Statistical mechanics of weakly nonlinear optical multimode gases

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.

On the maximization of entropy in the process of thermalization of highly multimode nonlinear beams

We present a direct experimental confirmation of the maximization of entropy which accompanies the thermalization of a highly multimode light beam, upon its nonlinear propagation in standard graded-index (GRIN) optical fibers.

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.

Dalton's law of partial optical thermodynamic pressures in highly multimoded nonlinear photonic systems

We show that in highly multimoded nonlinear photonic systems, the optical thermodynamic pressures emerging from different species of the optical field obey Dalton's law of partial pressures. In multimode settings, the optical thermodynamic pressure is defined as the conjugate to the extensive variable associated with the system's total number of modes and is directly related to the actual electrodynamic radiation forces exerted at the physical boundaries of the system. Here, we extend this notion to photonic configuration supporting different species of the optical field. Under thermal equilibrium conditions, we formally derive an equation that establishes a direct link between the partial thermodynamic pressures and the electrodynamic radiation pressures exerted by each polarization species. Our theoretical framework provides a straightforward approach for quantifying the total radiation pressures through the system's thermodynamic variables without invoking the Maxwell stress tensor formalism. In essence, we show that the total electrodynamic pressure in such arrangements can be obtained in an effortless manner from initial excitation conditions, thus avoiding time-consuming simulations of the utterly complex multimode dynamics. To illustrate the validity of our results, we carry out numerical simulations in multimoded nonlinear optical structures supporting two polarization species and demonstrate excellent agreement with the Maxwell stress tensor method.

Nature of Optical Thermodynamic Pressure Exerted in Highly Multimoded Nonlinear Systems

The theory of optical thermodynamics provides a comprehensive framework that enables a self-consistent description of the intricate dynamics of nonlinear multimoded photonic systems. This theory, among others, predicts a pressurelike intensive quantity (p[over ^]) that is conjugate to the system's total number of modes (M)-its corresponding extensive variable. Yet at this point, the nature of this intensive quantity is still nebulous. In this Letter, we elucidate the physical origin of the optical thermodynamic pressure and demonstrate its dual essence. In this context, we rigorously derive an expression that splits p[over ^] into two distinct components, a term that is explicitly tied to the electrodynamic radiation pressure and a second entropic part that is responsible for the entropy change. We utilize this result to establish a formalism that simplifies the quantification of radiation pressure under nonlinear equilibrium conditions, thus eliminating the need for a tedious evaluation of the Maxwell stress tensor. Our theoretical analysis is corroborated by numerical simulations carried out in highly multimoded nonlinear optical structures. These results may provide a novel way in predicting and controlling radiation pressure processes in a variety of nonlinear electromagnetic settings.

Thermalization of the Ablowitz-Ladik lattice in the presence of non-integrable perturbations

We investigate the statistical mechanics of the photonic Ablowitz-Ladik lattice, the integrable version of the discrete nonlinear Schrödinger equation. In this regard, we demonstrate that in the presence of perturbations, the complex response of this system can be accurately captured within the framework of optical thermodynamics. Along these lines, we shed light on the true relevance of chaos in the thermalization of the Ablowitz-Ladik system. Our results indicate that when linear and nonlinear perturbations are incorporated, this weakly nonlinear lattice will thermalize into a proper Rayleigh-Jeans distribution with a well-defined temperature and chemical potential, in spite of the fact that the underlying nonlinearity is non-local and hence does not have a multi-wave mixing representation. This result illustrates that in the supermode basis, a non-local and non-Hermitian nonlinearity can in fact properly thermalize this periodic array in the presence of two quasi-conserved quantities.

Coherence properties of light in highly multimoded nonlinear parabolic fibers under optical equilibrium conditions

We study the coherence characteristics of light propagating in nonlinear graded-index (GRIN) multimode fibers after attaining optical thermal equilibrium conditions. The role of optical temperature on the spatial mutual coherence function and the associated correlation area is systematically investigated. In this respect, we show that the coherence properties of the field at the output of a multimode nonlinear fiber can be controlled through its optical thermodynamic properties.

Universality of light thermalization in multimoded nonlinear optical systems

Recent experimental studies in heavily multimoded nonlinear optical systems have demonstrated that the optical power evolves towards a Rayleigh-Jeans (RJ) equilibrium state. To interpret these results, the notion of wave turbulence founded on four-wave mixing models has been invoked. Quite recently, a different paradigm for dealing with this class of problems has emerged based on thermodynamic principles. In this formalism, the RJ distribution arises solely because of ergodicity. This suggests that the RJ distribution has a more general origin than was earlier thought. Here, we verify this universality hypothesis by investigating various nonlinear light-matter coupling effects in physically accessible multimode platforms. In all cases, we find that the system evolves towards a RJ equilibrium-even when the wave-mixing paradigm completely fails. These observations, not only support a thermodynamic/probabilistic interpretation of these results, but also provide the foundations to expand this thermodynamic formalism along other major disciplines in physics.

