Density ripples in expanding low-dimensional gases as a probe of correlations

Physical Review A - Atomic, Molecular, and Optical Physics

Volumn 80, Issue 3, 2009, Pages

  Imambekov, A ^a,b   Mazets, I E ^c,d   Petrov, D S ^e,f   Gritsev, V ^g   Manz, S ^c   Hofferberth, S ^h   Schumm, T ^c,i   Demler, E ^h   Schmiedmayer, J ^c  

^a Yale University   (United States)

^b RICE UNIVERSITY   (United States)

^c TU WIEN   (Austria)

^d IOFFE INSTITUTE   (Russian Federation)

^e UNIVERSITÉ PARIS SUD   (France)

^f KURCHATOV INSTITUTE   (Russian Federation)

^g UNIVERSITY OF FRIBOURG   (Switzerland)

^h HARVARD UNIVERSITY   (United States)

ⁱ UNIVERSITY OF VIENNA   (Austria)

Author keywords

[No Author keywords available]

Indexed keywords

ANALYTICAL EXPRESSIONS; ANALYTICAL RELATIONS; BALLISTIC EXPANSION; BEREZINSKII-KOSTERLITZ-THOULESS TRANSITION; BKT TRANSITION; BOSE GAS; CONFINED SYSTEMS; CORRELATION FUNCTION; DENSITY FLUCTUATION; FIRST-ORDER; QUANTUM FLUCTUATION; SELF-SIMILAR; SHAPE-CONTROLLED; TRANSVERSE CONFINEMENT; TWO-POINT; ULTRA-COLD;

BIOELECTRIC PHENOMENA; BOSONS; ELECTRON ENERGY ANALYZERS; GASES; LEAKAGE (FLUID); QUANTUM ELECTRONICS; THERMAL EXPANSION;

DENSITY OF GASES;

EID: 70249151041     PISSN: 10502947     EISSN: 10941622     Source Type: Journal    
DOI: 10.1103/PhysRevA.80.033604     Document Type: Article

References (158)

