Spatio-temporal equilibrium thermodynamics of guided optical waves at positive and negative temperatures

TL;DR

This work develops a (2+1)D spatio-temporal thermodynamic framework for incoherent light in multimode guided-wave systems, revealing distinct positive- and negative-temperature equilibria governed by dispersion sign ($\kappa$). In the anomalous (negative) dispersion regime, ST condensation emerges with macroscopic occupation of the fundamental spatial mode and strong spectral narrowing, culminating in complete ST condensation in the thermodynamic limit. In the normal (positive) dispersion regime, negative-temperature states exhibit inverted spatial mode populations while maintaining a temporal peak at the carrier frequency, with a phase transition to BE condensation at a negative critical temperature driven by increasing $\tilde{T}$. The results distinguish classical RJ from quantum BE behavior via both marginal distributions and near-field intensity, and they illuminate pathways to dual spatio-temporal beam cleaning and broader optical thermodynamics, while outlining experimental and theoretical extensions to disorder, nonparaxiality, and nonlinear regimes.

Abstract

Optical thermalization has been recently studied theoretically and experimentally in the 2D spatial evolution of (quasi-)monochromatic light waves propagating in multimode fibers. In this work, we investigate the spatio-temporal equilibrium properties of incoherent multimode optical waves through the analysis of the (2+1)D Bose-Einstein thermal distribution and the corresponding classical Rayleigh-Jeans approximation. In the anomalous dispersion regime, the spatio-temporal equilibrium is characterized by positive temperatures. In this regime, we show that as the number of modes of the waveguide increases, the fundamental spatial mode becomes macroscopically populated, while its temporal spectrum undergoes significant narrowing, ultimately leading to complete (2+1)D spatio-temporal condensation in the thermodynamic limit. In the normal dispersion regime, the spatio-temporal equilibrium is characterized by negative temperature states that exhibit a hybrid character: the spatial equilibrium displays an inverted modal population, whereas the temporal spectrum remains peaked around the fundamental (carrier) optical frequency. In this regime, we predict that spatio-temporal light waves exhibit a phase transition to Bose-Einstein condensation at negative temperatures, which occurs by increasing the temperature above a negative critical value. Our work opens new avenues for future research, including the possibility for a dual spatio-temporal beam cleaning through full spatio-temporal light condensation, and lay the groundwork for the development of spatio-temporal optical thermodynamics.

Figures (6)

