Theory of Superfluidity and Drag Force in the One-Dimensional Bose

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 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 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-Sébastien 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-Sébastien 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. In Section 3, we derive an expres- boundary conditions quantize it in units of 2π/L,where sion for the drag force through the dynamic structure L is the ring circumference. Classification of all the ex- factor within linear response theory and show that the citations can be done [29] in the same manner as for the notion of the drag force generalizes the Landau criterion Tonks–Girardeau gas of non-interacting fermions (γ → of superfluidity. The values of the drag force in various +∞). Thus, in order to create an elementary particle-like limiting regimes are calculated in Section 4. In the sub- excitation, one needs to add a quasimomentum beyond sequent section, the Luttinger liquid theory is exploited the occupied Fermi segment. By contrast, for a hole-like to describe the drag force at small impurity velocities. excitation, one needs to remove a quasimomentum lying An exact method for obtaining the drag force for a finite inside the Fermi segment. All the excitations can be con- number of bosons, exploiting the algebraic Bethe ansatz, structed from the above elementary excitations. Due to is briefly discussed in Section 5.3. In Section 5.4 we con- the conservation of total number of particles, the number sider a simple interpolation formula for the DSF, which of particle-like excitations coincides with that of hole-like works for arbitrary strength of the interparticle interac- excitations. The low-lying spectrum of N = nL bosons tions. In the subsequent section, we show that another as shown in Fig. 1 has local minima [30] at the supercur- approach to the one-dimensional Bose gas, the theory of rent states I (I =0, 1, 2,...)withmomentapI =2πnI phase slip transitions [23], reproduces the same power- and excitation energies law behaviour for the drag force as the Luttinger liquid 2 εI = pI /(2Nm)(2) theory does. In Section 5.6, the drag force in the limit of infinite interactions is obtained analytically beyond lin- These correspond to Galilean transformations of the ear response theory. The decay of ring currents is exam- ground state with velocities vI = pI /(Nm). The min- ined in Section 6. In Section 7.1, the zero-temperature ima do not depend on interactions and tend to zero in phase diagram is obtained for the superfluid–insulator the limit of large system size at constant density. The transition of the one-dimensional Bose gas in moving first supercurrent state is also called the umklapp exci- shallow lattices for arbitrary values of velocity, filling tation [24] by analogy with periodic lattices because it factor, and strength of boson interactions. The motion can be reached from the ground state by imparting the of the 1D Bose gas in random potentials and the re- momentum Kr to each particle, where Kr =2π/L is lated phase diagram at zero temperature are considered the reciprocal wave vector in the ring geometry. As ex- in Section 7.2. Finally, in the conclusion the results and plained above, this changes the total momentum while prospects are briefly discussed. preserving the internal state of the system. Suppose that the gas is put into rotation with linear velocity vI , and after that, is braked with an artificial 2 Landau criterion of superfluidity and Hess– macroscopic “obstacle”, e.g., created by a laser beam Fairbank effect [31]. In the reference frame where the gas is at rest, the obstacle moves with the velocity vI . In a superfluid Model — We model cold bosonic atoms in a waveguide- we expect to see no energy dissipation, and the drag like micro trap by a simple 1D gas of N bosons with point force is zero (the current is persistent). Otherwise one interactions of strength gB > 0 (i.e., the Lieb–Liniger can observe decay of the current. It follows from energy (LL) model [24]) conservation that the transitions from the ground state N 2 2 caused by the moving obstacle with velocity v lie on the − ∂ − H = 2 + gB δ(xi xj )(1)line ω = vk in the energy–momentum plane. According 2m ∂xi i=1 1ivc. However, Girardeau (TG) gas [25, 26]. In this limit, it can be in 3D we also have similar supercurrent states, which mapped onto an ideal Fermi gas since infinite contact apparently leads to the paradoxical absence of current repulsions emulate the Pauli principle. In the opposite metastability. The solution to this is that we need to limit of small γ, we recover the Bogoliubov model of consider not only the spectrum but also probabilities of weakly interacting bosons. For an overview of theoreti- excitations. Below we argue that in the 3D case, the cal approaches to the one-dimensional Bose gas, see the probability to excite supercurrents is vanishingly small, recent review [27]. while in the 1D case it depends on the strength of bosonic Alexander Yu. Cherny, Jean-Sébastien Caux, and Joachim Brand, Front. Phys., 2012, 7(1) 57

the wave functions. We emphasize that this procedure is equivalent to the method used in this paper, because the Galilean transformation implies a phase gradient of the wave functions observed in the frame of the moving walls, and the gradient leads to twisted boundary conditions. It is also consistent with the deﬁnition of the superﬂuid component in three dimensions through the Fourier-transformed autocorrelation function of the transverse current [6, 10].

Fig. 1 Schematic of the excitation spectrum of the 1D Bose gas in a perfectly isotropic ring. The supercurrent states I lie on the parabola 2k2/(2Nm)(dotted line). Excitations occur in the shaded area; the discrete structure of the spectrum is not shown for simplicity. The blue (dark) area represents single particle–hole excitations [29]. Motion of the impurity with respect to the gas causes transitions from the ground state to the states lying on the straight (red) line. Reproduced from Ref. [32], Copyright c 2009 The American Physical Society. interactions. Hess–Fairbank effect — When the walls of a toroidal container are set into rotation adiabatically with a small tangential velocity vD, a superfluid stays at rest while a normal fluid follows the container. This effect leads to a nonclassical rotational inertia of superfluid systems, which can be used to determine the superfluid fraction [6, 10]. For the 1D Bose gas, rotation of the annular trap amounts to shifting the excitation spectrum to ε − vDp as shown in Fig. 2. It is assumed that an unspecified relaxation mechanism allows the system to relax to the ground state in the frame where the trap is at rest.The low-lying LL excitation spectrum is a convex function of Fig. 2 (a) Low-lying excitation spectrum εmin(k)forN =10 momentum for 0 p p1 [29], and, hence, the momen- particles for 1D repulsive bosons. At γ 1, the low-lying spec- tum zero state remains the ground state for |vD| 0 [33, 34]. Ac- a moving trap. Shown are the low-energy excitations of the 1D cording to this equilibrium property which is completely Bose gas in the moving frame εmin(k) − vDk,calculatedfrom determined by the low-lying energy spectrum [1], the 1D the Bethe-ansatz equations [24] for different values of the coupling Bose gas has a 100% superfluid fraction and zero rota- strength (compare with εmin(k) of Fig. 1). Inset: The velocity of the gas at equilibrium changes abruptly at values of driving veloc- tional inertia at zero temperature. The same results were ity vD/vc =1, 3, 5,..., since the gas occupies the state with lowest obtained by using instanton techniques [35] and within energy. In particular, the system is at rest when the driving velocity is less than vc = vF/N (perfect Hess–Fairbank effect). Here, Luttinger liquid theory for a finite number of bosons [34, 2 2 kF ≡ πn, vF = kF/m,andvF ≡ kF/(2m). Reproduced from 36]. Ref. [38], Copyright c 2010 Siberian Federal University. It is the convexity of the low-lying excitation spectrum between the supercurrent states (see Fig. 2) Note that the Bose–Fermi mapping for the TG gas that allows us to obtain this result without nu- implies that periodic boundary conditions for the Bose merical calculations. The minima of energy can be gas only translate into periodic boundary conditions for reached only in the supercurrent states [see Fig. 2(b)], the Fermi gas for odd N but into antiperiodic ones for whose energy are known analytically with Eq. (2). In even N [37], in contrast to the true Fermi gas with pe- the usual way (see, e.g., Ref. [34]), the superfluid riodic conditions. One can see that for even N,thereis component can be defined and obtained numerically no Hess–Fairbank effect in the true Fermi gas (although through the second derivative of the phase ϕ in the the TG gas always shows the Hess–Fairbank effect). This twisted boundary conditions ψ(x1,...,xj +L,...,xN )= is due to the instability of fermions at the Fermi point, exp(iϕ)ψ(x1,...,xj ,...,xN ), j =1,...,N, imposed on which results from the degeneracy of the ground state 58 Alexander Yu. Cherny, Jean-Sébastien Caux, and Joachim Brand, Front. Phys., 2012, 7(1)

