Cusp solitons mediated by a topological nonlinearity - part 2

For an introduction see part 1, which I wrote when our paper was uploaded to arXiv.

After a few rounds of review our paper was published in Chaos last week!

With so much negativity surrounding peer review nowadays, it is important to highlight when it works well. Our original manuscript mentioned a few possible experimental realizations of our new model, without going into specifics - it seemed like a hard problem which could not be solved before Harvey was due to graduate. So we left it for future work.

The referees correctly pointed out that a vague discussion of possible implementations is not very helpful - this hard problem is then delegated to the reader. Could we provide specifics?

We could not at the time of writing the manuscript. But the referee comments inspired (forced?) us to try harder. During our search we stumbled upon a very nice experiment published in Physical Review Letters: Nonlinear Non-Hermitian Skin Effect and Skin Solitons in Temporal Photonic Feedforward Lattices

We realized that the feedforward nonlinearity used in this work could be readily adapted to realize our model. We added the recipe to our manuscript as an appendix, giving strong evidence for the experimental feasibility of observing our results!

Peer review isn't meant to be just a rubber stamp or a hard rejection; it should also be a way to help present the research in the best possible way. While "open science" and "open peer review" are trending topics, anonymity gives referees the safety to ask hard (or annoying) questions. 

Topological Photonics: Limitations and Possibilities

Last week our Perspective article "Limitations and possibilities of topological photonics" was published in Nature Reviews Physics. As the title suggests, we address some overblown claims of topological robustness frequently made in the literature and clarify in which areas topological protection can play a useful role for applications in photonics.

We first thought of writing such an article in July 2023, in response to several papers somehow being published in high impact journals despite their central claims being based on a misunderstanding of the nature of topological protection and robustness in the systems they studied. For example, claims of "topologically enhanced" or "topologically protected" localization are generally unfounded, given that the localization length is generally determined by the width of the band gap, a non-topological quantity.

Another common problem we wanted to address was the frequent use of comparisons between trivial and non-trivial structures to claim various forms of topological "enhancement". Sadly, such claims also frequently appear in top journals. As we discuss in the article, such a comparison ends up being meaningless because trivial and non-trivial structures host modes with differing dimensionality. For example, in 2D structures the edge modes (localized along the 1D boundary of the system) will naturally give a stronger light localization than a trivial 2D structure without any edge states. However, there are many ways to create edge states that do not require complicated topologically non-trivial designs. What matters is whether unidirectional chiral edge states (which are unique to topologically non-trivial systems) offer some advantage compared to non-chiral states, appearing either as trivial edge states or, more simply, as bulk states of a one-dimensional system. This kind of fair comparison is surprisingly rare in the literature - the most prominent example I know of is the 2014 paper "Topologically Robust Transport of Photons in a Synthetic Gauge Field". 

Unfortunately, this methodology was not widely adopted, and there was little progress on the hard problem of demonstrating quantitative performance enhancements of topological designs compared to state-of-the-art non-topological designs; for example, we had to wait until 2023 to see a rigorous comparison between scattering in valley Hall and non-topological photonic crystal waveguides. In this work, the non-topological W1 photonic crystal waveguide had lower scattering losses in the slow light regime.

Promoters of topological photonics may argue that such a comparison is also unfair, given that the W1 photonic crystal waveguide design is the result of years of testing, experimentation, and optimization, whereas the valley Hall design is much newer, with the potential for further optimization. This point brings me to the "possibilities" of topological photonics we discuss in our article: a topologically non-trivial band structure should not be the end of the design process. Rather, topological bands provide a unique starting point for further optimization, for example by guaranteeing the creation of localized modes near the middle of a band gap. Before the advent of topological band theory we did not have a systematic way to do this!

In the next phase of research in topological photonics, the focus will not be on demonstrating ever more exotic topological phenomena in increasingly more complicated setups. Rather, we should be aiming to integrate this new design tool with other approaches such as fine-tuning or inverse design to move from proofs of concept to genuinely better devices. Photonic crystal waveguides and fibers, integrated lasers, and frequency combs are three areas ripe for further breakthroughs, in my opinion. Watch this space for more on these topics!