1. M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature (London) 415, 39 (2002). 10.1038/415039a
2. U. Schneider, L. Hackermüller, S. Will, Th. Best, I. Bloch, T. A. Costi, R. W. Helmes, D. Rasch, and A. Rosch, Science 322, 1520 (2008). 10.1126/science.1165449
3. A. Perrin, H. Chang, V. Krachmalnicoff, M. Schellekens, D. Boiron, A. Aspect, and C. I. Westbrook, Phys. Rev. Lett. 99, 150405 (2007). 10.1103/PhysRevLett.99.150405
4. S. Hofferberth, I. Lesanovsky, B. Fischer, T. Schumm, and J. Schmiedmayer, Nature (London) 449, 324 (2007). 10.1038/nature06149
5. S. Hofferberth, I. Lesanovsky, T. Schumm, A. Imambekov, V. Gritsev, E. Demler, and J. Schmiedmayer, Nat. Phys. 4, 489 (2008). 10.1038/nphys941
6. V. L. Berezinskii, Sov. Phys. JETP 32, 493 (1971)
7. V. L. Berezinskii, Sov. Phys. JETP 34, 610 (1972).
8. J. M. Kosterlitz and D. J. Thouless, J. Phys. C 6, 1181 (1973) 10.1088/0022-3719/6/7/010
9. J. M. Kosterlitz, J. Phys. C 7, 1046 (1974). 10.1088/0022-3719/7/6/005
10. Z. Hadzibabic, P. Krüger, M. Cheneau, B. Battelier, and J. Dalibard, Nature (London) 441, 1118 (2006). 10.1038/nature04851
11. 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
12. T. Jeltes, J. M. McNamara, W. Hogervorst, W. Vassen, V. Krachmalnicoff, M. Schellekens, A. Perrin, H. Chang, D. Boiron, A. Aspect, and C. I. Westbrook, Nature (London) 445, 402 (2007). 10.1038/nature05513
13. S. Fölling, F. Gerbier, A. Widera, O. Mandel, T. Gericke, and I. Bloch, Nature (London) 434, 481 (2005). 10.1038/nature03500
14. T. Rom, Th. Best, D. van Oosten, U. Schneider, S. Fölling, B. Paredes, and I. Bloch, Nature (London) 444, 733 (2006). 10.1038/nature05319
15. A. Öttl, S. Ritter, M. Köhl, and T. Esslinger, Phys. Rev. Lett. 95, 090404 (2005). 10.1103/PhysRevLett.95.090404
16. F. Schreck, L. Khaykovich, K. L. Corwin, G. Ferrari, T. Bourdel, J. Cubizolles, and C. Salomon, Phys. Rev. Lett. 87, 080403 (2001). 10.1103/PhysRevLett.87.080403
17. A. Görlitz, J. M. Vogels, A. E. Leanhardt, C. Raman, T. L. Gustavson, J. R. Abo-Shaeer, A. P. Chikkatur, S. Gupta, S. Inouye, T. Rosenband, and W. Ketterle, Phys. Rev. Lett. 87, 130402 (2001). 10.1103/PhysRevLett.87. 130402
18. T. Kinoshita, T. Wenger, and D. S. Weiss, Science 305, 1125 (2004). 10.1126/science.1100700
19. B. Paredes, A. Widera, V. Murg, O. Mandel, S. Fölling, I. Cirac, G. V. Shlyapnikov, T. W. Hänsch, and I. Bloch, Nature (London) 429, 277 (2004). 10.1038/nature02530
20. J.-B. Trebbia, J. Esteve, C. I. Westbrook, and I. Bouchoule, Phys. Rev. Lett. 97, 250403 (2006). 10.1103/PhysRevLett.97.250403
21. A. H. van Amerongen, J. J. P. van Es, P. Wicke, K. V. Kheruntsyan, and N. J. van Druten, Phys. Rev. Lett. 100, 090402 (2008). 10.1103/PhysRevLett.100. 090402
22. I. Bouchoule, N. J. Van Druten, and C. I. Westbrook, e-print arXiv:0901.3303.
23. T. Kinoshita, T. Wenger, and D. S. Weiss, Phys. Rev. Lett. 95, 190406 (2005). 10.1103/PhysRevLett.95.190406
24. B. Laburthe Tolra, K. M. O'Hara, J. H. Huckans, W. D. Phillips, S. L. Rolston, and J. V. Porto, Phys. Rev. Lett. 92, 190401 (2004). 10.1103/PhysRevLett.92.190401
