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.

The response function and the underlying truth

At the age of seven or eight, I recall experiencing a recurring feeling of vulnerability. Being the only child of immigrants, with all of my aunts, uncles, and grandparents overseas - most of them trapped behind the iron curtain of the powerful Soviet Union of the early 1960s - I could sense my precarious existence. The thought of being orphaned was often on my mind.

This angst lead me through various imagined scenarios. What if both my parents had died, but were surreptitiously replaced by artificial beings who were programed to react to all events exactly as would my real parents? If I hurt myself, they would hug me with facial expressions of sympathy; and, would provide the same level of emotional and material comforts as my real parents. I concluded that I would probably be just as happy. If there were no test that could betray their artificial nature, how would I know otherwise?

The scientific method works in much the same way. Controlled experiments are designed to observe the reactions of a system under various conditions to deduce its "essence." The observed behavior under the influence of an external stimulus is called a response function, and provides a window into the inner workings of everything. The characterization of my parents in terms of the sum total of all possible responses to all my possible actions defined their being. The old expression "we are what we do" is not far from saying that a full characterization of our response function defines us. We are all judged by our works - be it our compassion, diligence, intelligence, empathy, selflessness, greed, selfishness, etc.

Thermodynamics serves as an instructive example from physics. The heat capacity of a material is a response function that quantifies the change in temperature (response) per unit of heat supplied (input). Other examples are a material's compressibility; in optics, the nonlinear susceptibility; and in material structure determinations, the scattering cross-section. All of these quantities are measured by poking things and observing how they respond, from which we deduce the underlying properties.

Such simple experiments have lead to some of the deepest concepts developed by the human mind. For example, statistical mechanics seeks to explain all of thermodynamics in terms of the motions of a large number of tiny particles. Since materials responded to all experiments as predicted under the assumption of the existence of small and undetectable particles, early twentieth century scientists begrudgingly accepted the notion that for all practical purposes, materials were made of such stuff.

Whether the result of great intellect or luck, the particle picture turned out to be accurate. New experiments were designed that more directly confirmed the existence of molecules, atoms, and subatomic particles. For example, bubble chambers yield tracks of the paths of elementary particles -- from which their masses and charges can be determined. By the end of the twentieth century, the scanning tunneling microscope (STM) was developed, making it possible to not only directly image atoms and molecules, but atoms could be picked up and placed on a smooth surface at will. One early demonstration of this technique spelled out the letters IBM by intentionally arranging a group of atoms on a flat crystal surface.

This approach we call science, which views phenomena from new angles, leads to an ever more precise characterization of our world, leading us closer to what I would call the truth. The picture becomes more and more in focus as we become more clever in our prodding and poking.

Due to its complexity, studies of the human mind until modern times had been limited to measuring its response function in much the same way as I defined my parents. Breakthroughs in physics, biochemistry and new technologies now allow us to probe the brain directly with noninvasive methods. While an individual is responding to old-fashioned stimuli such as hunger, lust, satisfaction, and deep thought, the neural activity of the brain can be mapped. The STM brought us pictures of atoms while brain-scanning technology has brought us images of thought and emotion.

One can rightly argue that seeing an image of the brain while the subject is having an experience in response to stimuli does not imply that thoughts and experience reside in the complex firing of neurons. Rather, it could be merely a byproduct of thought. A definitive test would be the deliberate creation of thought, emotion, or sensation using a direct stimulus of the brain. Indeed, researchers have been able to reproduce not only simple sensations, but have been able to make subjects feel spirituality, unity with the universe, and a divine presence.

As experimental technology becomes more sophisticated, and researchers more clever, we are finding direct evidence that those qualities that define our humanness and spirituality are all the result of material processes. Ironically, my father is almost 95 years old, so I am yet to be orphaned. The fact that he is defined by his response function and I by mine does not detract from the meaning derived from human interactions. This knowledge enriches my life with the understanding of how the material world is interconnected and I value the incredible privilege of being a locus of material that has arranged itself in a way to give me consciousness and loved ones, friends and collegues with whom I can share my experiences.

