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 $q\to 1$, and those of a compass state with the appearance of chessboard-type interference patterns when $q \to 0$.

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