Two-level systems with relaxation.

A method is presented for the coherent control of two-level systems when T2 relaxation is significant. The Bloch equations are rewritten as an equation of motion of the stereographic projection, Gamma, of the spin vector. This allows a Schur-type iteration used for the design of shaped pulses in magnetic resonance and coherent optics to be extended to include the effect of T2. In general, the effect of T2 on Gamma cannot be completely compensated for, although in practice it can be to a high degree. An example is presented of a driving field that produces a coherent superposition (no on-diagonal elements of the density matrix) over a chosen band of frequencies, in the presence of relaxation.