September  2017, 10(3): 553-571. doi: 10.3934/krm.2017022

Dimension reduction for dipolar Bose-Einstein condensates in the strong interaction regime

1. 

Department of Mathematics, National University of Singapore, Singapore 119076, Singapore

2. 

IRMAR, Université de Rennes 1 and INRIA-Rennes, IPSO Project, 35042 Rennes, France

Received  October 2015 Revised  September 2016 Published  December 2016

Fund Project: This work was supported by the Academic Research Fund of Ministry of Education of Singapore grant No. R-146-000-223-112 (W.B.), by the ANR-FWF Project Lodiquas ANR-11-IS01-0003 and by the ANR project Moonrise ANR-14-CE23-0007-01 (L.L.T. and F.M.)

We study dimension reduction for the three-dimensional Gross-Pitaevskii equation with a long-range and anisotropic dipole-dipole interaction modeling dipolar Bose-Einstein condensation in a strong interaction regime. The cases of disk shaped condensates (confinement from dimension three to dimension two) and cigar shaped condensates (confinement to dimension one) are analyzed. In both cases, the analysis combines averaging tools and semiclassical techniques. Asymptotic models are derived, with rates of convergence in terms of two small dimensionless parameters characterizing the strength of the confinement and the strength of the interaction between atoms.

Citation: Weizhu Bao, Loïc Le Treust, Florian Méhats. Dimension reduction for dipolar Bose-Einstein condensates in the strong interaction regime. Kinetic & Related Models, 2017, 10 (3) : 553-571. doi: 10.3934/krm.2017022
References:
[1]

W. BaoN. Ben Abdallah and Y. Cai, Gross-Pitaevskii-Poisson equations for dipolar Bose-Einstein condensate with anisotropic confinement, SIAM J. Math. Anal., 44 (2012), 1713-1741. doi: 10.1137/110850451. Google Scholar

[2]

W. Bao and Y. Cai, Mathematical theory and numerical methods for Bose-Einstein condensation, Kinet. Relat. Mod., 6 (2013), 1-135. doi: 10.3934/krm.2013.6.1. Google Scholar

[3]

W. BaoY. Cai and H. Wang, Efficient numerical methods for computing ground states and dynamics of dipolar Bose-Einstein condensates, J. Comput. Phys., 229 (2010), 7874-7892. doi: 10.1016/j.jcp.2010.07.001. Google Scholar

[4]

W. BaoL. Le Treust and F. Méhats, Dimension reduction for anisotropic Bose-Einstein condensates in the strong interaction regime, Nonlinearity, 28 (2015), 755-772. doi: 10.1088/0951-7715/28/3/755. Google Scholar

[5]

N. Ben AbdallahY. CaiC. Castella and F. Méhats, Second order averaging for the nonlinear Schrödinger equation with strongly anisotropic potential, Kinet. Relat. Models, 4 (2011), 831-856. doi: 10.3934/krm.2011.4.831. Google Scholar

[6]

N. Ben AbdallahF. Castella and F. Méhats, Time averaging for the strongly confined nonlinear Schrödinger equation, using almost-periodicity, J. Differ. Equations, 245 (2008), 154-200. doi: 10.1016/j.jde.2008.02.002. Google Scholar

[7]

Y. CaiM. RosenkranzZ. Lei and B. Bao, Mean-field regime of trapped dipolar Bose-Einstein condensates in one and two dimensions, Phys. Rev. A, 82 (2010), 043623. Google Scholar

[8]

R. Carles, Semi-classical Analysis for Nonlinear Schrödinger Equations, World Scientific, 2008.Google Scholar

[9]

R. CarlesP. A. Markowich and C. Sparber, On the Gross-Pitaevskii equation for trapped dipolar quantum gases, Nonlinearity, 21 (2008), 2569-2590. doi: 10.1088/0951-7715/21/11/006. Google Scholar

[10]

T. Cazenave, Semilinear Schrödinger Equations, vol. 10, AMS Bookstore, 2003.Google Scholar

[11]

A. Griesmaier, J. Werner, S. Hensler, J. Stuhler and T. Pfau, Bose-Einstein Condensation of Chromium, Phys. Rev. Lett., 2005.Google Scholar

[12]

B. Helffer, Théorie Spectrale Pour Des Opérateurs Globalement Elliptiques, Société mathématique de France, 1984. Google Scholar

[13]

L. LahayeC. MenottiL. SantosM. Lewenstein and T. Pfau, The physics of dipolar bosonic quantum gases, Rep. Prog. Phys., 72 (2009), 126401. Google Scholar

[14]

M. LuN.Q. BurdickS.H. Youn and B.H. Lev, A strongly dipolar Bose-Einstein condensate of dysprosium, Phy. Rev. Lett., 107 (2011), 190401. Google Scholar

[15] C.J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases, Cambridge University Press, 2002. Google Scholar
[16] L.P. Pitaevskii and S. Stringari, Bose-Einstein Condensation, Clarendon Press, Oxford, 2003. Google Scholar
[17]