Thermal control of the topological edge flow in nonlinear photonic lattices

The chaotic evolution resulting from the interplay between topology and nonlinearity in photonic systems generally forbids the sustainability of optical currents. Here, we systematically explore the nonlinear evolution dynamics in topological photonic lattices within the framework of optical thermodynamics. By considering an archetypical two-dimensional Haldane photonic lattice, we discover several prethermal states beyond the topological phase transition point and a stable global equilibrium response, associated with a specific optical temperature and chemical potential. Along these lines, we provide a consistent thermodynamic methodology for both controlling and maximizing the unidirectional power flow in the topological edge states. This can be achieved by either employing cross-phase interactions between two subsystems or by exploiting self-heating effects in disordered or Floquet topological lattices. Our results indicate that photonic topological systems can in fact support robust photon transport processes even under the extreme complexity introduced by nonlinearity, an important feature for contemporary topological applications in photonics.

The nonlinear evolution dynamics in topological photonic lattices is systematically investigated within the framework of optical thermodynamics. This approach allows for the precise prediction of topological currents even under the extreme complexity introduced by nonlinearity.

Thermalization Dynamics of Nonlinear Non-Hermitian Optical Lattices

We develop a rigorous theoretical framework based on principles from statistical mechanics that allows one to predict the equilibrium response of classical non-Hermitian arrangements in the weakly nonlinear regime. In this respect, we demonstrate that a pseudo-Hermitian configuration can always be driven into thermal equilibrium when a proper nonlinear operator is paired with the linear Hamiltonian of the system. We show that, in this case, the system will thermodynamically settle into an irregular pattern that does not resemble any known statistical distribution. Interestingly, this stable equilibrium response is associated with a Rayleigh-Jeans law when viewed within an appropriately transformed space that displays unitary dynamics. By considering a non-Hermitian Su-Schrieffer-Heeger chain, our results indicate that the thermodynamic equilibrium will always favor the edge modes instead of the ground state, in stark contrast to conventional nonlinear Hermitian configurations. Moreover, non-Hermitian lattices are shown to exhibit unusually high heat capacities, potentially acting as optical heat reservoirs to other Hermitian systems, by employing only a small number of sites and low power levels.

Statistical mechanics of beam self-cleaning in GRIN multimode optical fibers

Since its first demonstration in graded-index multimode fibers, spatial beam self-cleaning has attracted a growing research interest. It allows for the propagation of beams with a bell-shaped spatial profile, thus enabling the use of multimode fibers for several applications, from biomedical imaging to high-power beam delivery. So far, beam self-cleaning has been experimentally studied under several different experimental conditions. Whereas it has been theoretically described as the irreversible energy transfer from high-order modes towards the fundamental mode, in analogy with a beam condensation mechanism. Here, we provide a comprehensive theoretical description of beam self-cleaning, by means of a semi-classical statistical mechanics model of wave thermalization. This approach is confirmed by an extensive experimental characterization, based on a holographic mode decomposition technique, employing laser pulses with temporal durations ranging from femtoseconds up to nanoseconds. An excellent agreement between theory and experiments is found, which demonstrates that beam self-cleaning can be fully described in terms of the basic conservation laws of statistical mechanics.

Thermalization of Light's Orbital Angular Momentum in Nonlinear Multimode Waveguide Systems

We show that the orbital angular momentum (OAM) of a light field can be thermalized in a nonlinear cylindrical multimode optical waveguide. We find that upon thermal equilibrium, the maximization of the optical entropy leads to a generalized Rayleigh-Jeans distribution that governs the power modal occupancies with respect to the discrete OAM charge numbers. This distribution is characterized by a temperature that is by nature different from that associated with the longitudinal electromagnetic momentum flow of the optical field. Counterintuitively and in contrast to previous results, we demonstrate that even under positive temperatures, the ground state of the fiber is not always the most populated in terms of power. Instead, because of OAM, the thermalization processes may favor higher-order modes. A new equation of state is derived along with an extended Euler equation resulting from the extensivity of the entropy itself. By monitoring the nonlinear interaction between two multimode optical wave fronts with opposite spins, we show that the exchange of angular momentum is dictated by the difference in OAM temperatures, in full accord with the second law of thermodynamics. The theoretical analysis presented here is corroborated by numerical simulations that take into account the complex nonlinear dynamics of hundreds of modes. Our results may pave the way toward high-power optical sources with controllable orbital angular momenta, and at a more fundamental level, they may open up opportunities in drawing parallels with other complex multimode nonlinear systems like rotating atomic clouds.

Equilibrium and nonequilibrium description of negative temperature states in a one-dimensional lattice using a wave kinetic approach

We predict negative temperature states in the discrete nonlinear Schödinger (DNLS) equation as exact solutions of the associated wave kinetic equation. Within the wave kinetic approach, we define an entropy that results monotonic in time and reaches a stationary state, that is consistent with classical equilibrium statistical mechanics. We also perform a detailed analysis of the fluctuations of the actions at fixed wave numbers around their mean values. We give evidence that such fluctuations relax to their equilibrium behavior on a shorter timescale than the one needed for the spectrum to reach the equilibrium state. Numerical simulations of the DNLS equation are shown to be in agreement with our theoretical results. The key ingredient for observing negative temperatures in lattices characterized by two invariants is the boundedness of the dispersion relation.