Decoherence in Ion Trap Quantum Computers

The intrinsic decoherence from vibrational coupling of the ions in the Cirac-Zoller quantum computer [Phys. Rev. Lett. 74, 4091 (1995)] is considered. Starting from a state in which the vibrational modes are at a temperature $T$, and each ion is in a superposition of an excited and a ground state, an adiabatic approximation is used to find the inclusive probability $P\left(t\right)$ for the ions to evolve as they would without the vibrations, and for the vibrational modes to evolve into any final state. An analytic form is found for $P\left(t\right)$ at $T=0\mathrm{}$, and the decoherence time is found for all $T$. The decoherence is found to be quite small, even for 1000 ions.