A few posts back, I discussed Monte Carlo studies that shed light on many unexplained observations. For those of you interested in seeing a preprint, it can be found on the physics archives at: http://lanl.arxiv.org/PS_cache/arxiv/pdf/1101/1101.1041v1.pdf. The manuscript is now under review at the Journal of the Optical Society of America B.
I am still excited by our observations because of the breadth of understanding that has resulted. While this paper may not be appreciated by a large number of people, it is certainly on my list of top 5 papers that I have published over my career. Ironically, while my most highly-cited papers report solid science that has been useful to many other researchers, I prefer to judge my work on the degree to which it opens my mind; and, the awe/wonder that it elicits. Ideas that paint the universe with the broadest of brushes are king. This recent work is an intellectual creation that, like a brilliant child, has grown beyond its creators. I marvel at all that it continues to teach us as well as its ability to inspire new lines of research.
The abstract and conclusion says it all:
ABSTRACT: Studies aimed at understanding the global properties of the hyperpolarizabilities have focused on identifying universal properties when the hyperpolarizabilities are at the fundamental limit. These studies have taken two complimentary approaches: (1) Monte Carlo techniques that statistically probe the full parameter space of the Schrodinger Equation using the sum rules as a constraint; and, (2) numerical optimization studies of the first and second hyperpolarizability where models of the scalar and vector potentials are parameterized and the optimized parameters determined, from which universal properties are investigated. Here, we employ an energy spectrum constraint on the Monte Carlo method to bridge the divide between these two approaches. The results suggest an explanation for the origin of the factor of 20-30 gap between the best molecules and the fundamental limits and establishes the basis for the three-level ansatz.
CONCLUSION
Classifying Monte Carlo simulations using an energy spectrum function resolves several long-standing questions. First, our work shows the centrality of energy spacing in determining the intrinsic nonlinear response. While a broad range of transition moments are observed in atoms and molecules, the energy spacing - as characterized by the energy parameter, E, varies little between systems. Indeed, the importance of the energy parameter in attaining larger hyperpolarizabilities has been demonstrated in several experimental studies.[6, 37]
...
Monte Carlo calculation using the energy classification scheme have bridged the divide between Monte Carlo simulations and potential energy optimization studies. The power of the Monte-Carlo technique lies in the fact that all possible Hilbert spaces are probed, leading to very broad and fundamental relationships. Using energy classifications allows the parameter space to be reduced to subsets that describe atoms and molecules. Future refinements may lead to more specific design guidelines for making improved molecules for a variety of applications. The potential for discovering new fundamental science with this approach is of equal importance.
Showing posts with label monte carlo. Show all posts
Showing posts with label monte carlo. Show all posts
Monday, January 17, 2011
Tuesday, December 14, 2010
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.
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.
Tuesday, July 20, 2010
Recent paper accepted
All papers are written with expectations that they will be appreciated by the scientific community. That's why authors are indignant when a manuscript gets rejected by reviewers before it can see the light of day in a scientific journal.
Recently, Shoresh submitted a manuscript to JOSA B (see http://lanl.arxiv.org/PS_cache/arxiv/pdf/1006/1006.1320v2.pdf for a preprint). In this work, he used Monte Carlo calculations, which were first implemented by my son, to let the computer role the dice to randomly determine transition moments and matrix elements of a hypothetical quantum system. To make the results consistent with quantum mechanics, we used a procedure that constrains the choices to be consistent with the sum rules. By rolling the dice millions of times, we can get a feeling for the properties of a much broader range of systems than one could synthesize in the laboratory.
In a paper that we published three years ago, we used this approach to study the hyperpolarizability. In the recent work, we aplied the approach to the second hyperpolarizability. The calculations yielded the same kind of surprising result, which this time around were a tad bit less unexpected, and that is that the Monte Carlo approach seems to give certain properties that are not observed experimentally nor predicted theoretically using standard Hamiltonians. The upshot is that some very exotic systems with a large second hyperpolarizabilities may be lurking out there - yet to be discovered.
Every time I submit a new paper, I fret that my long string of acceptances will come to an end. Sure, I have had manuscripts that got rejected, but eventually they get published once I fix some relatively minor error. W were delighted to have another acceptance. The reviewers' summaries follow this post. The detailed comments are technical in nature and have been omitted. The bottom line is that the paper is slated to be published in the next couple of months. While we are pleased, we are already working on the next two papers, each which extend our work to the next level. The more we learn, the longer term our goals become.
Recently, Shoresh submitted a manuscript to JOSA B (see http://lanl.arxiv.org/PS_cache/arxiv/pdf/1006/1006.1320v2.pdf for a preprint). In this work, he used Monte Carlo calculations, which were first implemented by my son, to let the computer role the dice to randomly determine transition moments and matrix elements of a hypothetical quantum system. To make the results consistent with quantum mechanics, we used a procedure that constrains the choices to be consistent with the sum rules. By rolling the dice millions of times, we can get a feeling for the properties of a much broader range of systems than one could synthesize in the laboratory.
In a paper that we published three years ago, we used this approach to study the hyperpolarizability. In the recent work, we aplied the approach to the second hyperpolarizability. The calculations yielded the same kind of surprising result, which this time around were a tad bit less unexpected, and that is that the Monte Carlo approach seems to give certain properties that are not observed experimentally nor predicted theoretically using standard Hamiltonians. The upshot is that some very exotic systems with a large second hyperpolarizabilities may be lurking out there - yet to be discovered.
Every time I submit a new paper, I fret that my long string of acceptances will come to an end. Sure, I have had manuscripts that got rejected, but eventually they get published once I fix some relatively minor error. W were delighted to have another acceptance. The reviewers' summaries follow this post. The detailed comments are technical in nature and have been omitted. The bottom line is that the paper is slated to be published in the next couple of months. While we are pleased, we are already working on the next two papers, each which extend our work to the next level. The more we learn, the longer term our goals become.
Reviewer comments appear here:
Reviewer 1
This manuscript is closely related to similar work on the first
hyperpolarizability by two of the authors (ref. 13) which sought
to understand the gap between the theoretical maximum
hyperpolarizability and the highest values measured experimentally.
The present paper extends this approach to the second
hyperpolarizability, γ, whose theoretical maximum value was derived
in ref. 2. In theory, the second hyperpolarizability in the zero-
frequency limit depends only upon the energies of the excited
states of the system and the transition dipole moments connecting
these states with each other and with the ground state. (Actually
the usual expression involves both transition dipole moments and
permanent dipole moments, but it can be transformed to one that
involves only transition dipoles as shown in ref. 19.) There are
also fundamental sum rules that relate these quantities. In this
manuscript, the authors use a Monte Carlo method to randomly sample
different combinations of excited state energies and transition
dipole moments, always requiring that they be constrained by the
general sum rules. The goal is to gain some insight into the
physical parameters needed to produce γ values near the theoretical
limit. The largest second hyperpolarizabilities found by this method
do approach the analytically calculated theoretical maximum.
Furthermore, these results indicate that the largest γ values are
found when only three states—the ground state and two excited
states dominate.
The manuscript is clearly written and well motivated. While the
derivations refer to much previous work from the corresponding
author and his co-workers, it is not necessary to have read those
papers to understand the results presented here. The results do
not provide much help to those trying to design real molecules with
large second hyperpolarizabilities. However, they are certainly
interesting and this is a worthwhile addition to the literature of
this field from one of its
most original thinkers.
There are a few typos/misspellings/grammatical errors that do not
interfere with the readability of the paper.
Reviewer 2
This article presents very interesting new results and should
be published as soon as possible provided that some minor
changes/clarifications are addressed...
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