Cusp solitons mediated by a topological nonlinearity

Harvey just finished what should be the last paper of his PhD studies: Cusp solitons mediated by a topological nonlinearity

Harvey's PhD project studied the intersection between topological data analysis (TDA) techniques and nonlinear and many-body quantum dynamics. His first paper devised a TDA-based pipeline for detecting the emergence of quantum chaos in a periodically-driven nonlinear Kerr cavity. He followed this up with a demonstration of many-body quantum scar detection using topology-based dimensional reduction.

These works, while very nice, were ultimately using TDA to recover known physics. We really want to find examples where TDA can unveil new physics. This is a hard problem. Where to look? And what counts as "new"?

The easier solution for us was to insert TDA "by hand" into a nonlinear model, and see what came out of it.

For our testbed we took the nonlinear Schrodinger equation, frequently used to model nonlinear waves in various platforms. In the usual nonlinear Schrodinger equation, the conserved energy is the Hamiltonian,

$$ H = \int dx \left[ \frac{1}{2} |\partial_x \psi |^2 - \frac{g}{2} |\psi|^4 \right] $$

The second term, responsible for the nonlinear dynamics, can be interpreted as an intensity-dependent potential of depth $\frac{g}{2}|\psi|^2$. We looked at what would happen if we replaced this term with a quantity obtained using TDA. When dealing with one-dimensional functions, such as intensity profiles $|\psi(x)|^2$, TDA frequently uses sublevel set persistent homology, characterizing shape in terms of the persistence of local maxima and minima. We used the total persistence of these features as an energy penalty term, leading to

$$ H^{\prime} = \int dx \left[ \frac{1}{2} |\partial_x \psi|^2 - \alpha \mathrm{sgn}( \partial_x |\psi|^2 ) (\partial_x |\psi|^2) \right]  $$

Deriving the equations of motion, we found that this topological energy penalty gives rise to effective $\delta$ function potentials at the local maxima and minima of intensity, which act to enhance or suppress local maxima, depending on the sign of the nonlinear coefficient $\alpha$. We then studied the resulting nonlinear dynamics, including the focusing of Gaussian and flat-top beams.

The dynamics are very different from the regular nonlinear Schrodinger equation with focusing nonlinearity, where such a flat top beam would quickly break up into a collection of tightly-focused bright solitons. In this case, since the nonlinearity is proportional to the intensity gradient, its influence is mainly limited to the edges of the flat-top beam. 

We also uncovered some interesting connections to the physics of nonlocal nonlinear systems. Specifically, our "topological nonlinearity", when regularized, resembles a weakly nonlocal nonlinearity with a vanishing local part. Such nonlinearity leads to cusp solitons, as was previously studied in the context of plasma physics!

We hope to follow up this study with investigations of similar "topological" nonlinearities and potential experimental realizations. In the present work we speculated that similar nonlinearities may arise in the context of fluid-mediated nonlinearities and lattices undergoing Floquet modulation, but demonstrating such implementations explicitly remains an open problem for us.

Flatbands: then and now

We published a review article on flatband fine-tuning and its photonic applications in Nanophotonics last week! This follows up on our earlier perspective on photonic flatbands published in APL Photonics in 2018.

How has the field changed in 6 years?

In 2018, we identified promising areas for future research where flatbands had not yet been extensively explored yet: coupled resonator lattices, circuit QED, and photonic crystals.

For the case of coupled resonators, the idea of synthetic dimensions (considering coupling in the frequency domain rather than space) has since emerged as a new direction for non-Hermitian and topological photonics, with the ability to fine-tune short- and long-range hoppings to realize flat band lattices using coupled optical fiber loops.

Circuit QED now sees broad interest as a platform for quantum simulation, especially for studying lattices on hyperbolic space. 

Flatband photonic crystals have received a great amount of attention, driven especially by the rise of moire materials which exhibit flat bands at "magic" twist angles. This breakthrough in condensed matter physics inspired the development of theory (see Phys. Rev. Lett. 126, 136101 (2021), Phys. Rev. Lett. 126, 223601 (2021), and Phys. Rev. Research 4, L032031 (2022), for example), with applications to photonic crystal lasers and shaping free electron radiation being actively explored. 

