Weak gravitational field effects on Bell tests with massive particles

We study the generation of entangled states of two massive particles endowed with internal degrees of freedom using Franson and Hugged interferometric arrays in the presence of a weak gravitational field. Each internal degree of freedom plays the role of a clock. We consider an expansion of the gravitational field up to second order in powers of Delta h/R, where Delta h is the height of the arrays and R is the radius of the Earth. We study the violation of the Clauser-Horne-Shymony-Holt (CHSH) inequality, showing that the maximum value of the CHSH function, for a suitable choice of local phases, is Sigma(max) = 2 root 2 cos(2) (Delta tau Delta E/ 2h), where Delta tau is the gravitational time delay induced by the gravitational field in the trajectories of the particles and Delta E is the difference in energy between the levels of a two-level clock. If the local phases cannot be controlled, then the CHSH function is modulated by a harmonic function of the gravitationally induced phase shift. The same results holds for the Hugged array but for the second order approximation. If we consider only a first-order approximation, the gravity-induced phase shift vanishes, then the local phases do not depend on the gravitational field, and consequently the maximum value of the CHSH function Sigma(max) is reached. These results contrast with the case of photons, in the absence of frequency dispersion, in a weak gravitational field, where the CHSH function reaches its maximum value of 2 root 2. We compute the logarithmic negativity and show that it does not normally vanish, indicating that the generated states are entangled even if they do not violate the CHSH inequality. We show that an increase in the number of clocks per particle and in the number of levels of each clock leads to an exponential decrease in the maximum value of the CHSH functional and in the logarithmic negativity. We also consider the presence of momentum dispersion in the particle state. Interestingly, in the absence of internal degrees of freedom and to a first-order approximation, the Hugged array can lead to the maximum value of the CHSH function even in the presence of momentum dispersion, while the Franson array is affected by an exponential decay of the maximum value of the CHSH function.