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

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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.

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References

  1. Breuer, H.P., Petruccione, F.: The Theory of Open Quantum Systems, 1st edn. Oxford University Press, Oxford (2002)

    Google Scholar 

  2. Trimborn, F., Witthaut, D., Wimberger, S.: J. Phys. B 41(17), 171001 (2008)

    Article  ADS  Google Scholar 

  3. Witthaut, D., Trimborn, F., Hennig, H., Kordas, G., Geisel, T., Wimberger, S.: Phys. Rev. A 83, 063608 (2011)

    Article  ADS  Google Scholar 

  4. Dast, D., Haag, D., Cartarius, H., Wunner, G.: Phys. Rev. A 90, 052120 (2014)

    Article  ADS  Google Scholar 

  5. Anglin, J.R., Vardi, A.: Phys. Rev. A 64, 013605 (2001)

    Article  ADS  Google Scholar 

  6. Vardi, A., Anglin, J.R.: Phys. Rev. Lett 86, 568 (2001)

    Article  ADS  Google Scholar 

  7. Abdullaev, F.K., Konotop, V.V., Salerno, M., Yulin, A.V.: Phys. Rev. E 82, 056606 (2010)

    Article  ADS  Google Scholar 

  8. Bludov, Y.V., Konotop, V.V.: Phys. Rev. A 81, 013625 (2010)

    Article  ADS  Google Scholar 

  9. Moiseyev, N.: Non-Hermitian Quantum Mechanics. Cambridge University Press, Cambridge (2011)

    Book  MATH  Google Scholar 

  10. Kagan, Y., Muryshev, A.E., Shlyapnikov, G.V.: Phys. Rev. Lett 81, 933 (1998)

    Article  ADS  Google Scholar 

  11. Mostafazadeh, A.: J. Math. Phys 43, 205 (2002)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  12. Mostafazadeh, A.: J. Math. Phys. 43, 2814 (2002)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  13. Mostafazadeh, A.: J. Math. Phys. 43, 3944 (2002)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  14. Bender, C.M., Boettcher, S.: Phys. Rev. Lett. 80, 5243 (1998)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  15. Bender, C.M., Boettcher, S., Meisinger, P.N.: J. Math. Phys. 40, 2201 (1999)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  16. Bender, C.M.: Rep. Prog. Phys. 70, 947 (2007)

    Article  ADS  Google Scholar 

  17. Klaiman, S., Günther, U., Moiseyev, N.: Phys. Rev. Lett. 101, 080402 (2008)

    Article  MathSciNet  ADS  Google Scholar 

  18. Schindler, J., Li, A., Zheng, M.C., Ellis, F.M., Kottos, T.: Phys. Rev. A 84, 040101 (2011)

    Article  ADS  Google Scholar 

  19. Bittner, S., Dietz, B., Günther, U., Harney, H.L., Miski-Oglu, M., Richter, A., Schäfer, F.: Phys. Rev. Lett. 108, 02410 (2012)

    Article  Google Scholar 

  20. Mayteevarunyoo, T., Malomed, B.A., Reoksabutr, A.: Phys. Rev. E 88, 022919 (2013)

    Article  ADS  Google Scholar 

  21. Ru̇ter, C.E., Makris, K.G., El-Ganainy, R., Christodoulides, D.N., Segev, M., Kip, D.: Nat. Phys. 6, 192 (2010)

    Article  Google Scholar 

  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)

    Article  ADS  Google Scholar 

  23. Peng, B., Özdemir, .K., Rotter, S., Yilmaz, H., Liertzer, M., Monifi, F., Bender, C.M., Nori, F., Yang, L.: Science 346, 328 (2014)

    Article  ADS  Google Scholar 

  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)

    Article  Google Scholar 

  25. Jones, H.F.: Phys. Rev. D 78, 065032 (2008)

    Article  ADS  Google Scholar 

  26. Cartarius, H., Haag, D., Dast, D., Wunner, G.: J. Phys. A 45, 444008 (2012)

    Article  MathSciNet  ADS  Google Scholar 

  27. Cartarius, H., Wunner, G.: Phys. Rev. A 86, 013612 (2012)

    Article  ADS  Google Scholar 

  28. Konotop, V.V., Zezyulin, D.A.: Opt. Lett. 39 (5), 1223 (2014)

    Article  ADS  Google Scholar 

  29. Midya, B.: Nonlinear Dyn. (2014). doi:10.1007/s11071-014-1674-9

  30. Graefe, E.M.: J. Phys. A 45, 444015 (2012)

    Article  MathSciNet  ADS  Google Scholar 

  31. Dast, D., Haag, D., Cartarius, H., Wunner, G., Eichler, R., Main, J.: Fortschr. Physik 61, 124 (2013)

    Article  ADS  Google Scholar 

  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. Dast, D., Haag, D., Cartarius, H., Main, J., Wunner, G.: J. Phys. A 46, 375301 (2013)

    Article  MathSciNet  Google Scholar 

  34. Haag, D., Dast, D., Löhle, A., Cartarius, H., Main, J., Wunner, G.: Phys. Rev. A 89, 023601 (2014)

    Article  ADS  Google Scholar 

  35. Kreibich, M., Main, J., Cartarius, H., Wunner, G.: Phys. Rev. A 87(R), 051601 (2013)

    Article  ADS  Google Scholar 

  36. Single, F., Cartarius, H., Wunner, G., Main, J.: Phys. Rev. A 90, 042123 (2014)

    Article  ADS  Google Scholar 

  37. Gericke, T., Wurtz, P., Reitz, D., Langen, T., Ott, H.: Nat. Phys. 4, 949 (2008)

    Article  Google Scholar 

  38. Robins, N.P., Figl, C., Jeppesen, M., Dennis, G.R., Close, J.D.: Nat. Phys. 4, 731 (2008)

    Article  Google Scholar 

  39. Schimeczek, C., Wunner, G.: Comp. Phys. Comm. 185(2), 614 (2014)

  40. de Boor, C.: SIAM J. Numer. Anal. 14(3), 441 (1977)

    Article  MATH  MathSciNet  ADS  Google Scholar 

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Haag, D., Dast, D., Cartarius, H. et al. Finite element Calculations of \(\mathcal {P}\mathcal {T}\)-Symmetric Bose-Einstein Condensates. Int J Theor Phys 54, 4100–4109 (2015). https://doi.org/10.1007/s10773-014-2481-2

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  • DOI: https://doi.org/10.1007/s10773-014-2481-2

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