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arXiv:1401.2354 · Acta Polytechnica

STABILITY OF BOSE-EINSTEIN CONDENSATES IN A PT-SYMMETRIC DOUBLE-δ POTENTIAL CLOSE TO BRANCH POINTS

Andreas Löhle, Holger Cartarius, Daniel Haag, Dennis Dast, Jörg Main, Günter Wunner

Published Apr 30, 2014• 7 citations
arXivHTMLPDF

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 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-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.

DOI
10.14311/ap.2014.54.0133

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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-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.
We investigate dipolar Bose-Einstein condensates in a complex external double-well potential that features combined parity and time-reversal symmetry. On the basis of the Gross-Pitaevskii equation we study the effects of the long-ranged anisotropic dipole-dipole interaction on ground and excited states by the use of a time-dependent variational approach. We show that the property of a similar nondipolar condensate of possessing real energy eigenvalues in certain parameter ranges is preserved despite the inclusion of this nonlinear interaction. Furthermore, we present states that break the PT\mathcal{PT}PT symmetry and investigate the stability of the distinct stationary solutions. In our dynamical simulations we reveal a complex stabilization mechanism for PT\mathcal{PT}PT-symmetric as well as for PT\mathcal{PT}PT-broken states which are, in principle, unstable with respect to small perturbations.
We investigate the mean-field dynamics of a Bose-Einstein condensate described by the Gross-Pitaevskii equation (GPE) in a double-well potential with particle gain and loss, rendering the system PT\mathcal{PT}PT symmetric. The stationary solutions of the system show a change from elliptically stable behavior to hyperbolically unstable behavior caused by the appearance of PT\mathcal{PT}PT-broken solutions of the GPE and influenced by the nonlinear interaction. The dynamical behavior is visualized using the Bloch sphere formalism. However, the dynamics is not restricted to the surface of the sphere due to the nonlinear and non-Hermitian nature of the system.
We investigate the Gross-Pitaevskii equation for a Bose-Einstein condensate in a PT symmetric double-well potential by means of the time-dependent variational principle and numerically exact solutions. A one-dimensional and a fully three-dimensional setup are used. Stationary states are determined and the propagation of wave function is investigated using the time-dependent Gross-Pitaevskii equation. Due to the nonlinearity of the Gross-Pitaevskii equation the potential dependson the wave function and its solutions decide whether or not the Hamiltonian itself is PT symmetric. Stationary solutions with real energy eigenvalues fulfilling exact PT symmetry are found as well as PT broken eigenstates with complex energies. The latter describe decaying or growing probability amplitudes and are not true stationary solutions of the time-dependent Gross-Pitaevskii equation. However, they still provide qualitative information about the time evolution of the wave functions.
Abstract We investigate the bifurcation structure of stationary states in a dipolar Bose–Einstein condensate located in an external   -symmetric potential. The imaginary part of this external potential allows for the effective description of in- and outcoupling of particles. To unveil the complete bifurcation structure and the properties of the exceptional points we perform an analytical continuation of the Gross–Pitaevskii equation, which is used to describe the system. We use an elegant and numerically efficient method for the analytical continuation of the Gross–Pitaevskii equation with dipolar interactions containing bicomplex numbers. The Bose–Einstein condensate with dipole interaction shows a much richer bifurcation scenario than a condensate without long-range interactions. The inclusion of analytically continued states can also explain property changes of bifurcation points which were hidden before, and allows for the examination of the properties of the exceptional points associated with the branch points. With the new analytically continued states we are able to prove the existence of an exceptional point of fifth order.
Eigenvalue structure of a Bose–Einstein condensate in a PT\mathcal {PT}PT-symmetric double well
We study a Bose–Einstein condensate in a -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 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 analyse 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.
Abstract The existence and stability of gap solitons in a quasi-one-dimensional Bose–Einstein condensate with three-body interactions loaded in a PT-symmetric potential are investigated numerically. Under the mean-field approximation, the dynamical behaviors of the system are described by a cubic-quintic Gross–Pitaevskii equation. Firstly, we obtained the band-gap structures via linearizing the GPE. The PT-symmetric of the system will be broken if the magnitude of the imaginary part of the external potential exceeds a critical value. Secondly, various of gap solitons were found by the Newton-Conjugate-Gradient method. Finally, the stability properties of gap solitons were investigated through the linear stability analysis and the direct long-time nonlinear dynamical evolution. The results indicated that the stability of on-site and off-site solitons was remarkably influenced by three-body interactions. There exists unstable on-site gap solitons when the PT-symmetric potential is taken into account. Additionally, the magnitude of the imaginary part of the potential also affects the profile and stability of the gap solitons. When it increases, the symmetry of the solitons will be broken.