The huge growth of interest in flat bands in photonic crystals and related platforms such as metasurfaces has been quite remarkable. It is driven by the realization that one does not need to carefully control symmetries or suppress long range couplings, guided by simple tight binding models for flat bands, to design them. Rather, a sufficiently complex system supporting parameter fine-tuning is all that you need to realize flat bands! Equipped with this knowledge, our latest review is timely in that it covers novel phenomena that can emerge in fine-tuned flat band systems. 

Observing strongly-coupled Mie polaritons using water droplets

Mie theory, the analytical solution for electromagnetic wave scattering off a spherical particle, provides a powerful approach for understanding scattering spectra in terms of different multipole resonances. While the assumption of spherical symmetry is often merely an approximation, Mie theory can nevertheless give useful insights in more realistic settings such as resonances of cylindrical high refractive index nanopillars.

One setting where spherical scatterers arise quite naturally is in liquids with high surface tension, which promotes the formation of spherical droplets. Remarkably, for the case of water droplets with radii of a few microns, the Mie resonances coincide with the infrared stretching and bending vibrational resonances of the H2O molecule! This leads to strong coupling between electromagnetic and vibrational degrees of freedom leading to the formation of polaritons, as reported in recent work published in Physical Review Letters: Self-Hybridized Vibrational-Mie Polaritons in Water Droplets.

Observing the key signature of strong coupling - Rabi splitting between upper and lower polariton resonances (corresponding to electromagnetic and vibrational oscillations being in or out of phase) - using water droplets is complicated by the non-uniform droplet sizes. Thus, the measured scattering spectrum involved not just a few resonances at specific frequencies, but a distribution of different resonance frequencies dependent on the particles' sizes.

To overcome this, the authors of the study also measured the scattering spectra of droplets of heavy water, where the vibrational modes become red-shifted due to the increased mass of the deuterium atoms. The authors observed that the absorption peaks associated with the strong coupling between vibrational and electromagnetic resonances are also red-shifted.

In addition to applications to the spectra of water droplets in the atmosphere, it will be interesting to explore similar strong coupling phenomena in other high surface tension liquids and applications to polariton chemistry, whereby strong coupling between electromagnetic and molecular degrees of freedom shows promise as a means of controlling rates of chemical reactions.

Transformer quantum states: more than meets the eye?

The large language models that have boomed in power and popularity over the last year are based on a neural network architecture called the transformer. Transformers were originally designed to efficiently learn complicated long range correlations arising in natural language processing and are now being applied to other areas, including many-body quantum physics, where they being used as flexible variational quantum states. 

Interest in neural network quantum states has grown rapidly since 2017, when Carleo and Troyer showed that a neural network architecture called the Restricted Boltzmann Machine could be trained to find the ground state wavefunction of the transverse-field Ising and antiferromagnetic Heisenberg models. One limitation with this original work was the difficulty of computing expectation values of the ground state using this architecture, since the trained network takes as its input a spin configuration and returns the corresponding probability amplitude, meaning that time-consuming Monte-Carlo sampling is needed to evaluate expectation values.

Monte-Carlo sampling can be avoided using different parameterizations of the many-body quantum state. For example, the autoregressive quantum state encodes the many-body wavefunction $\Psi(\boldsymbol{s})$ as a product of conditional probability distributions:

$$\Psi(\boldsymbol{s}) = \prod_{i=1}^N \psi_i(s_1 | s_1,...,s_{i-1}) = \psi_1(s_1) \psi_2(s_2|s_1) \psi_3 (s_3 | s_2 s_1) ...\psi_N(s_N| s_{N-1}...s_2 s_1) $$

One can thereby draw unbiased samples from the many-body ground state by first drawing the first spin, $s_1$, according to the learned probability distribution $\psi_1(s_1)$, followed by $s_2$ according to the conditional probability distribution $\psi_2(s_2 | s_1)$, and so on until a complete spin configuration is obtained. However, there is a conservation of misery in that we avoid Monte Carlo sampling but instead need to learn an exponential number of conditional probability distributions! Luckily, it is empirically observed that a single neural network is able to encode all this information, if we allow it to take as an additional input a hidden vector $h_i$ that encodes information as to the previously-drawn spins. This gives the neural network an autoregressive structure, in that the output from one pass is sequentially fed back into its input.

