The broad success of optimally controlling quantum systems with external
fields has been attributed to the favorable topology of the underlying control
landscape, where the landscape is the physical observable as a function of the
controls. The control landscape can be shown to contain no suboptimal trapping
extrema upon satisfaction of reasonable physical assumptions, but this
topological analysis does not hold when significant constraints are placed on
the control resources. This work employs simulations to explore the topology
and features of the control landscape for pure-state population transfer with a
constrained class of control fields. The fields are parameterized in terms of a
set of uniformly spaced spectral frequencies, with the associated phases acting
as the controls. Optimization results reveal that the minimum number of phase
controls necessary to assure a high yield in the target state has a special
dependence on the number of accessible energy levels in the quantum system,
revealed from an analysis of the first- and second-order variation of the yield
with respect to the controls. When an insufficient number of controls and/or a
weak control fluence are employed, trapping extrema and saddle points are
observed on the landscape. When the control resources are sufficiently
flexible, solutions producing the globally maximal yield are found to form
connected `level sets' of continuously variable control fields that preserve
the yield. These optimal yield level sets are found to shrink to isolated
points on the top of the landscape as the control field fluence is decreased,
and further reduction of the fluence turns these points into suboptimal
trapping extrema on the landscape. Although constrained control fields can come
in many forms beyond the cases explored here, the behavior found in this paper
is illustrative of the impacts that constraints can introduce.Comment: 10 figure