Proper condensates

TL;DR

This paper introduces a new way to characterize Bose-Einstein condensates by focusing on regular excitations with bounded numbers, providing a clearer and unambiguous definition of proper condensates compared to traditional occupation number methods.

Contribution

It proposes the concept of proper condensates based on regular excitations, offering a precise framework for analyzing condensates beyond occupation number criteria.

Findings

Proper condensates are unambiguously defined using regular wave functions.

The method is demonstrated on non-interacting trapped states and their thermodynamic limits.

The concept aligns with and extends the Onsager-Penrose criterion for identifying condensates.

Abstract

In this article a novel characterization of Bose-Einstein condensates is proposed. Instead of relying on occupation numbers of a few dominant modes, which become macroscopic in the limit of infinite particle numbers, it focuses on the regular excitations whose numbers stay bounded in this limit. In this manner, subspaces of global, respectively local regular wave functions are identified. Their orthogonal complements determine the wave functions of particles forming proper (infinite) condensates in the limit. In contrast to the concept of macroscopic occupation numbers, which does not sharply fix the wave functions of condensates in the limit states, the notion of proper condensates is unambiguously defined. It is outlined, how this concept can be used in the analysis of condensates in models. The method is illustrated by the example of trapped non-interacting ground states and their multifarious thermodynamic limits, differing by the structure of condensates accompanying the Fock vacuum. The concept of proper condensates is also compared with the Onsager-Penrose criterion, based on the analysis of eigenvalues of one-particle density matrices. It is shown that the concept of regular wave functions is useful there as well for the identification of wave functions forming proper condensates.