The autoregressive quantum states were inspired by the autoregressive neural networks developed for natural language processing tasks. The performance and scalability of autoregressive neural networks is limited by the need for sequential processing to generate a single sample, the potential for vanishing gradients making it difficult to train the network, and a bias of the learned probability distribution to the recently-sampled spins. More advanced formulations based on masked convolutional networks are able to generate entire configurations using a single evaluation of the network, but still encounter issues with trainability and encoding distributions exhibiting complex correlations.

Then along came the transformer architecture. The innovation here was the inclusion of multiple parameterized transformations to the input data that can be trained to pick out different features and (long-range) correlations – the multi-head attention. Once the key features are identified by the multi-head attention, a relatively simple feed-forward neural network is sufficient to compute the output probability. The success at this architecture for language modelling is now inspiring many studies on applications to many-body physics.

One line of research is exploring transformer neural networks as a flexible ansatz capable of describing families of many-body quantum systems, exemplified by the paper "Transformer quantum state: A multipurpose model for quantum many-body problems." In this work, the transformer neural network was trained to learn the many-body ground states of a family of Ising models. Thus, it takes as its input model parameters (e.g. the applied magnetic field strength), and then draws samples of ground state spin configurations. The model can also extrapolate to predict properties of ground states not included in the training data, albeit with lower accuracy, particularly when attempting to extrapolate across a phase transition. The network can also be "inverted" to perform a maximum likelihood estimation of the system's parameters given a few spin configurations drawn from its ground state, analogous to shadow tomography of many-body quantum states.

A second line of research is exploring the use of transformers as a means of accurately computing ground state energies from specific strongly-correlated model Hamiltonians, such as the Shastry-Sutherland model, see for example the paper "Transformer Variational Wave Functions for Frustrated Quantum Spin Systems". In this case, an architecture called the vision transformer is trained to learn the complex correlations present in the ground state. The biggest challenge is training the network, which is particularly difficult for complex-valued networks, however recent work has shown that a two-stage architecture that applies a real-valued transformer followed by a complex fully-connected neural network can be trained more easily.

What next for this hot topic? Better training methods or more easily-trainable transformer architectures are needed, since in training data for quantum many-body systems is much harder to come by than web-scraped training data for large language models. Future research on applications of classical transformer neural networks will likely be divided between problem-specific models tailored to solve certain hard many-body problems, and less accurate general purpose "foundation" models which may be useful for generating initial guesses for other more precise iterative methods. Beyond this, quantum and quantum-inspired generalisations of the transformer architecture are also cool directions to watch!

Towards fault-tolerant quantum computing with Rydberg atoms

 I'm a bit late to the party, but finally managed to get a chance to read the paper "Logical quantum processor based on reconfigurable atom arrays" by Harvard, QuEra, and collaborators, which hit the headlines a few weeks ago. My thoughts:

• Sadly many articles covering the paper gloss over the important distinction between error detection and error correction: QuEra's press release, The Harvard Gazette, EurekaAlert, and others. Optics & Photonics News provides more balanced coverage. The impressively high (above break-even) fidelities demonstrated in the paper require post-selection, discarding experimental runs where errors were detected. The post-selection probability is as low as 0.04% for the largest system sizes studied, and will get exponentially smaller for bigger circuits. The bottom line: scaling up to a useful size needs integration of error correction.
• How to integrate error correction? One needs to process the error detection measurements in real-time and then apply correcting gates to the qubits while the circuit is being run. Figure 4 of the paper does demonstrate implementation of measurement-dependent feedforward operations, but not yet integrated with error decoding and correction operations. This seems to be in principle an engineering challenge that can be solved with more hard work.
• Scaling up to more qubits and deeper circuits will require continuous pumping and replenishment of Rydberg atoms. Otherwise, the circuit width will be limited by the finite success probability for trapping each atom, and the depth by the ~10s trapping lifetime.
• The quantum processor architecture, involving separate storage, processing, and readout zones, as well as the ability to execute gates with arbitrary connectivity and in parallel using just a few structured laser beams, looks much more promising for scalability compared to superconducting quantum processors.

