  In this article, we study optimal control of dynamics in a linear chain of
three spin 1/2, weakly coupled with unequal Ising couplings. We address the
problem of time-optimal synthesis of multiple spin quantum coherences. We
derive time-optimal pulse sequence for creating a desired spin order by
computing geodesics on a sphere under a special metric. The solution to the
geodesic equation is related to the nonlinear oscillator equation and the
minimum time to create multiple spin order can be expressed in terms of an
elliptic integral. These techniques are used for efficient creation of multiple
spin coherences in Ising spin-chains with unequal couplings.
