  We have made significant progress on extending lattice QCD calculation of the
polarizability of the neutron and other hadrons to include the effects of
charged dynamical quarks. This is done by perturbatively reweighting the
charges of the sea quarks to couple them to the background field. The dominant
challenge in such a calculation is stochastic estimation of the weight factors,
and we discuss the difficulties in this estimation. Here we use an extremely
aggressive dilution scheme with N = 124,416 sources per configuration to reduce
the stochastic noise to a manageable level. We find that \alpha_E = 2.70(55) *
10^-4 fm^3 for the neutron on one ensemble. We show that low-mode substitution
can be used in tandem with dilution to construct an even better estimator, and
introduce the offdiagonal matrix element mapping technique for predicting
estimator quality.