There is more discussion over at Shtetl-Optimized.

Physics meets machine learning and AI

Machine learning research of interest to physicists can be broadly divided into two categories: using machine learning tools to solve physics problems, and using ideas from physics to improve machine learning techniques.

An example of the former is the transformer neural networks used in the design of large language models such as ChatGPT. The ability of the transformer neural network architecture to efficiently learn long-ranged correlations in data is also useful for variational methods for finding ground states of strongly-correlated quantum many-body systems. Two papers demonstrating this approach were published in Physical Review B and Physical Review Letters earlier this year.

Popular image generation tools such as Dall-E and Stable Diffusion (which I wrote about previously) are based on time-reversing a diffusion process to generate desired samples from noise. This approach is heavily inspired by techniques from non-equilibrium statistical mechanics published in Physical Review E in 1997.

Another pressing issue in machine learning and AI is how to understand the emergent properties of large language models as their size or training time is scaled up. This is a problem that physicists are well-posed to tackle using techniques from statistical physics, random matrix theory, and the theory of phase transitions, which have recently been applied to shallow neural network models in a few different studies:

Memorizing without overfitting: Bias, variance, and interpolation in overparameterized models

Learning through atypical phase transitions in overparameterized neural networks

Grokking phase transitions in learning local rules with gradient descent

Droplets of Good Representations: Grokking as a First Order Phase Transition in Two Layer Networks

I'm sure we'll see a growing number of theoretical physicists becoming involved in this exciting area of research in the coming years.

From structured light to structured waves

Introductory textbooks on quantum mechanics and electromagnetism typically use plane waves, standing waves formed from a superposition of two counter-propagating plane waves, or twin slit interference to illustrate concepts such as the role of boundary conditions, phase velocity, energy transport, interference, and so on. However, these special cases actually too simple to capture the full breadth of wave physics. A nice example of this is shown in Figure 1 of the review article "Singular Optics: Optical Vortices and Polarization Singularities", which was one of my first introductions to the field:

(a) Twin slit interference results in alternating bright and dark fringes, the latter corresponding to lines of vanishing intensity. (b) In three slit interference the intensity only vanishes at points, which correspond to singularities of the field's phase in (c). Panel (d) shows a close up of a pair of oppositely-charged phase singularities.

In the 2000s and early 2010s, studies of the peculiarities of these kinds of structured wave fields were focused on optics, where the availability of devices such as spatial light modulators made phenomena associated with structured waves conveniently accessible. There is now growing interest in structured waves in other wave systems including electron beams, acoustics, water waves, and condensed matter physics. Recent developments in this direction are summarised in a Journal of Optics article, "Roadmap on structured waves," that was just published today. Thanks to Konstantin Bliokh for the invitation to contribute to this article and the mammoth effort of collating and coordinating the work of the 49 coauthors!

Quantum chemistry with subspace states: the conclusion

 Just over a year ago I wrote about a paper on quantum machine learning using subspace states, which inspired a project we undertook on applications of similar quantum states to variational quantum circuits for quantum chemistry and condensed matter physics. Over the weekend our manuscript was published in Physical Review A!

We were fortunate to have three knowledgeable referees who gave constructive and insightful comments on the original manuscript. We heavily revised the manuscript compared to the original arXiv preprint to not only improve the presentation, but also emphasize the broader applicability of the subspace space approach, specifically the ability to prepare correlated fermionic ansatz states beyond pairwise correlations. Our approach can yield substantially shallower quantum circuits for solving problems where the electron density (number of electrons d / number of orbitals used N) is small, for example when trying to extrapolate finite basis set calculations to the complete basis set limit. This is illustrated in the figure below, taken from the paper:

Estimated two-qubit gate depth per occupied mode d to prepare an N-mode Slater determinant and pairwise-correlated ansatz states using subspace states, compared to existing d-independent and linear in N approaches.

arXiv highlights

Quantum-noise-limited optical neural networks operating at a few quanta per activation

Suitably-trained optical neural networks can still perform classification tasks accurately using low intensity light with a low signal to noise ratio. This suggests that specialized light-based analogue hardware for machine learning may offer a route towards reducing the enormous energy consumption of neural networks!

