  Superconducting qubits are sensitive to a variety of loss mechanisms which
include dielectric loss from interfaces. The calculation of participation near
the key interfaces of planar designs can be accomplished through an analytical
description of the electric field density based on conformal mapping. In this
way, a two-dimensional approximation to coplanar waveguide and capacitor
designs produces values of the participation as a function of depth from the
top metallization layer as well as the volume participation within a given
thickness from this surface by reducing the problem to a surface integration
over the region of interest. These quantities are compared to finite element
method numerical solutions, which validate the values at large distances from
the coplanar metallization but diverge near the edges of the metallization
features due to the singular nature of the electric fields. A simple
approximation to the electric field energy at shallow depths (relative to the
waveguide width) is also presented that closely replicates the numerical
results based on conformal mapping and those reported in prior literature.
These techniques are applied to the calculation of surface participation within
a transmon qubit design, where the effects due to shunting capacitors can be
easily integrated with those associated with metallization comprising the local
environment of the qubit junction.
