A correspondence with a colleague on our nonlinear topology/geometry project

We are not resting on our laurels, but rather are continuing to work on new ideas and extending our work to more general cases.  In the process, we call get confused about the physics of the system, and try to find ways to picture what is going on that gives insights into how a system may behave under various conditions.

In working on this project, I realize that I spend lots of time writing emails to colleagues and students about various aspects of the work.  An email may come in at 10:30 at night because my collaborator is hot to figure something out and wants insights.  I too write emails at weird hours asking students for more details on something they observed in an effort to test my newest crazy ideas.  This is not a line of work for those who want to sleep soundly.  Being excited about physics is the best stimulant.

Anyway, today I took a two-hour break to play floor hockey and found a series of emails from a collaborator who is very excited (as am I) on the new direction of our work and the new physics that is implied.   Since we are moving into new territories, many things are not well mapped out, so we have to navigate partially by instinct.  I occasionally like to post excerpts from emails to give readers a sense of the kinds of exchanges that we have in the process of working on a project.  Here is my response to an exchange about a new quantum graph that we are working on, as shown above.

An excerpt from my email response (edited for typos and names removed):

First, I think there is a big difference in taking the limit as the prong length goes to zero and actually having a zero prong length.  In the former, I think that you get a broken vertex, because as the wire gets shorter, the end of the wire is getting closer to the vertex, so indeed the wavefunction on the prong must get vanishing small as the prong length goes to zero.  Then, this will look identically to a break in the loop.  Of course, when there is no prong, you get the traveling wave solutions.  So it is interesting that the limiting case yields a very different result than the case without the prong - again a statement about topology.  As long as the prong is there, no matter how short, the topology is of a loop-star.  Get rid of a prong, and then it is just a loop!


I am a little confused about what you are saying with regards to the ground state energy.  So, let me just make a point that may not be related at all to what you are saying.  If I have a prong with zero wavefunction, the loop cannot have a constant wavefunction, i.e. the wavefunction with n=0.  By continuity, the amplitude of the constant wavefunction will be zero, so there is no particle in the system.  So, I believe that in a loop, you have the zero-energy wavefunction but as long as a prong is there, you will not have a zero-energy wave function.


However, you bring up an interesting possibility of a wavefunction in which there is zero wavefunction in the prong and a standing wave in the loop in which there is a node at the prong.  This seems to be alright in terms of the continuity of the wavefunctions, and probability current is certainly conserved.  So unless there is another constraint that I am not seeing, this looks reasonable.


The problem is that this may be a legitimate wavefunction, but not an energy eigenfunction and is therefore not a stationary state.  Forcing this kind of state would be like having a particle in a box wave function, let's say with multiple nodes.  Having the wavefunction oscillate to he left of the node, then zeroing it to the right of the node obeys all the boundary conditions, but, this is not an energy eigenstate.


So, I think I will stick with my original assessment, though now I have thought about this for an additional 10 minutes for a total of 15.  I am not confident in my view because I have not done anything with paper and pencil, just picturing things in my head - a very dangerous activity!


These kinds of discussions do not come out of the graduate students until the point that they are getting ready to graduate.  Only then does a light bulb turn on, which makes them get it.  Once they become very useful and great fun in terms of being intellectually stimulating, they leave for greener pastures.  Then they can start to challenge and stimulate their new advisers.

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.