  A physical artificial complex-valued neuron is formed by four-wave mixing in
a homogeneous three-dimensional Bose-Einstein condensate. Bragg beamsplitter
pulses prepare superpositions of three plane-waves states as an input- and the
fourth wave as an output signal. The nonlinear dynamics of the non-degenerate
four-wave mixing process leads to Josephson-like oscillations within the closed
four-dimensional subspace and defines the activation function of a neuron. Due
to the high number of symmetries, closed form solutions can be found by
quadrature and agree with numerical simulation. The ideal behaviour of an
isolated four-wave mixing setup is compared to a situation with additional
population of rogue states. We observe a robust persistence of the main
oscillation. As an application for neural learning of this physical system, we
train it on the XOR problem. After $100$ training epochs, the neuron responds
to input data correctly at the $10^{-5}$ error level.
