# Pseudo-Magnetism in Photonics: From a Mathematical Theory to Experiment

While magnetic fields deflect the path of charged particles, such as electrons, they do not directly influence the path of light. In 2021, **Professor Michael I. Weinstein ** (APAM - Applied Mathematics and Mathematics Dept.), in a collaboration with **Professor Mikael C. Rechtsman **and his graduate student **Jonathan Guglielmon** in the Physics Department at Penn State University, GR&W, published a mathematical theory [3] that demonstrated how to engineer a 2D non-magnetic photonic crystal in which photons of light would move in a manner analogous to the motion of electrons under the influence of a magnetic field.

GR&W took motivation from the analogous effect known in condensed matter physics for the 2D material *graphene*, a one atom thick arrangement of carbon atoms which has the symmetries of a honeycomb tiling of the plane. After its discovery [4], it was observed that appropriately strained graphene induces electrons to behave as though they flowed in the presence of an out-of-plane magnetic field exhibiting Landau level electronic spectra with high density of electronic states.

*Is such an effect possible for photons as well?* It turns out that the *tight binding *mathematical model (a discrete low energy approximation) on which the condensed matter predictions for graphene rest, does not apply to 2D and 3D photonic crystals. A continuum theory applicable to Maxwell’s equations of electromagnetism was needed, and this is what GR&W put forward in [3] for general scalar wave equations. Its extension to incorporate the vectorial effects of Maxwell equations was carried out as part of the experimental work in [2]. The full vectorial theory is needed and gives an excellent match with experiment.

In particular, GR&W showed that a non-uniformly deformed (strained) photonic (or other wave-propagating) continuous medium with honeycomb spatial symmetries, gives rise to effective magnetic and effective electric fields, which would influence the propagation of light waves. The dynamics of spatially- and frequency-concentrated “wave-packets” of light is described by a system of Dirac equations with the effective field potentials. When a strained pattern is chosen to produce a constant perpendicular effective magnetic field, *photonic Landau levels* are induced in the structure. These are infinitely degenerate (in practice, very highly degenerate) states of light that can be used to enhance light-matter interactions. They could potentially be used to more efficiently generate quantum light in the form of entangled pairs of photons, or increase the emission from quantum emitters such as quantum dots or crystalline defect centers. Further, this work suggests strategies for inducing topological phenomena in photonic and other wave systems, which are analogous to those in quantum materials such as topological insulators.

GR&W’s predictions [3] were recently experimentally observed by the Penn State group of M.C. Rechtsman. An excellent match with theory is achieved with no free free fitting parameters! These results are reported on in an article which has very recently appeared in *Nature Photonics*; see Barsukova et. al., [2]. TThe results of independent experiments, also confirming the theory were published by the group of E. Verhagen in the Netherlands [1].

*Funding for this research was provided by the Office of Naval Research Multidisciplinary University Research Initiatives program, the Air Force Office of Scientific Research Multidisciplinary University Research Initiatives program, the U.S. National Science Foundation, the Kaufman Foundation, the Packard Foundation and a Simons Foundation Math + X Investigator Award.*

**References**

[1] R. Barczyk, L. Kuipers, and E. Verhagen. “Observation of Landau levels and chiral edge states in photonic crystals through pseudomagnetic fields induced by synthetic strain”. *Nature Photonics *(2024). https://doi.org/10.1038/s41566-024-01412-3

[2] M. Barsukova, F. Grisé, Z. Zhang, S. Vaidya, J. Guglielmon, M. I. Weinstein, L. He, B. Zhen, R. McEntaffer, and M. C. Rechtsman. “Direct Observation of Landau Levels in Silicon Photonic Crystals”.*Nature Photonics *(2024). https://doi.org/10.1038/s41566-024-01425-y

[3] J. Guglielmon, M. C. Rechtsman, and M. I. Weinstein. “Landau levels in strained two-dimensional photonic crystals”. *Phys. Rev. A* 103 (2021), p. 013505. https://doi.org/10.1103/PhysRevA.103.013505

[4] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov. “Two-dimensional gas of massless Dirac fermions in graphene”. *Nature *438 (2005), pp. 197–200. https://doi.org/10.1038/nature04233