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

Multiple dark solitons in Bose-Einstein condensates at finite temperatures

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.
Citation: P.G. Kevrekidis, Dimitri J. Frantzeskakis. Multiple dark solitons in Bose-Einstein condensates at finite temperatures. Discrete and 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), 2926-2929. doi: 10.1103/PhysRevLett.86.2926.

[2]

I. Aranson and V. Steinberg, Stability of multicharged vortices in a model of superflow, Phys. Rev. B, 53 (1996), 75-78. doi: 10.1103/PhysRevB.53.75.

[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), 496-501. doi: 10.1038/nphys962.

[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), 671-680.

[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), 5198-5201. doi: 10.1103/PhysRevLett.83.5198.

[6]

Th. Busch and J. R. Anglin, Motion of dark solitons in trapped Bose-Einstein Condensates, Phys. Rev. Lett., 84 (2000), 2298-2301. doi: 10.1103/PhysRevLett.84.2298.

[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), 033625.

[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), R139-R202.

[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), 023605.

[10]

S. Choi, S. A. Morgan and K. Burnett, Phenomenological damping in trapped atomic Bose-Einstein condensates, Phys. Rev. A, 57 (1998), 4057-4060. doi: 10.1103/PhysRevA.57.4057.

[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), 174101. doi: 10.1103/PhysRevLett.104.174101.

[12]

S. P. Cockburn and N. P. Proukakis, The stochastic Gross-Pitaevskii equation and some applications, Laser Phys., 19 (2009), 558-570. doi: 10.1134/S1054660X09040057.

[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), 97-101. doi: 10.1126/science.287.5450.97.

[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), 663-668. doi: 10.1126/science.1062527.

[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), 160405. doi: 10.1103/PhysRevLett.99.160405.

[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), 3220-3224. doi: 10.1103/PhysRevA.60.3220.

[17]

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

[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), 053608. doi: 10.1103/PhysRevA.66.053608.

[19]

R. Graham, Decoherence of Bose-Einstein condensates in traps at finite temperature, Phys. Rev. Lett., 81 (1998), 5262-5265. doi: 10.1103/PhysRevLett.81.5262.

[20]

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

[21]

B. Jackson, C. F. Barenghi and N. P. Proukakis, Matter wave solitons at finite temperatures, J. Low Temp. Phys., 148 (2007), 387-391. doi: 10.1007/s10909-007-9410-1.

[22]

B. Jackson, N. P. Proukakis and C. F. Barenghi, Dark-soliton dynamics in Bose-Einstein condensates at finite temperature, Phys., Rev. A, 75 (2007), 051601. doi: 10.1103/PhysRevA.75.051601.

[23]

T. Kapitula and P. G. Kevrekidis, Bose-Einstein condensates in the presence of a magnetic trap and optical lattice, Chaos, 15 (2005), 037114. doi: 10.1063/1.1993867.

[24]

T. Kapitula, P. G. Kevrekidis and B. Sandstede, Counting eigenvalues via the Krein signature in infinite-dimensional Hamiltonian system, Physica D, 195 (2004), 263-282. doi: 10.1016/j.physd.2004.03.018.

[25]

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

[26]

Yu. S. Kivshar and W. Królikowski, Lagrangian approach for dark solitons, Opt. Commun., 114 (1995), 353-362. doi: 10.1016/0030-4018(94)00644-A.

[27]

Yu. S. Kivshar and X. Yang, Perturbation-induced dynamics of dark solitons, Phys. Rev. E, 49 (1994), 1657-1670. doi: 10.1103/PhysRevE.49.1657.

[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), 3583-3590. doi: 10.1088/0953-4075/35/16/316.

[29]

M. D. Lee and C. W. Gardiner, Quantum kinetic theory. VI. The growth of a Bose-Einstein condensate, Phys. Rev. A, 62 (2000), 033606. doi: 10.1103/PhysRevA.62.033606.

[30]

E. J. M. Madarassy and C. F. Barenghi, Vortex dynamics in trapped Bose-Einstein condensate, J. Low Temp. Phys., 152 (2008), 122-135. doi: 10.1007/s10909-008-9811-9.

[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), 260402. doi: 10.1103/PhysRevLett.89.260402.

[32]

L. P. Pitaevskii, , Zh. Eksp. Teor. Fiz., 35 (1958), 408; [Sov. Phys. JETP, 35 (1959), 282].

[33]

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

[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), 130408. doi: 10.1103/PhysRevLett.93.130408.

[35]

R. Sásik, L. M. A. Bettencourt and S. Habib, Thermal vortex dynamics in a two-dimensional condensate, Phys. Rev. B, 62 (2000), 1238-1243. doi: 10.1103/PhysRevB.62.1238.

[36]

D. V. Skryabin, Energy of internal modes of nonlinear waves and complex frequencies due to symmetry breaking, Phys. Rev. E, 64 (2001), 055601. doi: 10.1103/PhysRevE.64.055601.

[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), 120406. doi: 10.1103/PhysRevLett.101.120406.

[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), 045601. doi: 10.1103/PhysRevA.76.045601.

[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), 063604. doi: 10.1103/PhysRevA.81.063604.

[40]

M. Tsubota, K. Kasamatsu and M. Ueda, Vortex lattice formation in a rotating Bose-Einstein condensate, Phys. Rev. A, 65 (2002), 023603; ibid Nonlinear dynamics of vortex lattice formation in a rotating Bose-Einstein condensate, 67 (2003), 033610.

