Quantum fluctuations in trapped time-dependent Bose-Einstein condensates

Physical Review A - Atomic, Molecular, and Optical Physics

Volumn 79, Issue 3, 2009, Pages

  Uhlmann, Michael ^a  

^a AUSTRALIAN NATIONAL UNIVERSITY   (Australia)

Author keywords

[No Author keywords available]

Indexed keywords

BIOELECTRIC PHENOMENA; BOSE-EINSTEIN CONDENSATION; FLUID DYNAMICS; QUANTUM ELECTRONICS; STEAM CONDENSERS;

BOGOLIUBOV THEORIES; BOGOLIUBOV- DE GENNES EQUATIONS; BOSE- EINSTEIN CONDENSATES; COUPLING TERMS; DIAGONALIZATION; DIFFERENT MODES; EFFECTIVE SPACES; EIGEN-MODE EXPANSIONS; FIELD OPERATORS; HYDRODYNAMIC REGIMES; INHOMOGENEITY; INTERACTION STRENGTHS; INTERMODE COUPLINGS; LOW ENERGIES; QUANTUM FLUCTUATIONS; QUASI ONE-DIMENSIONAL; QUASIPARTICLE; TIME DEPENDENTS; TRAP FREQUENCIES;

MATHEMATICAL OPERATORS;

EID: 62849104871     PISSN: 10502947     EISSN: 10941622     Source Type: Journal    
DOI: 10.1103/PhysRevA.79.033601     Document Type: Article

References (85)

