  The statistical features of the radiation emitted by Free-Electron Lasers
(FELs), either by Self-Amplified Spontaneous Emission (SASE-FELs) or by seeded
emission (seeded-FELs), are attracting increasing attention because of the use
of such light in probing high energy states of matter and their dynamics. While
the experimental studies conducted so far have mainly concentrated on
correlation functions, here we shift the focus towards reconstructing the
distribution of the occupation numbers of the radiation energy states. In order
to avoid the various drawbacks related to photon counting techniques when large
numbers of photons are involved, we propose a Maximum Likelihood reconstruction
of the diagonal elements of the FEL radiation states in the energy eigenbasis
based on the statistics of no-click events. The ultimate purpose of such a
novel approach to FEL radiation statistics is the experimental confirmation
that SASE-FEL radiation exhibits thermal occupation number statistics, while
seeded-FEL light Poissonian statistics typical of coherent states and thus of
laser light. In this framework, it is interesting to note that the outcome of
this work can be extended to any process of harmonic generation from a coherent
light pulse, unlocking the gate to the study of the degree to which the
original distinctive quantum features deduced from the statistical photon
number fluctuations are preserved in non-linear optical processes.
