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Abstract
Here we theoretically investigate the coherent interactions between the quantum emitters with magnetic dipole transitions and subwavelength all-dielectric resonators of whispering gallery modes (WGMs). We extend a semi-analytical method which can efficiently calculate the far-field spectrum of a general hybrid system. Then, a subwavelength sphere with refractive index around n = 3.5 is chosen as the dielectric resonator. Due to the high magnetic field enhancements of the WGMs of the sphere, strong couplings between magnetic quantum emitters and subwavelength WGMs can occur, where a clear Rabi splitting appears on the extinction spectrum of the hybrid system. The match between the relaxation times of the WGMs and emitters are important to efficiently achieve a strong enough coupling. The other parameters including the order of a WGM, the radius, the refractive index, the transition dipole moment and excitation intensity are also important factors that can affect the couplings. Our results pave the way for strong interactions between light and magnetic emitters mediated by subwavelength all-dielectric resonators.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Nanophotonic all-dielectric resonators (DRs) with high refractive index have recently drawn a lot of research interest [1–3] due to their important applications, such as metasurfaces [4], enhanced nonlinear optics [5], and optical nanoantennas [6–8]. A DR can exhibit strong far-field scattering responses which are usually accompanied by certain electric and/or magnetic near-field enhancements around the structure. These near-field enhancements could enable its interactions with other DRs [9–12] and plasmonic structures [13–16]. The couplings of different DRs can induce many phenomena including the hybridizations of electromagnetic modes [9] and Fano resonances [10–12]. The interactions of DRs and plasmonic structure can strongly modify their optical responses including the far-field scattering, and near-field enhancements. The near-field enhancements in DRs can also induce their couplings with the quantum emitters. The coupling strength between DRs and quantum emitters can be characterized by the Rabi splitting on the scattering spectrum of the coupled system [17–24]. In some hybrid systems, the emission properties are investigated [25–30], where the enhancements of decay rate and fluorescence have been obtained.
One important feature of DRs is that they can readily support magnetic resonances with magnetic near-field enhancements [1–3]. Thus, they can also be utilized to turn the interactions between light and magnetic quantum emitters which exhibit magnetic dipole (MD) transitions. Those MD transitions can be found in many rare-earth ions [31]. Their couplings with light are usually much weaker than that with electric emitters. The DRs could bring stronger couplings between magnetic emitters and light due to their magnetic field enhancement. The emission properties of MD emitters can be modified a lot with DRs [32–39] such as the decay rate and directionality. It has been shown that the coherent couplings between DRs and MD emitters can also induce a dip on the scattering [40]. The magnetic modes involved in those couplings are low-order resonances, where their magnetic field enhancements are not high enough (∼101). Thus, the coupling strengths are still weak. It is not intuitive to increase the near field enhancement in DRs as they are widely spread around the entire volume. One practical way is to find a resonant mode with a high quality (Q) factor. Some new concepts have been introduced in individual DRs to enhance the Q-factors, such as anapole and quasi- bound states in the continuum (quasi-BIC) modes, where the field enhancement can be increased several times [41–44]. Whispering gallery modes (WGMs) of micro-resonators have been shown to exhibit high-Q factors [45–48], while most of the reported WGMs are far beyond the subwavelength scale. Recently, it has been shown that WGMs of subwavelength DRs can support ultrahigh field enhancements [49] due to their well-known high-Q factors. The enhancement values are more than one order of magnitude larger than that of the modes mentioned above. The ultrahigh field enhancements indicate that WGMs may be used to boost the interactions between light and magnetic emitters.
Here we utilize and extend a semi-analytical method which can efficiently calculate the coherent coupling between a general DR and MD emitters. It is shown that strong coupling between subwavelength DRs of WGMs and MD emitters can be obtained. The DRs are spheres with a high refractive index around n = 3.5. The MD emitters are taken as two-level emitters. The electromagnetic multipole responses and the corresponding near fields can be calculated by the Mie theory [50]. For the subwavelength WGM of b7 order, the magnetic field enhancement reaches a high value of ∼1300 which is about two orders of magnitude higher than the common low-order modes. Thus, the strong coupling between the WGMs and MD emitters can be achieved, which is characterized by a relatively large enough Rabi splitting on the extinction spectrum. The match between the relaxation times of a WGM and the MD emitters are also an important factor to achieve strong enough coupling. The effects from the other system parameters such as the order of a WGM, radius and the refractive index will also be considered.
