  Selftrapping has been traditionally studied on the assumption that
quasiparticles interact with harmonic phonons and that this interaction is
linear in the displacement of the phonon. To complement recent semiclassical
studies of anharmonicity and nonlinearity in this context, we present below a
fully quantum mechanical analysis of a two-site system, where the oscillator is
described by a tunably anharmonic potential, with a square well with infinite
walls and the harmonic potential as its extreme limits, and wherein the
interaction is nonlinear in the oscillator displacement. We find that even
highly anharmonic polarons behave similar to their harmonic counterparts in
that selftrapping is preserved for long times in the limit of strong coupling,
and that the polaronic tunneling time scale depends exponentially on the
polaron binding energy. Further, in agreement, with earlier results related to
harmonic polarons, the semiclassical approximation agrees with the full quantum
result in the massive oscillator limit of small oscillator frequency and strong
quasiparticle-oscillator coupling.
