The Boundary Time Crystal as a light source for collectively enhanced sensing

Modern precision measurements, such as interferometry for detecting gravitational waves, rely on the estimation of optical phases encoded in light fields. Here, we propose to exploit the collectively enhanced output field of a driven-dissipative many-body quantum system as a light source in order to improve the precision of estimating optical phases. Pronounced temporal correlations of such output fields benefit the sensitivity of measurement protocols, which we show theoretically by employing a boundary time crystal as a light source. The fundamental bound on the precision of such estimation shows scaling with the number of constituents $N$ of the many-body system as $N^4$ while scaling linearly with the measurement time $T$. We discuss this scaling both from a perspective of the resources employed to build the light source and of the resources produced by the light source, namely the number of emitted photons and their correlations. We show that a protocol, in which the phase shifted light field is guided into an auxiliary replica system, which serves as a detector that is sensitive to non-trivial temporal correlations of light, can saturate the fundamental bound on precision at an optimal operating point.