Exceptional Points for Nonlinear Schroedinger Equations Describing Bose-Einstein Condensates of Ultracold Atomic Gases

G Wunner,J Main,H Cartarius,S Rau,P Koeberle

doi:10.14311/1418

Abstract

The coalescence of two eigenfunctions with the same energy eigenvalue is not possible in Hermitian Hamiltonians. It is, however, a phenomenon well known from non-hermitian quantum mechanics. It can appear, e.g., for resonances in open systems, with complex energy eigenvalues. If two eigenvalues of a quantum mechanical system which depends on two or more parameters pass through such a branch point singularity at a critical set of parameters, the point in the parameter space is called an exceptional point. We will demonstrate that exceptional points occur not only for non-hermitean Hamiltonians but also in the nonlinear Schroedinger equations which describe Bose-Einstein condensates, i.e., the Gross-Pitaevskii equation for condensates with a short-range contact interaction, and with additional long-range interactions. Typically, in these condensates the exceptional points are also found to be bifurcation points in parameter space. For condensates with a gravity-like interaction between the atoms, these findings can be confirmed in an analytical way.

Highlights

In 1924, Satyendra Nath Bose and Albert Einstein predicted that when the thermal de Broglie wavelength becomes of the order of the interparticle distance, bosons begin to “condense” into their ground state
We have demonstrated that “nonlinear versions” of exceptional points appear in bifurcating solutions of the Gross-Pitaevskii equations describing the Bose-Einstein condensation of ultracold atomic gases with attractive 1/r interaction and with dipoledipole interaction
The identification as exceptional points required a complex extension of the scattering length

Summary

Introduction

In 1924, Satyendra Nath Bose and Albert Einstein predicted that when the thermal de Broglie wavelength becomes of the order of the interparticle distance, bosons begin to “condense” into their ground state. Γ = hω0/Eu is the trapping frequency measured in the time scale given by the “Rydberg” energy Eu associated with the strength of the 1/r interaction, N is the particle number, and a/au the scattering length in units of the “Bohr” radius au. It can be shown [9] that the equation effectively depends only on two physical parameters, viz. Equations (1) and (2) are the starting point for the investigations below

Variational results

Numerical analysis of the bifurcation point for self-trapped condensates

Dipolar condensates

Numerical results

Summary