Dynamics and Bloch oscillations of mobile impurities in one-dimensional quantum liquids
DDynamics and Bloch oscillations of mobile impurities in one-dimensional quantumliquids
M. Schecter, D.M. Gangardt, and A. Kamenev School of Physics and Astronomy, University of Minnesota, Minneapolis, MN 55455 School of Physics and Astronomy, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK William I. Fine Theoretical Physics Institute and School of Physics and Astronomy,University of Minnesota, Minneapolis, MN 55455 (Dated: October 10, 2018)We study dynamics of a mobile impurity moving in a one-dimensional quantum liquid. Suchan impurity induces a strong non-linear depletion of the liquid around it. The dispersion relationof the combined object, called depleton, is a periodic function of its momentum with the period2 πn , where n is the mean density of the liquid. In the adiabatic approximation a constant externalforce acting on the impurity leads to the Bloch oscillations of the impurity around a fixed position.Dynamically such oscillations are accompanied by the radiation of energy in the form of phonons.The ensuing energy loss results in the uniform drift of the oscillation center. We derive exact resultsfor the radiation-induced mobility as well as the thermal friction force in terms of the equilibriumdispersion relation of the dressed impurity (depleton). These results show that there is a wide rangeof external forces where the (drifted) Bloch oscillations exist and may be observed experimentally. a r X i v : . [ c ond - m a t . qu a n t - g a s ] M a y I. INTRODUCTION
Recent experiments [1–4] achieved a substantial progress in fabricating and studying dilute impurities immersed inone-dimensional (1d) quantum liquids. Such liquids are formed by ultracold bosonic or fermionic atomic gases placedin 1d optical lattices [5–9] or using the magnetic confinement on atom chips [4]. A hyperfine state of a few atoms isthen switched locally with the help of the RF magnetic field pulse [1, 4]. As a result, the atoms become distinguishablefrom the rest of the quantum liquid and thus may be considered as mobile impurities. Another promising realizationis achieved by placing ions of Yb + , Ba + or Rb + in the Bose-Einstein condensate of neutral Rb atoms [2, 3].Remarkably, the impurities may be selectively acted upon with the help of external forces. In the case of neutralatoms the external force is gravity, uncompensated by the force from the vertical magnetic trap as the impurityatoms are created in magnetically untrapped states. In the case of an ion, the selective force is due to the appliedelectric field. These setups thus allow to drive the non-equilibrium dynamics of mobile impurities immersed in a 1dquantum liquid. By releasing the trap after a certain delay time allows monitoring the resulting density and velocitydistribution of impurities.It is important to emphasize that in this publication we are dealing with the case of impurities with finite bare mass M , not very different from the mass m of the particles in the background. The velocity of the impurity is thereforefree to change as a result of absorbed momentum. This situation is qualitatively different from that of static, infinitelymassive impurities [16–19].Dynamics of mobile impurities in quantum liquids is an old subject pioneered by Landau and Khalatnikov [10] andBardeen, Baym, Pines and Ebner [11, 12] in their studies of He atoms in superfluid He. These authors realized thatat a finite temperature the impurities experience a viscous force from the normal component of the liquid, even iftheir speed is less than the critical superfluid velocity. The mechanism leading to such a viscous force was shown tobe the Raman scattering of the thermal excitations of the superfluid i.e. thermal phonons scattering off the impurity He atoms. It was found [10, 12] that the corresponding friction coefficient scales with temperature as T in the lowtemperature limit. It was later discussed by Castro-Neto and Fisher [13] that in 1d the Raman mechanism leadsto the friction force proportional to T due to the phase-space considerations. The same mechanism governs thetemperature-induced acceleration of grey solitons [44]. Recently, two of the present authors [14, 43] showed that inaddition to phase-space effects the friction is extremely sensitive to the details of the interactions between impurityand the background liquid and vanishes for exactly solvable 1d models. The friction therefore may be rather small inthe spin-flip setup if the parameters are near the Yang-Gaudin [15] integrable case.Due to the one-dimensional kinematics the slowly moving impurity tends to deplete the host liquid creating densityand current gradients in its vicinity. At small momenta the dispersion of the impurity can still be described byquadratic law E = P / M ∗ with effective mass M ∗ of the impurity atom [20, 21]. In the weakly interacting regimeand sufficiently small momenta the depletion of the host liquid can be studied within the framework of Bogoliubovtheory. The resulting excitations, polarons, consist of impurities surrounded by a cloud of linear excitations of the Π Π Π Π P (cid:144) (cid:209) n E (cid:72) P (cid:76) FIG. 1. Schematic dispersion relation for a mobile impurity in a weakly interacting superfluid. The dispersion defines the lowerbound of the many-particle excitation spectrum. In the region above the dispersion (shaded gray) there exists a continuum ofmany-body states. The energy is measured in units of the maximum energy of soliton excitations in the superfluid background,shown by the dashed line. condensate and were studied in Ref. [22] and later in Refs. [23, 24]. However, even for a weakly interacting liquid,the induced depletion becomes essentially non-linear at larger momenta and/or strong impurity-liquid coupling. Thisbrings a qualitative change of the dispersion relation and we introduce a quasiparticle which we call depleton describingthe impurity dressed by the non-linear depletion cloud.The remarkable property of depleton dispersion is that it is a periodic function of the momentum P with the period2 π (cid:126) n , where n is the density of 1d host liquid. To explain this feature it is worth noticing that, being quadratic atsmall momenta, the depleton energy is less than any of the excitations of the host liquid (see Fig. 1). Indeed, thelow-energy excitations of the host liquid have a sound-like nature with the energy cP > P / M ∗ , where c is the speedof sound. Therefore the dressed impurity excitation provides the cheapest way for the system to accommodate a smallmomentum. This means that the depleton dispersion relation is defined as the lowest possible many-body excitationenergy of the system with a given momentum. An example of such dispersion is shown in Fig. 1. Above the depletonenergy there is a continuum of many-body excitations comprised of the moving impurity and a certain number ofphonons. This spectral edge is characterized by the power-low singularities of zero-temperature correlation functionsand was discussed in Refs. [21, 25–27].One may argue now that in the infinite system the ground-state energy with a given momentum is a periodic functionof the latter with the period 2 π (cid:126) n . Indeed, it is easy to see that the ground-state energy with momentum 2 π (cid:126) n vanishesin the thermodynamic limit. To this end consider a ring of length L where the spectrum of the momentum operator isquantized in units of 2 π (cid:126) /L . If the momentum of each particle is boosted by one quantized unit, the total momentumof the system is 2 π (cid:126) n , while the total energy vanishes thanks to the total mass diverging in the thermodynamic limit.We thus conclude that the ground-state for a given momentum P which is the dispersion relation E ( P ) of the dressedimpurity is a periodic function of momentum. Explicit examples are provided by exactly solvable models [25] andweakly interacting Bose-liquid corresponding to Fig. 1 and considered in details in Appendix C.The physics behind the periodic dispersion relation is in the transfer of momentum from the accelerated impurityto the supercurrent in the background liquid: similarly to the density depletion, the moving impurity creates thesharp phase drop Φ across it. To satisfy the periodic boundary conditions the rest of the liquid must sustain thephase gradient Φ /L , resulting in the supercurrent which carries momentum (cid:126) n Φ. While the supercurrent is absorbingmomentum, it does not contribute to energy. Indeed, as it was already mentioned, in the thermodynamic limit thebulk of the liquid is infinitely heavy and thus can accommodate any momentum at no energy cost. As a result, theenergy and momentum of the depleton core, being periodic functions of the phase drop Φ with the period 2 π , oscillateas functions of the total momentum with the period 2 π (cid:126) n , while the rest of the momentum goes into the supercurrent.A similar periodic dispersion relation would arise if the host liquid were considered as a rigid crystal with the latticeconstant a = 1 /n . Then the momentum interval between P = − (cid:126) πn and P = (cid:126) πn is nothing but the Brillouin zoneof such a crystal, while the impurity dispersion, discussed above, is its lowest Bloch band.Although the above considerations completely disregard the absence of the true long-range order in the 1d liquid(either superfluid or crystalline) and thus should be taken with care, the periodicity of the dispersion suggests thepossibility of Bloch oscillations of the impurity atom subject to an external force [14]. Indeed, if an external force F is applied to the impurity atom ( e.g. electric field is acting on an ion) the total momentum of the system changeslinearly with time, P = F t . For infinitesimal force this leads to adiabatic change of the energy E ( P ) and the velocity V = ∂E/∂P of the depleton which become periodic functions of time with the period 2 π (cid:126) n/F . It is quite remarkablethat in such a process, on average, the impurity does not accelerate; moreover, it does not even move. Instead, itchannels the momentum into the collective motion, i.e. , the supercurrent, of the liquid and in the process oscillatesaround a fixed