American Institute of Mathematical Sciences

Advanced
September  2016, 36(9): 5097-5118. doi: 10.3934/dcds.2016021

Dimension reduction for rotating Bose-Einstein condensates with anisotropic confinement

Florian Méhats 1, and  Christof Sparber 2,

1. 

IRMAR, Université de Rennes 1 and IPSO, INRIA Rennes, 35042 Rennes Cedex
2. 

Department of Mathematics, Statistics, and Computer Science, University of Illinois, Chicago, 851 South Morgan Street, Chicago, Illinois 60607

Received  July 2015 Revised  February 2016 Published  May 2016

We consider the three-dimensional time-dependent Gross-Pitaevskii equation arising in the description of rotating Bose-Einstein condensates and study the corresponding scaling limit of strongly anisotropic confinement potentials. The resulting effective equations in one or two spatial dimensions, respectively, are rigorously obtained as special cases of an averaged three dimensional limit model. In the particular case where the rotation axis is not parallel to the strongly confining direction the resulting limiting model(s) include a negative, and thus, purely repulsive quadratic potential, which is not present in the original equation and which can be seen as an effective centrifugal force counteracting the confinement.
Keywords: Gross-Pitaevskii equation, angular momentum operator., averaging, dimension reduction, Bose-Einstein condensation.
Mathematics Subject Classification: Primary: 35Q55; Secondary: 35B2.
Citation: Florian Méhats, Christof Sparber. Dimension reduction for rotating Bose-Einstein condensates with anisotropic confinement. Discrete & Continuous Dynamical Systems - A, 2016, 36 (9) : 5097-5118. doi: 10.3934/dcds.2016021
References:
[1]

P. Antonelli, R. Carles and J. D. Silva, Scattering for nonlinear Schrödinger equation under partial harmonic confinement,, Commun. Math. Phys., 334 (2015), 367.  doi: 10.1007/s00220-014-2166-y.  Google Scholar

[2]

P. Antonelli, D. Marahrens and C. Sparber, On the Cauchy problem for nonlinear Schrödinger equations with rotation,, Discrete Contin. Dyn. Syst., 32 (2012), 703.  doi: 10.3934/dcds.2012.32.703.  Google Scholar

[3]

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

[4]

W. Bao, P. A. Markowich, C. Schmeiser and R. M. Weishäupl, On the Gross-Pitaevskii equation with strongly anisotropic confinement: Formal asymptotics and numerical experiments,, Math. Models Meth. Appl. Sci., 15 (2005), 767.  doi: 10.1142/S0218202505000534.  Google Scholar

[5]

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

[6]

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

[7]

N. Ben Abdallah, F. Méhats, C. Schmeiser and R. M. Weishäupl, The nonlinear Schrödinger equation with a strongly anisotropic harmonic potential,, SIAM J. Math. Anal., 37 (2005), 189.  doi: 10.1137/040614554.  Google Scholar

[8]

R. Carles, Global existence results for nonlinear Schrödinger equations with quadratic potentials., Discrete Contin. Dyn. Syst., 13 (2005), 385.  doi: 10.3934/dcds.2005.13.385.  Google Scholar

[9]

R. Carles, Nonlinear Schrödinger equations with repulsive harmonic potential and applications,, SIAM J. Math. Anal., 35 (2003), 823.  doi: 10.1137/S0036141002416936.  Google Scholar

[10]

T. Cazenave, Semilinear Schrödinger Equations,, Courant Lecture Notes in Mathematics, (2003).   Google Scholar

[11]

F. Delebecque-Fendt and F. Méhats, An effective mass theorem for the bidimensional electron gas in a strong magnetic field,, Comm. Math. Phys., 292 (2009), 829.  doi: 10.1007/s00220-009-0868-3.  Google Scholar

[12]

S. Flügge, Practical Quantum Mechanics,, Classics in Mathematics, (1999).   Google Scholar

[13]

G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities,, Cambridge University Press, (1988).   Google Scholar

[14]

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

[15]

H. Kitada, On a construction of the fundamental solution for Schrödinger equations,, J. Fac. Sci. Univ. Tokyo Sec. IA, 27 (1980), 193.   Google Scholar

[16]

E. H. Lieb, R. Seiringer, J. P. Solovej and J. Yngvason, The Mathematics of the Bose Gas and its Condensation,, Oberwolfach Seminars, (2005).   Google Scholar

[17]

