  A method is introduced which allows to measure normal-ordered moments of the
displaced photon-number operator up to high orders. It is based on unbalanced
homodyne correlation measurements, the local oscillator being replaced by a
displaced dephased laser. The measured moments yield a simple approximation of
the full quantum state, which is obtained without the need of photon-number
resolution. Quantum properties of light are efficiently certified through the
direct detection of normal-ordered observables, which is illustrated for a
weakly squeezed vacuum and a single-photon-added thermal state.