25. Yu. Kagan, B. V. Svistunov, and G. V. Shlyapnikov, JETP Lett. 42, 209 (1985).
26. Y. Miroshnychenko, W. Alt, I. Dotsenko, L. Förster, D. Meschede, D. Schrader, M. Khudaverdyan, and A. Rauschenbeutel, Nature (London) 442, 151 (2006). 10.1038/442151a
27. Y. Colombe, T. Steinmetz, G. Dubois, F. Linke, D. Hunger, and J. Reichel, Nature (London) 450, 272 (2007). 10.1038/nature06331
28. K. D. Nelson, X. Li, and D. S. Weiss, Nat. Phys. 3, 556 (2007). 10.1038/nphys645
29. T. Gericke, P. Würtz, D. Reitz, T. Langen, and H. Ott, Nat. Phys. 4, 949 (2008). 10.1038/nphys1102
30. M. Wilzbach, D. Heine, S. Groth, X. Liu, T. Raub, B. Hessmo, and J. Schmiedmayer, Opt. Lett. 34, 259 (2009). 10.1364/OL.34.000259
31. D. Heine, M. Wilzbach, T. Raub, B. Hessmo, and J. Schmiedmayer, Phys. Rev. A 79, 021804 (R) (2009). 10.1103/PhysRevA.79.021804
32. D. S. Petrov, G. V. Shlyapnikov, and J. T. M. Walraven, Phys. Rev. Lett. 87, 050404 (2001). 10.1103/PhysRevLett.87.050404
33. S. Dettmer, D. Hellweg, P. Ryytty, J. J. Arlt, W. Ertmer, K. Sengstock, D. S. Petrov, G. V. Shlyapnikov, H. Kreutzmann, L. Santos, and M. Lewenstein, Phys. Rev. Lett. 87, 160406 (2001). 10.1103/PhysRevLett.87.160406
34. D. Hellweg, S. Dettmer, P. Ryytty, J. J. Arlt, W. Ertmer, K. Sengstock, D. S. Petrov, G. V. Shlyapnikov, H. Kreutzmann, L. Santos, and M. Lewenstein, Appl. Phys. B: Lasers Opt. 73, 781 (2001). 10.1007/s003400100747
35. H. Kreutzmann, A. Sanpera, L. Santos, M. Lewenstein, D. Hellweg, L. Cacciapuoti, M. Kottke, T. Schulte, K. Sengstock, J. J. Arlt, and W. Ertmer, Appl. Phys. B: Lasers Opt. 76, 165 (2003). 10.1007/s00340-002-1074-6
36. J. Viana Gomes, A. Perrin, M. Schellekens, D. Boiron, C. I. Westbrook, and M. Belsley, Phys. Rev. A 74, 053607 (2006). 10.1103/PhysRevA.74.053607
37. L. S. Cederbaum, A. I. Streltsov, Y. B. Band, and O. E. Alon, Phys. Rev. Lett. 98, 110405 (2007). 10.1103/PhysRevLett.98.110405
38. O. E. Alon, A. I. Streltsov, and L. S. Cederbaum, Phys. Lett. A 373, 301 (2009). 10.1016/j.physleta.2008.11.037
39. P. Öhberg and L. Santos, Phys. Rev. Lett. 89, 240402 (2002) 10.1103/PhysRevLett.89.240402
40. P. Pedri, L. Santos, P. Öhberg, and S. Stringari, Phys. Rev. A 68, 043601 (2003). 10.1103/PhysRevA.68.043601
41. M. Rigol and A. Muramatsu, Phys. Rev. Lett. 93, 230404 (2004) 10.1103/PhysRevLett.93.230404
42. M. Rigol and A. Muramatsu, Phys. Rev. Lett. 94, 240403 (2005) 10.1103/PhysRevLett.94.240403
43. M. Rigol and A. Muramatsu, Mod. Phys. Lett. B 19, 861 (2005). 10.1142/S0217984905008876
44. A. Minguzzi and D. M. Gangardt, Phys. Rev. Lett. 94, 240404 (2005). 10.1103/PhysRevLett.94.240404
45. M. Rigol, V. Dunjko, V. Yurovsky, and M. Olshanii, Phys. Rev. Lett. 98, 050405 (2007). 10.1103/PhysRevLett.98.050405
46. D. M. Gangardt and M. Pustilnik, Phys. Rev. A 77, 041604 (R) (2008). 10.1103/PhysRevA.77.041604
47. H. Buljan, R. Pezer, and T. Gasenzer, Phys. Rev. Lett. 100, 080406 (2008) 10.1103/PhysRevLett.100.080406
