  Recently, a superradiant phase transition first predicted theoretically in
the quantum Rabi model (QRM) has been verified experimentally. This further
stimulates the interest in the study of the process of phase transition and the
nature of the superradiant phase since the fundamental role of the QRM in
describing the interaction of light and matter, and more importantly, the QRM
contains rich physics deserving further exploration despite its simplicity.
Here we propose a scheme consisting of two successive diagonalization to
accurately obtain the ground-state and excited states wavefunctions of the QRM
in full parameter regime ranging from weak to deep-strong couplings. Thus one
is able to see how the phase transition happens and how the photons populate in
Fock space of the superradiant phase. We characterize the photon populations by
borrowing the distribution concept in random matrix theory and find that the
photon population follows a Poissonian-like distribution once the phase
transition happens and further exhibits the statistics of Gaussian unitary
ensemble as increasing coupling strength. More interestingly, the photons in
the excited states behave even like the statistics of Gaussian orthogonal
ensemble. Our results not only deepen understanding on the superradiant phase
transition but also provide an insight on the nature of the superradiant phase
of the QRM and related models.
