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arXiv:1306.3871 · Journal of Physics A Mathematical and Theoretical

Eigenvalue structure of a Bose–Einstein condensate in a $\mathcal {PT}$-symmetric double well

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

Published Aug 21, 2013• 63 citations
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Abstract

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.

DOI
10.1088/1751-8113/46/37/375301

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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 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.
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.
The understanding of nonlinear PT symmetric quantum systems, arising for example in the theory of Bose–Einstein 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 Bose–Einstein 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 ‘Quantum physics with non-Hermitian operators’.
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.
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.
The observation of PT\mathcal{PT}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\mathcal{PT}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\mathcal{PT}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\mathcal{PT}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\mathcal{PT}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\mathcal{PT}PT-symmetric double wells.