Interplay of Thermalization and Strong Disorder: Wave Turbulence Theory, Numerical Simulations, and Experiments in Multimode Optical Fibers

We address the problem of thermalization in the presence of a time-dependent disorder in the framework of the nonlinear Schrödinger (or Gross-Pitaevskii) equation with a random potential. The thermalization to the Rayleigh-Jeans distribution is driven by the nonlinearity. On the other hand, the structural disorder is responsible for a relaxation toward the homogeneous equilibrium distribution (particle equipartition), which thus inhibits thermalization (energy equipartition). On the basis of the wave turbulence theory, we derive a kinetic equation that accounts for the presence of strong disorder. The theory unveils the interplay of disorder and nonlinearity. It unexpectedly reveals that a nonequilibrium process of condensation and thermalization can take place in the regime where disorder effects dominate over nonlinear effects. We validate the theory by numerical simulations of the nonlinear Schrödinger equation and the derived kinetic equation, which are found in quantitative agreement without using any adjustable parameter. Experiments realized in multimode optical fibers with an applied external stress evidence the process of thermalization in the presence of strong disorder.

Figure 1

Dynamics dominated by strong disorder ${\mathcal{L}}_{\mathrm{kin}}^{\mathrm{RJ}}\gg {\mathcal{L}}_{\mathrm{kin}}^{\mathrm{eq}}$. The system irreversibly relaxes toward the equilibrium $w_j^{\mathrm{eq}}$. Evolutions of the fundamental mode $w_0(z)$ (a), and $w_2(z)$ (b), the energy $E(z)/(N{\beta }_0)$ (c), obtained from the numerical simulation of the NLS Eq. (2). Sixty-four realizations are reported with colored lines; the bold white line is the corresponding empirical average; the dashed black line is the prediction of the KE [Eq. (4)]. (d) Modal distribution $w_j$ in the initial condition ($z=0$, blue) and at $z{\beta }_0=2\times {10}^6$ for the NLS simulation (red), and the KE (black). Parameters: $L_{\mathrm{dis}}/L_{\mathrm{lin}}=7$, $L_{\mathrm{dis}}/L_{\mathrm{nl}}=4.1\times {10}^{-4}$, ${\ell }_c{\beta }_0=42$.

Figure 2

Thermalization precedes equilibrium relaxation. Same panels as in Fig. 1, but in the regime ${\mathcal{L}}_{\mathrm{kin}}^{\mathrm{RJ}}\lesssim {\mathcal{L}}_{\mathrm{kin}}^{\mathrm{eq}}$. The system exhibits an incipient process of RJ thermalization and nonequilibrium condensation characterized by a growth of the condensate amplitude $w_0(z)$ for $z{\beta }_0\lesssim 2\times {10}^5$. Disorder subsequently prevails, which induces a decay of $w_0(z)$ (and eventually brings the system to equilibrium $w_j^{\mathrm{eq}}$). Parameters: $L_{\mathrm{dis}}/L_{\mathrm{lin}}=7$, $L_{\mathrm{dis}}/L_{\mathrm{nl}}=0.033$, and ${\ell }_c{\beta }_0=167$.

Figure 3

Thermalization prevails over equilibrium relaxation. Same panels as in Fig. 1, but in the regime ${\mathcal{L}}_{\mathrm{kin}}^{\mathrm{RJ}}\ll {\mathcal{L}}_{\mathrm{kin}}^{\mathrm{eq}}$. The system exhibits a process of RJ thermalization and condensation characterized by a significant growth of the condensate amplitude $w_0(z)$ to the value predicted by the RJ distribution, $w_0^{\mathrm{RJ}}/N\simeq 0.55$ (horizontal dashed-dotted black line) (a). At variance with Figs. 1 and 2, the energy $E(z)$ is almost constant (c). The modes approach the RJ distribution $w_j^{\mathrm{RJ}}$ (green) (d). Parameters: $L_{\mathrm{dis}}/L_{\mathrm{lin}}=8.4$, $L_{\mathrm{dis}}/L_{\mathrm{nl}}=0.04$, and ${\ell }_c{\beta }_0=4\times {10}^3$.

Figure 4

Observation of light condensation with strong disorder. Measurements of the condensate fraction $w_0/N$ vs $E/N$ at small power [linear (LIN) regime] and high power [nonlinear (NL) regime], for a moderate (a), and a large (b), strength of random mode coupling. The solid line reports the prediction from the RJ theory, $w_0^{\mathrm{RJ}}/N$. In the absence of strong disorder (squares), $w_0/N$ increases as the power increases, and reaches the RJ prediction (solid line)—each color refers to a different value of $E/N$. In the presence of strong disorder (big circles), the energy $E/N$ increases due to disorder (squares are shifted to big circles of the same color), by $\mathrm{\Delta }\overline{E/N}\simeq 6\%$ (a), and $\mathrm{\Delta }\overline{E/N}\simeq 11\%$ (b). The big circles report the average over 10 different realizations of disorder (10 small circles for each color). RJ thermalization takes place in the presence of strong disorder (a), and it is quenched by further increasing the amount of disorder (b) [51].