A bead on a hoop rotating about a horizontal axis: a 1-D ponderomotive trap

TL;DR

This paper introduces a mechanical system mimicking an rf Paul ion trap, using a rotating hoop and bead to demonstrate particle trapping dynamics analogous to atomic ion confinement.

Contribution

It presents a simple, tangible mechanical analog of the rf Paul trap, including theoretical analysis and a working model to illustrate ion trapping principles.

Findings

The bead's motion satisfies the Mathieu equation.

The system effectively replicates the dynamics of trapped atomic ions.

Experimental model demonstrates the operation of an rf Paul trap.

Abstract

We describe a simple mechanical system that operates as a ponderomotive particle trap, consisting of a circular hoop and a frictionless bead, with the hoop rotating about a horizontal axis lying in the plane of the hoop. The bead in the frame of the hoop is thus exposed to an effective sinusoidally-varying gravitational field. This field's component along the hoop is a zero at the top and bottom. In the same frame, the bead experiences a time-independent centrifugal force that is zero at the top and bottom as well. The system is analyzed in the ideal case of small displacements from the minimum, and the motion of the particle is shown to satisfy the Mathieu equation. In the particular case that the axis of rotation is tangential to the hoop, the system is an exact analog for the rf Paul ion trap. Various complicating factors such as anharmonic terms, friction and noise are considered. A working model of the proposed system has been constructed, using a ball-bearing rolling in a tube along the outside of a section of a bicycle rim. The apparatus demonstrates in detail the operation of an rf Paul trap by reproducing the dynamics of trapped atomic ions and illustrating the manner in which the electric potential varies with time.