Diffusion in infinite and semi-infinite lattices with long-range coupling (Q2901854)

Consider the discrete Schrödinger equation NEWLINE\[NEWLINEi\frac{du_n}{dz}+\sum_{m\neq n} V_{n,m} u_m= 0,NEWLINE\]NEWLINE describing the evolution of an excitation along a one-dimensional infinite or semi-infinite periodic lattice, where \(u_n\) is the complex amplitude of the excitation at site \(n\) and \(z\) the evolution coordinate (e.g., ``time'' in the tight-binding model for electrons, or ``longitudinal distance'' for coupled waveguide arrays in optics). The matrix element \(V_{n,m}\) denotes the coupling between the sites \(n\) and \(m\). The matrix is periodic in space with \(V_{n,m}= V_{m,n}\) depending only on \(|n-m|\). The dynamical evolution of a pulse initially localized at no can be monitored by the mean square displacement NEWLINE\[NEWLINE\langle n^2\rangle= \sum_n(n- n_0)^2|u_n(z)|^2/\sum_n|u_n(z)|^2.NEWLINE\]NEWLINE For the infinite lattice, a closed form expression for \(\langle n^2\rangle\) is given, implying that the propagation is ballistic at all times, with a ``speed'' that depends on the ``smoothness'' of the dispersion relation. For the semi-finite lattice with initial pulse at the edge, an approximate form of \(\langle n^2\rangle\) is obtained, predicting ballistic propagation at long times. Several explicit examples are discussed.