Another semester and another new class - general relativity and thermodynamics

When I was an undergraduate at the University of Pennsylvania, I took a class in general relativity (GR). It was taught by the eccentric Professor Jeffrey Cohen, not to be confused with the equally eccentric Professor Michael Cohen.

Michael Cohen had instilled in me a deep appreciation for truly understanding physics. Just as Michael Cohen felt that he would never attain the depth of understanding commanded by his adviser, THE Richard Feynman , I too feel that I will never approach the physical intuition of Michael Cohen. It is fortunate that the singularities that we call great physicists are born with abilities far superior to their contemporaries.

After more than three decades have passed, I recall little from my undergraduate flirtation with GR. However, some of the mathematical formalism of differential forms has taunted me for much of my career. I recall Jeffrey Cohen mentioning a paper on the topic of the properties of a black hole in some complex geometry that took forty pages of derivations in an article that appeared in The Physical Review. Using the trickery of differential forms, he was able to solve the problem in just a few steps.

The trick was to formulate the problem in a coordinate independent way, then to project the results into the coordinate system that reflected the symmetry of the problem. In contrast, the Physical Review paper used the inelegant brute-force approach of picking the coordinate system up front, and then by necessity painstakingly plodding through all the messy mathematics.

Given the complexity of the problems that we work on as a matter of daily routine in our research, I am always looking for simplifying tools. In teaching various classes, my intention is to sneak in a little bit of differential forms to wet the appetites of my acolytes and to teach my old brain some new tricks. Furthermore, the geometric interpretation of the mathematics adds a deeper layer of understanding.

In the upcoming spring semester, I am teaching graduate statistical mechanics for my first time. As usual, preparing for a new course if filled with grand excitement. You can imagine my elation when I realized that a homework assignment in the textbook could be done with ease using differential geometry. Since then, it has been difficult for me to think about anything else.

The problem is a simple one that normally requires a bit of math. The student is to show that the 6N-dimensional volume element in phase space for an N-particle system is invariant under a canonical transformation. To put this into simple English, the problem seeks to show that a transformation of coordinates does not change the nature of the results. Be reformulating the problem so that the volume element is represented as a wedge product of what are called one-forms, the volume element is shown to be the same when the so-called Poisson bracket yields unity -- the requirement of a canonical transformation. Thus, the problem is solved without the need for messy mathematics.

This realization makes me feel like a kid at Christmas. Ironically, tonight is Christmas Eve, the focal point of my family's celebration. My father has made what may be his last trip to Pullman from Philadelphia. He is 94 and still lives on his own, drives a car, and prepares meals for senior citizens at the Ukrainian Cultural Center in Fox Chase, Pennsylvania. Though still vigorous, his body betrays the telltale signs of wear and tear due to old age. Both of my children are home for the holidays, and all the fragrance from the traditional Ukrainian foods simmering on the stove and in the oven permeate the house. As I write this post, my wife is busily making last-minute preparations.

It is fortunate for me that my family values my passion for physics, and allows me to occasionally be a recluse. Just a few minutes ago, my wife called out a query about my whereabouts. I simply answered, "I am excited about something." Though she undoubtedly had some mundane duty for me to perform, she immediately signaled her understanding of my state of mind, and left me alone. I am truly fortunate to be living with someone who shares in my passions.

The intensity and meaningfulness of spirituality that I derive from physics far exceeds all others, including the times in my distant past when I had embraced religion. As my family turns in for the night, I continue to sit at my desk, full of excitement in my new-found understanding, and looking forward to sharing this understanding with my family and my students. It is a truly privileged life that allows me to rekindles the child-like wonder of Christmas on a daily basis.

Corresponding with colleagues

As I have mentioned in previous posts, scientists are members of a large international community. I occasionally get emails from colleagues who have questions about my research or about a source of confusion. I too do not hesitate to write colleagues to ask them questions about topics that I am trying to understand. Being able to call upon a worldwide network of experts is one of the perks of being a scientist.

This morning, I got an email from a colleague who is many years my senior. Though retired, he is still actively pursuing research that is moving him into my areas of expertise. He was particularly interested in the suitability of using three-level models. He had recalled me stating in the past that such models are highly inaccurate, but more recently, saw some of my papers that relied on three-level models. He was also curious about the resurgence of an old classical model of the nonlinear-optical response proposed by Miller, and whether or not such models were meaningful. Below was my response.