Dissipative mean-field theory of IBM utility experiment

Another approach towards reproducing the results of IBM's kicked Ising model quantum simulation experiment, this time using mean field theory. The Appendix gives a simple rule of thumb for estimating the quantum volume of specific devices based on their two-qubit gate and readout fidelities and compares some different hardware providers.

Maximally-Localized Exciton Wannier Functions for Solids

Wannier functions - localized states constructed as a superposition of Bloch waves from an energy band of interest - are an important tool of the condensed matter physicists' trade. This work presents a method for constructing maximally-localized Wannier functions for multi-particle states, focusing on applications to excitons (electron-hole pairs).

Tensorized orbitals for computational chemistry

This work presents a tensor network-based compression of the matrix elements that need to be computed and stored when performing quantum chemistry calculations, based on Tensor Cross Interpolation. This is yet another example of how tools from quantum many-body physics can be used to speed up time-consuming computational tasks - no working quantum computer needed!

The localization landscape

Localization of waves due to destructive interference in disordered media - Anderson localization - has been a subject of intense investigation for more than 50 years. The original paper has now been cited more than 15,000 times according to Google Scholar. Being a property of linear, non-interacting Hamiltonians subjected to a random potential, one might think that everything there is to know about this problem would have already been studied to death a long time ago, with current work focusing on understanding peculiarities arising under special circumstances.

Delightfully, this is not the case! Back in February, this preprint caught my attention due to its keywords of many-body localization and persistent homology. Specifically, the authors have used persistent homology to characterize the shape of the "localization landscape" of a many-body Hamiltonian. 

What is the localization landscape?

The localization landscape was first proposed by Filoche and Mayboroda in an article in PNAS which, despite its broad applicability to understanding wave transport in a variety of settings (condensed matter, photonics, acoustics), did not attract as much interest as other arguably more specialized areas such as non-Hermitian systems or topological edge states. In short, the localization landscape u of a Hamiltonian $\mathcal{H}$ satisfying certain properties is the solution of the linear equation

$$\mathcal{H} u = \mathcal{1},$$

where $\mathcal{1}$ is a vector with all elements equal to 1. u corresponds to the steady-state response of the medium to a uniformly-distributed source.

Left: The localization landscape of a disordered two-dimensional wave medium. Right: Five lowest energy eigenstates of the Hamiltonian, which are localized to distinct peaks of the landscape bounded by lines of small $u$ (red). For more details, read the paper!

Remarkably, it can be shown that $u$ bounds the spatial extend of the low energy eigenmodes of $\mathcal{H}$. Thus, instead of solving the eigenvalue problem and plotting its low energy modes individually to find out where they are localized, plotting $u$ alone is enough to find all the effective low-energy valleys in a medium. Moreover, as a solution to a linear system of equations, $u$ is easier to obtain than the eigenstates themselves. A more rigorous discussion (including the properties that must be satisfied by $\mathcal{H}$ can be found in the PNAS article, which is a pleasant and accessible read.

Interest is now growing in localization landscapes, thanks to recent generalizations that can bound the spatial extent of modes residing in the middle of the energy spectrum, eigenvectors of real symmetric matrices and modes of many-body interacting quantum systems. The latter remarkably shows that useful landscape functions are not limited to low-dimensional wave media, but can also bound the spreading of wavefunctions in the high-dimensional Fock space of many-body quantum systems.

A natural question that arises from these recent works is whether the localization landscape might have some potential applications in the context of quantum computing and noisy intermediate-scale quantum (NISQ) processors. For example, finding the ground state of a generic many-body Hamiltonian is a computationally challenging problem, and methods for NISQ devices such as the variational quantum eigensolver may fail to converge due to the presence of vanishing gradients or sub-optimal local minima. 

Can the localization landscape offer an easier, more hardware-efficient way to sample from low-energy solutions of a hard-to-solve Hamiltonian? Watch this space to find out!

Reading the right papers

Students often find it particularly hard to tell which papers are worth an in-depth reading, which can be skimmed, and which are not essential to the current research project. Since this is something that is usually only learned through experience, examples can be helpful for building intuition.

