Elliptical ion traps and trap arrays for quantum computation

The properties of a rf quadrupole trap, the elliptical ion trap, are derived. Elliptical traps can confine large numbers of ions in the Lamb-Dicke regime due to a hitherto unrecognized mechanism unique to one-dimensional Coulomb crystals, implicit in the theories of Dubin and Schiffer. This follows from a linear crystal stability condition, which uniquely relates the crystal size to ellipticity, and a micromotion relation, which reveals a 1/5-root dependence on the number of trapped ions. Elliptical traps offer several advantages over linear traps in the Cirac-Zoller model of quantum computing, both for initial tests and as a potential method of creating a full-scale quantum computer. Numerical solutions of a one-electrode structure show that microscopic elliptical traps, each containing $\approx 100$ ions, can be constructed at a density of 100 traps/${\mathrm{cm}}^2,$ making possible arrays containing $>{10}^6$ ions in the Lamb-Dicke regime for precision spectroscopy or quantum computation.