Phase space analysis of quantum states is a newly developed topic in quantum optics. In this work we present Wigner phase space distributions for the two-mode binomial state produced by quantum entanglement between a vacuum state and a number state in a beamsplitter. By using two new binomial formulas involving two-variable Hermite polynomials and the so-called entangled Wigner operator, we find that the analytical Wigner function for the binomial state |ξ〉q ≡ D(ξ) |q, 0〉 is related to a Laguerre polynomial, i.e.,
$ W\left (\sigma _{,}\gamma \right ) =\frac {(-1)^{q}e^{-\left \vert \gamma \right \vert ^{2}-\left \vert \sigma \right \vert ^{2}}}{\pi ^{2}}L_{q}\left (\left \vert \frac {-\varsigma (\sigma -\gamma )+\sigma ^{\ast }+\gamma ^{\ast }} {\sqrt {1+|\varsigma |^{2}}}\right \vert ^{2}\right ) $and its marginal distributions are proportional to the module-square of a single-variable Hermite polynomial. Also, the numerical results show that the larger number sum q of two modes lead to the stronger interference effect and the nonclassicality of the states |ξ〉q is stronger for odd q than for even q.
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