  A formula for the sandwiched relative $\alpha$-entropy
$\widetilde{D}_\alpha(\rho\vert\vert\sigma)=\frac{1}{\alpha-1}\, \ln\,{\rm
Tr}\,
\left(\sigma^{\frac{1-\alpha}{2\alpha}}\,\rho\,\sigma^{\frac{1-\alpha}{2\alpha}}\right)^\alpha$
for $0 < \alpha < 1$, of two $n$ mode gaussian states $\rho$, $\sigma$ in the
boson Fock space $\Gamma(\mathbb{C}^n)$ is presented. This computation
extensively employs the $\mathcal{E}_2$-parametrization of gaussian states in
$\Gamma(\mathbb{C}^n)$ introduced in J. Math. Phys. {\bf 62} (2021), 022102.
