  Some models of quantum gravity predict that the very structure of spacetime
is `frothy' due to quantum fluctuations. Although the effect is expected to be
tiny, if these spacetime fluctuations grow over a large distance, the initial
state of a photon, such as its energy, is gradually washed out as the photon
propagates. Thus, in these models, even the most monochromatic light source
would gradually disperse in energy due to spacetime fluctuations over large
distances. In this paper, we use science verification observations obtained
with ESPRESSO at the Very Large Telescope to place a novel bound on the growth
of spacetime fluctuations. To achieve this, we directly measure the width of a
narrow Fe II absorption line produced by a quiescent gas cloud at redshift
z=2.34, corresponding to a comoving distance of ~5.8 Gpc. Using a heuristic
model where the energy fluctuations grow as sigma_E / E = (E/E_P)^alpha, where
E_P = 1.22 x 10^28 eV is the Planck energy, we rule out models with alpha <
0.634, including models where the quantum fluctuations grow as a random walk
process (alpha = 0.5). Finally, we present a new formalism where the
uncertainty accrued at discrete spacetime steps is drawn from a continuous
distribution. We conclude, if photons take discrete steps through spacetime and
accumulate Planck-scale uncertainties at each step, then our ESPRESSO
observations require that the step size must be at least >10^13.2 L_P, where
L_P is the Planck length.
