We study the time evolution of SU(1,1) coherent states in an exactly solvable model of a single-mode quantized electromagnetic field in a four-photon medium that has previously been shown to give squeezed states from ordinary coherent states. If the SU(1,1) coherent state initially exhibits squeezing in the
X1 quadrature, the squeezing is revoked. We also show that for certain choices of phase in the initial state, enhanced squeezing may appear in the
X2 quadrature. Numerical results are also presented for the case when the model is not exactly solvable.
We introduce nonlinear canonical transformations that yield effective, solvable Hamiltonians of multiphoton down-conversion processes, and we define the associated non-Gaussian multiphoton squeezed states as the coherent states of the multiphoton Hamiltonians. We study in detail the four-photon processes and the associated non-Gaussian four-photon squeezed states. The realization of squeezing, the behaviour of the field statistics, and the structure of the phase space distributions show that these states realize a natural four-photon generalization of the two-photon squeezed states.