Theory of superfluidity and drag force in the one-dimensional Bose gas

Abstract

The one-dimensional Bose gas is an unusual superfluid. In contrast to higher spatial dimensions, the existence of non-classical rotational inertia is not directly linked to the dissipationless motion of infinitesimal impurities. Recently, experimental tests with ultracold atoms have begun and quantitative predictions for the drag force experienced by moving obstacles have become available. This topical review discusses the drag force obtained from linear response theory in relation to Landau’s criterion of superfluidity. Based upon improved analytical and numerical understanding of the dynamical structure factor, results for different obstacle potentials are obtained, including single impurities, optical lattices and random potentials generated from speckle patterns. The dynamical breakdown of superfluidity in random potentials is discussed in relation to Anderson localization and the predicted superfluid–insulator transition in these systems.

Keywords

Lieb–Liniger model / Tonks–Girardeau gas / Luttinger liquid / drag force / superfluidity / dynamical structure factor

References

[1] J. Leggett, Rev. Mod. Phys., 1999, 71(2): S318

[2] A. J. Leggett, Rev. Mod. Phys., 2001, 73(2): 307

[3] É. B. Sonin, Sov. Phys. Usp., 1982, 25: 409

[4] P. Nozières and D. Pines, The Theory of Quantum Liquids: Superfluid Bose Liquids, Redwood City: Addison-Wesley, 1990

[5] L. Pitaevskii and S. Stringari, Bose–Einstein Condensation, Oxford: Clarendon, 2003

[6] A. J. Leggett, Quantum Liquids: Bose Condensation and Cooper Pairing in Condensed-Matter Systems, Oxford: Oxford University Press, 2006

[7] N. N. Bogoliubov, Quasi-Expectation Values in Problems of Statistical Machanics, New York: Gordon and Breach, 1961

[8] P. C. Hohenberg, Phys. Rev., 1967, 158(2): 383

[9] G. B. Hess and W. M. Fairbank, Phys. Rev. Lett., 1967, 19: 216

[10] A. J. Leggett, Phys. Fenn., 1973, 8: 125

[11] A. G. Sykes, M. J. Davis, and D. C. Roberts, Phys. Rev. Lett., 2009, 103(8): 085302

[12] S. Gupta, K. W. Murch, K. L. Moore, T. P. Purdy, and D. M. Stamper-Kurn, Phys. Rev. Lett., 2005, 95(14): 143201

[13] C. Ryu, M. F. Andersen, P. Cladé, V. Natarajan, K. Helmerson, and W. D. Phillips, Phys. Rev. Lett., 2007, 99(26): 260401

[14] S. E. Olson, M. L. Terraciano, M. Bashkansky, and F. K. Fatemi, Phys. Rev. A, 2007, 76(6): 061404

[15] S. Palzer, C. Zipkes, C. Sias, and M. Köhl, Phys. Rev. Lett., 2009, 103(15): 150601

[16] J. Catani, G. Lamporesi, D. Naik, M. Gring, M. Inguscio, F. Minardi, A. Kantian, and T. Giamarchi, arXiv:1106.0828, 2011

[17] L. Fallani, L. De Sarlo, J. E. Lye, M. Modugno, R. Saers, C. Fort, and M. Inguscio, Phys. Rev. Lett., 2004, 93(14): 140406

[18] C. D. Fertig, K. M. O’Hara, J. H. Huckans, S. L. Rolston, W. D. Phillips, and J. V. Porto, Phys. Rev. Lett., 2005, 94(12): 120403

[19] J. Mun, P. Medley, G. K. Campbell, L. G. Marcassa, D. E. Pritchard, and W. Ketterle, Phys. Rev. Lett., 2007, 99(15): 150604

[20] J. Billy, V. Josse, Z. Zuo, A. Bernard, B. Hambrecht, P. Lugan, D. Clément, L. Sanchez-Palencia, P. Bouyer, and A. Aspect, Nature, 2008, 453(7197): 891

[21] G. Roati, C. D’Errico, L. Fallani, M. Fattori, C. Fort, M. Zaccanti, G. Modugno, M. Modugno, and M. Inguscio, Nature, 2008, 453(7197): 895

[22] B. Deissler, M. Zaccanti, G. Roati, C. D’Errico, M. Fattori, M. Modugno, G. Modugno, and M. Inguscio, Nat. Phys., 2010, 6(5): 354

[23] V. A. Kashurnikov, A. I. Podlivaev, N. V. Prokof’ev, and B. V. Svistunov, Phys. Rev. B, 1996, 53(19): 13091

[24] E. H. Lieb and W. Liniger, Phys. Rev., 1963, 130(4): 1605

[25] L. Tonks, Phys. Rev., 1936, 50(10): 955

[26] M. Girardeau, J. Math. Phys., 1960, 1(6): 516

[27] M. A. Cazalilla, R. Citro, T. Giamarchi, E. Orignac, and M. Rigol, arXiv:1101.5337, 2011, to appear in Rev. Mod. Phys.