It seems that research goes through cycles, and I have already observed at least one period! Let me address the three-level model first. One of the problems is that there are two definitions of a three-level model, and I am guilty of not being careful enough in stressing the distinction in my papers. The three-level model can be either a first approximation or it can be an exact theoretical construct, as I describe below.

Most researchers who apply the three-level model do so in the spirit of a first approximation. At certain wavelengths and for certain molecules, this may be a good approximation. However, many researchers apply this approximation with impunity. They get away with it since almost everybody does it because of its simplicity. In analogy, using the two-level model for beta can be highly inaccurate but it has formed the basis of organic NLO for decades. We have been working on a paper for several years, which shows that a three-level model can predict the linear and NLO response of AF455. In that paper, we show that the absolute TPA spectrum, beta, and linear absorption are all described by a three-level model without any adjustable parameters. Each parameter in the model is separately measured. We also have a physical interpretation of why such a simple model works. This may be a rare case where the three-level model works.

There are many papers that have shown that more states are required to get the correct magnitude of the nonlinear response. On the other end of the spectrum are the quantum calculations, which sometimes use more than 100 states to predict a single number. Given that there are many ways for 100 states to lead to the same number, I also take exception with using such models to gain an understanding of the origin of the NLO response. The bottom line is that the NLO response is a complex phenomenon.

In a paper with my former student Perez-Moreno, we showed that even when on the two-photon resonance in a two-photon absorption experiment, higher excited states can also make significant contributions. This is most likely the source of the comment from Eric.

Then there is the three-level ansatz, which I originally introduced when calculating the fundamental limits: when the nonlinear response of a molecule is at the fundamental limit, only three state contribute. Thus, to be at the limit, there can be no transitions to other states, otherwise, oscillator strength would be sucked away from the dominant states, and the nonlinear response would be suboptimal. That is the topic of my paper in Nonlinear Optics Quantum Optics.

Because of this state of affairs, I think that the signal-to-noise ratio is very low in NLO research. Thus, my research has migrated away from studying specific molecules to trying to understand properties of quantum systems with nonlinearities that are near the fundamental limit. This work has lead to the identification of certain universal properties of a quantum system at the limit, which hints at ways of making better molecules.

Now your main point. The Miller formula is based on a classical oscillator, i.e. Equations 13 and 14 in the paper that you have attached. Thus, it misses all of the intricacies in the SOS expressions, such as resonances at many wavelengths and the presence of continuum states. Therefore I think that its applicability is limited. Any dispersion model with a couple of parameters would do equally well at describing the data. If the goal of the research is to come up with a method to approximate the dispersion of a simple system such as a diatomic molecule, then this approach may be acceptable. But, such work does not lead to a fundamental understanding of the origin of the NLO response at the level that you are seeking.

My interest in the NLO response of simple quantum systems covers air molecules since there are relatively simple to analyze. Let me know more specifically what you have in mind, and perhaps we can work on this together.

With regards to the Physics Reports paper, my activation barrier is in preparing the outline. I get the feeling that the paper will be very different once we actually get around to doing a literature review. But, I have now added this to my near-term to-do list. My new-year's resolution once again is to give you an outline.

I too wish you and your family a Merry Christmas and a prosperous and healthy New Year.

Scientists continue to learn from each other over their lifetimes through the scientific literature, conferences, and correspondence. In the age of the internet, we are connected to each other literally at the speed of light. Compare our times to those of the great scientists of the seventeenth through the nineteenth century, who had to often wait months to get a response to a letter. While there is a downside to technology, such as the wildfire propagation of errors and misinformation, having access to the world from my desk makes me appreciate the age in which I live.

Fellowships and Awards

I was just notified that I have been elected Fellow of SPIE, the International Society for Optics and Photonics. In the past, I was very negative about awards. As a young assistant professor, I got a note from the dean of the College of Sciences that he was planing to nominate me for an award. My response to him is reproduced below.

Dear Lee:

Thanks for thinking about awards that would be appropriate for my research area. While I appreciate your confidence that I am a reasonable candidate, I frankly dislike the concept of an award in the sciences and would rather not be nominated. There are many reasons for my distaste for prizes: they distort the motivation for research; they stifle interaction between researchers who worry too much about getting credit for results; they encourage form and marketing over substance; and they consume resources that would be better spent on science. I therefore ask that I not be nominated for an award.

