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

Abstract

Following the paper by M. Combescure [Ann. Phys. (NY) 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 ≪ n max ∼ 100, provided that the dimensionless constant characterizing the strength of the repulsive potential is large enough, g * ∼ 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.

Evolution of Gaussian wave packet and nonadiabatic geometrical phase for the time-dependent singular oscillator

“…On the other hand, the number of exactly solvable quantum time-dependent problems is very restricted, one of the rare examples admitting exact solutions of the Schrödinger equation and have been studied intensively lately [12][13][14][15][16][17][18][19] is the quantum time-dependent generalized singular oscilator…”

Exact solutions for the general nonstationary oscillator with a singular perturbation

“…Recently a considerable attention has been paid in the literature [1,2,3,4,5] to the (nonstationary) singular oscillator, i.e. the particle with mass m in the harmonic plus an inverse harmonic potential…”

“…The potential (1) has vide applications in molecular and solid state physics: the radial motion of such systems as the hydrogen atom, the n-dimensional oscillator, the charged particle in an uniform magnetic plus electric field with scalar potential proportional to 1/(x 2 + y 2 ) and the N identical particles interacting pairwise with potential energy V ij = (mω 2 /2)(x i −x j ) 2 +g/(x i −x j ) 2 can be reduced [12,13,15] to the case of SO. The potential (1) was recently applied to describe a two-ion trap [1].…”

On a dynamical symmetry group of the relativistic linear singular oscillator

“…2 /8m, so d is a real positive number [3]. Using c − and c + , one can construct a dynamical symmetry algebra of the linear singular oscillator (11).…”

“…-The singular harmonic oscillator is one of the rare exactly solvable problems in non-relativistic quantum mechanics [1,2]. This model is useful to explain many phenomena, such as description of interacting many-body systems [3], diatomic [4] and polyatomic [5] molecules, spin chains [6], quantum Hall effect [7], fractional statistics and anyons [8]. In spite of many interesting papers devoted to the study of the non-relativistic singular harmonic oscillator model [9], the number of works studying relativistic approachs to singular oscillator exact solution is still rather few [10].…”