  In this paper, we explore the connection between the curvature of the
background De Sitter space-time with the spectroscopic study of entanglement of
two atoms. Our set up is in the context of an Open Quantum System (OQS), where
the two atoms, each having two energy levels and represented by Pauli spin
tensor operators projected along any arbitrary direction. The system mimics the
role of a pair of freely falling Unruh De-Witt detectors, which are allowed to
non-adiabatically interact with a conformally coupled massless probe scalar
field which has the role of background thermal bath. The effective dynamics of
this combined system takes into account of the non-adiabatic interaction, which
is commonly known as the Resonant Casimir Polder Interaction (RCPI) with the
thermal bath. Our analysis reveals that the RCPI of two stable entangled atoms
in the quantum vacuum states in OQS depends on the de Sitter space-time
curvature relevant to the temperature of the thermal bath felt by the static
observer. We also find that, in OQS, RCPI produces a new significant
contribution appearing in the effective Hamiltonian of the total system and
thermal bath under consideration. We find that the Lamb Shift is characterized
by a decreasing inverse square power-law behavior, $L^{-2}$, when inter atomic
Euclidean distance, $L$, is much larger than a characteristic length scale,
$k$, which is the inverse surface gravity of the background De Sitter space. If
the background space-time would have been Minkowskian this shift decreases as,
$L^{-1}$, and is independent of temperature. Thus, we establish a connection
between the curvature of the De Sitter space-time with the Lamb Shift
spectroscopy.