foranevennumberoffermions. impurity v q/mi in one dimension. It yields for +∞ | |2 the drag force Fv = −∞ dqq Vi(q) S(q, qv)/L.Us- ing the properties of the DSF S(q, ω)=S(−q, ω)= 3 Drag force as a generalization of Landau’s eβωS(q, −ω), which follow from its definition (4), we criterion of superfluidity obtain +∞ In order to study frictionless motion, let us consider an 2 Fv = dqq |Vi(q)| S(q, qv)[1 − exp(−βqv)]/L impurity of mass mi, moving with velocity v in a medium 0 of particles. For the major part of this topical review, (8) we will treat the interaction between the impurity and This is the most general form of the drag force within Bose gas perturbatively in linear response theory and linear response theory. The form of the impurity interac- thus reduce the problem to calculating properties of the tion potential can be of importance as, e.g., in the cases integrable 1D Bose gas, although in general the system of shallow lattices and random potentials discussed in will lose integrability when adding an impurity. Using Sections 7.1 and 7.2, respectively. When the impurity Fermi’s golden rule, one can easily show (see Appendix interaction is of short-range type, we can replace it with A) that the resulting friction leads to an energy loss per good accuracy by a contact interaction Vi(r)=giδ(x), unit time given by which leads to Vi(q)=gi and yields at zero temperature D d q 2 [39] E˙ = − |Vi(q)| n (2π)D−1 2 +∞ q2 q2 2gi nm 2 (q · v − )S(q, q · v − ) Fv ≡ fv = g dqqS(q, qv)/L (9) 2mi 2mi 2 i · (3) 0 N In the first equality, we introduce the dimensionless drag Here V (q) is the Fourier transform of the interaction po- i force f with the help of the “natural” unit 2g2nm/2. tential V (r) between the impurity and the particles, D is v i i Its physical nature will be discussed below in Sections the spatial dimension, n is the density of particles (num- 4.1 and 5.6. ber of particles per D-dimensional volume), and S(q, ω) The notion of drag force generalizes the Landau cri- is the dynamical structure factor (DSF) of the medium. terion of superfluidity. Indeed, the integral in Eq. (9) is It is given by the definition [5] taken along the line ω = qv in the ω–q plane. If the exci- −1 −βEm 2 S(q, ω)=Z e |m|δρˆq|n| δ(ω −En +Em) tation spectrum lies completely above the line then the n,m integral vanishes, as one can see from the DSF definition (4) (5). On the other hand, the integral can be infinitesimally small or vanish even if the spectrum lies below the line Z − with = m exp( βEm) being the partition func- but the excitation probabilities, given by the correspond- tion and β being the inverse temperature. Here δρˆq = ing matrix elements Vi(q)0|δρˆq|n, are suppressed. The −iq·rj − q j e NΔ( ) is the Fourier component of the op- drag force can vanish even if the system is not super- q q erator of the density fluctuations, Δ( )=1at =0and fluid in principle, but excitations are not accessed with a q Δ( ) = 0 otherwise. At zero temperature, the structure given potential. This happens when the matrix elements factor takes a simpler form 0|δρˆq|n are not small but the Fourier transform of the 2 impurity potential Vi(q) takes non-zero values only in a S(q, ω)= |0|δρˆq|n| δ(ω − En + E0)(5) n finite region of q-space. We consider such an interesting case in Sections 7.1 and 7.2 below. The DSF relates to the time-dependent density correla- Thus, determining the drag force within linear re- tor through the Fourier transformation sponse theory is reduced to the problem of calculating D dtd r ωt−q·r the dynamic structure factor. Below we consider various S(q, ω)=N ei( )δρˆ(r,t)δρˆ(0, 0)/n (2π)D approximations for the dynamic structure factor and the (6) associated drag force. where δρˆ(r,t) ≡ j δ(r − rj(t)) − n is the operator of the density fluctuations. The DSF obeys the f-sum rule 4 The drag force in different regimes [5] +∞ 4.1 Large impurity velocities dωωS(q, ω)=Nq2/(2m)(7) −∞ Let us consider the case of contact interactions with

The drag force is deﬁned by the formula E˙ = −Fv · v. the impurity, Vi(r)=giδ(x). At large impurity velocity In this paper, we will use the expression (3) for a heavy v πn/m, the main contribution to the integral in Eq. Alexander Yu. Cherny, Jean-S´ebastien Caux, and Joachim Brand, Front. Phys., 2012, 7(1) 59

