  In this paper, we study the design of pulse sequences for NMR spectroscopy as
a problem of time optimal control of the unitary propagator. Radio frequency
pulses are used in coherent spectroscopy to implement a unitary transfer of
state. Pulse sequences that accomplish a desired transfer should be as short as
possible in order to minimize the effects of relaxation and to optimize the
sensitivity of the experiments. Here, we give an analytical characterization of
such time optimal pulse sequences applicable to coherence transfer experiments
in multiple-spin systems. We have adopted a general mathematical formulation,
and present many of our results in this setting, mindful of the fact that new
structures in optimal pulse design are constantly arising. Moreover, the
general proofs are no more difficult than the specific problems of current
interest. From a general control theory perspective, the problems we want to
study have the following character. Suppose we are given a controllable right
invariant system on a compact Lie group, what is the minimum time required to
steer the system from some initial point to a specified final point? In NMR
spectroscopy and quantum computing, this translates to, what is the minimum
time required to produce a unitary propagator? We also give an analytical
characterization of maximum achievable transfer in a given time for the two
spin system.
