International Journal of Theoretical Physics

, Volume 54, Issue 11, pp 4100–4109 | Cite as

Finite element Calculations of \(\mathcal {P}\mathcal {T}\)-Symmetric Bose-Einstein Condensates

  • Daniel Haag
  • Dennis Dast
  • Holger Cartarius
  • Günter Wunner
Article
  • 146 Downloads

Abstract

\(\mathcal {P}\mathcal {T}\)-symmetric systems have been intensively studied in optical waveguides, where the \(\mathcal {P}\mathcal {T}\) symmetry is achieved by pumping and absorption processes. In such systems the \(\mathcal {P}\mathcal {T}\) symmetry leads to a wide range of effects promising technical and scientific applications. By analogy, balanced gain and loss of particles in Bose-Einstein condensates (BEC) can be described by introducing a \(\mathcal {P}\mathcal {T}\)-symmetric imaginary potential into the Gross-Pitaevskii equation (GPE). This equation can be solved numerically by various methods including the finite element approach. We apply this method to the GPE with arbitrary complex potentials and explicitly solve a double-well potential with shifted barriers.

Keywords

Bose-Einstein condensation Finite element method \(\mathcal {P}\mathcal {T}\) symmetry 

References

  1. 1.
    Breuer, H.P., Petruccione, F.: The Theory of Open Quantum Systems, 1st edn. Oxford University Press, Oxford (2002)Google Scholar
  2. 2.
    Trimborn, F., Witthaut, D., Wimberger, S.: J. Phys. B 41(17), 171001 (2008)CrossRefADSGoogle Scholar
  3. 3.
    Witthaut, D., Trimborn, F., Hennig, H., Kordas, G., Geisel, T., Wimberger, S.: Phys. Rev. A 83, 063608 (2011)CrossRefADSGoogle Scholar
  4. 4.
    Dast, D., Haag, D., Cartarius, H., Wunner, G.: Phys. Rev. A 90, 052120 (2014)CrossRefADSGoogle Scholar
  5. 5.
    Anglin, J.R., Vardi, A.: Phys. Rev. A 64, 013605 (2001)CrossRefADSGoogle Scholar
  6. 6.
    Vardi, A., Anglin, J.R.: Phys. Rev. Lett 86, 568 (2001)CrossRefADSGoogle Scholar
  7. 7.
    Abdullaev, F.K., Konotop, V.V., Salerno, M., Yulin, A.V.: Phys. Rev. E 82, 056606 (2010)CrossRefADSGoogle Scholar
  8. 8.
    Bludov, Y.V., Konotop, V.V.: Phys. Rev. A 81, 013625 (2010)CrossRefADSGoogle Scholar
  9. 9.
    Moiseyev, N.: Non-Hermitian Quantum Mechanics. Cambridge University Press, Cambridge (2011)MATHCrossRefGoogle Scholar
  10. 10.
    Kagan, Y., Muryshev, A.E., Shlyapnikov, G.V.: Phys. Rev. Lett 81, 933 (1998)CrossRefADSGoogle Scholar
  11. 11.
    Mostafazadeh, A.: J. Math. Phys 43, 205 (2002)MATHMathSciNetCrossRefADSGoogle Scholar
  12. 12.
    Mostafazadeh, A.: J. Math. Phys. 43, 2814 (2002)MATHMathSciNetCrossRefADSGoogle Scholar
  13. 13.
    Mostafazadeh, A.: J. Math. Phys. 43, 3944 (2002)MATHMathSciNetCrossRefADSGoogle Scholar
  14. 14.
    Bender, C.M., Boettcher, S.: Phys. Rev. Lett. 80, 5243 (1998)MATHMathSciNetCrossRefADSGoogle Scholar
  15. 15.
    Bender, C.M., Boettcher, S., Meisinger, P.N.: J. Math. Phys. 40, 2201 (1999)MATHMathSciNetCrossRefADSGoogle Scholar
  16. 16.
    Bender, C.M.: Rep. Prog. Phys. 70, 947 (2007)CrossRefADSGoogle Scholar
  17. 17.
    Klaiman, S., Günther, U., Moiseyev, N.: Phys. Rev. Lett. 101, 080402 (2008)MathSciNetCrossRefADSGoogle Scholar
  18. 18.