location. As a result, no energy is transferred, on average, to the system from the external potential.This spectacular phenomenon is present at zero temperature and under an infinitesimal external force. Both finitetemperature and a finite force complicate the picture in a substantial way. The aim of this paper is to clarify theinfluence of these two factors on the observability of the Bloch oscillations. In brief, our conclusions are as follows:at a sufficiently low temperature there is a parametrically wide range of external forces F min ( T ) < F < F max , wherethe Bloch oscillations are observable . Contrary to the adiabatic picture, they are accompanied by a drift and exhibitcertain amplitude and period renormalization.The drift manifests itself in the appearance of an average velocity V D , superimposed on top of the periodic Blochoscillations. It is a linear function of force at small forces, allowing to define the impurity mobility σ , as V D = σF . As aresult, the total energy of the system increases ( i.e. the system is heated) with the rate F V D = σF . This energy goesto the emitted long wavelength phonons, which run away from the impurity with the sound velocity c . The maximalforce can then be estimated as F max ≈ c/σ . At a larger force the drift velocity exceeds c , leading to Cherenkovradiation of phonons. The phonons take a substantial part of the momentum and thus ruin the Bloch oscillationmechanism, discussed above. The impurity motion is then either incoherent drift, or an unlimited acceleration,depending on the parameters.We derive the exact analytic expression for the drift mobility σ expressed in terms of the equilibrium dispersionrelation of the impurity. It is worth noticing that the mobility σ is not the linear response property, despite the linearrelation between the drift velocity and the external force. Indeed, the dynamics of depleton in this regime is the driftsuperimposed with the essentially non-linear pattern of Bloch oscillations in which the particle explores the entirerange of impurity momenta and energy. It is therefore not immediately obvious that the mobility σ may be expressedin terms of the equilibrium properties. Nevertheless we prove that such a relation does exist.The true linear response is associated with the thermally induced Landau-Khalatnikov friction force F fr ( V ) ∝ − T V ,which arises due to the Raman scattering of phonons discussed above. It provides the lower limitation on the externallyapplied force F min ( T ) = F fr ( V c ), where V c is the maximal equilibrium velocity given by the maximum slope of thedepleton dispersion, Fig. 1. Indeed, at smaller external forces the velocity saturates and therefore the depletondynamics is confined to the small momenta and Bloch oscillations do not occur.The friction coefficient in the small momentum regime was discussed in Ref. [14] and found to be vanishing in exactlysolvable cases. The reason behind it is the presence of infinite number of the conservation laws, which prevent a non-equilibrium state from thermalization. Here we extend those calculations to the case where the impurity explores theentire range of momenta and derive an exact result for the full momentum dependence of the friction force. Quitenaturally, it vanishes in exactly solvable cases too.The outline of the paper is as follows: in Section II we give a qualitative description of the depleton quasiparticlesand the mechanism of coupling to linear sound excitations, or phonons, of the background liquid. Section III is devotedto the formal derivation of the depleton Lagrangian based on superfluid thermodynamics. We discuss coupling of thedepleton with the phonon subsystem and derive a set of stochastic equations of motion for the impurity dynamicsin Section IV. These equation are then used to study radiation losses and derive expression of depleton mobility inSection V and thermal friction in Section VI. The main results are summarized in Section VII. Technical details aredelegated to Appendices. II. QUALITATIVE ANALYSIS
The key to understanding the impurity dynamics is in its interactions with the phonons of the host liquid. Thisproblem is rather non-trivial even if the impurity is weakly coupled to the liquid. Indeed, no matter how weakthe interactions are, the impurity develops local depletion, which become appreciable when the impurity momentumapproaches πn (we set (cid:126) = 1 throughout the rest of the paper). To visualize this process it is useful to assume asemiclassical picture of the background, valid for weakly interacting Bose liquid. In this regime, to accommodate thetotal momentum πn the depletion cloud takes the form of the dark soliton [28]. The dark soliton is an essentiallynon-linear mesoscopic object, which includes a large number of particles and a complete depletion of the liquid density.It is exactly the soliton formation which is responsible for channeling momentum into the supercurrent and thus forthe Bloch oscillations. It is also the soliton which determines the interactions of the impurity with the dynamicallyinduced phonons. Therefore the non-equilibrium dynamics of the quantum impurity cannot be separated from thedynamics of the essentially non-linear soliton-like depletion cloud.What makes the problem analytically tractable is the scale separation between the spatial extent of the local soliton-like cloud and the characteristic phonon wavelength. The former is given by the healing length ξ = 1 /mc . The latterappears to be much longer than ξ , if the temperature is sufficiently low and the external force is not too strong. Onecan thus separate the near-field mesoscopic region, which contains the quantum impurity and its depletion cloud, fromthe far-field region, supporting the radiation emitted by the impurity. Since the depletion is restored exponentiallyat the healing length away from the impurity, the precise position of the boundary between the near-field and thefar-field regions is of no importance.Because of the wide difference in their spatial scales, the impurity together with its entire non-linear depletioncloud represents a dynamic point-like scatterer for the long wavelength phonons. From the viewpoint of such phononsany point scatterer may be entirely described by two phase shifts. These two phase shifts are the discontinuities ofthe phonon displacement and momentum fields across the scatterer. They may be expressed through the numberof depleted particles N and the phase drop Φ across the depleton quasiparticle. Therefore out of many degrees offreedom of the near-field region only N and Φ interact with the phononic sub-system.What remains is to describe the dynamics of the local depletion cloud with certain fixed values of N and Φ. Solutionof this latter problem is facilitated by the fact that the characteristic equilibration rate of the cloud, estimated as1 /τ = c/ξ , is much faster than the relevant phononic frequencies. As a result, the cloud may be treated as being in thestate of local equilibrium , conditioned to certain values of the slow collective coordinates N and Φ. The fact that thereare two such slow variables is due to the presence of the two conservation laws: number of particles and momentum.The fast internal equilibration of the near-field region is therefore conditioned by the instantaneous values of the twoconserved quantities.The slow change (compared to the fast time scale of τ ) in the number of depleted particles N and the phase dropΦ are due to the fact that the local chemical potential µ and the local current j at the position of the impurity areboth affected by the state of the global phononic sub-system. As a result, one may express the Lagrangian of thenear-field region as a function of N and Φ through the equilibrium thermodynamic potential which is a function of µ and j . This latter function may be independently measured or analytically evaluated in certain limiting cases andfor exactly solvable models.These considerations allow one to separate the local, non-linear but equilibrium problem, from the global, non-equilibrium but linear one. The latter statement implies that the host liquid sufficiently far away from the impuritymay be treated as the linear i.e. Luttinger liquid [29, 30]. This is certainly an approximation which disregards thepossibility of the moving impurity to emit non-linear excitations, such as grey solitons or shock waves. The train ofsolitons emitted by the impurity moving with a constant supercritical velocity was indeed observed in simulations ofRef. [17]. The kinematics of this process suggests that it is only possible if the drift velocity is close to the speed ofsound c . We therefore assume that as long as F < F max , one may disregard solitons emission and treat the liquid awayfrom the impurity as the linear one. This is essentially the same criterion, which allows us to separate the depletioncloud from the long wavelength phonons.Adopting these approximations, one is able to integrate out the phononic degrees of freedom, characterizing theliquid away from the impurity. It reduces the problem to the dynamics of the impurity described by its coordinate X ( t ) and momentum P ( t ) along with the dynamics of its near-field depletion cloud fully described by the two collectivecoordinates i.e. number of depleted particles N ( t ) and the phase drop Φ( t ). We derive an effective action written interms of such an extended set of degrees of freedom. Such an action leads to the coupled system of quantum Langevinequations governing the dynamics of the depleton.Away from equilibrium, for F > F min , the equations of motion yield the pattern of Bloch oscillations. Thedeterministic part of these equations provides with the information about drift velocity, amplitude and shape of thevelocity oscillations, as well as their period. The stochastic part results in a certain dephasing of the oscillations.It is interesting to notice that the stochastic part is manifestly different from the equilibrium noise, prescribed bythe fluctuation-dissipation theorem. As a consequence, the exactly integrable models loose their special status andtheir non-equilibrium dynamics appears to be not qualitatively different from the dynamics of generic non-integrablemodels.