2. Semi-analytical method for a DR-MD emitter coupled system
We now consider the optical resonance of hybrid nanostructure consisting of a general dielectric resonator (DR) and single two-level quantum emitter with magnetic dipole transition. The analytical expressions are similar to the hybrids consisting of nanocavities with electric or magnetic dipole resonances [40,51,52], while the nanophotonic structures involved here are general DRs with higher electromagnetic modes and the emitter is of magnetic dipole transition. The Hamiltonian for the MD emitter in the coupled system can be expressed as ${H_{MD}} = \hbar {\omega _0}{\hat{a}^\dagger }\widehat {a\; } - {\mu _{MD}}{B_{MD}}\hat{a} - {\mu _{MD}}B_{MD}^\ast {\hat{a}^\dagger }$, where ${\hat{a}^\dagger }$ and $\hat{a}\; \; $ are the creation and annihilation operators of the excited state, respectively. ${\mu _{MD}}\; $ and ${\omega _0}$ are the matrix element of the transition and resonance frequency of the MD emitter, respectively. ${B_{MD}}$ is the total magnetic field felt by the MD emitter, which consists of the magnetic field of the excitation plane wave and the magnetic field excited by the DR ${B_{MD}} = {B_0} + X{m_{\textrm{DR}}}$, where ${m_{\textrm{DR}}}$ is the magnetic multipolar moment of the DR, and X represents the proportional coefficient between the magnetic field produced by the DR and its magnetic multipolar moment. Similarly, the response of the DR is a multipole moment, and the equivalent magnetic field felt at the origin of the multipole moment can be written as ${B_{DR}} = {B_0} + Y{m_{\textrm{MD}}}$, where $Y{m_{\textrm{MD}}}$ is the contribution of the MD emitter, ${m_{\textrm{MD}}}\; $ is the magnetic dipole moment of the MD emitter, and Y represents the proportional coefficient between the equivalent magnetic field felt by DR and the dipole moment of the MD emitter. The ${m_{\textrm{MD}}}$ can be written as ${\mu _{\textrm{MD}}}({{\rho_{12}} + {\rho_{21}}} )$ by using the density matrix. If many MD emitters (rare-earth ion cluster) are considered and the number of the ions is N, the magnetic dipole moment of the ion cluster can be written as $N{\mu _{\textrm{MD}}}({{\rho_{12}} + {\rho_{21}}} )$. The magnetic dipole moment of magnetic ions can also be expressed as ${m_{MD}} = \mu _0^{ - 1}{B_{MD}}{\alpha _{MD}}$, where ${\alpha _{MD}}$ is the magnetic dipole polarizability of the magnetic ions. Thus, ${B_{MD}}$ can be expressed as, ${B_{MD}}(t) = \frac{\hbar }{{{\mu _{MD}}}}[{({\Omega + G{{\bar{\rho }}_{21}}} ){e^{ - i\omega t}} + ({{\Omega ^ \ast } + {G^ \ast }{{\bar{\rho }}_{12}}} ){e^{i\omega t}}} ]$ where $\Omega = \frac{{{\mu _{MD}}}}{\hbar }(\textrm{1} + X\frac{{{\alpha _M}}}{{{\mu _0}}})\frac{{{B_\textrm{0}}}}{\textrm{2}},$ $G = \frac{{\mu _{MD}^\textrm{2}XYN{\alpha _M}}}{{{\mu _0}\hbar }},$ ${\alpha _M}$ is the magnetic multipole polarizability of the DR, ${\bar{\rho }_{12}}$ and ${\bar{\rho }_{21}}$ are the density matrix.
The matrix elements satisfy the master equation, $\frac{{d{\rho _{ij}}}}{{dt}} = \frac{i}{\hbar }{[{\rho ,{H_{QE}}} ]_{ij}} - {\Gamma _{ij}}{\rho _{ij}},$ where ${\Gamma _{22}} ={-} {\Gamma _{11}} = 1/{T_1}$ and ${\Gamma _{12}} = {\Gamma _{21}} = 1/{T_2}$. T1 and T2 are the longitudinal and transverse homogeneous relaxation times, respectively. By solving the master equation with the rotating wave approximation under steady state situation [40], one can obtain the magnetic dipole polarizability of magnetic ions as
The extinction cross section of magnetic ion cluster can be expressed as
The extinction section of the DR can be divided into two parts. One is the contribution from the resonant multipolar modes of DR which are coupling with the magnetic ions, and the other is the contribution of other modes which is assumed to be unaffected by ions. Therefore, the extinction cross section of the coupled DR can be written as (see Supplement 1)
3. Results and discussion
Let us consider the optical interaction between a practical subwavelength all-dielectric nanosphere and MD emitters (Fig. 1(a)). The structure is excited by a plane wave along the z-axis with the polarization direction of the x-axis. Silicon is chosen as material for the sphere, and its refractive index is taken to be n = 3.5 in our considered wavelength range [55]. With a diameter of D = 1140 nm, the individual sphere shows a WGM mode at λ0=1200 nm corresponding to the b7 order in the Mie theory (see Supplement 1). Figure 1(b) shows the magnetic field enhancement on the y-z plane of the b7 WGM obtained by the Mie theory. The highest value reaches B/B0 = 1300, which is much larger than that the common low-order modes in all-dielectric nanocaivties. Such a large field enhancement is beneficial for the coupling between all-dielectric cavities and MD emitters.
