  Coupled micro- and nanomechanical resonators are of significant interest
within a number of areas of research, ranging from synchronisation, nonlinear
dynamics and chaos, to quantum sensing and transduction. Building upon our work
on soft-clamped membrane resonators, here we present a study on phononic
dimers, consisting of two defects embedded in a phononic crystal membrane.
These devices exhibit widely tunable (2-100 kHz) inter-defect coupling
strengths, leading to delocalized hybrid modes with mechanical Qf-products >
10^14 Hz at room temperature,ensuring low thermomechanical force noise. The
mode splitting exhibits a strong dependence on the dimer orientation within the
crystal lattice, as well as the spatial separation between the two defects.
Given the importance of dynamic range for sensing applications, we characterize
the relevant mechanical nonlinearities, specifically the self- and
cross-Duffing parameters, as well as self and cross-nonlinear dampings. This
work establishes soft-clamped resonators with engineered spatial and spectral
multi-mode structure as a versatile mechanical platform both in the classical
and quantum regimes. Applications in microwave-to-optical transduction and
magnetic resonance force microscopy are particularly attractive prospects.