III. LAGRANGIAN OF THE MOBILE IMPURITY
Let us first consider the background liquid in the absence of impurity employing hydrodynamical descriptionproposed by Popov [29]. Its Lagrangian is expressed in terms of the slowly varying chemical potential µ and density n as an integral of the local thermodynamic pressure L ( µ, n ) = (cid:90) dx p ( µ, n ) = (cid:90) dx (cid:104) µn − e ( n ) (cid:105) . (1)Here e ( n ) is the energy density of the liquid. In the thermodynamic equilibrium the density is a function of thechemical potential, given by a solution of the following equation: µ = µ ( n ) = ∂e /∂n , which is a result of theminimization of this functional with respect to n . This way one defines the grandcanonical thermodynamic potentialof the host liquid as Ω ( µ ) = − L ( µ, n ( µ )) . (2)For a uniform system the Lagrangian and the corresponding thermodynamic potential are both proportional to thelength of the system.Consider now an impurity of mass M , having a coordinate X and moving through the liquid with velocity V = ˙ X ,as measured in the laboratory reference frame. It is convenient to choose the reference frame where the impurityis at rest and the liquid flows with the velocity − V , as shown in Fig. 2. In this co-moving frame the impurityexperiences the supercurrent j (cid:48) and the chemical potential µ (cid:48) . Hereafter primes denote physical quantities definedin the co-moving frame to distinguish them from the corresponding quantities in the laboratory frame. We employGalilean transformation into the moving frame which gives j (cid:48) = − nV, µ (cid:48) = µ + mV / . (3)Together with the Galilean transformation of the energy density e (cid:48) = e + mV /
2, the transformation (3) combinedwith Eqs. (1) and (2) show the invariance of the background grandcanonical potential Ω (cid:48) ( µ (cid:48) ) = Ω ( µ ). As expectedfrom the Galilean invariance, the latter is independent of the velocity V or the supercurrent j (cid:48) .We introduce now the impurity into the flowing liquid maintaining both j (cid:48) and µ (cid:48) fixed and let it equilibrate.Its motion distorts the host liquid density and velocity fields, forming the depletion cloud moving along with the FIG. 2. Transformation to moving frame. impurity. The grandcanonical potential increases by an amount Ω (cid:48) d ( j (cid:48) , µ (cid:48) ) = E (cid:48) d − µ (cid:48) N d , where E (cid:48) d and N d are thecorresponding changes in energy and number of particles. Using the Galilean invariance one can relate the energy E (cid:48) d = E d − P d V + mN d V / E d and momentum P d induced by the moving impurity in the laboratoryframe. This fact and relations (3) allows one to identify, in the spirit of Popov’s approach, the Lagrangian of thedepletion cloud with the negative change of the grandcanonical potential L d ( V, n ) = P d V − E d + µN d = − E (cid:48) d + µ (cid:48) N d = − Ω (cid:48) d ( j (cid:48) , µ (cid:48) ) . (4)This relation is quite remarkable as the left hand side describes dynamics of the polarization cloud moving with thevelocity V , while its right hand side is the thermodynamic quantity. The link between them comes from the Galileantransformation, Eqs. (3). The relation between the Lagrangian and the grandcanonical potential, Eq. (4), can beviewed as a generalization of the Popov relation, Eq. (2), to the case of mobile impurities.Two remarks are in order. First, in assuming the thermodynamic equilibrium at nonzero supercurrent j (cid:48) flowingthrough the impurity we rely on the superfluidity. Second, we note that the increase in energy E (cid:48) d , momentum P d ,number of particles N d and the grandcanonical potential Ω (cid:48) d due to the presence of one single impurity are finite sizecorrections to the corresponding extensive quantities. A. Collective degrees of freedom of the depleton
Variations of the so far fixed parameters j (cid:48) and µ (cid:48) of the background liquid induce changes in the thermodynamicpotential of the depletion cloud. It can be written with the help of the corresponding response functions asdΩ (cid:48) d = Φ d j (cid:48) + N d µ (cid:48) , Φ = ∂ j (cid:48) Ω (cid:48) , N = ∂ µ (cid:48) Ω (cid:48) . (5)The response to the variation of the chemical potential N = − N d is identified with the number of particles expelled from the liquid by the impurity (hence the minus sign). The response Φ to the change of the supercurrent j (cid:48) isthe superfluid phase and has no analogy in classical thermodynamics. In the state of the global thermodynamicequilibrium both Φ and N are rigidly locked to j (cid:48) and µ (cid:48) and, consequently, to V and n . This is denoted by writingΦ = Φ ( V, n ) and N = N ( V, n ). These functions can be obtained from the derivatives of the Lagrangian defined inEq. (4) as described in the next subsection.In the nonequilibrium situations, where the supercurrent and chemical potential fluctuate it is convenient to treatΦ and N as independent variables . We perform the standard Legendre transformation to a new thermodynamicpotential, H d (Φ , N ) = Ω (cid:48) d − j (cid:48) Φ − µ (cid:48) N, d H d = − j (cid:48) dΦ − µ (cid:48) d N. (6)The independent variables Φ and N describe the state of the depleted liquid in the immediate vicinity of the impuritywhich may or may not be in equilibrium with globally imposed j (cid:48) and µ (cid:48) . In equilibrium situation H d (Φ , N ) does notcontain any additional information with respect to the thermodynamic potential Ω (cid:48) d ( j (cid:48) , µ (cid:48) ), which is in turn relatedto the depletion cloud Lagrangian L d ( V, n ) by Eq. (4). The aim of introducing H d (Φ , N ) is to allow for interactionsof the depletion cloud with the long wavelength phonons. As we shall see in Section IV the latter may change thenumber of particles and the momentum of the depletion cloud, forcing it to equilibrate to some new values of Φ and N . Before turning to phonons, it is instructive to rewrite down the impurity Lagrangian (4) by substituting into itEq. (6) and considering Φ and N as independent vaiables, L d = 12 M V − j (cid:48) Φ − µ (cid:48) N − H d (Φ , N ) . (7)Expressing j (cid:48) , µ (cid:48) through V , n using Eq. (3), we finally obtain the Lagrangian of the depleton L ( V, Φ , N ) = 12 ( M − mN ) V + nV Φ − µN − H d (Φ , N ) , (8)where µ = µ ( n ) is the equilibrium chemical potential of the host liquid in the laboratory frame and we have addedthe bare kinetic energy proportional to the impurity mass M . The momentum of the depleton is obtained by thestandard procedure P = ∂L∂V = ( M − mN ) V + n Φ , (9)The last term describes the supercurrent momentum n Φ stored in the background. The first term is proportional tothe reduced mass M − mN , expressing the fact that the mass mN is removed from the local vicinity of the movingimpurity. This quantity should not be confused with the effective mass M ∗ given by the curvature the equilibriumdispersion Eq. (15) or obtained from the equilibrium dynamics in Eq. (19).We can use Eq. (9) to express velocity of the depleton as a function of its momentum,˙ X = V = V ( P, Φ , N ) = P − n Φ M − mN . (10)Combining Eq. (10) and the Lagrangian (8) leads to the Hamiltonian H ( P, Φ , N ) = P V − L = 12 ( P − n Φ) M − mN + µN + H d (Φ , N ) . (11)This Hamiltonian generates the following equations of motion˙ P = 0 (12)0 = − ∂ Φ H = nV − ∂ Φ H d (13)0 = − ∂ N H = − µ − mV / − ∂ N H d (14)in addition to Eq. (10). The first equation is the momentum conservation expected in the absence of external forcesand for homogeneous background. Two other equations are in fact static constraints . This is a manifestation of thealready mentioned fact that without phonons Φ and N are rigidly locked constants and do not have an independentdynamics. B. Dynamics of depleton in the absence of phonons and equilibrium values of collective variables
As we shall see in Section IV, the coordinates canonically conjugated to Φ, N are phononic displacement andphase at the location of the impurity. In the absence of these degrees of freedom the only consistent solution ofEqs. (13), (14) is static relations Φ = Φ ( P, n ), N = N ( P, n ). Substituting them back into the Hamiltonian (11)leads to the equilibrium dispersion relation of the dressed impurity (depleton), H ( P, Φ ( P, n ) , N ( P, n )) = E ( P, n ) . (15)The corresponding “equilibrium Lagrangian” can be obtained as L ( V, n ) =
