Dynamics of vortex dipoles in anisotropic Bose-Einstein condensates
Abstract
We study the motion of a vortex dipole in a Bose-Einstein condensate confined to an anisotropic trap. We focus on a system of ODEs describing the vortices' motion, which is in turn a reduced model of the Gross-Pitaevskii equation describing the condensate's motion. Using a sequence of canonical changes of variables, we reduce the dimension and simplify the equations of motion. In this study, we uncover two interesting regimes. Near a family of periodic orbits known as guiding centers, we find that the dynamics is essentially that of a pendulum coupled to a linear oscillator, leading to stochastic reversals in the overall direction of rotation of the dipole. Near the separatrix orbit in the isotropic system, we find other families of periodic, quasi-periodic, and chaotic trajectories. In a neighborhood of the guiding center orbits, we derive an explicit iterated map that simplifies the problem further. Numerical calculations are used to illustrate the phenomena discovered through the analysis. Using the results from the reduced system, we are able to construct complex periodic orbits in the original, PDE, mean-field model for Bose-Einstein condensates, which corroborates the phenomenology observed in the reduced dynamical equations.
- Authors:
-
- New Jersey Inst. of Technology, Newark, NJ (United States)
- Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
- San Diego State Univ., San Diego, CA (United States)
- Publication Date:
- Research Org.:
- Los Alamos National Laboratory (LANL), Los Alamos, NM (United States)
- Sponsoring Org.:
- USDOE
- OSTI Identifier:
- 1233170
- Report Number(s):
- LA-UR-14-28136
Journal ID: ISSN 1536-0040; TRN: US1600426
- Grant/Contract Number:
- AC52-06NA25396
- Resource Type:
- Accepted Manuscript
- Journal Name:
- SIAM Journal on Applied Dynamical Systems
- Additional Journal Information:
- Journal Volume: 14; Journal Issue: 2; Journal ID: ISSN 1536-0040
- Publisher:
- Society for Industrial and Applied Mathematics
- Country of Publication:
- United States
- Language:
- English
- Subject:
- 74 ATOMIC AND MOLECULAR PHYSICS; 97 MATHEMATICS AND COMPUTING; Vortex dynamics; nonlinear Schrodinger equation; Gross-Pitaevskii equation; Bose-Einstein condensates; Hamiltonian ODEs
Citation Formats
Goodman, Roy H., Kevrekidis, P. G., and Carretero-González, R. Dynamics of vortex dipoles in anisotropic Bose-Einstein condensates. United States: N. p., 2015.
Web. doi:10.1137/140992345.
Goodman, Roy H., Kevrekidis, P. G., & Carretero-González, R. Dynamics of vortex dipoles in anisotropic Bose-Einstein condensates. United States. https://doi.org/10.1137/140992345
Goodman, Roy H., Kevrekidis, P. G., and Carretero-González, R. Tue .
"Dynamics of vortex dipoles in anisotropic Bose-Einstein condensates". United States. https://doi.org/10.1137/140992345. https://www.osti.gov/servlets/purl/1233170.