Fig. 1. (a) Schematic of a hybrid system under study. (b) Magnetic field distribution of the b7 WGM of an individual Si sphere in the y-z plane. (c) The extinction spectra of the coupled system (red), coupled DR (blue) and coupled ion cluster (black) under the weak excitation limit.
The hybrid system consists of the above Si nanosphere and an ion cluster which is placed at a point with the highest magnetic field enhancement of the b7 WGM. The matrix element for the MD transition is 0.3${\mu _B}$ (${\mu _B}$ is the Bohr magneton). The number of ions is N = 28000. First, we consider the coupling under weak excitation limit, where the excitation ${I_0} = 10W/c{m^2}$ satisfies $\mu _{MD}^2{|{{B_{MD}}} |^2}{T_1}{T_2}/{\hbar ^2} < < 1$. The responses of the MD and the hybrid are in linear region. The relaxation time T1 and T2 of the ions are both taken to be T = 200 ps (T1 = T2 = T). We note that the T1 may be different from T2, especially under the room-temperature condition, where T1 can be much larger than T2. However, T1 does not affect the coupled MD emitters and the coupled system under the weak excitation limit ($\mu _{MD}^2{|{{B_{MD}}} |^2}{T_1}{T_2}/{\hbar ^2} < < 1$). This is because the term which contains T1 is omitted in Eq. (1). Thus, it is safe to assume that T1 = T2 = T in the linear region. Figure 1(c) shows the corresponding extinction spectrum of the hybrid obtained by the semi-analytical method. The contribution of the other modes which is assumed to be unaffected by ions is about 1.4${\times} {10^{ - 12}}{m^2}$. The cross section of the coupled magnetic ions is more than 3${\times} $105 times higher than that of the individual ions at resonance (see Supplement 1). This enhancement is much higher than the one with a magnetic-dipole-resonant DR (∼200 times) [40], indicating that the coupling in this system is much stronger than the later. The extinction spectrum of the whole coupled system, which consists of the contributions from the coupled sphere and coupled ions, exhibits a clear Rabi splitting, indicating a strong coupling between the magnetic ions and the WGM. It can be checked that the extinction spectra obtained by the semi-analytical method under weak excitation limit agrees well with that from direct FDTD simulations (see Supplement 1). This confirms that X equals Y in our method. Note that it takes several days to calculate the spectrum of a common hybrid system with the FDTD method due to the multi-scale feature of the system, while such a hybrid can be calculated instantly by our method.
We now study the effect from the relaxation time T of the ions on the Rabi splitting. Figure 2(a) shows the extinction spectra with the relaxation time T varying from 100 ps to 500 ps. The other parameters are the same as that in Fig. 1(a). The Rabi splitting width ΩR first increases and gradually becomes saturated around the relaxation time of the b7 WGM. The relative splitting width ΩR/γ shows a similar behavior, where γ represents the larger decay rate of the magnetic ions γMD and WGM γDR (γ = maximal {γMD, γDR}). When the relaxation time of ions is much larger than that of the WGM, although the Rabi splitting will not increases with T, the dip becomes relatively more obvious and the dip value approaches 0 eventually (see Supplement 1). The behavior of the Rabi splitting width ΩR can be explained by the fact that the ions are spectrally fully involved in the couplings when T is a large enough value. The coupling strength will not change as the µMD is fixed. Thus, the splitting ΩR becomes saturated with a large enough T (γMD < γDR). This situation is ruined when the γMD > γDR, where the spectral width of the coupled ions follows that of the WGM (see Supplement 1), and it is smaller than that of the individual ions. Thus, from the point view of the spectral overlapping, the ions are only partially involved in the coupling. This factor will make the coupling relatively weaker. As for the dip value, it can be checked that the dip value satisfies ${({{\sigma_{\textrm{ext}}}} )_{DR}}_{({\omega = {\omega_\textrm{0}}} )} = k\textbf{Im}({\alpha _M}){\left( {\frac{\textrm{1}}{{\textrm{1} + A}}} \right)^\textrm{2}}$, where $A = \frac{{|Y ||B |N\mu _{MD}^2{T_2}}}{{\hbar {B_0}}}$ (see Supplement 1). Thus it becomes smaller with T and approaches 0 gradually.
