Integrable Families of Hard-Core Particles with Unequal Masses in a One-Dimensional Harmonic Trap

N L Harshman,A G Volosniev,A S Dehkharghani,N T Zinner,Maxim Olshanii,Steven Glenn Jackson

doi:10.1103/physrevx.7.041001

Abstract

We show that the dynamics of particles in a one-dimensional harmonic trap with hard-core interactions can be solvable for certain arrangements of unequal masses. For any number of particles, there exist two families of unequal mass particles that have integrable dynamics, and there are additional exceptional cases for three, four and five particles. The integrable mass families are classified by Coxeter reflection groups and the corresponding solutions are Bethe ansatz-like superpositions of hyperspherical harmonics in the relative hyperangular coordinates that are then restricted to sectors of fixed particle order. We also provide evidence for superintegrability of these Coxeter mass families and conjecture maximal superintegrability.

Highlights

The complexity of interacting quantum systems can be partially tamed by extrapolating from solvable models, especially in one dimension [1,2,3]
Prominent examples include the Lieb-Liniger model with zero-range contact interactions in free space [4], the Tonks-Girardeau gas with hard-core contact interactions [5], the Calogero-Moser (CM) model with inverse square interactions either free or in a harmonic trap [6,7], and the extended family of Calogero-Sutherland-Moser (CSM) models [8]
Interest in one-dimensional models has surged because of experiments with ultracold atoms trapped in tight wave guides with interactions controlled by Feshbach and confinement-induced resonances [9,10]. These systems are well described by a one-dimensional model with contact interactions [11]

Summary

INTRODUCTION

The complexity of interacting quantum systems can be partially tamed by extrapolating from solvable models, especially in one dimension [1,2,3]. For sectors with Coxeter tiling symmetry, the energies can be calculated algebraically and all excited states can be expressed as orthogonal polynomials times the ground state; this property is called exact solvability [45,46] These solutions are constructed by Betheansatz-like superpositions, not of plane waves, but of spherical (or hyperspherical) harmonics. The unitary (hard-core) limit of equal-mass particles in a harmonic trap with contact interactions is a maximally superintegrable system isomorphic to one limit of the CM model [7,52,53]. We discuss the connections to quantum billiards and the consequences of integrability for thermalization in ultracold atomic gases

MODEL AND SYMMETRY

EXACT SOLVABILITY AND BETHE-ANSATZ INTEGRABILITY

LIOUVILLE INTEGRABILITY AND SUPERINTEGRABILITY

EXPERIMENTAL OUTLOOK