  In this paper, we study the quantum properties of the three-mode squeezed
operator. This operator is constructed from the optical parametric oscillator
based on the three concurrent $\chi^{(2)}$ nonlinearities. We give a complete
treatment for this operator including the symmetric and asymmetric
nonlinearities cases. The action of the operator on the number and coherent
states are studied in the framework of squeezing, second-order correlation
function, Cauchy-Schwartz inequality and single-mode quasiprobability function.
The nonclassical effects are remarkable in all these quantities. We show that
the nonclassical effects generated by the asymmetric case--for certain values
of the system parameters--are greater than those of the symmetric one. This
reflects the important role for the asymmetry in the system. Moreover, the
system can generate different types of the Schr\"odinger-cat states.
