  We study the energy spectrum of a graphene bilayer in the presence of
transverse electric and magnetic fields. We find that the resulting Landau
levels exhibit a nonmonotonic dependence on the electric field, as well as
numerous level crossings. This behavior is explained using quasiclassical
quantization rules that properly take into account the pseudospin of the
quasiparticles. The pseudospin generates the Berry phase, which leads to a
shift in energy quantization and results in a pseudo-Zeeman effect. The latter
depends on the electric field, alternates in sign among the two valleys, and
also reduces the band gap. Analytic formulas for other pseudospin-related
quantities, such as the anomalous Hall conductivity, are derived and compared
with prior theoretical work.
