Abstract
The Gross-Pitaevskii equation for a Bose-Einstein condensate in a \(\mathcal {PT}\)-symmetric double-well potential is investigated theoretically. An in- and outcoupling of atoms is modelled by an antisymmetric imaginary potential rendering the Hamiltonian non-Hermitian. Stationary states with real energies and \(\mathcal {PT}\)-symmetric wave functions are found, which proves that Bose-Einstein condensates are a good candidate for a first experimental verification of a \(\mathcal {PT}\)-symmetric quantum system. Time-resolved calculations demonstrate typical effects only observable in \(\mathcal {PT}\)-symmetric potentials, viz. an oscillation of the condensate’s probability density between these wells with an oscillation frequency critically depending on the strength of the in- and outcoupling. \(\mathcal {PT}\)-broken eigenstates with complex energy eigenvalues are also solutions of the time-independent Gross-Pitaevskii equation but are not true stationary states of its time-dependent counterpart. The comparison of a one-dimensional and a three-dimensional calculation shows that it is possible to extract highly precise quantitative results for a fully three-dimensional physical setup from a simple one-dimensional description.
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References
Guo, A., Salamo, G.J., Duchesne, D., Morandotti, R., Volatier-Ravat, M., Aimez, V., Siviloglou, G.A., Christodoulides, D.N.: Observation of \({\cal PT}\)-symmetry breaking in complex optical potentials. Phys. Rev. Lett. 103, 093902 (2009)
Rüter, C.E., Makris, K.G., El-Ganainy, R., Christodoulides, D.N., Segev, M., Kip, D.: Observation of parity-time symmetry in optics. Nat. Phys. 6, 192 (2010)
Bender, C.M., Boettcher, S.: Real spectra in non-Hermitian Hamiltonians having \({\cal PT}\) symmetry. Phys. Rev. Lett. 80, 5243 (1998)
Jakubský, V., Znojil, M.: An explicitly solvable model of the spontaneous \({\cal PT}\)-symmetry breaking. Czech. J. Phys. 55, 1113–1116 (2005)
Ruschhaupt, A., Delgado, F., Muga, J.G.: Physical realization of \({\cal PT}\)-symmetric potential scattering in a planar slab waveguide. J. Phys. A 38, L171 (2005)
Mostafazadeh, A.: Delta-function potential with a complex coupling. J. Phys. A 39, 13495 (2006)
Musslimani, Z., Makris, K.G., El-Ganainy, R., Christodoulides, D.N.: Optical solitons in \({\cal PT}\) periodic potentials. Phys. Rev. Lett. 100, 30402 (2008)
Musslimani, Z.H., Makris, K.G., El-Ganainy, R., Christodoulides, D.N.: Analytical solutions to a class of nonlinear Schrödinger equations with \({\cal PT}\)-like potentials. J. Phys. A 41, 244019 (2008)
Klaiman, S., Günther, U., Moiseyev, N.: Visualization of branch points in \({\cal PT}\)-symmetric waveguides. Phys. Rev. Lett. 101, 080402 (2008)
Graefe, E.M., Günther, U., Korsch, H.J., Niederle, A.E.: A non-Hermitian \({\cal PT}\) symmetric Bose-Hubbard model: eigenvalue rings from unfolding higher-order exceptional points. J. Phys. A 41, 255206 (2008)
Graefe, E.M., Korsch, H.J., Niederle, A.E.: Mean-field dynamics of a non-Hermitian Bose-Hubbard dimer. Phys. Rev. Lett. 101, 150408 (2008)
Makris, K.G., El-Ganainy, R., Christodoulides, D.N., Musslimani, Z.H.: Beam dynamics in \({\cal PT}\) symmetric optical lattices. Phys. Rev. Lett. 100, 103904 (2008)
Jones, H.F.: Interface between Hermitian and non-Hermitian Hamiltonians in a model calculation. Phys. Rev. D 78, 065032 (2008)
Mostafazadeh, A., Mehri-Dehnavi, H.: Spectral singularities, biorthonormal systems and a two-parameter family of complex point interactions. J. Phys. A 42, 125303 (2009)
