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

Universality classes of thermalization and energy diffusion

In this paper, we show that classical lattices can be classified into two universality classes for thermalization, based solely on the properties of their eigenmodes. This discovery is a consequence of our systematic multiwave quasiresonance analysis, a tool developed to this end. Lattices with extended modes belong to one class that can thermalize within a finite time, even when the nonlinearity strength is very weak, provided the system size is sufficiently large. In contrast, lattices with purely localized modes fall into another class. For these systems, the scaling behavior of thermalization time shifts stepwise from low-order to progressively higher-order quasiresonances as nonlinear strength decreases, implying that thermalization may become unattainable within a reasonable time for sufficiently weak nonlinearity strength. Furthermore, we show that the real-space energy diffusion behavior of the two classes is qualitatively different as well.

Coupled Thermal and Power Transport of Optical Waveguide Arrays: Photonic Wiedemann-Franz Law and Rectification Effect

In isolated nonlinear optical waveguide arrays, simultaneous conservation of longitudinal momentum flow ("internal energy") and optical power ("particle number") of the optical modes enables study of coupled thermal and particle transport in the negative temperature regime. Based on exact numerical simulation and rationale from Landauer formalism, we predict generic photonic version of the Wiedemann-Franz law in such systems, with the Lorenz number L∝|T|^{-2}. This is rooted in the spectral decoupling of thermal and particle current, and their different temperature dependence. In addition, in asymmetric junctions, relaxation of the system toward equilibrium shows apparent asymmetry for positive and negative biases, indicating rectification behavior. This Letter illustrates the possibility of simulate nonequilibrium transport processes using optical networks, in parameter regimes difficult to reach in natural condensed matter or atomic gas systems. It also provides new insights in manipulating power and momentum flow of optical waves in artificial waveguide arrays.

Wave turbulence and the kinetic equation beyond leading order

We derive a scheme by which to solve the Liouville equation perturbatively in the nonlinearity, which we apply to weakly nonlinear classical field theories. Our solution is a variant of the Prigogine diagrammatic method and is based on an analogy between the Liouville equation in infinite volume and scattering in quantum mechanics, described by the Lippmann-Schwinger equation. The motivation for our work is wave turbulence: A broad class of nonlinear classical field theories are believed to have a stationary turbulent state-a far-from-equilibrium state, even at weak coupling. Our method provides an efficient way to derive properties of the weak wave turbulent state. A central object in these studies, which is a reduction of the Liouville equation, is the kinetic equation, which governs the occupation numbers of the modes. All properties of wave turbulence to date are based on the kinetic equation found at leading order in the weak nonlinearity. We explicitly obtain the kinetic equation to next-to-leading order.

Optical Kinetic Theory of Nonlinear Multimode Photonic Networks

Recent experimental developments in multimode nonlinear photonic circuits (MMNPCs), have motivated the development of an optical thermodynamic theory that describes the equilibrium properties of an initial beam excitation. However, a nonequilibrium transport theory for these systems, when they are in contact with thermal reservoirs, is still terra incognita. Here, by combining Landauer and kinematics formalisms we develop a universal one-parameter scaling theory that describes the whole transport behavior from the ballistic to the diffusive regime, including both positive and negative optical temperature scenarios. We also derive a photonic version of the Wiedemann-Franz law that connects the thermal and power conductivities. Our work paves the way toward a fundamental understanding of the transport properties of MMNPCs and may be useful for the design of all-optical cooling protocols.

Random matrix model of Kolmogorov-Zakharov turbulence

We introduce and study a random matrix model of Kolmogorov-Zakharov turbulence in a nonlinear purely dynamical finite-size system with many degrees of freedom. For the case of a direct cascade, the energy and norm pumping takes place at low energy scales with absorption at high energies. For a pumping strength above a certain chaos border, a global chaotic attractor appears with a stationary energy flow through a Hamiltonian inertial energy interval. In this regime, the steady-state norm distribution is described by an algebraic decay with an exponent in agreement with the Kolmogorov-Zakharov theory. Below the chaos border, the system is located in the quasi-integrable regime similar to the Kolmogorov-Arnold-Moser theory and the turbulence is suppressed. For the inverse cascade, the system rapidly enters a strongly nonlinear regime where the weak turbulence description is invalid. We argue that such a dynamical turbulence is generic, showing that it is present in other lattice models with disorder and Anderson localization. We point out that such dynamical models can be realized in multimode optical fibers.

