We study the normal state of the 2D attractive Hubbard model using quantum Monte Carlo simulations. We show that singlet pairing correlations develop above
Tc, and the normal state of a short coherence length superconductor deviates from a canonical Fermi liquid. In the intermediate U regime, the spin susceptibility ${\mathrm{\ensuremath{\chi}}}_{\mathit{s}}$ is strongly temperature dependent, and the low-frequency spectral weight, as measured by the NMR relaxation rate 1/
T1T, is shown to track ${\mathrm{\ensuremath{\chi}}}_{\mathit{s}}$. This provides a simple, qualitative explanation for the spin-gap behavior observed in several high-
Tc systems.
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.