Localized surface plasmon resonances (LSPRs) arising from the collective electron oscillation at the surfaces of metallic nanoparticles have the ability to concentrate light into the nanometric scale, produce highly localized fields, and promote light–matter interactions [1], which enables their use in a wide range of nanophotonic devices, such as nanolasers [2,3] and nanosensors [4,5], as well as in Purcell enhancement [6,7]. In recent years, LSPR-based strong coupling (SC) characterized by the realization of Rabi oscillation and the formation of anti-crossing behavior (new hybrid states) has become the focus of research, divided into three main categories. In the first category, the realization of SC between gap plasmons supported by metal nanoparticle-on-mirror structures (LSPR–mirror coupled modes) and different kinds of quantum emitters, including semiconductor quantum dots (QDs) [8], molecules [9–14], J-aggregates [15–17], and two-dimensional materials [18–22], has been intensively explored. In particular, more recently it has been observed experimentally that the SC between gap plasmons and a single molecule [23,24] or single QD [25,26] successfully forms part-light/part-matter states (mixed states) at the quantum limit, which paves the road toward quantum entangled single-photon sources. In the second category, the SC between LSPR or magnetic plasmons and optical modes (waveguide modes or Fabry–Perot nanocavity modes) enables high electric and magnetic intensity enhancement [27,28] and assisted hot-electron transfer [28], which has important applications in photovoltaic devices [29,30]. In addition, the SC between LSPR or magnetic plasmons and propagating surface plasmon polaritons (SPPs) (the third category) can also achieve high electric and magnetic intensity enhancement [31,32], assisted hot-electron transfer [33], enhanced dephasing time [34], and improved sensor performance [35,36].

The common shortcoming of LSPR is the low quality factor ($Q$-factor) caused by intrinsic metal absorption (nonradiative) and radiative (scattering) losses, which severely shortens the lifetime of only 7–8 fs and subsequently limits the interaction time with the matter [37–39]. In other words, for LSPR-based SC systems, the LSPR corresponds to a short-lifetime oscillator ($\sim 7–8\mathrm{fs}$), while the quantum emitter (QD, molecule, J-aggregate, two-dimensional material), the optical mode (waveguide mode, Fabry–Perot nanocavity mode), and the SPP correspond to a long-lifetime oscillator. As a result, the Rabi splitting energies of LSPR-based SC systems are limited to less than 320 meV [8–36]. To the best of our knowledge, the largest Rabi splitting, with a record value of up to 700 meV, has only been demonstrated between plasmonic Fabry–Perot cavity modes and dye molecules [40], which reveals that using high-$Q$ plasmonic modes could be an effective way for improving Rabi splitting.

In this paper, we aim to explore the SC effect between two high-$Q$ plasmonic modes in a spherical hyperbolic metamaterial (HMM) cavity formed by seven alternating silver/dielectric layers wrapping around a dielectric nanosphere core. First, we theoretically demonstrate a very large Rabi splitting energy of up to 805 meV arising from the SC between a high-$Q$ plasmonic whispering gallery mode (WGM) and high-$Q$ cavity plasmon resonance in the spherical HMM cavity. In addition, the new hybrid modes formed by the SC are demonstrated to exhibit a longer lifetime of up to 71.9–81.6 fs and achieve electric field intensity enhancement at both the dielectric core and the dielectric layers. More importantly, by simply adjusting the dielectric core size, the SC is demonstrated to occur at the same dielectric core index and to be tuned across a broad spectral range (from visible to near-IR).

