Transition to sub-Planck structures through the superposition of q-oscillator stationary states

We investigate the superposition of four different quantum states based on the $q$-oscillator. These quantum states are expressed by means of Rogers-Szeg"o polynomials. We show that such a superposition has the properties of the quantum harmonic oscillator when $q\to 1$, and those of a compass state with the appearance of chessboard-type interference patterns when $q\to 0$.