In this work, a comprehensive experimental investigation of the photon statistics in random fiber laser (RFL) is conducted using the classical temporal second-order correlation function, and its unique evolutionary behavior is unveiled for the first time.

In this study, we investigate the emergence of the quantum boomerang effect in discrete-time quantum walks (DTQWs) subjected to random phase disorder. Our analysis shows that this effect can arise solely from the bias induced by the coin degree of freedom of the DTQW, without requiring external bias or asymmetry. We explore the evolution of the mean position of the quantum walker, denoted as X(t) , under various initial conditions of the walker and quantum coin operators. The results indicate a significant dependence of the observed phenomena on the choice of initial state, enabling the selective induction of the quantum boomerang effect in both or only one portion of the wavepacket associated with specific internal states. By varying the quantum coin parameter θ , we find that the maximum mean position follows a power-law decay near the Pauli-Z coin, characterized by X_Max∼θ ^-2 . Additionally, we identify a scaling behavior X_Max∼ W^-2 , which is consistent with the localization length observed in disordered quantum systems.

As a novel optical platform, random fiber laser (RFL) exhibits multiple metastable energy landscapes due to its disordered feedback mechanism and nonlinear effects, which make it an ideal platform for the study of nonequilibrium statistical physics and multiple nonlinear optical effects. We systematically explores the progress of cutting-edge research on RFL from three key perspectives: complex physical properties, time-domain dynamics, and frequency-domain dynamics in conjunction with theoretical models. In particular, the refinement of the modeling of Rayleigh scattering promotes the study of the replica symmetry breaking phenomenon of RFL. On the basis of this theoretical study, the article elucidates the innovative applications of RFL in several fields, especially providing new possibilities in the field of laser-driven inertial confinement fusion. Finally, the article explores the challenges and possibilities of RFL integration with cutting-edge fields, such as neural networks, and provides an outlook on its prospects for multidisciplinary fusion applications.

We propose and demonstrate a magnetic field sensing approach using a deep learning technique applied to light scattering images. A multi-headed convolutional neural network is trained to predict magnetic field intensity from scattering patterns captured by a CCD camera under different scattering conditions. We employed images generated by conventional laser and random fiber laser illumination sources. The magnetic field can affect the polarization and absorption properties of the medium, besides affecting light scattering, which introduces subtle yet learnable variations in the resultant speckle images. While these variations are imperceptible to human vision, particularly in the low-field regime, the application of deep learning acts to bolster the magnetic field sensor based on scattering images, showing high accuracy in results. Shannon entropy is introduced to quantify subtle differences between distribution patterns associated with different magnetic fields. Furthermore, we demonstrate a low-cost alternative using images generated with a conventional laser pointer, which also yields high accuracy.

Lévy α-stable distributions have important applications in the study of diverse stochastic phenomena, including anomalous diffusion and long-range correlations. Here we revisit the absence of general elementary closed-form expressions for Lévy α-stable distributions with arbitrary parameters. We address this problem from a new angle, based on a Mellin transform approach to the characteristic function. After expressing the Lévy density function as a complex-plane integral, we analyze the pole structure of the integrand in terms of the distribution parameters α and β. In general, the pole structure rules out elementary representations. However, through a remarkable cancellation of poles for specific combinations of α and β, we show that such representations arise only in the known cases of Gaussian, Cauchy-Lorentz, and Lévy-Smirnov densities. To our knowledge, this is the first time that the Mellin transform has been used to systematically investigate the absence of general elementary expressions for Lévy distributions.

ABSTRACT We investigate the relationship between Shannon entropy and Lévy statistics in a random fiber laser based on a dye‐doped polymer optical fiber with magnetic core–shell nanoparticles as scatterers. Spectral measurements acquired below and above the lasing threshold, under varying magnetic fields, are statistically analyzed. Shannon entropy analysis reveals that the intensity fluctuations of each random laser mode, which show the emergence of random laser emission peaks, result in low entropy values. We find a positive correlation between Shannon entropy and the stability index of the Lévy distribution of output intensities for all magnetic fields and pump energies investigated. As an application, we implement a machine learning approach to classify and predict the magnetic field and pump energy from individual spectra when that information is not available. When the network receives only the spectra as input, the prediction accuracy of the model reaches 69%, but when Shannon entropy is also included as an additional feature, the accuracy increases to 98%. These results demonstrate that Shannon entropy provides relevant complementary information for both the physical characterization of random laser dynamics and the development of efficient real‐time optical sensing systems.

Complex systems exhibit rich equilibrium states, yet the universal principles governing these systems remain unrevealed, motivating the search for novel experimental platforms. Random fiber lasers (RFLs), which generate partially-coherent light-wave through feedback from Rayleigh scattering, provide a photonic realization of such systems. Here we report a comprehensive theoretical and experimental investigation of photon statistics for RFLs based on classical second-order temporal correlation function g^(2)(τ), revealing unique statistical properties and introduce a two-dimensional framework for controlling photon statistics. Remarkably, we establish a unified landscape between photon correlation, intensity statistics governed by Levy statistics, and phase transitions with replica symmetry breaking. This multifaceted relationship, observed for the first time, bridges disordered photonics with statistical physics of complex system. Our results offer new pathways for engineering laser emission with controllable photon statistics, and more broadly, this work positions RFLs as a fertile land for exploring emergent behaviors in disordered systems.

