Observation of replica symmetry breaking in disordered nonlinear wave propagation

Abstract: A landmark of statistical mechanics, spin-glass theory describes critical phenomena in disordered systems that range from condensed matter to biophysics and social dynamics. The most fascinating concept is the breaking of replica symmetry: identical copies of the randomly interacting system that manifest completely different dynamics. Replica symmetry breaking has been predicted in nonlinear wave propagation, including Bose-Einstein condensates and optics, but it has never been observed. Here, we report the experimental evidence of replica symmetry breaking in #optical-wave #propagation, a phenomenon that emerges from the interplay of disorder and nonlinearity. When mode interaction dominates light dynamics in a disordered optical waveguide, different experimental realizations are found to have an anomalous overlap intensity distribution that signals a transition to an optical glassy phase. The findings demonstrate that nonlinear propagation can manifest features typical of #spin-glasses and provide a novel platform for testing #so-far unexplored fundamental physical theories for complex systems. Introduction Disorder in physical systems can introduce frustration among its interacting constituents. This implies that a large number of energetically equivalent and ergodically separated states are accessible in phase space. The condition in which these states dominate dynamics is generally known as the glassy state 1,2. From the equilibrium perspective of the spin-glass theory, the transition to a glassy phase is signaled by replica symmetry breaking (RSB), that is, a change in the statistical distribution of the overlap between measurements in different realizations of the dynamics 3,4. This Parisi overlap is the order parameter that indicates the transition to a RSB phase dominated by an energetic landscape. In spite of huge theoretical efforts, the replica breaking scenario has been only recently observed in photonics in the spectral features of random lasers 5,6,7,8 and multimodal laser emission9. In the latter case, although structural disorder is absent, frustration between modes emerges directly in their interaction. In random lasers, a spin-glass approach to the cavity modes of the electromagnetic field predicted how the competition of quenched disorder and nonlinearity induces glassy mode-locked regimes with many degenerate lasing states 10,11,12,13,14,15. The resulting RSB phenomenology has also been found to be robust with respect to an averaging over different realizations of the disorder16. However, at variance with the case of nonlinear propagation, lasing dynamics require the influx of external energy 17. The fact that a closed Hamiltonian system, and, in particular, a nonlinear wave propagation, can support RSB is still an open question with great relevance in fields such as nonlinear optics, polaritonics and Bose–Einstein condensates 13. Generally, the dynamics of disordered nonlinear waves admits a non-equilibrium statistical mechanics description in terms of wave turbulence18,19,20, which involves phenomena such as wave condensation 21,22,23 and strong turbulence of coherent and incoherent structures 24,25,26,27,28,29,30,31,32,33,34. In particular, interacting localized modes propagating in nonlinear disordered media35,36,37 and multiple laser filaments in gases38,39 may lead to optical states whose complexity resembles glassy phases. This suggests that the statistical properties of coupled nonlinear optical waves can be investigated statically by means of Hamiltonian models with quenched disordered interactions. In this respect, fields propagating under a generalized nonlinear Schrödinger equation (GNLSE) with disorder should sustain a transition to a glassy behavior. According to theoretical predictions, replica symmetry breaking for equivalent realizations of the optical dynamics takes place for increasing nonlinearity, even for small degrees of disorder13. Here we report the observation of the breaking of replica symmetry in nonlinear optical propagation. Shot-to-shot fluctuations and spatial correlations of the optical field are investigated in a photorefractive disordered slab waveguide across the transition that leads from coherent to optically turbulent propagation, where strong variations of the speckle pattern and the degree of spatial correlations set in. In remarkable agreement with the general theories of spin-glasses, the Parisi overlap probability distribution function undergoes a radical change into a double-peaked distribution as the nonlinearity exceeds a threshold value. Replica symmetry breaking here indicates a global locking of several spatial modes so that completely anticorrelated states may emerge from equivalent conditions, the signature that different metastable states underlie dynamics. Results Nonlinear optical propagation in a disordered waveguide To investigate RSB in nonlinear propagation we make use of the large optical nonlinearity of disordered ferroelectric crystals in proximity of their structural phase transition 28,37,40,41. Specifically, we exploit a disordered micrometric-sized photorefractive slab waveguide of potassium-lithium-tantalate-niobate (Fig. 1a, b). The experimental geometry of our setup is sketched in Fig. 1a and detailed in Methods. This system has been also recently used to experimentally demonstrate light beams undergoing antidiffraction and negative mass dynamics 42. In our experiments, spatial inhomogeneities spontaneously arising in the slab constitute a weak linear disordered optical potential whose modes are mainly delocalized. The linear interaction between these optical modes is weak and nonlinearity is needed to couple them all. As shown in Fig. 1c and detailed in Methods, distinct realizations of the experiment present a different spatial distribution of disorder. Structural disorder changes from one replica ("shot") to another whereas it is fixed on the time scale of each single realization of the dynamics. Therefore, the replica symmetry breaking phenomenology we report hereafter has to be considered as averaged over different realizations of the disorder in analogy with results in non-static random laser systems16. The waveguide itself excludes dynamics along the transverse coordinates y, simplifying data analysis and allowing us to adopt a one-dimensional #GNLSE model for wave propagation (see Supplementary Note 1). In our experiment, the degree of disorder is fixed by the experimental conditions so that the strength of the nonlinearity plays the role of an inverse temperature13. The strength of the nonlinearity is controlled by the time the waveguide is exposed to the input light beam (Supplementary Note 2). This relies on the nature of the photorefractive nonlinearity, that is noninstantaneous and accumulates in time as a photogenerated space-charge field builds up43. Observations at different times correspond to beam propagation for increasing nonlinearity, up to saturation. The macroscopic time scale of the process is fixed by the input power and the applied voltage 28. more... References: https://www.nature.com/articles/s41467-017-01612-2#ref-CR1 ARTICLE PDF https://www.nature.com/articles/s41467-017-01612-2.pdf