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

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