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

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