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Two and four-level systems in magnetic fields restricted in time

We describe some new exact solutions for two- and four-level systems. In all the cases, external fields have a restricted behavior in time. First, we consider two types of new solutions for one-spin equation, one of them is in a external magnetic field that acts during a finite time interval. A new solution for two interacting spins is found in the case when the field difference between the external fields in each spin vary adiabatically, vanishing on the time infinity. The latter system can be identified with a quantum gate realized by two coupled quantum dots. The probability of the Swap operation for such a gate can be explicitly expressed in terms of special functions. Using the obtained expressions, we construct plots for the Swap operation for some parameters of the external magnetic field and interaction function.