Quantum entanglement distribution with hybrid parity gate

We propose a scheme for entanglement distribution among different single atoms trapped in separated cavities. In our scheme, by reflecting an input coherent optical pulse from a cavity with a single trapped atom, a controlled phase-shift gate between the atom and the coherent optical pulse is achieved. Based on this gate and homodyne detection, we construct an $n$-qubit parity gate and show its use for distribution of a large class of entangled states in one shot, including the GHZ state $\left\vert GHZ_{n}\right\rangle$, W state $\left\vert W_{n}\right\rangle$, Dicke state $\left\vert D_{n,k}\right\rangle$ and certain sums of Dicke states $$. We also show such distribution could be performed with high success probability and high fidelity even in the presence of channel loss.