This is a fascinating notion to me, and while generally I get things I have no idea what foundation I would need to understand this. What exactly is going on?
The scale of this experiment is pretty big in quantum terms, so a semi-classical explanation is not inaccurate. The electron is behaving like water filling a bowl. The height of the bowl floor represents how much potential energy the electron needs to reach that point in space.
Suppose that the bowl is invisible, and detailed knowledge of its shape is of scientific interest for one reason or another. Here, it's interesting because they want to fine-tune their ability to spatially and electromagnetically control individual electrons so that they can explore "spintronics".
One way to measure the shape of the bowl is to measure the shape of the electron "fluid" filling the bowl. They've developed some technique for doing that. I haven't read the details.
That only gives you information about the bottommost part of the bowl, where the fluid lies. By applying voltages to the electrodes, the authors can raise the potential energy needed for the electron to hang out at one end of the bowl relative to the other. This effectively tips the bowl, and the electron "fluid" re-distributes itself, allowing them to measure the shape of the bowl at locations other than the bottom.
That's all I get from the article. You'd have to read the linked original paper to learn more. A traditional undergraduate series in classical mechanics, electromagnetics, waves, and quantum mechanics is very helpful, but cutting-edge research is never communicated in the same terms that are used in undergraduate teaching. There are going to be some jargon barriers no matter what. Happy reading!
Quantum mechanics mostly but with impacts in quantum computing. If you come from the computing side and want to tangentially back into quantum mechanics from the quantum computing side, I would highly recommend Andy Matuschak and Michael Nielsen's highly approachable intro to quantum computing which goes into surprising depth - https://quantum.country/qcvc
Once you feel like you have a good grasp, just dive into the deep end (why not?) with a pictorial survey of the geometry of quantum states - https://arxiv.org/pdf/1901.06688.pdf
Then see where the gaps are and decide if it's worth digging in deeper to understand this better :-)
Undergrad level quantum mechanics is enough to have a basic idea of what is going on. The electron position is calculated as a time- and space- dependent probability density that can also be time-independent if certain conditions are met. This probability density is proportional to the magnitude squared of the wavefunction. The wavefunction is a solution to a partial differential equation like Schrodinger's equation given a specified Hamiltonian and boundary conditions. In this experiment, the potential energy portion of the Hamiltonian (total energy) is controlled via externally applied electric fields. Designing the quantum dot to control the spatial pattern of these electric fields is the secret sauce of the experiment that is not so easily understood at an undergrad level, but as a first pass you can consider canonical problems like the quantum harmonic oscillator. It's been a while, so others should feel free to correct me if I said anything inaccurate.
