  We study the magnetic field induced Hofstadter butterfly in twisted bilayer
graphene (TBG) in various kinds of situations. First, we study the equilibrium
case and identify the interlayer hopping processes that are most crucial for
the appearance of a Hofstadter butterfly. Surprisingly, the hopping processes
that are important for the appearance of the Hofstadter butterfly can be
categorized as AA stacking type - that is interlayer hoppings between
equivalent sublattices. This is in contrast to AB/BA-type hoppings that are
important for the appearance of flat bands in magic angle TBG and were
discussed in [Phys. Rev. Lett. 122, 106405 (2019)]. We also find that if
AB-type interlayer-hopping processes are turned off the resulting model is
chiral but differs from the model discussed in \cite{Tarnopolsky}. Therefore,
TBG has two separate chiral limits - one of them is important to understand the
formation of flat bands and the other for the Hofstadter butterfly. Taking this
as motivation we discuss how the role of AA-type hoppings in combination with
lattice relaxation effects can make individual Landau levels slightly harder to
resolve in an experimental setting than one would expect from a non-relaxed
lattice setting. Finally, we consider the impact of different forms of light on
the fractal structure of the butterfly. Particularly, we study the impact of
circularly polarized light and longitudinal light originating from a waveguide.
As the system is exposed to circularly polarized light we find butterflies with
increasingly pronounced asymmetry with respect to energy $E=0$. This is due to
the introduction of a gap term that breaks the chiral symmetries for both of
the two chiral limits mentioned above. Lastly, we study the effect of
longitudinal light that can be produced at the exit of a waveguide, in a
slightly simplified model. Here, we find ...