In order to further understand these sharp resonant modes supported by the spherical HMM cavity, we perform the near-field profiles at the selected energies using an analytical Mie solution. Figures 1(c)–1(h) show the electric field intensity ($|E/E_0{|}^2$) distributions calculated at the resonances ${\mathrm{WGM}}_{1,1}$ ($E\sim 0.602\mathrm{eV}$), ${\mathrm{WGM}}_{1,2}$ ($E\sim 0.873\mathrm{eV}$), ${\mathrm{WGM}}_{1,3}$ ($E\sim 1.305\mathrm{eV}$), ${\mathrm{TM}}_{1,1}$ ($E\sim 2.827\mathrm{eV}$), ${\mathrm{WGM}}_{2,1}$ ($E\sim 1.021\mathrm{eV}$), and ${\mathrm{TM}}_{2,1}$ ($E\sim 3.217\mathrm{eV}$), respectively. It is clear in Figs. 1(c)–1(h) that the electric fields for all resonant modes are tightly confined in the different dielectric layers (WGMs) or the dielectric core (TM modes) of the HMM cavity, and their intensities are all greatly enhanced with factors from 101 to 2550. In addition, the electric field distributions of ${\mathrm{WGM}}_{1,1}$, ${\mathrm{WGM}}_{1,2}$, and ${\mathrm{WGM}}_{1,3}$ shown in Figs. 1(c)–1(e) are found to exhibit a similar twofold symmetry, which reveals that these three resonances correspond to the excitations of a dipolar WGM with an angular mode number of $l=1$. The difference in the field distribution is that the electric fields for ${\mathrm{WGM}}_{11}$, ${\mathrm{WGM}}_{1,2}$, and ${\mathrm{WGM}}_{1,3}$ are mainly concentrated in the first ($d_1$), second ($d_2$), and third dielectric layers ($d_3$) of the HMM cavity, respectively. This confirms that ${\mathrm{WGM}}_{1,1}$, ${\mathrm{WGM}}_{1,2}$, and ${\mathrm{WGM}}_{1,3}$ are the first-order ($m=1$), second-order ($m=2$), and third-order modes ($m=3$), respectively, from the dipolar WGMs ($l=1$). For the resonance ${\mathrm{TM}}_{1,1}$ ($E\sim 2.827\mathrm{eV}$), the electric fields are mainly localized within the dielectric core, showing a single-lobed feature [Fig. 1(f)], which corresponds to the excitations of electric dipolar cavity plasmons ($l=1$) in the HMM cavity. Similar to the analysis above, the electric field intensity distributions for both ${\mathrm{WGM}}_{2,1}$ and ${\mathrm{TM}}_{2,1}$ modes should have the same fourfold symmetry feature ($l=2$), and should mainly be concentrated within the first dielectric layer ($d_1$) and the dielectric core, as displayed in Figs. 1(g) and 1(h), respectively. The above results clearly demonstrate that both the plasmonic WGMs and the cavity plasmon modes can be excited in the HMM cavity. In addition, the sharpness of the WGMs and TM modes in the absorption spectrum reveals a long lifetime of $\tau \sim 14–84\mathrm{fs}$ (Fig. 6 in Appendix A). As a result, the spherical HMM cavity can provide us with an ideal platform for investigating the coupling effect between two high-$Q$ (long-lifetime) plasmonic modes.

To further reveal the physical origins of the observed anti-crossing effect between ${\mathrm{WGM}}_{1,3}$ and ${\mathrm{TM}}_{1,1}$ mode, we use a semiclassical coupled two-oscillator model (CTM) in the following expression [34,36]: $$E_{\text{hybrid}}^{\pm }=\frac{E_{\mathrm{TM}}+E_{\mathrm{WGM}}}{2}\pm \sqrt{\frac{\mathrm{\Delta }}{2}}+\frac{{(E_{\mathrm{TM}}-E_{\mathrm{WGM}})}^2}{4},$$(2)where $\mathrm{\Delta }$ is the coupling strength and $E_{\mathrm{WGM}}$ and $E_{\mathrm{TM}}$ are the resonant energies of the uncoupled ${\mathrm{WGM}}_{1,3}$ and ${\mathrm{TM}}_{1,1}$ mode, respectively. To accurately obtain the resonant energies of the uncoupled ${\mathrm{WGM}}_{1,3}$ and ${\mathrm{TM}}_{1,1}$ mode, the coupling effect between these two modes should be completely excluded. As has been demonstrated above, most of the electric fields of the WGMs are tightly concentrated within the dielectric layers of the HMM cavity (Fig. 1). This feature reveals that the resonant energies of the WGMs should be insensitive to the core materials, even for metals. In addition, if we set the dielectric core of the HMM cavity as a solid silver nanosphere, the cavity plasmons cannot be supported. As a result, the coupling effect between cavity plasmons and WGMs can be completely excluded. The resonant energy of the uncoupled ${\mathrm{WGM}}_{1,3}$ is extracted as 1.444 eV by changing the dielectric core of the HMM cavity to a same-sized solid silver nanosphere, which is displayed as the horizontal white dashed line in Fig. 2(a). Similarly, the resonant energy of the uncoupled ${\mathrm{TM}}_{1,1}$ mode can be obtained via a metallic nanoshell, which cannot support plasmonic WGMs. The cavity plasmons supported by the metallic shell are highly dependent on both the dielectric core and the metal layer thickness [45]. For a silver nanoshell with a thicker silver layer (more than 60 nm), the cavity plasmons cannot be efficiently excited by the visible and near-IR light because the silver thickness is much greater than the optical skin depth in the current spectral range [45]. For a thinner silver layer (less than 15 nm), the light confinement ability is weak, and thus it cannot effectively concentrate the resonant energy within the dielectric core of the silver nanoshell, which also results in a broad feature in the far-field spectrum due to the large radiative loss [45]. Therefore, the resonant energy of an uncoupled ${\mathrm{TM}}_{1,1}$ mode is extracted in a silver nanoshell by choosing an appropriate value for the silver thickness (30 nm) for the same dielectric core size ($r_{\mathrm{core}}=50\mathrm{nm}$). The resonant energies of uncoupled ${\mathrm{TM}}_{1,1}$ modes as a function of $n_{\mathrm{core}}$ are shown by the oblique white dashed line in Fig. 2(a).

