American Institute of Mathematical Sciences

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October  2011, 4(5): 1199-1212. doi: 10.3934/dcdss.2011.4.1199

Multiple dark solitons in Bose-Einstein condensates at finite temperatures

P.G. Kevrekidis 1, and  Dimitri J. Frantzeskakis 2,

1. 

University of Massachusetts, Lederle Graduate Research Tower, Department of Mathematics and Statistics, Amherst, MA 01003
2. 

Department of Physics, University of Athens, Panepistimiopolis, Zografos, Athens 15784, Greece

Received  September 2009 Revised  October 2009 Published  December 2010

We study analytically, as well as numerically, single- and multiple-dark matter-wave solitons in atomic Bose-Einstein condensates at finite temperatures. Our analysis is based on the study of the dissipative Gross-Pitaevskii equation, which incorporates a phenomenological damping term accounting for the interaction of the condensate with the thermal cloud. We illustrate how the negative Krein sign eigenmodes (associated with the the single- or multiple-dark soliton states) can give rise to Hopf bifurcations and oscillatory instabilities, whose ensuing dynamics is also elucidated. In all cases, the finite-temperature induced dynamics results in soliton decay, and the system eventually asymptotes to the ground state.
Keywords: finite temperature., Bose-Einstein condensates, Dark solitons.
Mathematics Subject Classification: Primary: 35Q51; Secondary: 35A3.
Citation: P.G. Kevrekidis, Dimitri J. Frantzeskakis. Multiple dark solitons in Bose-Einstein condensates at finite temperatures. Discrete & Continuous Dynamical Systems - S, 2011, 4 (5) : 1199-1212. doi: 10.3934/dcdss.2011.4.1199
References:
[1]

B. P. Anderson, P. C. Haljan, C. A. Regal, D. L. Feder, L. A. Collins, C. W. Clark and E. A. Cornell, Watching dark solitons decay into vortex rings in a Bose-Einstein condensate,, Phys. Rev. Lett., 86 (2001), 2926.  doi: 10.1103/PhysRevLett.86.2926.  Google Scholar

[2]

I. Aranson and V. Steinberg, Stability of multicharged vortices in a model of superflow,, Phys. Rev. B, 53 (1996), 75.  doi: 10.1103/PhysRevB.53.75.  Google Scholar

[3]

C. Becker, S. Stellmer, P. Soltan-Panahi, S. Dörscher, M. Baumert, E.-M. Richter, J. Kronjäger, K. Bongs and K. Sengstock, Oscillations and interactions of dark and dark-bright solitons in Bose-Einstein condensates,, Nature Phys., 4 (2008), 496.  doi: 10.1038/nphys962.  Google Scholar

[4]

K. Bongs, S. Burger, S. Dettmer, D. Hellweg, J. Arlt, W. Ertmer and K. Sengstock, Coherent manipulation and guiding of Bose-Einstein condensates by optical dipole potentials,, C. R. Acad. Sci. Paris, 2 (2001), 671.   Google Scholar

[5]

S. Burger, K. Bongs, S. Dettmer, W. Ertmer, K. Sengstock, A. Sanpera, G. V. Shlyapnikov and M. Lewenstein, Dark solitons in Bose-Einstein condensates,, Phys. Rev. Lett., 83 (1999), 5198.  doi: 10.1103/PhysRevLett.83.5198.  Google Scholar

[6]

Th. Busch and J. R. Anglin, Motion of dark solitons in trapped Bose-Einstein Condensates,, Phys. Rev. Lett., 84 (2000), 2298.  doi: 10.1103/PhysRevLett.84.2298.  Google Scholar

[7]

R. Carretero-González, B. P. Anderson, P. G. Kevrekidis, D. J. Frantzeskakis and C. N. Weiler, Dynamics of vortex formation in merging Bose-Einstein condensate fragments,, Phys. Rev. A, 77 (2008).   Google Scholar

[8]

R. Carretero-González, P. G. Kevrekidis and D. J. Frantzeskakis, Nonlinear waves in Bose-Einstein condensates: Physical relevance and mathematical techniques,, Nonlinearity, 21 (2008).   Google Scholar

