Generation of three-mode nonclassical vibrational states of ions

Various quantum effects associated with quanta of the center-of-mass motion of ions and atoms have been observed experimentally (1) thanks to advances in ion and atom cool-ing and trapping. In addition to opening additional experi-mental routes for testing fundamentals of quantum mechan-ics, this allows one to prepare different nonclassical states of the ion or atom vibration (see, e.g., (2)). Such nonclassical states prove to be promising and realistic candidates forbuilding quantum logic gates in quantum computers (3) because of negligible quantum decoherences: the environmental influence on the ion or atom vibrational modes is extremely weak. Many kinds of nonclassical states of light have thus been extended to the motion of ions (see, e.g., (4 -7)). Schemes for producing nonclassical states of ion motion have mainly concerned a two-dimensional (2D) trap and, hence, the states are of two-mode nature. Recently, three-mode nonclassical states called trio coherent states were introduced (8,9) generalizing the known two-mode pair coherent state (10). Let a_x ,a_y ,a_z be the annihilation opera-tors of the quantum of the ion quantized vibration along the x ,y ,z axes, then the vibrational trio coherent state of the ion, uj ,p ,q&, is defined as the joint eigenstate of the three operators

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