  Motional heating of ions in micro-fabricated traps is a challenge hindering
experimental realization of large-scale quantum processing devices. Recently a
series of measurements of the heating rates in surface-electrode ion traps
characterized their frequency, distance, and temperature dependencies, but our
understanding of the microscopic origin of this noise is still vague. In this
work we develop a theoretical model for the electric field noise which is
associated with a random distribution of adsorbed atoms on the trap electrode
surface. By using first principle calculations of the fluctuating dipole
moments of the adsorbed atoms we evaluate the distance, frequency and
temperature dependence of the resulting electric field fluctuation spectrum.Our
theory calculates the noise spectrum beyond the standard scenario of two-level
fluctuators, by incorporating all the relevant vibrational states. The $1/f$
noise is shown to commence at roughly the frequency of the fundamental phonon
transition rate and the $d^{-4}$ dependence with distance of the ion from the
electrode surface is established.
