  Noise induced decoherence is one of the main threats to large-scale quantum
computation. In an attempt to assess the noise affecting a qubit we go beyond
the standard steady-state solution of the transmission through a qubit-coupled
cavity in input-output theory by including dynamical noise in the description
of the system. We solve the quantum Langevin equations exactly for a noise-free
system and treat the noise as a perturbation. In the long-time limit the
corrections may be written as a sum of convolutions of the noise power spectral
density with an integration kernel that depends on external control parameters.
Using the convolution theorem, we invert the corrections and obtain relations
for the noise spectral density as an integral over measurable quantities.
Additionally, we treat the noise exactly in the dispersive regime, and again
find that noise characteristics are imprinted in the long-time transmission in
convolutions containing the power spectral density.
