  Ultra-cold fermionic atoms, having two valence electrons, exhibit a
distinctive internal state structure, wherein the nuclear spin becomes
decoupled from the electronic degrees of freedom in the ground electronic
state. Consequently, the nuclear spin states are well isolated from the
environment, rendering these atomic systems an opportune platform for quantum
computation and quantum simulations. Coupling with off-resonance light is an
essential tool to selectively and coherently manipulate the nuclear spin
states. In this paper, we present a systematic derivation of the effective
Hamiltonian for the nuclear spin states of ultra-cold fermionic atoms due to
such an off-resonance light. We obtain compact expressions for the scalar,
vector and tensor light shifts taking into account both linear and quadratic
contributions to the hyperfine splitting. The analysis has been carried out
using the Green operator approach and solving the corresponding Dyson equation.
Finally, we analyze different scenarios of light configurations which lead to
the vector- and tensor-light shifts, as well as the pure spin-orbit coupling
for the nuclear spin.
