  We study a system of two cavities each encapsulating a qubit and an
oscillator degrees of freedom. An ultrastrong interaction strength between the
qubit and the oscillator is assumed, and the photons are allowed to hop between
the cavities. A partition of the time scale between the fast moving oscillator
and the slow moving qubit allows us to set up an adiabatic approximation
procedure where we employ the delocalized degrees of freedom to diagonalize the
Hamiltonian. The time evolution of the $N00N$-type initial states now
furnishes, for instance, the reduced density matrix of a bipartite system of
two qubits. For a macroscopic size of the $N00N$ component of the initial state
the sudden death of the entanglement between the qubits and its continued null
value are prominently manifest as the information percolates to the qubits
after long intervals. For the low photon numbers of the initial states the
dynamics produces almost maximally entangled two-qubit states, which by
utilizing the Hilbert-Schmidt distance between the density matrices, are
observed to be nearly pure generalized Bell states.