E. Lieb and R. Seiringer, Derivation of the Gross-Pitaevskii equation for rotating Bose gases,, Comm. Math. Phys., 264 (2006), 505.  doi: 10.1007/s00220-006-1524-9.  Google Scholar

[18]

C. J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases,, Cambridge University Press, (2002).   Google Scholar

[19]

L. P. Pitaevskii and S. Stringari, Bose-Einstein Condensation,, Clarendon Press, (2003).   Google Scholar

[20]

N. Tzvetkov and N. Visciglia, Small data scattering for the nonlinear Schrödinger equation on product spaces,, Commun. Partial Differ. Equ., 37 (2012), 125.  doi: 10.1080/03605302.2011.574306.  Google Scholar

show all references

References:
[1]

P. Antonelli, R. Carles and J. D. Silva, Scattering for nonlinear Schrödinger equation under partial harmonic confinement,, Commun. Math. Phys., 334 (2015), 367.  doi: 10.1007/s00220-014-2166-y.  Google Scholar

[2]

P. Antonelli, D. Marahrens and C. Sparber, On the Cauchy problem for nonlinear Schrödinger equations with rotation,, Discrete Contin. Dyn. Syst., 32 (2012), 703.  doi: 10.3934/dcds.2012.32.703.  Google Scholar

[3]

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

[4]

W. Bao, P. A. Markowich, C. Schmeiser and R. M. Weishäupl, On the Gross-Pitaevskii equation with strongly anisotropic confinement: Formal asymptotics and numerical experiments,, Math. Models Meth. Appl. Sci., 15 (2005), 767.  doi: 10.1142/S0218202505000534.  Google Scholar

[5]

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

[6]

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

[7]

N. Ben Abdallah, F. Méhats, C. Schmeiser and R. M. Weishäupl, The nonlinear Schrödinger equation with a strongly anisotropic harmonic potential,, SIAM J. Math. Anal., 37 (2005), 189.  doi: 10.1137/040614554.  Google Scholar

[8]

R. Carles, Global existence results for nonlinear Schrödinger equations with quadratic potentials., Discrete Contin. Dyn. Syst., 13 (2005), 385.  doi: 10.3934/dcds.2005.13.385.  Google Scholar

[9]

R. Carles, Nonlinear Schrödinger equations with repulsive harmonic potential and applications,, SIAM J. Math. Anal., 35 (2003), 823.  doi: 10.1137/S0036141002416936.  Google Scholar

[10]

T. Cazenave, Semilinear Schrödinger Equations,, Courant Lecture Notes in Mathematics, (2003).   Google Scholar

[11]

F. Delebecque-Fendt and F. Méhats, An effective mass theorem for the bidimensional electron gas in a strong magnetic field,, Comm. Math. Phys., 292 (2009), 829.  doi: 10.1007/s00220-009-0868-3.  Google Scholar

[12]

S. Flügge, Practical Quantum Mechanics,, Classics in Mathematics, (1999).   Google Scholar

[13]

G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities,, Cambridge University Press, (1988).   Google Scholar

[14]

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

[15]

H. Kitada, On a construction of the fundamental solution for Schrödinger equations,, J. Fac. Sci. Univ. Tokyo Sec. IA, 27 (1980), 193.   Google Scholar

[16]

E. H. Lieb, R. Seiringer, J. P. Solovej and J. Yngvason, The Mathematics of the Bose Gas and its Condensation,, Oberwolfach Seminars, (2005).   Google Scholar

[17]

E. Lieb and R. Seiringer, Derivation of the Gross-Pitaevskii equation for rotating Bose gases,, Comm. Math. Phys., 264 (2006), 505.  doi: 10.1007/s00220-006-1524-9.  Google Scholar

[18]

C. J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases,, Cambridge University Press, (2002).   Google Scholar

[19]

L. P. Pitaevskii and S. Stringari, Bose-Einstein Condensation,, Clarendon Press, (2003).   Google Scholar

[20]

N. Tzvetkov and N. Visciglia, Small data scattering for the nonlinear Schrödinger equation on product spaces,, Commun. Partial Differ. Equ., 37 (2012), 125.  doi: 10.1080/03605302.2011.574306.  Google Scholar

[1]

Dong Deng, Ruikuan Liu. Bifurcation solutions of Gross-Pitaevskii equations for spin-1 Bose-Einstein condensates. Discrete & Continuous Dynamical Systems - B, 2019, 24 (7) : 3175-3193. doi: 10.3934/dcdsb.2018306
[2]

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
[3]

