the ‘benchmark’ solution, obtained from the eigenvalues of the associated Hamiltonian, by using Eq. (\[eq:cps\_eigs\]). The eigenvalues of the Hamiltonians of Eqs. (\[eq:HamiltonianMatrices\]) have been calculated numerically for the set of parameters of Fig. \[fig:CPSlong\].

It is evident that the ‘benchmark’ solutions offers a reasonably good estimate for the size of the CPS. The exact dynamics driven by the master equation (\[eq:mastercoll2\]) presents an additional oscillatory behavior both on a short time scale (transient processes), and on a long-time scale. The longer time scale comes from the fact that to induce the cross-Kerr nonlinearity, one has to detune the fields away from the dark resonance. This detuning is very small and is seen in the oscillations on a long time-scale. As both probe and trigger fields are detuned by the same amount, only one frequency of long-time oscillations is observed.

In Fig. \[fig:CPSlong\] (), fidelities (averaged over all possible two-qubit initial states) are shown in two cases. Both are calculated by using Eq. (\[eq:fid\]), but they differ in the way $\rho_f(t)$ is defined. The solid line in Fig. \[fig:CPSlong\] () refers to the *unconditional* fidelity ${\mathcal F}(t)$, which is calculated from Eq. (\[eq:fid\]) by taking $\rho_f(t) = {\rm Tr}_{at}\{\rho(t)\}$, where 