Contributions to the study of time dependent oscillators in Paul traps. Semiclassical approach

TL;DR

This paper explores the quantum and semiclassical dynamics of ions in Paul traps, establishing equivalences between Hamilton equations and Hill equations, and deriving exact solutions and invariants for the system.

Contribution

It introduces a semiclassical approach to analyze time-dependent oscillators in Paul traps, providing new insights into quantum states, invariants, and exact solutions for trapped ions.

Findings

Quantum states are Fock states.

Exact solutions are quasienergy states.

An adiabatic invariant is introduced.

Abstract

We investigate quantum dynamics for an ion confined within an oscillating quadrupole field, starting from two well known and elegant approaches. It is established that the Hamilton equations of motion, in both Schr\"{o}dinger and Heisenberg representations, are equivalent to the Hill equation. One searches for a linear independent solution associated to a harmonic oscillator (HO). An adiabatic invariant, which is also a constant of motion, is introduced based on the Heisenberg representation. Thus, the state of the non-autonomous system can be determined at any subsequent moment of time. The quantum states for trapped ions are demonstrated to be Fock (number) states, while the exact solutions of the Schr\"{o}dinger equation for a trapped ion are exactly the quasienergy states. Semiclassical dynamics is also investigated for many-body systems of trapped ions, where the wavefunction associated to the Schr\"{o}dinger equation is prepared as a Gauss package multiplied by a Hermite polynomial. We also discuss time evolution for the system under investigation and supply the propagator.