Phase Space Analysis of the Two-mode Binomial State Produced by Quantum Entanglement in a Beamsplitter

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 σ , γ = ( − 1 ) q e − γ 2 − σ 2 π 2 L q − ς ( σ − γ ) + σ ∗ + γ ∗ 1 + | ς | 2 2 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 .