For the first time,we derive the compact forms of normalization factors for photon-added（-subtracted） two-mode squeezed thermal states by using the P-representation and the integration within an ordered product of operators（IWOP） technique.It is found that these two factors are related to the Jacobi polynomials.In addition,some new relationships for Jacobi polynomials are presented.
In this Letter, a new fractional entangling transformation （FRET） is proposed, which is generated in the entangled state representation by a unitary operator exp{iθ（ab^＋ ＋ a^＋ b）} where a（b） is the Bosonic annihilate operator. The operator is actually an entangled one in quantum optics and differs evidently from the separable operator, exp（iθ（a^＋a＋ b^＋ b）}, of complex fractional Fourier transformation. The additivity property is proved by employing the entangled state representation and quantum mechanical version of the FRET. As an application, the FrET of a two-mode number state is derived directly by using the quantum version of the FRET, which is related to Hermite polynomials.
We investigate the nonclassical properties of arbitrary number photon annihilation-then-creation operation （AC） and creation-then-annihilation operation （CA） to the thermal state （TS）, whose normalization factors are related to the polylog- arithm function. Then we compare their quantum characters, such as photon number distribution, average photon number, Mandel Q-parameter, purity and the Wigner function. Because of the noncommutativity between the annihilation operator and the creation operator, the ACTS and the CATS have different nonclassical properties. It is found that nonclassical properties are exhibited more strongly after AC than after CA. In addition we also examine their non-Gaussianity. The result shows that the ACTS can present a slightly bigger non-Gaussianity than the CATS.
