For a couple months now, we have been struggling with calculations of the nonlinear-optical response of quantum wires. Our idea is to build up complex structures by connecting together pieces of straight wire segments. The problem is that the sum rules appear to have pathologies. But in reality, the problems lie in the way that we idealize the wire.
As I discussed in a previous post, the case of the quantum rotor is a specific example that had been treated rigorously by Stavros Fallieros. I had an idea of how to apply a similar argument to a straight section of wire. The upshot is that along a wire, the sum rules hold. The problem with an idealized one-dimensional wire is that the wave function is by definition confined to the wire, and therefore vanishes outside. By the Heisenberg uncertainty principle, a particle that is confined in that way must have an infinite transverse momentum, implying an infinite energy state.
If these infinite energy states are included, the sum rules are obeyed. I came up with a simple textbook approach that models transverse confinement with a Dirac delta function potential in the limit when the strength of the delta function is infinite. While I had not solved the full problem, I wrote up the concept in a file LaTeX where I wrote out the form of the solutions. Then, I passed the document along to my students for them to do the hard part: evaluating infinite sums of complicated expressions in the limit when various parameters are large and small. Since there are no loopholes in the way the sum rules are derived, I am confident that this approach will work.
The other day, after I emailed this file to the students, we had a spirited debate. They disagreed with my approach and gave all sorts of counterarguments to prove me wring. They constructed special cases that seemed airtight arguments against my approach. But slowly, they became convinced; not because I am the expert, but because my argument is sound. This is one of the most satisfying aspects of the community of science. In the end, reason wins. This time, I may have been vindicated, but I have made enough mistakes in the past to not be overly dejected when I am proven wrong. Being scientists requires us to admit error.
As an update to one of our papers that was initially rejected, in the process of responding to a substantive comment made by one of the reviewers, David Watkins found an intrinsic hyperpolarizability that exceeds unity - an impossibility, according to my theory. Though my theory has been tested over an over again using different computational techniques under a broad range of conditions, I panic when it appears that I might have missed something. David and I sent many emails back and forth on the topic, trying to understand if somehow the sum rules were being violated by the new case under study. To my delight, David found and fixed a couple of bugs in his code, which solved the problem. The results for this new case is now consistent with all our other calculations.
On another front, three students from my nonlinear optics class and I had finished a nice paper on cascading at the beginning of the summer. As I had reported in a previous post, just prior to submitting the paper, we had found a case where the fundamental limits were exceeded. Since then, we have tried all sorts of approaches to reconcile the problem, but to no avail. Nathan, the lead student on the project, believes that cascading is a way to beat the limits. However, based on general principles, I know the limits must hold. And it's not that I want my theory to be true, but, based on general arguments, the cascading results - which are a special case - must agree with the more general theory. If a specific case appears to violate the more general one, it is incumbent upon us to track down the source of the inconsistency. In other words, we have to specifically show how this case falls outside the realm of the theory. At this point, cascading seems to be formulated in a way that makes it a simple subset of the more general theory.
I have been writing much about theory, but our experimental work has been going well. Shiva has built a beautiful temperature-controlled chamber that will allow him to do experiments from temperatures well bellow ambient to over 100 C. Since the temperature-dependence of a measurement provides a window into the energetics of a process, we hope the new experiments will provide us with a clue as to the metastable species involved that usher self heal self healing of a molecule upon photodegradation. In parallel, I am trying to work out a general theory of self-healing based on our past observations. The real test of this theory will be its power to predict the behavior of new observations as better experiments push the envelope of our knowledge. As a sneak preview, the theory includes a recovery process that is akin to stimulated emission, but in the case of dye recovery, has to do with coupling between the guest molecules and phonons in the host polymer.
Prabodh, a new graduate student in our group is specializing in making a large variety of samples so that he can study how the dopant and polymer host affect the healing process. Ben is doing a a series of experiments to optically image the damaged areas to better pin down the population dynamics, and he is building a new experiment that will allow us to determine the absorption spectrum at each point in the damage region. The combination of new samples and new measurements will provide valuable complimentary data that will undoubtedly aid us in unraveling the puzzle of self healing.
Nathan is getting additional data on the photomechanical response that appears to be consistent with our models. While the results are giving us insights into the new class of liquid crystal elastomeric materials, our conclusions appear to be at odds with those of our collaborators, who supplied us with the samples. I am confident that we will eventually reach a consensus because the truth always bubbles to the top. Even if we are proven right, we most likely have only part of the answer. More interesting mysteries are undoubtedly lurking at the next layer of depth.
Xianjun is in the process of calculating the response of Photomechanical Optical Devices (PODs), with the goal of predicting how they will behave when acting in series. This is a highly nonlinear problem, with complex solutions. At this stage, we are still struggling with the relatively simple things, like the response of a single nonlinear etalon. More complex systems will require us to consider more subtle issues and to be clever in our approximations to solving the full problem. In parallel, Xianjun is starting experiments to burn Bgragg gratings in polymer optical fibers with the goal of making and characterizing PODs. Measurements will play an indispensable part in developing our numerical models.
I hve more to write, but our flight to Amsterdam is boarding. This long trip will eventually lead us to Budapest, where I am giving a plenary lecture on self healing and photo mechanical effects. I apologize for any typos that resulted from my haste, and will write about the meeting upon my return.