The eigenvector set of ︱φ2(x,r1,r2)with three compatible operators(1+2+3,p2+p3-2p1 and p3-p2) is constructed by virtue of Radon transformation of the Wigner operator and the technique of integration within an ordered product(IWOP) of operators.Its entanglement property is then revealed by deriving its standard Schmidt decomposition.︱φ2(x,r1,r2) makes up a new quantum mechanical representation.A new three-mode squeezing operator is found by using this entangled state representation.
For the first time, we derive the photon number cumulant for two-mode squeezed state and show that its cumulant expansion leads to normalization of two-mode photon subtracted-squeezed states and photon added- squeezed states. We show that the normalization is related to Jacobi polynomial, so the cumulant expansion in turn represents the new generating function of Jacobi polynomial.
