  Observation of quantum phenomena in cryogenic, optically cooled mechanical
resonators has been recently achieved by a few experiments based on cavity
optomechanics. A well-established experimental platform is based on a thin film
stoichiometric ($ Si_3 N_4 $) nanomembrane embedded in a Fabry-Perot cavity,
where the coupling with the light field is provided by the radiation pressure
of the light impinging on the membrane surface. Two crucial parameters have to
be optimized to ensure that these systems work at the quantum level: the
cooperativity $ C$ describing the optomechanical coupling and the product $ Q
\times \nu$ (quality factor - resonance frequency) related to the decoherence
rate. A significant increase of the latter can be obtained with high
aspect-ratio membrane resonators where uniform stress dilutes the mechanical
dissipation. Furthermore, ultra-high $Q \times \nu$ can be reached by
drastically reducing the edge dissipation via clamp-tapering and/or by
soft-clamping, virtually a clamp-free resonator configuration. In this work, we
investigate, theoretically and experimentally, the edge loss mechanisms
comparing two state-of-the-art resonators built by standard
micro/nanofabrication techniques. The corresponding results would provide
meaningful guidelines for designing new ultra-coherent resonating devices.
