Phase properties of second harmonics generated by different initial fields

The joint probability distribution P (θ a , θ b ) for the phases θ a and θ b of the fundamental and second-harmonic modes is calculated, according to the Pegg-Barnett hermitian phase formalism, for different initial states of the fundamental mode undergoing second harmonic generation. The evolution of P (θ a , θ b ) is shown for four different initial states of the field (coherent, squeezed vacuum, Fock, and chaotic). Phase properties of the resulting fields are shortly discussed.