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

TL;DR

This paper explores how superposing four q-oscillator states transitions from quantum harmonic oscillator behavior to sub-Planck structures resembling compass states with chessboard interference patterns as q varies.

Contribution

It demonstrates the transition of superposed q-oscillator states from harmonic oscillator properties to sub-Planck structures, revealing new insights into quantum state superpositions.

Findings

Superposition exhibits harmonic oscillator properties at q→1

Transition to compass states with chessboard patterns at q→0

Shows continuous transition between quantum states based on q-value

Abstract

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 $q \to 0$.