[9]

R. Carretero-González, N. Whitaker, P. G. Kevrekidis and D. J. Frantzeskakis, Vortex structures formed by the interference of sliced condensates,, Phys. Rev. A, 77 (2008).   Google Scholar

[10]

S. Choi, S. A. Morgan and K. Burnett, Phenomenological damping in trapped atomic Bose-Einstein condensates,, Phys. Rev. A, 57 (1998), 4057.  doi: 10.1103/PhysRevA.57.4057.  Google Scholar

[11]

S. P. Cockburn, H. E. Nistazakis, T. P. Horikis, P. G. Kevrekidis, N. P. Proukakis and D. J. Frantzeskakis, Matter-wave dark solitons: Stochastic vs. analytical results,, Phys. Rev. Lett, 104 (2010).  doi: 10.1103/PhysRevLett.104.174101.  Google Scholar

[12]

S. P. Cockburn and N. P. Proukakis, The stochastic Gross-Pitaevskii equation and some applications,, Laser Phys., 19 (2009), 558.  doi: 10.1134/S1054660X09040057.  Google Scholar

[13]

J. Denschlag, J. E. Simsarian, D. L. Feder, C. W. Clark, L. A. Collins, J. Cubizolles, L. Deng, E. W. Hagley, K. Helmerson, W. P. Reinhardt, S. L. Rolston, B. I. Schneider and W. D. Phillips, Generating solitons by phase engineering of a Bose-Einstein condensate,, Science, 287 (2000), 97.  doi: 10.1126/science.287.5450.97.  Google Scholar

[14]

Z. Dutton, M. Budde, C. Slowe and L. V. Hau, Observation of quantum shock waves created with ultra-compressed slow light pulses in a Bose-Einstein Condensate,, Science, 293 (2001), 663.  doi: 10.1126/science.1062527.  Google Scholar

[15]

P. Engels and C. Atherton, Stationary and nonstationary fluid flow of a Bose-Einstein condensate through a penetrable barrier,, Phys. Rev. Lett., 99 (2007).  doi: 10.1103/PhysRevLett.99.160405.  Google Scholar

[16]

P. O. Fedichev, A. E. Muryshev and G. V. Shlyapnikov, Dissipative dynamics of a kink state in a Bose-condensed gas,, Phys. Rev. A, 60 (1999), 3220.  doi: 10.1103/PhysRevA.60.3220.  Google Scholar

[17]

D. J. Frantzeskakis, Dark solitons in atomic Bose-Einstein condensates: from theory to experiments,, J. Phys. A: Math. Theor., 43 (2010).  doi: 10.1088/1751-8113/43/21/213001.  Google Scholar

[18]

D. J. Frantzeskakis, G. Theocharis, F. K. Diakonos, P. Schmelcher and Yu. S. Kivshar, Interaction of dark solitons with localized impurities in Bose-Einstein condensates,, Phys. Rev. A, 66 (2002).  doi: 10.1103/PhysRevA.66.053608.  Google Scholar

[19]

R. Graham, Decoherence of Bose-Einstein condensates in traps at finite temperature,, Phys. Rev. Lett., 81 (1998), 5262.  doi: 10.1103/PhysRevLett.81.5262.  Google Scholar

[20]

B. Jackson and N. P. Proukakis, Finite-temperature models of Bose-Einstein condensation,, J. Phys. B: At. Mol. Opt. Phys., 41 (2008).  doi: 10.1088/0953-4075/41/20/203002.  Google Scholar

[21]

B. Jackson, C. F. Barenghi and N. P. Proukakis, Matter wave solitons at finite temperatures,, J. Low Temp. Phys., 148 (2007), 387.  doi: 10.1007/s10909-007-9410-1.  Google Scholar

[22]

B. Jackson, N. P. Proukakis and C. F. Barenghi, Dark-soliton dynamics in Bose-Einstein condensates at finite temperature,, Phys., 75 (2007).  doi: 10.1103/PhysRevA.75.051601.  Google Scholar

[23]

T. Kapitula and P. G. Kevrekidis, Bose-Einstein condensates in the presence of a magnetic trap and optical lattice,, Chaos, 15 (2005).  doi: 10.1063/1.1993867.  Google Scholar

