Not all triangles are the same

Just the other day I wrote about a revelation I had about the self healing process, a hot topic in our lab these days. As often happens, the first impression is simplistic and not quite right, but eventually, we hopefully converge on the truth. However, my fallacy of yesterday gave me insights today, which I continue to pursue.

On another front, we have completed a new paper that we are submitting to Physical Review Letters, the highest impact physics journal. I always have reservations about sending a manuscript to a journal just because it is prestigious. What counts is the quality of the paper. On the other hand, if our work is as significant as we believe, then appearing in a top journal will give it more visibility.

I am excited by the science, and the possibility that we may have started a new branch of study. At the heart of our work are calculations of the optical nonlinearity of quantum wires. This in itself is totally new (to the best of our knowledge), but we are taking the elevator down a level to the realm of fundamental science. For those more practically minded, our work may also have some useful applications.

Science is often focused on a particular thing. While a researcher may be interested in solving the problem of global warming -- a grand problem, the actual work may involve studying the behavior of a particular kind of electrode dipped in a specific chemical. In fact, many groups around the world may be studying exactly the same thing, trying to work out a detail that could make a battery store 5% more energy. Such a leap would indeed be important.

Rather than focusing on details, our work is painting a big picture. I like ideas that have broad influence; views of the world from unique perspectives; and unexpected results on topics that have not crossed anyone's mind, but that resonate with all scientists as being really neat.

Our new work falls beyond the typical boundaries of what others are doing. We are interested in the abstract concept of how the shape (geometry) and topology of an object determine its optical properties. These ideas go beyond specific molecules or materials. To allow us to focus on the basics, we need to remove other complications. To that end, we study what is often called a toy model -- one that brings out the qualities of interest and suppresses the rest. In our case, we are considering structures made of connected wire segments that carry a sole electron.

Consider a continuous loop of wire in the shape of a triangle. If we deform this triangle into other triangles with differing edge lengths and angles, we find that the nonlinearity changes smoothly and not a hell of a lot. In fact, deform the triangle into a quadrangle and then into a quintangle, and nothing much new happens. Any closed loop, independent of the shape, is of the same topology. Thus, we might conclude that the geometry has little effect on the nonlinear response.

A bent wire that does not form a loop is of a different topology. So, consider the simple experiment of a triangle whose nonlinear-optical response is being measured. Now cut a vertex of the triangle so that two of the edges no longer touch. This is still a triangle but its topology has changed. Interestingly, the nonlinear response is found to be profoundly different with the snip of the wire cutters. Thus a change in topology for fixed geometry leads to a dramatic change of the nonlinear-optical response.

This work has applications in the design of better materials because it suggests that taking a molecule (modeled as a wire) and lopping off just a single bond could yield a dramatic improvement. Or, our work could inform nano-technologists on how to make better quantum wires.

We have only evaluated a small number of shapes, including loops made into triangles, quadrangles, quintangles, bent wires, split triangles, and star graphs. Star graphs, which are lines radiating from a central point, represent a topology that yields the larges hyperpolarizability.

To sample the space of all possible shapes, we let the computer randomly pick triangles, quadrangles, quitangles, star graphs, and whatever other shape we can squeeze in. Then we can see what is possible. With enough random tries -- we usually run our simulations over tens of thousands of configurations -- we can test the influence of any parameter, such as topology.

Below is a plot of the first (left) and second (right) hyperpolarizability, which tells us how strongly two and three photons interact with a molecule. Included are triangles (red), simple quadrilaterals (with no crossing edges - green), and all quadrilaterals (blue). Each point (and there are 10,000 here of each color), represents one configuration. A casual glance at the pattern reveals that geometrical effects do not make a big difference. To see the effects of topology, you'll have to read our paper on The Physics Archives.



I find this work really neat (and I hope the reviewers will agree) because we are sampling a very fundamental property of a molecule in terms of some very simple mathematical concepts that go back hundreds to thousands of years. The ancient Greeks heard the music of the spheres in planetary motion using a the metaphorical geometric ear. In our work, we can literally see the effects with light on our eyes when the system's structure changes so ever subtly. And, we get to enjoy a vision of the underlying process with the minds eye as portrayed in very pretty and colorful plots.