  The dynamics of a weakly anharmonic superconducting qubit in a complex
electromagnetic environment is generally well-described by an effective
multimode Kerr Hamiltonian at sufficiently weak excitation. This Hamiltonian
can be embedded in a master equation with losses determined by the details of
the electromagnetic environment. Recent experiments indicate, however, that
when a superconducting circuit is driven with microwave signals the observed
relaxation rates appear to be substantially different from expectations based
on the electromagnetic environment of the qubit alone. This issue is a limiting
factor in the optimization of superconducting qubit readout schemes. We claim
here that an effective master equation with drive-power dependent parameters is
an efficient approach to model such quantum dynamics. In this sequence of
papers, we derive effective master equations, whose parameters exhibit
nonlinear dependence on the excitation level of the circuit as well as the
electromagnetic environment of the qubit. We show that the number
non-conserving terms in the qubit nonlinearity generally lead to a
renormalization of dissipative parameters of the effective master equation,
while the number conserving terms give rise to a renormalization of the system
frequencies. Here, in Part I, we consider the excitation-relaxation dynamics of
a transmon qubit that is prepared in a certain initial state, but is not driven
otherwise. A unitary transformation technique is introduced to study the
renormalization of i) qubit relaxation due to coupling to a generic bath and
ii) Purcell decay. Analytic expressions are provided for the dependence of the
nonlinear dissipative terms on the details of the electromagnetic environment
of the qubit. The perturbation technique based on unitary transformations
developed here is generalized to the continuously driven case in Part II.
