  Recent years have witnessed a resurgence of interest in nonlinear multimode
optical systems where a host of intriguing effects have been observed that are
impossible in single-mode settings. While nonlinearity can provide a rich
environment where the chaotic power exchange among thousands of modes can lead
to novel behaviors, at the same time, it poses a major challenge in terms of
understanding and harnessing these processes to advantage. Over the years,
statistical models have been developed to macroscopically describe the response
of these complex systems. One of the cornerstones of these theoretical
formalisms is the prediction of a photon-photon mediated thermalization process
that leads to a Rayleigh-Jeans distribution of mode occupations. Here we report
the use of mode-resolved measurement techniques to make the first direct
observations of thermalization to a Rayleigh-Jeans power distribution in a
multimode optical fiber. We experimentally demonstrate that the underlying
system Hamiltonian remains invariant during propagation while power
equipartition takes place among degenerate groups of modes - all in full accord
with theoretical predictions. Our results may pave the way toward a new
generation of high-power optical sources whose brightness and modal content can
be controlled using principles from thermodynamics and statistical mechanics.
