In 1935, Einstein, Podolsky, and Rosen (EPR) conceived a Gedankenexperiment in which two particles are entangled through interactions, spatially separated, and measured. Under the classical assumption of local realism, they showed that the measurement correlations predicted by quantum mechanics for this scenario lead to a violation of the Heisenberg uncertainty principle. This contradiction, later denominated EPR paradox, revealed that the completeness of quantum mechanics is not compatible with the local realist description of nature that characterises classical physics.

Although the EPR paradox has been observed between systems consisting of few particles, this has not yet been achieved between larger systems: The entanglement of macroscopic objects has already been demonstrated, but the measured correlations were not strong enough to demonstrate the EPR paradox. However, the presence of entanglement of the EPR type in many-particle systems has been shown by measuring correlations within single systems.

In this thesis I describe an EPR experiment with two spatially separated massive many-particle systems: In close analogy to the original Gedankenexperiment, we entangle about 1400 atoms in a two-component Rubidium-87 Bose-Einstein condensate (BEC) via tunable collisional interactions and coherently split them into two separate condensates. Our splitting technique preserves the overlap and coherence between the components in each of the split BECs, allowing us to individually manipulate them. The entanglement inherited from the initial system results in measurement correlations between the two BECs that are strong enough to show the EPR paradox.

Our work shows that the conflict between quantum mechanics and local realism does not disappear when the size of the involved systems is increased to $ {sim 10^3} $ atoms. In addition to this, EPR entanglement – in conjunction with the spatial separation and individual addressability of the two systems demonstrated in our experiment – is a valuable resource for quantum metrology and quantum information processing with many-particle systems.