Quantifying the mesoscopic quantum coherence of approximate NOON states and spin-squeezed two-mode Bose-Einstein condensates

We examine how to signify and quantify the mesoscopic quantum coherence of approximate two-mode NOON states and spin-squeezed two-mode Bose-Einstein condensates (BEC). We identify two criteria that verify a nonzero quantum coherence between states with quantum number different by n. These criteria negate certain mixtures of quantum states, thereby signifying a generalized n-scopic Schr\"odinger cat-type paradox. The first criterion is the correlation ̂ \^a^n ̂ \^bⁿ≠0 (here ̂ \^a and ̂ \^b are the boson operators for each mode). The correlation manifests as interference fringes in n-particle detection probabilities and is also measurable via quadrature phase amplitude and spin-squeezing measurements. Measurement of ̂ \^a^n ̂ \^bⁿ enables a quantification of the overall nth order quantum coherence, thus providing an avenue for high efficiency verification of high-fidelity photonic NOON states. The second criterion is based on a quantification of the measurable spin-squeezing parameter ξN. We apply the criteria to theoretical models of NOON states in lossy interferometers and double-well trapped BECs. By analyzing existing BEC experiments, we demonstrate generalized atomic ``kitten'' states and atomic quantum coherence with $n⪆10$ atoms.