Generation of entangled coherent states for distant Bose-Einstein condensates via electromagnetically induced transparency

We propose a method to generate entangled coherent states between two spatially separated atomic Bose-Einstein condensates (BECs) via the technique of electromagnetically induced transparency (EIT). Two strong coupling laser beams and two entangled probe laser beams are used to cause two distant BECs to be in EIT states and to generate an atom-photon entangled state between probe lasers and distant BECs. The two BECs are initially in unentangled product coherent states while the probe lasers are initially in an entangled state. Entangled states of two distant BECs can be created through the performance of 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 is used as 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 states (or quasi-Bell-states) and a pair of atomic entangled coherent states (or quasi-Bell-states).

Figure 1

The schematic diagram of creating entangled coherent states for two distant BECs via EIT. Two pairs of coupling and probe lasers are applied to two BECs, respectively. The entangled source produces two entangled probe lasers. After leaving two BECs, the two outgoing probe lasers enter a Bell-state analyzer to perform projective measurements.

Figure 2

Three-level $\Lambda$-type atoms coupled to two laser fields with the detunings ${\Delta }_1$ and ${\Delta }_2$, respectively.

Figure 3

The schematic diagram of creating an entangled single-photon state given by Eq. (17) from two-pairs of two-mode squeezed vacuum states by using two beam splitters and two on-off photon detectors.