Thanks again for being supportive of our work.

Best Regards
Mark G. Kuzyk

In my older age, I recognize that it is a great privilege to have an employer who supports my addiction to physics, and that awards reflect well on the institution. I truly appreciate my position at Washington State University, where I am encouraged to feed my passions and to infect others around me.

The Rosetta Stone of nonlinear optics

This morning, my decade-long quest of intense research to develop a deeper understanding of the nonlinear-optical response has taken a giant leap forward. It all started as a calculation in the fall of 1999 to determine the fundamental limits of the nonlinear response of a quantum system, a question that had burned inside my sole since graduate school. And finally, a decade later, it is all starting to make sense. It is rare moments such as these, punctuating the excitement of discovery, that makes the many years of hard work worthwhile.

Over a decade ago, while on sabbatical in the fall semester, I finally had some time to sit peacefully with paper and pencil in an effort to answer that burning question, "Is there a limit to the nonlinear-optical response?" Many people had made hand-waving estimates based on all sorts of assumptions. My goal was to use rigorous calculations without assumptions to get a result that would universally hold for any quantum system.

The precess itself was exhilarating. I had many false starts based on false assumptions and mathematical errors. When I was finally on the right track, the calculation was messy and tedious. As I plodded along, the equations slowly got simpler and simpler, shedding off this term and that. Along the way, I had several terms with infinities, a sure sign of trouble; but, I persevered. As the equations simplified, I noticed with excitement that the infinite terms canceled. Finally, I was left with a simple but beautiful equation. I stared at it with admiration. This was perhaps the first time in my life that I felt I had made a truly fundamental discovery. At that moment, I felt that my life was complete.

However, an interesting result is not always sufficient for a publication. I needed to connect this work with reality. So, I used tabulations of measurements to show that all molecules that had ever been measured fell below my calculated limit. I then submitted my paper to the best physics journal, Physical Review Letters, and waited for what would certainly be accolades from the reviewers. Instead, I got mixed reviews, but in the end, the paper got accepted and published. I had expected that my paper would cause a sensation, but after a couple of nice emails from leaders in the field, it got little notice. Instead, some chemists approached my work with animosity. Who was I to say that there was a limit to what was possible?

At that point, I moved on to other projects, which occupied my time. A couple years later, two developments got me back into the game of investigating the ramifications of the sum rules and fundamental limits. First, I had found an error in my program that I had used to plot the curve representing the fundamental limit. (My theory was correct.) After correcting the plot, I found that the best known molecules fell a factor of 30 short of the fundamental limit. This gap gave researchers a milestone to beat, and even today, researchers that refer to my original papers do so on the basis that it shows that there is room for improvement. The second development was that two quantum chemists wrote a comment on my PRL paper. While I believe that I successfully answered their criticisms in my rebuttal (which also appeared in PRL), the more important consequence was that it got me thinking about new ideas. At the same time, a Canadian group nano-engineered a material that breached the factor-or-thirty gap. In a press release from their university, they made the first reference to The Kuzyk Gap. So, my name got associated with the theory not by academicians but by Madison-Avenue types.

The history of my work has taken many turns. The next big leap resulted from meeting David Watkins at the Wine Bar in Pullman. He was the brother-in-law of the mother of one of my daughter's friends. Over a couple bottles of red wine, it quickly became apparent that David, a mathematician, was an expert in the calculations that I wanted to implement. In fact, he wrote a textbook on the topic. The basic idea was that we would try to make toy models of quantum systems to understand what properties lead to a large nonlinear response. This work led to our proposal that conjugation of modulation (basically, making speed bumps in molecules to trip up the electrons) was the way to optimize the nonlinear response. Later, in work with my collaborators in Belgium and in China, we used this design principle to demonstrate a molecule with a world-record nonlinear-optical response. This work got all sorts of recognition worldwhide.

To put all of this in perspective, my calculations of the limits are very general, and they apply to any quantum system. All of the molecules ever made are but a negligible fraction of the total. The work with real molecules and the calculations using toy models don't even scratch the surface of possibilities. A few years ago, I bought my son a laptop computer with the understanding that he would apply his newly-acquired skills to do some modeling for me.