The Physics of Limits

I spend lots of time thinking - a trance-like state where ideas flow. The feeling is similar when reading about physics, solving problems, doing calculations or randomly following the meanderings of the mind. It brings far greater pleasures than drinking or socializing, though it does not replace the need for human interactions. Exchanging ideas with others is just as fulfilling.

Undoubtedly, the naturally occurring peptide substances in the brain that act as neurotransmitters and appear in abundance while thinking are responsible for the euphoria that is associated with thought. This is augmented by the great satisfaction of new insights that are gained in the process. Strenuous physical activity releases natural endorphins that bring a similar feeling of pleasure. Perhaps these chemical triggers fuel my passion/addiction for physics and ice hockey.

One of our research areas is in fundamental limits of the nonlinear susceptibility. The limits that we calculate follow from the laws of physics. A while back, I got intrigued by the idea that the laws of physics might be derivable from a formulation in terms of limits, or more precisely, constraints. As usual, the idea is not fully original.

In a way, some of the laws of physics are already formulated in this way. For example, the entropy cannot decrease; so, there is a lower limit for the change in entropy. Then there is the upper limit of the speed of light, a crucial constraint from which special relativity follows. The uncertainty principle, which does not allow certain pairs of properties to be simultaneously measured with infinite precision, is yet another constraint. And, that fact that energy is a constant is a very stringent limit; it cannot increase or decrease.

Since physicists have been thinking about these problems for a long time, there are probably few new ideas that would provide a novel approach to physics. However, I still have this gnawing feeling that there is something interesting lurking behind this approach. For example, since the sum rules are a direct consequence of the Schrodinger equation, then perhaps under a constraint, the sum rules could be used to generate general physical principles. Such a formalizing might have unexpected consequences that could lead to the prediction of new and unexpected phenomena.

Since I have been busy with other things, I have not had time to develop this idea, and probably never will. Instead, I will occasionally tinker with paper and pencil to get my neurotransmitters flowing without delusions of success with an occasional vigorous game of hockey to add a tad of spice.

Me as a reviewer and a complainer about modern-day publication practices

In the past, I have complained about reviewers who have evaluated my papers. However, it is a time-consuming job with almost no rewards, so I do appreciate their efforts. It is a service that we are all expected to provide. Given the time others have spent on my papers, I feel obligated to return the favor.

The most rewarding reviews are those from which I learn. I spent this morning reviewing a paper by some very distinguished scientists in my field. While I admit to the possibility that I may be wrong, I believe that there are serious issues with their paper, which will require attention before it is suitable for publication.

As usual, one activity leads my mind jumping around to other thoughts. This paper is an example of one in a series that is trying to simultaneously correct errors in the literature while introducing new science. This got me thinking again about the curse of information overload.

I am concerned that modern-day science, with the huge number of venues available for disseminating research results, is producing too much information along with lots of junk. The signal to noise levels are dropping while the whole system is bursting at the seems. People are becoming more specialized and less aware of other work. Since researchers are being judged on numbers of publications and citations, they overload existing top journals with so many papers that editors often cut good papers based on arbitrary guidelines. More second-tier journals are popping up to meet the growing demands by authors.

It's getting difficult for me to find useful information in the literature. For example, when searching electronically using very specific keywords, I get too many irrelevant hits that take forever to sort. It is also frustrating to have done what I believe to be great work in the past, only for it to be ignored for 20 years. Even more annoying is seeing the same identical topic of my research appearing many years later in Nature or Physical Review Letters with no citations to my papers. It is even more irksome when the modern work is but a subset of my original research, yet gets lots of recognition.

Sometimes, I send these modern-day authors reprints of my older papers. Some will respond apologetically pleading ignorance of my research, then continue not to cite my work. Others ignore my emails. These are indicators of a system that is not serving its purpose in producing research that serves society.

