Spatio-temporal thermalization and adiabatic cooling of guided light waves

TL;DR

This work develops a spatio-temporal wave turbulence framework to describe 3D thermalization of classical light in multimode Kerr waveguides. By formulating a unidirectional propagation equation (UPE) and a corresponding wave turbulence kinetic equation, the authors show that the continuous temporal degree of freedom enables efficient spatio-temporal thermalization through plentiful quasi-resonances. A central result is an intrinsic adiabatic cooling mechanism, with local RJ temperatures decaying as $T^{loc}(z) \sim z^{-1/7}$ while the spectral window expands as $ω_c^{loc}(z) ∼ z^{1/7}$, driving beam cleaning and condensation into the fundamental mode. These findings illuminate nonequilibrium Hamiltonian dynamics of light and suggest routes to full 3D condensation and coherent light generation in conservative optical media.

Abstract

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.

Figures (7)

• Figure 1: Quasi-resonances. (a) Schematic visualization of the waveguide. (b) Number of mode quadruplets $\{m,p,q,r\}$ verifying a resonance condition in the spatial-only $|\Delta \beta^S_{mpqr}| L_{\rm nl} < \delta$ (blue line) and in the ST $|\Delta \beta^{ST}_{mpqr}| L_{\rm nl} < \delta$ (red line) cases. The temporal degree of freedom leads to a dramatic enhancement of the number of quasi-resonances. Inset: Modal dispersion relations ${\tilde{\beta}}_m(\omega)$ (in mm$^{-1}$) used to compute the resonant quadruplets in the ST case. Parameters: $L_{\rm nl}=0.3$m, $\tau_0=\sqrt{|\kappa_2| L_{\rm nl}}$, ${\omega}_c=25/\tau_0$ (the waveguide configuration is described in the Appendix, Sec. IV).
• Figure 2: Spatio-temporal thermalization. (a) Simulation of NLSE (\ref{['eq:upe']}): Evolution of spatial modal occupation $N_m^{ST}/N$, showing the relaxation to the equilibrium RJ distribution Eq.(\ref{['eq:n_rj_general']}) (dashed black line). (b) Evolution of the distance ${\cal D}^{ST}(z)$ to equilibrium, whose decrease to zero evidences ST thermalization. (c) This irreversible process is characterized by a monotonous growth of entropy, as described by the $H-$theorem of the wave turbulence kinetic Eq.(\ref{['eq:kin']}). The distance and entropy evolutions are in contrast with those of the spatial case, see Figs. \ref{['fig:3']}(b)-(c). Temporal spectrum $|b_m(\omega, z)|^2$ of the fundamental mode $m=0$ (d), intermediate mode $m=5$ (e), highest mode ($m=9$) (f), at $z=0$ (dark blue) and $z = 5000 L_{\rm nl}$ (red), showing thermalization to RJ spectra (dashed light blue). Parameters: step-index waveguide supporting $M=10$ modes (see Fig. \ref{['fig:1']}(b)), with anomalous dispersion and defocusing nonlinearity, $\tau_0=\sqrt{|\kappa_2| L_{\rm nl}}$, $L_{\rm nl}=0.3$m, ${\omega}_c=25/\tau_0$, $\sigma_\omega=1/\tau_0$, see the Appendix.
• Figure 3: Pure spatial dynamics: Frozen thermalization. (a) Simulation of Eq.(\ref{['eq:nls']}): Evolution of the spatial spectrum $N_m^{S}/N$ starting from the same initial condition as in the ST simulation in Fig. \ref{['fig:2']}. The thermalization process is frozen, as evidenced by the distance ${\cal D}^{S}(z)$ to RJ equilibrium (b), and the entropy (c), which, in contrast to the ST case of Fig. \ref{['fig:2']}(b)-(c), maintain a constant value at long times. Because of the large fluctuations of individual realizations, an average has been taken over 14 realizations.
• Figure 4: Local-equilibrium route to ST thermalization and adiabatic cooling. (a) Mode-integrated temporal spectrum of the field $\sum_m |b_m(\omega,z)|^2$ at $z=0$ (light blue), $z=2079\,L_{\rm nl}$ (dark blue), and local RJ equilibrium distribution over the reduced frequency window $[-{\omega}_{c}^{loc}(z),{\omega}_{c}^{loc}(z)]$ (orange). (b) Modal population $N_m^{loc}/N$ computed from the local RJ equilibrium (circles), and modal population $N_m^{ST}/N$ in NLSE simulation (solid lines). Corresponding evolutions during propagation of local frequency cut-off $\omega_c^{loc}(z)$ (c), and local temperature $T^{loc}(z)$ (d): Results of NLSE simulations (squares) are fitted by a power-law $\sim z^\alpha$ (red lines), showing quantitative agreement with the theory Eq.(\ref{['eq:scaling']}) (dashed black lines). The decrease in $T^{loc}(z)$ reflects an adiabatic cooling, which drives a spatial beam condensation characterized by the growth of $N_0^{loc}(z)/N$ in (b). Parameters: step-index waveguide supporting $M = 5$ modes, with anomalous dispersion and defocusing nonlinearity, $L_{\rm nl}=0.06$m, ${\omega}_c=25/\tau_0$, $\sigma_\omega = 0.4/\tau_0$.
• Figure 5: Spatial case: Frozen thermalization. The blue lines report the numerical simulation of the kinetic Eqs.(\ref{['eq:n_cl']}-\ref{['eq:J_0']}) showing the spectrum $n_m^S(z)$ at $z=6000 L_{\rm nl}$ (a), and corresponding evolutions of the distance to the RJ equilibrium ${\cal D}^S(z)$ (b), and the entropy (c). The orange lines in (a)-(b)-(c) report the numerical simulations of Eq.(\ref{['eq:nls']}) governing the spatial modal amplitudes $b_m^S(z)$: Because of the large fluctuations, an average over 14 realizations has been taken by starting from the same initial spectrum (solid black line), with different realizations of the random phases. A good agreement between the kinetic Eqs.(\ref{['eq:n_cl']}-\ref{['eq:J_0']}) and the spatial model Eq.(\ref{['eq:nls']}) is obtained without using adjustable parameters. The dashed black line in (a) reports the expected RJ equilibrium spectrum. The kinetic Eq.(\ref{['eq:n_cl']}-\ref{['eq:J_0']}) then explains the frozen thermalization of the purely spatial dynamics $b_m^S(z)$ [Eq.(\ref{['eq:nls']})] discussed through the Fig. 3, as confirmed by the evolutions of the distance ${\cal D}^S(z)$ and the entropy in panels (b)-(c).