[24]

T. Kapitula, P. G. Kevrekidis and B. Sandstede, Counting eigenvalues via the Krein signature in infinite-dimensional Hamiltonian system,, Physica D, 195 (2004), 263.  doi: 10.1016/j.physd.2004.03.018.  Google Scholar

[25]

P. G. Kevrekidis, D. J. Frantzeskakis and R. Carretero-González R (eds.), "Emergent Nonlinear Phenomena in Bose-Einstein Condensates: Theory and Experiment,", Springer, (2007).   Google Scholar

[26]

Yu. S. Kivshar and W. Królikowski, Lagrangian approach for dark solitons,, Opt. Commun., 114 (1995), 353.  doi: 10.1016/0030-4018(94)00644-A.  Google Scholar

[27]

Yu. S. Kivshar and X. Yang, Perturbation-induced dynamics of dark solitons,, Phys. Rev. E, 49 (1994), 1657.  doi: 10.1103/PhysRevE.49.1657.  Google Scholar

[28]

C. K. Law, P. T. Leung and M.-C. Chu, Quantum fluctuations of coupled dark solitons in a trapped Bose-Einstein condensate,, J. Phys. B: At. Mol. Opt. Phys., 35 (2002), 3583.  doi: 10.1088/0953-4075/35/16/316.  Google Scholar

[29]

M. D. Lee and C. W. Gardiner, Quantum kinetic theory. VI. The growth of a Bose-Einstein condensate,, Phys. Rev. A, 62 (2000).  doi: 10.1103/PhysRevA.62.033606.  Google Scholar

[30]

E. J. M. Madarassy and C. F. Barenghi, Vortex dynamics in trapped Bose-Einstein condensate,, J. Low Temp. Phys., 152 (2008), 122.  doi: 10.1007/s10909-008-9811-9.  Google Scholar

[31]

A. A. Penckwitt, R. J. Ballagh and C. W. Gardiner, Nucleation, growth and stabilization of Bose-Einstein condensate vortex lattices,, Phys. Rev. Lett., 89 (2002).  doi: 10.1103/PhysRevLett.89.260402.  Google Scholar

[32]

L. P. Pitaevskii, ,, Zh. Eksp. Teor. Fiz., 35 (1958).   Google Scholar

[33]

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

[34]

N. P. Proukakis, N. G. Parker, C. F. Barenghi and C. S. Adams, Parametric driving of dark solitons in atomic Bose-Einstein condensates,, Phys. Rev. Lett., 93 (2004).  doi: 10.1103/PhysRevLett.93.130408.  Google Scholar

[35]

R. Sásik, L. M. A. Bettencourt and S. Habib, Thermal vortex dynamics in a two-dimensional condensate,, Phys. Rev. B, 62 (2000), 1238.  doi: 10.1103/PhysRevB.62.1238.  Google Scholar

[36]

D. V. Skryabin, Energy of internal modes of nonlinear waves and complex frequencies due to symmetry breaking,, Phys. Rev. E, 64 (2001).  doi: 10.1103/PhysRevE.64.055601.  Google Scholar

[37]

S. Stellmer, C. Becker, P. Soltan-Panahi, E.-M. Richter, S. Dörscher, M. Baumert, J. Kronjäger, K. Bongs and K. Sengstock, Collisions of dark solitons in elongated Bose-Einstein condensates,, Phys. Rev. Lett., 101 (2008).  doi: 10.1103/PhysRevLett.101.120406.  Google Scholar

[38]

G. Theocharis, P. G. Kevrekidis, M. K. Oberthaler and D. J. Frantzeskakis, Dark matter-wave solitons in the dimensionality crossover,, Phys. Rev. A, 76 (2007).  doi: 10.1103/PhysRevA.76.045601.  Google Scholar

[39]

G. Theocharis, A. Weller, J. P. Ronzheimer, C. Gross, M. K. Oberthaler, P. G. Kevrekidis and D. J. Frantzeskakis, Multiple atomic dark solitons in cigar-shaped Bose-Einstein condensates,, Phys. Rev. A, 81 (2009).  doi: 10.1103/PhysRevA.81.063604.  Google Scholar

