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

TL;DR

This paper demonstrates that second order traps in the control landscape of a three-level $$-system are not true local maxima, and discusses implications for optimization algorithms, supported by numerical analysis.

Contribution

It reveals that second order traps are not local maxima and highlights the need for higher-order optimization methods in quantum control landscapes.

Findings

Second order traps are not local maxima.

Objective growth near traps can be up to 4th order.

Gradient methods may struggle near traps, requiring advanced algorithms.

Abstract

We show that the second order traps in the control landscape for a three-level $\Lambda$-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 $\varepsilon(t)=0$ second order trap, performed using the GRAPE and BFGS algorithms.