Quantifying the mesoscopic quantum coherence of approximate NOON states and spin-squeezed two-mode Bose-Einstein condensates
Bogdan Opanchuk, L. Rosales-Zárate, Run Yan Teh, M. D. Reid · Physical Review A · 2016
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 $\ensuremath{\langle}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}^{\ifmmode\dagger\else\textdagger\fi{}n}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{b}}^{n}\ensuremath{\rangle}\ensuremath{ e}0$ (here $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}$ and $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{b}$ 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 $\ensuremath{\langle}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}^{\ifmmode\dagger\else\textdagger\fi{}n}{\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{b}}^{n}\ensuremath{\rangle}$ enables a quantification of the overall $n\text{th}$ 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 ${\ensuremath{\xi}}_{N}$. 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.