My review reminded me of the past era of more responsibility in publishing. Perhaps I view the past with unfounded fondness. However, I can atest to the fact that the authors of the manuscript that I have just reviewed are interested in the seeking truth. I therefore feel confident that they will carefully consider my comments and will only move forward with a revised manuscript if they are certain that they can make a real contribution to the field.

Below, I include a copy of my review for all to read. I, of course, will not reveal the identity of the authors, nor the journal to which this paper has been submitted. I take the risk of being exposed as the reviewer, but, I am sure that they will have already guessed my identity based on the flavor of my review; and, I will not deny being the reviewer if asked. Having gotten this off my chest, I need to get back to writing a proposal and grading homework. Perhaps I can then squeeze in a few moments to think about physics, and achieve the bliss that accompanies such thoughts.

And now, finally, the review:

The authors do some combinatorial wizardry to determine the coefficients of the various orders of the nonlinear birefringence. I am not willing to check all of the math, but from what I have checked, I trust that this is done correctly. However, I have a serious concern that may invalidate the approach, as I describe below.

The fundamental property of a material is its nonlinear susceptibility, not the nonlinear birefringence. The nonlinear susceptibility is what governs the physics of light/light interactions while the nonlinear birefringence is the quantity measured. They are related through the constitutive relationship D = epsilon E = E + 4 pi P. In the process of relating the two, one takes a square root of a power series in the field with the susceptibilities as coefficients. The crux of what I believe to be the fallacy of this paper is that n_m is related only to chi^(n+1). In the process of doing the expansion of the square root, one gets cross terms that are products of various lower-orders of the nonlinear susceptibilities that coincidentally may look like expressions that one sees in cascading calculations. The authors have in effect only expanded the square root to the first term. I believe that if the calculations are done properly, then it may be impossible to define unique constants of proportionality. However, under certain approximations, it may be able to define unique constants in the spirit of the authors' original intention.

A second problem along these lines is the neglect of the imaginary parts of the susceptibility. While experiments are off-resonance, there is always a small imaginary part. The cross-terms that I mention above can include products of imaginary parts that give a real response. Since it is possible that effects due to the imaginary part may get large for higher-order susceptibilities, they also need to be considered in the calculation. The fact that in practice, higher-order susceptibilities are by necessity more resonantly enhanced is a problem with applying this theory to real experiments at ultra-high intensities, and should be mentioned.

Nonlinear dichroism can also lead to polarization rotation, a common way of measuring the nonlinear birefringence. This contribution might also be large in practical experiments. While a good experimentalist would take this into account, I am concerned that a blind application of your results could lead to the unintended consequence of more junk in the literature. Thus, if not accounted for specifically, I would suggest adding at least a cautionary note.

I believe that the above issue needs to be carefully addressed before the manuscript is reconsidered for publication.

As a more minor point, In the introduction, the authors mention that the proportionality constant depends on the number of eigenmodes. I usually associate this factor with the number of degenerate frequencies. Is it true that if the frequencies are the same in a pump-probe geometry one gets the factor of 2/3? I thought that if the prorogation directions are different in the non-collinear polarization geometry, the effect shows up in the tensor properties of the susceptibility. That is, if the polarizations are different, then one is probing that particular component of the nonlinear susceptibility tensor. In any case, the meaning of eigenmode needs to be clarified or the sentence needs to be reworded. The use of the expression "eigenmode" in the rest of the paper may also need to be reconsidered.

I find the issue of cross terms to be a major one. If the authors choose to argue that the results hold in some limiting cases, that might diminish the relevance of the paper in loss of generality. If the calculation includes the terms that I believe to be missing, the coefficients will no longer be well defined. In either case, I believe that this paper may need major revision before it is suitable for publication. If I am wrong in my assessment, then the paper may be suitable for publication after minor revisions. In this scenario, it would be useful if the authors provided a more detailed explanation of the relationship between the nonlinear birefringence and the nonlinear susceptibility.