  It has recently been pointed out that phases of matter with intrinsic
topological order, like the fractional quantum Hall states, have an extra
dynamical degree of freedom that corresponds to quantum geometry. Here we
perform extensive numerical studies of the geometric degree of freedom for the
simplest example of fractional quantum Hall states -- the filling $\nu=1/3$
Laughlin state. We perturb the system by a smooth, spatially dependent metric
deformation and measure the response of the Hall fluid, finding it to be
proportional to the Gaussian curvature of the metric. Further, we generalize
the concept of coherent states to formulate the bulk off-diagonal long range
order for the Laughlin state, and compute the deformations of the metric in the
vicinity of the edge of the system. We introduce a "pair amplitude" operator
and show that it can be used to numerically determine the intrinsic metric of
the Laughlin state. These various probes are applied to several experimentally
relevant settings that can expose the quantum geometry of the Laughlin state,
in particular to systems with mass anisotropy and in the presence of an
electric field gradient.
