# Levy Statistics and the Glassy Behavior of Light in Random Fiber Lasers
## 3 Results
### Theoretical Framework
In systems with null or weak leakage, in which the off-diagonal contribution in \(g_{k_{1}k_{2}}^{(2)}\) is negligible, the coefficient rates of amplification \((\gamma_{k})\) and radiation loss \((\alpha_{k})\) are related to the real part of the diagonal coupling through \(g_{kk}^{(2)R}=Re\{g_{kk}^{(2)}\}=\alpha_{k}-\gamma_{k}\). On the other hand, the quartic coupling \(g_{k_{1}k_{2}k_{3}k_{4}}^{(4)}\) is associated with the modulation of the nonlinear \(\chi^{(3)}\)-susceptibility with a random spatial profile [73; 74; 75; 76; 77; 78; 79; 80]. The spatial disorder in the scattering medium generally makes the explicit calculation of the quadratic and quartic couplings in Equation (2) rather difficult. Consequently, in [73; 74; 75; 76; 77; 78; 79; 80], these couplings were considered as quenched Gaussian variables, with probability distributions independent of the mode combinations \(\{k_{1}k_{2}\}^{\prime}\) and \(\{k_{1}k_{2}k_{3}k_{4}\}^{\prime}\), respectively. In addition, in the mean-field approach of refs. [73; 74; 75; 76; 77; 78; 79; 80], the above frequency-matching constraints were relaxed, implying that all modes interact unrestrictedly. In the case in which the total optical intensity, \(I=\sum_{k}c_{k}|a_{k}|^{2}\), is a constant, with time-independent prefactors \(c_{k}\), the real part \(H^{R}\) of the Functional (2) is shown in [78; 80] to become analogous to the Hamiltonian of the magnetic \(p\)-spin model with a spherical constraint [83]. Indeed, the \(p\)-spin Hamiltonian also presents a sum of quadratic (\(p=2\)) and quartic (\(p=4\)) interaction terms, just as in Equation (2), with the couplings drawn from Gaussian distributions [83]. Moreover, in the \(p\)-spin model, the sum of the squared spin variables is a constant (spherical constraint), as also happens to the total optical intensity \(I\) in the photonic system. This photonic-to-magnetic analogy is indeed relevant, since it allows identifying the amplitudes of the modes with the spin variables, and the excitation (pump) energy in the photonic system with the inverse temperature in the magnetic one. In this sense, we comment below that the typical magnetic phases exhibited by the disordered \(p\)-spin model can also find an analogous counterpart in the phase diagram of an RL system. Once the disordered Hamiltonian \(H^{R}\), given by the real part of Equation (2), has been built in terms of mode amplitudes that are spin analogues, a repertoire of statistical-physics-based analytical techniques to treat disordered magnetic systems becomes immediately available to the photonic system. In particular, the so-called replica trick [84] can be readily applied to \(H^{R}\). This approach essentially consists of considering identical copies (i.e., replicas) of the system in order to compute the powers \(Z^{n}\) of the partition function, while calculating the free energy from the limit expression \(\ln Z=\lim_{n\to 0}(Z^{n}-1)/n\). At the end, the phase diagram of the RL system is obtained as a function of the input pumping rate and disorder strength [77; 78; 79]. Remarkably, the physical equivalence (or symmetry) among these replicas can be broken in some circumstances, as discussed below. Following this photonic-to-magnetic analogy, the photonic phases identified in the RL system maintain some resemblance to the magnetic behaviors of the \(p\)-spin model. Indeed, we next describe the main properties of the four photonic regimes obtained in the phase diagram of refs. [77; 78; 79], namely: incoherent wave, mode-locking laser, phase-locking wave, and spin-glass RL behavior. In the incoherent-wave regime, which occurs for low input powers and any disorder strength, the modes oscillate incoherently in an uncorrelated way. In this case, the system operates in a regime with amplified spontaneous emission. According to the analogy above, this phase is similar to the paramagnetic behavior in spin systems at high temperatures and for any degree of disorder, in which the uncorrelated spin directions are random and present fast dynamics.