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The integrable nonlinear Schr\"odinger equation (NLSE) is a fundamental modelof nonlinear science which also has important consequences in engineering. Thepowerful framework of the periodic inverse scattering transform (IST) providesa description of the nonlinear phenomena modulational instability andFermi-Pasta-Ulam-Tsingou (FPUT) recurrence in terms of exact solutions. Itassociates the complex nonlinear dynamics with invariant nonlinear spectraldegrees of freedom that may be used to encode information. While optical fiberis an ideal testing ground of its predictions, maintaining integrability oversufficiently long distances to observe recurrence, as well as synthesizing andmeasuring the field in both amplitude and phase on the picosecond timescales oftypical experiments is challenging. Here we report on the experimentalrealization of FPUT recurrence in terms of an exact space-time-periodicsolution of the integrable NLSE in a testbed for optical communicationexperiments. The complex-valued initial condition is constructed by means ofthe finite-gap integration method, modulated onto the optical carrier driven byan arbitrary waveform generator and launched into a recirculating fiber loopwith periodic amplification. The measurement with an intradyne coherentreceiver after a predetermined number of revolutions provides a non-invasivefull-field characterization of the space-time dynamics. The recurrentspace-time evolution is in close agreement with theoretical predictions over adistance of 9000 km. Nonlinear spectral analysis reveals an invariant nonlinearspectrum. The space-time scale exceeds that of previous experiments on FPUTrecurrence in fiber by three orders of magnitude.

Experimental realization of Fermi-Pasta-Ulam-Tsingou recurrence in a long-haul optical fiber transmission system