Quasiparticle tunneling: P(E) theory

In this note, we calculate the dissipation due to the tunneling of quasiparticles using the environmental P (E) theory[1] to properly account for coupling to the qubit from the charge transfer. We will first calculate the damping rate for a inductor-capacitor resonator using environmental theory; the prediction for phase qubits, for example, will be similar since qubit and harmonic oscillator matrix elements are nearly identical. We will then consider the tunneling of quasiparticles, which has tunneling of charge carriers that is both electronand hole-like. Standard environmental theory gives a probability P (E) for the environment to absorb energy from a tunneling event with charge transfer q