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In the transmon qubit we expect from conservation of momentum and energy acoupling between the plasma oscillations and the vibrations of the underlyinglattice. Specifically, the electron velocities and their kinetic energy densityare boosted by the underlying lattice vibrations. We consider this effect in arepresentative transmon comprising two semi-circular superconducting chargeislands joined by a Josephson junction. In particular, we solve the Fouriertransform of a two-dimensional radial current density having inversionsymmetry. The resulting spectral density is ohmic but also scales quadraticallywith the critical current I_c and logarithmically with the size of thetransmon: J(w) ~ I_c^2 w log(kR). We make positive-definite Born-Markovapproximations in a generalized Fermi's Golden Rule and estimate thephonon-induced dephasing rate is negligible compared to current experiments.

Transmon-phonon coupling of plasma oscillations and lattice vibrations