  We investigated the estimation of an unknown Gaussian process (containing
displacement, squeezing and phase-shift) applied to a matter system. The state
of the matter system is not directly measured; instead, we measure an optical
mode which interacts with the system. We propose an interferometric setup
exploiting a beam-splitter-type of light-matter interaction with homodyne
detectors and two methods of estimation. We demonstrate the superiority of the
interferometric setup over alternative non-interferometric schemes.
Importantly, we show that even limited coupling strength and a noisy matter
system are sufficient for very good estimation. Our work opens the way to many
future investigations of light-matter interferometry for experimental platforms
in quantum metrology of matter systems.
