A correspondence with a colleague on our nonlinear topology/geometry project

We are not resting on our laurels, but rather are continuing to work on new ideas and extending our work to more general cases.  In the process, we call get confused about the physics of the system, and try to find ways to picture what is going on that gives insights into how a system may behave under various conditions.

In working on this project, I realize that I spend lots of time writing emails to colleagues and students about various aspects of the work.  An email may come in at 10:30 at night because my collaborator is hot to figure something out and wants insights.  I too write emails at weird hours asking students for more details on something they observed in an effort to test my newest crazy ideas.  This is not a line of work for those who want to sleep soundly.  Being excited about physics is the best stimulant.

Anyway, today I took a two-hour break to play floor hockey and found a series of emails from a collaborator who is very excited (as am I) on the new direction of our work and the new physics that is implied.   Since we are moving into new territories, many things are not well mapped out, so we have to navigate partially by instinct.  I occasionally like to post excerpts from emails to give readers a sense of the kinds of exchanges that we have in the process of working on a project.  Here is my response to an exchange about a new quantum graph that we are working on, as shown above.

An excerpt from my email response (edited for typos and names removed):

First, I think there is a big difference in taking the limit as the prong length goes to zero and actually having a zero prong length.  In the former, I think that you get a broken vertex, because as the wire gets shorter, the end of the wire is getting closer to the vertex, so indeed the wavefunction on the prong must get vanishing small as the prong length goes to zero.  Then, this will look identically to a break in the loop.  Of course, when there is no prong, you get the traveling wave solutions.  So it is interesting that the limiting case yields a very different result than the case without the prong - again a statement about topology.  As long as the prong is there, no matter how short, the topology is of a loop-star.  Get rid of a prong, and then it is just a loop!


I am a little confused about what you are saying with regards to the ground state energy.  So, let me just make a point that may not be related at all to what you are saying.  If I have a prong with zero wavefunction, the loop cannot have a constant wavefunction, i.e. the wavefunction with n=0.  By continuity, the amplitude of the constant wavefunction will be zero, so there is no particle in the system.  So, I believe that in a loop, you have the zero-energy wavefunction but as long as a prong is there, you will not have a zero-energy wave function.


However, you bring up an interesting possibility of a wavefunction in which there is zero wavefunction in the prong and a standing wave in the loop in which there is a node at the prong.  This seems to be alright in terms of the continuity of the wavefunctions, and probability current is certainly conserved.  So unless there is another constraint that I am not seeing, this looks reasonable.


The problem is that this may be a legitimate wavefunction, but not an energy eigenfunction and is therefore not a stationary state.  Forcing this kind of state would be like having a particle in a box wave function, let's say with multiple nodes.  Having the wavefunction oscillate to he left of the node, then zeroing it to the right of the node obeys all the boundary conditions, but, this is not an energy eigenstate.


So, I think I will stick with my original assessment, though now I have thought about this for an additional 10 minutes for a total of 15.  I am not confident in my view because I have not done anything with paper and pencil, just picturing things in my head - a very dangerous activity!


These kinds of discussions do not come out of the graduate students until the point that they are getting ready to graduate.  Only then does a light bulb turn on, which makes them get it.  Once they become very useful and great fun in terms of being intellectually stimulating, they leave for greener pastures.  Then they can start to challenge and stimulate their new advisers.

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.

A pendent necklace and a new insight about self-healing molecules

In a recent post on our research on self healing, I discussed our new theory, which is posted in the Physics Archives (see it here). The paper has been accepted for publication in the Journal of Chemical Physics and will appear soon.

We used lots of data as input to construct the model, which took years to complete. Data that seemed to support one model initially would later be contradicted by additional data. Over time, the model evolved into a coherent picture as more hypotheses were eliminated by experiments. Finally, we had a model that fit the data AND had as its cornerstone the formation of domains of molecules that together, would help a damaged molecule heal.

There is no direct evidence for domain formation, though the behavior of all the experiments to date are consistent with this model, and only this model. Remove the domains and the predicitve power of the theory is lost. The burning question pertains to the nature of the domains. What are they? Are they clumps of molecules or molecules that are somehow stuck to the same polymer chain? What is the nature of the force that keeps the domains together, and how is it that a domain of healthy molecules acts to promote healing in a damaged one?

We may be closer to an answer.

The lab is in a wonderful buzz of activity with lots of new measurements -- always an exciting time. There are bold new hypotheses based on initial data that generalize our model, followed by letdowns after new data or a more detailed calculation proves us wrong. The process is highly stimulating. I can just smell it; something new and wonderful is brewing.

In the midst of all this activity, I found myself sitting at my computer writing my conference paper for SPIE, where I will give a couple of papers in August. I completed writing the introduction and then explained our new model. What next? I needed something new that did not detract from the presentations of my students. So, I drew the molecular structures of the polymer and the molecules, and started to play with them, rotating this one this way and that one here, etc.

In less than a few minutes, I realized again that a molecule could stick to a polymer through what is called hydrogen bond -- an attractive force between a hydrogen molecule and in this case, an oxygen, very much like the forces found between water molecules. This thought had crossed my mind in the past, and is indeed a motivation for a subset of projects. However, having all this jumbled data running around my head made me realize that Shiva, my coauthor on the theory paper, had already determined the three parameters of our model, one of which is the force that binds the molecule to a domain. If the molecules are sticking to the polymer chain through a hydrogen bond, the hydrogen bond energy should have the same value as the corresponding parameter in the model.

This is an excellent example of a model that we built to explain the data is now guiding us in figuring out what is going on.

