  The first measurement of the impact-parameter dependent angular anisotropy in
the decay of coherently photoproduced $\rho^0$ mesons is presented. The
$\rho^0$ mesons are reconstructed through their decay into a pion pair. The
measured anisotropy corresponds to the amplitude of the $\cos(2\phi)$
modulation, where $\phi$ is the angle between the two vectors formed by the sum
and the difference of the transverse momenta of the pions, respectively. The
measurement was performed by the ALICE Collaboration at the LHC using data from
ultraperipheral Pb$-$Pb collisions at a center-of-mass energy of
$\sqrt{s_{\mathrm{NN}}}~=~5.02$ TeV per nucleon pair. Different
impact-parameter regions are selected by classifying the events in
nuclear-breakup classes. The amplitude of the $\cos(2\phi)$ modulation is found
to increase by about one order of magnitude from large to small impact
parameters. Theoretical calculations, which describe the measurement, explain
the $\cos(2\phi)$ anisotropy as the result of a quantum interference effect at
the femtometer scale that arises from the ambiguity as to which of the nuclei
is the source of the photon in the interaction.