[40]

M. Tsubota, K. Kasamatsu and M. Ueda, Vortex lattice formation in a rotating Bose-Einstein condensate,, Phys. Rev. A, 65 (2002).   Google Scholar

[41]

A. Weller, J. P. Ronzheimer, C. Gross, J. Esteve, M. K. Oberthaler, D. J. Frantzeskakis, G. Theocharis and P. G. Kevrekidis, Experimental observation of oscillating and interacting matter-wave dark solitons,, Phys. Rev. Lett., 101 (2008).  doi: 10.1103/PhysRevLett.101.130401.  Google Scholar

show all references

References:
[1]

B. P. Anderson, P. C. Haljan, C. A. Regal, D. L. Feder, L. A. Collins, C. W. Clark and E. A. Cornell, Watching dark solitons decay into vortex rings in a Bose-Einstein condensate,, Phys. Rev. Lett., 86 (2001), 2926.  doi: 10.1103/PhysRevLett.86.2926.  Google Scholar

[2]

I. Aranson and V. Steinberg, Stability of multicharged vortices in a model of superflow,, Phys. Rev. B, 53 (1996), 75.  doi: 10.1103/PhysRevB.53.75.  Google Scholar

[3]

C. Becker, S. Stellmer, P. Soltan-Panahi, S. Dörscher, M. Baumert, E.-M. Richter, J. Kronjäger, K. Bongs and K. Sengstock, Oscillations and interactions of dark and dark-bright solitons in Bose-Einstein condensates,, Nature Phys., 4 (2008), 496.  doi: 10.1038/nphys962.  Google Scholar

[4]

K. Bongs, S. Burger, S. Dettmer, D. Hellweg, J. Arlt, W. Ertmer and K. Sengstock, Coherent manipulation and guiding of Bose-Einstein condensates by optical dipole potentials,, C. R. Acad. Sci. Paris, 2 (2001), 671.   Google Scholar

[5]

S. Burger, K. Bongs, S. Dettmer, W. Ertmer, K. Sengstock, A. Sanpera, G. V. Shlyapnikov and M. Lewenstein, Dark solitons in Bose-Einstein condensates,, Phys. Rev. Lett., 83 (1999), 5198.  doi: 10.1103/PhysRevLett.83.5198.  Google Scholar

[6]

Th. Busch and J. R. Anglin, Motion of dark solitons in trapped Bose-Einstein Condensates,, Phys. Rev. Lett., 84 (2000), 2298.  doi: 10.1103/PhysRevLett.84.2298.  Google Scholar

[7]

R. Carretero-González, B. P. Anderson, P. G. Kevrekidis, D. J. Frantzeskakis and C. N. Weiler, Dynamics of vortex formation in merging Bose-Einstein condensate fragments,, Phys. Rev. A, 77 (2008).   Google Scholar

[8]

R. Carretero-González, P. G. Kevrekidis and D. J. Frantzeskakis, Nonlinear waves in Bose-Einstein condensates: Physical relevance and mathematical techniques,, Nonlinearity, 21 (2008).   Google Scholar

[9]

R. Carretero-González, N. Whitaker, P. G. Kevrekidis and D. J. Frantzeskakis, Vortex structures formed by the interference of sliced condensates,, Phys. Rev. A, 77 (2008).   Google Scholar

[10]

S. Choi, S. A. Morgan and K. Burnett, Phenomenological damping in trapped atomic Bose-Einstein condensates,, Phys. Rev. A, 57 (1998), 4057.  doi: 10.1103/PhysRevA.57.4057.  Google Scholar

[11]

S. P. Cockburn, H. E. Nistazakis, T. P. Horikis, P. G. Kevrekidis, N. P. Proukakis and D. J. Frantzeskakis, Matter-wave dark solitons: Stochastic vs. analytical results,, Phys. Rev. Lett, 104 (2010).  doi: 10.1103/PhysRevLett.104.174101.  Google Scholar

[12]