P V − E by expressing the momentumwith the help of V = ∂E/∂P . In most situations it is rather these quantities and not the “internal energy” H d (Φ , N )represent the physical input about the dynamics of the dressed impurity. They can be obtained from solving theequilibrium problem for the impurity moving with the constant momentum P or velocity V , through the liquid withthe asymptotic density n . Below we show explicitly how collective variables Φ and N and the corresponding energy H d can be obtained from the knowledge of E ( P, n ) or L ( V, n ).We start with the situation when velocity V is a control parameter. In this case finding H d (Φ , N ) amounts toperform the Legendre transformation Eq. (6) by exploiting the definition (4) of the Lagrangian L d = L − M V / (cid:48) d and using the pair ( V, n ) instead of the thermodynamicalvariables ( j (cid:48) , µ (cid:48) ). To achieve this goal we use Eqs. (3) to relate the corresponding partial derivatives by the lineartransform (cid:18) ∂ V ∂ n (cid:19) = (cid:18) − n mV − V mc /n (cid:19) (cid:18) ∂ j (cid:48) ∂ µ (cid:48) (cid:19) . (16)Here we have used the relation ∂µ/∂n = mc /n between the compressibility and the sound velocity c . Using thedefinitions Eqs. (5) and Eqs. (16) we can express the derivatives of L ( V, n ) in terms of collective variables Φ , N∂L∂V − M V = P − M V = n Φ − mV N (17) ∂L∂n = V Φ − mc n N. (18)Solving these equations yield equilibrium values, Φ ( V, n ) and N ( V, n ). Eq. (17) can be otherwise obtained bysimply substituting Φ ( V, n ) and N ( V, n ) into the definition of the momentum, Eq. (9). This is a consequence ofthe equations of motion (13), (14). The quantities involving second derivatives of the Lagrangian do not enjoy thisproperty. The most obvious case is the effective mass M ∗ = ∂P∂V = M − mN − mV ∂N ∂V + n ∂ Φ ∂V , (19)which differs from the expression M − mN obtained by taking partial derivative of Eq. (9) with respect to velocity.The equilibrium relations Φ ( V, n ) and N ( V, n ) can be inverted to find velocity V (Φ , N ), and density n (Φ , N )for given values of Φ and N . Substituting them into Eq. (8) gives H d (Φ , N ) = − L ( V , n ) + M V n V Φ − (cid:18) µ ( n ) + mV (cid:19) N . (20)Conversely, we can use the momentum P as a control parameter. Again, using equations of motion Eqs (13), (14)one is able to show the equivalence of the derivatives ∂H ( P, Φ , N ) /∂n = ∂E ( P, n ) /∂n and ∂H ( P, Φ , N ) /∂P = ∂E ( P, n ) /∂P . The latter defines the velocity V ( P, n ). Using these facts and differentiating explicitly Eq. (11) wehave the following system of equations V ( P, n ) = P − n Φ M − mN ; ∂E ( P, n ) ∂n = mc n N − V ( P, n )Φ , (21)which are equivalent to Eqs. (17), (18) by virtue of the fact that ( ∂L/∂n ) V = − ( ∂E/∂n ) P . Solving Eqs. (21)yield equilibrium values Φ ( P, n ), N ( P, n ) as functions of P and n . Next, we invert these relations to obtain P (Φ , N ) , n (Φ , N ) as functions of Φ , N . Using Eq. (15) and Eq. (11) we obtain the expression for the core energy H d (Φ , N ) = E ( P , n ) − µ ( n ) N −
12 ( P − n Φ) M − mN . (22)in terms of the dispersion E ( P, n ) and its derivatives.To illustrate this procedure we use two cases, where the energy H d (Φ , N ) possesses a simple analytic form. One isthe grey soliton in weakly interacting Bose-Einstein condensate describing a massless, M = 0, impurity propagating ina weakly interacting Bose liquid with the coupling constant g . The standard results [28, 31] for the soliton dynamicsare provided in Appendix B and lead to H d (Φ , N ) = 18 mg N (cid:34) − (cid:18) sin Φ2 (cid:19) − (cid:35) . (23)Another example is provided by a strongly interacting impurity [21, 32]. In this case the number of expelled particles N is almost independent on the state of the impurity and may be considered as a non-dynamic constant. Theremaining dependence of energy on the superfluid phase Φ has a standard Josephson form H d (Φ) = − E J cos Φ . (24)The Josephson energy E J = (cid:126) nV c is expressed through the corresponding critical velocity V c , which in this case ismuch smaller than the sound velocity c .Expressions (20) or (22) provide the core energy of the locally equilibrium depletion cloud as a function of its slowvariables Φ and N . This procedure emphasizes the fact that introduction of Φ and N does not rely on the semiclassicalinterpretation of the condensate wavefunction. In fact, they may be defined even away from the semiclassical weaklyinteracting regime, where the phase of the condensate as well as its depletion are not well defined. IV. COUPLING TO PHONONS
An external force F acting on the impurity drives the system away from equilibrium, making the impurity radiateenergy and momentum. For a sufficiently weak force such a radiation takes the form of long wavelength phonons, i.e. small deviations of density ρ ( x, t ) and velocity u ( x, t ) fields from their equilibrium values, see Fig. 3. Below weshow how coupling to phonons can be formulated in terms of the collective variables Φ , N . It turns out that thisprocedure is based solely on the principles of gauge and Galilean invariance, i.e. conservation of number of particlesand momentum, and leads to universal results. A. Hydrodynamic description of phononic bath
We start by considering the Lagrangian, governing dynamics of the phonon fields in the bulk of the liquid. Tothis end it is convenient to parameterize them by introducing the superfluid phase ϕ ( x, t ) and the displacement field ϑ ( x, t ) such that u = ∂ x ϕ/m and ρ = ∂ x ϑ/π . The dynamics of these variables can be described following the methodof Popov [29] by considering slow change of the density n → n + ρ ( x, t ) and, independently, the change of the chemicalpotential, µ → µ − ˙ ϕ ( x, t ) − mu ( x, t )2 . (25)Substituting them into Eq. (1) yields the Lagrangian of phonons, L ph = (cid:90) dx (cid:104) p ( µ ( x, t ) , n ( x, t )) − p ( µ, n ) (cid:105) = (cid:90) dx (cid:20) − ρ ˙ ϕ − m ( n + ρ ) u − (cid:16) e ( n + ρ ) − e ( n ) − µρ (cid:17)(cid:21) . (26)For nonzero phononic fields, the impurity is subject to the modified local supercurrent and chemical potential inthe co-moving reference frame µ (cid:48) = µ − ˙ ϕ − mu m ( V − u ) µ ( n ) + mV − ( ˙ ϕ + V ∂ x ϕ ) ; (27) j (cid:48) = − ( n + ρ )( V − u ) = − nV − π ( ˙ ϑ + V ∂ x ϑ ) , (28)where the phonon variables are taken at the instantaneous spatial position, X ( t ) of the impurity. Equation (27)follows from Eq. (25) and the fact that in the presence of the background flow u = ∂ x ϕ/m the velocity of impuritywith respect to the liquid is changed to V − u . To derive expression (28) for the modified supercurrent we have usedthe continuity equation in the form ˙ ϑ/π = − ( n + ρ ) u . This relation is an exact statement, which follows from thegauge invariance and is valid for any configuration of the fields.Substituting the modified supercurrent and chemical potential, Eqs. (27), (28), into Eq. (7) and subtracting thecorresponding equilibrium values, results in the following universal form of the interaction Lagrangian, L int = 1 π Φ dd t ϑ ( X, t ) + N dd t ϕ ( X, t ) . (29)It is full time derivative d / d t = ∂ t + ˙ X∂ x = ∂ t + V ∂ x which enters the interaction term, as follows from Eqs. (27), (28).The interaction Lagrangian, Eq. (29) provides dynamics of the collective coordinates N and Φ. It shows that thecorresponding canonical momenta are the phonon degrees of freedom at the location of the impurity, i.e ϕ ( X, t ) and ϑ ( X, t ) correspondingly. Through the gradient terms in Eq. (26) these two local variables are connected to the phononfields elsewhere and it is the dynamical properties of these phonons which determine the behavior of the impurity. Forexample if the spectrum of phonons is discrete, one expects coherent oscillations of few modes. In the infinite systemthe continuous spectrum of the background modes leads to dissipation similar to that of Caldeira-Leggett model [33].