We apply the phenomenological renormalization group to resting-state fMRI time series of brain activity in a large population. By recursively coarse graining the data, we compute scaling exponents for the series variance, log probability of silence, and largest covariance eigenvalue. The scaling exponents clearly exhibit linear interdependencies in the form of scaling relations and inherent variability of values closely related to the structure of correlations of brain activity. The scaling relations between the exponents are derived analytically. We find a significant correlation of exponents with clinical (gray matter volume) and behavioral (cognitive performance) traits. Akin to scaling relations near critical points in thermodynamics, our results suggest that this interdependency is intrinsic to brain organization, and may also exist in other complex systems.

We revisit the problem of random search walks in the two-dimensional (2D) space between concentric absorbing annuli, in which a searcher performs random steps until finding either the inner or the outer ring. By considering step lengths drawn from a power-law distribution, we obtain the exact analytical result for the search efficiency η in the ballistic limit, as well as an approximate expression for η in the regime of searches starting far away from both rings, and the scaling behavior of η for very small initial distances to the inner ring. Our numerical results show good overall agreement with the theoretical findings. We also analyze numerically the absorbing probabilities related to the encounter of the inner and outer rings and the associated Shannon entropy. The power-law exponent marking the crossing of such probabilities (equiprobability) and the maximum entropy condition grows logarithmically with the starting distance. Random search walks inside absorbing annuli are relevant, since they represent a mean-field approach to conventional random searches in 2D, which is still an open problem with important applications in various fields.

We investigate the statistical properties of higher-order soliton fission resulting from the propagation of partially coherent input pulses, based on numerical solutions of the one-dimensional Nonlinear Schr & ouml;dinger Equation (NLSE). The findings reveal a frustrating competition for energy among the multiple solitons generated during the fission process, characterized by spin-glass-like correlations and supported by the manifestation of Replica Symmetry Breaking (RSB) in the temporal profile of the propagating pulse. This behavior arises from the interplay between nonlinear (NL) compression and random temporal phase fluctuations, which redistribute energy among the generated solitons while preserving the soliton order parameter. The evolution of the Parisi overlap parameter |qmax| shows a non-monotonic behavior: it gradually decays after successive fissions and re-emerges in a second RSB manifestation, reflecting transitions analogous to those observed in random lasers. These results provide key insights into pulse propagation dynamics in the presence of disorder, and open new avenues for controlling soliton interactions in NL optical systems, with potential applications in photonics and telecommunications.

While replica symmetry breaking (RSB) was originally formulated in spin-glass theory, its photonic counterpart has been realized in the last decade. The existence of a photonic glassy RSB regime has been demonstrated in several photonic platforms characterised by high disorder, such as colloidal random lasers and random fiber lasers. However, the emergence of RSB phase with weak disorder in a standard passive mode-locking regime has not been experimentally demonstrated. Here, we report such experimental demonstration in an ultrafast fiber laser. In contrast to the photonic glassy-like RSB phase in random lasers, the intensity fluctuations of the optical modes in the pulsating passive mode-locking regime originate from the interplay of the nonlinearity and competition of gain among modes, leading to different sets of activated and frustrated modes associated with each pulse. We study the underlying mechanism by characterizing the phase stability of pulses in the mode-locking regimes, demonstrating phase stability in the stable mode-locking regime, periodic oscillation in the pulsating mode locking regime with periodic pulses, and chaotic evolution in the aperiodic pulsating regime. We interpret the latter as an unobserved RSB regime in the aperiodic pulsating phase of passive mode-locked ultrafast lasers. Originally developed in spin-glass physics, replica symmetry breaking (RSB) has recently found a photonic counterpart in disordered laser systems. The study reports the first experimental observation of replica symmetry breaking (RSB) in the aperiodic pulsating regime of a standard passive mode-locked ultrafast fiber laser.

While a thermodynamic description of light has been very useful in discussing light propagation in multimode nonlinear waveguides, it is often stated that such a description is not simple for free-space beams due to the onset of an ultraviolet catastrophe. Herein it is proposed that the beam width and divergence angle are natural cutoffs that remove the singular behavior. Also, the beam propagation (or quality) factor can be associated with a statistically defined temperaturelike parameter. In the high-temperature limit, the beam wavefront is described by a Gaussian Schell-model source, being strongly multimode with a specklelike intensity pattern. In the lowtemperature limit the beam has a wandering Gaussian profile, showing strong correlations and anticorrelations for the intensity's Pearson correlation coefficient, with good agreement between numerical simulations and experimental data. Hence, the present results may subsidize the emerging discussions of thermodynamical descriptions in a wider range of optical systems, such as propagation in turbulent media or biological tissues. Moreover, this work's description is not limited to optical systems, since it is founded on a simple unbounded scalar linear system described using a complete orthonormal basis. Therefore, our findings may also provide further conceptual connections with properties of other systems, such as acoustic waves and quantum particles.