The idea was simple, yet powerful. He would use Monte Carlo techniques to try to sample the whole universe of possibilities by randomly picking the properties of a quantum system under the constraints of the sum rules. By repeating this process millions of times, he could build a picture of the essential features of a quantum system that leads to a hyperpolarizability at the fundamental limit. This led to a whole set of new results as well as confirmed the validity of my models. The problem with the Monte Carlo approach is that it gives such general results that it is difficult to connect them to real systems.

We started a project more recently to classify the Monte Carlo simulations according to the energy-level spacing of the system. For example, molecules, on average, have an energy spectrum that becomes more dense at higher energies. In an atom, the energy of state n is proportional to the reciprocal of n squared, while in a molecule, it might vary as the reciprocal of n cubed. Being very busy this semester, I had put off writing the paper. But now that I am writing the paper and thinking deeply about the results, I am finding that this approach is making many profound connections with lots of our previous work. I have also found, with great relief, that it appears that a decade ago, I was more clever than I had imagined.

In those calculations, I made one assumption, which we have not been able to prove but appears to be correct. The assumption can be stated as follows: when a quantum system has a hyperpolarizability at the limit, only three states contribute. This assumption was not arbitrary, but based on intuition, which I argued as follows. A two-state system optimizes the polarizability without approximation. The result is based on the simple fact that the effect gets diluted when shared between multiple states. The hyperpolarizability is a much more complex quantity, and such an argument does not obviously hold. In addition, the sum rules - the holey grail of quantum mechanics - demand that at least three states are required. Putting these two facts together made me settle on three-states because dilution effects are minimized while the sum rules are obeyed. This is referred to as a the three-level ansatz. In German, an ansatz is basically a guess. It is common for physicists to make such guesses, then checking if the consequences are consistent with experiment.

In our most recent Monte Carlo simulations, the three-level ansatz is seen to be obeyed in all energy classes. Furthermore, as the energy classes are smoothly varied from decreasing to increasing energy density, the nonlinearities behave in a way that is predicted by our models. What is even more astonishing is that this behavior is observed even for system with more than three-levels. So, results that were calculated for the specific case of molecules with large nonlinearities also seem to hold for systems with 80 states. Furthermore, the present work resolves puzzles that arose in our toy models and sheds light on the reasons underlying the factor of thirty gap between theory and experiments.

It is unusual for one piece of work to resolve so many issues. Ironically, Shoresh did this work many months ago and has been bugging me to work on the manuscript. I few days ago, I felt this to be a solid piece of work that needed to be published before we moved on to the really interesting research. In the process of bringing all the results together for publication, I have experienced a moment of clarity that unifies all of the seemingly sloppy pieces. It was a moment to savor and to share on my blog.

But alas, I must get back to working on the manuscript and preparing for my lectures for next semester. Perhaps when I look back to this moment, I will chuckle at my naivety. The fact that we can look back at simpler times attests to our steady progress, jumping from one wrung to another on the ladder of knowledge and understanding. The calisthenics alone make the process fulfilling, but moments such as this one are rare and precious, deserving of quiet celebration.

Nostalgia - a childhood of looking up to the heavens

I still get a chill down my spine when I see the dark blue sky after sunset, with bright stars, planets, and the moon decorating the view. As a youngster, I would spend hours staring at the sky, wondering what treasured lay hidden just out of view. In my junior year of high school, I finally got a respectable 6" telescope for gazing at the heavens.

The article on “Meteors in the Telescope” by Allan M. MacRobert in the July 2005 issue of Sky and telescope jogged a memory that I may have had taken a photograph of a meteor through my 6” reflector about 30 years prior. This got me frantically looking through piles and piles of family photos that were randomly tossed in several shoe boxes. After an evening of searching (and of course, pausing occasionally to marvel at pictures of me with hair or being awed by the incredible cuteness of my children), I finally found it! I was disappointed to see that the 3” by 4” photo showed signs of heavy abuse.

This had been my very first attempt at taking a photograph through a telescope, and Saturn was my target. It took me quite some time to center the planet in the telescope. Then I faced the frustration of mounting my camera and aligning it with the eyepiece. To make matters worse, the telescope was incredibly wobbly. However, nothing was worse than trying to focus the image through the finder of my 35mm camera. Finally, my brand new 6” reflector from Edmund Scientific was ready to go, so I pulled the trigger release. It took me all evening to get one shot, and I was mighty tired in school the next day from sleep deprivation.