FIG. 3. Impurity propagating in local environment. B. Linear phonons and transformation to chiral fields
Away from this interaction region the excitations of the liquid may be considered as linear ones. Therefore onemay keep only the quadratic terms in the phononic Lagrangian Eq. (26). For the kinetic energy in Eq. (26) one thusretains the leading term mnu /
2, while the potential energy is expanded as e ( n + ρ ) − e ( n ) − µρ (cid:39) ( ∂µ/∂n ) ρ / mc / n ) ρ , using thermodynamic relation between the compressibility and the sound velocity c . As a result, oneobtains the quadratic Luttinger liquid Lagrangian L ph = 1 π (cid:90) dx (cid:20) − ∂ x ϑ∂ t ϕ − c K ( ∂ x ϑ ) − cK ∂ x ϕ ) (cid:21) . (30)Besides the sound velocity c the Lagrangian in Eq. (30) is characterized by the dimensionless Luttinger parameter K = πn/mc , which depends on the degree of correlations in the host liquid [30]. For a liquid of weakly repulsive bosonsthe Luttinger parameter K is large, K (cid:29) K = 1 by increasing therepulsive interactions or decreasing the density of the particles [34]. The combination πc/K = mc /n entering Eq. (30)contains information about interactions between the particles in the liquid, while the combination cK/π = cκ = n/m is independent of the interactions as a consequence of the Galilean invariance.To gain an additional insight into the physics of the depleton–phonon interaction, the phononic fields can bedecomposed into a doublet of right- and left-moving chiral components with the help of the linear transformation (cid:18) ϑ/πϕ (cid:19) = T χ = T (cid:18) χ + χ − (cid:19) , T = 1 √ (cid:18) √ κ √ κ √ κ − √ κ (cid:19) , (31)where κ = K/π = n/mc . In terms of the chiral fields the Lagrangian, Eq. (30) splits into a sum of two independentcontributions, L ph [ χ ] = 12 (cid:90) dx (cid:2) χ + ( ∂ x ∂ t + c∂ x ) χ + + χ − ( − ∂ x ∂ t + c∂ x ) χ − (cid:3) = 12 (cid:90) dx χ † D − χ . (32)The matrix of inverse phonon propagator is defined by its Fourier representation, δ ( x ) δ ( t ) D − = (cid:90) d q π d ω π e iqx − iωt D − ( q, ω ) , D − ( q, ω ) = (cid:18) q ( ω − cq ) 00 − q ( ω + cq ) (cid:19) . (33)The equation of motion following from the Lagrangian (32) dictate a simple coordinate and time dependence, χ ± ( x, t ) = χ ± ( x ∓ ct ). Using this fact one can show that for uniformly moving reference point X = V t one hasdd t χ ± ( X, t ) = ( V ∓ c ) ∂ x χ ± ( X, t ) . (34)This property is in fact a statement about correlation functions of the fields χ ± calculated with the Gaussian action,Eq. (32) which enforces classical equations of motion; the path integration is performed over arbitrary configurationsof the fields. The interaction term, Eq. (29) may be conveniently rewritten by introducing chiral collective variablesΛ = (cid:18) Λ + Λ − (cid:19) = T † (cid:18) Φ N (cid:19) . (35)In the static limit the quantities Λ ± are proportional, up to factor of √ π , to the chiral phase shifts δ ± introduced inRefs.[26, 35]. In presence of external force F they acquire the dynamics, which is governed by the total Lagrangianof the depleton interacting with the phonons L tot = P ˙ X − H ( P, Λ) + U ( X ) − ˙Λ † ( t ) χ ( X, t ) + L ph [ χ ] (36)where the impurity Hamiltonian H ( P, Λ) is obtained from Eq. (11) by using the linear relation, Eq. (35) betweenthe chiral phase shifts Λ and collective variables Φ and N , and L ph is given by Eq. (32). Here U ( X ) is an externalpotential acting on the impurity atom only and F = − ∂U/∂X is the external force. C. Integrating out the phonons
We have now all necessary ingredients for describing dynamics of impurity coupled through the interaction term,Eq. (29) to the phononic bath. The presence of the impurity is felt by phonons through time-dependent boundaryconditions at x = X ( t ) parameterized by the collective variables N ( t ) and Φ( t ), or equivalently by chiral phases Λ ± ( t ).1Here we simplify even further our description by solving phononic linear equations of motion for any variation of thesecollective variables and substituting the obtained solution back into the action. This procedure leads to dynamics ofthe impurity expressed in terms of collective variables only and is equivalent to exact integration of Gaussian phononicaction.To this end we employ the Keldysh formalism [36, 37] and extend the dynamical variables X ( t ), P ( t ), Λ ± ( t ) as wellas phononic fields χ ± ( x, t ) to forward and backward parts of the closed time contour t → t ± . Performing Keldyshrotation, we write them as X ( t ± ) = X cl ± X q /
2, Λ( t ± ) = Λ cl ± Λ q /
2, and χ ( t ± ) = χ cl ± χ q / L int = − ˙Λ † cl ( t ) χ q ( X cl , t ) − ˙Λ † q ( t ) χ cl ( X cl , t ) − ˙Λ † cl ( t ) X q ( t ) ∂ x χ cl ( X cl , t ) (37)up to terms linear in X q . The advantage of chiral fields introduced in Eq. (31) is that one can use the property (34)together with classical trajectory X cl = V t to simplify the interactions, Eq. (37) as L int = − ˙Λ † cl χ q + (cid:16) Λ † q + X q ˙Λ † cl V − (cid:17) d t χ cl (38)where we have introduced the matrix V − = (cid:18) c − V − c + V (cid:19) . (39)The interaction term, Eq. (38) is linear in phononic fields so that a Gaussian integration with quadratic action,Eq. (36) can be performed by standard methods as explained in Appendix A. It leads to quadratic, though a nonlocalin time effective action for the collective variables, S eff = − (cid:90) d t ˙Λ † cl ( t ) (cid:104) Λ q ( t ) + V − ˙Λ cl ( t ) X q ( t ) (cid:105) − (cid:90) d t d t (cid:48) (cid:104) Λ † q ( t ) + X q ( t ) ˙Λ † cl ( t ) V − (cid:105) ∂ t F ( t − t (cid:48) ) (cid:104) Λ q ( t (cid:48) ) + V − ˙Λ cl ( t (cid:48) ) X q ( t (cid:48) ) (cid:105) , (40)where, assuming thermal equilibrium of the phononic subsystem, the matrix F ( t ) is related by inverse Fourier transformto the matrix F ( ω ) = (cid:32) coth ω T +
00 coth ω T − (cid:33) , (41)of thermal distribution of the chiral bosons with the temperatures T ± = T (1 ∓ V /c ) modified by the correspondingDoppler shifts.One should supplement the action Eq. (40) with the Keldysh analogue of the action corresponding to the depletonHamiltonian, Eq. (11), S = (cid:90) dt (cid:104)(cid:16) ˙ X cl − ∂ P H (cid:17) P q − (cid:16) ˙ P cl − ∂ X U (cid:17) X q − ∇ Λ H · Λ q (cid:105) , (42)where we kept only terms linear in the quantum components and H = H ( P cl , Λ cl ). Notice that quadratic terms inquantum fields are absent in Eq. (42) while cubic and higher orders are omitted in the spirit of the semiclassicalapproximation.The second line in the effective action Eq. (40) may be split with the help of the Hubbard-Stratonovich transforma-tion, which introduces two real uncorrelated Gaussian noises ξ + ( t ) , ξ − ( t ). Their correlation matrix in the frequencyrepresentation takes the standard Ohmic form (see, e.g. [33]), (cid:68) ξ ( ω ) ξ † ( ω ) (cid:69) = ω F ( ω ) , ξ = (cid:18) ξ + ξ − (cid:19) (43)The action, Eq. (40) becomes local in time, S eff = − (cid:90) d t (cid:16) ˙Λ † cl − ξ † (cid:17) (cid:16) Λ q + V − ˙Λ cl X q (cid:17) . (44)2Now the entire semiclassical action is linear in quantum components and integration over them enforces the delta-functions of the equation of motions. While Eq. (10) remains intact, due to the absence of P q in the effective action(44), Eqs. (12), (13) and (14) are modified by the phonons:˙ P = F −
12 ˙Λ † V − ˙Λ + ξ † V − ˙Λ , (45)12 ˙Λ = −∇ Λ H + ξ , (46)where we have dropped subscripts for clarity. The obtained equations include additional dissipative terms involvingtime derivatives of the collective variables Λ. They also include fluctuations coming from the pair of Gaussian noises ξ ± ( t ) correlated accordingly to Eq. (43). V. DEPLETON DYNAMICS AT ZERO TEMPERATURE. RADIATIVE CORRECTIONS
