Tests of Macro-realism and Quantum Weak Values for NOON States and Bose-Einstein condensates

Leggett and Garg suggested testing macroscopic realism by comparing the predictions of quantum mechanics with those based on two premises: macroscopic realism per se and non-invasive measurability: The premises imply an inequality that can be violated for macroscopic quantum superposition states [1]. We construct tests of macro-realism for NOON states [2,3]. These mesoscopic superposition states may be realised optically, or can be created dynamically by a two-mode Josephson Hamiltonian that includes the nonlinearity associated with, for example, Bose-Einstein condensates. To test the Leggett-Garg inequality, one requires a non-invasive measurement. We propose such a measurement for NOON and BEC states [2,3]. The proposed measurement may be realised for atomic systems using an ac Stark shift. For small measurement interaction, we demonstrate weak values, where post-selected results exceed the eigenvalue range [4]. We link our results to violation of a Bell inequality.
[1] A. J. Leggett and A. Garg, Phys. Rev. Lett. 54, 857 (1985)
[2] B. Opanchuk, L. Rosales-Zarate, R. Y Teh and M. D Reid, Phys. Rev. A 94, 062125 (2016)
[3] L. Rosales-Zarate, B. Opanchuk, Q. Y. He and M. D Reid, arXiv [quant-ph]:1612.05726
[4] Y. Aharonov, D. Albert and L. Vaidmann, Phys. Rev. Lett. 60, 1351 (1988)