Quantum control landscape for a $\mathrm{\Lambda }$-atom in the vicinity of second order traps

We show that the second order traps in the control landscape for a three-level $\mathrm{\Lambda }$-system found in our previous work {\it Phys. Rev. Lett.} {\bf 106}, 120402 (2011) 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 $\epsilon (t)=0$ second order trap, performed using the GRAPE and BFGS algorithms.