Einstein-Podolsky-Rosen argument and Bell inequalities for Bose-Einstein spin condensates
F. Lalo\"{e} and W. J. Mullin

We discuss the properties of two Bose-Einstein condensates in different spin states, represented quantum mechanically by a double Fock state. Individual measurements of the spins of the particles are performed in transverse directions (perpendicular to the spin quantization axis), giving access to the relative phase of the two macroscopically occupied states. Before the first spin measurement, the phase is completely undetermined; after a few measurements, a more and more precise knowledge of its value emerges under the effect of the quantum measurement process. This naturally leads to the usual notion of a quasi-classical phase (Anderson phase) and to an interesting transposition of the EPR (Einstein-Podolsky-Rosen) argument to macroscopic physical quantities.\ The purpose of this article is to discuss this transposition, as well as situations where the notion of a quasi-classical phase is no longer sufficient to account for the quantum results, and where significant violations of Bell type inequalities are predicted. Quantum mechanically, the problem can be treated exactly: the probability for all sequences of results can be expressed in the form of a double integral, depending on all parameters that define the experiment (number of particles, number and angles of measurements).\ We discuss the differences between this case and the usual two-spin case.\ We discuss the effect of the many parameters that the experimenters can adjust for their measurements, starting with a discussion of the effect of the angles of measurement (the \textquotedblleft settings\textquotedblright), and then envisaging various choices of the functions that are used to obtain violation of BCHSH inequalities. We then discuss how the \textquotedblleft sample bias loophole\textquotedblright\ (often also called \textquotedblleft efficiency loophole\textquotedblright) can be closed in this case, by introducing a preliminary sequence of measurements to localize the particles into \textquotedblleft measurement boxes\textquotedblright.\ We finally show how that the same non-local effects can be observed with distinguishable spins.

© 2008 The American Physical Society.