Bosons in Disc-Shaped Traps: From 3D to 2D

TL;DR

This paper rigorously analyzes the ground state of a dilute Bose gas confined in a strongly anisotropic trap, deriving effective 2D models and identifying conditions for Bose-Einstein condensation.

Contribution

It provides a mathematically rigorous derivation of the effective 2D behavior and Gross-Pitaevskii functional for a 3D Bose gas under strong confinement, including arbitrary short-range interactions.

Findings

Derivation of effective 2D scattering length $a_{2D}$.

Identification of regimes where 2D Gross-Pitaevskii functional applies.

Proof of Bose-Einstein condensation under certain coupling conditions.

Abstract

We present a mathematically rigorous analysis of the ground state of a dilute, interacting Bose gas in a three-dimensional trap that is strongly confining in one direction so that the system becomes effectively two-dimensional. The parameters involved are the particle number, $N\gg 1$, the two-dimensional extension, $\bar L$, of the gas cloud in the trap, the thickness, $h\ll \bar L$ of the trap, and the scattering length $a$ of the interaction potential. Our analysis starts from the full many-body Hamiltonian with an interaction potential that is assumed to be repulsive, radially symmetric and of short range, but otherwise arbitrary. In particular, hard cores are allowed. Under the premisses that the confining energy, $\sim 1/h^2$, is much larger than the internal energy per particle, and $a/h\to 0$, we prove that the system can be treated as a gas of two-dimensional bosons with scattering length $a_{\rm 2D}= h\exp(-(\hbox{\rm const.)}h/a)$. In the parameter region where $a/h\ll |\ln(\bar\rho h^2)|^{-1}$, with $\bar\rho\sim N/\bar L^2$ the mean density, the system is described by a two-dimensional Gross-Pitaevskii density functional with coupling parameter $\sim Na/h$. If $|\ln(\bar\rho h^2)|^{-1}\lesssim a/h$ the coupling parameter is $\sim N |\ln(\bar\rho h^2)|^{-1}$ and thus independent of $a$. In both cases Bose-Einstein condensation in the ground state holds, provided the coupling parameter stays bounded.