Exact dynamics of finite Glauber-Fock lattices

Phys. Rev. A 84, 053845 (2011)

This is one of those papers that just pop out with a life of their own. Last year, there was a theoretical paper—one of the authors was Hector, my PhD supervisor—where they described the so-called Glauber-Fock lattice—a semi-infinite one-dimensional, coupled waveguide array with couplings varying as the square root of the position—. There, they showed that one could give an analytical close form evolution for classical fields by creatively mapping each waveguide to a number state and playing with the resulting algebra. Later, this year they published a nice experimental paper on the topic.

A few months ago—for reasons that now seem alien to me but I'm sure I will try it again and again in the future—I thought that studying the finite version of the system could be a good way to make new friends; I couldn't be further from the truth.

Anyway, I got fun with the math and the analysis of this system and in the end I collected my marbles and sent it to PRA. I  got the best reviewer I have ever had, he/she helped me a lot in clarifying the exposition and results.  Also the comments from Changsuk and Rafa help me a lot to get the paper to its latter form.

Experimental systems: I have my heart on photonic waveguides, they have already been built and tested by the Jenna group. It seems like Robert Keil and Alexander Szameit from Jena can build any configuration that one can think about.

Major result: By using the method of minors, the polynomial related to the eigenvalue problem is shown to be the N-th Hermite polynomial—where N is the size of the finite lattice. Once the spectra is found, the j-th component of the k-th eigenvectors is easily calculated as the j-th Hermite polynomial evaluated at the k-th eigenvalue.

With the analytical solution at hand, it is possible to calculate whatever you want. 

The Physics: The evolution given by the aforementioned result is such that the system acts like an almost perfect mirror for input close to the zeroth waveguide. This is shown in the paper explicitly for single photon single- or multi-waveguide input as well as two-photon single- or multi-waveguide input—the graphics are damn big, sorry for that—.

Curious things I learned:

• If one is patient, it is possible to write an analytical closed form—a radical form—for all the roots of the first ten Hermite polynomials. If one gets Mathematica, one can get the roots of higher order polynomials in radical form.
• For some reason that I still don't understand but that I documented extensively numerically, the last component of  all the normalized eigenvectors is always the same.
• PRA copyeditors don't like passive voice, they changed all my "(...) was shown." Sorry, I promise I will stop using it.

Non-Academic things I learned:

• My mother was right when she told me: "Fool me once, shame on you. Fool me twice, shame on me."
• I'm still and idealistic fool that believes in people and I will keep being one. Great people are by far a majority in academy.

Well, I hope you can find some use for the results in the paper and, as always, drop me a line, I am always glad to discuss or try to help whenever possible. Citations are welcome! 

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