(9) comes from the high momentum region of the DSF, where vF ≡ πn/m is the sound velocity in the TG which can be calculated analytically [40]. Indeed, at large regime. The result is represented in Fig. 3(a). The TG velocities, the momentum transfer from the impurity to gas is obviously not superfluid. the particles is big enough to neglect the interparticle interactions. Then one can use the DSF values of the ideal gas [5, 40] 2q2 2pq S(q, ω)= np(1 ± np q)δ ω − − + 2m m p 2q2 2pq npδ ω − − (10) p 2m m † Here np ≡apap is the average occupation numbers of particles, and the plus is taken for bosons and the minus for fermions. The second equality in Eq. (10) is due to the large momentum q mv πn,whichleads to npnp+q 0 for arbitrary values of p. Substituting Eq. (10) into Eq. (9) yields the value of the drag force 2 2 2gi mn/ , which can be used as the force unit. It is not difficult to see that this result is valid beyond linear response theory, given by Eq. (9). Indeed, at sufficiently large particle velocity, the initial particle momentum can be neglected during the scattering. Then in the reference frame where the impurity remains at rest, the relative particle momentum is q mv. Therefore, the reflection coefficient is determined by the squared absolute value of the scattering amplitude in the Born ap- Fig. 3 The dimensionless drag force (9) versus impurity velocity 2 2 4 2 proximation m gi /( q ). Each particle, being reflected, in various approximations, c is the speed of sound. (a) Tonks– transfers momentum 2q. The total number of scattered Girardeau and Bogoliubov regimes. (b) The random phase ap- particles per unit time is nv = nq/m. The product of proximations (RPA) at non-zero temperatures. The curve becomes smoother when the temperature grows. the last three quantities yields the value of momentum transfer per unit time, that is, the drag force. This quantity, independent of the wave vector, is nothing else but 4.3 The drag force in the Bogoliubov regime the limiting value of the drag force, obtained above. The opposite regime to the TG gas is the limit of weak 4.2 The Tonks–Girardeau regime interactions γ 1 (the Bogoliubov regime). The crucial point in Bogoliubov theory [43] is the developed Bose– Let us first investigate the drag force for the TG gas Einstein condensate. In spite of the absence of Bose– (γ → +∞), having the same structure factor as the ideal Einstein condensation in one dimension [8, 44], the up- Fermi gas. The DSF at zero temperature is well-known per dispersion curve ω+(k) is well described at small γ in the thermodynamic limit [41, 42] [29] by the Bogoliubov relation εF kF S(k, ω) = (11) 2 2 N 4k ωk = Tk +4TkεFγ/π (13) for ω−(k) ω ω (k), and zero otherwise. Here + 2 2 2 where Tk = k /(2m) is the free-particle energy spec- ω±(k)=|2kFk ± k |/(2m) are the limiting dispersions that bound quasiparticle–quasihole excitations (see Fig. trum. The Bogoliubov theory also yields the correct val- ≡ ≡ 2 2 ues of the ground state energy and chemical potential. 4). By definition, kF πn and εF kF/(2m)are the Fermi wave vector and energy of the TG gas, re- This paradox can be explained [45] by a strong singu- spectively. As follows from Eq. (11) for the DSF, the larity of the DSF near ω+ in the Bogoliubov regime. transition probability from the ground state is inversely As a result, it is localized almost completely within a proportional to the momentum transfer but does not de- small vicinity of the upper branch. This tendency can be seen even at γ = 10 (Fig. 4). Thus, the behavior of the pend on the excitation energies within the borders ω±(k). Simple integration in Eq. (9) then yields DSF simulates the δ-function spike, which appears due to the Bose–Einstein condensate. One can simply put v/vF, 0 v vF SBog(k, ω)=Cδ(ω − ωk) and determine the constant C fv = (12) 1,v vF from the f-sum rule (7) 60 Alexander Yu. Cherny, Jean-Sébastien Caux, and Joachim Brand, Front. Phys., 2012, 7(1)

Tk for ω−(k) ω ω+(k), and zero elsewhere. The symbol SBog(k, ω)=N δ(ω − ωk) (14) ωk O(x) denotes terms of order x or smaller. The limiting The drag force is then obtained analytically: it is a dispersions in the strong-coupling regime take the form step function 2 |2kFk ± k | 4 1 ω±(k)= 1 − + O (17) 0, 0 v c 2m γ γ2 fv = (15) 1,v c From the linear part of the dispersion (17) at small shown in Fig. 3(a). The sound√ velocity in the Bogoliubov momentum ω±(k) ck, we obtain the well-known re- regime is given by c = vF γ/π. At non-zero tempera- sult [29, 34] for the sound velocity at zero temperature 2 tures the result does not change. This result was ﬁrst c = vF(1 − 4/γ)+O(1/γ ). obtained in Ref. [39] by means of the GP equation. Note In the same manner as in Section 4.2, we derive from that the supersonic impurity motion in three dimensions Eqs. (9) and (16) was earlier studied by the same method in Ref. [46]. ⎧ ⎨⎪ ln(v/c) v 1+8 , 0 v c fv = γ c (18) ⎩⎪ 1,v c

As expected, the linear approximation (18) works badly for small values of impurity velocity. This is due to the anomalous behaviour of the DSF within the linear approximation in vicinity of the umklapp excitation point at ω =0,k =2kF. Indeed, Eq. (18) leads to unphysical negative values of the drag force at suﬃciently small impurity velocities. For this reason, we need a more careful examination of the drag force in this regime.

Fig. 4 Numerical values of the DSF (4) for the coupling parameter γ = 10 [60]. The dimensionless value of the rescaled 5 Theoretical approaches DSF S(k, ω)εF/N is shown in shades of gray between zero (white) and 1.0 (black). The upper and lower solid (blue) lines represent the dispersions ω+(k)andω−(k), respectively, limiting the single 5.1 Random phase approximation near the Tonks– “particle–hole” excitations in the Lieb–Liniger model at T =0. Girardeau regime The dispersions are obtained numerically by solving the system of integral equations [29]. The gray scale plot of the DSF demon- strates that the main contribution to the DSF comes from the The DSF in the RPA was calculated and described in single particle-hole excitations, lying inside the region ω−(k) details in Section 4.2 of Ref. [42]. The RPA is based on ω ω+(k) (see also Fig. 1). Only one point at k = kG,shownin the Hartree–Fock scheme, which is appropriate at suf- full (red) circle, contributes to the integral when the perturber is a shallow cosine potential with a reciprocal vector kG. Reproduced ficiently large interactions between bosons γ>8. The from Ref. [32], Copyright c 2009 The American Physical Society. equation for the drag force within the RPA is derived in Appendix B. 4.4 The linear approximation near the Tonks– The RPA approximation has some advantages. It al- Girardeau regime ways gives positive values of the drag force and is applicable at non-zero temperatures. The results are shown For finite γ, the model can also be mapped onto a Fermi in Fig. 3(b). The RPA scheme can be extended to non- gas [37] with local interactions, inversely proportional to homogeneous systems beyond the local density approxi- gB [41, 42, 47, 48]. Using the explicit form of the interac- mation [50]. However, the RPA scheme also works badly tions, one can develop the time-dependent Hartree–Fock in the vicinity of the umklapp excitation. As a conse- scheme [41, 42] in the strong-coupling regime with small quence, the formula for the drag force (B7), obtained parameter 1/γ. Approximations of the linear response within the RPA, does not reproduce the correct power- functions on this level are known as Random Phase ap- law behaviour at small velocities (see discussion in Sec- proximation (RPA) with exchange or generalized RPA tions. 5.2 and 5.4 below.). Another disadvantage of the [4, 49]. The scheme yields the correct expansion of the scheme is that the drag force as a function of velocity DSF up to the first order [41, 42] has an unphysical peak near v = c at zero temperatures, 2 2 which becomes quite pronounced for γ 15. This is an ε k 8 1 ω − ω− 1 S(k, ω) F = F 1+ + ln + O apparent artifact of RPA. In the next sections we con- 2 − 2 2 N 4k γ 2γ ω+ ω γ sider much better approximations that work for all values (16) of the interaction strength and impurity velocities. Alexander Yu. Cherny, Jean-Sébastien Caux, and Joachim Brand, Front. Phys., 2012, 7(1) 61