@article{osti_1233170,
title = {Dynamics of vortex dipoles in anisotropic Bose-Einstein condensates},
author = {Goodman, Roy H. and Kevrekidis, P. G. and Carretero-González, R.},
abstractNote = {We study the motion of a vortex dipole in a Bose-Einstein condensate confined to an anisotropic trap. We focus on a system of ODEs describing the vortices' motion, which is in turn a reduced model of the Gross-Pitaevskii equation describing the condensate's motion. Using a sequence of canonical changes of variables, we reduce the dimension and simplify the equations of motion. In this study, we uncover two interesting regimes. Near a family of periodic orbits known as guiding centers, we find that the dynamics is essentially that of a pendulum coupled to a linear oscillator, leading to stochastic reversals in the overall direction of rotation of the dipole. Near the separatrix orbit in the isotropic system, we find other families of periodic, quasi-periodic, and chaotic trajectories. In a neighborhood of the guiding center orbits, we derive an explicit iterated map that simplifies the problem further. Numerical calculations are used to illustrate the phenomena discovered through the analysis. Using the results from the reduced system, we are able to construct complex periodic orbits in the original, PDE, mean-field model for Bose-Einstein condensates, which corroborates the phenomenology observed in the reduced dynamical equations.},
doi = {10.1137/140992345},
journal = {SIAM Journal on Applied Dynamical Systems},
number = 2,
volume = 14,
place = {United States},
year = {Tue Apr 14 00:00:00 EDT 2015},
month = {Tue Apr 14 00:00:00 EDT 2015}
}
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Works referenced in this record:
Discrete Symmetry and Stability in Hamiltonian Dynamics
journal, June 2011
- Bountis, Tassos; Chechin, George; Sakhnenko, Vladimir
- International Journal of Bifurcation and Chaos, Vol. 21, Issue 06
A Melnikov Method for Homoclinic Orbits with Many Pulses
journal, September 1998
- Camassa, Roberto; Kovačič, Gregor; Tin, Siu-Kei
- Archive for Rational Mechanics and Analysis, Vol. 143, Issue 2
Bose-Einstein condensates with vortices in rotating traps
journal, January 1999
- Castin, Y.; Dum, R.
- The European Physical Journal D, Vol. 7, Issue 3
Dynamics of Vortices in Two-Dimensional Bose–Einstein Condensates
journal, April 2002
- Chang, Shu-Ming; Lin, Wen-Wei; Lin, Tai-Chia
- International Journal of Bifurcation and Chaos, Vol. 12, Issue 04
MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs
journal, June 2003
- Dhooge, A.; Govaerts, W.; Kuznetsov, Yu. A.
- ACM Transactions on Mathematical Software, Vol. 29, Issue 2
Vortices in a trapped dilute Bose-Einstein condensate
journal, March 2001
- Fetter, Alexander L.; Svidzinsky, Anatoly A.
- Journal of Physics: Condensed Matter, Vol. 13, Issue 12
Rotating trapped Bose-Einstein condensates
journal, May 2009
- Fetter, Alexander L.
- Reviews of Modern Physics, Vol. 81, Issue 2
Vortex dynamics in spin-orbit-coupled Bose-Einstein condensates
journal, February 2014
- Fetter, Alexander L.
- Physical Review A, Vol. 89, Issue 2
Real-Time Dynamics of Single Vortex Lines and Vortex Dipoles in a Bose-Einstein Condensate
journal, September 2010
- Freilich, D. V.; Bianchi, D. M.; Kaufman, A. M.
- Science, Vol. 329, Issue 5996
Chaotic scattering in solitary wave interactions: A singular iterated-map description
journal, June 2008
- Goodman, Roy H.
- Chaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 18, Issue 2
Hamiltonian Hopf bifurcations and dynamics of NLS/GP standing-wave modes
journal, September 2011
- Goodman, Roy
- Journal of Physics A: Mathematical and Theoretical, Vol. 44, Issue 42
Kink-Antikink Collisions in the phi^4 Equation: The n-Bounce Resonance and the Separatrix Map
journal, January 2005
- Goodman, Roy H.; Haberman, Richard
- SIAM Journal on Applied Dynamical Systems, Vol. 4, Issue 4
Chaotic Scattering and the -Bounce Resonance in Solitary-Wave Interactions
journal, March 2007
- Goodman, Roy H.; Haberman, Richard
- Physical Review Letters, Vol. 98, Issue 10
Self-trapping and Josephson tunneling solutions to the nonlinear Schrödinger / Gross-Pitaevskii equation
journal, January 2015
- H. Goodman, Roy; L. Marzuola, Jeremy; I. Weinstein, Michael
- Discrete & Continuous Dynamical Systems - A, Vol. 35, Issue 1
Poincaré, celestial mechanics, dynamical-systems theory and “chaos”
journal, September 1990
- Holmes, Philip
- Physics Reports, Vol. 193, Issue 3
A tale of two distributions: from few to many vortices in quasi-two-dimensional Bose–Einstein condensates
journal, August 2014
- Kolokolnikov, T.; Kevrekidis, P. G.; Carretero-González, R.
- Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 470, Issue 2168
Dynamics of three noncorotating vortices in Bose-Einstein condensates
journal, April 2014
- Koukouloyannis, V.; Voyatzis, G.; Kevrekidis, P. G.
- Physical Review E, Vol. 89, Issue 4
Size and dynamics of vortex dipoles in dilute Bose-Einstein condensates
journal, January 2011
- Kuopanportti, Pekko; Huhtamäki, Jukka A. M.; Möttönen, Mikko
- Physical Review A, Vol. 83, Issue 1
Dynamics for Ginzburg-Landau vortices under a mixed flow
journal, January 2009
- Kurzke, Matthias; Melcher, Christof; Moser, Roger
- Indiana University Mathematics Journal, Vol. 58, Issue 6
Chaotic behavior of three interacting vortices in a confined Bose-Einstein condensate
journal, June 2014
- Kyriakopoulos, Nikos; Koukouloyannis, Vassilis; Skokos, Charalampos
- Chaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 24, Issue 2
Imprinting Vortices in a Bose-Einstein Condensate using Topological Phases
journal, October 2002
- Leanhardt, A. E.; Görlitz, A.; Chikkatur, A. P.
- Physical Review Letters, Vol. 89, Issue 19
Vortex dipole in a trapped two-dimensional Bose-Einstein condensate
journal, May 2008
- Li, Weibin; Haque, Masudul; Komineas, Stavros
- Physical Review A, Vol. 77, Issue 5
On the Existence of a Slow Manifold
journal, August 1986
- Lorenz, Edward N.
- Journal of the Atmospheric Sciences, Vol. 43, Issue 15
Stationary States of a Rotating Bose-Einstein Condensate: Routes to Vortex Nucleation
journal, May 2001
- Madison, K. W.; Chevy, F.; Bretin, V.
- Physical Review Letters, Vol. 86, Issue 20
Vortex Formation in a Stirred Bose-Einstein Condensate
journal, January 2000
- Madison, K. W.; Chevy, F.; Wohlleben, W.
- Physical Review Letters, Vol. 84, Issue 5
Long time dynamics near the symmetry breaking bifurcation for nonlinear Schrödinger/Gross-Pitaevskii equations
journal, January 2010
- L. Marzuola, Jeremy; I. Weinstein, Michael
- Discrete & Continuous Dynamical Systems - A, Vol. 28, Issue 4
Vortices in a Bose-Einstein Condensate
journal, September 1999
- Matthews, M. R.; Anderson, B. P.; Haljan, P. C.
- Physical Review Letters, Vol. 83, Issue 13
Small numbers of vortices in anisotropic traps
journal, May 2009
- McEndoo, S.; Busch, Th.
- Physical Review A, Vol. 79, Issue 5
Vortex dynamics in anisotropic traps
journal, July 2010
- McEndoo, S.; Busch, Th.
- Physical Review A, Vol. 82, Issue 1
Guiding-center dynamics of vortex dipoles in Bose-Einstein condensates
journal, July 2011
- Middelkamp, S.; Torres, P. J.; Kevrekidis, P. G.
- Physical Review A, Vol. 84, Issue 1
Periodic solutions near equilibria of symmetric Hamiltonian systems
journal, May 1988
- Montaldi, J. A.; Roberts, R. M.; Stewart, Ian Nicholas
- Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, Vol. 325, Issue 1584, p. 237-293
Existence of nonlinear normal modes of symmetric Hamiltonian systems
journal, August 1990
- Montaldi, J.; Roberts, M.; Stewart, I.
- Nonlinearity, Vol. 3, Issue 3
Stability of nonlinear normal modes of symmetric Hamiltonian systems
journal, August 1990
- Montaldi, J.; Roberts, M.; Stewart, I.
- Nonlinearity, Vol. 3, Issue 3
Stationary vortex clusters in nonrotating Bose-Einstein condensates
journal, March 2005
- Möttönen, M.; Virtanen, S. M. M.; Isoshima, T.
