  Kerr nonlinearity in the form of self- and cross-phase modulation imposes a
fundamental limitation to the capacity of wavelength division multiplexed (WDM)
optical communication systems. Digital back-propagation (DBP), that requires
solving the inverse-propagating nonlinear Schr\"odinger equation (NLSE), is a
widely adopted technique for the mitigation of impairments induced by Kerr
nonlinearity. However, multi-channel DBP is too complex to be implemented
commercially in WDM systems. Recurrent neural networks (RNNs) have been
recently exploited for nonlinear signal processing in the context of optical
communications. In this work, we propose multi-channel equalization through a
bidirectional vanilla recurrent neural network (bi-VRNN) in order to improve
the performance of the single-channel bi-VRNN algorithm in the transmission of
WDM M-QAM signals. We compare the proposed digital algorithm to full-field DBP
and to the single channel bi-RNN in order to reveal its merits with respect to
both performance and complexity. We finally provide experimental verification
through a QPSK metro link, showcasing over 2.5 dB optical signal-to-noise ratio
(OSNR) gain and up to 43% complexity reduction with respect to the
single-channel RNN and the DBP.