5.2 Luttinger liquid theory By comparing the first-order expansion (16) in the vicinity of the umklapp point with Eq. (20) and us- Luttinger liquid theory allows us to correctly describe the ing the expansion (21), one can easily obtain [45] the low-energy excitations of a 1D system of particles whose model-dependent coefficient at large but finite interac- interactions are independent of the velocities. It faith- tions when K − 1 1 fully handles the nonlinear effects of these interactions π A (K)= 1 − (1 + 4 ln 2) (K − 1) + O (K − 1)2 and yields values of the DSF at low energies for arbi- 1 4 trary coupling strength [30, 39] and thus can be used to (23) calculate the drag force at small values of the impurity velocity. Using the DSF (20) predicted by Luttinger theory and The behaviour of the DSF in the vicinity of the umk- equation (9) for the drag force, we arrive at the expres- lapp excitation is related to the long-range asymptotics sion for the drag force at small velocities v c [39] of the time-dependent density–density correlator. The √ 2 2 2K Γ (K) gi n v Fv = π A1(K) 2K (24) asymptotics for kFx 1andt x/c Γ (K +1/2) v c δρˆ(x, t)δρˆ(0, 0) K 1 1 In the TG regime, this formula yields the same values of − + n2 4π2n2 (x − ct)2 (x + ct)2 the drag force as Eq. (12). Equation (24) gives us the

cos(2kFx) universal exponent of the power-law behaviour of the +A1(K) K K (19) ∼ 2K−1 n2 (x2 − c2t2) drag force at small velocities: Fv v .Thesame result was obtained in Ref. [36]. While the non-universal can be justified by generalizing Haldane’s method [30] coefficient A1(K) is now known for arbitrary strength of or using perturbation theory [51, 52]or quantum inverse interactions [57–59], its actual expression is too unwieldy scattering method [53] or conformal field theory [54, 55]. to be considered here. In Section 5.4 we prefer to con- The first two terms are related to the behaviour of the sider another approach, which allows us to determine it DSF in vicinity of ω =0,q = 0, while the third term approximately. is related to the umklapp excitations. Substituting the third term of Eq. (19) into Eq. (6) yields the DSF in the 5.3 The algebraic Bethe ansatz and ABACUS vicinity of the “umklapp” point (k =2kF =2πn, ω =0) [39] The exact integrability of the Lieb–Liniger model now permits the direct numerical calculation of dynamical 2K 2 K−1 S(k, ω) nc ω ω−(k) = A (K) 1 − correlation functions such as the DSF [60] for systems 2 2 1 2 N ω mc ω with finite numbers of particles using the ABACUS al- (20) gorithm [61]. The strategy consists in using the Lehmann representation (5). The eigenstates themselves are con- for ω ω−(k), and zero otherwise. Within Luttinger- structed by the Bethe Ansatz. The state norms [62, 63] liquid theory, the dispersion is linear near the umklapp and matrix element of the density operator [64, 65] be- point: ω−(k) c|k−2πn|. By definition, K ≡ πn/(mc) ing exactly known, the sum over intermediate states can is the Luttinger parameter. For repulsive bosons, the be taken starting from the dominant few-particle states value of parameter K lies between 1 (TG gas) and +∞ and let run until a satisfactory saturation of sum rules (ideal Bose gas). The value of the Luttinger parameter is achieved. The results for the DSF take the form il- in the strong-coupling regime lustrated in Fig. 4. For generic values of γ,mostofthe K =1+4/γ + O(1/γ2) (21) signal is concentrated between the dispersions ω+(k)and ω−(k) of single particle/hole (Type 1/2 in Lieb’s nota- is derived with the expression for the sound velocity ob- tion) excitations. The signal is by construction identi- tained in Section 4.4. The coefficients in Eqs. (19) and cally zero below ω−(k) (since no eigenstates are found (20) are related by [56] there); above ω+(k), the signal is carried by multiple 2K particle–hole excitations and is very weak in view of the Γ 2(K) 2K A (K)= A (K) (22) small matrix elements of the density operator between 1 2π π 1 these states and the ground state. with Γ (K) being the gamma function. The coefficient We use data for N = 150 particles (γ = 5 and 20), A1(K) is non-universal, and is explicitly known for the N = 200 (γ =1)andN = 300 (γ =0.25). The f- Lieb–Liniger model [57, 58]. We will make use of its value sum rule saturations at k =2kF were 99.64% (γ = 20), in two limiting cases: A1(K)=π/4atK =1and 97.81% (γ =5),99.06% (γ =1)and99.08% (γ =0.25), 1−3K 2 A1(K) 4 exp(−2γcK)π/Γ (K)forK 1 [39, the saturation getting better at smaller momenta. The 56], where γc =0.5772 ... is the Euler constant. drag force is computed from the numerical DSF data in 62 Alexander Yu. Cherny, Jean-Sébastien Caux, and Joachim Brand, Front. Phys., 2012, 7(1) the following way. Since the DSF in finite size is given by discrete but densely distributed δ-function peaks, we consider the integral ∞ qv g2 F I (v) ≡ i dq dωS(q, ω) L 0 0 2 ∞ gi | | | |2 − = dq 0 δρˆq n Θ( qv En +E0) L 0 n (25) whose derivative with respect to v simply gives the drag force (9). Here Θ is the Heaviside step function. The integral in ω conveniently gets rid of all energy δ-functions in the expression for the DSF, and this integrated quantity is readily computed without need for smoothing using the raw ABACUS data for the DSF. The derivative with respect to v can then be taken numerically by fitting an interpolating polynomial to the data points for F I (v)in the vicinity of the velocity for which the drag force needs to be calculated. The resulting data for the drag force are illustrated in Fig. 5(a). The advantage of this method is that the reliability of the results is more or less independent of the value of the interaction parameter, in the sense that states and matrix elements can be individually constructed irrespective Fig. 5 The dimensionless drag force versus the velocity (relative to the sound velocity) of the impurity at various values of the cou- of what γ is. The remaining issues are the distribution of pling parameter γ. (a) The solid (blue) lines represent the force correlation weight among the excitations, and how this obtained with Eqs. (9) and (26), open circles are the numerical affects the speed of convergence. The limit of very small data obtained using ABACUS [60]. (b) The values of the dimen- γ is the easiest to treat, since only very few states in sional drag force obtained from the interpolation formula (26) [solid (blue) lines] are compared with that of the analytical formula (29) the vicinity of the Type 1 mode are of importance. The for small velocities [dashed (red) lines]. DSF then becomes almost a δ-function in the Type 1 dispersion relation, and the drag force tends to the step at the borders of the spectrum √ω+(k)andω−(k). We function as expected in this limit. The opposite case of also put by definition α ≡ 1+1/ K. The most general infinite γ is also straightforward since then only single way of obtaining ω±(k), μ±(k), and K is to solve numer- particle–hole excitations have non-negligible matrix ele- ically the corresponding integral equations of Refs. [29] ments, the drag force becomes a constant between Type and [67], respectively. 1 and 2 dispersions, and the drag force acquires a con- It follows from energy and momentum conservation stant slope. For interaction values between these two ex- that S(k, ω) is exactly equal to zero below ω−(k)for tremes, many eigenstates must be summed over for good 0 k 2πn. In the other regions of ω>ω+ and convergence, and the DSF takes on a nontrivial lineshape ω<ω− (for k>2πn), possible contributions can arise making the drag force take the positive-curvature shapes due to coupling to multi-particle excitations [29]. How- in Fig. 5(a). ever, these contributions are known to vanish in the Tonks–Girardeau (γ →∞) and Bogoliubov (γ → 0) lim- 5.4 An effective approximation for the dynamic its and are found to be very small numerically for finite structure factor and drag force interactions [60]. The exponents μ± are non-negative [69]. As a conse- In Refs. [45, 66], an interpolating expression was sug- quence, the DSF diverges at the upper branch ω+.Atthe gested for the DSF: lower branch ω−, the DSF shows a continuous transition

