  We investigate the holographic bound utilizing a homogeneous, isotropic, and
non-relativistic neutral hydrogen gas present in the de Sitter space.
Concretely, we propose to employ de Sitter holography intertwined with quantum
deformation of the hydrogen atom using the framework of quantum groups.
Particularly, the $\mathcal U_q(so(4))$ quantum algebra is used to construct a
finite-dimensional Hilbert space of the hydrogen atom. As a consequence of the
quantum deformation of the hydrogen atom, we demonstrate that the Rydberg
constant is dependent on the de Sitter radius, $L_\Lambda$. This feature is
then extended to obtain a finite-dimensional Hilbert space for the full set of
all hydrogen atoms in the de Sitter universe. We then show that the dimension
of the latter Hilbert space satisfies the holographic bound. We further show
that the mass of a hydrogen atom $m_\text{atom}$, the total number of hydrogen
atoms at the universe, $N$, and the retrieved dimension of the Hilbert space of
neutral hydrogen gas, $\text{Dim}{\mathcal H}_\text{bulk}$, are related to the
de Sitter entropy, $S_\text{dS}$, the Planck mass, $m_\text{Planck}$, the
electron mass, $m_\text{e}$, and the proton mass $m_\text{p}$, by
$m_\text{atom}\simeq m_\text{Planck}S_\text{dS}^{-\frac{1}{6}}$, $N\simeq
S_\text{dS}^\frac{2}{3}$ and $\text{Dim}{\mathcal
H}_\text{bulk}=2^{\frac{m_\text{e}}{m_\text{p}}\alpha^2S_\text{dS}}$,
respectively.
