Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1605.01927 · Journal of Physics A Mathematical and Theoretical

Bifurcations and exceptional points in a -symmetric dipolar Bose–Einstein condensate

Robin Gutöhrlein, Holger Cartarius, Jörg Main, Günter Wunner

Published Nov 7, 2016• 15 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1751-8113/49/48/485301

Related papers

Bifurcations and exceptional points in dipolar Bose–Einstein condensates

Robin Gutöhrlein, Jörg Main, Holger Cartarius +1 more

Bose–Einstein condensates are described in a mean-field approach by the nonlinear Gross–Pitaevskii equation and exhibit phenomena of nonlinear dynamics. The stationary states can undergo bifurcations in such a way that two or more eigenvalues and the corresponding wavefunctions coalesce at critical values of external parameters. For example, in condensates without long-range interactions a stable and an unstable state are created in a tangent bifurcation when the scattering length of the contact interaction is varied. At the critical point, the coalescing states show the properties of an exceptional point. In dipolar condensates fingerprints of a pitchfork bifurcation have been discovered by Rau et al (2010 Phys. Rev. A 81 031605). We present a method to uncover all states participating in a pitchfork bifurcation, and investigate in detail the signatures of exceptional points related to bifurcations in dipolar condensates. For the perturbation by two parameters, namely the scattering length and a parameter breaking the cylindrical symmetry of the harmonic trap, two cases leading to different characteristic eigenvalue and eigenvector patterns under cyclic variations of the parameters need to be distinguished. The observed structures resemble those of three coalescing eigenfunctions obtained by Demange and Graefe (2012 J. Phys. A: Math. Theor. 45 025303) using perturbation theory for non-Hermitian operators in a linear model. Furthermore, the splitting of the exceptional point under symmetry breaking in either two or three branching singularities is examined. Characteristic features are observed when one, two or three exceptional points are encircled simultaneously.

arXiv:1302.5615•Journal of Physics A Mathematical and Theoretical•Jul 5, 2013•38 citations

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

Dennis Dast, Daniel Haag, Holger Cartarius +2 more

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.

arXiv:1306.3871•Journal of Physics A Mathematical and Theoretical•Aug 21, 2013•63 citations

Bifurcations, order and chaos in the Bose–Einstein condensation of dipolar gases

Patrick Köberle, Holger Cartarius, Tomaž Fabčič +2 more

We apply a variational technique to solve the time-dependent Gross–Pitaevskii equation for Bose–Einstein condensates in which an additional dipole–dipole interaction between the atoms is present with the goal of modelling the dynamics of such condensates. We show that universal stability thresholds for the collapse of the condensates correspond to bifurcation points where always two stationary solutions of the Gross–Pitaevskii equation disappear in a tangent bifurcation, one dynamically stable and the other unstable. We point out that the thresholds also correspond to 'exceptional points', i.e. branching singularities of the Hamiltonian. We analyse the dynamics of excited condensate wave functions via Poincaré surfaces of section for the condensate parameters and find both regular and chaotic motion, corresponding to (quasi-) periodically oscillating and irregularly fluctuating condensates, respectively. Stable islands are found to persist up to energies well above the saddle point of the mean-field energy, alongside collapsing modes. The results are applicable when the shape of the condensate is axisymmetric.

arXiv:0802.4055•New Journal of Physics•Feb 9, 2009•23 citations

Discovery of exceptional points in the Bose-Einstein condensation of gases with attractive 1/r interaction

Holger Cartarius, Jörg Main, Günter Wunner

The extended Gross-Pitaevskii equation for the Bose-Einstein condensation of gases with attractive $1/r$ interaction has a second solution which is born together with the ground state in a tangent bifurcation. At the bifurcation point both states coalesce, i.e., the energies and the wave functions are identical. We investigate the bifurcation point in the context of exceptional points, a phenomenon known for linear non-Hermitian Hamiltonians. We point out that the mean field energy, the chemical potential, and the wave functions show the same behavior as an exceptional point in a linear, nonsymmetric system. The analysis of the analytically continued Gross-Pitaevskii equation reveals complex waves at negative scattering lengths below the tangent bifurcation. These solutions are interpreted as a decay of the condensate caused by an absorbing potential.

arXiv:0711.4786•Physical Review A•Jan 25, 2008•49 citations

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

Andreas Löhle, Holger Cartarius, Daniel Haag +3 more

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.

arXiv:1401.2354•Acta Polytechnica•Apr 30, 2014•7 citations

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

Rüdiger Fortanier, Dennis Dast, Daniel Haag +4 more

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.

arXiv:1403.6742•Physical Review A•Jun 9, 2014•36 citations
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.
Bose–Einstein condensates are described in a mean-field approach by the nonlinear Gross–Pitaevskii equation and exhibit phenomena of nonlinear dynamics. The stationary states can undergo bifurcations in such a way that two or more eigenvalues and the corresponding wavefunctions coalesce at critical values of external parameters. For example, in condensates without long-range interactions a stable and an unstable state are created in a tangent bifurcation when the scattering length of the contact interaction is varied. At the critical point, the coalescing states show the properties of an exceptional point. In dipolar condensates fingerprints of a pitchfork bifurcation have been discovered by Rau et al (2010 Phys. Rev. A 81 031605). We present a method to uncover all states participating in a pitchfork bifurcation, and investigate in detail the signatures of exceptional points related to bifurcations in dipolar condensates. For the perturbation by two parameters, namely the scattering length and a parameter breaking the cylindrical symmetry of the harmonic trap, two cases leading to different characteristic eigenvalue and eigenvector patterns under cyclic variations of the parameters need to be distinguished. The observed structures resemble those of three coalescing eigenfunctions obtained by Demange and Graefe (2012 J. Phys. A: Math. Theor. 45 025303) using perturbation theory for non-Hermitian operators in a linear model. Furthermore, the splitting of the exceptional point under symmetry breaking in either two or three branching singularities is examined. Characteristic features are observed when one, two or three exceptional points are encircled simultaneously.
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.
We apply a variational technique to solve the time-dependent Gross–Pitaevskii equation for Bose–Einstein condensates in which an additional dipole–dipole interaction between the atoms is present with the goal of modelling the dynamics of such condensates. We show that universal stability thresholds for the collapse of the condensates correspond to bifurcation points where always two stationary solutions of the Gross–Pitaevskii equation disappear in a tangent bifurcation, one dynamically stable and the other unstable. We point out that the thresholds also correspond to 'exceptional points', i.e. branching singularities of the Hamiltonian. We analyse the dynamics of excited condensate wave functions via Poincaré surfaces of section for the condensate parameters and find both regular and chaotic motion, corresponding to (quasi-) periodically oscillating and irregularly fluctuating condensates, respectively. Stable islands are found to persist up to energies well above the saddle point of the mean-field energy, alongside collapsing modes. The results are applicable when the shape of the condensate is axisymmetric.
The extended Gross-Pitaevskii equation for the Bose-Einstein condensation of gases with attractive 1/r1/r1/r interaction has a second solution which is born together with the ground state in a tangent bifurcation. At the bifurcation point both states coalesce, i.e., the energies and the wave functions are identical. We investigate the bifurcation point in the context of exceptional points, a phenomenon known for linear non-Hermitian Hamiltonians. We point out that the mean field energy, the chemical potential, and the wave functions show the same behavior as an exceptional point in a linear, nonsymmetric system. The analysis of the analytically continued Gross-Pitaevskii equation reveals complex waves at negative scattering lengths below the tangent bifurcation. These solutions are interpreted as a decay of the condensate caused by an absorbing potential.
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.