α α μ− to zero for any finite value of γ except for the specific (ω − ω−) S(k, ω)=C α α μ (26) point γ =+∞ (or K = 1) of the Tonks-Girardeau gas, (ω − ω ) + + where the DSF remains finite but has a discontinuous for ω−(k) ω ω+(k), and S(k, ω)=0otherwise. transition to zero at both boundaries ω− and ω+. Here, μ+(k)andμ−(k) are the exact exponents [67, 68] The normalization constant C depends on momentum but not on frequency and is determined from the f-sum ∓μ±(k) S(k, ω) ∼ ω − ω±(k) (27) rule (7). The expression (26) is applicable for all ranges Alexander Yu. Cherny, Jean-Sébastien Caux, and Joachim Brand, Front. Phys., 2012, 7(1) 63 of the parameters k, ω,andγ with increasing accuracy phase slip transitions, changing the phase winding num- at large γ [45]. ber. One can write down explicitly the low-energy Hamil- The parameter α is needed to reconcile√ √ the limiting tonian [23] that consists of the supercurrent states and value of the exponent μ−(2πn)=2 K( K − 1) in the phonon excitations, interacting due to a static impurity. vicinity of the umklapp point [67] and the Luttinger the- The interaction Vi(x) between the impurity and bosons ory predictions, given by Eq. (20). Now one can see from can be rewritten in terms of the supercurrent states and (26) that the phonons. One can then directly derive the transi- K− tion probability per unit time between the supercurrent ω2( 1),k=2πn S(k, ω) ∼ (28) states I and I from Fermi’s golden rule [71]. The result μ−(k) (ω − ω−) ,k=2 πn reads at zero temperature [see Eq. (8) of Ref. [71]] 2 3/2 2α Thus, the suggested formula (26) is consistent with both |ggin| 4π α ξII WII = (30) 1 II the Luttinger liquid behavior at the umklapp point and Γ [1 + α]Γ [ 2 + α] ξ γ0 Imambekov’s and Glazman’s power-law behavior in its 2 where α ≡ (I − I ) K and ξII ≡ εI − εI .Thetran- vicinity, as it should be. A more detailed discussion can sition probability (30) is determined up to an overall be found in Ref. [45]. Similar approximations for the DSF factor, because the dimensionless parameters g and γ, of 1D Bose gas, confined in a harmonic trap, are consid- which are of the order of 1, cannot be defined exactly ered in Ref. [70]. from the long-range effective Hamiltonian. The energy The drag force can now be calculated by means of Eqs. 0 is a characteristic high-energy cutoff for the phonon (9) and (26) for arbitrary strength of interactions and ar- spectrum. bitrary velocities. The results are shown in Fig. 5(a). For the Ith supercurrent decay at zero temperature For the important question whether persistent cur- due to the static impurity, the dominant contribution rents may exist at all, the small velocity regime is most comes from the transition to the (I − 1)th state, and one relevant, which is dominated by transitions near the can write for the energy loss per unit time first supercurrent state (umklapp point at ω =0and | |2 3/2 2K k =2k ). The drag force in this regime has a power-law ggin 4π K ξI,I−1 F E˙ = WI,I− ξI,I− = ∼ 2K−1 1 1 1 dependence on the velocity Fv v for v c,as Γ (1+K)Γ ( 2 +K) γ0 first found by Astrakharchik and Pitaevskii [39]. From (31) Eqs. (9) and (26) we obtain For large winding number I,wehavetherelation K− K F πε v 2 1 4ε 2 ξI,I−1 2πn vI , resulted from Eq. (2). By putting the f ≡ v F 2K F v 2 3 energy cutoff 0 to be equal to 2πn c and remembering gi kF vF ω+(2kF) the definition of the drag force |E˙ | = Fvv,weobtain Γ 1+ 2K − μ (2k ) × α + F from Eq. (31) 2K Γ α Γ 1 − μ+(2kF) K 4π3/2 1 v 2 1 | |2 Γ 1+μ−(2k ) Γ 1+ Fv = ggin (32) × F α Γ (K) Γ ( 1 + K) v γc 1 (29) 2 Γ 1+μ−(2k )+ F α 2K−1 Thus, we come to the dependence Fv ∼ v at small where√ √Γ (x) is Euler’s Gamma-function, and μ−(2kF)= velocities, in agreement with the previous results (24) 2 K( K − 1) [67]. This formula is valid for arbitrary and (29). coupling constant and works even in the Bogoliubov regime at γ 1. In practice, Eq. (29) works well up 5.6 Direct calculation for the Tonks–Girardeau gas be- to v 0.1c [see Fig. 5(b)]. The strong suppression of yond linear response theory the drag force in the Bogoliubov regime appears because of the large exponent value 2K − 1 1 and the large In this section we consider the TG gas, or the ideal gas of argument 2K/α 1 of the gamma function in the de- spin-polarized fermions. In order to find the drag force, nominator. Certainly, drag force values of order 10−40 lie we choose a frame of reference, in which the massive beyond realizable experiments and as a matter of fact, impurity is at rest. Moving particles are then scattered tell us about superfluidity in this regime. by the impurity potential Vi(x)=giδ(x). For an ideal Fermi gas the scattering process for each particle is in- 5.5 Drag force from the phase slip transitions dependent of the scattering of the other particles. The reflection coefficient for a particle with wave vector k is The quantum number I, describing the supercurrent easily determined from the one-body Schrödinger equa- 2 2 2 2 4 2 with energy (2), is nothing else but the phase√ wind- tion: r(k)=m gi /(m gi + k ). If dNk is the number ing number of the original Bose field Ψ = neiΦ [30]. of particles with wave vectors lying between k and k+dk, The transitions between the supercurrent states are the then these particles transfer momentum to the impurity 64 Alexander Yu. Cherny, Jean-Sébastien Caux, and Joachim Brand, Front. Phys., 2012, 7(1)

2 2 2 per unit time dF =2 nr(k)k dNk/m. Since in the cho- 2 dk 2 Fv = n(k − mv/)sign(k)r(k)k (33) sen frame of reference, the momentum distribution is the m 2π Fermi distribution shifted by mv,wecanwritedownthe where n(k)=1for−πn k πn, and zero otherwise. explicit expression of the drag force The integral is readily taken ⎧ − ⎪ γi π(1 ξ) − π(1 + ξ) 2 ⎨ ξ + arctan arctan , 0 ξ 1 2mngi 2π γi γi Fv = (34) 2 ⎪ γ π(ξ − 1) π(1 + ξ) ⎩ 1+ i arctan − arctan ,ξ 1 2π γi γi