After we perform the CTM model, the resonant energies of the two hybrid modes are predicted and displayed with the red lines with circles in Fig. 2(a). Comparing the CTM model (red/circle lines) with the theoretical results (Mie theory) shows good agreement [Fig. 2(a)], which successfully reproduces the anti-crossing behavior of ${\mathrm{WGM}}_{1,3}$ and ${\mathrm{TM}}_{1,1}$ mode. As a result, the normal Rabi splitting energy is obtained at $E({\mathrm{TM}}_{1,1})=E({\mathrm{WGM}}_{1,3})$ or $n_{\mathrm{core}}=4.3$, with an ultralarge value of up to 629 meV ($\hslash \mathrm{\Omega }=629\mathrm{meV}$) displayed in Fig. 2(b), which corresponds to a large coupling strength of $\mathrm{\Delta }=0.198{(\mathrm{eV})}^2$ ($\hslash \mathrm{\Omega }=\sqrt{2\mathrm{\Delta }}$). It should be pointed out that the refractive index of several semiconductor materials can reach a value of 4.3, such as Si, Ge, and GaAs [46,47]. We also note that $\hslash \mathrm{\Omega }\gg ({\mathrm{\Gamma }}_{\mathrm{TM}}+{\mathrm{\Gamma }}_{\mathrm{WGM}})/2=28\mathrm{meV}$ as required for SC. It is also clearly seen in Fig. 2(a) that the ${\mathrm{TM}}_{1,1}$ resonance can also result in anti-crossing behavior with ${\mathrm{WGM}}_{1,2}$ mode when shifting close to the resonant energy of ${\mathrm{WGM}}_{1,2}$ at a large $n_{\mathrm{core}}$ ($\sim 6$), while the Rabi splitting energy of ${\mathrm{TM}}_{1,1}$ and ${\mathrm{WGM}}_{1,2}$ is much smaller than that of ${\mathrm{TM}}_{1,1}$ and ${\mathrm{WGM}}_{1,3}$ due to the fact that the electric field spatial overlap between ${\mathrm{TM}}_{1,1}$ mode and ${\mathrm{WGM}}_{1,3}$ is much larger than that of ${\mathrm{WGM}}_{1,2}$ and ${\mathrm{WGM}}_{1,1}$ both at the dielectric core and in the dielectric layers (Fig. 8 in Appendix A). In Fig. 2(b), we further find that two hybrid modes formed by SC exhibit ultralong lifetimes of 71.9 fs (${\mathit{E}}_{\text{hybrid}}^{-}=1.1117\mathrm{eV}$) and 81.6 fs (${\mathit{E}}_{\text{hybrid}}^{+}=1.7405\mathrm{eV}$), which are about 2 times larger than those of the uncoupled ${\mathrm{WGM}}_{1,3}$ and ${\mathrm{TM}}_{1,1}$ mode (Fig. 6 in Appendix A).

In the following, we further demonstrate that the normal Rabi splitting energy ($\hslash \mathrm{\Omega }$) or the coupling strength ($\mathrm{\Delta }$) is mainly determined by the inner silver thickness ($s_4$). As $s_4$ increases from 15 nm to 25 nm and to 35 nm, not only does the Rabi splitting energy (or coupling strength $\mathrm{\Delta }$) decrease quickly from 629 meV $[\mathrm{\Delta }=0.198{(\mathrm{eV})}^2]$ to 335 meV $[\mathrm{\Delta }=0.056{(\mathrm{eV})}^2]$ and to 185 meV $[\mathrm{\Delta }=0.017{(\mathrm{eV})}^2]$, but the $n_{\mathrm{core}}$ also increases rapidly from 4.3 to 5.0 and to 5.5, respectively. The values of 5.0 and 5.5 obviously exceed the refractive index of most materials in the visible to the near-IR spectral range (Fig. 9 in Appendix A).

