Weakly bound states of two- and three-boson systems in the crossover from two to three dimension

TL;DR

This paper investigates how weakly bound two- and three-boson states evolve when transitioning from three to two dimensions using a squeezed dimension model, revealing a clear crossover and energy ratio transitions.

Contribution

It introduces a simple momentum-space Schrödinger equation for boson systems in a dimensional crossover and numerically analyzes the resulting bound state behaviors.

Findings

Three-boson states either disappear or merge into 2D states during the crossover.

Sharp transitions in energy ratios occur from 2D to 3D regimes.

The derived equation simplifies the study of dimensional interpolation in bosonic systems.

Abstract

The spectrum and properties of quantum bound states is strongly dependent on the dimensionality of space. How this comes about and how one may theoretically and experimentally study the interpolation between different dimensions is a topic of great interest in different fields of physics. In this paper we study weakly bound states of non-relativistic two and three boson systems when passing continuously from a three (3D) to a two-dimensional (2D) regime within a 'squeezed dimension' model. We use periodic boundary conditions to derive a surprisingly simple form of the three-boson Schr{\"o}dinger equation in momentum space that we solve numerically. Our results show a distinct dimensional crossover as three-boson states will either disappear into the continuum or merge with a 2D counterpart, and also a series of sharp transitions in the ratios of three-body and two-body energies from being purely 2D to purely 3D.