Abstract

Two ions in a rf quadrupole (Paul) trap represent a pair of driven oscillators which are nonlinearly coupled by their electrostatic repulsion. Numerical integration of the equations of motion shows that this system can undergo a transition from periodic to chaotic motion when the Mathieu equation parameter q, which is proportional to the trap voltage, is varied. The transition to chaos entails considerable energy transfer from the rf trapping field to the ions and results in a diffuse hot cloud. Experimental observations of two trapped Ba⁺ ions show a transition from a compact ordered state to a diffuse cloud when q exceeds ~0.85, as predicted by the calculations; we interpret this as a transition from periodic to chaotic motion. This is the first quantitative agreement between calculations and experiment for a phase transition in a trapped ion system. We note that laser cooling introduces additional control parameters into the equations of motion, allowing an explanation of the melting of ion crystals observed by Diedrich et al.¹

© 1988 Optical Society of America