  Strong nonlinear interactions between quantized excitations are an important
resource for quantum technologies based on bosonic oscillator modes. However,
most electromagnetic and mechanical nonlinearities arising from intrinsic
material properties are far too weak compared to dissipation in the system to
allow for nonlinear effects to be observed on the single-quantum level. To
overcome this limitation, electromagnetic resonators in both the optical and
microwave frequency regimes have been coupled to other strongly nonlinear
quantum systems such as atoms and superconducting qubits, allowing for the
demonstration of effects such as photon blockade and coherent quantum protocols
using the Kerr effect. Here, we demonstrate the realization of the
single-phonon nonlinear regime in a solid-state mechanical system. The
single-phonon anharmonicity in our system exceeds the decoherence rate by a
factor of 6.8, allowing us to use the lowest two energy levels of the resonator
as a mechanical qubit, for which we show initialization, readout, and a
complete set of direct single qubit gates. Our work adds another unique
capability to a powerful quantum acoustics platform for quantum simulations,
sensing, and information processing.