S. P. Cockburn and N. P. Proukakis, The stochastic Gross-Pitaevskii equation and some applications,, Laser Phys., 19 (2009), 558.  doi: 10.1134/S1054660X09040057.  Google Scholar

[13]

J. Denschlag, J. E. Simsarian, D. L. Feder, C. W. Clark, L. A. Collins, J. Cubizolles, L. Deng, E. W. Hagley, K. Helmerson, W. P. Reinhardt, S. L. Rolston, B. I. Schneider and W. D. Phillips, Generating solitons by phase engineering of a Bose-Einstein condensate,, Science, 287 (2000), 97.  doi: 10.1126/science.287.5450.97.  Google Scholar

[14]

Z. Dutton, M. Budde, C. Slowe and L. V. Hau, Observation of quantum shock waves created with ultra-compressed slow light pulses in a Bose-Einstein Condensate,, Science, 293 (2001), 663.  doi: 10.1126/science.1062527.  Google Scholar

[15]

P. Engels and C. Atherton, Stationary and nonstationary fluid flow of a Bose-Einstein condensate through a penetrable barrier,, Phys. Rev. Lett., 99 (2007).  doi: 10.1103/PhysRevLett.99.160405.  Google Scholar

[16]

P. O. Fedichev, A. E. Muryshev and G. V. Shlyapnikov, Dissipative dynamics of a kink state in a Bose-condensed gas,, Phys. Rev. A, 60 (1999), 3220.  doi: 10.1103/PhysRevA.60.3220.  Google Scholar

[17]

D. J. Frantzeskakis, Dark solitons in atomic Bose-Einstein condensates: from theory to experiments,, J. Phys. A: Math. Theor., 43 (2010).  doi: 10.1088/1751-8113/43/21/213001.  Google Scholar

[18]

D. J. Frantzeskakis, G. Theocharis, F. K. Diakonos, P. Schmelcher and Yu. S. Kivshar, Interaction of dark solitons with localized impurities in Bose-Einstein condensates,, Phys. Rev. A, 66 (2002).  doi: 10.1103/PhysRevA.66.053608.  Google Scholar

[19]

R. Graham, Decoherence of Bose-Einstein condensates in traps at finite temperature,, Phys. Rev. Lett., 81 (1998), 5262.  doi: 10.1103/PhysRevLett.81.5262.  Google Scholar

[20]

B. Jackson and N. P. Proukakis, Finite-temperature models of Bose-Einstein condensation,, J. Phys. B: At. Mol. Opt. Phys., 41 (2008).  doi: 10.1088/0953-4075/41/20/203002.  Google Scholar

[21]

B. Jackson, C. F. Barenghi and N. P. Proukakis, Matter wave solitons at finite temperatures,, J. Low Temp. Phys., 148 (2007), 387.  doi: 10.1007/s10909-007-9410-1.  Google Scholar

[22]

B. Jackson, N. P. Proukakis and C. F. Barenghi, Dark-soliton dynamics in Bose-Einstein condensates at finite temperature,, Phys., 75 (2007).  doi: 10.1103/PhysRevA.75.051601.  Google Scholar

[23]

T. Kapitula and P. G. Kevrekidis, Bose-Einstein condensates in the presence of a magnetic trap and optical lattice,, Chaos, 15 (2005).  doi: 10.1063/1.1993867.  Google Scholar

[24]

T. Kapitula, P. G. Kevrekidis and B. Sandstede, Counting eigenvalues via the Krein signature in infinite-dimensional Hamiltonian system,, Physica D, 195 (2004), 263.  doi: 10.1016/j.physd.2004.03.018.  Google Scholar

[25]

P. G. Kevrekidis, D. J. Frantzeskakis and R. Carretero-González R (eds.), "Emergent Nonlinear Phenomena in Bose-Einstein Condensates: Theory and Experiment,", Springer, (2007).   Google Scholar

[26]

Yu. S. Kivshar and W. Królikowski, Lagrangian approach for dark solitons,, Opt. Commun., 114 (1995), 353.  doi: 10.1016/0030-4018(94)00644-A.  Google Scholar

[27]

