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arXiv:1403.6742 · Physical Review A

Dipolar Bose-Einstein condensates in a PT -symmetric double-well potential

Rüdiger Fortanier, Dennis Dast, Daniel Haag, Holger Cartarius, Jörg Main, Günter Wunner, Robin Gutöhrlein

Published Jun 9, 2014• 36 citations
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Abstract

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 $\mathcal{PT}$ symmetry and investigate the stability of the distinct stationary solutions. In our dynamical simulations we reveal a complex stabilization mechanism for $\mathcal{PT}$-symmetric as well as for $\mathcal{PT}$-broken states which are, in principle, unstable with respect to small perturbations.

DOI
10.1103/physreva.89.063608

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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 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.
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.
Abstract The existence of 𝒫𝒯 symmetric wave functions describing Bose‐Einstein condensates in a one‐dimensional and a fully three‐dimensional double‐well setup is investigated theoretically. When particles are removed from one well and coherently injected into the other the external potential is 𝒫𝒯 symmetric. We solve the underlying Gross‐Pitaevskii equation by way of the time‐dependent variational principle (TDVP) and show that the 𝒫𝒯 symmetry of the external potential is preserved by both the wave functions and the nonlinear Hamiltonian as long as eigenstates with real eigenvalues are obtained. 𝒫𝒯 broken solutions of the time‐independent Gross‐Pitaevskii equation are also found but have no physical relevance. To prove the applicability of the TDVP we compare its results with numerically exact solutions in the one‐dimensional case. The linear stability analysis and the temporal evolution of condensate wave functions demonstrate that the 𝒫𝒯 symmetric condensates are stable and should be observable in an experiment.
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.
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.
An analytical approximation method to clarify the symmetry-breaking stationary solutions to the Gross–Pitaevskii equation with symmetric double-well external potential is presented. By this method, we can understand that each symmetry-breaking solution bifurcates from a symmetry-preserving solution as one varies the coupling constant of the nonlinear interaction, and predict the bifurcation point. Also, the method gives a good approximation to the energy eigenvalues of the lower states, symmetry-breaking as well as symmetry-preserving, as long as the nonlinear interaction is not too strong.