provides an efficient magnetic field enhancement for the ion cluster. Thus, the absorption (extinction) cross section of the ion cluster can be significantly enhanced. Quantitatively, the resonant extinction of coupled ion cluster based on Eq. (7) becomes under the weak excitation condition $μ_{MD}^{2} {| B_{MD} |}^{2} T_{1} T_{2} / ℏ^{2} ≪ 1$ . $B_{MD 0}$ is close to $B_{1}$ and it is much larger than $B_{0}$ [ $B_{1}$ is the magnetic field at the center of the individual Si sphere at the frequency $ω_{0}$ , $B_{1} = (1 + X α_{M} / μ_{0}) B_{0}$ ]. Thus, the resonant extinction of the coupled ions is ${| B_{MD 0} |}^{2} / {| B_{0} |}^{2}$ times higher than that of the individual ions ( $k N μ_{0} μ_{MD}^{2} T_{2} / ℏ$ ). The FWHM of the extinction spectrum [Eq. (8)] is reduced to It is slightly larger than the individual case $2 / T_{2}$ (the value of $B_{MD}$ will be shown later in Fig. 2). The extinction spectra of the coupled Si sphere and the hybrid structure are shown in Fig. 2(b). Pronounced dips appear on the spectra of the hybrid structure and the coupled Si sphere, respectively. The inset shows the extinction spectrum of the whole system for a wider energy regime. We will discuss the dips in detail later. Here, it is interesting to note that the results in the linear regime can be well reproduced by the common finite-difference time-domain (FDTD) simulations, where the ions are taken as a nanosphere which has a classical magnetic Lorentz model for its permeability (see the relevant results in Appendix B).

Figure 2.Extinction spectra of a coupled system in the weak and strong light intensity regimes. (a), (c) The extinction spectra of the individual and coupled ion cluster in the weak (a) and strong (c) light intensity regime. The insets are the population difference spectra of the individual ions and the coupled ions. (b), (d) The extinction spectra of the coupled Si sphere and the hybrid structure in the weak (b) and strong (d) light intensity regime. The insets are the extinction spectra for a wider frequency regime.

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We now turn to a high light intensity $I_{0} = 10^{8} W / {cm}^{2}$ ( $B_{0} = 9.1453 \times 10^{- 2} T$ ). Figure 2(c) shows extinction spectra of the individual and coupled ion clusters. The resonant extinction (absorption) cross section shows a weak increment (about 5 times). For $ω = ω_{0}$ , the value $Δ$ for individual ions is much closer to 0 compared to that with weak excitations. And the $Δ$ of coupled ions is close to 0, which means that it is more difficult for the ions to be excited compared to the case of weak excitation. Thus, the ions cannot efficiently absorb more light even when there is enhanced magnetic field around it. For such a high light intensity, we have $μ_{MD}^{2} {| B_{MD} |}^{2} T_{1} T_{2} / ℏ^{2} ≫ 1$ , and the expression for the resonant extinction of a coupled ion cluster based on Eq. (7) can now be written as The FWHM of the extinction spectrum [Eq. (8)] is reduced to It is larger than that of the individual ions ( $2 / T_{2}$ ) and the linear case [Eq. (11)] as $μ_{MD}^{2} {| B_{MD} |}^{2} T_{1} T_{2} / ℏ^{2} ≫ 1$ . Note that the high intensity excitation ( $μ_{MD}^{2} {| B_{MD} |}^{2} T_{1} T_{2} / ℏ^{2} ≫ 1$ ) condition is a special situation of the nonlinear regime ( $μ_{MD}^{2} {| B_{MD} |}^{2} T_{1} T_{2} / ℏ^{2} > 0.1$ ). Figure 2(d) shows extinction spectra of the coupled Si sphere and the whole system. The variation of the extinction value for the coupled system (or coupled Si sphere) is much smaller than the linear case, while the spectrum shows a typical Fano-like line shape.

