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Microwave penetration depth $\lambda$ and surface resistance at 27 GHz aremeasured in high quality crystals of KOs$_2$O$_6$. Firm evidence forfully-gapped superconductivity is provided from $\lambda(T)$. Below the secondtransition at $T_{\rm p}\sim 8$ K, the superfluid density shows a step-likechange with a suppression of effective critical temperature $T_{\rm c}$.Concurrently, the extracted quasiparticle scattering time shows a steepenhancement, indicating a strong coupling between the anomalous rattling motionof K ions and quasiparticles. The results imply that the rattling phonons helpto enhance superconductivity, and that K sites freeze to an ordered state withlong quasiparticle mean free path below $T_{\rm p}$.

Effects of Rattling Phonons on the Quasiparticle Excitation and Dynamics in the Superconducting $\beta$-Pyrochlore KOs$_2$O$_6$