Sub Shot-Noise Phase Sensitivity with a Bose-Einstein Condensate Mach-Zehnder Interferometer

Bose Einstein Condensates, with their coherence properties, have attracted wide interest for their possible application to ultra precise interferometry and ultra weak force sensors. Since condensates, unlike photons, are interacting, they may permit the realization of specific quantum states needed as input of an interferometer to approach the Heisenberg limit, the supposed lower bound to precision phase measurements. To this end, we study the sensitivity to external weak perturbations of a representative matter-wave Mach-Zehnder interferometer whose input are two Bose-Einstein condensates created by splitting a single condensate in two parts. The interferometric phase sensitivity depends on the specific quantum state created with the two condensates, and, therefore, on the time scale of the splitting process. We identify three different regimes, characterized by a phase sensitivity $\Delta \theta$ scaling with the total number of condensate particles $N$ as i) the standard quantum limit $\Delta \theta \sim 1/N^{1/2}$, ii) the sub shot-noise $\Delta \theta \sim 1/N^{3/4}$ and the iii) the Heisenberg limit $\Delta \theta \sim 1/N$. However, in a realistic dynamical BEC splitting, the 1/N limit requires a long adiabaticity time scale, which is hardly reachable experimentally. On the other hand, the sub shot-noise sensitivity $\Delta \theta \sim 1/N^{3/4}$ can be reached in a realistic experimental setting. We also show that the $1/N^{3/4}$ scaling is a rigorous upper bound in the limit $N \to \infty$, while keeping constant all different parameters of the bosonic Mach-Zehnder interferometer.