The variation from a dip to a Fano-like line shape for the extinction spectrum of the coupled Si sphere (or the whole structure) can be understood by considering the magnetic polarizability of the ions $α_{MD}$ and the total magnetic field felt by the ions $B_{MD}$ . Based on Eq. (9), the variation of the extinction of the coupled Si sphere is mainly related to $Re ({Y m}_{MD})$ and $Im ({Y m}_{MD})$ . And it can be easily checked that $B_{0}$ is much higher than $Re ({Y m}_{MD})$ and $Im ({Y m}_{MD})$ . Therefore, the main factor that causes the change of the extinction of the Si sphere is $Re ({Y m}_{MD})$ , where $Y$ is fixed for a given system, and it is dominated by a real value (see Table 1). The magnetic dipole moment of the ions is dependent on the polarizability of the ions and the total magnetic field felt by the ions $m_{MD} = μ_{0}^{- 1} B_{MD} α_{MD}$ . Figures 3(a)–3(c) show the spectra of $α_{MD}$ , $B_{MD}$ , and $m_{MD}$ under a weak excitation. The $Im (α_{MD})$ and $Re (α_{MD})$ show a peak and an antisymmetric line shape, respectively. The magnitude of $Im (α_{MD})$ is higher than that of $Re (α_{MD})$ , and $Im (B_{MD})$ is several times larger than $Re (B_{MD})$ . Thus, $Re (m_{MD})$ is mainly dependent on $Im (α_{MD}) \times Im (B_{MD})$ , which has a dip line shape. The negative value of $Re (m_{MD})$ means that $Re (m_{MD})$ and the excitation field $B_{0}$ are out of phase, and their destructive coupling leads to the appearance of a dip on the spectrum of the coupled Si sphere (and the whole system). For a strong excitation, the $Im (α_{MD})$ and $Re (α_{MD})$ also show a peak and an antisymmetric line shape, respectively, while $Re (α_{MD})$ is much larger than $Im (α_{MD})$ [Fig. 3(d)]. The $B_{MD}$ shows similar behavior [Fig. 3(e)] to that with the weak excitation. Thus, $Re (m_{MD})$ is now mainly dependent on $Re (α_{MD}) \times Re (B_{MD})$ , whose line shape is determined by $Re (α_{MD})$ as $B_{MD}$ is hardly changed with frequency around the $ω_{0}$ . As a result, the extinction line of the coupled Si sphere has a Fano-like line shape which is similar to $Re (α_{MD})$ [Fig. 3(f)]. Here, it can also be seen that the $| B_{MD} |$ is close to $B_{1}$ and it is much larger than the excitation field $B_{0}$ under both the weak and strong excitations.

Figure 3.Magnetic polarizabilities $α_{MD}$ , magnetic fields $B_{MD}$ , and magnetic dipole moments $m_{MD}$ for the ionic cluster. (a), (d) The magnetic polarizabilities $α_{MD}$ in the weak and strong light intensity regimes. (b), (e) The magnetic fields $B_{MD}$ in the weak and strong light intensity regimes. (c), (f) The magnetic dipole moments $m_{MD}$ in the weak and strong light intensity regimes.

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Figures 4(a) and 4(b) show the extinction spectra of the coupled ion cluster and hybrid structure with different excitation intensities, respectively. The resonant extinction cross section of the coupled ion cluster decreases with the light intensity in the nonlinear regime. This is because the absorption of the ions is becoming saturated with high excitation intensity as discussed before. Quantitatively, the resonant extinction is approaching $\sim k N ℏ μ_{0} / B_{0}^{2} T_{1}$ , which will decrease with $B_{0}$ . The FWHM of the spectrum is also getting larger as expected in the nonlinear regime [Eq. (13)]. In Fig. 4(b), the extinction spectrum of the hybrid structure varies gradually from a dip to a Fano resonance shape with the light intensity. Moreover, the response spectrum becomes broader and the variation becomes smaller, which are in consistent with the responses of the ion cluster.

Figure 4.Extinction spectra of (a) the ion cluster and (b) the whole system with different light intensities.