48. H. Buljan, R. Pezer, and T. Gasenzer, Phys. Rev. Lett. 102, 049903 (E) (2009). 10.1103/PhysRevLett.102.049903
49. D. Jukić, R. Pezer, T. Gasenzer, and H. Buljan, Phys. Rev. A 78, 053602 (2008). 10.1103/PhysRevA.78.053602
50. D. Jukić, B. Klajn, and H. Buljan, Phys. Rev. A 79, 033612 (2009). 10.1103/PhysRevA.79.033612
51. A. del Campo and J. G. Muga, Europhys. Lett. 74, 965 (2006). 10.1209/epl/i2006-10061-5
52. M. Girardeau, J. Math. Phys. 1, 516 (1960). 10.1063/1.1703687
53. N. Bogoliubov, J. Phys. (Moscow) 11, 23 (1947).
54. L. Pitaevskii and S. Stringari, Bose-Einstein Condensation (Oxford University Press, New York, 2003).
55. C. J. Pethick and H. Smith, Bose-Einstein Condensation in Dilute Gases (Cambridge University Press, Cambridge, England, 2001).
56. 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
57. Ph. 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
58. J. L. Roberts, N. R. Claussen, J. P. Burke, Jr., C. H. Greene, E. A. Cornell, and C. E. Wieman, Phys. Rev. Lett. 81, 5109 (1998). 10.1103/PhysRevLett.81.5109
59. N. D. Mermin and H. Wagner, Phys. Rev. Lett. 17, 1133 (1966) 10.1103/PhysRevLett.17.1133
60. P. C. Hohenberg, Phys. Rev. 158, 383 (1967). 10.1103/PhysRev.158.383
61. S. Coleman, Commun. Math. Phys. 31, 259 (1973). 10.1007/BF01646487
62. P. D. Naselsky, D. I. Novikov, and I. D. Novikov, The Physics of the Cosmic Microwave Background (Cambridge University Press, Cambridge, England, 2006).
63. J. C. Mather, Rev. Mod. Phys. 79, 1331 (2007) 10.1103/RevModPhys.79.1331
64. G. F. Smoot, Rev. Mod. Phys. 79, 1349 (2007). 10.1103/RevModPhys.79.1349
65. T. Padmanabhan, Structure Formation in the Universe (Cambridge University Press, Cambridge, England, 1993).
66. L. J. Garay, J. R. Anglin, J. I. Cirac, and P. Zoller, Phys. Rev. Lett. 85, 4643 (2000). 10.1103/PhysRevLett.85.4643
67. G. E. Volovik, Universe in a Helium Droplet (Oxford University Press, Oxford, 2003).
68. P. O. Fedichev and U. R. Fischer, Phys. Rev. A 69, 033602 (2004). 10.1103/PhysRevA.69.033602
69. C. Barceló, S. Liberati, and M. Visser, Living Rev. Relativ. 8, 12 (2005).
70. M. Uhlmann, Y. Xu, and R. Schützhold, New J. Phys. 7, 248 (2005). 10.1088/1367-2630/7/1/248
71. R. Balbinot, A. Fabbri, S. Fagnocchi, A. Recati, and I. Carusotto, Phys. Rev. A 78, 021603 (R) (2008). 10.1103/PhysRevA.78.021603
72. I. Carusotto, S. Fagnocchi, A. Recati, R. Balbinot, and A. Fabbri, New J. Phys. 10, 103001 (2008). 10.1088/1367-2630/10/10/103001
73. M. Uhlmann, Phys. Rev. A 79, 033601 (2009). 10.1103/PhysRevA.79.033601
74. 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
75. L. Cacciapuoti, D. Hellweg, M. Kottke, T. Schulte, W. Ertmer, J. J. Arlt, K. Sengstock, L. Santos, and M. Lewenstein, Phys. Rev. A 68, 053612 (2003). 10.1103/PhysRevA.68.053612
76. F. Gerbier, J. H. Thywissen, S. Richard, M. Hugbart, P. Bouyer, and A. Aspect, Phys. Rev. A 67, 051602 (R) (2003). 10.1103/PhysRevA.67.051602
77. S. Richard, F. Gerbier, J. H. Thywissen, M. Hugbart, P. Bouyer, and A. Aspect, Phys. Rev. Lett. 91, 010405 (2003). 10.1103/PhysRevLett.91.010405
