Entanglement in parametric down-conversion with phase mismatch at finite temperature

In this work the degree of entanglement of two-photon states generated by parametric-down conversion with a phase mismatch is computed in terms of the temperature of the nonlinear medium where the interaction takes place. The minimum interaction time, named {\it birth time of entanglement}, needed before entanglement is attained, is computed in terms of the temperature and of the phase mismatch. Our results show that, for each value of the phase mismatch, there is a critical temperature above which entanglement disappears. We are able to identify the regions of parameters, temperature and mismatch, where the states are entangled and where they are separable. We also propose a entanglement witness based in a dynamical invariant of the nonlinear interaction.