Fig. 2. (a) Extinction spectra of the coupled system with different relaxation times T of the ions. The other parameters are the same as that in Fig. 1(a). (b) The Rabi splitting width ΩR of the extinction spectrum of the hybrid system and relative splitting width ΩR/γ as functions of T. γ = maximal {γMD, γDR}. The dashed line shows the position of the relaxation time for the b7 WGM. (c) Extinction spectra of the coupled system with different modes of WGM. (d) The Rabi splitting width ΩR and relative splitting width ΩR/γ with different modes of WGM. The other parameters are the same as that in Fig. 1(a).
The influence from the relaxation time of the WGM mode on the coupling of the hybrid system is also considered (Fig. 2(c)), where the different WGMs (b6, b7 and b8) exhibit different relaxation times. Figure 2(d) shows the Rabi splitting width ΩR and the relative Rabi splitting ΩR/γ as a function of the order of the WGM. Among these hybrids, only the one with the b7 WGM can achieve a strong coupling (ΩR/γ∼1). The relative Rabi splitting ΩR/γ becomes even smaller with higher WGMs. It is easily to understand the weaker coupling for a lower order of WGM, where the magnetic field enhancement is much smaller than that of the b7 WGM, and the coupling is weaker correspondingly. For a higher order of WGM with γMD > γDR, this situation is the same as discussed above that the ions are only partially involved in the coupling from the point view of spectral overlapping. This will make the coupling relatively weaker. From Fig. 2, it is seen that the match between the relaxation times of ions and WGMs is an important factor for achieving strong enough coupling.
We now consider the ability of the WGMs in the coupled system under the condition that the relaxation time of ions T is adjusted to a long enough value. From Fig. 2, the T can be taken to be a little larger than the relaxation time of the WGM and thus it will not affect the Rabi splitting ΩR. The relaxation times of ions can be turned by varying the surrounding temperature. The working wavelength is fixed around λ0=1200 nm. The different WGMs are obtained by varying the size of the sphere. Figure 3(a) shows the required number of ions to achieve the strong coupling ΩR/γ = 1 as a function of the order of WGM. The required number shows approximately an exponential decay with the order of WGM, and it approaches to 1 around b11 for n = 3.5. The dielectric sphere with a higher refractive index can excite a relatively stronger magnetic field enhancement for the same order of WGM. Besides, the dielectric sphere has a smaller radius, which means the interaction coefficient X(Y) between the dielectric sphere and magnetic ions is larger. Thus, the required number for the same order of WGM is smaller, and it reaches 1 at b8 which is around the subwavelength region.
Fig. 3. (a) The minimal number N of magnetic ions to reach the strong coupling ΩR/γ = 1 as a function of the order of WGM. The red and black dots represent with the case with the refractive index n = 3.5 and n = 4, respectively. (b) The minimal number N of magnetic ions to reach the strong coupling ΩR/γ = 1 as a function of the order of WGM.
The size of the DR also affects the coupling strength a lot even with the same order of WGM. Figure 3(b) shows the minimal number N of magnetic ions required to achieve the strong coupling ΩR/γ = 1 as a function of the sphere radius. The refractive index and mode order are n = 3.5 and b7, respectively. The required number increases with the radius of the sphere. This is induced by the fact that by increasing the radius, the magnetic field enhancement value keeps the same, while the scattering value increases correspondingly. The interaction coefficient X(Y) decreases with the radius, and thus the number of magnetic ions required to achieve the strong coupling becomes larger correspondingly.
We have focused on the cases with a fixed dipole transition moment µMD. Now let us turn to the situation with different µMD. Figure 4(a) shows the extinction spectra of the hybrid system with varying the µMD under the weak excitation limit. The Rabi splitting width ΩR increases with µMD as expected. This means that if the ions with a larger µMD are involved in the coupling, the required number of ions to achieve the strong coupling ΩR/γ = 1 will become smaller. For example, the required number is 4 times smaller if the µMD becomes twice larger. The splitting depth of the extinction spectrum becomes deeper as indicated by the expression for $({{\sigma_{\textrm{ext}}}} )_{DR(\omega = {\omega _0})}$ before. The situation with high excitation intensity is also considered, where the condition $\mu _{MD}^2{|{{B_{MD}}} |^2}{T_1}{T_2}/{\hbar ^2} < < 1$ is not satisfied, the coupled system turns to the nonlinear response region. The splitting of the coupled system becomes weaker with the intensity (Fig. 4(b)) due to the saturation effect [40].