1. I. Bloch, J. Dalibard, and W. Zwerger, Rev. Mod. Phys. 80, 885 (2008). 10.1103/RevModPhys.80.885
2. L. Pitaevskii and S. Stringari, Bose-Einstein Condensation (Oxford University Press, Oxford, UK, 2003)
3. A. J. Leggett, Rev. Mod. Phys. 73, 307 (2001). 10.1103/RevModPhys.73.307
4. M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature (London) 415, 39 (2002); 10.1038/415039a
5. F. Gerbier, S. Fölling, A. Widera, O. Mandel, and I. Bloch, Phys. Rev. Lett. 96, 090401 (2006); 10.1103/PhysRevLett.96.090401
6. S. Fölling, A. Widera, T. Müller, F. Gerbier, and I. Bloch, Phys. Rev. Lett. 97, 060403 (2006). 10.1103/PhysRevLett.97.060403
7. M. P. A. Fisher, P. B. Weichman, G. Grinstein, and D. S. Fisher, Phys. Rev. B 40, 546 (1989) 10.1103/PhysRevB.40.546
8. D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, Phys. Rev. Lett. 81, 3108 (1998) 10.1103/PhysRevLett.81.3108
9. D. Jaksch and P. Zoller, Ann. Phys. (N.Y.) 315, 52 (2005) 10.1016/j.aop.2004.09.010
10. R. Schützhold, M. Uhlmann, Y. Xu, and U. R. Fischer, Phys. Rev. Lett. 97, 200601 (2006) 10.1103/PhysRevLett.97.200601
11. C. Kollath, A. M. Läuchli, and E. Altman, Phys. Rev. Lett. 98, 180601 (2007). 10.1103/PhysRevLett.98.180601
12. G. Volovik, The Universe in a Helium Droplet (Oxford University Press, Oxford, UK, 2003).
13. C. Barceló, S. Liberati, and M. Visser, Living Rev. Relativ. 8, 12 (2005).
14. U. R. Fischer, Mod. Phys. Lett. A 19, 1789 (2004). 10.1142/ S0217732304015099
15. L. J. Garay, J. R. Anglin, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 85, 4643 (2000). 10.1103/PhysRevLett.85.4643
16. S. Wüster, Phys. Rev. A 78, 021601 (R) (2008) 10.1103/PhysRevA.78. 021601
17. S. Wüster and C. M. Savage, Phys. Rev. A 76, 013608 (2007). 10.1103/PhysRevA.76.013608
18. W. G. Unruh, Phys. Rev. Lett. 46, 1351 (1981) 10.1103/PhysRevLett.46.1351
19. W. G. Unruh, Phys. Rev. D 51, 2827 (1995). 10.1103/PhysRevD.51.2827
20. Artificial Black Holes, edited by, M. Novello, M. Visser and, G. Volovik, (World Scientific, Singapore, 2002).
21. P. Jain, S. Weinfurtner, M. Visser, and C. W. Gardiner, Phys. Rev. A 76, 033616 (2007). 10.1103/PhysRevA.76.033616
22. S. Weinfurtner, A. White, and M. Visser, Phys. Rev. D 76, 124008 (2007). 10.1103/PhysRevD.76.124008
23. C. Barceló, S. Liberati, and M. Visser, Phys. Rev. A 68, 053613 (2003) 10.1103/PhysRevA.68.053613
24. C. Barceló, S. Liberati, and M. Visser, Int. J. Mod. Phys. D 12, 1641 (2003) 10.1142/S0218271803004092
25. C. Barceló, S. Liberati, and M. Visser, Class. Quantum Grav. 18, 1137 (2001) 10.1088/0264-9381/18/6/312
26. C. Barceló, S. Liberati, and M. Visser, Class. Quantum Grav. 18, 3595 (2001). 10.1088/0264-9381/18/17/313
27. U. R. Fischer and R. Schützhold, Phys. Rev. A 70, 063615 (2004). 10.1103/PhysRevA.70.063615
28. M. Uhlmann, Yan Xu, and R. Schützhold, New J. Phys. 7, 248 (2005). 10.1088/1367-2630/7/1/248
29. P. O. Fedichev and U. R. Fischer, Phys. Rev. A 69, 033602 (2004). 10.1103/PhysRevA.69.033602
30. Y. Kurita and T. Morinari, Phys. Rev. A 76, 053603 (2007) 10.1103/PhysRevA.76.053603
31. Y. Kurita, M. Kobayashi, T. Morinari, M. Tsubota, and H. Ishihara, e-print arXiv:0810.3088.
32. M. Visser and S. Weinfurtner, Phys. Rev. D 72, 044020 (2005). 10.1103/PhysRevD.72.044020
33. S. W. Hawking, Nature (London) 248, 30 (1974) 10.1038/248030a0
34. S. W. Hawking, Commun. Math. Phys. 43, 199 (1975). 10.1007/BF02345020
35. A. R. Liddle and D. H. Lyth, Cosmological Inflation and Large-Scale Structure (Cambridge University Press, Cambridge, UK, 2000).
36. N. D. Birrell and P. C. Q. Davies, Quantum Fields in Curved Space (Cambridge University Press, Cambridge, UK, 1982).
37. S. Fölling, F. Gerbier, A. Widera, O. Mandel, T. Gericke, and I. Bloch, Nature (London) 434, 481 (2005). 10.1038/nature03500
38. D. Hellweg, L. Cacciapuoti, M. Kottke, T. Schulte, K. Sengstock, W. Ertmer, and J. J. Arlt, Phys. Rev. Lett. 91, 010406 (2003) 10.1103/PhysRevLett. 91.010406
39. S. Hofferberth, I. Lesanovsky, B. Fischer, T. Schumm, and J. Schmiedmayer, Nature (London) 449, 324 (2007) 10.1038/nature06149
40. S. Hofferberth, Nat. Phys. 4, 489 (2008). 10.1038/nphys941
41. S. B. Papp, J. M. Pino, R. J. Wild, S. Ronen, C. E. Wieman, D. S. Jin, and E. A. Cornell, Phys. Rev. Lett. 101, 135301 (2008). 10.1103/PhysRevLett.101. 135301
42. M. Schellekens, R. Hoppeler, A. Perrin, J. Viana Gomes, D. Boiron, A. Aspect, and C. I. Westbrook, Science 310, 648 (2005). 10.1126/science.1118024
43. J. Esteve, J. B. Trebbia, T. Schumm, A. Aspect, C. I. Westbrook, and I. Bouchoule, Phys. Rev. Lett. 96, 130403 (2006). 10.1103/PhysRevLett.96.130403
44. A. Griffin, Phys. Rev. B 53, 9341 (1996) 10.1103/PhysRevB.53.9341
