  Cooper pairs in the superconductor are a natural source of spin entanglement.
The existing proposals of the Cooper pair splitter can only realize a low
efficiency of entanglement production, and its size is constrained by the
superconducting coherence length. Here we show that a long-range Cooper pair
splitter can be implemented in a normal metal-superconductor-normal metal (NSN)
junction by driving a supercurrent in the S. The supercurrent results in a band
gap modification of the S, which significantly enhances the crossed Andreev
reflection (CAR) of the NSN junction and simultaneously quenches its elastic
cotunneling. Therefore, a high entanglement production rate close to its
saturation value can be achieved by the inverse CAR. Interestingly, in addition
to the conventional entangled electron states between opposite energy levels,
novel entangled states with equal energy can also be induced in our proposal.
