Forming complex neurons by four-wave mixing in a Bose-Einstein condensate

A physical artificial complex-valued neuron is formed by four-wave mixing in a homogeneous three dimensional Bose-Einstein condensate. Bragg beam-splitter pulses prepare superpositions of three plane-wave states as an input and the fourth wave as an output signal. The nonlinear dynamics of the nondegenerate 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 the numerical simulation. The ideal behavior 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.