Fast phase gates with trapped ions
  We implement faster-than-adiabatic two-qubit phase gates using smooth state-dependent forces. The forces are designed to leave no final motional excitation, independently of the initial motional state in the harmonic, small-oscillations limit. They are simple, explicit functions of time and the desired logical phase of the gate, and are based on quadratic invariants of motion and Lewis-Riesenfeld phases of the normal modes. 
M. Palmero, S. Mart\'inez-Garaot, D. Leibfried, D. J. Wineland, and J.   G. Muga
quant-ph