  We investigate the time evolution and asymptotic behavior of a system of two
two-level atoms (qubits) interacting off-resonance with a single mode radiation
field. The two atoms are coupled to each other through dipole-dipole as well as
Ising interactions. An exact analytic solution for the system dynamics that
spans the entire phase space is provided. We focus on initial states that cause
the system to evolve to entanglement sudden death (ESD) between the two atoms.
We find that combining the Ising and dipole-dipole interactions is very
powerful in controlling the entanglement dynamics and ESD compared with either
one of them separately. Their effects on eliminating ESD may add up
constructively or destructively depending on the type of Ising interaction
(Ferromagnetic or anti-Ferromagnetic), the detuning parameter value, and the
initial state of the system. The asymptotic behavior of the ESD is found to
depend substantially on the initial state of the system, where ESD can be
entirely eliminated by tuning the system parameters except in the case of an
initial correlated Bell state. Interestingly, the entanglement, atomic
population and quantum correlation between the two atoms and the field
synchronize and reach asymptotically quasi-steady dynamic states. Each one of
them ends up as a continuous irregular oscillation, where the collapse periods
vanish, with a limited amplitude and an approximately constant mean value that
depend on the initial state and the system parameters choice. This indicates an
asymptotic continuous exchange of energy (and strong quantum correlation)
between the atoms and the field takes place, accompanied by diminished ESD for
these chosen setups of the system. This system can be realized in spin states
of quantum dots or Rydberg atoms in optical cavities, and superconducting or
hybrid qubits in linear resonators.
