Relating the curvature of De Sitter Universe to Open Quantum Lamb Shift Spectroscopy

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.