2 Here we denote γi ≡ gim/( n), and ξ ≡ v/vF.There- excitations above a lower supercurrent state while still sults are shown in Fig. 6. As discussed in Section 4.1, the conserving the total energy. The elementary excitations limiting value of the drag force at large velocities coin- are quasiparticle–quasihole excitations in the Bethe– cides with that obtained within linear response theory. Ansatz wave function [29]. Both in energy and character At small impurity coupling γi 1, Eq. (34) reproduces most of these excitation lie between the phonon-like ω+ the linear response formula (12) [see Fig. 3(a)]. branch and the ω− branch, which is related to dark soli- tons (see Fig. 4). Assuming that these excitations have little eﬀect on successive transitions, we estimate [32] the decay of the center-of-mass velocity v by the classical equation Nmv˙ = −Fv(v), where Fv is given by Eqs. (9) and (26). This equation was integrated numerically and the result is shown in Fig. 8. At the initial supersonic velocity, where the drag force is saturated [see Fig. 5(a)] the supercurrent experiences constant deceleration. For v c the drag force decreases and consequently the deceleration slows down. For the TG gas we ﬁnd an analytical solution for exponential decay v(t)=v0 exp(−t/τ) for v0 vF. In the weakly-interacting regime, the decay Fig. 6 The diagram shows the dimensionless drag force (9) for may be slow compared to experimental time scales. the TG gas at various values of the impurity coupling parameter 2 γi ≡ gim/( n). The results are obtained beyond the linear response approximation. Note that the absolute value of the drag 2 force Fv is proportional to γi .

6 Consquence of drag force: Velocity damping

In the presence of an obstacle as, e.g., shown in Fig. 7, a ring current can decay due to successive transitions to supercurrent states with smaller momentum. Starting in one of the local minima of the excitation spectrum as seen in Fig. 1, the kinetic energy of the center-of- Fig. 8 Decay of the ring current velocity of 1D bosons from the initial velocity of 1.1vF at t = 0. The solid (blue) and dashed (red) mass translation will be transformed into elementary lines represent the results obtained with the approximate formula 3 2 and ABACUS, respectively. The time scale is τ = Nπ /(2mgi ). Reproduced from Ref. [32], Copyright c 2009 The American Phys- ical Society.

7 Drag force in extended potentials

7.1 1D bosons in a moving shallow lattice Fig. 7 An experimental scheme to observe current decays. An artificial “impurity” is created by optical methods. It can be switched Equation (8) can be verified experimentally for differ- on adiabatically or abruptly. ent types of obstacles: for Vi(x)=giδ(x)allthepoints Alexander Yu. Cherny, Jean-Sébastien Caux, and Joachim Brand, Front. Phys., 2012, 7(1) 65

at the transition line contribute to the drag force, while 9 and 10. The ﬁlling factor α =1(kG =2kF) corre- for the periodic potential with the spatial period a only sponds to the Mott insulator state in a deep lattice. As a set of discrete points in the ω–k plane do [32, 38]. can be seen from Fig. 4, at this value of the reciprocal This is simply due to the fact that a periodic potential vector the DSF is almost independent of ω when γ 1, has only a discrete set of Fourier components with mo- and the drag force takes non-zero values for arbitrary menta jkG with j being integer and kG ≡ 2π/a being v ω+(kG)/kG (see Fig. 9). the reciprocal lattice vector. For instance, we have two nonzero components kG and −kG for optical lattice potential Vi(x)=gL cos(2πx/a) | |2 2 − Vi(q) /L = πgL δ(q kG)+δ(q + kG) /2 (35) in the thermodynamic limit (n =const,L →∞). The ﬁlling factor of the lattice, that is, the number of particles per site, is given by α =2πn/kG, because the total number of lattice periods equals LkG/(2π). Substituting Eq. (35) into the general expression for the drag force (8), we obtain at zero temperature

2 Fv = πgLkGS(kG,kGv)/2 (36)

The values of the obtained drag force can easily be es- timated with the ω–k diagram for the DSF (see Fig. 4). Note that the drag force is now proportional to the total number of particles, by contrast to the point-like impurity case, described by Eq. (9). This is because the lattice potential is non-local. Let us recall that the model considered in this paper assumes periodic boundary conditions. However, according to the general principles of statistical mechanics, the boundary conditions do not play a role in the thermodynamic limit. Hence, the obtained formula (36) can be exploited at sufficiently large number of particles even in the case of a cigar-shaped quasi-1D gas of bosons. This equation gives us the momentum transfer per unit time from a moving shallow lattice, which can be measured experimentally [17–19]. In the case of Bogoliubov and TG regimes, the drag force admits analytic solutions. As discussed in Section Fig. 9 Zero temperature phase diagram for superfluid–isolator 4.3, at small γ the upper dispersion curve ω+(k)isde- transition of the Bose gas in a moving shallow lattice: dimension- scribed well by the Bogoliubov relation (13), and the 2 2 less drag force Fv /(gLmL) versus the lattice velocity (in units behavior of the DSF simulates the δ-function spike in vF) and the interaction strength γ at various values of the filling accordance with Eq. (14). We derive from Eqs. (14) and factor α. The dimensionless values are represented in shades of (36) gray between zero (white)and1.0(black). The solid (blue) lines correspond to the DSF borders ω+(k)andω−(k), respectively. g2 mL 1 1 γ F = L δ ξ − + (37) By contrast, at small γ its non-zero values practically v 2 2αξ α2 π2 localize in the vicinity of v = ω+(kG)/kG,asshownin where by definition ξ ≡ v/vF. Fig. 10. Then superfluidity breaks down when the point In the TG regime, discussed in Section 4.2, the DSF is ω = vkG and k = kG lies exactly on the Bogoliubov dis- given by Eq. (11) with the limiting dispersions ω± being persion curve (see Fig. 4). Taking into account that the known analytically. Then Eqs. (11) and (36) yield Bogoliubov dispersion is very close to the free particle 2 2 2 one, we obtain for the break point kGv = kG/(2m). F = g mL/(4 )[Θ(ξ − ξ ) − Θ(ξ − ξ−)] (38) v L + One can see that this point coincides with the point of where we put by definition ξ± ≡|1 ± 1/α|. dynamical instability for bosons in the cosine shallow The values of the drag force (36) obtained from the lattice. Indeed, the dynamical instability appears at a interpolating expression for DSF (26) is shown in Figs. value of lattice velocity v corresponding to a negative 66 Alexander Yu. Cherny, Jean-Sébastien Caux, and Joachim Brand, Front. Phys., 2012, 7(1)

was examined in terms of quantum phase slips, discussed in Section 5.5. So, both the quasiparticle and quantum phase slip descriptions lead to the same results.

