Morphology of three-body quantum states from machine learning

The relative motion of three impenetrable particles on a ring, in our case two identical fermions and one impurity, is isomorphic to a triangular quantum billiard. Depending on the ratio kappa of the impurity and fermion masses, the billiards can be integrable or non-integrable (also referred to in the main text as chaotic). To set the stage, we first investigate the energy level distributions of the billiards as a function of 1/kappa is an element of [0, 1] and find no evidence of integrable cases beyond the limiting values 1/kappa = 1 and 1/kappa = 0. Then, we use machine learning tools to analyze properties of probability distributions of individual quantum states. We find that convolutional neural networks can correctly classify integrable and non-integrable states. The decisive features of the wave functions are the normalization and a large number of zero elements, corresponding to the existence of a nodal line. The network achieves typical accuracies of 97%, suggesting that machine learning tools can be used to analyze and classify the morphology of probability densities obtained in theory or experiment.

Automated summary

The study shows that the relative motion of three impenetrable particles on a ring is isomorphic to a quantum billiard, which can be classified into integrable and non-integrable states using machine learning tools. The decisive features of the wave functions for classification are normalization and a large number of zero elements, with the network achieving typical accuracies of 97%.