Evolution of the entanglement of the N00N-type of states in a coupled two cavity system via an adiabatic approximation

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.