7.2 1D Bose gas in a moving random potential

Recently, direct observation of Anderson localization with a controlled disorder was reported in a one- dimensional waveguide [20]. An initially trapped Bose gas is released into a one-dimensional waveguide, where it is exposed to an artificially created speckled potential. The wave function of the spreading atoms can be directly measured by optical methods. Anderson localization was initially predicted for a system of non-interacting particles [79]. Modern experimental technique in cold atoms allows to control the strength of interparticle interactions and the parameters of speckled potentials as well (see, e.g., reviews [80, 81]). Delo- calization of a disordered bosonic system by repulsive interactions was observed [22]. Thus, the interplay be- Fig. 10 The same diagram as shown in Fig. 9, but here the drag tween interactions and disorder remains a fundamental force is represented as a function of velocity and inverse filling fac- problem in this field. Considerable attention has recently tor. been given to this problem in many papers [82–91]. Most of the papers consider weakly-interacting bosons or the curvature of the Bloch dispersion curve at the momen- regime of infinite interactions (TG gas). By using the tum mv/. This is because in the reference frame where generalized Landau criterion of superfluidity, introduced the lattice is at rest, particles’ momenta are shifted by in Section 3), one can study the behaviour of a 1D Bose the value mv/, and the effective mass, given by the sec- gas in weak random potentials and obtain the superfluid- ond derivative of the curve in this point, becomes neg- insulator phase diagram for arbitrary strength of boson ative. This implies that the effective kinetic energy of interactions. particles and the interaction energy have different signs, It is convenient to consider random potentials by intro- which immediately leads to the instability of the repul- ducing an ensemble of various potentials. Then the ran- sive Bose gas at sufficiently large numbers of particles. dom potential features can be obtained by averaging over For a shallow lattice, the Bloch dispersion curve is very the ensemble. One of the most important characteristics close to that of free particles, except for momenta near of random potentials, created with speckled laser beams, the edge of the Brillouin zone k = ±kG/2, where the is their correlation function Vi(x)Vi(x ) = g(x − x ), dispersion has a negative curvature (see, e.g., Ref. [72]). where ··· stands for the average over the random po- Hence, the condition for the dynamical instability takes tential ensemble. For an arbitrary potential profile, the the form kG/2=mv/. This coincides with the above drag force is calculated with Eq. (8). Taking the average condition for the break point, obtained within the gen- of this equation with respect to the ensemble, we obtain eralized Landau criterion of superfluidity, suggested in the drag force acting from the moving random potentials. Section 3. A similar analysis can also be carried out for At zero temperature, we derive the TG gas. The frictionless motion at some values of the parame- 2kC Fv = dkkg(k)S(k, kv) (39) ters v, γ, α is consistent with the presence of persistent 0 currents in the 1D Bose–Hubbard model [73–78]. As dis- ≡| |2 cussed in Section 3, the drag force can be considered as Here g(k) Vi(k) /L is the Fourier transform of a measure for superfluidity in the absence of the order the correlation function g(x). The integration in (39) paramenter. Then Fig. 9 represents the phase diagrams is limited, because for speckled laser beams, the func- in the v − γ and v − 1/α planes. They are similar to tion g(k) always has a finite support due to the limited that of Ref. [75]. One can see from the diagrams that aperture of the diffusion plate generating the random | | there is no sharp transition from superfluid to isolated phase [92, 93]. So, g(k)=0for k > 2kC.Foresti- phase in 1D, which is consistent with the experimental mations, we take a realistic correlation function [92, 93] 2 − |k|σr − |k|σr ≡ findings of Ref. [19]. Note that in paper [78], superfluidity g(k)=πVR σrΘ 1 2 1 2 .Hereσr 1/kC Alexander Yu. Cherny, Jean-Sébastien Caux, and Joachim Brand, Front. Phys., 2012, 7(1) 67

is the random potential correlation length, depending on where we introduce the notations λ0 ≡ 2/(πnσr), λ± ≡ 2 1 λ++λ− (λ0−λ−) the parameters of the experimental device. It is usually 2|ξ ± 1|, f ≡ (λ − λ−)(1 − ), f ≡ . 1 4 + 2λ0 2 8λ0 of the order of 1 μm [81]. Thus, we have three parameters The obtained results are in accordance with the exis- governing the phase diagram: the potential velocity, the tence of a mobility edge for a particle moving in a random interaction strengh, and the correlation length. potential with a finite correlation length σr.Inthiscase, Like the lattice potential, the random potential is non- the mobility edge is given by [83, 85] kmob = kC ≡ 1/σr. local, and, as a consequence, the drag force (39) is pro- If |k| >kC , then the k-wave propagation is not sup- portional to the total number of particles. The results pressed, while in the opposite case, the particle wave obtained with the interpolating formula (26) are shown function is localized (Anderson localization), which leads in Fig. 11. As for the shallow lattice potentials, the figure to the particle immobility. In the TG regime, the gas is can be treated as the zero-temperature phase diagram for equivalent to the ideal Fermi gas, where the mobile par- the superfluid–insulator transion: superfluidity assumes ticles lie in the vicinity of the Fermi points. In the ref- zero or strongly suppressed values of the drag force. erence frame where the random potential is at rest, the absolute value of momentum of the lowest Fermi level is | − | given by kF = kF mv .WhenkF >kC ,thesystem should be superfluid. This is consistent with Eq. (39), which yields zero value of the drag force for kF >kC . Indeed, the value 2kF corresponds to the cross point of the dispersion curves for the TG gas and the transition line: ω±(2kF)=2kFv,whereω± is given by Eq. (17) with infinite γ.Once2kF > 2kC then the value of drag force is zero, because the DSF is zero within the limits of integration in Eq. (39) (see Fig. 4).

