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The normal-ordered form of the composition of the exponential of a quadratic annihilation operator and its adjoint is derived by means of a new method based on quantum white noise calculus. Our method consists of two steps: (i) to derive a system of differential equations consisting of quantum white noise derivatives and Wick products; and (ii) to solve explicitly the system of differential equations.

Calculating Normal-Ordered Forms in Fock Space by Quantum White Noise Derivatives