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We present a universal algorithm for an efficient deterministic preparationof an arbitrary two--mode bosonic state. In particular, we discuss in detailpreparation of entangled states of a two-dimensional vibrational motion of atrapped ion via a sequence of laser stimulated Raman transitions. Our formalismcan be generalized for multi-mode bosonic fields. We examine stability of ouralgorithm with respect to a technical noise.

Quantum-state synthesis of multimode bosonic fields: Preparation of arbitrary states of two-dimensional vibrational motion of trapped ions