Morphology of three-body quantum states from machine learning

Abstract

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 κ 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/κ ∈ [0, 1] and find no evidence of integrable cases beyond the limiting values 1/κ = 1 and 1/κ = 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 and experiment.

Probing the edge between integrability and quantum chaos in interacting few-atom systems

“…There are theoretical examples in the literature of quantum systems with only 3 or 4 interacting particles that already exhibit chaotic properties. They include the cesium atom, which has 4 valence electrons [16]; systems composed of 4 particles of unequal masses in a harmonic trap [17] and 3 particles with unequal masses on a ring [18]; 4 or 3 excitations in spin-1/2 chains with short-range [19] or long-range couplings [20]; and even spin-1/2 chains with only 3 sites [21]. In the context of thermalization due to chaos, we also find works that obtained the Fermi-Dirac distribution in systems with only 4 particles [22][23][24][25][26].…”