45. S. Giorgini, Phys. Rev. A 61, 063615 (2000). 10.1103/PhysRevA.61.063615
46. M. J. Davis, S. A. Morgan, and K. Burnett, Phys. Rev. Lett. 87, 160402 (2001) 10.1103/PhysRevLett.87.160402
47. P. B. Blakie and M. J. Davis, Phys. Rev. A 72, 063608 (2005) 10.1103/PhysRevA.72.063608
48. P. B. Blakie, A. S. Bradley, M. J. Davis, R. J. Ballagh, and C. W. Gardiner, Adv. Phys. 57, 363 (2008). 10.1080/00018730802564254
49. S. Wüster, B. J. Da̧browska-Wüster, A. S. Bradley, M. J. Davis, P. B. Blakie, J. J. Hope, and C. M. Savage, Phys. Rev. A 75, 043611 (2007). 10.1103/PhysRevA.75.043611
50. Dimensionless units can be introduced through the initial trap frequency ω0 and the atom mass m. All energies are then measured in units of ω0, the dimensionless time is defined through t= ω0 τ, and the characteristic length a0 =1/ m ω0, where =1.
51. E. Tiesinga, B. J. Verhaar, and H. T. C. Stoof, Phys. Rev. A 47, 4114 (1993) 10.1103/PhysRevA.47.4114
52. S. Inouye, M. R. Andrews, J. Stenger, H.-J. Miesner, D. M. Stamper-Kurn, and W. Ketterle, Nature (London) 392, 151 (1998) 10.1038/32354
53. P. Courteille, R. S. Freeland, D. J. Heinzen, F. A. van Abeelen, and B. J. Verhaar, Phys. Rev. Lett. 81, 69 (1998). 10.1103/PhysRevLett.81.69
54. Y. Castin and R. Dum, Phys. Rev. Lett. 77, 5315 (1996) 10.1103/PhysRevLett.77.5315
55. Yu. Kagan, E. L. Surkov, and G. V. Shlyapnikov, Phys. Rev. A 54, R1753 (1996). 10.1103/PhysRevA.54.R1753
56. M. Girardeau and R. Arnowitt, Phys. Rev. 113, 755 (1959) 10.1103/PhysRev.113.755
57. C. W. Gardiner, Phys. Rev. A 56, 1414 (1997) 10.1103/PhysRevA.56.1414
58. M. D. Girardeau, Phys. Rev. A 58, 775 (1998). 10.1103/PhysRevA.58.775
59. E. P. Gross, Nuovo Cimento 20, 454 (1961) 10.1007/BF02731494
60. L. P. Pitaevskii, Zh. Eksp. Teor. Fiz. 40, 646 (1961);
61. [L. P. Pitaevskii, Sov. Phys JETP 13, 451 (1961)].
62. N. N. Bogoliubov, J. Phys. (Moscow) 11, 23 (1947)
63. P. G. de Gennes, Superconductivity of Metals and Alloys (Benjamin, New York, 1966).
64. A similar expansion of the linear fluctuations is also considered in. There, the authors expanded the linear field operator χ of a static/stationary condensate into eigenmodes and not the phase and density fluctuations χ ±. This leads to complex mode functions vn and un, while hn± are real.
65. A. L. Fetter, Ann. Phys. (N.Y.) 70, 67 (1972) 10.1016/0003-4916(72)90330- 2
66. M. Lewenstein and L. You, Phys. Rev. Lett. 77, 3489 (1996) 10.1103/PhysRevLett.77.3489
67. M. Naraschewski and R. J. Glauber, Phys. Rev. A 59, 4595 (1999). 10.1103/PhysRevA.59.4595
68. R. Schützhold, Phys. Rev. Lett. 97, 190405 (2006). 10.1103/PhysRevLett.97.190405
69. The eigenfunctions of density and phase fluctuations δ and δ Φ are defined slightly differently from those of χ +: χ + = hn+ X n+ = (hn / 0) δ n =δ / 0, χ - = hn- X n- = 0 hn δ Φ = 0 δ Φ. Noting that x 0 0 the center of the trap, the eigenfunctions hn+ and hn (and similarly hn- and hn) must be approximately proportional for low excitations. Hence, the Xn± for Ωn μ0 indeed represent the proper density and phase eigenmodes δ n and δ Φ n up to a constant prefactor 0 ±1/2.
70. R. M. Wald, General Relativity (University of Chicago Press, Chicago, IL, 1984).
71. C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation (Freeman, San Francisco, 1973).
72. M. Visser, Lorentzian Wormholes: From Einstein to Hawking (Springer, New York, 1996).
73. S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Spacetime (Cambridge University Press, Cambridge, UK, 1973).
74. M. Visser, Class. Quantum Grav. 15, 1767 (1998). 10.1088/0264-9381/15/6/ 024
75. R. Schützhold, Lect. Notes Phys. 718, 5 (2007) 10.1007/3-540-70859- 6-2
76. R. Schützhold, Class. Quantum Grav. 25, 114011 (2008). 10.1088/0264-9381/25/11/114011
77. C. M. Dion and E. Cancès, Phys. Rev. E 67, 046706 (2003). 10.1103/PhysRevE.67.046706
78. Handbook of Mathematical Functions, edited by, M. Abramowitz, and, I. A. Stegun, (Dover, New York, 1970).
79. S. Sachdev, Quantum Phase Transitions (Cambridge University Press, Cambridge, UK, 2000).
80. L. E. Sadler, J. M. Higbie, S. R. Leslie, M. Vengalattore, and D. M. Stamper-Kurn, Nature (London) 443, 312 (2006) 10.1038/nature05094
81. H. Saito and M. Ueda, Phys. Rev. A 72, 023610 (2005) 10.1103/PhysRevA.72. 023610
82. H. Saito, Y. Kawaguchi, and M. Ueda, Phys. Rev. Lett. 96, 065302 (2006) 10.1103/PhysRevLett.96.065302
83. A. Lamacraft, Phys. Rev. Lett. 98, 160404 (2007) 10.1103/PhysRevLett.98. 160404
84. M. Uhlmann, R. Schützhold, and U. R. Fischer, Phys. Rev. Lett. 99, 120407 (2007). 10.1103/PhysRevLett.99.120407
85. S. Giorgini, L. P. Pitaevskii, and S. Stringari, Phys. Rev. Lett. 80, 5040 (1998). 10.1103/PhysRevLett.80.5040