  The application of a uniform background magnetic field makes standard quark
operators utilising gauge-covariant Gaussian smearing inefficient at isolating
the ground state nucleon at nontrivial field strengths. In the absence of QCD
interactions, Landau modes govern the quark energy levels. There is evidence
that residual Landau mode effects remain when the strong interaction is turned
on. Here we introduce novel quark operators constructed from the
two-dimensional $U(1)$ Laplacian eigenmodes that describe the Landau levels of
a charged particle on a periodic finite lattice. These eigenmode-projected
quark operators provide enhanced precision for calculating nucleon energy
shifts in a magnetic field. Using asymmetric source and sink operators, we are
able to encapsulate the predominant effects of both the QCD and QED
interactions in the interpolating fields for the neutron. The neutron magnetic
polarizability is calculated using these techniques on the $32^3 \times 64$
dynamical QCD lattices provided by the PACS-CS Collaboration. In conjunction
with a chiral effective-field theory analysis, we obtain a neutron magnetic
polarizability of $\beta^n = 2.05(25)(19) \times 10^{-4}$ fm$^3$, where the
numbers in parentheses describe statistical and systematic uncertainties.
