Front. Phys., 2012, 7(1): 54–71 DOI 10.1007/s11467-011-0211-2 REVIEW ARTICLE Theory of superfluidity and drag force in the one-dimensional Bose gas Alexander Yu. Cherny1, Jean-S´ebastien Caux2, Joachim Brand3,† 1Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 141980, Dubna, Moscow region, Russia 2Institute for Theoretical Physics, Science Park 904, University of Amsterdam, 1090 GL Amsterdam, The Netherlands 3Centre for Theoretical Chemistry and Physics and New Zealand Institute for Advanced Study, Massey University, Private Bag 102904 North Shore, Auckland 0745, New Zealand E-mail: †[email protected] Received June 30, 2011; accepted August 18, 2011 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 quanti- tative 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 Lan- dau’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 PACS numb ers 03.75.Kk, 03.75.Hh, 05.30.Jp, 67.10.-d Contents gas beyond linear response theory 63 6 Consquence of drag force: Velocity damping 64 1 Introduction 54 7 Drag force in extended potentials 64 2 Landau criterion of superfluidity and 7.1 1D bosons in a moving shallow lattice 64 Hess–Fairbank effect 56 7.2 1D Bose gas in a moving random potential 66 3 Drag force as a generalization of Landau’s 8 Conclusions 67 criterion of superfluidity 58 Acknowledgements 68 4 The drag force in different regimes 58 Appendix A: General formula for the drag force 4.1 Large impurity velocities 58 from Fermi’s golden rule 68 4.2 The Tonks–Girardeau regime 59 Appendix B: The drag force in the RPA 4.3 The drag force in the Bogoliubov regime 59 approximation 68 4.4 The linear approximation near the B.1 The general expression 68 Tonks–Girardeau regime 60 B.2 The drag force for small velocities at zero 5 Theoretical approaches 60 temperature 69 5.1 Random phase approximation near the References and notes 69 Tonks–Girardeau regime 60 5.2 Luttinger liquid theory 61 5.3 The algebraic Bethe ansatz and ABACUS 61 1 Introduction 5.4 An effective approximation for the dynamic structure factor and drag force 62 Superfluidity in a neutral gas or liquid is not easily 5.5 Drag force from the phase slip transitions 63 defined but rather understood as a complex cluster of 5.6 Direct calculation for the Tonks–Girardeau phenomena. The associated properties may include fric- c Higher Education Press and Springer-Verlag Berlin Heidelberg 2012 Alexander Yu. Cherny, Jean-S´ebastien Caux, and Joachim Brand, Front. Phys., 2012, 7(1) 55 tionless flow through thin capillaries, suppression of the this is called the drag force. The question of metastabil- classical inertial moment, metastable currents, quantized ity then becomes equivalent to the drag force of a small circulation (vortices), Josephson effect (coherent tunnel- and heavy impurity that is dragged through the rest- ing), and so on (see, e.g., Refs. [1–6]). There is a close and ing gas. In the scope of this topical review, we consider deep analogy between superfluidity in a neutral system mainly infinitesimal impurities and calculate the lowest and superconductivity in a charged system [4, 6]. order terms in linear response of the interacting gas to The 3D weakly-interacting Bose gas has all the su- perturbation by the impurity. The authors of Ref. [11] perfluid properties mentioned above, which can be in- followed a different approach by calculating the effects ferred from the existence of an order parameter repre- of finite impurities on the flow of a weakly-interacting sented by the wave function of the Bose–Einstein conden- Bose–Einstein condensate. sate (BEC). By contrast, there is no BEC in a repulsive In spite of rapid progresses in experimental techniques 1D Bose gas even at zero temperature in the thermody- along this line [12–14], so far no conclusive experimen- namic limit, provided that interactions are independent tal data on the drag force or metastability of ring cur- of particle velocities [7, 8]. This can be easily proved us- rents in the 1D Bose gas is available, and thus, this is ing the Bogoliubov “1/q2” theorem [7]. This predicts a one of the outstanding fundamental questions remain- 1/q2 divergence at small momentum in the average oc- ing about the properties of ultra-cold Bose gases [2]. Not cupation number nq for nonzero temperature and 1/q long ago, an experiment along this line was carried out divergence for zero temperature. Nevertheless, the exis- [15], in which the