Quantum manifestations of classical periodic orbits in a square billiard: Formation of vortex lattices

We extend the presentation of the SU(2) coherent states to analytically construct the wave function concentrated on high-order classical periodic orbits in a square billiard. With the constructed wave function, the localization of the wave pattern is found to be very efficient. We also analyze the vortices arising from the singular points of the quantum phase for the constructed coherent states. It is found that the wave interference gives rise to the appearance of vortex lattices in the probability current density associated with the high-order periodic orbits. Moreover, the topological charge of the vortex is in general nonintegral except for the periodic orbits with the same winding number to the sides of the square.