Decoherence in ion trap quantum computers

Cirac and Zoller [Phys. Rev. Lell. 74, 4091 (1995)] have proposed a concrete "hardware" scheme for quantum computation entailing N vibrationally coupled trapped ions. Starting from an initial state in which the normal modes are at a temperature T, and each ion is in a superposition of a ground and an excited state, we calculate the inclusive probability P(t) for the ions to evolve as they would in the absence of coupling, and any final vibrational mode state. An adiabatic approximation exploiting the low vibrational frequencies (10⁸ Itz) vs. the electronic ionic transitions (10¹⁴ Hz) is used. An analytic form is found for P(f) at T = 0, and decoherence times are found for all T and N ≈ 1000. Implications for actual realization of such computers are discussed.