Weizhu Bao, Yongyong Cai. Mathematical theory and numerical methods for Bose-Einstein condensation. Kinetic & Related Models, 2013, 6 (1) : 1-135. doi: 10.3934/krm.2013.6.1
[4]

Anne de Bouard, Reika Fukuizumi, Romain Poncet. Vortex solutions in Bose-Einstein condensation under a trapping potential varying randomly in time. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 2793-2817. doi: 10.3934/dcdsb.2015.20.2793
[5]

Xiaoyu Zeng, Yimin Zhang. Asymptotic behaviors of ground states for a modified Gross-Pitaevskii equation. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5263-5273. doi: 10.3934/dcds.2019214
[6]

Patrick Henning, Johan Wärnegård. Numerical comparison of mass-conservative schemes for the Gross-Pitaevskii equation. Kinetic & Related Models, 2019, 12 (6) : 1247-1271. doi: 10.3934/krm.2019048
[7]

Brahim Alouini, Olivier Goubet. Regularity of the attractor for a Bose-Einstein equation in a two dimensional unbounded domain. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 651-677. doi: 10.3934/dcdsb.2014.19.651
[8]

Joshua Cape, Hans-Christian Herbig, Christopher Seaton. Symplectic reduction at zero angular momentum. Journal of Geometric Mechanics, 2016, 8 (1) : 13-34. doi: 10.3934/jgm.2016.8.13
[9]

Liren Lin, I-Liang Chern. A kinetic energy reduction technique and characterizations of the ground states of spin-1 Bose-Einstein condensates. Discrete & Continuous Dynamical Systems - B, 2014, 19 (4) : 1119-1128. doi: 10.3934/dcdsb.2014.19.1119
[10]

Georgy L. Alfimov, Pavel P. Kizin, Dmitry A. Zezyulin. Gap solitons for the repulsive Gross-Pitaevskii equation with periodic potential: Coding and method for computation. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1207-1229. doi: 10.3934/dcdsb.2017059
[11]

Roy H. Goodman, Jeremy L. Marzuola, Michael I. Weinstein. Self-trapping and Josephson tunneling solutions to the nonlinear Schrödinger / Gross-Pitaevskii equation. Discrete & Continuous Dynamical Systems - A, 2015, 35 (1) : 225-246. doi: 10.3934/dcds.2015.35.225
[12]

Brahim Alouini. Finite dimensional global attractor for a Bose-Einstein equation in a two dimensional unbounded domain. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1781-1801. doi: 10.3934/cpaa.2015.14.1781
[13]

Brahim Alouini. Long-time behavior of a Bose-Einstein equation in a two-dimensional thin domain. Communications on Pure & Applied Analysis, 2011, 10 (6) : 1629-1643. doi: 10.3934/cpaa.2011.10.1629
[14]

José A. Carrillo, Katharina Hopf, Marie-Therese Wolfram. Numerical study of Bose–Einstein condensation in the Kaniadakis–Quarati model for bosons. Kinetic & Related Models, 2020, 13 (3) : 507-529. doi: 10.3934/krm.2020017
[15]

Norman E. Dancer. On the converse problem for the Gross-Pitaevskii equations with a large parameter. Discrete & Continuous Dynamical Systems - A, 2014, 34 (6) : 2481-2493. doi: 10.3934/dcds.2014.34.2481
[16]

Ko-Shin Chen, Peter Sternberg. Dynamics of Ginzburg-Landau and Gross-Pitaevskii vortices on manifolds. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 1905-1931. doi: 10.3934/dcds.2014.34.1905
[17]

Thomas Chen, Nataša Pavlović. On the Cauchy problem for focusing and defocusing Gross-Pitaevskii hierarchies. Discrete & Continuous Dynamical Systems - A, 2010, 27 (2) : 715-739. doi: 10.3934/dcds.2010.27.715
[18]

E. Norman Dancer. On a degree associated with the Gross-Pitaevskii system with a large parameter. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 1835-1839. doi: 10.3934/dcdss.2019120
[19]

Philipp Bader, Sergio Blanes, Fernando Casas, Mechthild Thalhammer. Efficient time integration methods for Gross-Pitaevskii equations with rotation term. Journal of Computational Dynamics, 2019, 6 (2) : 147-169. doi: 10.3934/jcd.2019008
[20]

Xuguang Lu. Long time strong convergence to Bose-Einstein distribution for low temperature. Kinetic & Related Models, 2018, 11 (4) : 715-734. doi: 10.3934/krm.2018029

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (42)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences