Macroscopicity of quantum superposition

Abstract: In this paper, we define a universal measure of macroscopicity $\beta$ in the form of, ({\em experimentally observed coherence time})/({\em characteristic time required to repeatably distinguish the components of a quantum superposition state}), to divide the quantum/classical (Q-C) boundary. The analysis shows that, for a matter-wave interferometer, covering systems from electron to macromolecular, the measure of its macroscopicity is expressed by $\beta=p\theta d/8\hbar$, and it consistently yields $\beta<1$ by investigating the coherence data reported in the literature. The result implies that limited advancement has been made toward exploring the Q-C boundary in such experiments. Additionally, for a Ramsey-like interferometer, the measure of its macroscopicity is expressed as $\beta=8\pi^3\alpha\nu_0r_s^2\nu_s^2T/ (9c^2)$. After analyzing the reported experimental data, we find that a maximum value of $\beta \sim 300$ is achieved in the vertical atomic fountain experiments. Therefore, searching the Q-C boundary in the atomic fountain may be a more proper direction.