Spatio-temporal thermalization and adiabatic cooling of guided light waves

We propose and theoretically characterize three-dimensional spatio-temporal thermalization of a continuous-wave classical light beam propagating along a multi-mode optical waveguide. By combining a non-equilibrium kinetic approach based on the wave turbulence theory and numerical simulations of the field equations, we anticipate that thermalizing scattering events are dramatically accelerated by the combination of strong transverse confinement with the continuous nature of the temporal degrees of freedom. In connection with the blackbody catastrophe, the thermalization of the classical field in the continuous temporal direction provides an intrinsic mechanism for adiabatic cooling and, then, spatial beam condensation. Our results open new avenues in the direction of a simultaneous spatial and temporal beam cleaning.

💡 Research Summary

In this work the authors present a comprehensive theoretical study of three‑dimensional (2 D transverse + 1 D temporal) thermalization of a continuous‑wave classical light field propagating in a multimode optical waveguide. Building on the analogy between paraxial propagation and a fluid of light, they model the evolution of the slowly varying envelope ψ(t, r, z) with a nonlinear Schrödinger equation (NLSE) that includes transverse diffraction, a guiding potential V(r), group‑velocity dispersion (κ²), and Kerr nonlinearity (γ₀). By expanding ψ on the discrete set of guided transverse modes uₘ(r) with eigenvalues βₘ, the field is described by mode amplitudes bₘ(ω, z) that depend on the continuous frequency offset ω from the carrier. The resulting evolution equation (2) shows that the linear dispersion becomes ω‑dependent, ˜βₘ(ω)=βₘ−κ²ω², while the nonlinear coupling is mediated by spatial overlap integrals Wₘₚqᵣ.

A central insight is that thermalization relies on resonant four‑wave mixing among quartets of modes. In a purely spatial (monochromatic) setting the discrete β‑spectrum typically yields large phase‑mismatch Δβ_S, suppressing resonances unless special symmetries are imposed. When the temporal degree of freedom is retained, the continuous ω‑dependence of ˜βₘ provides an extra tuning knob: for any quartet {m,p,q,r} one can choose frequencies {ω, ω₁, ω₂, ω₃} that minimize the detuning Δ˜β_ω₁ω₂ω₃, making |Δβ_ST| Lₙₗ ≪ 1. Figure 1(b) quantifies this effect, showing that the number of quasi‑resonant quartets in the spatio‑temporal (ST) case exceeds the spatial‑only case by several orders of magnitude. This dramatically accelerates the approach to equilibrium.

To describe the statistical evolution, the authors employ wave‑turbulence theory. In the weak‑nonlinearity limit they derive a hybrid discrete‑continuous kinetic equation (3) for the spectral densities nₘ(ω, z)=⟨|bₘ(ω, z)|²⟩. The equation conserves particle number N, momentum P, and kinetic energy E, and obeys an H‑theorem guaranteeing monotonic entropy growth S(z)=−∑ₘ∫dω ln nₘ. Its stationary solution is the Rayleigh‑Jeans (RJ) distribution
n_RJ,ₘ(ω)=T /