  Understanding gluon density distributions and how they are modified in nuclei
are among the most important goals in nuclear physics. In recent years,
diffractive vector meson production measured in ultra-peripheral collisions
(UPCs) at heavy-ion colliders has provided a new tool for probing the gluon
density. In this Letter, we report the first measurement of $J/\psi$
photoproduction off the deuteron in UPCs at the center-of-mass energy
$\sqrt{s_{_{\rm NN}}}=200~\rm GeV$ in d$+$Au collisions. The differential cross
section as a function of momentum transfer $-t$ is measured. In addition, data
with a neutron tagged in the deuteron-going Zero-Degree Calorimeter is
investigated for the first time, which is found to be consistent with the
expectation of incoherent diffractive scattering at low momentum transfer.
Theoretical predictions based on the Color Glass Condensate saturation model
and the gluon shadowing model are compared with the data quantitatively. A
better agreement with the saturation model has been observed. With the current
measurement, the results are found to be directly sensitive to the gluon
density distribution of the deuteron and the deuteron breakup, which provides
insights into the nuclear gluonic structure.
