Interaction of arrays of emitters with photons is now a subject of active research. This is driven by a recent emergence of highly coherent artificial emitter platforms, such as those based on cold atoms, superconducting qubits, and semiconductor quantum dots. While the basic physical effects in these systems, for example, Dicke superradiance, have been known for decades, I am not aware of any systematic modern considerations, especially suitable for teaching purposes. I believe that one of the best ways to learn theoretical physics is to solve problems. So, I am compiling a set of problems and solutions describing various cooperative effects in the scattering of light from the resonant structures, mostly focusing on 1D arrays.
These problems elucidate the formation of collective superradiant and subradiant modes, formation of collective polaritonic states, Bragg scattering, and so on. Most of the problems are focused on the classical optics regime. Almost no previous knowledge of quantum mechanics is required. Classical electrodynamics at the undergraduate level should be mostly sufficient.
As of early 2024, the set of problems is being actively updated. I would be very grateful for any ideas for more problems and for finding misprints. For each of the problems posted below there are prepared solutions available on demand by email.
List of problems
Collective modes and Green functions
Resonant light scattering an emitter coupled to the waveguide link to pdf
Interference in light scattering from two emitters link to pdf
Coupled dipole equations for light scattering on an array link to pdf
Dispersion law of polaritons in an array of emitters link to pdf
Dispersion law of polaritons in an array of emitters, chirally coupled to the waveguide link to pdf
Destructive interference for two coupled emitters link to pdf
Collective eigenmodes in Bragg-spaced array of emitters link to pdf
Collective eigenmodes in an array of emitters depending on the period link to pdf
Decay rate of collective subradiant states link to pdf
Decay rate of most subradiant states by Fermi Golden Rule link to pdf
Reciprocity for the transmission coefficients link to pdf
Polariton reflection coefficient from the edge of the array link to pdf
Nonconjugated orthogonality for non-Hermitian Hamiltonian link to pdf
transfer matrices
Transfer matrix via reflection and transmission coefficients link to pdf
Transfer matrix via the reflection and transmission coefficients link to pdf
Reflection and transmission coefficients via the transfer matrix link to pdf
Dispersion law in transfer matrix method link to pdf
Polariton dispersion law in the transfer matrix method link to pdf
Polariton dispersion law in the Bragg structure (numerical) link to pdf
Polariton dispersion law in the Bragg structure (analytical) link to pdf
Polariton dispersion law in effective medium approximation link to pdf
Reflection from multilayered structure (numerical) link to pdf
Reflection from multilayered structure vs period (numerical) link to pdf
Some literature
semiconductor quantum wells
E. L. Ivchenko, A. I. Nesvizhskii, and S. Jorda, Bragg Reflection of Light from Quantum-Well Structures, Phys. Solid State 36, 1156 (1994) pdf
E. L. Ivchenko, Chapter 3 in Optical Spectroscopy of Semiconductor Nanostructures (Alpha Science International, Harrow, UK (2005) pdf
superconducting qubits
A. F. van Loo, A. Fedorov, K. Lalumiere, B. C. Sanders, A. Blais, and A. Wallraff, Photon-Mediated Interactions between Distant Artificial Atoms, Science 342, 1494 (2013) web
atoms
Y.-X. Zhang and K. Mølmer, Theory of Subradiant States of a One-Dimensional Two-Level Atom Chain, Phys. Rev. Lett. 122, 203605 (2019) web
review
A.S. Sheremet, M.I. Petrov, I.V. Iorsh, A.V. Poshakinskiy, and A.N. Poddubny, Waveguide quantum electrodynamics: Collective radiance and photon-photon correlations Rev. Mod. Phys. 95, 015002 (2023) web pdf