Electric field noise above surfaces: a model for heating rate scaling law in ion traps

We present a model for the scaling laws of the electric field noise spectral density as a function of the distance, $d$, above a conducting surface. Our analytical approach models the patch potentials by introducing a correlation length, $\zeta$, of the electric potential on the surface. The predicted scaling laws are in excellent agreement with two different classes of experiments (cold trapped ions and cantilevers), that span at least four orders of magnitude of $d$. According to this model, heating rate in miniature ion traps could be greatly reduced by proper material engineering.