  This paper is devoted to the derivation of $L^2$ - $L^2$ decay estimates for
the solution of the homogeneous linear damped wave equation on the Heisenberg
group $\mathbf{H}_n$, for its time derivative and for its horizontal gradient.
Moreover, we consider the improvement of these estimates when further
$L^1(\mathbf{H}_n)$ regularity is required for the Cauchy data. Our approach
will rely strongly on the group Fourier transform of $\mathbf{H}_n$ and on the
properties of the Hermite functions that form a maximal orthonormal system for
$L^2(\mathbb{R}^n)$ of eigenfunctions of the harmonic oscillator.