After getting my pictures developed, I found an out-of-focus and overexposed image of the planet. However, I was proud of having caught what looked like a meteor. So, I taped the photo to the refrigerator. The years of re-taping the photo after falling under the fridge many times led to the fingerprints, tape marks, and sticky areas. Years of exposure to sunlight caused the colors to fade into a magenta hue with splashes of purple.

I recall having seen a flash of light in the camera’s finder on taking the shot. This could have been a memory that I conveniently added later upon getting the photo. For posterity, I scanned the legendary photo it into my computer at high resolution and carefully inspected it for clues. Could this have been an artifact?

Upon casual inspection, I believe that I indeed had caught a meteor. The head is clearly round, so it’s not a ghost image of Saturn. Furthermore, the small dim ghost of Saturn just below the image is clear evidence that the telescope jittered during the shot. Similarly, there is a small ghost of the meteor, just below and to the left of it - implying that it was not an artifact added at the film lab. The fact that the meteor’s ghost is displaced to the left implies that the meteor had moved after the ghost image was recorded. Though the image of the meteor looks suspiciously like a comet, I do not recall seeing one at the eyepiece. Given its brightness, I am sure that I would not have missed it. And the ghost image suggests that the object recorded was moving pretty quickly.

So, I believe that in my first attempt at astro-photography through a telescope, I nabbed a meteor in its tracks. I have never seen such a marvelously delicate tail in any meteor photographs (note that I did no processing after scanning the image aside from adjusting the color slightly to match the original print). Has anyone ever seen a similar image? I would be interested in hearing from anyone with similar pictures, or anyone with insights that support or refute my claim.

Almost a decade ago, I got back into astro imaging, though for the last few years, I have been too busy to use any of my telescopes. The advantage of being older is having the capital to buy decent equipment. For reference, I include a more recent shot (from 2004) that I took with a webcam coupled to my telescope. Telescopes have come of long way, not to mention the ubiquitous use of CCD cameras rather than film. I should get out more often!

Aliens?

Last November, 80% of Denver voters rejected Initiative 300 to "adopt an initiated ordinance to require the creation of an extraterrestrial affairs commission to help ensure the health, safety and cultural awareness of Denver residents and visitors in relation to potential encounters or interactions with extraterrestrial intelligent beings or their vehicles"

While on its surface, Jeff Peckman's initiative is not unreasonable (more about that later), a quick perusal of his website (see http://extracampaign.org/) revels that he believes that extraterrestrials routinely visit the earth and that the government and its minions are involved in a conspiracy to keep the public in the dark. Rather than preparing for the possibility of an ET visit, Peckman's intention was to use a mandated extraterrestrial affairs commission to expose his belief in an ET cover-up.

I am often asked by non scientists if I believe in extraterrestrial life. From the scientific viewpoint, my answer is that there is no evidence for ET life. However, the existence of life beyond the earth is a testable hypothesis and therefore fits comfortably within the realm of science.

In the middle of the 20th century, Miller and Urey recreated in a glass vessel what were believed to be the conditions on the younger earth: water, methane ammonia and hydrogen in the presence of electrical sparks to simulate lightening. In a recent (2008) reanalysis of the data reported by Johnson et. al. in Science, these simple experiments showed that 22 amino acids - the basis of life on earth - were formed. Many other experiments since then have shown that amino acids can be formed under a variety of conditions both on the earth and in space. Thus, on the basis of our current knowledge of chemistry, astro-biology and extrasolar planetary systems, the existence of life is not only plausible but highly likely.

While there may be many worlds out there that have simple life forms, only a tiny fraction of them would evolve intelligent life forms. Even so, the numbers of stellar systems are astronomically large, so it is likely that alien intelligent life forms exist.

Some of my friends might argue that a belief in alien life forms is akin to a belief in God. Not so. I would argue that these two beliefs are significantly distinct. My arguments about the plausibility of ET life were based on our knowledge of terrestrial life, general science, and simple extrapolation. However, there is no scientific evidence for God. In fact, the Judeo-Christian God prohibits humans from putting Him to the test. On many levels, God can only exist through belief not through the scientific method. This does not imply that God cannot exist, only that there is no rational method for proof.

Given what we know, it is unlikely that we are alone in the universe.