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An analytical construction of a wave function with localization in classicalperiodic orbits in an elliptic billiard has been achieved by appropriatelysuperposing nearly coherent states expressed as products of Mathieu functions.We analyze and discuss the rotational and librational regimes of motion in theelliptic billiard. Simplified line equations corresponding to the classicaltrajectories can be extracted from the quantum coherent state as an integralequation involving angular Mathieu functions. The phase factors appearing inthe integrals are connected to classical initial positions and velocitycomponents. We analyze the probability current density, the phase maps, and thevortex distributions of the coherent states for both rotational and librationalmotions. The coherent state may represent traveling and standing trajectoriesinside the elliptic billiard.

Classical periodic orbits from coherent states in mesoscopic quantum elliptic billiards