Fig. 4. (a) Extinction spectra of the coupled system with varying the MD matrix element. The other parameters are the same as that in Fig. 1. (b) Extinction spectra of the coupled system with different light intensities. The other parameters are the same as that in Fig. 1.
Besides the strong magnetic field enhancement, WGMs also support strong electric field enhancements. In fact the couplings between the WGMs and the quantum emitters with electric dipole transitions have been studied a lot [55–58]. The previous cavities are mainly in micro-scale, and they are usually much large than the working wavelength. Thus, the WGMs are of much higher orders before. Here we shall demonstrate the coupling in the hybrid system with a subwavelength WGM. The analytical treatment is quite similar to that with magnetic emitters above. Here one needs to replace the magnetic field and µMD with the electric field and μe respectively, where μe represent the electric dipole moment of the emitter. Figure 5(a) shows the electric field enhancement distribution of the b7 WGM on the y-z plane. The electric field enhancement reaches 350 times. Figure 5(b) shows the extinction spectrum of the coupled systems consisting of the sphere and a single molecule whose transition dipole moment is 3.8 Debye. The relaxation time of the molecule is Te = 100ps which is smaller than that of the WGM. The system is under the weak excitation limit. There is a clear splitting on the extinction spectra of the hybrid system. It can be confirmed that the splitting width here is larger than the width of the molecule, indicating that the strong coupling appears in the hybrid system. The coupling becomes weaker with decreasing the relaxation time of the molecule (Fig. 5(c)). This can be understood by the explanations for the similar results in Fig. 2(a). The coupling is much weaker than the plasmonic cases [59–61] when the relaxation time of the molecule is short enough although the field enhancement here is large. This is mainly due to the fact that the spectral width of the WGM is now much narrower than that of the individual molecule, while the plasmon responses still covers all of the spectral response of the molecule.
Fig. 5. (a) Electric field enhancement distribution of the b7 WGM on the y-z plane. The parameters are the same as that in Fig. 1(a). (b) The extinction spectra of the coupled system (red), coupled sphere (blue) and coupled molecule (black). (b) The extinction spectra of the coupled system with different relaxation time of the emitters Te.
4. Conclusion
In conclusion, we have extended a semi-analytical method which can efficiently calculate the extinction spectrum of a general coupled system of a DR and MD emitters. Specifically, we investigated the DRs with WGMs as they hold high magnetic field enhancements. Strong couplings between MD emitters and WGMs can be achieved with proper parameters, where coupling is much more efficient than that with the common low-order modes like magnetic dipole resonance of a DR. The match between the relaxation times of a WGM and the MD emitters (γMD ≈ γDR) is important for efficient couplings as it will affect the spectral overlapping for their interaction. For a given WGM, a longer relaxation time T of the ions (γMD < γDR) will not make the Rabi splitting larger, while the required temperature becomes lower. For a given ion cluster, the relative Rabi splitting ΩR/γ becomes smaller when the relaxation time of the WGM is either larger (γMD >γDR) or smaller (γMD < γDR) than the ions. The other parameters such as the order of a WGM, the radius and refractive index, the µMD and excitation intensity can also affect the coupling behavior a lot. Additionally, we also consider the coupling between a b7 WGM with a common single molecule which has an electric dipole transition. Strong couplings can be obtained with long enough relaxation time of the molecule (∼100 ps). When the relaxation time of the molecule is much smaller, the coupling becomes much weaker, and it is relatively less efficient than the plasmonic system. Our results demonstrate the possibility to achieve strong couplings between light and MD emitters mediated by DRs with WGMs in subwavelength scale, which may find important applications in quantum optics. As for the experimental realization, there have been some attempts to prepare sphere DRs with magnetic emitters [32]. This may be considered to realize our system. Also one may consider subwavelength DRs with other geometries such as disks which also support high-Q WGMs [49]. As the both the WGM and the emitter have narrow spectral widths, their spectral overlap may be a challenge in experiments. Thus, it requires accurate fabrications [57,58]. On the other hand, one may fabricate many DRs with varied sizes to match the strict spectral overlap.
Funding
National Natural Science Foundation of China (11704416).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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