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After a penetrating analysis of the known number–phase commutative relations we find that the common eigenvector |η of two particles' relative coordinate and total momentum also embodies the entanglement with respect to the correlative amplitude–operational phase. Based on the fact that the number-difference operator is the canonical conjugate to the Noh–Fougeres–Mandel operational phase operator, the corresponding entanglement is also briefly discussed.

The generalized Wigner operator and Wigner distribution function in phase space corresponding to the field-intensity-dependent operators is constructed in the nonlinear coherent state representation. The generalized Wigner theorem is also derived. The whole discussion is based on the technique of integration within an ordered product of the field-intensity-dependent operators.

We introduce the dual vector | z,f (n)* of a single-mode nonlinear coherent state (NCS) by using the contour integral form of δ-function, and the explicit form of | z,f (n)* is obtained in enlarged Hilbert space. The nonlinear Fock states and their contour representation are deduced. A new completeness relation composed of | z,f (n)* and NCS in contour integral form is derived and its application in constructing a complex P-representation of density operators is shown.

We introduce the nonlinear SU(3) charged and hypercharged bosonic coherent state in three-mode Fock space. It can be further recast into a compact exponential form. The fermionic case is also discussed.

We study even- and odd-displaced squeezed states, which were proposed by us as 1/Ne(D(z)+D(-z))S(r) mod 0) and 1/N0(D(z)-D(-z))S(r) mod 0). We find: (i) when z is real, the 2Nth moments in both states are larger than in the ordinary squeezed state; (ii) for the same z= mod z mod , the two states can not exhibit stronger squeezing than the squeezed state in the same order; (iii) under the condition mod z mod e-r square root 2>>1, the even (odd)-displaced squeezed state can respectively exhibit stronger (4k-2) (4k)-order (k=1,2,3,...) squeezing than the squeezed state.