Coherent Transport in Photonic Lattices: A Survey of Recent Analytic Results

Abstract

Abstract. The analytic specifications of photonic lattices with fractional revival (FR) and perfect state transfer (PST) are reviewed. The approach to their design which is based on orthogonal polynomials is highlighted. A compendium of analytic models with PST is offered. New results on their FR properties are included. The nearest-neighbour approximation is adopted in most of the review; one analytic example with next-to-nearest neighbour interactions is also presented.

Analytic "Newton's cradles" with perfect transfer and fractional revival

“…Note that this choice of parameters means that we have c = a + 1 2 , and in this case, the para-Racah polynomials are actually orthogonal on a single quadratic lattice and reduce to the dual-Hahn polynomials [12]. It is interesting to observe that when c = 1 2 , the chain is "smooth" in a sense, but when c = 1 2 , there is some sort of discontinuity in the middle for the masses and spring constants. When c < 1 2 , the middle masses and spring constants tend to become smaller, whereas when c > 1 2 they tend to become bigger, this is depicted in figure 1.…”

“…Both phenomena are of importance in quantum information. It has been shown how to engineer analytic spin chains or photonic lattices (see [1] for a review and references) which dynamically enact these processes. Their design relies on the theory of orthogonal polynomials in the context of inverse spectral problems [2].…”

A classical model for perfect transfer and fractional revival based on $q$-Racah polynomials

“…Much effort has been put in the design of quantum spin chains exhibiting perfect state transfer (PST) [1,2,3,4], that is chains allowing the end-to-end transfer of a quantum state. The main interest in such quantum wires has to do with the fact that the state transport is realized by the dynamics of the device thereby minimizing the need for external controls and their decoherence effects.…”

Perfect state transfer in a spin chain without mirror symmetry

“…Analytic solutions are always attractive in view of their elegance and because they provide users with closed formulas [20]. Finding systems with specific transport properties that are amenable to exact treatment is hence quite pertinent.…”