Sub-Planck structures and phase-space sensitivity of circular states

The superposition of coherent states has attracted extensive and intensive attention, exemplified by intriguing instances such as Schr"odinger cat states and compass states. In this study, we derive analytical expressions of the Wigner functions and phase-space displacement sensitivity of circular states, which are superpositions of coherent states distributed equidistantly on a circle. The visualization of these phase-space functions exhibits fascinating quantum kaleidoscopic patterns, and intuitively reveals the internal coherence and symmetric structure in these states. Moreover, we unveil the inherent sub-Planck structure of circular states in terms of the zeros of sensitivity functions.