  Phase lag associated with coherent control where an excited system decays
into more than one product channel has been subjected to numerous
investigations. Although previous theoretical studies have treated the phase
lag across resonances in model calculations, quantitative agreement has never
been achieved between the theoretical model and experimental measurements of
phase lag from the $\omega-2\omega$ ionization of atomic barium
\cite{PhysRevLett.98.053001,PhysRevA.76.053401}, suggesting that a toy model
with phenomenological parameters is inadequate to describe the observed phase
lag behavior. Here the phase lag is treated quantitatively from a multichannel
coupling formulation, and our calculations based on multichannel quantum defect
and $R$-matrix treatment achieves good agreement with the experimental
observations. Our treatment also develops formulas to describe the effects of
hyperfine depolarization on multiphoton ionization processes, and further,
identifies resonances between $Ba^{+}$ $5d_{3/2}$ and $5d_{5/2}$ thresholds
that have apparently never been experimentally observed and classified.