Our goal is to discuss non-equilibrium solutions of the equations of motion in the presence of a constant externalforce F . Neglecting for a moment fluctuation terms and using transformation Eq. (35) the equations of motion,Eqs. (45), (46) can be rewritten in terms of the collective variables Φ and N as follows,˙ P = F − (cid:16) ˙Φ , ˙ N (cid:17) TV − T † (cid:18) ˙Φ˙ N (cid:19) = F − cc − V (cid:18) κV c ˙Φ + ˙Φ ˙ N + V κc ˙ N (cid:19) (47) κ ˙Φ2 = − ∂ Φ H = nV − ∂ Φ H d (48)˙ N κ = − ∂ N H = − mV / − µ ( n ) − ∂ N H d . (49)The rate of energy radiated by phonons is obtained by taking derivative with respect to time of the total impurityenergy, W = ˙ H − F ˙ X = V ( ˙ P − F ) − ˙Λ † · ∇ Λ H = V ( ˙ P − F ) + ˙Φ ∂ Φ H + ˙ N ∂ N H. (50)Using Eq. (10) and equations of motion either in the form (45–46) or (47–49) we obtain W = −
12 ˙Λ † (cid:2) + V V − (cid:3) ˙Λ = − c c − V (cid:18) κ + Vc ˙Φ ˙ N + 12 κ ˙ N (cid:19) , (51)The dissipation of momentum, Eq. (47) and energy, Eq. (51) is a generalization of Eqs. (22),(23) in Ref.[38] (up to afactor of two), where they were derived in the context of dynamics of grey solitons.According to Eqs. (12)–(14), in the absence of the external force F = 0 there is a family of stationary solutionsof the equations of motion, which are characterized by a constant velocity V below some critical velocity V c . Thissolutions describe the dissipationless motion of the impurity consistent with the superfluidity. Indeed, by neglectingthe fluctuation terms we effectively put the temperature to be zero, thus making the one-dimensional liquid superfluid.On the first glance one can just solve the set of the evolutionary equations (10), (47)–(49) to fully describe theimpurities dynamics. One needs to be careful, though, because Eqs. (48), (49) correspond to the motion in the vicinityof the maximum of the Hamiltonian H and therefore exhibit runaway instability. The further look at this instabilityshows that its characteristic rate is of the order mc ∼ µ , which is well outside the frequency range of applicabilityof the theory developed above. In fact this high-frequency instability of Φ and N evolution is a direct analog ofthe well-known spurious self-acceleration of charges due to the back reaction of the electromagnetic field [39]. Therecipe to overcome it is, of course, also well-known: instead of trying to solve equations of motion directly, one shouldperturbatively find how radiation corrections influence the dynamics [40]. This strategy offers a convenient analyticalapproach to treat the dynamics described by Eqs. (10) and (47)–(49). Below we apply it to study modifications toBloch oscillations which arise due to the phonon radiation. A. Radiative corrections to Bloch oscillations
At zero temperature with a sufficiently small applied force one expects the system to adiabatically stay in theground state with total momentum P , while the latter is changed by the external force P = F t . In this zeroth3approximation, the motion is nothing but a tracing of the dispersion relation E ( P, n ) and the phononic subsystemgains no share of the work done on the system by F . The velocity is simply V (0) ( t ) = V ( F t, n ) = ∂E ( P, n ) /∂P (cid:12)(cid:12)(cid:12) P = F t . (52)Since the dispersion relation displays a periodic behavior, one immediately obtains velocity Bloch oscillations withperiod τ (0)B = 2 πn/F , amplitude V c and zero drift. In reality, the slow acceleration of the impurity over the course ofa Bloch cycle gives rise to a soft radiation of low energy phonons, which serve to renormalize the period, amplitudeand drift from the zeroth order approximations.To study the corrections to the depleton trajectory, let us assume it exhibits a steady-state motion such that V ( t + τ B ) = V ( t ), N ( t + τ B ) = N ( t ) and Φ( t + τ B ) = Φ( t )+2 π . Then it follows from Eq. (9) that P ( t + τ B ) = P ( t )+2 πn .Here τ B is the, a priori unknown, true period of the motion, not to be confused with the zeroth approximation τ (0)B .To find it we integrate Eq. (47) over a single Bloch cycle2 πn = (cid:90) τ B dt ˙ P = F τ B − (cid:90) τ B dt c − V (cid:16) κV ˙Φ + 2 c ˙Φ ˙ N + κ − V ˙ N (cid:17) = τ B ( F − F rad ) (53)where F rad is the average radiative friction force exerted on the impurity over a single Bloch cycle. Since the radiativefrictional force tends to reduce the applied force, Eq. (53) indicates that the true period of oscillation is larger thanthe zeroth approximation τ (0)B .The work of the external force per unit time is given by F V = ˙ E ( P, n ) − W. (54)The first term in the r.h.s. of this equation is the reversible change in energy of the impurity, while the second term,owing to Eq. (51), is the rate of energy channeled into the phonon system. We average Eq. (54) over a single Blochcycle, noticing that (cid:104) ˙ E (cid:105) = 0 due to the periodicity of the dispersion relation. The remaining term corresponds to thepower radiated into phononic bath and leads to the drift velocity : V D = −(cid:104) W (cid:105) /F. (55)Assuming the energy pumped into the phonon system per Bloch cycle to be small we use the bare trajectories. V ( P, n ), Φ ( P, n ), N ( P, n ). Using the fact that dt = dP/F one can show that V D = σF. (56)Here σ is the T = 0 mobility of the impurity, given by the average over the Brillouin zone σ = 12 πn πn (cid:90) − πn dP (cid:18) c c − V (cid:19) (cid:34) κ (cid:18) ∂ Φ ∂P (cid:19) + V c (cid:18) ∂N ∂P (cid:19) (cid:18) ∂ Φ ∂P (cid:19) + 12 κ (cid:18) ∂N ∂P (cid:19) (cid:35) . (57)It was mentioned in Sec. III B that the equilibrium functions Φ ( P, n ) and N ( P, n ) can be obtained directly frompartial derivatives of the equilibrium dispersion relation E ( P, n ). Since V = ∂E/∂P , the mobility may be expressedentirely in terms of E ( P, n ) and the Luttinger parameter K = πκ .The fact that the mobility can be expressed through equilibrium properties is reminiscent to the Kubo linearresponse formulation. It is crucial to mention that they are not a linear response property in the Kubo sense. TheKubo linear response takes place at finite temperature, see Section VI. At T = 0 the liquid is superfluid and theimpurity undergoes Bloch oscillations with the amplitude V c at arbitrarily small external force F . It means thatthe response is essentially non-linear. The mobility σ describes the average (over one period) shift of the oscillationcenter due to the energy radiated in the course of such non-linear oscillations. The fact that it may be fully expressedthrough the equilibrium properties is rather remarkable on its own right.The result in Eq. (56) holds for sufficiently weak external perturbation F < F max . One can estimate F max bycomparing the corresponding drift velocity with the velocity of sound. From Eq. (56) we have F max = c/σ. (58)As the force increases the separation of length and energy scales used to define the depleton dynamics cannot bejustified. In other words for a strongly perturbed system the equilibrium dispersion relation E ( P, n ) ceases to be ameaningful concept.Below we use a model of an impurity coupled via delta-function interaction with strength G to the backgroundparticles to illustrate the dynamical properties of a depleton discussed above. We discuss two regimes: the strongcoupling regime, where the interactions in the liquid can be of arbitrary strength and the weak coupling regime wherethe dynamics of the depletion cloud is governed by Gross-Pitaevskii equation. In the latter case one is restricted to aweakly interacting bosonic background.4 mc t V (cid:72) t (cid:76) (cid:144) c F (cid:61) (cid:61) (cid:61) (cid:61) (cid:61) (cid:61) (cid:61) (cid:61) FIG. 4. Velocity as a function of time for various forces listed in the legend ( F in units of F max = 2 nmc ). Here κ = 20, G/c = 20 and M = 40 m . The dashed lines correspond to the drift velocity plotted in Fig. 5. One notices that as F increases,the drift velocity and frequency of oscillations increases, while the velocity amplitude (as measured from V D ) decreases.