I got on the internet and searched for hydrogen boding and found a table of numbers. The energy between a hydrogen and oxygen was one of the first values listed, at 0.3 eV. Then I nervously clicked through the directory tree on my computer to find its measured value. As I scrolled to the table with the results, my eyes focused on the value of the lambda parameter -- 0.29 eV with an uncertainty of 0.01. The two matched!

It is not often that things work out this easily, so I considered the next question, and that was how self-healing is mediated by molecules attached to a chain. A polymer with molecules connected by hydrogen bonding looks a lot like a necklace (polymer) with pendents (molecules) thrown on the night dresser as shown in the figure below.The hypothesis that I proposed is as follows. (a) When a molecule absorbs a photon, (b) it breaks into two fragments that are charged. There is evidence from earlier work that charged species are involved. One of the fragments is fixed in place by the polymer and (c-e) the other hops from molecule to molecule along the chain (f) until it finds its mate and recombines.

An alternative explanation is that the attached fragment attracts a small fragment from a neighboring molecule. The neighboring molecule then attracts a fragment from its neighbor, and so on, which propogates down chain like a wave of fans at a stadium until the original damaged piece combines with an adjacent fragment. The more molecules in the domain (i.e number of molecules attached to a polymer chain), the bigger the chance that there is a contiguous path for the fragment to find a mate.

This is indeed an exciting time. In addition to this work, there are other very exciting developments that I will post in the near future. Breakthroughs can be addictive. I can't wait for the next one!

An excuse for UPS late delivery - Derailment

I just realized that a package I had expected to arrive a few days ago had not yet arrived, so I clicked on my UPS tracking number, and here is a screen shot of what I found. The train with my package got derailed. I never saw this excuse before!

Zeroing in on the cause of self healing

As I have mentioned in the past, one of our biggest projects seeks to develop an understanding of the mysterious self healing process following damage to a molecule by a zap of light. Recently, a former graduate and I developed a model of the healing process that hinges on the formation of domains of molecules. Members of these domains are highly cooperative: they accelerate the healing of a damaged molecule in proportion to the size of the group and they prevent their comrades from being damaged. This behavior is as strange from the sociological perspective as it is from the underlying physics. Why do the molecules aggregate and how does their community enhance healing and prevent physical damage?

We have gone out on a limb and made what I believe is a bold assertion; that there are forces between the molecules that cause them to aggregate, and that these same forces are responsible for healing. Such an assertion would be just a wild guess if it were not for lots of data that we find to be consistent with our model. With only three parameters, our data fits the model as a function of temperature, concentration, time, and intensity. The model also makes predictions beyond our present experimental capabilities, so it will gain acceptance only if it holds up to future scrutiny.

When submitting something this interesting (at least to us) that may go past the present paradigms (Shiva got some lifted eyebrows and jaw dropping during an interview talk, which turned to nods of approval after he presented supporting evidence), one always worries that the work will not be understood. There are many examples of Nobel-prizewinning work being rejected by a journal. In our case, the first journal did not even send the paper out to review, claiming that our work was not appropriate. How can a physics paper not be appropriate to a physics journal?

Of course, I have no illusions that this is a Nobel-prizewinning paper, but if the underlying mechanism is found to be new, it could very well end up being a significant achievement for whoever makes this discovery.

Rather than fight the editor, back in mid May, we sent the paper to a second journal of equal quality. Then we waited. I was still concerned that the reviewers may not see the importance of the work. But alas, they accepted it on the first pass, suggesting only minor revisions. And it was also incredibly fast given the nature of our paper. The first reviewer summarizes the paper as follows,

"This interesting manuscript continues the authors' work aimed at discovering the mechanism behind the observation of self-healing of photoluminescence in chromophore doped polymers. The authors have proposed a phenomenological model for their observations that is able to predict aspects of the time, temperature, concentration and intensity dependence. The model focuses on the formation of dye domains in the polymer and studies the dynamics of these..."

Then (s)he goes on,

"While these are interesting results, the manuscript could be more satisfying if the authors did more to understand the physical mechanisms behind the model. Some well-considered speculation on the materials physics in the conclusions would suffice. "

We tried to hold back on speculation, but this review gives us an opportunity to present what we think is happening. Incidentally, the reviewer is right that we need to work more on the mechanisms, which is exactly what we are doing now. We are already getting data that is pointing at the mechanism, but its still too premature to mention.

The second reviewer made no suggestions for revisions and believes that the paper is in good shape in its present form. (S)he writes,

"In this paper authors present a model on photodegradation/self-healing kinetics of dye molecules doped in a polymer matrix. This investigation is an extension of their previous work. Using phenomenological arguments the authors generalize their model. They allow (implicitly) for association of dye molecules which form correlated domains interacting with the polymer matrix. A healing rate is assumed to be proportional to the number of undamaged molecules in a correlated region and a decay rate is proportional to the intensity normalized to the correlation volume. The model proposed by the authors predicts decay and recovery of the population of doped molecules. The results of the theory are successfully tested with experimental data.

"The paper is generally well written and contains several interesting results. I recommend it to be published as it stands..."

The next step will be to determine the physical significance of these parameters. I am excited by the prospects that we may be looking at some very new physics because this process is like no other that I have ever seen. As I sit at my computer bogged down with lots of administrative tasks, new physics is in the air. I hope to be able to get back with pencil and paper to work on the next set of ideas. But first I need to work on some proposals so that we have the resources to do lots of wonderful work in the future. And as penance for writing proposals, I also have some that I need to review. Similarly, I have a pileup of papers to review.

Hopefully in my next post I will report on even more interesting physics. On another project, something very exciting is brewing. Again, new physics! Until then, ...