Yu. S. Kivshar and X. Yang, Perturbation-induced dynamics of dark solitons,, Phys. Rev. E, 49 (1994), 1657.  doi: 10.1103/PhysRevE.49.1657.  Google Scholar

[28]

C. K. Law, P. T. Leung and M.-C. Chu, Quantum fluctuations of coupled dark solitons in a trapped Bose-Einstein condensate,, J. Phys. B: At. Mol. Opt. Phys., 35 (2002), 3583.  doi: 10.1088/0953-4075/35/16/316.  Google Scholar

[29]

M. D. Lee and C. W. Gardiner, Quantum kinetic theory. VI. The growth of a Bose-Einstein condensate,, Phys. Rev. A, 62 (2000).  doi: 10.1103/PhysRevA.62.033606.  Google Scholar

[30]

E. J. M. Madarassy and C. F. Barenghi, Vortex dynamics in trapped Bose-Einstein condensate,, J. Low Temp. Phys., 152 (2008), 122.  doi: 10.1007/s10909-008-9811-9.  Google Scholar

[31]

A. A. Penckwitt, R. J. Ballagh and C. W. Gardiner, Nucleation, growth and stabilization of Bose-Einstein condensate vortex lattices,, Phys. Rev. Lett., 89 (2002).  doi: 10.1103/PhysRevLett.89.260402.  Google Scholar

[32]

L. P. Pitaevskii, ,, Zh. Eksp. Teor. Fiz., 35 (1958).   Google Scholar

[33]

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

[34]

N. P. Proukakis, N. G. Parker, C. F. Barenghi and C. S. Adams, Parametric driving of dark solitons in atomic Bose-Einstein condensates,, Phys. Rev. Lett., 93 (2004).  doi: 10.1103/PhysRevLett.93.130408.  Google Scholar

[35]

R. Sásik, L. M. A. Bettencourt and S. Habib, Thermal vortex dynamics in a two-dimensional condensate,, Phys. Rev. B, 62 (2000), 1238.  doi: 10.1103/PhysRevB.62.1238.  Google Scholar

[36]

D. V. Skryabin, Energy of internal modes of nonlinear waves and complex frequencies due to symmetry breaking,, Phys. Rev. E, 64 (2001).  doi: 10.1103/PhysRevE.64.055601.  Google Scholar

[37]

S. Stellmer, C. Becker, P. Soltan-Panahi, E.-M. Richter, S. Dörscher, M. Baumert, J. Kronjäger, K. Bongs and K. Sengstock, Collisions of dark solitons in elongated Bose-Einstein condensates,, Phys. Rev. Lett., 101 (2008).  doi: 10.1103/PhysRevLett.101.120406.  Google Scholar

[38]

G. Theocharis, P. G. Kevrekidis, M. K. Oberthaler and D. J. Frantzeskakis, Dark matter-wave solitons in the dimensionality crossover,, Phys. Rev. A, 76 (2007).  doi: 10.1103/PhysRevA.76.045601.  Google Scholar

[39]

G. Theocharis, A. Weller, J. P. Ronzheimer, C. Gross, M. K. Oberthaler, P. G. Kevrekidis and D. J. Frantzeskakis, Multiple atomic dark solitons in cigar-shaped Bose-Einstein condensates,, Phys. Rev. A, 81 (2009).  doi: 10.1103/PhysRevA.81.063604.  Google Scholar

[40]

M. Tsubota, K. Kasamatsu and M. Ueda, Vortex lattice formation in a rotating Bose-Einstein condensate,, Phys. Rev. A, 65 (2002).   Google Scholar

[41]

A. Weller, J. P. Ronzheimer, C. Gross, J. Esteve, M. K. Oberthaler, D. J. Frantzeskakis, G. Theocharis and P. G. Kevrekidis, Experimental observation of oscillating and interacting matter-wave dark solitons,, Phys. Rev. Lett., 101 (2008).  doi: 10.1103/PhysRevLett.101.130401.  Google Scholar

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Colin Guillarmou, Antônio Sá Barreto. Inverse problems for Einstein manifolds. Inverse Problems & Imaging, 2009, 3 (1) : 1-15. doi: 10.3934/ipi.2009.3.1

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