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

Following the paper by Combescure [Ann. Phys. (N.Y.) 204, 113 (1990)], we apply the quantum singular time-dependent oscillator model to describe the relative one-dimensional motion of two ions in a trap. We argue that the model can be justified for low-energy excited states with the quantum numbers $n\ensuremath{\ll}{n}_{\mathrm{max}}\ensuremath{\sim}100$, provided the dimensionless constant characterizing the strength of the repulsive potential is large enough ${g}_{*}\ensuremath{\sim}{10}^{5}$. Time-dependent Gaussian-like wave packets generalizing odd coherent states of the harmonic oscillator and excitation number eigenstates are constructed. We show that the relative motion of the ions, in contradistinction to its center-of-mass counterpart, is extremely sensitive to the time dependence of the binding harmonic potential since the large value of ${g}_{*}$ results in a significant amplification of the transition probabilities between energy eigenstate even for slow time variations of the frequency.