• Figure 1: ST equilibria at positive and negative temperatures. Equilibrium distribution (\ref{['eq:BE_to_RJ']}) for a step-index multimode waveguide, in the anomalous dispersion regime ($\kappa<0$) leading to positive temperatures, ${\tilde{T}}>0$ (left column), and in the normal dispersion regime ($\kappa>0$) leading to negative temperatures, ${\tilde{T}}<0$ (right column). At variance with ${\tilde{T}}>0$ where the spatial and temporal spectra are peaked in the fundamental spatial mode $m=0$ at frequency $\omega=0$, for ${\tilde{T}}<0$ the equilibrium is featured by an inverted population of spatial modes, with temporal spectra still peaked at $\omega=0$. The equilibrium distributions are plotted at ${\tilde{T}}=1.3{\tilde{T}}_c$ [left column: (a-b-c)], and ${\tilde{T}}=-2|{\tilde{T}}_c|$ [right column: (d-e-f)]. The plots on the first row report the ST distribution $n_m(\omega)$ from Eq.(\ref{['eq:BE_to_RJ']}), the plots on the second and third rows display cross-sections of the corresponding distributions in the top row: temporal spectra $n_m(\omega)$ for different modes with eigenvalues $\beta_m$ (2nd row); spatial mode distribution integrated in frequency, ${\tilde{n}}_m=(2\pi c_0)^{-1}\int n_m(\omega) d\omega$ (3rd row). Parameters are given in the text.
• Figure 2: Condensation at positive temperatures. Convergence to the thermodynamic limit for ${\tilde{T}}>0$ in a step-index waveguide: (a) chemical potential vs temperature, ${\tilde{\mu}}({\tilde{T}})$; (b) spatial condensate fraction vs temperature, $\rho_0({\tilde{T}})/\rho$. The solid lines refer to the computation of the discrete sums beyond the thermodynamic limit, from Eq.(\ref{['eq:rho_beyond_TL']}) for (a), and from Eq.(\ref{['eq:rho_0_beyond_TL']}) for (b). By increasing the number of modes $M$ (or the waveguide surface $S$) while keeping the photon density $\rho/S=$const and $V_0=$const, the curves approach the thermodynamic limit (dashed blue line), which are obtained from Eq.(\ref{['eq:rho_TL_00']}) for (a), and from Eq.(\ref{['eq:frac_spat_cond_TL']}) for (b). In the thermodynamic limit, ${\tilde{\mu}}({\tilde{T}})$ in (a), and $\rho_0({\tilde{T}})/\rho$ in (b), correspond to the (2+1)D ST condensation: both curves display a singular cusped behavior at ${\tilde{T}} = {\tilde{T}}_c$, where ${\tilde{\mu}}=\beta_{0}$ for $0<{\tilde{T}} \le {\tilde{T}}_c$, and $\rho_0=0$ for ${\tilde{T}} > {\tilde{T}}_c$, where the critical temperature ${\tilde{T}}_c$ is given by Eq.(\ref{['eq:T_c_pos']}). Parameters are given in the text.
• Figure 3: Spatial mode distribution and temporal spectra in the condensed regime at positive temperatures. Equilibrium distribution (\ref{['eq:BE_to_RJ']}) for a step-index waveguide at ${\tilde{T}}=0.6 {\tilde{T}}_c$ for $M=17$ [left column: (a-b-c)], and $M=954$ [right column: (d-e-f)]. The plots on the first row report the ST distribution $n_m(\omega)$ from Eq.(\ref{['eq:BE_to_RJ']}), the plots on the second and third rows display cross-sections of the corresponding distributions in the top row: temporal spectra $n_m(\omega)$ for different modes $m$ (2nd row); spatial mode distribution integrated in frequency, ${\tilde{n}}_m=(2\pi c_0)^{-1}\int n_m(\omega) d\omega$ (3rd row). By increasing the number of modes $M$, we observe a significant spectral narrowing of the fundamental mode (compare (b) and (e)), as well as a macroscopic population of the fundamental spatial mode $m=0$ (compare (c) and (f)). Parameters are given in the text.
• Figure 4: BE vs RJ regimes. Relation between the optical power $P$ and the critical temperature $\tilde{T}_{c}$ to condensation, from Eq.(\ref{['eq:rho_grin_Tpos']}) (blue line), for a parabolic multimode waveguide. For large powers $P \gg 0.1$W, the photon density can be approximated by the RJ limit (dashed orange line, ${\tilde{T}}_c \sim P$). By decreasing the power, the photon density follows the BE expression (\ref{['eq:rho_grin_Tpos']}) with $V_0 \to \infty$ (dashed green line, ${\tilde{T}}_c \sim P^{2/5}$). Transverse 2D intensity distribution ${\tilde{\rho}}(r)$ (with $r=|{\bm r}_\perp|$) (b), spatial mode distribution ${\tilde{\rho}}_S(\beta)$ (c), and temporal spectrum ${\tilde{\rho}}_T(\omega)$ (d), computed with the BE law (solid lines), and the corresponding RJ approximation (dotted lines), at ${\tilde{T}}={\tilde{T}}_c$. As seen in panels (b) to (d), at large power ($P=3 \, W$), the distributions are well approximated by the RJ law, while at small power ($P=0.1 \, W$) noticeable deviations emerge. Parameters are given in the text.
• Figure 5: Condensation at negative temperatures. Convergence to the thermodynamic limit for ${\tilde{T}}<0$ in a parabolic waveguide: (a) chemical potential vs temperature, ${\tilde{\mu}}({\tilde{T}})$; (b) condensate fraction in the highest mode group vs temperature, $\rho_G({\tilde{T}})/\rho$. The solid lines refer to the computation of the discrete sums beyond the thermodynamic limit, from Eq.(\ref{['eq:rho_beyond_TL_neg']}) (a), and from Eq.(\ref{['eq:spat_rho_0_beyond_TL']}) (b). By increasing the number of modes $M$ (or the waveguide surface $S$) while keeping the photon density $\rho/S=$const and $V_0=$const, the curves approach the thermodynamic limit (dashed blue lines), which are obtained from Eq.(\ref{['eq:rho_gen_Tc_neg']}) for (a), and from Eq.(\ref{['eq:gen_cond_fract_neg']}) for (b). In the thermodynamic limit, ${\tilde{\mu}}({\tilde{T}})$ in (a), and $\rho_{G}({\tilde{T}})/\rho$ in (b), correspond to the (2+1)D ST condensation: both curves display a singular cusped behavior at ${\tilde{T}} = {\tilde{T}}_c <0$, where ${\tilde{\mu}}=\beta_{\rm max}=V_0$ for ${\tilde{T}}_c \le {\tilde{T}} \le 0$, and $\rho_G=0$ for ${\tilde{T}} < {\tilde{T}}_c <0$. Parameters are given in the text.