78. M. Hugbart, J. Retter, F. Gerbier, A. Varon, S. Richard, J. Thywissen, D. Clement, P. Bouyer, and A. Aspect, Eur. Phys. J. D 35, 155 (2005). 10.1140/epjd/e2005-00056-2
79. P. Cladé, C. Ryu, A. Ramanathan, K. Helmerson, and W. D. Phillips, Phys. Rev. Lett. 102, 170401 (2009). 10.1103/PhysRevLett.102.170401
80. E. Altman, E. Demler, and M. D. Lukin, Phys. Rev. A 70, 013603 (2004). 10.1103/PhysRevA.70.013603
81. 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
82. A. Polkovnikov, E. Altman, and E. Demler, Proc. Natl. Acad. Sci. U.S.A. 103, 6125 (2006). 10.1073/pnas.0510276103
83. V. Gritsev, E. Altman, E. Demler, and A. Polkovnikov, Nat. Phys. 2, 705 (2006). 10.1038/nphys410
84. A. Imambekov, V. Gritsev, and E. Demler, Phys. Rev. A 77, 063606 (2008). 10.1103/PhysRevA.77.063606
85. A. Imambekov, V. Gritsev, and E. Demler, in Ultracold Fermi Gases, Proceedings of the International School of Physics "Enrico Fermi", 2006 (IOS, Amsterdam, The Netherlands, 2007).
86. V. N. Popov, Theor. Math. Phys. 11, 565 (1972) 10.1007/BF01028373
87. Functional Integrals in Quantum Field Theory and Statistical Physics (Reidel, Dordrecht, 1983).
88. D. S. Petrov, G. V. Shlyapnikov, and J. T. M. Walraven, Phys. Rev. Lett. 85, 3745 (2000). 10.1103/PhysRevLett.85.3745
89. C. Mora and Y. Castin, Phys. Rev. A 67, 053615 (2003) 10.1103/PhysRevA.67.053615
90. Y. Castin, J. Phys. IV 116, 89 (2004). 10.1051/jp4:2004116004
91. C. Gils, L. Pollet, A. Vernier, F. Hebert, G. G. Batrouni, and M. Troyer, Phys. Rev. A 75, 063631 (2007). 10.1103/PhysRevA.75.063631
92. C. N. Yang and C. P. Yang, J. Math. Phys. 10, 1115 (1969). 10.1063/1.1664947
93. K. B. Efetov and A. I. Larkin, Sov. Phys. JETP 42, 390 (1975).
94. F. D. M. Haldane, Phys. Rev. Lett. 47, 1840 (1981). 10.1103/PhysRevLett. 47.1840
95. M. A. Cazalilla, J. Phys. B 37, S1 (2004). 10.1088/0953-4075/37/7/051
96. J. O. Andersen, U. Al Khawaja, and H. T. C. Stoof, Phys. Rev. Lett. 88, 070407 (2002). 10.1103/PhysRevLett.88.070407
97. U. Al Khawaja, J. O. Andersen, N. P. Proukakis, and H. T. C. Stoof, Phys. Rev. A 66, 013615 (2002). 10.1103/PhysRevA.66.013615
98. D. S. Petrov, D. M. Gangardt, and G. V. Shlyapnikov, J. Phys. IV 116, 5 (2004). 10.1051/jp4:2004116001
99. A. Lenard, J. Math. Phys. 5, 930 (1964) 10.1063/1.1704196
100. H. G. Vaidya and C. A. Tracy, Phys. Rev. Lett. 42, 3 (1979) 10.1103/PhysRevLett.42.3
101. H. G. Vaidya and C. A. Tracy, Phys. Rev. Lett. 43, 1540 (1979). 10.1103/PhysRevLett.43.1540
102. E. H. Lieb and W. Liniger, Phys. Rev. 130, 1605 (1963) 10.1103/PhysRev.130.1605
103. E. H. Lieb, Phys. Rev. 130, 1616 (1963). 10.1103/PhysRev.130.1616
104. D. M. Gangardt and G. V. Shlyapnikov, Phys. Rev. Lett. 90, 010401 (2003) 10.1103/PhysRevLett.90.010401
105. D. M. Gangardt and G. V. Shlyapnikov, New J. Phys. 5, 79 (2003). 10.1088/1367-2630/5/1/379
106. K. V. Kheruntsyan, D. M. Gangardt, P. D. Drummond, and G. V. Shlyapnikov, Phys. Rev. Lett. 91, 040403 (2003) 10.1103/PhysRevLett.91.040403
107. K. V. Kheruntsyan, D. M. Gangardt, P. D. Drummond, and G. V. Shlyapnikov, Phys. Rev. A 71, 053615 (2005). 10.1103/PhysRevA.71.053615
