Mass-Imbalanced Atoms in a Hard-Wall Trap: An Exactly Solvable Model Associated with D6 Symmetry

Liu, Yanxia; Qi, Fan; Zhang, Yunbo; Chen, Shu

doi:10.1016/j.isci.2019.11.018

“…However, our numerical analysis does not find any traces of solvability beyond the two limiting cases, and supports the widely accepted idea that almost any one-dimensional problem with mass imbalance is non-integrable (notable exceptions include Refs. [44][45][46][47][48]). Therefore, in this work we refer to systems with 1/κ = 0 and 1 as integrable, in the sense that they can be analytically solved using the Bethe ansatz (cf.…”

Section: Formulationmentioning

confidence: 99%

Morphology of three-body quantum states from machine learning

Huber,

Marchukov,

Hammer

et al. 2021

Preprint

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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.

show abstract

Morphology of three-body quantum states from machine learning

Huber

¹

,

Marchukov

²

,

Hammer

³

et al. 2021

View full text Add to dashboard Cite

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 or experiment.

show abstract