This paper studies the interaction of a four-lever cascade atom with two-mode cavity fields in which the lower mode is initially empty. It is shown that there exist long time dipper squeezing and alternate squeezing, as well as strong antibunching and superbunching. The relation between the squeezing and superbunching is disclosed, the nonclassical squeezing and antibunching may occur at the same time. This nonlinear properties originate from the effect of complicated superposition states.