propagation of spin impurity atoms tence of BEC is neither a sufficient nor necessary condi- through a strongly interacting one-dimensional Bose gas tion for superfluidity [1, 6], and a one-dimensional sys- was observed in a cigar-shaped geometry. The motion tem of bosons may be superfluid under some conditions. of the center-of-mass position of the wave packet is de- However, whether a system is superfluid or not depends scribed fairly well by the drag force, calculated with the very much on how superfluidity is defined, because one- dynamic structure factor of the Bose gas in the regime dimensional systems may exhibit only some but not all of infinite boson interactions. In the recent experiment of the superfluid phenomena. of Ref. [16], the dynamics of light impurities in a bath of Here we study superfluidity in an atomic gas of re- bosonic atoms was investigated and the decay of breath- pulsive spinless bosons in the 1D regime of very nar- ing mode oscillations was observed. In another line of row ring confinement. The investigations focus mainly experiments, atoms were subjected to a moving optical on the metastability of the circulating-current states in lattice potential and the momentum transfer was mea- various regimes. However, we also discuss another im- sured [17–19]. This implies that one can experimentally portant aspect of superfluidity relevant to a 1D system, obtain the drag force of a specific external potential act- which is the non-classical moment of inertia or Hess– ing on the gas. Experiments were also done with ultra- Fairbank effect [9] and the quantization of circulation. cold atoms in random and pseudo-random potentials. As we argue below, a perfect Hess–Fairbank effect and The direct observation of Anderson localization was re- quantization of circulation occur for the homogeneous ported in Refs. [20–22]. In particular, spreading of a 1D gas of repulsive spinless bosons in one dimension, while Bose gas in artificially created random potentials was ex- metastability of currents does not, in general. Note that perimentally investigated [20]. Below we show that the the Hess–Fairbank effect is much easier to investigate superfluid–insulator phase diagram of such a system can than metastability of current because of its “equilibrium” be obtained by calculating the drag force. nature [6, 10]. Indeed, it can be explained with the prop- The notion of drag force turns out to be theoretically erties of the low-lying energy excitation spectrum of the fundamental, because it generalizes Landau’s famous cri- system due to the ability of the system to relax to the terion of superfluidity. According to Landau, an obstacle ground state in the reference frame where the walls (i.e., in a gas, moving with velocity v, may cause transitions the trapping potentials) are at rest (see Section 2). The from the ground state of the gas to excited states lying metastability of currents is a much more complicated on the line ε = pv in the energy–momentum space. If all phenomenon to study, because at sufficiently large gas the spectrum is above this line, the motion cannot excite velocities, the system is obviously not in the ground state the system, and it is thus superfluid. However, it is also but in a metastable state. In order to study such an ef- possible that even when the line intersects the spectrum, fect, one needs to understand transitions between states, the transition probabilities to these states are strongly which presents a more intricate problem. suppressed due to boson interactions or to the specific Ideally, in order to study the decay of ring currents in kind of external perturbing potential. In this case, the a controlled manner, the gas should be kept in a ring or drag force gives us a quantitative measure of superfluid- a torus-like geometry with defined defects. The defects ity. may cause transitions to the states of lower energies, thus This paper is organized as follows. The basic model of leading to energy dissipation, related to a friction force; the 1D Bose gas considered in this paper is introduced 56 Alexander Yu. Cherny, Jean-S´ebastien Caux, and Joachim Brand, Front. Phys., 2012, 7(1) in Section 2. In the subsequent section, we study the Landau criterion — In the LL model the total mo- Hess–Fairbank effect and its relation to the Landau cri- mentum is a good quantum number [28], and periodic terion of superfluidity.