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I have been reading the article review for Optical Quantum Memory (http://arxiv.org/abs/1002.4659) when I came across this section about DLCZ protocol. I don't understand about Beam splitter. For what I can interpret, the beam splitter overlaps both photons which emitted simultaneously by two atomic ensembles. My question is how does the beam splitter mixes two photons together so they become entangled?

P.S. My English is not good. I am sorry.

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You have $2$ atomic ensembles.

For each atomic ensemble, the ground state is $|\psi_0\rangle$, and the excited state is $|\psi_1\rangle$. The excited state is signaled by an idler photon.

Without beam splitter, if you detect only one photon (for the whole set), you know which atomic ensemble is excited, so you have the state $|\psi_0\rangle |\psi_1\rangle$ or the state $|\psi_1\rangle |\psi_0\rangle$

However, if you merge the optical path of the two idlers photons (with a beam splitter), and if you detect a photon, you are not able to say which atomic ensemble is in the excited state. The global state is entangled :

$$|\psi\rangle = \frac{1}{\sqrt{2}} ( |\psi_0\rangle |\psi_1\rangle + e^{i\phi} |\psi_1\rangle |\psi_0\rangle) \tag{4}$$ where $\phi$ is a phase factor due to the difference of optical path.

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