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.

Figure 1

Negative temperatures and inverted modal population. (a) RJ equilibrium distribution $n_p^{\mathrm{RJ}}$ for positive temperature $T>0$ ($E<E_{*}$), where low-order modes are more populated, and negative temperatures $T<0$ ($E>E_{*}$), featured by an inverted modal population, while for $1/T\to 0$ ($E=E_{*}$), $n_p^{\mathrm{RJ}}=\mathrm{const}$. (b) Relative entropy $S$ vs energy $E$, showing that $1/T=(\partial S/\partial E{)}_{N,M}<0$ requires $E>E_{*}$. (c) Temperature $T$ vs energy $E$. Negative temperatures $T<0$ occur for $E>E_{*}$ with $E_{*}/E_{\min }\simeq 6.33$ (vertical dashed black line). The vertical purple lines in (b) and (c) denote the six values of $E$ considered in Fig. 2.

Figure 2

Rayleigh-Jeans thermalization to NT equilibriums. Experimental modal distributions averaged over realizations $n_p^{\exp }=⟨|a_p^{\exp }{|}^2⟩$ at the fiber input (blue), and at the fiber output (red). Also shown is the corresponding RJ equilibrium distribution $n_p^{\mathrm{RJ}}$ (green). Note the quantitative agreement between $n_p^{\mathrm{RJ}}$ and the experimental output distribution $n_p^{\exp }$ (red). The six panels correspond to different values of $E$, or equivalently different $T$; see the six vertical purple lines in Figs. 1 and 1. The modal distribution peaked on the lowest mode for (a) $T>0$ gets inverted for (b)–(f) $T<0$. An average over $\simeq 35$ realizations of speckle beams is considered for each panel. The fiber modes are sorted from the fundamental one (${\beta }_0$) to the highest-mode group (ninefold degenerate with ${\beta }_{\max }=9{\beta }_0$). Degenerate modes are equally populated at equilibrium, leading to a staircase distribution $n_p^{\mathrm{RJ}}$.

Figure 3

Energy flows in mode space. Experimental energy distributions averaged over 50 realizations ${ϵ}_p^{\exp }={\beta }_p{n}_p^{\exp }$, at the fiber input (blue) and output (red). The arrow indicates the energy flow to low-order modes. Also shown is the corresponding RJ equilibrium distribution ${ϵ}_p^{\mathrm{RJ}}={\beta }_p{n}_p^{\mathrm{RJ}}$ (green line), which is in quantitative agreement with the experimental output distribution (red).

Figure 4

Oscillating radial intensity distribution at NT. Intensity distribution $I^{\exp }(\mathit{r})$ averaged over the realizations and over angle as a function of the radial distance $|\mathit{r}|$ (red). Note the quantitative agreement with the theoretical RJ intensity distribution $I^{\mathrm{RJ}}(\mathit{r})$ in Eq. (3) (dashed green). The oscillating behavior of the intensity distribution is a signature of the NT equilibrium. Inset: corresponding 2D intensity averaged over the realizations (the radius of the circle is the fiber radius).

Figure 5

Macroscopic population of the highest energy level. Fraction of power ${\tilde{n}}_g/N$ into the highest mode group $g=9$ vs energy $E/E_{\min }$. Experimental measurements at the fiber output: The blue circles refer to the linear regime (small power), and the red circles to the nonlinear regime (high power). The green line denotes the RJ equilibrium theory. By increasing the energy, the power goes to the highest energy level, ${\tilde{n}}_g/N\to 1$ as $E/E_{\min }\to 9$.