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Following the paper by M. Combescure [Ann. Phys. (NY) 204, 113 (1990)], weapply the quantum singular time dependent oscillator model to describe therelative one dimensional motion of two ions in a trap. We argue that the modelcan be justified for low energy excited states with the quantum numbers $n\lln_{max}\sim 100$, provided that the dimensionless constant characterizing thestrength of the repulsive potential is large enough, $g_*\sim 10^5$. Timedependent Gaussian-like wave packets generalizing odd coherent states of theharmonic oscillator, and excitation number eigenstates are constructed. We showthat the relative motion of the ions, in contradistinction to its center ofmass counterpart, is extremely sensitive to the time dependence of the bindingharmonic potential, since the large value of $g_*$ results in a significantamplification of the transition probabilities between energy eigenstate evenfor slow time variations of the frequency.Comment: 19 pages, LaTeX, 5 eps-figures, to appear on Phys. Rev. A, one  reference correcte

physical review. a, atomic, molecular, and optical physics/physical review, a, atomic, molecular, and optical physics

Quantum singular oscillator as a model of a two-ion trap: An amplification of transition probabilities due to small-time variations of the binding potential