Universal Correlations in a Nonlinear Periodic 1D System

When a periodic 1D system described by a tight-binding model is uniformly initialized with equal amplitudes at all sites, yet with completely random phases, it evolves into a thermal distribution with no spatial correlations. However, when the system is nonlinear, correlations are spontaneously formed. We find that for strong nonlinearities, the intensity histograms approach a narrow Gaussian distributed around their mean and phase correlations are formed between neighboring sites. Sites tend to be out of phase for a positive nonlinearity and in phase for a negative one. Most impressively, the field correlation takes a universal shape independent of parameters. These results are relevant to bosonic gas in 1D optical lattices as well as to nonlinear optical waveguide arrays, which are used to demonstrate experimentally some of the features of this equilibrium state.

Figure 1

Experimental measurement of output light intensities from a waveguide array, when input fields with equal amplitudes, yet random phases, are coupled to several adjacent waveguides. The measured intensity values of many random-phase realizations are shown as histograms for linear propagation (blue squares) and high-intensity, nonlinear propagation (red circles). The output intensity distribution is exponential-like in the linear propagation and becomes narrow, Gaussian-like in the nonlinear case. Inset: Two sets of measurements for an input with the same phase realization after linear (top panel) and nonlinear (bottom panel) propagation, showing that the intensity fluctuations are reduced in the nonlinear case.

Figure 2

Simulation results of the DNLSE with uniform intensities and random phases. (a) Intensity histogram, (b) field correlation, and (c) phase-difference histogram for a linear system ($\Gamma =0$), exhibiting exponential intensity distribution and uncorrelated fields and phases. (d) Intensity histograms for $\Gamma =\pm 20$ (blue circles) and $\Gamma =\pm 200$ (red circles). The distributions are narrower at higher nonlinearities, yet independent of the sign of $\Gamma$. (e) The universal field correlation function is identical for both nonlinear values, but depends on their sign: blue for positive (focusing, $\Gamma >0$), red for negative nonlinearities. (f) Phase-difference histograms also approach a universal distribution, concentrating around $\pi$ for positive nonlinearity and 0 for the negative case. The theoretical lines in (d)–(f) are predictions of Eqs. (7, 8).

Figure 3

Maps of the potential $V_n$ induced by the fluctuating fields for (a) $\Gamma =20$ and (b) $\Gamma =200$ and the corresponding correlation maps $Y(k,\delta \zeta )$ (c), (d). The maps show a section of 40 waveguides (vertically) and a propagation of $\zeta =1$. Note the faster dynamics for the higher nonlinearity, evident also in the correlation map. For easy viewing, the potentials are normalized to their peaks, although the values in (b) are about 3 times higher than in (a).

Figure 4

The diffusion of a weak probe in the potential induced by the high-intensity fluctuating field with $\Gamma =20$ and $\Gamma =200$. (a) the averaged probe width as a function of ${\zeta }^{1/2}$. (b) The averaged probe profile at $\zeta =2000$.