Even teeny weeny discoveries are great fun

This morning, in the process of editing a paper, I made a small discovery. We have developed a new mathematical framework for determining the properties of quantum graphs in terms of the properties of the pieces. This work provides simple identities on the pieces that will allow us to determine general principles form the ground up rather than having to calculate the properties of the full graph.

I have to run because my wife is calling me to lunch.

Here is my email to my collaborators.

Your introduction to edge states was perfect. I liked the physical approach that leads to the formalism. In fact, its clarity was instrumental in allowing me to make a minor discovery (see below).

With regards to the paper, EDGE STATES ARE WONDERFUL! I am taking a break for lunch now, but FYI, I have been working soley on the appendix because I have done what I think is a really neat calculation which uses the power of the edge state. If you recall, in the past, we used the fact that the sum over all of the edges yields the full sum rule. However, it turns out that there are sum rules on each edge! The edge state formalism has allowed me to do this very easily. The result is given by Equation A20 of the geometry. I have pretty much dropped everything to work on this, but I will need to get back to preparing my talks for SPIE since I still have lots to do.

I suggest the following. I still need to reread the appendix because I was making changes while calculating -- never a good thing in terms of introducing typos. I will work on this after lunch. In the meantime, please check the appendix and let me know if I made any errors. The result is so logical that it seams right. I will then alternate between working on my talks and working on the paper.

Most likely, I will not go in to work today so that I can finish the paper in time to be posted on the archives tonight. Even these small discoveries are great fun.

After almost a year of waiting, finally some good news!

These are tough times for any programs that are funded by the U.S. government. Last summer I was contacted by NSF for some clarifications on my proposal, which I provided and resulted in the program manager's approval. In my experience, official notification to our university about an award comes no more than a few weeks after this first contact. In this case, we heard nothing for months.

While on an NSF panel in the fall semester, I visited with the program manager to inquire about the disposition of my proposal. He told me that the paperwork had been signed and that the documents "were on the Directors desk" awaiting final signatures. However, congressional battles on funding to NSF put proposals on hold until a resolution was in sight.

I was getting concerned that this proposal would never be funded. However, a couple weeks ago I was contacted by an accountant with questions about a budget issue. It took only a few emails to resolve the issue, so again, I waited. While I have not received official word in the form of a binding award letter, the NSF website now shows my proposal as "awarded." The reviews are also posted.

Excerpts from the panel summary follow.

Objective: The objective of this program is to invent new approaches for manipulating quantum systems in a way that enhances their nonlinear-optical response.

Intellectual Merit:
The intellectual merit of this proposal is very high and may lead to discovery of universal properties that will enable optimization of materials for a given task, such as nonlinear optical response. The PI's approach is to use sum rules in conjunction with numerical optimization and Monte Carlo studies to broadly understand those issues that are most important in making an optimized material. This will include development of fundamental quantum mechanical concepts that build an understanding of the performance limit of optical materials and practical methods for attaining the limit. The knowledge will guide chemists and nanotechnologists in designing new materials and nanostrucutures. The PI is very well qualified to carry out the proposed research. If successful, the proposed effort will lead to transformative paradigms enabling physicists and materials scientists to synthesize novel functional materials and to generate novel photonic devices.

Broader Impacts:
This proposal has the potential to make transformative advancements in the development of novel materials for photonic applications. The fundamental knowledge will serve as a guide in optimizing a material for a given task, such as nonlinear optical response. The work applies to any system based on the interaction between light and matter including molecules, inorganic materials, nanoparticles, smart materials, nanowires, etc. Education and outreach efforts are very strong and will broaden participation of Native American and under-represented groups and undergraduate students in the research program. The PI also plans to promote undergraduate participation and interaction with high school students through online interactive resources. Numerical codes that will be developed as part of the program work will be distributed over the web so that students can participate in research.

Summary:
The panel considers this is an excellent proposal with strong technical and education components. If successful the research will have a transformative impact on development of fundamental quantum mechanical concepts to synthesize novel functional materials.
Reviewer Ratings: E, E, V,

Panel Recommendation:
The proposal was placed in the Highly Recommended [HR]] category by the panel. The Program Directors concur with the panel opinion as expressed in the panel summary with respect to both the Intellectual Merit and the Broader Impacts criteria.

We finally got it right (we hope)

It's been three years since Juefei Zhou finished his Ph.D. research that culminated in a nice piece of collaborative work with the group of Koen Clays of the University of Leuven in Belgium. The research used a combination of theory and experiments to determine all the parameters needed to predict the full wavelength dependence of the two-photon absorption cross section. The beauty of the approach is that the theory uses the Thomas Kuhn sum rules to significantly reduce the number of parameters required to describe a molecule. This reduced set of parameters was determined from two experiments - a linear absorption spectrum and the measurement of the hyperpolarizability at just one wavelength.

Given that nonlinear-optical quantum calculations are notoriously inaccurate; and, independent measurements (such as first and second hyperpolarizability measurements) often disagree, we were elated that our approach led to a global agreement between all quantities using just one small set of parameters. The only problem was that our theory was wrong. We had made a false assumption. Thus, our manuscript was placed on the back burner.

A year later, in 2009, I spent a summer in Belgium, and used a combination of symmetry arguments and sum rules to show that our equations turned out to be correct, but for very different reasons. As we were applying the final touches to the manuscript when I returned back to Pullman, Xavi found an error. After days of intense debate, we found a way to correct the mistake and submitted a revised manuscript to Physical Review A.

A very sharp reviewer caught what appeared to be a fatal error. Our symmetry arguments were correct, but they implied an additional condition that rendered our approach untenable. For the next 12 months, we were haunted by a model that was wrong yet seemed to fit the data perfectly well.

At the beginning of the summer of 2010, Shengting was getting frustrated with a laser that refused to work properly, and asked for a theoretical project as a diversion. I suggested that he learn group theory and apply it to fixing the model. While Shengting was making good progress in both learning group theory and developing a plan of attack to address our problems, Xavi arrived from Belgium for a six-week stay. During my trip to Budapest, they had found a solution, albeit with a few holes.

A day before I returned, Koen Clays arrived in Pullman, and spent some time discussing the problem with Xavi and Shengting. As a result, they got closer to a solution. When I got back to Pullman (a day late because of a missed connection in Amsterdam), the four of us met to discus the problem and the proposed fix. Xavi acted as the spokesperson and very animatedly described the approach on the blackboard. Within an hour, everything fell into place. Not only did the original mathematical form of the theory turn out to be true, the underlying physics was even more beautiful than we had imagined. This project has led to new ideas that will take us into novel areas of research that we hope will make a closer connection between our theory, which is a bit esoteric, and real molecules.

It was worth the wait. This paper will add a significant new paradigm to the body of knowledge that seeks to more deeply understand the nonlinear-optical response of complex molecules. The path of our research took us through exhilarating highs and unbearable lows. Hopefully, our models are finally right. If not, the self-correcting process of the scientific method will eventually lead us, or someone else, closer to the truth.