  In the absence of a complete theory of quantum gravity, phenomenological
models built upon minimal assumptions have been explored for the analysis of
possible quantum effects in gravitational systems. Implications of a
superposition of geometries have been considered in such models, including the
occurrence of processes with indefinite order. In a gravitational quantum
switch, in particular, the order of operations applied by two agents on a
target system is entangled with the state of the geometry. We consider a model
describing the superposition of geometries produced by distinct arrangements of
spherical mass shells, and show that a protocol for the implementation of a
gravitational quantum switch can be formulated in such a system. The geometries
in superposition are identical in an exterior region outside a given radius,
and differ within such a radius. The exterior region provides a classical frame
from which the superposition of geometries in the interior region can be
probed. One of the agents crosses the interior region and becomes entangled
with the geometry, which is explored as a resource for the implementation of
the quantum switch. Novel features of the protocol include the superposition of
nonisometric geometries, the existence of a region with a definite geometry,
and the fact that the agent that experiences the superposition of geometries is
in free fall, preventing information on the global geometry to be obtained from
within its laboratory.
