Wigner functions for two-photon and three-photon entangled states

The space-time structure of two-photon and three-photon entangled states is analysed and the Wigner functions for these states are developed. The analysis of the Wigner functions is made in real position-momentum space where position goes with time in a retarded frame and the momentum of the photon in free space is proportional to frequency. The entanglement of the photons in frequencies and wave vectors is due to the phase matching conditions. By the use of narrow-band filters the dominant entanglement effect is due to phase matching conditions in frequencies, and under this condition the description of the three-photon entangled state becomes analogous to that of the two-photon. The motion of the entangled photons is seperated into the centre of mass motion and the relative motion between the photons, including two or three degrees of freedom for the two-photon and three-photon, respectively. Under the approximation of narrow-band filters we get for each degree of freedom a Gaussian wave packet in position momentum with the corresponding time frequency in retarded frames.