Symmetries of three harmonically trapped particles in one dimension

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.

Distinguishability, degeneracy, and correlations in three harmonically trapped bosons in one dimension

“…As we show below, the interaction term, V int , has certain discrete rotational symmetry in the X-Y plane. In addition to these symmetries of the Hamiltonian, the symmetrization of the identical A bosons imposes an additional constrain on the wave functions, as they have to be invariant under the interchange of the two A atoms [45]. Let us see how this discrete symmetry of V int together with the symmetrization condition over the identical bosons permit us to grasp some properties of the wave functions of the system.…”

“…as introduced in [45,47]. In these variables, Hamiltonian (1) becomes H = H cm + H rel + V int , with…”

“…All elements of the group C 6v are isomorphic to a permutation of the three atoms (see [45]), and therefore to a spatial transformation. For example, the permutation of the distinguishable atom with coordinate y with one A atom, say x 1 , is isomorphic to the spatial transformation…”

“…For example, how to relate spatial symmetries with the energy spectra in this system, and the differences with a system of three indistinguishable atoms were discussed in Refs. [45,46]. Analytical expressions for the wave function, which are exact for some limiting values of the inter-and intraspecies interactions, have been also recently obtained, together with the relationship of the system with anyonic particles, which show fractional statistics [47,48].…”

“…We also show that they are in agreement with the discrete group theory results of Refs. [45,46]. We study the coherence and correlations of the system as the interactions are tuned.…”