108. V. V. Cheianov, H. Smith, and M. B. Zvonarev, Phys. Rev. A 73, 051604 (R) (2006) 10.1103/PhysRevA.73.051604
109. V. V. Cheianov, H. Smith, and M. B. Zvonarev, J. Stat. Mech.: Theory Exp. (2006) P08015. 10.1088/1742-5468/2006/08/P08015
110. J.-S. Caux and P. Calabrese, Phys. Rev. A 74, 031605 (R) (2006). 10.1103/PhysRevA.74.031605
111. J.-S. Caux, P. Calabrese, and N. A. Slavnov, J. Stat. Mech.: Theory Exp. (2007) P01008. 10.1088/1742-5468/2007/01/P01008
112. A. Imambekov and L. I. Glazman, Phys. Rev. Lett. 100, 206805 (2008). 10.1103/PhysRevLett.100.206805
113. A. Yu. Cherny and J. Brand, J. Phys.: Conf. Ser. 129, 012051 (2008) 10.1088/1742-6596/129/1/012051
114. A. Y. Cherny and J. Brand, Phys. Rev. A 79, 043607 (2009). 10.1103/PhysRevA.79.043607
115. V. Gritsev, T. Rostunov, and E. Demler, e-print arXiv:0904.3221.
116. G. E. Astrakharchik and S. Giorgini, Phys. Rev. A 68, 031602 (R) (2003). 10.1103/PhysRevA.68.031602
117. P. D. Drummond, P. Deuar, and K. V. Kheruntsyan, Phys. Rev. Lett. 92, 040405 (2004). 10.1103/PhysRevLett.92.040405
118. M. Krämer, C. Tozzo, and F. Dalfovo, Phys. Rev. A 71, 061602 (R) (2005). 10.1103/PhysRevA.71.061602
119. I. Bouchoule, K. V. Kheruntsyan, and G. V. Shlyapnikov, Phys. Rev. A 75, 031606 (R) (2007). 10.1103/PhysRevA.75.031606
120. A. G. Sykes, D. M. Gangardt, M. J. Davis, K. Viering, M. G. Raizen, and K. V. Kheruntsyan, Phys. Rev. Lett. 100, 160406 (2008). 10.1103/PhysRevLett.100. 160406
121. P. Deuar, A. G. Sykes, D. M. Gangardt, M. J. Davis, P. D. Drummond, and K. V. Kheruntsyan, Phys. Rev. A 79, 043619 (2009). 10.1103/PhysRevA.79.043619
122. S. Burger, F. S. Cataliotti, C. Fort, P. Maddaloni, F. Minardi, and M. Inguscio, Europhys. Lett. 57, 1 (2002) 10.1209/epl/i2002-00532-1
123. D. Rychtarik, B. Engeser, H.-C. Nägerl, and R. Grimm, Phys. Rev. Lett. 92, 173003 (2004) 10.1103/PhysRevLett.92.173003
124. Z. Hadzibabic, S. Stock, B. Battelier, V. Bretin, and J. Dalibard, Phys. Rev. Lett. 93, 180403 (2004) 10.1103/PhysRevLett.93.180403
125. N. L. Smith, W. H. Heathcote, G. Hechenblaikner, E. Nugent, and C. J. Foot, J. Phys. B 38, 223 (2005). 10.1088/0953-4075/38/3/007
126. P. Krüger, Z. Hadzibabic, and J. Dalibard, Phys. Rev. Lett. 99, 040402 (2007). 10.1103/PhysRevLett.99.040402
127. Z. Hadzibabic, P. Krüger, M. Cheneau, S. P. Rath, and J. Dalibard, New J. Phys. 10, 045006 (2008). 10.1088/1367-2630/10/4/045006
128. V. Schweikhard, S. Tung, and E. A. Cornell, Phys. Rev. Lett. 99, 030401 (2007). 10.1103/PhysRevLett.99.030401
129. W. Kane and L. Kadanoff, Phys. Rev. 155, 80 (1967) 10.1103/PhysRev.155.80
130. W. Kane and L. Kadanoff, J. Math. Phys. 6, 1902 (1965). 10.1063/1.1704738
131. D. S. Petrov, M. Holzmann, and G. V. Shlyapnikov, Phys. Rev. Lett. 84, 2551 (2000). 10.1103/PhysRevLett.84.2551
132. A. Posazhennikova, Rev. Mod. Phys. 78, 1111 (2006). 10.1103/RevModPhys. 78.1111
133. M. Holzmann, G. Baym, J.-P. Blaizot, and F. Laloë, Proc. Natl. Acad. Sci. U.S.A. 104, 1476 (2007) 10.1073/pnas.0609957104
