Parametric feedback cooling of librations of a nanodiamond in a Paul trap: Towards matter-wave interferometry with massive objects

Quantum mechanics (QM) and General relativity (GR), also known as the theory of gravity, are the two pillars of modern physics. A matter-wave interferometer with a massive particle can test numerous fundamental ideas, including the spatial superposition principle - a foundational concept in QM - in completely new regimes, as well as the interface between QM and GR, e.g., testing the quantization of gravity. Consequently, there exists an intensive effort to realize such an interferometer. While several paths are being pursued, we focus on utilizing nanodiamonds (NDs) as our particle, and a spin embedded in the ND together with Stern-Gerlach forces, to achieve a closed loop in space-time. There is a growing community of groups pursuing this path [1]. We are posting this technical note (as part of a series of seven such notes) to highlight our plans and solutions concerning various challenges in this ambitious endeavor, hoping this will support this growing community. Here, we present a theoretical study concerning the impact of rotations of the ND on the interferometric contrast. We have previously shown that for a first-generation Stern-Gerlach interferometer with an ND composed of 10^7 atoms, it is sufficient to cool the center of mass to milli-Kelvin temperatures. In this work, we similarly show that rotation does not have to be cooled to the ground state, and cooling to hundreds of rotational phonons is good enough. We describe and simulate parametric feedback cooling of librational modes of a charged ND levitated in a Paul trap. The cooling is performed by modulating the electric field of the trap. We examine the dependence of the efficiency of cooling on the electric potential and the shape of the object. We show that the required libration temperatures should be within reach in the very near future. We would be happy to make more details available upon request.