Ramezani, H., Kottos, T., El-Ganainy, R., Christodoulides, D.N.: Unidirectional nonlinear \({\cal PT}\)-symmetric optical structures. Phys. Rev. A 82, 043803 (2010)
Makris, K.G., El-Ganainy, R., Christodoulides, D.N., Musslimani, Z.H.: \({\cal PT}\)-symmetric optical lattices. Phys. Rev. A 81, 063807 (2010)
Mehri-Dehnavi, H., Mostafazadeh, A., Batal, A.: Application of pseudo-Hermitian quantum mechanics to a complex scattering potential with point interactions. J. Phys. A 43, 145301 (2010)
Moiseyev, N.: Non-Hermitian Quantum Mechanics. Cambridge University Press, Cambridge (2011)
El-Ganainy, R., Makris, K.G., Christodoulides, D.N., Musslimani, Z.H.: Theory of coupled optical \({\cal PT}\)-symmetric structures. Opt. Lett. 32, 2632 (2007)
Gross, E.P.: Structure of a quantized vortex in Boson systems. Nuovo Cimento 20, 454 (1961)
Pitaevskii, L.P.: Vortex lines in an imperfect Bose gas. Sov. Phys. JETP 13, 451 (1961)
Graefe, E.M., Korsch, H.J., Niederle, A.E.: Quantum-classical correspondence for a non-Hermitian Bose-Hubbard dimer. Phys. Rev. A 82, 013629 (2010)
Cartarius, H., Wunner, G.: Model of a \({\cal PT}\)-symmetric Bose-Einstein condensate in a \(\delta \)-function double-well potential. Phys. Rev. A 86, 013612 (2012)
Cartarius, H., Haag, D., Dast, D., Wunner, G.: Nonlinear Schrödinger equation for a \({\cal PT}\)-symmetric delta-function double well. Journal of Physics A 45, 444008 (2012)
Dast, D., Haag, D., Cartarius, H., Wunner, G., Eichler, R., Main, J.: A Bose-Einstein condensate in a \({\cal PT}\) symmetric double well. Fortschritte der Physik 61, 124–139 (2013)
Shin, Y., Jo, G.B., Saba, M., Pasquini, T.A., Ketterle, W., Pritchard, D.E.: Optical weak link between two spatially separated Bose-Einstein condensates. Phys. Rev. Lett. 95, 170402 (2005)
Gati, R., Albiez, M., Fölling, J., Hemmerling, B., Oberthaler, M.: Realization of a single Josephson junction for Bose-Einstein condensates. Appl. Phys. B 82, 207 (2006)
Heller, E.J.: Classical S-matrix limit of wave packet dynamics. J. Chem. Phys. 65, 4979 (1976)
Heller, E.J.: Frozen Gaussians: A very simple semiclassical approximation. J. Chem. Phys. 75, 2923 (1981)
McLachlan, A.D.: A variational solution of the time-dependent Schrödinger equation. Mol. Phys. 8, 39 (1964)
Rau, S., Main, J., Köberle, P., Wunner, G.: Pitchfork bifurcations in blood-cell-shaped dipolar Bose-Einstein condensates. Phys. Rev. A 81, 031605(R) (2010)
Rau, S., Main, J., Wunner, G.: Variational methods with coupled Gaussian functions for Bose-Einstein condensates with long-range interactions. I. General Concept. Phys. Rev. A 82, 023610 (2010)
Rau, S., Main, J., Cartarius, H., Köberle, P., Wunner, G.: Variational methods with coupled Gaussian functions for Bose-Einstein condensates with long-range interactions. II. Applications. Phys. Rev. A 82, 023611 (2010)
Graefe, E.M.: Stationary states of a \({\cal PT}\) symmetric two-mode Bose-Einstein condensate. J. Phys. A 45, 444015 (2012)
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Dast, D., Haag, D., Cartarius, H., Wunner, G., Eichler, R., Main, J. (2016). Description of Bose-Einstein Condensates in \(\mathcal {PT}\)-Symmetric Double Wells. In: Wunner, G., Pelster, A. (eds) Selforganization in Complex Systems: The Past, Present, and Future of Synergetics. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-27635-9_9
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DOI: https://doi.org/10.1007/978-3-319-27635-9_9
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