Symmetries of Three Harmonically-Trapped Particles in One Dimension

Abstract

We present a method for solving trapped few-body problems and apply it to three equal-mass particles in a one-dimensional harmonic trap, interacting via a contact potential. By expressing the relative Hamiltonian in Jacobi cylindrical coordinates, i.e. the two-dimensional version of three-body hyperspherical coordinates, we discover an underlying ${\mathrm{C}}_{6v}$ symmetry. This symmetry simplifies the calculation of energy eigenstates of the full Hamiltonian in a truncated Hilbert space constructed from the trap Hamiltonian eigenstates. Particle superselection rules are implemented by choosing the relevant representations of ${\mathrm{C}}_{6v}$. We find that the one-dimensional system shows nearly the full richness of the three-dimensional system, and can be used to understand separability and reducibility in this system and in standard few-body approximation techniques.

27 pages, 5 figures, 6 tables, 37 references, 4 footnotes, 1 article; v2 has revised introduction and results sections as well as typos corrected

References in corpus (26)

• Universality in Few-body Systems with Large Scattering Length
• One dimensional Bosons: From Condensed Matter Systems to Ultracold Gases
• The unitary gas in an isotropic harmonic trap: symmetry properties and applications
• Few-nucleon systems in translationally invariant harmonic oscillator basis
• Few-body physics with ultracold atomic and molecular systems in traps
• One-dimensional Bosons in Three-dimensional Traps
• Analytical solution of the bosonic three-body problem
• No-core shell model in an effective-field-theory framework
• Effective Theory for Trapped Few-Fermion Systems
• Universal correlations of trapped one-dimensional impenetrable bosons
• The few-body problem for trapped bosons with large scattering length
• Level crossing in the three-body problem for strongly interacting fermions in a harmonic trap
• An effective field theory approach to two trapped particles
• Three and Four Harmonically Trapped Particles in an Effective Field Theory Framework
• Analytic Harmonic Approach to the N-body problem
• Energy spectrum of harmonically trapped two-component Fermi gases: Three- and Four-Particle Problem
• Production of three-body Efimov molecules in an optical lattice
• S-wave quantum entanglement in a harmonic trap
• Effective renormalized multi-body interactions of harmonically confined ultracold neutral bosons
• Three particles in an external trap: Nature of the complete J=0 spectrum
• Theoretical description of two ultracold atoms in finite 3D optical lattices using realistic interatomic interaction potentials
• Three-fermion problems in optical lattices
• Universal Two-Body Spectra of Ultracold Harmonically Trapped Atoms in Two and Three Dimensions
• Universal few-body physics in a harmonic trap
• Short Range Scaling Laws of Quantum Gases With Contact Interactions
• Efimov Trimers in a Harmonic Potential

Cited by in corpus (44)

• One-dimensional mixtures of several ultracold atoms: a review
• Fermionization of two-component few-fermion systems in a one-dimensional harmonic trap
• Few-body Bose gases in low dimensions -- a laboratory for quantum dynamics
• Distinguishability, degeneracy and correlations in three harmonically trapped bosons in one-dimension
• Quenching small quantum gases: Genesis of the orthogonality catastrophe
• Quantum correlations and spatial localization in one-dimensional ultracold bosonic mixtures
• Spectroscopy for a Few Atoms Harmonically-Trapped in One Dimension
• One-dimensional Fermi gas with a single impurity in a harmonic trap: Perturbative description of the upper branch
• Non-Equilibrium Thermodynamics of Harmonically Trapped Bosons
• One-Dimensional Traps, Two-Body Interactions, Few-Body Symmetries: I. One, Two, and Three Particles
• A variational approach to repulsively interacting three-fermion systems in a one-dimensional harmonic trap
• Strongly interacting mesoscopic systems of anyons in one dimension
• Impenetrable Mass-Imbalanced Particles in One-Dimensional Harmonic Traps
• Fractional energy states of strongly-interacting bosons in one dimension
• One-Dimensional Traps, Two-Body Interactions, Few-Body Symmetries: II. $N$ Particles
• Three interacting atoms in a one-dimensional trap: A benchmark system for computational approaches
• Exploring the few- to many-body crossover using cold atoms in one dimension
• SU(N) Fermions in a One-Dimensional Harmonic Trap
• Anyons from Three-Body Hard-Core Interactions in One Dimension
• Strongly correlated one-dimensional Bose-Fermi quantum mixtures: symmetry and correlations
• Ultracold few fermionic atoms in needle-shaped double wells: spin chains and resonating spin clusters from microscopic Hamiltonians emulated via antiferromagnetic Heisenberg and t-J models
• Analytical treatment of the interaction quench dynamics of two bosons in a two-dimensional harmonic trap
• An interpolatory ansatz captures the physics of one-dimensional confined Fermi systems
• Time-periodic quantum states of weakly interacting bosons in a harmonic trap
• Probing the edge between integrability and quantum chaos in interacting few-atom systems
• Entanglement of an Impurity in a Few-Body One-Dimensional Ideal Bose System
• Energy level splitting for weakly interacting bosons in a harmonic trap
• A modular implementation of an effective interaction approach for harmonically trapped fermions in 1D
• Correlation properties of a three-body bosonic mixture in a harmonic trap
• Hyperspherical Treatment of Strongly-Interacting Few-Fermion Systems in One Dimension
• Ground-states of spin-1 bosons in asymmetric double-wells
• Computation of local exchange coefficients in strongly interacting one-dimensional few-body systems: local density approximation and exact results
• Comparison of renormalized interactions using one-dimensional few-body systems as a testbed
• Are smooth pseudopotentials a good choice for representing short-range interactions?
• Dynamics of small trapped one-dimensional Fermi gas under oscillating magnetic fields
• Connecting scrambling and work statistics for short-range interactions in the harmonic oscillator
• The Goldilocks model of separable, zero-range, few-body interactions in one-dimensional harmonic traps
• Beyond braid statistics: Constructing a lattice model for anyons with exchange statistics intrinsic to one dimension
• Bose-like few-fermion systems
• One-Dimensional Quantum Systems - From Few to Many Particles
• Phase-space localization at the lowest Landau level
• Graded-index optical fiber emulator of an interacting three-atom system: illumination control of particle statistics and classical non-separability
• Efficient determination of eigenenergies and eigenstates of $N$ ($N=3$--$4$) identical 1D bosons and fermions under external harmonic confinement
• On the symmetry of three identical interacting particles in a one-dimensional box