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–Szegö polynomials. We show that such a superposition has the properties of the quantum harmonic oscillator when q→1q→1, and those of a compass state with the appearance of chessboard-type interference patterns when q→0q→0.