- Physical Review A, Vol. 71, Issue 3
Dynamics of a Few Corotating Vortices in Bose-Einstein Condensates
journal, May 2013
- Navarro, R.; Carretero-González, R.; Torres, P. J.
- Physical Review Letters, Vol. 110, Issue 22
Observation of Vortex Dipoles in an Oblate Bose-Einstein Condensate
journal, April 2010
- Neely, T. W.; Samson, E. C.; Bradley, A. S.
- Physical Review Letters, Vol. 104, Issue 16
Observation of Superfluid Flow in a Bose-Einstein Condensed Gas
journal, September 2000
- Onofrio, R.; Raman, C.; Vogels, J. M.
- Physical Review Letters, Vol. 85, Issue 11
Normal form for the symmetry-breaking bifurcation in the nonlinear Schrödinger equation
journal, November 2012
- Pelinovsky, D. E.; Phan, T. V.
- Journal of Differential Equations, Vol. 253, Issue 10
Erratum to: Etude des surfaces asymptotiques
journal, January 1890
- Poincaré, H.
- Acta Mathematica, Vol. 13, Issue 1-2
Vortex Nucleation in a Stirred Bose-Einstein Condensate
journal, November 2001
- Raman, C.; Abo-Shaeer, J. R.; Vogels, J. M.
- Physical Review Letters, Vol. 87, Issue 21
Overcritical Rotation of a Trapped Bose-Einstein Condensate
journal, January 2001
- Recati, A.; Zambelli, F.; Stringari, S.
- Physical Review Letters, Vol. 86, Issue 3
Vortex Formation by Merging of Multiple Trapped Bose-Einstein Condensates
journal, March 2007
- Scherer, David R.; Weiler, Chad N.; Neely, Tyler W.
- Physical Review Letters, Vol. 98, Issue 11
Three-vortex configurations in trapped Bose-Einstein condensates
journal, September 2010
- Seman, J. A.; Henn, E. A. L.; Haque, M.
- Physical Review A, Vol. 82, Issue 3
Hierarchy of bifurcations in the truncated and forced nonlinear Schrödinger model
journal, March 2005
- Shlizerman, Eli; Rom-Kedar, Vered
- Chaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 15, Issue 1
Dynamic Instability of a Rotating Bose-Einstein Condensate
journal, October 2001
- Sinha, Subhasis; Castin, Yvan
- Physical Review Letters, Vol. 87, Issue 19
Impact of anisotropy on vortex clusters and their dynamics
journal, January 2011
- Stockhofe, J.; Middelkamp, S.; Kevrekidis, P. G.
- EPL (Europhysics Letters), Vol. 93, Issue 2
Dynamics of vortex dipoles in confined Bose–Einstein condensates
journal, August 2011
- Torres, P. J.; Kevrekidis, P. G.; Frantzeskakis, D. J.
- Physics Letters A, Vol. 375, Issue 33
Spontaneous vortices in the formation of Bose–Einstein condensates
journal, October 2008
- Weiler, Chad N.; Neely, Tyler W.; Scherer, David R.
- Nature, Vol. 455, Issue 7215
Preparing topological states of a Bose–Einstein condensate
journal, October 1999
- Williams, J. E.; Holland, M. J.
- Nature, Vol. 401, Issue 6753
Exploring vortex dynamics in the presence of dissipation: Analytical and numerical results
journal, April 2014
- Yan, D.; Carretero-González, R.; Frantzeskakis, D. J.
- Physical Review A, Vol. 89, Issue 4
Exploring rigidly rotating vortex configurations and their bifurcations in atomic Bose-Einstein condensates
journal, October 2013
- Zampetaki, A. V.; Carretero-González, R.; Kevrekidis, P. G.
- Physical Review E, Vol. 88, Issue 4
Works referencing / citing this record:
Dynamics of vortex quadrupoles in nonrotating trapped Bose-Einstein condensates
journal, July 2016
- Yang, Tao; Hu, Zhi-Qiang; Zou, Shan
- Scientific Reports, Vol. 6, Issue 1