Cooperatively enhanced precision of hybrid light-matter sensors

Abstract

We consider a hybrid system of matter and light as a sensing device and quantify the role of cooperative effects. The latter generically enhance the precision with which modifications of the effective light-matter coupling constant can be measured. In particular, considering a fundamental model of $N$ qubits coupled to a single electromagnetic mode, we demonstrate that the ultimate bound for the precision shows double-Heisenberg scaling: $\mathrm{\ensuremath{\Delta}}\ensuremath{\theta}\ensuremath{\propto}1/(Nn)$, with $N$ and $n$ the number of qubits and photons, respectively. Moreover, even using classical states and measuring only one subsystem, a Heisenberg-times-shot-noise scaling, i.e., $1/(N\sqrt{n})$ or $1/(n\sqrt{N})$, is reached. As an application, we show that a Bose-Einstein condensate trapped in a double-well optical lattice within an optical cavity can in principle be used to detect the gravitational acceleration $g$ with the relative precision of $\mathrm{\ensuremath{\Delta}}g/g\ensuremath{\sim}{10}^{\ensuremath{-}4}\phantom{\rule{4pt}{0ex}}{\mathrm{Hz}}^{\ensuremath{-}1/2}$. The analytical approach presented in this study takes into account the leakage of photons through the cavity mirrors, and allows one to determine the sensitivity when $g$ is inferred via measurements on atoms or photons.