Optimal sizes of dielectric microspheres for cavity QED with strong coupling

Abstract

The whispering gallery modes (WGMs) of quartz microspheres are investigated for the purpose of strong coupling between single photons and atoms in cavity quantum electrodynamics (cavity QED). Within our current understanding of the loss mechanisms of the WGMs, the saturation photon number ${n}_{0}$ and critical atom number ${N}_{0}$ cannot be minimized simultaneously, so that an ``optimal'' sphere size is taken to be the radius for which the geometric mean $\sqrt{{n}_{0}{N}_{0}},$ is minimized. While a general treatment is given for the dimensionless parameters used to characterize the atom-cavity system, detailed consideration is given to the ${D}_{2}$ transition in atomic cesium at ${\ensuremath{\lambda}}_{0}=852\mathrm{nm}$ using fused-silica microspheres, for which the maximum coupling coefficient ${g}_{a}/(2\ensuremath{\pi})\ensuremath{\approx}750\mathrm{MHz}$ occurs for a sphere radius $a=3.63\ensuremath{\mu}\mathrm{m}$ corresponding to the minimum for ${n}_{0}\ensuremath{\approx}6.06\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}.$ By contrast, the minimum for ${N}_{0}\ensuremath{\approx}9.00\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$ occurs for a sphere radius of $a=8.12\ensuremath{\mu}\mathrm{m},$ while the optimal sphere size for which $\sqrt{{n}_{0}{N}_{0}}$ is minimized occurs at $a=7.83\ensuremath{\mu}\mathrm{m}.$ On an experimental front, we have fabricated fused-silica microspheres with radii $a\ensuremath{\sim}10\ensuremath{\mu}\mathrm{m}$ and consistently observed quality factors $Q>~0.8\ifmmode\times\else\texttimes\fi{}{10}^{7}.$ These results for the WGMs are compared with corresponding parameters achieved in Fabry-Perot cavities to demonstrate the significant potential of microspheres as a tool for cavity QED with strong coupling.