Spectral-temporal-spatial customization via modulating multimodal nonlinear pulse propagation

Multimode fibers (MMFs) are gaining renewed interest for nonlinear effects due to their high-dimensional spatiotemporal nonlinear dynamics and scalability for high power. High-brightness MMF sources with effective control of the nonlinear processes would offer possibilities in many areas from high-power fiber lasers, to bioimaging and chemical sensing, and to intriguing physics phenomena. Here we present a simple yet effective way of controlling nonlinear effects at high peak power levels. This is achieved by leveraging not only the spatial but also the temporal degrees of freedom during multimodal nonlinear pulse propagation in step-index MMFs, using a programmable fiber shaper that introduces time-dependent disorders. We achieve high tunability in MMF output fields, resulting in a broadband high-peak-power source. Its potential as a nonlinear imaging source is further demonstrated through widely tunable two-photon and three-photon microscopy. These demonstrations provide possibilities for technology advances in nonlinear optics, bioimaging, spectroscopy, optical computing, and material processing.

Statistics of modal condensation in nonlinear multimode fibers

Optical pulses traveling through multimode optical fibers encounter the influence of both linear disturbances and nonlinearity, resulting in a complex and chaotic redistribution of power among different modes. In our research, we explore the phenomenon where multimode fibers reach stable states marked by the concentration of energy into both single and multiple sub-systems. We introduce a weighted Bose-Einstein law, demonstrating its suitability in describing thermalized modal power distributions in the nonlinear regime, as well as steady-state distributions in the linear regime. We apply the law to experimental results and numerical simulations. Our findings reveal that, at power levels situated between the linear and soliton regimes, energy concentration occurs locally within higher-order modal groups before transitioning to global concentration in the fundamental mode within the soliton regime. This research broadens the application of thermodynamic principles to multimode fibers, uncovering previously unexplored optical states that exhibit characteristics akin to optical glass.

Quasi-periodic spectro-temporal pulse breathing in a femtosecond-pumped tellurite graded-index multimode fiber

We report on the multidimensional characterization of femtosecond pulse nonlinear dynamics in a tellurite glass graded-index multimode fiber. We observed novel multimode dynamics of a quasi-periodic pulse breathing which manifests as a recurrent spectral and temporal compression and elongation enabled by an input power change. This effect can be assigned to the power dependent modification of the distribution of excited modes, which in turn modifies the efficiency of involved nonlinear effects. Our results provide indirect evidence of periodic nonlinear mode coupling occurring in graded-index multimode fibers thanks to the modal four-wave-mixing phase-matched via Kerr-induced dynamic index grating.

Observation of Light Thermalization to Negative-Temperature Rayleigh-Jeans Equilibrium States in Multimode Optical Fibers

Although the temperature of a thermodynamic system is usually believed to be a positive quantity, under particular conditions, negative-temperature equilibrium states are also possible. Negative-temperature equilibriums have been observed with spin systems, cold atoms in optical lattices, and two-dimensional quantum superfluids. Here we report the observation of Rayleigh-Jeans thermalization of light waves to negative-temperature equilibrium states. The optical wave relaxes to the equilibrium state through its propagation in a multimode optical fiber-i.e., in a conservative Hamiltonian system. The bounded energy spectrum of the optical fiber enables negative-temperature equilibriums with high energy levels (high-order fiber modes) more populated than low energy levels (low-order modes). Our experiments show that negative-temperature speckle beams are featured, in average, by a nonmonotonic radial intensity profile. The experimental results are in quantitative agreement with the Rayleigh-Jeans theory without free parameters. Bringing negative temperatures to the field of optics opens the door to the investigation of fundamental issues of negative-temperature states in a flexible experimental environment.