Optimal cooling of multiple levitated particles: Theory of far-field wavefront shaping

The opportunity to manipulate small-scale objects pushes us to the limits of our understanding of physics. Particularly promising in this regard is the interdisciplinary field of levitation, in which light fields can be harnessed to isolate nanoparticles from their environment by levitating them optically. When cooled towards their motional quantum ground state, levitated systems offer the tantalizing prospect of displaying mesoscopic quantum properties. While the interest in levitation has so far been focused mainly on manipulating individual objects with simple shapes, the field is currently moving towards the control of more complex structures, such as those featuring multiple particles or different degrees of freedom. Unfortunately, current cooling techniques are mostly designed for single objects and thus cannot easily be multiplexed to address such coupled many-body systems. Here we present an approach based on the spatial modulation of light in the far field to cool multiple nano-objects in parallel. Our procedure is based on the experimentally measurable scattering matrix and on its changes with time. We demonstrate how to compose from these ingredients a linear energy-shift operator, whose eigenstates are identified as the incoming wavefronts that implement the most efficient cooling of complex moving ensembles of levitated particles. Submitted in parallel with Hüpfl et al. [Phys. Rev. Lett. 130, 083203 (2023)], this article provides a theoretical and numerical study of the expected cooling performance as well as of the robustness of the method against environmental parameters.