  We propose a hybrid quadratic estimator to measure cross correlations between
gravitational lensing of the cosmic microwave background (CMB) and differential
screening effects arising from fluctuations in the electron column density,
such as could arise from patchy reionization. The hybrid quadratic estimators
are validated by simulated data sets with both Planck and CMB-Stage 4 (CMB-S4)
instrumental properties and found to be able to recover the cross-power spectra
with almost no biases. We apply this technique to Planck 2015 temperature data
and obtain cross-power spectra between gravitational lensing and differential
screening effects. Planck data alone cannot detect the
patchy-reionization-induced cross-power spectrum but future experiment like
CMB-S4 will be able to robustly measure the expected signal and deliver new
insights on reionization.
