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.