  In the transmon qubit we expect from conservation of momentum and energy a
coupling between the plasma oscillations and the vibrations of the underlying
lattice. Specifically, the electron velocities and their kinetic energy density
are boosted by the underlying lattice vibrations. We consider this effect in a
representative transmon comprising two semi-circular superconducting charge
islands joined by a Josephson junction. In particular, we solve the Fourier
transform of a two-dimensional radial current density having inversion
symmetry. The resulting spectral density is ohmic but also scales quadratically
with the critical current I_c and logarithmically with the size of the
transmon: J(w) ~ I_c^2 w log(kR). We make positive-definite Born-Markov
approximations in a generalized Fermi's Golden Rule and estimate the
phonon-induced dephasing rate is negligible compared to current experiments.