[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), 130401. doi: 10.1103/PhysRevLett.101.130401.

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), 2926-2929. doi: 10.1103/PhysRevLett.86.2926.

[2]

I. Aranson and V. Steinberg, Stability of multicharged vortices in a model of superflow, Phys. Rev. B, 53 (1996), 75-78. doi: 10.1103/PhysRevB.53.75.

[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), 496-501. doi: 10.1038/nphys962.

[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), 671-680.

[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), 5198-5201. doi: 10.1103/PhysRevLett.83.5198.

[6]

Th. Busch and J. R. Anglin, Motion of dark solitons in trapped Bose-Einstein Condensates, Phys. Rev. Lett., 84 (2000), 2298-2301. doi: 10.1103/PhysRevLett.84.2298.

[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), 033625.

[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), R139-R202.

[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), 023605.

[10]

S. Choi, S. A. Morgan and K. Burnett, Phenomenological damping in trapped atomic Bose-Einstein condensates, Phys. Rev. A, 57 (1998), 4057-4060. doi: 10.1103/PhysRevA.57.4057.

[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), 174101. doi: 10.1103/PhysRevLett.104.174101.

[12]

S. P. Cockburn and N. P. Proukakis, The stochastic Gross-Pitaevskii equation and some applications, Laser Phys., 19 (2009), 558-570. doi: 10.1134/S1054660X09040057.

[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), 97-101. doi: 10.1126/science.287.5450.97.

[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), 663-668. doi: 10.1126/science.1062527.

[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), 160405. doi: 10.1103/PhysRevLett.99.160405.

[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), 3220-3224. doi: 10.1103/PhysRevA.60.3220.

[17]

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

[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), 053608. doi: 10.1103/PhysRevA.66.053608.

[19]

R. Graham, Decoherence of Bose-Einstein condensates in traps at finite temperature, Phys. Rev. Lett., 81 (1998), 5262-5265. doi: 10.1103/PhysRevLett.81.5262.

[20]

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

[21]

B. Jackson, C. F. Barenghi and N. P. Proukakis, Matter wave solitons at finite temperatures, J. Low Temp. Phys., 148 (2007), 387-391. doi: 10.1007/s10909-007-9410-1.

[22]

B. Jackson, N. P. Proukakis and C. F. Barenghi, Dark-soliton dynamics in Bose-Einstein condensates at finite temperature, Phys., Rev. A, 75 (2007), 051601. doi: 10.1103/PhysRevA.75.051601.

[23]

T. Kapitula and P. G. Kevrekidis, Bose-Einstein condensates in the presence of a magnetic trap and optical lattice, Chaos, 15 (2005), 037114. doi: 10.1063/1.1993867.

[24]

T. Kapitula, P. G. Kevrekidis and B. Sandstede, Counting eigenvalues via the Krein signature in infinite-dimensional Hamiltonian system, Physica D, 195 (2004), 263-282. doi: 10.1016/j.physd.2004.03.018.

[25]

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

[26]

Yu. S. Kivshar and W. Królikowski, Lagrangian approach for dark solitons, Opt. Commun., 114 (1995), 353-362. doi: 10.1016/0030-4018(94)00644-A.

[27]

Yu. S. Kivshar and X. Yang, Perturbation-induced dynamics of dark solitons, Phys. Rev. E, 49 (1994), 1657-1670. doi: 10.1103/PhysRevE.49.1657.

[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), 3583-3590. doi: 10.1088/0953-4075/35/16/316.

[29]

M. D. Lee and C. W. Gardiner, Quantum kinetic theory. VI. The growth of a Bose-Einstein condensate, Phys. Rev. A, 62 (2000), 033606. doi: 10.1103/PhysRevA.62.033606.

[30]

E. J. M. Madarassy and C. F. Barenghi, Vortex dynamics in trapped Bose-Einstein condensate, J. Low Temp. Phys., 152 (2008), 122-135. doi: 10.1007/s10909-008-9811-9.

[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), 260402. doi: 10.1103/PhysRevLett.89.260402.

[32]

L. P. Pitaevskii, , Zh. Eksp. Teor. Fiz., 35 (1958), 408; [Sov. Phys. JETP, 35 (1959), 282].

[33]

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

[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), 130408. doi: 10.1103/PhysRevLett.93.130408.

[35]

R. Sásik, L. M. A. Bettencourt and S. Habib, Thermal vortex dynamics in a two-dimensional condensate, Phys. Rev. B, 62 (2000), 1238-1243. doi: 10.1103/PhysRevB.62.1238.

[36]

D. V. Skryabin, Energy of internal modes of nonlinear waves and complex frequencies due to symmetry breaking, Phys. Rev. E, 64 (2001), 055601. doi: 10.1103/PhysRevE.64.055601.

[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), 120406. doi: 10.1103/PhysRevLett.101.120406.

[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), 045601. doi: 10.1103/PhysRevA.76.045601.

[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), 063604. doi: 10.1103/PhysRevA.81.063604.

[40]

M. Tsubota, K. Kasamatsu and M. Ueda, Vortex lattice formation in a rotating Bose-Einstein condensate, Phys. Rev. A, 65 (2002), 023603; ibid Nonlinear dynamics of vortex lattice formation in a rotating Bose-Einstein condensate, 67 (2003), 033610.

[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), 130401. doi: 10.1103/PhysRevLett.101.130401.

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