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.

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