Title: Tuneable Gaussian entanglement in levitated nanoparticle arrays

Abstract: Nanoparticles trapped in optical tweezers emerged as an interesting platform for investigating fundamental effects in quantum physics. The ability to shape the optical trapping potential using spatial light modulation and quantum control of their motion using coherent scattering to an optical cavity mode predispose them for emulating a range of physical systems and studying quantum phenomena with massive objects. To extend these capabilities of levitated nanoparticles to quantum many-body systems, it is crucial to develop feasible strategies to couple and entangle multiple particles either directly or via a common optical bus. Here, we propose a variable and deterministic scheme to generate Gaussian entanglement in the motional steady state of multiple levitated nanoparticles using coherent scattering to multiple cavity modes. Coupling multiple nanoparticles to a common optical cavity mode allows cooling of a collective Bogoliubov mode to its quantum ground state; cooling multiple Bogoliubov modes (enabled by trapping each particle in multiple tweezers such that each tweezer scatters photons into a separate cavity mode) removes most thermal noise, leading to strong entanglement between nanoparticles. We present numerical simulations for three nanoparticles showing great tuneability of the generated entanglement with realistic experimental parameters. Our proposal thus paves the way towards creating complex quantum states of multiple levitated nanoparticles for advanced quantum sensing protocols and many-body quantum simulations.