Last week paper...

I've decided to take advantage of my after-lunch laziness and write the first entry on the Last Week Paper series...

Last week, the daily diagonalization—hat tip to Carlos for showing me his word to describe the fast reading of a paper and which I will steal from now on—brought up some interesting papers.  In particular, Stefano Longhi's invited paper on "Classical simulation of relativistic quantum mechanics in periodic optical structures" caught my eye as he mentions the photonic analogue of relativistic zitterbewegung in a binary waveguide array. This brought to my mind an old and nice theoretical paper that is very simple to follow and replicate:

Controllable Scattering of a Single Photon inside a One-Dimensional Resonator Waveguide  

L. Zhou, Z. R. Gong, Y.-X. Liu, C. P. Sun, and F. Nori

Physical Review Letters 101, 100501 (2008)

The authors study the scattering of a single-photon wave-packet in a one-dimensional homogeneously-coupled resonator waveguide—constructed with a semi-infinite superconducting transmission line resonators—containing a single, highly-tunable scatterer target—a superconducting charge placed at the zeroth resonator qubit—.

The Hamiltonian of the systems is equivalent with a tight-binding boson model with well-known dispersion relation, Omega_{k} = omega - 2 xi cos(l k), that behaves:

• In the low-energy regime, long wavelengths lambda >> l, the spectrum is quadratic, Omega_{k} = omega - 2 xi + xi k^2.
• At the matching condition, lambda ~ 4 l, the spectrum is linear, Omega_{k} = omega - 2 xi + 2 xi k. 

The crux to calculate the reflection and transmission amplitudes of the scattered photon is finding the stationary states of the non-linear spectra: Omega_{k} = omega - 2 xi cos(l k). Low-energy limit calculations are also derived resulting in a low-energy field theory with a continuous effective Hamiltonian—field operators are continuous.

With their theoretical results and some experimentally-feasible parameters, they show that the scattering process of a single-photon in this system is a total reflection when the incident photon resonates with the qubit; in the off-resonance case, larger couplings between qubit and resonator gives larger reflection amplitudes.

When I first read this paper not two weeks ago, my first question was: What would happen if you have a bi-chromatic coupled resonators array? Well, I found the answer in subsection 2.1 of Stefano's paper:

Classical simulation of relativistic quantum mechanics in periodic
optical structures
S. Longhi
Applied Physics B 104, 453 (2011)

There (subsection 2.1), the author shows that light transport in a two-component optical super-lattice for Bloch waves simulates the temporal dynamics of the relativistic Dirac equation. Basically, the dispersion relation for a two-mode super-lattice is easy to calculate. The two bands supported by the lattice present a narrow gap. Near the edges of the Brillouin zone—maxima of the lower band and minima of the upper band—, the bands mimic the typical hyperbolic energy-momentum dispersion relation for a relativistic massive particle described by Dirac's equation.

In the rest of the article, Longhi talks about other relativistic phenomena that could be simulated classically with photonic lattices; Klein tunneling, vacuum decay and pair production, Dirac oscillator, Kroning-Penney model, and non-Hermitian relativistic wave equations.

If you are interested in coupled cavities arrays both papers are nice for slow consumption that may fuel creativity.