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The effects from the MD matrix element are investigated. We first consider the weak light intensity $I_{0} = 10^{4} W / {cm}^{2}$ . The resonant extinction cross section of the ions first increases and then decreases with the $μ_{MD}$ from $0.1 μ_{B}$ to $1.2 μ_{B}$ [Fig. 5(a)]. This can be understood based on the expression of the resonant extinction of the ions ${(σ_{ext})}_{MD}^{1} \approx ({k N T}_{2} μ_{0} μ_{MD}^{2} {| B_{MD 0} |}^{2}) / ℏ B_{0}^{2}$ [Eq. (10)], and it is in proportion to $μ_{MD}^{2} {| B_{MD 0} |}^{2}$ . $| B_{MD 0} |$ decreases with $μ_{MD}$ . When $μ_{MD}$ is small (e.g., $0.1 μ_{B}$ ), the coupling effect between the ion cluster and Si sphere is relatively small. The variation of $| B_{MD 0} |$ is small (see the relevant results in Appendix B), so the resonant extinction increases with $μ_{MD}$ . When $μ_{MD}$ is larger than $0.9 μ_{B}$ , $| B_{MD 0} |$ is approaching 0 and the decreasing of $| B_{MD 0} |$ is a dominant factor. Thus, the resonant extinction decreases. The dip on the extinction spectrum of the hybrid structure becomes deeper and broader with $μ_{MD}$ . The variation of the responses of the coupled Si sphere is mainly related to the $Re ({Y m}_{MD})$ , and the $Re (m_{MD})$ depends largely on $Im (α_{MD}) \times Im (B_{MD})$ in the linear region as discussed before. $α_{MD}$ is in proportion to $μ_{MD}^{2}$ [Eq. (6)]. The variation of $μ_{MD}^{2}$ is larger than that of $B_{MD}$ even for a large $μ_{MD}$ . So the variation of the extinction spectrum of the hybrid structure increases with $μ_{MD}$ . $B_{MD 1} - B_{MD 0}$ ( $B_{MD 0}$ ) increases (decreases) with $μ_{MD}$ (Appendix B), so the FWHM [Eq. (11)] $(2 / T_{2}) \sqrt{2 {| 1 + (B_{MD 1} - B_{MD 0}) / B_{MD 0} |}^{2} - 1}$ will also increase.

Figure 5.Extinction spectra of the ion cluster and the whole system with varying (a)–(d) the MD matrix element $μ_{MD}$ and (e)–(h) the number of ions $N$ . (a), (c) The extinction spectra of the coupled ion cluster in the weak (a) and strong (c) light intensity regime. (b), (d) The extinction spectra of the hybrid structure in the weak (b) and strong (d) light intensity regime. Panels (e)–(h) show the same contents as that in panels (a)–(d), respectively, with varying the $N$ .

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Figures 5(c) and 5(d) show the responses of the coupled ions and the hybrid structure with a high light intensity $I_{0} = 10^{8} W / {cm}^{2}$ , respectively. Different from the linear region, the resonant extinction of the ions hardly changes with the $μ_{MD}$ , because Eq. (10) for resonant extinction of ions in the high light intensity regime still holds here ( $μ_{MD}^{2} {| B_{MD} |}^{2} T_{1} T_{2} / ℏ^{2} ≫ 1$ ) and it is independent of $μ_{MD}$ . The spectrum of the ions becomes broader with $μ_{MD}$ , because its FWHM $Δ ω \approx 2 μ_{MD} | B_{MD 1} | \sqrt{T_{1} T_{2}} / T_{2} ℏ$ [Eq. (13)] increases significantly with $μ_{MD}$ as $B_{MD 1}$ is very close to $B_{1}$ in the $μ_{MD}^{2} {| B_{MD} |}^{2} T_{1} T_{2} / ℏ^{2} ≫ 1$ limit and it almost does not change with $μ_{MD}$ (Fig. 3 and Appendix B). For the hybrid structure, the antisymmetric Fano-like line shape becomes more pronounced with $μ_{MD}$ [Fig. 5(d)]. The variation of the extinction of the hybrid system is mainly dependent on $Re (m_{MD}) \approx μ_{0}^{- 1} \times Re (α_{MD}) \times Re (B_{MD})$ . For the typical positions of $ω_{0} \pm Δ ω / 2$ , the $Re (m_{MD}) \approx \mp {N T}_{2} μ_{0} μ_{MD} Re (B_{MD 1}) / (2 | B_{MD 1} | \sqrt{T_{1} T_{2}})$ will increase with $μ_{MD}$ as $B_{MD 1}$ almost does not change with $μ_{MD}$ .