M. RosenkranzY. Cai and W. Bao, Effective dipole-dipole interactions in multilayered dipolar Bose-Einstein condensates, Phys. Rev. A, 88 (2013), 013616. Google Scholar

[18]

L. SantosG. ShlyapnikovP. Zoller and M. Lewenstein, Bose-Einstein condensation in trapped dipolar gases, Phys. Rev. Lett., 85 (2000), 1791-1797. Google Scholar

[19]

S. Yi and L. You, Trapped atomic condensates with anisotropic interactions, Phys. Rev. A, 61 (2000), 041604. Google Scholar

[20]

S. Yi and L. You, Trapped condensates of atoms with dipole interactions, Phys. Rev. A, 63 (2000), 053607. Google Scholar

show all references

References:
[1]

W. BaoN. Ben Abdallah and Y. Cai, Gross-Pitaevskii-Poisson equations for dipolar Bose-Einstein condensate with anisotropic confinement, SIAM J. Math. Anal., 44 (2012), 1713-1741. doi: 10.1137/110850451. Google Scholar

[2]

W. Bao and Y. Cai, Mathematical theory and numerical methods for Bose-Einstein condensation, Kinet. Relat. Mod., 6 (2013), 1-135. doi: 10.3934/krm.2013.6.1. Google Scholar

[3]

W. BaoY. Cai and H. Wang, Efficient numerical methods for computing ground states and dynamics of dipolar Bose-Einstein condensates, J. Comput. Phys., 229 (2010), 7874-7892. doi: 10.1016/j.jcp.2010.07.001. Google Scholar

[4]

W. BaoL. Le Treust and F. Méhats, Dimension reduction for anisotropic Bose-Einstein condensates in the strong interaction regime, Nonlinearity, 28 (2015), 755-772. doi: 10.1088/0951-7715/28/3/755. Google Scholar

[5]

N. Ben AbdallahY. CaiC. Castella and F. Méhats, Second order averaging for the nonlinear Schrödinger equation with strongly anisotropic potential, Kinet. Relat. Models, 4 (2011), 831-856. doi: 10.3934/krm.2011.4.831. Google Scholar

[6]

N. Ben AbdallahF. Castella and F. Méhats, Time averaging for the strongly confined nonlinear Schrödinger equation, using almost-periodicity, J. Differ. Equations, 245 (2008), 154-200. doi: 10.1016/j.jde.2008.02.002. Google Scholar

[7]

Y. CaiM. RosenkranzZ. Lei and B. Bao, Mean-field regime of trapped dipolar Bose-Einstein condensates in one and two dimensions, Phys. Rev. A, 82 (2010), 043623. Google Scholar

[8]

R. Carles, Semi-classical Analysis for Nonlinear Schrödinger Equations, World Scientific, 2008.Google Scholar

[9]

R. CarlesP. A. Markowich and C. Sparber, On the Gross-Pitaevskii equation for trapped dipolar quantum gases, Nonlinearity, 21 (2008), 2569-2590. doi: 10.1088/0951-7715/21/11/006. Google Scholar

[10]

T. Cazenave, Semilinear Schrödinger Equations, vol. 10, AMS Bookstore, 2003.Google Scholar

[11]

A. Griesmaier, J. Werner, S. Hensler, J. Stuhler and T. Pfau, Bose-Einstein Condensation of Chromium, Phys. Rev. Lett., 2005.Google Scholar

[12]

B. Helffer, Théorie Spectrale Pour Des Opérateurs Globalement Elliptiques, Société mathématique de France, 1984. Google Scholar

[13]

L. LahayeC. MenottiL. SantosM. Lewenstein and T. Pfau, The physics of dipolar bosonic quantum gases, Rep. Prog. Phys., 72 (2009), 126401. Google Scholar

[14]

M. LuN.Q. BurdickS.H. Youn and B.H. Lev, A strongly dipolar Bose-Einstein condensate of dysprosium, Phy. Rev. Lett., 107 (2011), 190401. Google Scholar

[15] C.J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases, Cambridge University Press, 2002. Google Scholar
[16] L.P. Pitaevskii and S. Stringari, Bose-Einstein Condensation, Clarendon Press, Oxford, 2003. Google Scholar
[17]

M. RosenkranzY. Cai and W. Bao, Effective dipole-dipole interactions in multilayered dipolar Bose-Einstein condensates, Phys. Rev. A, 88 (2013), 013616. Google Scholar

[18]

L. SantosG. ShlyapnikovP. Zoller and M. Lewenstein, Bose-Einstein condensation in trapped dipolar gases, Phys. Rev. Lett., 85 (2000), 1791-1797. Google Scholar

[19]

S. Yi and L. You, Trapped atomic condensates with anisotropic interactions, Phys. Rev. A, 61 (2000), 041604. Google Scholar

[20]

S. Yi and L. You, Trapped condensates of atoms with dipole interactions, Phys. Rev. A, 63 (2000), 053607. Google Scholar

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