[28] In this paper we use the linear momentum and coordinates and velocities. The angular momentum and angle and angular velocity can easily be written as L_z = pL/(2π), ϕ = 2πx/L, ω_z = 2πv/L, respectively.

[29] E. H. Lieb, Phys. Rev., 1963, 130(4): 1616

[30] F. D. M. Haldane, Phys. Rev. Lett., 1981, 47: 1840, note a misprint in Eq. (7) for the density-density correlator: the sign before the second therm should be minus.

[31] C. Raman, M. Köhl, R. Onofrio, D. S. Durfee, C. E. Kuklewicz, Z. Hadzibabic, and W. Ketterle, Phys. Rev. Lett., 1999, 83(13): 2502

[32] A. Y. Cherny, J. S. Caux, and J. Brand, Phys. Rev. A, 2009, 80(4): 043604

[33] M. Ueda and A. J. Leggett, Phys. Rev. Lett., 1999, 83(8): 1489

[34] M. A. Cazalilla, J. Phys. B, 2004, 37(7): S1

[35] H. P. Büchler, V. B. Geshkenbein, and G. Blatter, Phys. Rev. Lett., 2001, 87(10): 100403

[36] R. Citro, A. Minguzzi, and F. W. J. Hekking, Phys. Rev. B, 2009, 79(17): 172505

[37] T. Cheon and T. Shigehara, Phys. Rev. Lett., 1999, 82(12): 2536

[38] A. Y. Cherny, J. S. Caux, J. Brand, and J. Sib, Fed. Univ. Math. Phys., 2010, 3: 289

[39] G. E. Astrakharchik and L. P. Pitaevskii, Phys. Rev. A, 2004, 70: 013608, note misprints in Eq. (20) for the drag force in this paper.

[40] P. C. Hohenberg and P. M. Platzman, Phys. Rev., 1966, 152(1): 198

[41] J. Brand and A. Y. Cherny, Phys. Rev. A, 2005, 72(3): 033619

[42] A. Y. Cherny and J. Brand, Phys. Rev. A, 2006, 73(2): 023612

[43] N. N. Bogoliubov, J. Phys. USSR, 1947, 11: 23, reprinted in Ref. [49]

[44] N. N. Bogoliubov, Lectures on Quantum Statistics, Vol. 2, New York: Gordon and Breach, 1970

[45] A. Y. Cherny and J. Brand, Phys. Rev. A, 2009, 79(4): 043607

[46] D. L. Kovrizhin and L. A. Maksimov, Phys. Lett. A, 2001, 282(6): 421

[47] M. D. Girardeau and M. Olshanii, Phys. Rev. A, 2004, 70(2): 023608

[48] B. E. Granger and D. Blume, Phys. Rev. Lett., 2004, 92(13): 1733202

[49] D. Pines (Ed.), The Many-Body Problem, New York: W. A. Benjamin, 1990

[50] M. Gattobigio, J. Phys. B, 2006, 39(10): S191

[51] V. N. Popov, Theor. Math. Phys., 1972, 11: 565

[52] V. N. Popov, Functional Integrals in Quantum Field Theory and Statistical Physics, Dordrecht: Reidel, 1983

[53] V. E. Korepin, N. M. Bogoliubov, and A. G. Izergin, Quantum Inverse Scattering Method and Correlation Functions, Cambridge: Cambridge University Press, 1993, see Section XVIII.2

[54] A. D. Mironov and A. V. Zabrodin, Phys. Rev. Lett., 1991, 66(5): 534

[55] A. O. Gogolin, A. A. Nersesyan, and A. M. Tsvelik, Bozonization and Strongly Correlated Systems, Cambridge: Cambridge University Press, 1998

[56] G. E. Astrakharchik, Quantum Monte Carlo Study of Ultracold Gases, <DissertationTip/>, Universit‘a degli Studi di Trento, 2004, Section 1.7.4

[57] N. Kitanine, K. K. Kozlowski, J. M. Maillet, N. A. Slavnov, and V. Terras, J. Stat. Mech.: Th. Exp., 2009, 2009: P04003

[58] A. Shashi, L. I. Glazman, J. S. Caux, and A. Imambekov, arXiv:1010.2268, 2010

[59] A. Shashi, L. I. Glazman, J. S. Caux, and A. Imambekov, Phys. Rev. B, 2011, 84(4): 045408

[60] J. S. Caux and P. Calabrese, Phys. Rev. A, 2006, 74(3): 031605

[61] J. S. Caux, J. Math. Phys., 2009, 50(9): 095214

[62] M. Gaudin, La fonction d’onde de Bethe., Collection du Commissariat a l’Energie Atomique. Serie Scientifique. Paris etc.: Masson. XVI, p330, 1983

