  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 nano-particles from their environment by levitating them optically.
When cooled down towards their motional quantum ground state, levitated systems
offer the tantalizing prospect of displaying mesoscopic quantum properties.
Currently restricted to single objects with simple shapes, the interest in
levitation is currently moving towards the manipulation 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 down 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 arxiv:2103.12592,
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.