Consider the first paper from my PhD research, Pseudospin and nonlinear conical diffraction in Lieb lattices, published in Physical Review A. With the benefit of hindsight, this turned out to be a Good Paper, with multiple experimental groups exploring some of the ideas in the following years. Why did it have an impact?

The research project didn't start by reading a bunch of papers and getting a new idea. The idea arose from talking to people - experimental collaborators, and one of the eventual co-authors (Omri), who had recently finished his PhD on the theory of wave propagation in graphene-like honeycomb photonic lattices. 

I was asked to see whether any of the ideas in his thesis could be feasibly investigated by our experimental collaborators. Honeycomb lattices being hard to do in their setup at the time, they wanted to know whether similar phenomena might be observable in a square lattice. Similar to how one can remove a period-doubled lattice from the triangular lattice to create a honeycomb lattice, removing sites from an ordinary square lattice yields a face-centred square lattice with intersecting bands. Great!

As is so often the case in research, we were not the first to have this idea, and actually in the preceding few years several groups had been exploring the properties of this lattice, motivated by huge interest in the electronic properties of graphene (Refs. [7,8,10,11,12,13] in the paper). These works were all published in the Physical Review, not "high impact" venues such as Nature / PRL, probably because referees thought it would be difficult to reproduce this model in an experiment. Being background material, an in-depth reading of all these papers was not required - we just needed to know roughly what they did and how they did it to understand how novel our results were.

In these papers we not only found the now commonly-used name for this lattice (the Lieb lattice), but also learned about how its properties were of interest in the context of cold atoms / BECs and electronic properties of materials. Lucky for us, we could not find any papers studying this lattice from the point of view of photonics, meaning that we had something novel! But on the other hand, we clearly couldn't just take these existing results (based on tight binding models) and do exactly the same using a "photonic" tight binding model without our work ending up being merely incremental and forgettable. Therefore we considered a few photonics-specific extensions:

(1) Wave propagation dynamics in the nonlinear regime, translating the analysis in one of Omri's recent papers (Ref. [11]) to the Lieb lattice setting. This one I had to read and re-read in detail to fully understand the analytical and numerical simulation tools used.

(2) Understanding the coupling between the different angular momentum degrees of freedom in our system. This similarly involved an extension of previous results by others for the honeycomb lattice (Ref. [18]) to the Lieb lattice setting. We also had to carefully read and understand this paper.

(3) Photonics-specific simulations not limited to a tight binding approximation and using experimentally-feasible parameters similar to those used in our collaborators' recent work (Ref. [26]).

In summary:

• Talk to experts early on to find out what the real important problems are and whether they have any that you are in a position to solve.
• Once you have an approximate solution or plan of attack, you need to check the literature to understand its importance and relevance to other work. At this stage you will often encounter papers with ideas very similar to yours.
• Identify your niche and expand on the novel points of your work, usually building on a few specific related papers that need to be carefully read and understood.
• It is usually easier to first solve a specific problem a single expert is having, and then figure out how your solution generalizes. The reverse approach - solving a problem in generality before considering specific examples - should only be attempted with extreme caution.

Suppression of modulational instability in valley-Hall waveguides

Posting has become infrequent due to some urgent deadlines and talk preparations over the last few weeks. Lots of great stuff has appeared on arXiv in May which I'm hoping to read and perhaps post about later. 

In the meantime, we also have a preprint out:

Self-steepening-induced stabilization of nonlinear edge waves at photonic valley-Hall interfaces

Several previous works demonstrated instabilities of topological edge states in the presence of weak nonlinearities, both numerically and analytically, often via reduction to a 1D nonlinear Schrodinger equation (NSE). We show here that if you go to stronger intensities, higher-order nonlinear effects (essentially arising from an intensity dependence of the effective Kerr nonlinearity strength) described by a modified nonlinear Schrodinger equation (MNSE) can stabilize the edge states! The phase diagram below nicely summarizes our central result:

I prepared some slides on this and our earlier analyses of nonlinear Dirac models describing topological edge states. The slides including some background material on photonic crystals and topological photonics are available here!