Quasiperiodic dynamics in bose-einstein condensates in periodic lattices and superlattices

Journal of Nonlinear Science

Volumn 17, Issue 1, 2007, Pages 59-83

  Van Noort, M ^a   Porter, Mason A ^b   Yi, Y ^c   Chow, S N ^c  

^a IMPERIAL COLLEGE LONDON   (United Kingdom)

^b California Institute of Technology   (United States)

^c GEORGIA INSTITUTE OF TECHNOLOGY   (United States)

Author keywords

Aubry Mather theory; Bose Einstein condensates; Hamiltonian dynamics; KAM theory

Indexed keywords

EID: 84867976637     PISSN: 09388974     EISSN: 14321467     Source Type: Journal    
DOI: 10.1007/s00332-005-0723-4     Document Type: Article

References (66)

1. B. P. Anderson and M. A. Kasevich. Macroscopic quantum interference from atomic tunnel arrays. Science, 282(5394):1686-1689, 1998. (Pubitemid 28549279)
2. M. H. Anderson, J. R. Ensher,M. R. Matthews, C. E.Wieman, and E. A. Cornell. Observation of Bose-Einstein condensation in a dilute atomic vapor. Science, 269(5221):198-201, 1995.
3. V. I. Arnol'd. Small denominators and problems of stability of motion in classical and celestial mechanics. Uspehi Mat. Nauk, 186 (114)):91-192, 1963. Translated in Russian Math. Surveys.
4. S. Aubry and P. Y. Le Daeron. The discrete Frenkel-Kontorova model and its extensions. I. Exact results for the ground-states. Physica D, 8(3):381-422, 1983.
5. G. D. Birkhoff. Proof of Poincaŕe's geometric theorem. Trans. Amer. Math. Soc., 14(1):14-22, 1913.
6. G. D. Birkhoff. An extension of Poincaŕe's last geometric theorem. Acta Mathematica, 47:297-311, 1925.
7. H. W. Broer, I. Hoveijn, M. van Noort, C. Sim'o, and G. Vegter. The parametrically forced pendulum: A case study in 1 1 2 degree of freedom. J. Dynam. Diff. Eq., 16(4):897-947, 2004. (Pubitemid 40133381)
8. H. W. Broer, M. van Noort, and C. Sim'o. Existence and measure of 2-quasiperiodicity in Hamiltonian one-and-a-half degree of freedom systems. In Equadiff-International Conference on Differential Equations, Hasselt 2003, pages 595-600. World Scientific, 2005.
9. J. C. Bronski, L. D. Carr, R. Carretero-Gonźalez, B. Deconinck, J. N. Kutz, and K. Promislow. Stability of attractive Bose-Einstein condensates in a periodic potential. Phys. Rev. E, 64(056615), 2001.
10. J. C. Bronski, L. D. Carr,B. Deconinck, and J. N.Kutz. Bose-Einstein condensates in standing waves: The cubic nonlinear Schr̈odinger equation with a periodic potential. Phys. Rev. Lett., 86(8):1402-1405, February 2001. (Pubitemid 32873600)
11. J. C. Bronski, L. D. Carr, B. Deconinck, J. N. Kutz, and K. Promislow. Stability of repulsive Bose-Einstein condensates in a periodic potential. Phys. Rev. E, 63(036612), 2001.
12. R. Carretero-Gonźalez and K. Promislow. Localized breathing oscillations of Bose-Einstein condensates in periodic traps. Phys. Rev. A, 66(033610), 2002.
13. F. S. Cataliotti, L. Fallani, F. Ferlaino, C. Fort, P. Maddaloni, and M. Inguscio. Superfluid current disruption in a chain of weakly coupled Bose-Einstein condensates. New J. Phys., 5(71), 2003.
14. G. Chong, W. Hai, and Q. Xie. Spatial chaos of trapped Bose-Einstein condensate in onedimensional weak optical lattice potential. Chaos, 14(2):217-223, 2004. (Pubitemid 38834907)
15. S.-N. Chow, M. van Noort, and Y. Yi. Quasiperiodic dynamics in Hamiltonian 1 1 2 degree of freedom systems far from integrability. J. Diff. Eq., 212(2):366-393, 2005.
16. V. P. Chua and M. A. Porter. Spatial resonance overlap in Bose-Einstein condensates in optical superlattices. Int. J. Bifurc. Chaos, 16:945-959.
17. F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari. Theory of Bose-Einstein condensation in trapped gases. Rev. Mod. Phys., 71(3):463-512, 1999. (Pubitemid 129330877)
18. K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle. Bose-Einstein condensation in a gas of sodium atoms. Phys. Rev. Lett., 75(22):3969-3973, 1995.
