  We study the problem of maximizing population transfer efficiency in the
STIRAP system for the case where the dissipation rate of the intermediate state
is much higher than the maximum amplitude of the control fields. Under this
assumption, the original three-level system can be reduced to a couple of
equations involving the initial and target states only. We find the control
fields which maximize the population transfer to the target state for a given
duration $T$, without using any penalty involving the population of the lossy
intermediate state, but under the constraint that the sum of the intensities of
the pump and Stokes pulses is constant, so the total field has constant
amplitude and the only control parameter is the mixing angle of the two fields.
In the optimal solution the mixing angle changes in the bang-singular-bang
manner, where the initial and final bangs correspond to equal instantaneous
rotations, while the intermediate singular arc to a linear change with time. We
show that the optimal angle of the initial and final rotations is the unique
solution of a transcendental equation where duration $T$ appears as a
parameter, while the optimal slope of the intermediate linear change as well as
the optimal transfer efficiency are expressed as functions of this optimal
angle. The corresponding optimal solution recovers the counterintuitive
pulse-sequence, with nonzero pump and Stokes fields at the boundaries. We also
show with numerical simulations that, transfer efficiency values close to the
optimal derived using the approximate system, can also be obtained with the
original STIRAP system using dissipation rates comparable to the maximum
control amplitude.
