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Stability of linear and non-linear lambda and tripod systems in the presence of amplitude damping

We present the stability analysis of the dark states in the adiabatic passage for the linear and non-linear lambda and tripod systems in the presence of amplitude damping (losses). We perform an analytic evaluation of the real parts of eigenvalues of the corresponding Jacobians, the non-zero eigenvalues of which are found from the quadratic characteristic equations, as well as by the corresponding numerical simulations. For non-linear systems, we evaluate the Jacobians at the dark states. Similarly to the linear systems, here we also find the non-zero eigenvalues from the characteristic quadratic equations. We reveal a common property of all the considered systems showing that the evolution of the real parts of eigenvalues can be split into three stages. In each of them the evolution of the stimulated Raman adiabatic passage (STIRAP) is characterized by different effective dimension. This results in a possible adiabatic reduction of one or two degrees of freedom.