  We show that highly confined superfluid films are extremely nonlinear
mechanical resonators, offering the prospect to realize a mechanical qubit.
Specifically, we consider third-sound surface waves, with nonlinearities
introduced by the van der Waals interaction with the substrate. Confining these
waves to a disk, we derive analytic expressions for the cubic and quartic
nonlinearities and determine the resonance frequency shifts they introduce. We
predict single-phonon shifts that are three orders of magnitude larger than in
current state-of-the-art nonlinear resonators. Combined with the exquisitely
low intrinsic dissipation of superfluid helium and the strongly suppressed
acoustic radiation loss in phononic crystal cavities, we predict that this
could allow blockade interactions between phonons as well as
two-level-system-like behavior. Our work provides a new pathway towards extreme
mechanical nonlinearities, and towards quantum devices that use mechanical
resonators as qubits.