The MD matrix element of ion cluster $m_{MD}$ is also dependent on the number of ions $N$ . Thus, the effects from the number of ions $N$ are also investigated [Figs. 5(e)–5(h)]. For a weak light intensity $I_{0} = 10^{4} W / {cm}^{2}$ [Figs. 5(e) and 5(f)], the behaviors of the ions and the hybrid structure are similar to the case with varying the $μ_{MD}$ [Figs. 5(a) and 5(b)]. The results can also be explained in a similar way to that for Figs. 5(a) and 5(b). For a high light intensity $I_{0} = 10^{8} W / {cm}^{2}$ , the resonant extinction of ions increases with the number of ions $N$ [Fig. 5(g)], which is different from the case with varying the $μ_{MD}$ [Fig. 5(c)]. This is because the resonant extinction of ions [Eq. (12)] increases with $N$ while it is independent of $μ_{MD}$ . The FWHM of the extinction spectrum [Eq. (13)] is independent of $N$ , and $B_{MD 1}$ is very close to $B_{1}$ in the $μ_{MD}^{2} {| B_{MD} |}^{2} T_{1} T_{2} / ℏ^{2} ≫ 1$ limit (Appendix B). The variations of the extinction of the hybrid system [Fig. 5(h)] and the corresponding explanation are similar to that with varying the $μ_{MD}$ [Fig. 5(d)]. Here, it should be pointed out that for small $μ_{MD}$ or $N$ under weak excitation, although the variation of the spectra for the hybrid structure is quite small, the absorption of the ions is still greatly enhanced.

The lifetime of the ions may change with the external environment, for example, temperature ^[64,65]. Thus the influence from the transverse homogeneous lifetime $T_{2}$ and the longitudinal homogeneous lifetime $T_{1}$ is also considered. For simplicity, $T_{2}$ and $T_{1}$ are kept to be the same for each case. We also consider the weak and strong excitation cases. The other parameters are the same as that in Fig. 2. Figures 6(a) and 6(b) show the results with light intensity $I_{0} = 10^{4} W / {cm}^{2}$ . The resonant extinction of the ion cluster first increases and then decreases with the lifetimes of $T_{2}$ and $T_{1}$ varying from 0.1 ns to 1.6 ns. The resonant extinction [Eq. (10)] is proportional to $T_{2} {| B_{MD 0} |}^{2}$ , while ${| B_{MD 0} |}^{2}$ decreases with $T_{2}$ ( $T_{1}$ ) (Appendix B). For small $T_{2}$ ( $T_{1}$ ), the coupling effect ( $X$ , $Y$ ) is relatively small. The variation of $T_{2}$ ( $T_{1}$ ) is larger than that of ${| B_{MD 0} |}^{2}$ . So the resonant extinction increases with $T_{2}$ ( $T_{1}$ ). When $T_{2}$ ( $T_{1}$ ) is larger than 0.8 ns, the coupling is strong and ${| B_{MD 0} |}^{2}$ is approaching 0. The variation of ${| B_{MD 0} |}^{2}$ is larger than that of $T_{2}$ ( $T_{1}$ ). Thus, the resonant extinction decreases with $T_{2}$ ( $T_{1}$ ). The variation of the extinction of the coupled Si sphere is mainly related to $Re (m_{MD})$ . For the resonant position of $ω = ω_{0}$ , $Re (m_{MD}) \approx N μ_{0} μ_{MD}^{2} T_{2} Im (B_{MD 0}) / ℏ$ . The variation of $T_{2}$ ( $T_{1}$ ) is larger than that of $B_{MD 0}$ (Appendix B) for each $T_{2}$ , so the dip of the coupled Si and the corresponding hybrid structure becomes deeper with $T_{2}$ ( $T_{1}$ ). It is interesting to note that the ions show relatively large extinction cross sections which are comparable to that of the coupled Si sphere for large $T_{2}$ ( $T_{1}$ ).

