Typical observables in a two-mode Bose system

A class of $k$-particle observables in a two-mode system of Bose particles is characterized by typicality: if the state of the system is sampled out of a suitable ensemble, an experimental measurement of that observable (almost) always yields the same result. We investigate the general features of typical observables and the criteria to determine typicality and finally focus on the case of density correlation functions, which are related to spatial distribution of particles and interference.

Figure 1

In the asymptotic regime, the outcome of a density measurement in a two-plane-wave mode system is approximated by the function (47). The visibility $\sim O\left(1\right)$ and the period $\sim k_0^{-1}$ of the interference pattern are (almost) fixed by typicality, namely, by the typical correlation function (46). The offset $\varphi$ of the pattern, and thus the value of density at the origin, does not appear in the typical observable (46) and fluctuates randomly. The order $O\left(\sqrt{N}\right)$ of the density fluctuations refers to the case $n=O\left(N^{\alpha }\right)$ with $\alpha \le 3/4$ [see Eq. (43)].