Two-Level Systems with Relaxation

A method is presented for the coherent control of two-level systems when $T_2$ 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 $T_2$. In general, the effect of $T_2$ 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.