Figure 6.Extinction spectra of the ion cluster and the hybrid structure with different lifetimes $T_{1}$ and $T_{2} (T_{1} = T_{2})$ . (a), (c) The extinction spectra of the coupled ion cluster in the weak (a) and strong (c) light intensity regime. (b), (d) The extinction spectra of the hybrid structure in the weak (b) and strong (d) light intensity regime.

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For a strong light intensity $I_{0} = 10^{8} W / {cm}^{2}$ , the resonant extinction of the ion cluster decreases with the lifetime $T_{2}$ ( $T_{1}$ ) [Fig. 6(c)]. This is because the resonant extinction [Eq. (12)] is inversely proportional to $T_{1}$ . Its FWHM [Eq. (13)] does not change with the lifetime $T_{2}$ ( $T_{1}$ ). The Fano-like spectrum of the hybrid structure almost does not change with the lifetime [Fig. 6(d)], and the extinction is much larger than that of the ion cluster. The variation of the response of the Si sphere depends largely on $Re (α_{MD}) \times Re (B_{MD})$ with the strong excitation. For the typical positions of $ω = ω_{0} \pm Δ ω / 2$ , $Re (α_{MD}) \approx \mp {N T}_{2} μ_{0} μ_{MD} / (2 | B_{MD 1} | \sqrt{T_{1} T_{2}})$ , which are not changed with the lifetime $T_{2}$ ( $T_{1}$ ) (the $B_{MD}$ does not change with $T_{2}$ under the strong excitation, see Appendix B). Therefore, the Fano spectral response of the hybrid structure does not change with the lifetime $T_{2}$ ( $T_{1}$ ).

The magnetic field enhancement of an individual Si sphere varies with the location inside the structure. This will affect the coupling between the ions and the Si sphere. The magnetic field enhancement reaches maximum at the sphere center and decreases with the distance $d$ between the location and the center of the sphere. The corresponding coupling coefficients $X$ and $Y$ decrease with $d$ (see Table 1). Thus, the coupling strength in the hybrid structure becomes weaker with $d$ . We also consider the weak and strong excitation cases. Figures 7(b) and 7(c) show the extinction spectra of the coupled ion cluster and the hybrid structure with different distances $d$ . The other parameters are the same as that in Figs. 2(a) and 2(b). The light intensity $I_{0} = 10^{4} W / {cm}^{2}$ corresponds to the linear regime. The resonant extinction of the ion cluster [Eq. (10)] decreases with the distance $d$ . This is because the total magnetic field felt by the ions $B_{MD}$ , which is related to the $X$ and $Y$ , becomes smaller with the $d$ . The FWHM is almost invariable ( $| B_{MD 0} |$ is close to $| B_{MD 1} |$ ). The dip of the extinction spectrum of the hybrid structure becomes weaker with the distance $d$ as the destructive coupling strength between the ions and the Si sphere decreases with $d$ .

Figure 7.Extinction spectra of the ion cluster and the hybrid structure with different distance $d$ between the ion cluster and the center of the Si sphere. (a) Schematic of a hybrid structure with a distance $d$ . (b), (d) The extinction spectra of the coupled ion cluster in the weak (b) and strong (d) light intensity regime. (c), (e) The extinction spectra of the hybrid structure in the weak (c) and strong (e) light intensity regime.