19. J. Hecker Denschlag, J. E. Simsarian, H. Ḧaffner, C. McKenzie, A. Browaeys, D. Cho, K. Helmerson, S. L. Rolston, and W. D. Phillips. A Bose-Einstein condensate in an optical lattice. J. Phys. B: Atomic Molecular and Optical Physics, 35:3095-3110, 2002. (Pubitemid 34854562)
20. R. Dieckerhoff and E. Zehnder. Boundedness of solutions via the twist-theorem. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 14(1):79-95, 1987.
21. E. A. Donley, N. R. Claussen, S. L. Cornish, J. L. Roberts, E. A. Cornell, and C. E.Weiman. Dynamics of collapsing and exploding Bose-Einstein condensates. Nature, 412:295-299, 2001. (Pubitemid 32691836)
22. Y. Eksioglu, P. Vignolo, and M. P. Tosi. Matter-wave interferometry in periodic and quasiperiodic arrays. Optics Commun., 243(1-6):175-181, 2004.
23. N. Gemelke, E. Sarajlic, Y. Bidel, S. Hong, and S. Chu. Parametric amplification of matter waves in periodically translated optical lattices. Phys. Rev. Lett., 95(170404), 2005.
24. M. Greiner, O. Mandel, T. Esslinger, T. Ḧansch, and I. Bloch. Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms. Nature, 415(6867):39-44, 2002. (Pubitemid 34062442)
25. J. Guckenheimer and P. Holmes. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Number 42 in Applied Mathematical Sciences. Springer-Verlag, New York, 1983.
26. E.W. Hagley, L. Deng, M. Kozuma, J.Wen, K. Helmerson, S. L. Rolston, andW. D. Phillips. A well-collimated quasi-continuous atom laser. Science, 283(5408):1706-1709, 1999.
27. M. R. Herman. Sur les courbes invariantes par les diff'eomorphismes de l'anneau, vol. 1, volume 103-104 of Ast'erisque. Socíet'e Math'ematique de France, 1983.
28. M. R. Herman. Sur les courbes invariantes par les diff'eomorphismes de l'anneau, vol. 2, volume 144 of Ast'erisque. Socíet'e Math'ematique de France, 1986.
29. H. Huang. Destruction of invariant tori in pendulum-type equations. J. Diff. Eq., 146(1):67-89, 1998. (Pubitemid 128345173)
30. P. G.Kevrekidis, G. Theocharis, D. J. Frantzeskakis, and B. A. Malomed. Feshbach resonance management for Bose-Einstein condensates. Phys. Rev. Lett., 90:230401, 2003.
31. M. Levi. Quasiperiodic motions in superquadratic time-periodic potentials. Commun. Math. Phys., 143(1):43-83, 1991.
32. M. Levi and E. Zehnder. Boundedness of solutions for quasiperiodic potentials. SIAM J. Math. Anal., 26(5):1233-1256, 1995.
33. E. H. Lieb, R. Seiringer, J. P. Solovej, and J. Yngvason. The Mathematics of the Bose Gas and Its Condensation. Birk̈auser, Basel, Germany, 2005.
34. J. E. Littlewood. Unbounded solutions of an equation y + g(y) = p(t), with p(t) periodic and bounded, and g(y)/y→∞as y →∞. J. London Math. Soc., 41:491-507, 1966.
35. B. Liu. Boundedness for solutions of nonlinear Hill's equations with periodic forcing terms via Moser's twist theorem. J. Diff. Eq., 79(2):304-315, 1989.
36. P. J. Y. Louis, E. A. Ostrovskaya, and Y. S. Kivshar. Matter-wave dark solitons in optical lattices. J. Optics B: Quantum and Semiclassical Optics, 6:S309-S317, 2004.
37. P. J. Y. Louis, E. A. Ostrovskaya, C. M. Savage, and Y. S. Kivshar. Bose-Einstein condensates in optical lattices: Band-gap structure and solitons. Phys. Rev. A, 67(013602), 2003.
38. M. Machholm, A. Nicolin, C. J. Pethick, and H. Smith. Spatial period-doubling in Bose-Einstein condensates in an optical lattice. Phys. Rev. A, 69(043604), 2004.
39. R. S. MacKay. A criterion for nonexistence of invariant tori for Hamiltonian systems. Physica D, 36(1-2):64-82, 1989.
40. J. N. Mather. Existence of quasiperiodic orbits for twist homeomorphisms of the annulus. Topology, 21(4):457-467, 1982.
41. J. N. Mather. Nonexistence of invariant circles. Ergodic Theory Dynam. Syst., 4(2):301-309, 1984.
42. C. Menotti, A. Smerzi, and A. Trombettoni. Superfluid dynamics of a Bose-Einstein condensate in a periodic potential. New J. Phys., 5(112), 2003.
43. O. Morsch, J. H. M̈uller, M. Christiani, D. Ciampini, and E. Arimondo. Bloch oscillations and mean-field effects of Bose-Einstein condensates in 1D optical lattices. Phys. Rev. Lett., 87(140402), September 2001.