[63] V. E. Korepin, Commun. Math. Phys., 1982, 86(3): 391

[64] N. A. Slavnov, Teor. Mat. Fiz., 1989, 79: 232

[65] N. A. Slavnov, Teor. Mat. Fiz., 1990, 82: 273

[66] A. Y. Cherny and J. Brand, J. Phys.: Conf. Ser., 2008, 129: 012051

[67] A. Imambekov and L. I. Glazman, Phys. Rev. Lett., 2008, 100(20): 206805

[68] Note that the field theory predictions of [67] actually include a singularity also for ω>ω+(k), with a universal shoulder ratio. We neglect this here since it gives only a small correction to the results.

[69] We slightly change the notations: our ω±and±μ±correspond to ω_1,2 and μ_1,2 in Ref. [67], respectively. We also denote the density of particles n and the Fermi wavevector for quasiparticles q₀ instead of D and q used in Refs. [53, 67], respectively.

[70] V. N. Golovach, A. Minguzzi, and L. I. Glazman, Phys. Rev. A, 2009, 80(4): 043611

[71] Y. Kagan, N. V. Prokof’ev, and B. V. Svistunov, Phys. Rev. A, 2000, 61(4): 045601

[72] J. M. Ziman, Principles of the Theory of Solids, Cambridge: Cambridge University Press, 1972

[73] A. Trombettoni and A. Smerzi, Phys. Rev. Lett., 2001, 86(11): 2353

[74] E. Altman, A. Polkovnikov, E. Demler, B. I. Halperin, and M. D. Lukin, Phys. Rev. Lett., 2005, 95(2): 020402

[75] A. Polkovnikov, E. Altman, E. Demler, B. Halperin, and M. D. Lukin, Phys. Rev. A, 2005, 71(6): 063613

[76] J. Ruostekoski and L. Isella, Phys. Rev. Lett., 2005, 95(11): 110403

[77] A. R. Kolovsky, New J. Phys., 2006, 8(9): 197

[78] I. Danshita and C. W. Clark, Phys. Rev. Lett., 2009, 102(3): 030407

[79] P. W. Anderson, Phys. Rev., 1958, 109(5): 1492

[80] L. Sanchez-Palencia and M. Lewenstein, Nat. Phys., 2010, 6(2): 87

[81] G. Modugno, Rep. Prog. Phys., 2010, 73(10): 102401

[82] L. Sanchez-Palencia, Phys. Rev. A, 2006, 74(5): 053625

[83] L. Sanchez-Palencia, D. Clément, P. Lugan, P. Bouyer, G. V. Shlyapnikov, and A. Aspect, Phys. Rev. Lett., 2007, 98(21): 210401

[84] P. Lugan, D. Clément, P. Bouyer, A. Aspect, M. Lewenstein, and L. Sanchez-Palencia, Phys. Rev. Lett., 2007, 98(17): 170403

[85] P. Lugan, D. Clément, P. Bouyer, A. Aspect, and L. Sanchez-Palencia, Phys. Rev. Lett., 2007, 99(18): 180402

[86] T. Paul, P. Schlagheck, P. Leboeuf, and N. Pavloff, Phys. Rev. Lett., 2007, 98(21): 210602

[87] A. S. Pikovsky and D. L. Shepelyansky, Phys. Rev. Lett., 2008, 100(9): 094101

[88] G. Kopidakis, S. Komineas, S. Flach, and S. Aubry, Phys. Rev. Lett., 2008, 100(8): 084103

[89] G. M. Falco, T. Nattermann, and V. L. Pokrovsky, Phys. Rev. B, 2009, 80(10): 104515

[90] I. L. Aleiner, B. L. Altshuler, and G. V. Shlyapnikov, Nat. Phys., 2010, 6(11): 900

[91] J. Radić, V. Bačić, D. Jukić, M. Segev, and H. Buljan, Phys. Rev. A, 2010, 81(6): 063639

[92] J. W. Goodman, Statistical Properties of Laser Speckle Patterns, in: Laser Speckle and Related Phenomena, edited by J.-C. Dainty, Berlin: Springer-Verlag, 1975, pp 9–75

[93] D. Clément, A. F. Varón, J. A. Retter, L. Sanchez-Palencia, A. Aspect, and P. Bouyer, New J. Phys., 2006, 8(8): 165

[94] D. Pines and P. Nozières, The Theory of Quantum Liquids: Normal Fermi Liquids, New York: W. A. Benjamin, 1966, see Eq. (2.69)

[95] E. Timmermans and R. Côté, Phys. Rev. Lett., 1998, 80(16): 3419

RIGHTS & PERMISSIONS

2014 Higher Education Press and Springer-Verlag Berlin Heidelberg