    Schindler, J., Li, A., Zheng, M.C., Ellis, F.M., Kottos, T.: Phys. Rev. A 84, 040101 (2011)CrossRefADSGoogle Scholar
  19. 19.
    Bittner, S., Dietz, B., Günther, U., Harney, H.L., Miski-Oglu, M., Richter, A., Schäfer, F.: Phys. Rev. Lett. 108, 02410 (2012)CrossRefGoogle Scholar
  20. 20.
    Mayteevarunyoo, T., Malomed, B.A., Reoksabutr, A.: Phys. Rev. E 88, 022919 (2013)CrossRefADSGoogle Scholar
  21. 21.
    Ru̇ter, C.E., Makris, K.G., El-Ganainy, R., Christodoulides, D.N., Segev, M., Kip, D.: Nat. Phys. 6, 192 (2010)CrossRefGoogle Scholar
  22. 22.
    Guo, A., Salamo, G.J., Duchesne, D., Morandotti, R., Volatier-Ravat, M., Aimez, V., Siviloglou, G.A., Christodoulides, D.N.: Phys. Rev. Lett. 103, 093902 (2009)CrossRefADSGoogle Scholar
  23. 23.
    Peng, B., Özdemir, .K., Rotter, S., Yilmaz, H., Liertzer, M., Monifi, F., Bender, C.M., Nori, F., Yang, L.: Science 346, 328 (2014)CrossRefADSGoogle Scholar
  24. 24.
    Peng, B., Özdemir, Ṡ.K., Lei, F., Monifi, F., Gianfreda, M., Long, G.L., Fan, S., Nori, F., Bender, C.M., Yang, L.: Nat. Phys. 10, 394 (2014)CrossRefGoogle Scholar
  25. 25.
    Jones, H.F.: Phys. Rev. D 78, 065032 (2008)CrossRefADSGoogle Scholar
  26. 26.
    Cartarius, H., Haag, D., Dast, D., Wunner, G.: J. Phys. A 45, 444008 (2012)MathSciNetCrossRefADSGoogle Scholar
  27. 27.
    Cartarius, H., Wunner, G.: Phys. Rev. A 86, 013612 (2012)CrossRefADSGoogle Scholar
  28. 28.
    Konotop, V.V., Zezyulin, D.A.: Opt. Lett. 39 (5), 1223 (2014)CrossRefADSGoogle Scholar
  29. 29.
    Midya, B.: Nonlinear Dyn. (2014). doi:10.1007/s11071-014-1674-9
  30. 30.
    Graefe, E.M.: J. Phys. A 45, 444015 (2012)MathSciNetCrossRefADSGoogle Scholar
  31. 31.
    Dast, D., Haag, D., Cartarius, H., Wunner, G., Eichler, R., Main, J.: Fortschr. Physik 61, 124 (2013)CrossRefADSGoogle Scholar
  32. 32.
    Rodrigues, A.S., Li, K., Achilleos, V., Kevrekidis, P.G., Frantzeskakis, D.J., Bender, C.M.: Rom. Rep. Phys. 65, 5 (2013)Google Scholar
  33. 33.
    Dast, D., Haag, D., Cartarius, H., Main, J., Wunner, G.: J. Phys. A 46, 375301 (2013)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Haag, D., Dast, D., Löhle, A., Cartarius, H., Main, J., Wunner, G.: Phys. Rev. A 89, 023601 (2014)CrossRefADSGoogle Scholar
  35. 35.
    Kreibich, M., Main, J., Cartarius, H., Wunner, G.: Phys. Rev. A 87(R), 051601 (2013)CrossRefADSGoogle Scholar
  36. 36.
    Single, F., Cartarius, H., Wunner, G., Main, J.: Phys. Rev. A 90, 042123 (2014)CrossRefADSGoogle Scholar
  37. 37.
    Gericke, T., Wurtz, P., Reitz, D., Langen, T., Ott, H.: Nat. Phys. 4, 949 (2008)CrossRefGoogle Scholar
  38. 38.
    Robins, N.P., Figl, C., Jeppesen, M., Dennis, G.R., Close, J.D.: Nat. Phys. 4, 731 (2008)CrossRefGoogle Scholar
  39. 39.
    Schimeczek, C., Wunner, G.: Comp. Phys. Comm. 185(2), 614 (2014)Google Scholar
  40. 40.
    de Boor, C.: SIAM J. Numer. Anal. 14(3), 441 (1977)MATHMathSciNetCrossRefADSGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Daniel Haag
    • 1
  • Dennis Dast
    • 1
  • Holger Cartarius
    • 1
  • Günter Wunner
    • 1
  1. 1.1. Institut für Theoretische PhysikUniversität StuttgartStuttgartGermany

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