Quantum control landscape for a Λ-atom in the vicinity of second-order traps

We show that the second-order traps in the control landscape for a three-level Λ-system found in our previous work (Phys. Rev. Lett. 2011, 106, 120402) are not local maxima: there exist directions in the space of controls in which the objective grows. The growth of the objective is slow - at best 4th order for weak variations of the control. This implies that simple gradient methods would be problematic in the vicinity of second-order traps, where more sophisticated algorithms that exploit the higher order derivative information are necessary to climb up the control landscape efficiently. The theory is supported by a numerical investigation of the landscape in the vicinity of the ε(t)=0 second-order trap, performed using the GRAPE and BFGS algorithms.