44. J. Moser. Recent developments in the theory of Hamiltonian systems. SIAM Rev., 28(4):459-485, 1986. (Pubitemid 17481705)
45. J. Moser. Minimal foliations on a torus. In M. Giaquinta, editor, Topics in calculus of variations (Montecatini Terme, 1987), volume 1365 of Lecture Notes in Mathematics, pages 62-99. Springer-Verlag, New York, 1989.
46. J. Moser. Quasi-periodic solutions of nonlinear elliptic partial differential equations. Bol. Soc. Brasil. Mat. (N.S.), 20(1):29-45, 1989.
47. A. I. Neishtadt. Estimates in the Kolmogorov theorem on conservation of conditionally periodic motions. J. Appl. Math. Mech., 45(6):1016-1025, 1981.
48. A. I. Neishtadt, V. V. Sidorenko, and D. V. Treschev. Stable periodic motions in the problem on passage through a separatrix. Chaos, 7(1):2-11, 1997.
49. C. Orzel, A. K. Tuchman, M. L. Fenselau, M. Yasuda, and M. A. Kasevich. Squeezed states in a Bose-Einstein condensate. Science, 291(5512):2386, 2001.
50. P. Pedri, L. Pitaevskii, S. Stringari, C. Fort, S. Burger, F. S. Cataliotii, P. Maddaloni, F. Minardi, and M. Inguscio. Expansion of a coherent array of Bose-Einstein condensates. Phys. Rev. Lett., 87(22):220401, 2001.
51. S. Peil, J. V. Porto, B. Laburthe Tolra, J. M. Obrecht, B. E. King, M. Subbotin, S. L. Rolston, and W. D. Phillips. Patterned loading of a Bose-Einstein condensate into an optical lattice. Phys. Rev. A, 67(051603(R)), 2003.
52. M. A. Porter and P. Cvitanovíc. Modulated amplitude waves in Bose-Einstein condensates. Phys. Rev. E, 69(047201), 2004.
53. M. A. Porter and P. Cvitanovíc. A perturbative analysis of modulated amplitude waves in Bose-Einstein condensates. Chaos, 14(3):739-755, 2004. (Pubitemid 39443306)
54. M. A. Porter and P. G. Kevrekidis. Bose-Einstein condensates in superlattices. SIAM J. Appl. Dynam. Syst., 4(4):783-807, 2005.
55. M. A. Porter, P.G.Kevrekidis, R. Carretero-Gonźalez, and D. J. Frantzeskakis. Dynamics and manipulation of matter-wave solitons in optical superlattices. Phys. Lett. A, 352:210-215, 2006. (Pubitemid 43330749)
56. M. A. Porter, P. G. Kevrekidis, and B. A. Malomed. Resonant and non-resonant modulated amplitude waves for binary Bose-Einstein condensates in optical lattices. Physica D, 196(1-2):106-123, 2004.
57. J. V. Porto, S. Rolston, B. Laburthe Tolra, C. J. Williams, and W. D. Phillips. Quantum information with neutral atoms as qubits. Phil. Trans.: Math., Physical & Engin. Sci., 361(1808):1417-1427, 2003. (Pubitemid 37256204)
58. J. P̈oschel. The concept of integrability on Cantor sets for Hamiltonian systems. Celestial Mech., 28(1-2):133-139, 1982.
59. A. M. Rey, B. L. Hu, E. Calzetta, A. Roura, and C. Clark. Nonequilibrium dynamics of optical lattice-loaded BEC atoms: Beyond HFB approximation. Phys. Rev. A, 69:033610, 2004.
60. L. Salasnich, A. Parola, and L. Reatto. Periodic quantum tunnelling and parametric resonance with cigar-shaped Bose-Einstein condensates. J. Phys. B: Atomic Molecular and Optical Physics, 35(14):3205-3216, 2002. (Pubitemid 34854569)
61. A. Smerzi, A. Trombettoni, P. G. Kevrekidis, and A. R. Bishop. Dynamical superfluidinsulator transition in a chain of weakly coupled Bose-Einstein condensates. Phys. Rev. Lett., 89:170402, 2002.
62. Q. Thommen, J. C. Garreau, and V. Zehne. Classical chaos with Bose-Einstein condensates in tilted optical lattices. Phys. Rev. Lett., 91:210405, 2003.
63. K. G. H. Vollbrecht, E. Solano, and J. L. Cirac. Ensemble quantum computation with atoms in periodic potentials. Phys. Rev. Lett., 93(220502), 2004.
64. S. Wiggins. Global Bifurcations and Chaos. Analytical Methods, Volume 73 of Applied Mathematical Sciences. Springer-Verlag, New York, 1988.
65. B.Wu, R. B. Diener, and Q. Niu. Bloch waves and Bloch bands of Bose-Einstein condensates in optical lattices. Phys. Rev. A, 65(025601), 2002.
66. J. You. Invariant tori and Lagrange stability of pendulum-type equations. J. Diff. Eq., 85:54-65, 1990.