Theoretical foundations for exploring quantum optimal control of molecules

Significant progress has been made in achieving experimental laser control of molecules due to advances in ultrafast laser technology utilizing femtosecond pulse shaping capabilities. In addition, the advent of few-cycle femtosecond as well as attosecond pulses has made possible observation and manipulation of electronic and nuclear motions. In this chapter, we review theoretical foundations for exploring quantum optimal control of molecular dynamics driven by laser fields. The goal is to present the theoretical machinery for simultaneous control of electronic transitions and nuclear motion in a molecule. The presentation encompasses quantum optimal control theory (OCT), quantum control landscape (QCL) analysis, a two-point boundary-value quantumcontrol paradigm(TBQCP), and the TBQCP-based algorithms to search for optimal control fields. Optimal control schemes in the context of the TBQCP are formulated using the Born- Oppenheimer (BO) concept for the separation of electronic and nuclear degrees of freedom, generalized to include the electric field amplitude as an additional degree of freedom. The TBQCP formulations are tailored in the weak and strong field limits and in the very short-time (sub-femto/attosecond) limit, with the goal of providing a unified treatment of optimal control over molecular dynamics. Numerical examples are presented to demonstrate the utility of the TBQCP optimal control search algorithms.