1. Strong coupling regime
G/c (cid:29) The impurity expels a large number N (cid:29) N and use the Josephson form, Eq. (24) for the remaining dependence of energy on superfluid phase. Due to the factthat velocity is bound by the critical value V c the impurity is slow V ≤ V c (cid:28) c . Indeed, the calculation for weaklyinteracting bosonic background, Eq. (C9) (see also Ref.[41] ) gives V c = c / G (cid:28) c . Therefore, if the impurity is nottoo heavy, M ∼ m , we can neglect the second term in the momentum, Eq. (9) and we are left with the superfluidcontribution P = n Φ only. Using this fact in Eq. (57) gives the universal result for the mobility σ = κ (cid:126) n = Khn = 12 nmc , (59)where we restored (cid:126) = h/ π . Note that in the strong coupling limit, the mobility is independent of the impurityparameters and only depends on the parameters of the host liquid, namely the Luttinger parameter K and theasymptotic density n . This result has been obtained by Castro-Neto and Fisher, [13] by using the linear responseapproach.For the case of impenetrable bosons or free fermions corresponding to K = 1 one can use the analogy from theelectronic transport. Suppose the background is made of non-interacting fermions each carrying electric charge e . Inthe frame co-moving with the impurity a current I = enV D flows through the wire, whose quantum resistance is R .The latter is given by the Landauer formula, which for spinless case reads as R = h/e . The ohmic power transferredto the system, I R , must be supplied by the external force, F V D = hn V giving the result in Eq. (59) for K = 1.For K > R = h/Ke [16] leading again to Eq. (59). Notice that discussion of Ref. [42] claiminginteraction-independent mobility is not applicable here, since we always assume that the system length is much largerthan the characteristic wavelength of phonons.Scaling with K implies that the mobility is higher for weaker interactions and diverges in the free-boson limit. Tounderstand this result, notice that the moving impenetrable impurity experiences nV D collisions per unit time. Agiven collision results in the momentum transfer mV D to the boson of the gas. The balance of forces F = nmV thenimplies V D = F/ √ nmF leading to σ = 1 / √ nmF → ∞ as F →
0, in agreement with non-interacting limit of Eq. (59).Turning to the period of oscillations, Eq. (53) we use the relations V ≈ V D = κF/ n , ˙Φ ≈ F/n to obtain therenormalized period of Bloch oscillations, τ B ≈ πnF (cid:34) (cid:18) F nmc (cid:19) (cid:35) . (60)The characteristic force entering Eq. (60) coincides with the upper bound F max = 2 nmc obtained from Eq. (58)using the result (59) for the mobility. Indeed, the frequency of the motion ω ≈ F/n should be small compared to thetypical phonon frequency mc in order to justify the large scale separation employed in Sec. II.5 F (cid:144) F max V D (cid:144) c F (cid:144) F max V B (cid:144) c FIG. 5. Drift velocity V D (left panel) and amplitude V B (right panel) as functions of the applied force F corresponding to thesame set of parameters are as in Fig. 4. The solid curve on left panel is indistinguishable from the analytic prediction, the firstof Eqs. (64), while the dashed curve is the small force result Eq. (56) with mobility σ = κ/ (cid:126) n . On right panel the solid curveis numerics, the dashed curve is Eq. (D9), and the dotted curve corresponds to the small F limit of Eq. (65). To go beyond lowest order in F we devise a numerical approach to the strongly coupled impurity based on theJosephson form, Eq. (24) for the energy. Using it in Eq. (48) and recalling Eq. (47) gives,˙ P = F − κVc − V ˙Φ ; κ ˙Φ2 n = V − V c sinΦ , (61)where P is given by Eq. (9). The second of Eqs. (61) relates the change in Φ with the deviation of the impurityvelocity V from its equilibrium value V c sin Φ. The velocity V ( t ) is obtained by solving Eqs. (61) numerically and theresults depicted in Figs. 4 agree well with the ansatz (62). As one can see the precise choice of initial conditions isessentially irrelevant to the ensuing discussion, where we focus only on the asymptotic, steady state behavior of thefunctions Φ( t ) and V ( t ).After a sufficiently long time the system reaches the regime of steady Bloch oscillations, where its velocity is givenby V ( t ) = V D + V B cos( ω B t + δ ) . (62)Here the parameters V D , V B , ω B = 2 π/τ B and δ depend on the ratio of the external force F to the critical force F max .To find the drift velocity and period of Bloch oscillations we substitute the ansatz V = V D and ˙ P = n ˙Φ = ω B intoEqs. (61) to obtain a closed set of equation, nω B = F − κV D ω c − V ; ω B mc = V D /c (63)which are solved to give V D c = ω B mc = F max F (cid:115) (cid:18) FF max (cid:19) − . (64)Expanding the expression for ω B = 2 π/τ B for small F/F max one recovers the result (60) obtained by a differentmethod. The results for the drift velocity are plotted in Fig. 5a and are in excellent agreement with numericalsolution of Eqs. (61).The Bloch amplitude V B can also be estimated from Eqs. (61). In Appendix D it is shown that V B = V c (cid:40) − (cid:34) (cid:18) M − mNmκ (cid:19) (cid:35) (cid:18) FF max (cid:19) (cid:41) . (65)Thus, Eq. (65) predicts a decrease of the Bloch amplitude V B with increasing F (see Fig. 5b). Physically sensible,this result implies that as the force increases, the ideal equilibrium tracing of the dispersion relation is, to a degree,lost. This is due to the phononic subsystem gaining a proportionately larger share of the work done by the externalforce. Expression (65) is compared with results of the numerical solution of Eqs. (61) and the results presented in theright panel of Fig. 5b show a good agreement.6
2. Weakly interacting background bosons and weak coupling regime
G/c (cid:28) We deal with the case of the background made of weakly interacting bosons by using the Gross-Pitaevskii equationfor the background Bose liquid as explained in Appendix C. By calculating numerically the dispersion relation of theimpurity, see e.g.
