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.
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