The absorption efficiency spectra of the spherical HMM cavity at $n_{\mathrm{core}}=4.3$ with dielectric core radii of 30 nm, 40 nm, 60 nm, and 80 nm are shown in Fig. 5(b). For the HMM cavity with a relatively small dielectric core radius of $r_{\mathrm{core}}=30\mathrm{nm}$, a record normal Rabi splitting energy as large as 788 meV $[\mathrm{\Delta }=0.31{(\mathrm{eV})}^2]$ arising from the SC between ${\mathrm{WGM}}_{1,3}$ and ${\mathrm{TM}}_{1,1}$ mode is observed in the visible spectral range from 2.23 eV (556.1 nm) to 1.441 eV (860.5 nm), as indicated by the black line in Fig. 5(b). As $r_{\mathrm{core}}$ increases from 30 nm to 40 nm, to 60 nm, and to 80 nm, the Rabi splitting energy decreases from 788 meV (black line) to 713 meV (red line), to 552 meV (blue line), and to 428 meV (magenta line), respectively. It should be noted that the Rabi splitting energy for the HMM cavity with a relatively large dielectric core radius of $r_{\mathrm{core}}=80\mathrm{nm}$ still reaches a relatively high value of 428 meV, and a high-order cavity plasmon mode of ${\mathrm{TM}}_{1,2}$ appears at 2.813 eV in the current spectral range, which is shown by the magenta line in Fig. 5(b). In addition, the spectral ranges where the SC occurs are shifted from 2.23–1.441 eV (556.1–860.3 nm) to 1.965–1.252 eV (631–990.4 nm), to 1.554–1.003 eV (797.7–1236.4 nm), and to 1.273–0.844 eV (974.4–1469.1 nm) as $r_{\mathrm{core}}$ increases from 30 nm to 40 nm, to 60 nm, and to 80 nm [Fig. 5(b)], demonstrating that the SC between ${\mathrm{WGM}}_{1,3}$ and ${\mathrm{TM}}_{1,1}$ mode in the HMM cavity can be tuned across a broad spectral range by simply changing the dielectric core size.

Figure 5(c) displays the Rabi splitting energy in the HMM cavity as a function of $r_{\mathrm{core}}$ (from 10 nm to 80 nm), and presents a trend of first an increase and then a decrease, revealing a maximum Rabi splitting energy of 805 meV at $r_{\mathrm{core}}=20\mathrm{nm}$. As $r_{\mathrm{core}}$ is increased from 20 nm to 80 nm, the Rabi splitting energy linearly decreases from 805 meV to 428 meV, due to the fact that the longer-wavelength light has a smaller optical skin depth in silver, and the coupling strength (Rabi splitting) is thus reduced for the longer-wavelength modes supported by the larger $r_{\mathrm{core}}$ of the HMM cavity. For the smaller $r_{\mathrm{core}}$, the coupling strength (Rabi splitting) between ${\mathrm{WGM}}_{1,3}$ and ${\mathrm{TM}}_{1,1}$ mode in the HMM cavity is also limited when the resonant modes blueshift to the vicinity of the interband transition of the silver [42,43]. It is shown in Fig. 5(c) that the Rabi splitting energy decreases quickly from 805 meV at $r_{\mathrm{core}}=20\mathrm{nm}$ to 626 meV at $r_{\mathrm{core}}=10\mathrm{nm}$.

In summary, we have theoretically demonstrated a record Rabi splitting energy of up to 805 meV, arising from the SC between two high-$Q$ (long-lifetime) plasmonic resonances (${\mathrm{WGM}}_{1,3}$ and ${\mathrm{TM}}_{1,1}$ mode) in a spherical HMM cavity. The strong energy exchange between these two modes leads to the formation of new hybrid modes, which exhibit a longer lifetime ($\tau \sim 71.9–81.6\mathrm{fs}$) and great enhancement of the electric field intensity both at the dielectric core and in the dielectric layers in the HMM cavity. Importantly, by adjusting the dielectric core size, we have shown that the SC occurs at the same dielectric core index and can be tuned across a broad range from the visible to the near-IR spectral range (broadband nature). These results provide direction for achieving ultralong lifetimes of surface plasmons and may find potential applications in bright single-photon sources [48,49].

Acknowledgment. G. P. thanks the NUPTSF and the Double Innovation Project of Jiangsu Province.