Entanglement witness for combined atom interferometer-mechanical oscillator setup

Recent efforts to explore the role massive objects play at the intersection of quantum mechanics and gravity have focused on developing hybrid systems of quantum sensors coupled to dynamical macroscopic masses. These systems can serve as a versatile testbed for measuring the low energy quantum mechanical behavior of macroscopic masses and investigating the integration of quantum mechanics with gravity. This work builds on a recent experimental proposal by D. Carney et al. for entangling a trapped atom interferometer with a torsional pendulum to elucidate the behaviors of certain quantum theories of gravity [https://doi.org/10.1103/PRXQuantum.2.030330]. We construct an entanglement witness applicable to the stated interferometer-oscillator setup. Our entanglement witness is a combination of observables that measures correlations between the two systems; it has a bound which, if violated, determines that entanglement has been generated between the interferometer and the oscillator. We investigate how atomic dephasing and thermal noise affects the practical use of this entanglement witness in our actual system.