Hyperbolic Metamaterial Nano-Resonators Make Poor Single Photon Sources

We study the optical properties of quantum dipole emitters coupled to hyperbolic metamaterial nano-resonators using a semi-analytical quasinormal mode approach. We show that coupling to metamaterial nano-resonators can lead to significant Purcell enhancements that are nearly an order of magnitude larger than those of plasmonic resonators with comparable geometry. However, the associated single photon output $\beta$-factors are extremely low (around 10$\%$), far smaller than those of comparable sized metallic resonators (70$\%$). Using a quasinormal mode expansion of the photon Green function, we describe how the low $\beta$-factors are due to increased Ohmic quenching arising from redshifted resonances, larger quality factors, and stronger confinement of light within the metal. In contrast to current wisdom, these results suggest that hyperbolic metamaterial nano-structures make poor choices for single photon sources.