Lissajous dynamics of a quantum particle in a tilted two-dimensional discrete lattice

The quantum dynamics of a single particle in a discrete two-dimensional tilted lattice is analyzed from the perspective of the classical-quantum correspondence. Utilizing the fact that tilting the lattice results in oscillatory dynamics, we show how the parameters of the lattice and the initial state of the particle can be tuned so that during evolution the probability distribution does not change its shape, while its center follows the trajectory known in classical mechanics as Lissajous curves.