8 Conclusions

The absence of a well-defined order parameter makes the behavior of the 1D Bose gas rather unusual in compari- son to the 3D case. As shown in Section 2, the 1D Bose gas exhibits superfluid phenomena of equilibrium type (Hess–Fairbank effect, analogous to the Meissner effect in superconductivity) but in general does not show dynamic superfluid phenomena, such as persistent currents in a ring. Instead of a phase transition to full super- Fig. 11 Zero temperature phase diagram for superfluid–isolator fluidity as is known in 3D, the 1D Bose gas shows a transition of the Bose gas in a moving random potential: drag force smooth crossover and reaches the metastability of cur- 2 2 (in units of 2πmVR σrN/ ) versus the potential velocity (in units rents only in a weakly-interacting limit. In this case, the vF) and the inverse correlation length (in units kF). The dashed drag force, being a simple integral parameter, can be (blue) lines correspond to the intersection of the limiting dispersions ω± and the transition line: v = σrω±(2/σr)/2. On the left chosen as a quantitative measure of superfluidity. Super- to the lines, the drag force is zero, because within the integration fluidity assumes zero or strongly suppressed values of the limits of Eq. (39), the values of the DSF is zero (see Fig. 4). drag force for the Bose gas moving in different small external potentials. Which value of the drag force should In the Bogoliubov and TG regimes, one can find the be taken as the transition threshold becomes a question drag force analytically. For γ 1, we obtain from Eqs. of convention in 1D. (14) and (39) The drag force turns out to be fundamental, because it generalizes the Landau criterion of superfluidity. The F = F Θ(ξ − ξc)Θ(1 − t)(1 − t) (40) v 0 generalized Landau criterion, based only on energy and ≡ 2 2 ≡ momentum conservation, does work when the usual Lan- HereF0 2πmVR σrN/√ is a unit of force, t 2 2 πnσr ξ − ξc ,andξc = γ/π is the sound velocity (in dau criterion fails. This is because the drag force effec- units of vF) in the Bogoliubov regime. In the TG regime, tively involves not only the spectrum but the probability Eqs. (11) and (39) yield of transitions to excited states. A good example is the dynamical instability of weakly-interacting 1D bosons, − − − − Fv = F0[f1 +(f2 f1)Θ(λ+ λ0) f2Θ(λ− λ0)] moving in a shallow lattice. As shown in Section 7.1, the (41) generalized Landau criterion not only successfully pre- 68 Alexander Yu. Cherny, Jean-Sébastien Caux, and Joachim Brand, Front. Phys., 2012, 7(1) dicts the dynamical instability of the system but gives yields for q =0 the quantitative characteristics of the phase transition. 2 2 2 2 k ,n|H |k ,m = |V (q)| n|δρˆq|m /V (A4) It should be noted that the suggested approach has f in i an apparent disadvantage. Being based on the first order At nonzero temperature, the medium can be in an ar- time-dependent perturbation theory, the scheme cannot bitrary initial state |m with the statistical probability describe changes of the ground state caused by the per- exp(−βEm)/Z. In this case, the transition rate (A1) turbing potential. Despite of this fact, it can describe should be averaged over the statistical ensemble, and we well the superfluid–insulator phase diagram, when the arrive at system propagates through shallow lattices or random 2π |V (q)|2 2q2 potentials. q i q · v − w( )= 2 S q, (A5) V 2mi Acknowledgements The authors are grateful to Antony Here we use Eq. (A4) and the definition of the DSF (4). Leggett, Lev Pitaevskii, Hans–Peter Büchler, Andrey Kolovsky, Anatoly Polkovnikov, and Andrew Sykes for fruitful discussions. In order to obtain the energy loss per unit time, we J.-S. Caux acknowledges the support from the FOM foundation need to sum up the energy transfer (A3) weighted with of the Netherlands. AYC acknowledges a visiting grant from the the rate (A5) over all the final states INSTANS activity of the European Science Foundation and the 2 2 support from the JINR-IFIN-HH projects and thanks Massey Uni- ˙ − q q · v − q versity for their hospitality. J. Brand was supported by the Mars- E = w( ) (A6) 2mi den Fund Council (contract MAU0706) from Government funding, q administered by the Royal Society of New Zealand. Replacing the sum by the integral in the thermodynamic D D limit q → V/(2π) d q and subsituting Eq. (A5) Appendix A: General formula for the drag yield Eq. (3). force from Fermi’s golden rule

Once an impurity moves into a homogeneous medium Appendix B: The drag force in the RPA ap- with velocity v, it is scattered by the medium particles. proximation In general, the scattering leads to transitions with momentum and energy transfer and, consequently, to a fi- B.1 The general expression nite value of energy loss per unit time. It can be calculated within linear response theory [5, 39, 94] or from In this appendix, we will use the dimensionless variables ≡ ≡ ≡ ≡ Fermi’s golden rule [94, 95]. λ q/kF, ν ω/εF, ξ v/vF,wherekF πn, ≡ 2 2 ≡ During the scattering, the initial state of the composite εF kF/(2m), and vF kF/m are the Fermi wave system is assumed to be |k ,m = |k |m,where|m vector, energy, and velocity, respectively. Besides, it is in in ≡ is initial state of the medium, the incident particle state convenient to introduce the small parameter α 2/γ, √ is the plane wave |k =exp(ik · y)/ V with wave vector which can take the values 0 α 1/4 within the RPA k ,andV = LD is the D-dimensional volume. The wave scheme [42]. In terms of the new variables, Eq. (9) reads in vector is connected to the initial velocity by the relation F πε +∞ f ≡ v F = dλλs(λ, 2ξλ)(B1) v = kin/mi. The same notations are adopted for the v 2 3 gi kF 0 final state |k ,n = |k |n. The rate for the scattering f f ≡ process (that is, the transition probability per unit time) whereweputs(λ, ν) εFS(kFλ, εFν/ )/N . is given in the lowest order in the impurity interaction The DSF in the RPA approximation was calculated by Fermi’s golden rule and described in details in Section 4.2 of paper [42]. The result at zero temperature can be written in the form 2π 2 w(k , k )= k ,n|H|k ,m in f f in s(λ, ν)=s (λ, ν)+A(λ)δ(ν − ν (λ)) (B2) n reg 0 × − − δ(En + Tkf Em Tkin )(A1)with the regular part of the dimensionless DSF 2 2 − 2 − where Tk = k /(2m) is the energy dispersion for the ≡ λ (1 3α) (1 2α) sreg(λ, ν) 2 (B3) impurity. The momentum and energy transfer are given 4 (1 − 3α)2λ − αh ln f +[απh]2 by This part is localized in the same region |λ2 − 2λ|(1 − q k − k = in f (A2) 2α) ν (λ2 +2λ)(1 − 2α), as the DSF in the linear − q · v − 2 2 ω = Tkin Tkf = q /(2mi)(A3)approximation, given by Eqs. (16) and (17). Here we put by definition respectively. y 2 − 2 − 2 − 2 Performing the integration over in the matrix ele- ≡ ν (λ 2λ) (1 2α) f(λ, ν) 2 2 2 2 (B4) ment of the interaction Hamiltonian H = j Vi(|y−xj|) ν − (λ +2λ) (1 − 2α) Alexander Yu. Cherny, Jean-Sébastien Caux, and Joachim Brand, Front. Phys., 2012, 7(1) 69 and other terms; however, this approximation works only at λ2 α ν2 3α(1 − 8α/3) very small values of ξ. A much better approximation can h(λ, ν)≡(1 − 9α/4) − − (B5) 2 2 8λ2 (1 − 2α)2 be obtained by keeping the logarithmic term together with the zero-order terms in ξ: The quantities in Eq. (B2) are given by A(λ) ≡ (1 − 3α)2 ξ A(kFλ)/N , ν0(λ) ≡ ω0(kFλ)/εF with A(q)andω0(q) fv (B7) (1 − 5α/2)2 − α 2 ξ 2 (1 3 ) − 2α ln + α2π2 being defined exactly in Section 4.2 of Ref. [42]. By em- 1−5α/2 2ξc ploying the general formula (B1) and the RPA expression (B2), we can evaluate the drag force by numerical integration. The δ-function contribution appears as a re- References and notes sult of intersection of the curve ν0(λ) and the line 2ξλ in the ν–λ plane and does not play a role at small val- 1. A. J. Leggett, Rev. Mod. Phys., 1999, 71(2): S318 ues of the impurity velocity, see Fig. 12. One can easily 2. A. J. Leggett, Rev. Mod. 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