  We report the first measurements of cumulants, up to $4^{th}$ order, of
deuteron number distributions and proton-deuteron correlations in Au+Au
collisions recorded by the STAR experiment in phase-I of Beam Energy Scan (BES)
program at the Relativistic Heavy Ion Collider. Deuteron cumulants, their
ratios, and proton-deuteron mixed cumulants are presented for different
collision centralities covering a range of center-of-mass energy per nucleon
pair $\sqrt{s_{NN}}$~=~7.7 to 200~GeV. It is found that the cumulant ratios at
lower collision energies favor a canonical ensemble over a grand canonical
ensemble in thermal models. An anti-correlation between proton and deuteron
multiplicity is observed across all collision energies and centralities,
consistent with the expectation from global baryon number conservation. The
UrQMD model coupled with a phase-space coalescence mechanism qualitatively
reproduces the collision-energy dependence of cumulant ratios and
proton-deuteron correlations.
