Vortex and entanglement occurring in propagating states through coupled lossy waveguides

A separable input state consisting of an $n$ -photon Fock state and a coherent state propagating through coupled waveguides is investigated in detail. We obtained the analytical solutions for the state vector evolution, the wavefunction or probability distribution in the quadrature space and the $P$ -function in the phase space. It is proved that the propagating states may evolve into quantum vortex states even for coupled lossy waveguides by appropriately selecting the propagation time. Based on the analytical $P$ -function in phase space and the relative linear entropy for the propagating state, it is found that the propagating state may be entangled and non-classical. Specially, in absence of loss, the degree of entanglement only depends on the photon number $n$ of the input Fock state but is independent of the displacement parameter $\alpha $ associated with the input coherent state. Moreover, for coupled lossy waveguides the entanglement evolution can exhibit new features.