Abstract
A Bose-Einstein condensate trapped in a double-well potential, where atoms are incoupled to one side and extracted from the other, can in the mean-field limit be described by the nonlinear Gross-Pitaevskii equation (GPE) with a <em>PT</em> symmetric external potential. If the strength of the in- and outcoupling is increased two <em>PT</em> broken states bifurcate from the <em>PT</em> symmetric ground state. At this bifurcation point a stability change of the ground state is expected. However, it is observed that this stability change does not occur exactly at the bifurcation but at a slightly different strength of the in-/outcoupling effect. We investigate a Bose-Einstein condensate in a <em>PT</em> symmetric double-δ potential and calculate the stationary states. The ground state’s stability is analysed by means of the Bogoliubov-de Gennes equations and it is shown that the difference in the strength of the in-/outcoupling between the bifurcation and the stability change can be completely explained by the norm-dependency of the nonlinear term in the Gross-Pitaevskii equation.
Highlights
In quantum mechanics non-Hermitian Hamiltonians with imaginary potentials have become an important tool to describe systems with loss or gain effects [1]
Non-Hermitian PT symmetric Hamiltonians, i.e. Hamiltonians commuting with the combined action of the parity (P: x → −x, p → −p) and time reversal (T : x → x, p → −p, i → −i) operators, possess the interesting property that, in spite of the gain and loss, they can exhibit stationary states with real eigenvalues [2]
When the strength of the gain and loss contributions is increased typically pairs of these real eigenvalues pass through an exceptional point, i.e. a branch point at which both the eigenvalues and the wave functions are identical, and become complex and complex conjugate
Summary
In quantum mechanics non-Hermitian Hamiltonians with imaginary potentials have become an important tool to describe systems with loss or gain effects [1]. In contrast to its linear counterpart, the Gross-Pitaevskii equation possesses no PT broken states emerging at this exceptional point They already appear for lower strengths of the gain/loss parameter and bifurcate from one of the PT symmetric states in a pitchfork bifurcation. To do so we study a Bose-Einstein condensate in an idealised double-δ trap [8,9,10], a system of which already its linear counterpart helped to understand basic properties of PT symmetric structures [29,30,31,32,33] This system is described by the Gross-Pitaevskii equation, i.e. the contact interaction has the norm-dependent form ∝ |ψ(x, t)|2.
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