  The exploration of atomic fractional quantum Hall (FQH) states is now within
reach in optical-lattice experiments. While ground-state signatures have been
observed in a system realizing the Hofstadter-Bose-Hubbard model in a box
[Leonard et al., Nature 2023], how to access hallmark low-energy collective
modes remains a central open question in this context. We introduce a
spectroscopic scheme based on two interfering Laguerre-Gaussian beams, which
transfer a controlled angular momentum and energy to the system. The edge and
bulk responses to the probe are detected through local density measurements, by
tracking the transfer of atoms between the bulk and the edge of the FQH
droplet. This detection scheme is shown to simultaneously reveal two specific
signatures of FQH states: their chiral edge branch and their bulk magneto-roton
mode. We numerically benchmark our method by considering few bosons in the
$\nu=1/2$ Laughlin ground state of the Hofstadter-Bose-Hubbard model, and
demonstrate that these signatures are already detectable in realistic systems
of two bosons, provided that the box potential is larger than the droplet. Our
work paves the way for the detection of fractional statistics in cold atoms
through edge signatures.
