Quantum nonstationary oscillators: Invariants, dynamical algebras and coherent states via point transformations

TL;DR

This paper develops a method using point transformations to analyze nonstationary quantum oscillators, enabling the construction of solutions, integrals of motion, and coherent states, with applications to particles in electromagnetic traps.

Contribution

It introduces a novel approach employing point transformations to solve and analyze time-dependent quantum oscillators, providing explicit solutions and a framework for their algebraic structure.

Findings

Constructed solutions as deformations of stationary oscillators

Derived quantum integrals of motion without ansatz

Built a basis for superpositions and coherent states

Abstract

We consider the relations between nonstationary quantum oscillators and their stationary counterpart in view of their applicability to study particles in electromagnetic traps. We develop a consistent model of quantum oscillators with time-dependent frequencies that are subjected to the action of a time-dependent driving force, and have a time-dependent zero point energy. Our approach uses the method of point transformations to construct the physical solutions of the parametric oscillator as mere deformations of the well known solutions of the stationary oscillator. In this form, the determination of the quantum integrals of motion is automatically achieved as a natural consequence of the transformation, without necessity of any ansatz. It yields the mechanism to construct an orthonormal basis for the nonstationary oscillators, so arbitrary superpositions of orthogonal states are available to obtain the corresponding coherent states. We also show that the dynamical algebra of the parametric oscillator is immediately obtained as a deformation of the algebra generated by the conventional boson ladder operators. A number of explicit examples is provided to show the applicability of our approach.