The understanding of nonlinear PT symmetric quantum systems, arising for example in the theory of BoseāEinstein condensates in PT symmetric potentials, is widely based on numerical investigations, and little is known about generic features induced by the interplay of PT symmetry and nonlinearity. To gain deeper insights it is important to have analytically solvable toy models at hand. In the present paper the stationary states of a simple toy model of a PT symmetric system previously introduced in [1, 2] are investigated. The model can be interpreted as a simple description of a BoseāEinstein condensate in a PT symmetric double well trap in a two-mode approximation. The eigenvalues and eigenstates of the system can be explicitly calculated in a straightforward manner; the resulting structures resemble those that have recently been found numerically for a more realistic PT symmetric double delta potential. In addition, a continuation of the system is introduced that allows an interpretation in terms of a simple linear matrix model. This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to āQuantum physics with non-Hermitian operatorsā.
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 PT symmetric external potential. If the strength of the in- and outcoupling is increased two PT broken states bifurcate from the PT 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 PT symmetric double-Ī“ potential and calculate the stationary states. The ground stateās 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.
The observation of
PT symmetry in a coupled optical waveguide system that involves a complex refractive index has been demonstrated impressively in the experiment by R\"uter et al. [Nat. Phys. 6, 192 (2010)]. This is, however, only an optical analog of a quantum system, and it would be highly desirable to observe the manifestation of
PT symmetry and the resulting properties also in a real, experimentally accessible, quantum system. Following a suggestion by Klaiman et al. [Phys. Rev. Lett. 101, 080402 (2008)], we investigate a
PT-symmetric arrangement of a Bose-Einstein condensate in a double-well potential, where in one well cold atoms are injected while in the other particles are extracted from the condensate. We investigate, in particular, the effects of the nonlinearity in the Gross-Pitaevskii equation on the
PT properties of the condensate. To study these effects we analyze a simple one-dimensional model system in which the condensate is placed into two
PT-symmetric $\ensuremath{\delta}$-function traps. The analysis will serve as a useful guide for studies of the behavior of Bose-Einstein condensates in realistic
PT-symmetric double wells.