  We demonstrate a method for encoding Gottesman-Kitaev-Preskill (GKP)
error-correcting qubits with single ultracold atoms trapped in individual sites
of a deep optical lattice. Using quantum optimal control protocols, we
demonstrate the generation of GKP qubit states with 10 dB squeezing, which is
the current minimum allowable squeezing level for use in surface code error
correction. States are encoded in the vibrational levels of the individual
lattice sites and generated via phase modulation of the lattice potential.
Finally, we provide a feasible experimental protocol for the realization of
these states. Our protocol opens up possibilities for generating large arrays
of atomic GKP states for continuous-variable quantum information.
