  The integrable nonlinear Schr\"odinger equation (NLSE) is a fundamental model
of nonlinear science which also has important consequences in engineering. The
powerful framework of the periodic inverse scattering transform (IST) provides
a description of the nonlinear phenomena modulational instability and
Fermi-Pasta-Ulam-Tsingou (FPUT) recurrence in terms of exact solutions. It
associates the complex nonlinear dynamics with invariant nonlinear spectral
degrees of freedom that may be used to encode information. While optical fiber
is an ideal testing ground of its predictions, maintaining integrability over
sufficiently long distances to observe recurrence, as well as synthesizing and
measuring the field in both amplitude and phase on the picosecond timescales of
typical experiments is challenging. Here we report on the experimental
realization of FPUT recurrence in terms of an exact space-time-periodic
solution of the integrable NLSE in a testbed for optical communication
experiments. The complex-valued initial condition is constructed by means of
the finite-gap integration method, modulated onto the optical carrier driven by
an arbitrary waveform generator and launched into a recirculating fiber loop
with periodic amplification. The measurement with an intradyne coherent
receiver after a predetermined number of revolutions provides a non-invasive
full-field characterization of the space-time dynamics. The recurrent
space-time evolution is in close agreement with theoretical predictions over a
distance of 9000 km. Nonlinear spectral analysis reveals an invariant nonlinear
spectrum. The space-time scale exceeds that of previous experiments on FPUT
recurrence in fiber by three orders of magnitude.
