Infinite Barriers and Symmetries for a Few Trapped Particles in One Dimension

TL;DR

This paper explores the spectral properties of few interacting particles in one-dimensional traps, using symmetry analysis to identify universal features and emergent symmetries related to infinite barriers, with implications for ultracold atom systems.

Contribution

It introduces a symmetry-based framework to analyze few-particle, few-well systems, revealing emergent symmetries at limiting cases and their role in spectral degeneracies and integrability.

Findings

Symmetry analysis explains degeneracy patterns in energy spectra.

Emergent symmetries arise from infinite barriers in configuration space.

Results inform stable state control in ultracold atom experiments.

Abstract

This article investigates the properties of a few interacting particles trapped in a few wells and how these properties change under adiabatic tuning of interaction strength and inter-well tunneling. While some system properties are dependent on the specific shapes of the traps and the interactions, this article applies symmetry analysis to identify generic features in the spectrum of stationary states of few-particle, few-well systems. Extended attention is given to a simple but flexible three-parameter model of two particles in two wells in one dimension. A key insight is that two limiting cases, hard-core repulsion and no inter-well tunneling, can both be treated as emergent symmetries of the few-particle Hamiltonian. These symmetries are the mathematical consequences of infinite barriers in configuration space. They are necessary to explain the pattern of degeneracies in the energy spectrum, to understand how degeneracies are broken for models away from limiting cases, and to explain separability and integrability. These symmetry methods are extendable to more complicated models and the results have practical consequences for stable state control in few-particle, few-well systems with ultracold atoms in optical traps.