Preparation of entangled states of four remote atomic qubits in decoherence-free subspace

We propose how to prepare four remote logical qubits, with each in a separate cavity and encoded in the decoherence-free subspace by two atoms, into various entangled states. By means of the cavity-assisted photon scattering, we can link the four remote logical qubits in different cavities to be, respectively, W state, Greenberger-Horne-Zeilinger state, and cluster state, which are important in view of the distributed quantum information processing. The generation of the latter two entangled states can be theoretically generalized to many-qubit cases. Because our qubits are defined in dephasing-free subspace, our scheme is immune to dephasing during or even after the entanglement preparation.