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Finding the evolution of two level Hamiltonian is of great importance inquantum computation and quantum precision manipulation due to the requirementof quantum experiment control. However, the Schr\"odinger equation of anarbitrary time-dependent two level Hamiltonian is hardly solvable due to itsnon-commutativity Hamiltonian in different times. In this article, we expandand demonstrate an exact solution of Schr\"odinger equation respect to generaltwo level systems with a few limitations. This analytical solution has lots ofmanipulative parameters and a few boundary restrictions, which could drive manyapplications. Furthermore, we show the adaptive capacity of our scheme, whichdemonstrated the widely use of our scheme, and make it suitable for most ofexperiment Hamiltonian directly.

An Analytical Solution in Detuned Two Level Systems