Universal correlations after thermalization in periodic nonlinear systems

Abstract

The evolution of random fields with known statistical properties is relatively straightforward to analyze in the linear regime, but becomes considerably more involved when nonlinearity, or interactions, are dominant. Previous works have shown that statistical physics techniques can be applied to predict the evolution of such systems. Here we study the evolution of random fields in a one-dimensional lattice of optical waveguides in the presence of strong nonlinearities, using the discrete nonlinear Schrödinger equation. Extending the 2009 work by Silberberg et al (Phys. Rev. Lett. 102 233904), we assume input fields with random amplitudes and phases. We derive analytic expressions for the system's statistical properties at thermodynamic equilibrium. Specifically, expressions for the probability density functions of field intensities, of fields' phase differences, and an expression for the field correlations. We express these properties in terms of the moments of the assumed statistical excitations, and verify the results with simulations. Most interestingly, we find that at thermodynamic equilibrium, correlations are formed through the interaction between sites. These exponentially decaying fields' correlations take a universal form that is essentially independent of excitation amplitudes but visibly shrink with increased spread of the exciting amplitudes. Our results are valid not only to nonlinear discrete optical systems, but extend also to the evolution of bosonic atoms in optical lattices in the high-occupancy limit that are governed by the equivalent Gross-Pitaevskii equation.

Equilibrium temperatures of discrete nonlinear systems

“…Broadening the phase window from 0 to 2 will translate the system vertically on the phase diagram along a line from the line up to the line. By maximize system entropy (similar maximization is described in more detail in [37]) we have derived the following PDF expressions (superscript "cd" stands for "cold"): Fig. 2(a) and Fig.…”

“…(i) Temperatures in the strong correlations region.-On and just above the line, the strong field correlations characterizing systems in the cold zone persist [33], [37]. Therefore, for systems placed in this region the kinetic energy term [ 2 ] cannot be neglected and must be included in the formulation of the temperature.…”

“…Therefore, for systems placed in this region the kinetic energy term [ 2 ] cannot be neglected and must be included in the formulation of the temperature. As in the cold zone, both angle-associated entropy , and intensity-associated entropy contribute, and the system's entropy in the strong correlations region is given by the sum: [33], [37]. Expressions for the PDFs in this strong correlation region, and were derived ( [37] Eq.…”