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Figures 7(d) and 7(e) show the results for a high light intensity $I_{0} = 10^{8} W / {cm}^{2}$ . The resonant extinction of the ion cluster is almost unchanged with the $d$ . The reason is that its value [Eq. (12)] is independent of the $X$ and $Y$ . The FWHM of the ions [Eq. (13)] becomes smaller significantly. This is caused by the decreasing of $B_{MD}$ with $d$ . For the Si sphere, the variation of its extinction depends mainly on $Y Re (m_{MD})$ , where $Re (m_{MD}) \approx μ_{0}^{- 1} \times Re (α_{MD}) \times Re (B_{MD})$ in the nonlinear regime. For the typical positions of $ω = ω_{0} \pm Δ ω / 2$ , the $Re (m_{MD}) \approx \mp {N T}_{2} μ_{0} μ_{MD} Re (B_{MD 1}) / (2 | B_{MD 1} | \sqrt{T_{1} T_{2}})$ . It can be easily verified that the variation of $Re (B_{MD 1}) / B_{MD 1}$ is much smaller than that of $Y$ . Thus, the extinction of Si decreases with the decreasing of $Y$ .

The above calculations are based on the Si as the high refractive index material, and we also consider the other materials for the dielectric sphere. Figure 8 shows a case with GaP as the material. The GaP sphere has a refractive index of 3.5. The radius is taken to be 71 nm, so that the corresponding magnetic dipole resonance is near 520 nm too. The ion cluster is the same as that in Fig. 2. Figures 8(b)–8(e) show the extinction spectra of the GaP-based hybrid structure with weak and strong excitations. The line shapes of the response spectra are similar to that of the Si-based hybrid structure. This is because the GaP sphere also shows a magnetic dipole resonance similar to the Si sphere, and the magnetic field enhancement inside the GaP sphere is nearly the same as that of the Si sphere.

Figure 8.Extinction spectra of the GaP-based hybrid structure. (a) Schematic of the system with a GaP sphere. (b), (d) The extinction spectra of the coupled ion cluster in the weak (b) and strong (d) light intensity regime. (c), (e) The extinction spectra of the hybrid structure in the weak (c) and strong (e) light intensity regime.

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4. CONCLUSIONS

In conclusion, we have investigated the optical properties of hybrid structures consisting of dielectric nanospheres and quantum emitters with MD transitions. For a given hybrid structure, the extinction-cross-section spectra of the quantum emitters and dielectric nanospheres show linear or nonlinear behaviors depending on the incident light intensity. For a low light intensity, the extinction of the quantum emitters is greatly enhanced, and a dip appears on the extinction spectrum of the hybrid structure. For a high light intensity, the resonant extinction of the quantum emitters does not show obvious enhancement while the extinction spectrum is broadened. A Fano-like line shape appears on the extinction spectrum of the hybrid. The different spectral responses of the hybrid structure are highly related to the behaviors of the magnetic polarizabilities of the quantum emitters. The effects from the geometric and material parameters of the hybrid structure are considered, which include the MD matrix element, the number of ions $N$ , the relaxation time of ions, and the materials of the dielectric sphere. The optical responses of the coupled structures can be tuned by these parameters. Our results reveal the efficient couplings between MD transitions and magnetic modes of dielectric structures with considerable magnetic field enhancements. For the experimental realizations, the samples may be prepared based on chemical synthesis or laser ablation ^[23,27]. If the ions are uniformly distributed in the dielectric spheres, the measured results should be comparable to our predictions according to the results in Fig. 7. The measurement should be done under a low temperature environment and it requires a spectrometer with high spectral resolution. We expect that more efficient magnetic coupling effects can be obtained in other carefully designed dielectric nanostructure–MD emitter hybrids.

Category: Nanophotonics and Photonic Crystals

Received: Feb. 12, 2019

Accepted: Aug. 6, 2019

Published Online: Sep. 18, 2019

The Author Email: Zhong-Jian Yang (zjyang@csu.edu.cn)

DOI:10.1364/PRJ.7.001142