Teaching Wave Mechanics with a Modern Digital Toolkit

Andrew Forbes

University of the Witwatersrand, South Africa

At my university in South Africa, which surely is typical of many universities in the developing world, our optics undergraduate laboratory exercises are more or less confined to double-slit interference, some imaging, and a little Fourier optics. Quantum optics experiments do not exist in the sense that we think of this topic today, with quantum experiments inspired by advances a century ago, mostly ‘particle’ based. Entanglement is very much a topic of the textbook. Yet in my own research laboratory, we routinely create and control high-dimensional photonic entangled states, performing state-of-the-art quantum optics experiments. Why does it take such a long time for the advances in research to reach the teaching laboratories? At least in the developing world, one reason is equipment, and the skills required to implement an experiment, particularly of a quantum nature. Recently, we have tried to rectify this problem by, on the one hand, introducing a modern digital toolkit for modern optical experiments, and, on the other hand, exploiting the fact that quantum mechanics is a wave theory, like optics, and thus ‘wave mechanics,’ classical and quantum, can be taught with the same principles and the same toolkit. I would even go so far as to suggest that we teach a wave mechanics course that is a blend of optics and quantum mechanics.

In a drive towards inexpensive, fast, and digital, we have introduced digital micro-mirror devices (DMDs) to our repertoire of optical tools for teaching and research alike, offering best practices to get started with DMDs,¹ to accelerate the uptake. What can you do with such digital tools in the optical laboratory? Well, first you can digitize the traditional double-slit experiment, bringing an element of computation, digital control, and automation to the experiment—not to mention more entertainment and versatility in execution; e.g., the slits, as images displayed on the device, can be dynamically changed in all parameters such as size, geometry, separation, and so on, giving the students flexibility to explore and truly “experiment.” Furthermore, the tools allow one to generalize the traditional interference experiment to fringes in other geometries and other degrees of freedom,² such as radial and azimuthal fringes, and fringes in the polarization structure of light, as well as in the orbital angular momentum of light, highlighting that fringes are not restricted to intensity. The same tools can be used to digitally propagate light, digitally spatial filter light, and thus to demonstrate Fourier optics with a modern flavor.³

The aforementioned experiments are traditionally taught as part of a course on wave optics, resting largely on the superposition principle of waves. It is well known that the equation of motion for quantum states has a form analogous to that in paraxial optics, and recently the parallels have deepened, with a realization that many quantum principles can be demonstrated with classical light,⁴ including entanglement.⁵ Recently, we exploited this to produce a do-it-yourself teaching kit⁶ that allows the user to demonstrate Bell violations, quantum state tomography, density matrices, and so on, all with 3D-printed components and conventional laser light, e.g., from inexpensive laser pointers. This has proven instrumental in visualizing and working through ‘quantum’ measurements, and ‘quantum’ data, except without any ‘quantum’ complexity in the experiment itself. Here the objective is to make the theory come alive, to work with real data, but at the expense of not realizing a true quantum experiment. Where time and resources are tight, as they often are in large undergraduate classes, this approach has its place.

In our university, we are in the process of switching experiments over to this approach and have used our digital toolkit to great effect in teaching, as well as in outreach to local schools, inspiring the next generation by showcasing that optics is modern and alive.

There is a classical particle mechanics and classical wave mechanics, but only one quantum mechanics. Both optics and quantum mechanics can be taught in the overlapping region where the superposition principle holds supreme. Vector beams, with inhomogeneous polarization states, can be used to mimic entangled states.