  We show that when one draws a simple conformal loop ensemble (CLE$_\kappa$
for $\kappa \in (8/3,4)$) on an independent $\sqrt{\kappa}$-Liouville quantum
gravity (LQG) surface and explores the CLE in a natural Markovian way, the
quantum surfaces (e.g., corresponding to the interior of the CLE loops) that
are cut out form a Poisson point process of quantum disks. This construction
allows us to make direct links between CLE on LQG, asymmetric
$(4/\kappa)$-stable processes, and labeled branching trees. The ratio between
positive and negative jump intensities of these processes turns out to be
$-\cos (4 \pi / \kappa)$, which can be interpreted as a "density" of CLE loops
in the CLE on LQG setting. Positive jumps correspond to the discovery of a CLE
loop (where the LQG length of the loop is given by the jump size) and negative
jumps correspond to the moments where the discovery process splits the
remaining to be discovered domain into two pieces.
  Some consequences are the following: (i) It provides a construction of a CLE
on LQG as a patchwork/welding of quantum disks. (ii) It allows to construct the
"natural quantum measure" that lives in a CLE carpet. (iii) It enables us to
derive some new properties and formulas for SLE processes and CLE themselves
(without LQG) such as the exact distribution of the trunk of the general
asymmetric SLE$_\kappa(\kappa-6)$ processes.
  The present work deals directly with structures in the continuum and makes no
reference to discrete models, but our calculations match those for scaling
limits of O(N) models on planar maps with large faces and CLE on LQG. Indeed,
our L\'evy-tree descriptions are exactly the ones that appear in the study of
the large-scale limit of peeling of discrete decorated planar maps such as in
recent work of Bertoin, Budd, Curien and Kortchemski.
  The case of non-simple CLEs on LQG is the topic of another paper.
