Superpositions of distinct phase states by a nonlinear evolution

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Abstract
The harmonic-oscillator phase state that is subjected to a nonlinear evolution exhibits remarkable properties. The state evolves into a generalized phase state. A discrete superposition of distinct phase states can evolve, and one special state consists of a superposition of two phase states out of phase by π rad. The condition for the phase state to evolve into a finite superposition of phase states is intimately related to whether the scaled time of evolution is rational or irrational. The phase distribution for these phase states is presented, and ideal phase measurements of the evolved phase states are described. The diagonal elements of the nonlinear evolution operator in the phase-state basis are shown to be related to curlicues and Gaussian sums.