  In this paper, we propose a method to generate entangled coherent states
between two spatially separated atomic Bose-Einstein condensates (BECs) via the
technique of the electromagnetically induced transparency (EIT). Two strong
coupling laser beams and two entangled probe laser beams are used to make two
distant BECs be in EIT states and to generate an atom-photon entangled state
between probe lasers and distant BECs. The two BECs are initially in
un-entangled product coherent states while the probe lasers are initially in an
entangled state. Entangled states of two distant BECs can be created through
performing projective measurements upon the two outgoing probe lasers under
certain conditions. Concretely, we propose two protocols to show how to
generate entangled coherent states of the two distant BECs. One is a
single-photon scheme in which an entangled single-photon state serves the
quantum channel to generate entangled distant BECs. The other is a multiphoton
scheme where an entangled coherent state of the probe lasers is used as the
quantum channel. Additionally, we also obtain some atom-photon entangled states
of particular interest such as entangled states between a pair of optical Bell
(or quasi-Bell) states and a pair of atomic entangled coherent (or quasi-Bell)
states.