134. M. Holzmann and W. Krauth, Phys. Rev. Lett. 100, 190402 (2008) 10.1103/PhysRevLett.100.190402
135. M. Holzmann, M. Chevallier, and W. Krauth, EPL 82, 30001 (2008). 10.1209/0295-5075/82/30001
136. N. Prokof'ev, O. Ruebenacker, and B. Svistunov, Phys. Rev. Lett. 87, 270402 (2001). 10.1103/PhysRevLett.87.270402
137. N. Prokof'ev and B. Svistunov, Phys. Rev. A 66, 043608 (2002). 10.1103/PhysRevA.66.043608
138. J. P. Fernández and W. J. Mullin, J. Low Temp. Phys. 128, 233 (2002). 10.1023/A:1019632502660
139. C. Gies and D. A. W. Hutchinson, Phys. Rev. A 70, 043606 (2004). 10.1103/PhysRevA.70.043606
140. D. A. W. Hutchinson and P. B. Blakie, Int. J. Mod. Phys. B 20, 5224 (2006). 10.1142/S0217979206036302
141. R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals (McGraw-Hill, New York, 1965).
142. L. Mathey, E. Altman, and A. Vishwanath, Phys. Rev. Lett. 100, 240401 (2008) 10.1103/PhysRevLett.100.240401
143. L. Mathey, A. Vishwanath, and E. Altman, Phys. Rev. A 79, 013609 (2009). 10.1103/PhysRevA.79.013609
144. Note an extra factor 1 2 in front of the mass in Eq. 16, in contrast to Eq. 10.
145. One should note that, strictly speaking, | ρ (q) | 2 does not have to be positive since it is not an expectation value of a positive operator. According to our definition [Eq. 22] square of the operator dxexp (iqx) ψ † (x,t) ψ (x,t) differs from L | ρ (q) | 2 by a term which arises due to normal ordering present in the definition of g2 (x;t).
146. In the right-hand side of Eq. 22 the term -1 in the parentheses was introduced to compensate for the large distance behavior of g2 (x;t) and does not affect the spectrum at q-0.
147. A. Gogolin, A. Nersesyan, and A. Tsvelik, Bosonization and Strongly Correlated Systems (Cambridge University Press, Cambridge, England, 1998).
148. T. Giamarchi, Quantum Physics in One Dimension (Oxford University Press, New York, 2004).
149. H. F. Talbot, Philos. Mag. 9, 401 (1836).
150. M. S. Chapman, C. R. Ekstrom, T. D. Hammond, J. Schmiedmayer, B. E. Tannian, S. Wehinger, and D. E. Pritchard, Phys. Rev. A 51, R14 (1995). 10.1103/PhysRevA.51.R14
151. F. Gerbier, Europhys. Lett. 66, 771 (2004). 10.1209/epl/i2004-10035-7
152. Constant -1 in Eq. 52 is included for the convenience to make the integrand vanish at ry →, and its contribution disappears after integration over rx.
153. S. Manz (unpublished).
154. A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinski, Methods of Quantum Field Theory in Statistical Physics (Dover, New York, 1963).
155. E. Burovski, N. Prokof'ev, B. Svistunov, and M. Troyer, New J. Phys. 8, 153 (2006). 10.1088/1367-2630/8/8/153
156. M. B. Zvonarev, V. V. Cheianov, and T. Giamarchi, J. Stat. Mech.: Theory Exp. (2009) P07035. 10.1088/1742-5468/2009/07/P07035
157. V. I. Smirnov, A Course of Higher Mathematics (Pergamon, Oxford, 1964), Vol. IV, p. 24.
158. V. E. Korepin, N. M. Bogoliubov, and A. G. Izergin, Quantum Inverse Scattering Method and Correlation Functions (Cambridge University Press, Cambridge, England, 1993).