  We present a Heisenberg-Langevin formalism to study the effective dynamics of
a superconducting qubit coupled to an open multimode resonator, without
resorting to the rotating wave, two level, Born or Markov approximations. Our
effective equations are derived by eliminating resonator degrees of freedom
while encoding their effect in the Green's function of the electromagnetic
background. We account for the openness of the resonator exactly by employing a
spectral representation for the Green's function in terms of a set of
non-Hermitian modes. A well-behaved time domain perturbation theory is derived
to systematically account for the nonlinearity of weakly nonlinear qubits like
transmon. We apply this method to the problem of spontaneous emission,
capturing accurately the non-Markovian features of the qubit dynamics, valid
for any qubit-resonator coupling strength. Any discrete-level quantum system
coupled to the electromagnetic continuum is subject to radiative decay and
renormalization of its energy levels. When inside a cavity, these quantities
can be strongly modified with respect to vacuum. Generally, this modification
can be captured by including only the closest resonant cavity mode. In
circuit-QED architecture, with substantial coupling strengths, it is however
found that such rates are strongly influenced by far off-resonant modes. A
multimode calculation over the infinite set of cavity modes leads to
divergences unless an artificial cutoff is imposed. Previous studies have not
pointed out what the source of this divergence is. We show that unless the
effect of $A^2$ is accounted for up to all orders exactly, any multimode
calculations of circuit-QED quantities is bound to diverge. Subsequently, we
present the calculation of finite radiative corrections to qubit properties
that is free of an artificially introduced high frequency cut-off.
