Ordered space-time structures: Quantum carpets from Gaussian sum theory

The term "quantum carpet" can be observed in many closed quantum systems, where the evolution of a wave function exhibits a carpet-like pattern. Quantum carpet mechanisms are also akin to the classical interference patterns of light. Although the origins of quantum carpets have previously been studied by various researchers, many interesting details are still worth exploring. In this study, we present a unified framework for simultaneously analyzing three different features of quantum carpets: full revival, fractional revival, and diagonal canal. For the fractional revival feature, a complete formula is presented to explain its formation through Gaussian sum theory, in which all essential features, including phases and amplitudes, are captured analytically. We also reveal important relationships between the interference terms of diagonal canals and their geometric interpretations such that a better understanding of the development of diagonal canals can be supported.