Berezinskii-Kosterlitz-Thouless crossover in a photonic lattice

We show that a periodic two-dimensional (2D) photonic lattice with Kerr nonlinearity exhibits a Berezinskii-Kosterlitz-Thouless (BKT) crossover associated with a vortex-unbinding transition. We find that averaging over random initial conditions is equivalent to Boltzmann thermal averaging with the discrete nonlinear Schr$\stackrel{\mathrm{··}}{\mathrm{o}}$dinger Hamiltonian. By controlling the initial randomness we can continuously vary the effective temperature. Since this Hamiltonian is in the 2D $\mathit{XY}$ universality class, a BKT transition ensues. We verify this prediction using experimentally accessible observables and find good agreement between theory and simulations. This opens the possibility of experimental access to interesting phase transitions known in condensed matter using nonlinear optics.

Figure 1

Linear regression goodness-of-fit parameter ($R^2$) vs normalized temperature (both without units). In order to distinguish between the two types of correlation we used a linear fit of $\ln [C(r)]$ vs $\ln (r)$ where a good fit indicates power-law decay (blue squares), and a linear fit of $\ln [C(r)]$ vs $r$ where a good fit indicates exponential decay (red triangles). The transition occurs at $T=0.95\pm 0.04$ in accordance with the known transition point $T_c=0.89$ [18]. Inset (a) is 2D correlation image at $T<T_c$. Inset (b) is the 2D correlation image at $T>T_c$ (in both insets the intensity scale is logarithmic). For sample fits of $C(r)$ vs $r$ in log-log and semilogarithmic scale, see [16].

Figure 2

$\kappa$ (no units) as function of normalized temperature (no units) for three lattice sizes. The dashed gray line corresponds to $\frac{7}{8}$, which is expected just below the phase transition. Inset (a): Comparison between extremely nonlinear waveguide propagation (red $\times$’s) and an $\mathit{XY}$ Monte Carlo simulation (solid blue line) in a $64\times 64$ array. Inset (b): $\kappa$ vs the effective temperature calculated either by averaging over propagation distance (green dots) or by initial condition averaging (purple plus signs) for a $64\times 64$ array. Good agreement between the two averaging methods is clearly observed.

Figure 3

Average number of free vortices as a function of the normalized temperature (no units). Red circles, simulation results; solid blue line, theoretical fit.