Qutrit evolution and contraction of Lie algebra su (3)

The evolution of a three-level quantum system interacting with the environment with the symmetryof the Lie algebra su(3) (qutrit) is considered. Theevolution of qutrit is obtained when interacting withthe environment. The transverse relaxation of thequtrit (due to the diagonal Lindblad operators) hasbeen studied, leading to zero (with the same speedfor the real and imaginary parts) off-diagonal elements of the density matrix. The dynamics of qutritis obtained in the presence of both transverse andlongitudinal relaxation, i.e. when the Lindblad operators are off-diagonal and there are transitions inthe system between all energy levels. It is shownthat the addition of the longitudinal relaxation ofthe qutrit in addition to the conversion of the off-diagonal elements to zero, also affects the diagonalelements of its density matrix. As a result of dissipation, the quantum system becomes incoherent overtime. In both cases, initially non-commuting quantum observables in the process of evolution acquirethe properties of commuting classical observables,which algebraically manifests itself in the contraction of the algebra su(3).