Resonances in optical cavities have been used to manipulate light propagation, enhance light-matter interaction, modulate quantum states, and so on. However, in traditional cavities, the permittivity contrast in and out the cavity is not so high. Recently, zero-index materials (ZIMs) with unique properties and specific applications have attracted great interest. By putting optical cavity into ZIMs, the extreme circumstance with infinite permittivity contrast can be obtained. Here, we theoretically study Mie resonances of dielectric cavities embedded in ZIMs with $varepsilon approx 0$, or $mu approx 0$, or $(varepsilon,mu) approx 0$. Owing to ultrahigh contrast ratio of $varepsilon$ or $mu$ in and out the cavities, with fixed wavelength, a series of Mie resonances with the same angular mode number $l$ but with different cavity radii are obtained; more interestingly, its $2^l$-TM (TE) and $2^{l+1}$-TE (TM) modes have the same resonant solution for the cavity in $varepsilon approx 0$ ($mu approx 0$) material, and the resonance degeneracy also occurs between $2^l$-TM mode and $2^l$-TE mode for $(varepsilon,mu) approx 0$ material. We further use resonance degeneracy to modulate the Purcell effect of quantum emitter inside the cavity. The results of resonance nesting and degeneracy will provide an additional view or freedom to enhance the performance of cavity behaviors.
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