Elliptical ion traps for quantum computation

The case of arbitrary ellipticity is considered and two results for the crystal size and micromotion amplitude are derived. The first relation is a linear crystal stability condition, which shows that the maximum number of ions which can be confined is uniquely related to the ellipticity by a 7/3 power law. The micromotion in elliptical traps is far smaller than one would expect specifically that the micromotion amplitude depends approximately on the 1/5 power of the number of trapped ions. Micromotion is important because it prevents trapping in the Lamb-Dicke regime, required for quantum computation or for precision spectroscopy.