Eigenvalue structure of a Bose-Einstein condensate in a PT-symmetric double well

TL;DR

This paper investigates the eigenvalue structure of a nonlinear, non-Hermitian Bose-Einstein condensate in a PT-symmetric double well, revealing bifurcation scenarios and exceptional points including an EP4 through analytic continuation and numerical analysis.

Contribution

It extends PT symmetry to nonlinear systems via bicomplex numbers and identifies new bifurcation phenomena, including an EP4, with a matching linear matrix model.

Findings

Identification of EP2 and EP3 bifurcations in the system

Discovery of an EP4 involving four interacting modes

Excellent agreement between matrix model and numerical results

Abstract

We study a Bose-Einstein condensate in a PT-symmetric double-well potential where particles are coherently injected in one well and removed from the other well. In mean-field approximation the condensate is described by the Gross-Pitaevskii equation thus falling into the category of nonlinear non-Hermitian quantum systems. After extending the concept of PT symmetry to such systems, we apply an analytic continuation to the Gross-Pitaevskii equation from complex to bicomplex numbers and show a thorough numerical investigation of the four-dimensional bicomplex eigenvalue spectrum. The continuation introduces additional symmetries to the system which are confirmed by the numerical calculations and furthermore allows us to analyze the bifurcation scenarios and exceptional points of the system. We present a linear matrix model and show the excellent agreement with our numerical results. The matrix model includes both exceptional points found in the double-well potential, namely an EP2 at the tangent bifurcation and an EP3 at the pitchfork bifurcation. When the two bifurcation points coincide the matrix model possesses four degenerate eigenvectors. Close to that point we observe the characteristic features of four interacting modes in both the matrix model and the numerical calculations, which provides clear evidence for the existence of an EP4.