One-Dimensional Traps, Two-Body Interactions, Few-Body Symmetries: I. One, Two, and Three Particles

Abstract

This is the first in a pair of articles that classify the configuration space and kinematic symmetry groups for $N$ identical particles in one-dimensional traps experiencing Galilean-invariant two-body interactions. These symmetries explain degeneracies in the few-body spectrum and demonstrate how tuning the trap shape and the particle interactions can manipulate these degeneracies. The additional symmetries that emerge in the non-interacting limit and in the unitary limit of an infinitely strong contact interaction are sufficient to algebraically solve for the spectrum and degeneracy in terms of the one-particle observables. Symmetry also determines the degree to which the algebraic expressions for energy level shifts by weak interactions or nearly-unitary interactions are universal, i.e.\ independent of trap shape and details of the interaction. Identical fermions and bosons with and without spin are considered. This article sequentially analyzes the symmetries of one, two and three particles in asymmetric, symmetric, and harmonic traps; the sequel article treats the $N$ particle case.

v3: 58 pages (double-spaced), 8 figures, 3 tables. Heavily revised from v2, including the addition of an introductory subsection on terminology and notation for symmetries, abstract groups, and group representations. New references added. The sequel article to this paper is arXiv:1505.00659

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• Generalization of the Tavis-Cummings model for multi-level anharmonic systems
• SU(N) Fermions in a One-Dimensional Harmonic Trap
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• Topological Exchange Statistics in One Dimension
• Infinite Barriers and Symmetries for a Few Trapped Particles in One Dimension
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• Entanglement production by statistical operators
• Symmetry oscillations in strongly interacting one-dimensional mixtures
• Identical Wells, Symmetry Breaking, and the Near-Unitary Limit
• Experimentally accessible invariants encoded in interparticle correlations of harmonically trapped ultra-cold few-fermion mixtures
• Connecting scrambling and work statistics for short-range interactions in the harmonic oscillator
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• Beyond braid statistics: Constructing a lattice model for anyons with exchange statistics intrinsic to one dimension
• Ground-State Magnetization in Mixtures of a Few Ultra-Cold Fermions in One-Dimensional Traps
• Coincidence Structures and Hard-Core Few-Body Interactions
• Sum rules for spin-$1/2$ quantum gases in states with well-defined spins: II. Spin-dependent two-body interactions
• Graded-index optical fiber emulator of an interacting three-atom system: illumination control of particle statistics and classical non-separability
• Impact of dark states on the stationary properties of quantum particles with off-centered interactions in one dimension
• Necklace Ansatz for strongly repulsive spin mixtures on a ring
• Symmetry oscillations sensitivity to SU(2)-symmetry breaking in quantum mixtures