  A unique attribute of atomically thin quantum materials is the in-situ
tunability of their electronic band structure by externally controllable
parameters like electrostatic doping, electric field, strain, electron
interactions, and displacement or twisting of atomic layers. This unparalleled
control of the electronic bands has led to the discovery of a plethora of
exotic emergent phenomena. But despite its key role, there is currently no
versatile method for mapping the local band structure in advanced 2D materials
devices in which the active layer is commonly embedded in various insulating
layers and metallic gates. Utilizing a scanning superconducting quantum
interference device, we image the de Haas-van Alphen quantum oscillations in a
model system, the Bernal-stacked trilayer graphene with dual gates, which
displays multiple highly-tunable bands. By resolving thermodynamic quantum
oscillations spanning over 100 Landau levels in low magnetic fields, we
reconstruct the band structure and its controllable evolution with the
displacement field with unprecedented precision and spatial resolution of 150
nm. Moreover, by developing Landau level interferometry, we reveal
shear-strain-induced pseudomagnetic fields and map their spatial dependence. In
contrast to artificially-induced large strain, which leads to pseudomagnetic
fields of hundreds of Tesla, we detect naturally occurring pseudomagnetic
fields as low as 1 mT corresponding to graphene twisting by just 1 millidegree
over one {\mu}m distance, two orders of magnitude lower than the typical angle
disorder in high-quality twisted bilayer graphene devices. This ability to
resolve the local band structure and strain on the nanoscale opens the door to
the characterization and utilization of tunable band engineering in practical
van der Waals devices.
