  Recent progress in the development of superconducting circuits has enabled
the realization of interesting sources of nonclassical radiation at microwave
frequencies. Here, we discuss field quadrature detection schemes for the
experimental characterization of itinerant microwave photon fields and their
entanglement correlations with stationary qubits. In particular, we present
joint state tomography methods of a radiation field mode and a two-level
system. Including the case of finite quantum detection efficiency, we relate
measured photon field statistics to generalized quasi-probability distributions
and statistical moments for one-channel and two-channel detection. We also
present maximum-likelihood methods to reconstruct density matrices from
measured field quadrature histograms. Our theoretical investigations are
supported by the presentation of experimental data, for which microwave quantum
fields beyond the single-photon and Gaussian level have been prepared and
reconstructed.
