  We derive shortcuts to adiabaticity maximizing population transfer in a
three-level $\Lambda$ quantum system, using the spin to spring mapping to
formulate the corresponding optimal control problem on the simpler system of a
classical driven dissipative harmonic oscillator. We solve the spring optimal
control problem and obtain analytical expressions for the impulses, the
durations of the zero control intervals and the singular control, which are the
elements composing the optimal pulse sequence. We also derive suboptimal
solutions for the spring problem, one with less impulses than the optimal and
others with smoother polynomial controls. We then apply the solutions derived
for the spring system to the original system, and compare the population
transfer efficiency with that obtained for the original system using numerical
optimal control. For all dissipation rates used, the efficiency of the optimal
spring control approaches that of the numerical optimal solution for longer
durations, with the approach accomplished earlier for smaller decay rates. The
efficiency achieved with the suboptimal spring control with less impulses is
very close to that of the optimal spring control in all cases, while that
obtained with polynomial controls lies below, and this is the price paid for
not using impulses, which can quickly build a nonzero population in the
intermediate state. The analysis of the optimal solution for the classical
driven dissipative oscillator is not restricted to the system at hand but can
also be applied in the transport of a coherent state trapped in a moving
harmonic potential and the transport of a mesoscopic object in stochastic
thermodynamics.