Modulation of scattering in random lasers (RLs) by magnetic fields has attracted much attention due to its rich physical insights. We fabricate magnetic gain polymer optical fiber to generate RLs. From macroscopic experimental phenomena, with the increase of the magnetic field strength, the magnetic transverse photocurrent exists in disordered multiple scattering of RLs and the emission intensity of RLs decreases, which is the experimental observation of photonic Hall effect (PHE) and photonic magnetoresistance (PMR) in RLs. At the microscopic level, based on the field dependence theory of magnetic disorder in scattered nanoparticles and the replica symmetry breaking theory, the magnetic-induced transverse diffusion of photons reduces the scattering disorder, and then decreases the intensity fluctuation disorder of RLs. Our work establishes a connection between the above two effects and RLs, visualizes the influence of magnetic field on RL scattering at the microscopic level, which is crucial for the design of RLs.

Spin glass theory, as a paradigm for describing disordered magnetic systems, constitutes a prominent subject of study within statistical physics. Replica symmetry breaking (RSB), as one of the pivotal concepts for the understanding of spin glass theory, means that under identical conditions, disordered systems can yield distinct states with nontrivial correlations. Random fiber laser (RFL) based on Rayleigh scattering (RS) is a complex disordered system, owing to the disorder and stochasticity of RS. In this work, for the first time, a precise theoretical model is elaborated for studying the photonic phase transition via the platform of RS-based RFL, in which we clearly reveal that, apart from the pump power, the photon phase variation in RFL is also an analogy to the temperature term in spin-glass phase transition, leading to a novel insight into the intrinsic mechanisms of photonic phase transition. In addition, based on this model and real-time high-fidelity detection spectral evolution, we theoretically predict and experimentally observe the mode-asymmetric characteristics of photonic phase transition in RS-based RFL. This finding contributes to a deeper understanding of the photonic RSB regime and the dynamics of RS-based RFL.

We report the first experimental demonstration of the replica symmetry breaking (RSB) phenomenon in a fiber laser system supporting standard mode-locking (SML) regime. Though theoretically predicted, this photonic glassy phase remained experimentally undisclosed so far. We employ an ytterbium-based mode-locked fiber laser with a very rich phase diagram. Two phase transitions are observed separating three different regimes: cw, quasi-mode-locking (QML), and SML. The regimes are intrinsically related to the distinct dynamics of intensity fluctuations in the laser spectra. We set the connection between the RSB glassy phase with frustrated modes and onset of L-shaped intensity distributions in the QML regime, which impact directly the replica overlap measure.

Space-fractional diffusion equations find widespread application in nature. They govern the anomalous dynamics of many stochastic processes, generalizing the standard diffusion equation to superdiffusive behavior. Strikingly, the solution of space-fractional diffusion equations on bounded domains is still an open problem. This is in part due to the difficulty of handling nonlocal boundary conditions ascribed to the space-fractional derivative, leading to the failure of standard methods. Here we revisit the space-fractional diffusion equation in one spatial dimension with bounded domains and present a solution in terms of weighted Jacobi polynomials. Calculated eigenvalues and eigenfunctions in the superdiffusive regime show remarkable agreement with results from numerical discretization of the space-fractional derivative operator and Monte Carlo simulations. To exemplify, we apply the proposed solution to obtain the exact mean residence time or mean first-passage time, first-passage-time distribution, and survival probability, in agreement with known results for the superdiffusive regime. The system of equations converges rather fast for the first eigensolutions, as is desirable for practical application purposes in superdiffusive phenomena.

The numerical hailstone sequences, or orbits, generated by the Collatz map have been disclosed to present relevant features commonly associated with complex systems. It is so despite the extreme simplicity of the arithmetic dynamical system iteration rule. Indeed, for a positive integer n , the Collatz map f reads $f(n) = n/2$ ( $f(n) = 3 n + 1$ ) for n even (odd). Seeking to elucidate this surprising fact, here we unveil distinct characteristics of stochastic-like behavior for collections of Collatz orbits by considering methods commonly employed to temporal series, as cryptography tests, power-spectrum, detrended fluctuation, auto-correlation and entropy measure. Besides confirming previous predictions that the Collatz orbits display some global properties of geometric Brownian motion, our results are likewise able to explain, at least heuristically, the reasons for so. In special, we show by means of comprehensive analysis that our findings cannot be ascribed to standard chaotic evolution. Moreover, we identify novel short- and mid-range correlations in the Collatz orbits. The Collatz map is hence a paradigmatic example of an arithmetic dynamical system which could also be regarded as displaying key characteristics of an arithmetic statistical physics system, explaining its dynamical richness.

In this work, the RSB phase transition of Raman random fiber laser (RFL) is experimentally demonstrated, and the relationship between different modes of Raman RFL and RSB phase transition is explored for the first time.