Fig. 1, we obtain the functions N and Φ for all values of the impurity-background coupling G .The results of numerical evaluation of Eq. (57) are shown in Fig. 6.Let us turn now to the case of weak coupling G (cid:28) c , where particles can tunnel through the semi-transparentimpurity. In such a case one expects the mobility to increase as the impurity becomes more transparent. For weakcoupling, the main contribution to the integral in Eq. (57) comes from the region where the velocity of the impurity ismaximal, i.e. V ∼ V c . It corresponds to the inflection points of the dispersion, Fig. 1. Beyond this point the dispersionfollows closely the dispersion of grey solitons, Eq. (B4) and one can use the expressions (B5) to estimate the mobility.As the momentum P c corresponding to maximum velocity V ( P c ) = V c is small we simplify Eqs. (B4), (B5) and write V (Φ) c (cid:39) − (cid:18) Φ2 (cid:19) ; P (Φ) n = Φ − sin Φ (cid:39) Φ P ) = (6 P/n ) / ; V ( P ) c = 1 − (6 P/n ) / . (67)Substituting these expressions into Eq. (57) we can estimate the mobility as σ ∼ κn (cid:90) P c dP − V /c (cid:18) ∂ Φ ∂P (cid:19) ∼ κn (cid:90) Λ dP/n ( P/n ) ∼ κn cG = 1 /nmG . (68)Here we have estimated the momentum cutoff Λ = P c /n ∼ G/c using Eq. (C7) for the critical velocity and the secondequation in (67). The coefficient in Eq. (68) is fixed from the numerics and was found to be very close to one for M = m . Interestingly, in the weak coupling limit σ is independent of the correlations within the liquid (providedthe liquid is weakly interacting, G (cid:28) c ), diverging in the limit of a completely transparent impurity. Thus, in theweak coupling limit, the upper critical force F max = nGmc corresponds to energy difference Gn per healing length ξ = 1 /mc . G (cid:144) c Σ n m c FIG. 6. Log-log plot of the zero temperature mobility σ as a function of the impurity coupling G in a weakly interactingsuperfluid (solid line). The dashed lines correspond to the asymptotic values of σ in the strong and weak coupling limits givenby Eqs. (59) and (68), respectively (with (cid:126) = 1). The Luttinger parameter for the liquid is taken to be κ = n/mc = 20, while M = m . VI. FINITE TEMPERATURE DYNAMICS OF DEPLETON. BACKSCATTERING OF THERMALPHONONS AND VISCOUS FRICTION
At a finite temperature, even in the absence of the external force, the collective coordinates fluctuate around theirequilibrium position Λ (0) ± , found from the condition ∂ Λ H = 0. Assuming these fluctuations are small, one may linearizeEq. (46) near its equilibrium point. Being transformed to the frequency representation, such linearized matrix equationof motion takes the form (cid:18) − iω Γ − (cid:19) Λ( ω ) = ξ ( ω ) , (69)where Γ is the Hessian matrix of the second derivatives of the impurity Hamiltonian at fixed momentum P . Its matrixelements are Γ − = (cid:18) Γ ++ Γ + − Γ − + Γ −− (cid:19) − = (cid:18) H Λ + Λ + H Λ + Λ − H Λ − Λ + H Λ − Λ − (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) Λ=Λ (0) (70)Solution of equation (69) takes the formΛ( ω ) = (cid:18) − iω Γ (cid:19) − Γ ξ ( ω ) = Γ ξ ( ω ) + iω Γ ξ ( ω ) + . . . , (71)where we consider it as a perturbative sequence in frequency, to avoid spurious instabilities mentioned in Section V.Substituting this solution into the right hand side of Eq. (45) and averaging it over the Gaussian noise (43), one findsfor the momentum loss rate˙ P = F fr = − (cid:90) dω π ω Tr (cid:110)(cid:10) ξ ( ω ) ξ † ( ω ) (cid:11)(cid:0) Γ † V − Γ − V − Γ † Γ (cid:1)(cid:111) = − (cid:90) dω π ω Tr (cid:110) F ( ω ) (cid:2) Γ † , V − (cid:3) Γ (cid:111) , (72)where we kept only leading order in frequency. The matrix-valued function F ( ω ) is odd in frequency, selecting only oddpowers of ω from the expression it is multiplied by. Equation (72) provides the expression for the viscous friction forceacting on the impurity from the normal component of the liquid. It can be identified with the Raman two-phononscattering mechanism [10, 14, 43, 44].Substituting the explicit form of the Ohmic noise correlator (41) and performing the frequency integration, onefinds for the friction force F fr = − (cid:12)(cid:12) Γ + − (cid:12)(cid:12) cc − V (cid:90) dω π ω (cid:18) coth ω T − − coth ω T + (cid:19) = − π (cid:12)(cid:12) Γ + − (cid:12)(cid:12) T c (cid:18) c + V c − V (cid:19) V . (73)The T dependence of the friction force was reported in [13] and is a direct consequence of the phase space fortwo-phonon scattering. It can be also viewed as a one-dimensional version of the Khalatnikov-Landau result [10]for the viscosity of liquid helium. It should be noted that the friction force (73) is proportional to the off-diagonalmatrix element Γ + − which represents backscattering of phonons by the depleton. It depends on the momentum P and, consequently velocity V of the mobile impurity in addition to the parameters of the impurity-backgroundinteractions. Below we derive the general expression for the backscattering amplitude and relate it to the equilibriumdispersion E ( P, n ) of the depleton.
A. Backscattering amplitude
The detailed information about microscopic impurity-liquid interactions is contained in the off-diagonal matrixelement Γ + − which represents the effective vertex for two-phonon scattering. In the spirit of our phenomenologicalapproach it may be expressed, as explained in Appendix E, through partial derivatives of the equilibrium valuesΦ ( V, n ) and N ( V, n ),Γ + − ( V, n ) = − κmM ∗ (cid:20) M − mNmn (cid:18) ∂ Φ ∂V + m ∂N ∂n (cid:19) + (cid:18) ∂ Φ ∂V ∂N ∂n − ∂ Φ ∂n ∂N ∂V (cid:19)(cid:21) . (74)Treating collective variables Φ = Φ ( P, n ) and N = N ( P, n ) as function of momentum rather than velocity andcalculating the corresponding derivatives as explained in Appendix E we find an alternative representationΓ + − ( P, n ) = − c (cid:18) Mm ∂ Φ ∂P + Φ ∂N ∂P − N ∂ Φ ∂P + ∂N ∂n (cid:19) . (75)8 G (cid:144) c (cid:72) (cid:71) (cid:43)(cid:45) m c (cid:76) FIG. 7. V = 0 and Φ = 0 ( i.e. , P = 0) dissipation rate as a function of the impurity coupling G for κ = 10 and M = m (solidline). The dashed line corresponds to the strong coupling G (cid:29) c limit, Eq. (76). The functions Φ and N are obtained from the equilibrium dispersion E ( P, n ) as explained in Section III B (see, e.g.
Eq. (21)). This relations can be used to obtain an equivalent representation for the backscattering amplitude Γ + − given by Eq. (E11) in Appendix E.Two remarks are in order. First, it should be noticed that the backscattering amplitude given by Eqs. (74), (75)depends on the velocity V or momentum P of the depleton. While the momentum dependence is unambiguous,there can be different regimes corresponding to the same velocity characterized by different values of the superfluidphase Φ. Second, it can be shown by calculating dispersion E ( P, n ) by Bethe Ansatz method that the off-diagonalmatrix element Γ + − vanishes identically for integrable mobile impurities, such as lowest excitation branch in Lieb-Liniger [45] or Yang-Gaudin [15] models and the soliton of the Gross-Pitaevskii equation (see Appendix B). Wepostpone the details of these exact calculations to another publication. Below we use the semiclassical approach validfor the weakly interacting bosonic background which provides results for backscattering amplitude in a wide range ofparameters relevant to experiments.
1. Strong coupling regime
G/c (cid:29) In the strong coupling regime the leading terms in the the number of depleted particles, the second equation in (C9)is a constant, N = 2 κ (cid:29) P = n Φ. These observationsimplify greatly the expression (75) for backscattering amplitude and one getsΓ + − = 1 mc (cid:20) − Mκm (cid:21) , (76)where we have used the fact that ∂κ/∂n = κ/ n . The expression (76) is momentum or velocity independent constant,which equals 1 /mc for a not too heavy impurity M (cid:28) κm .
2. Weak coupling regime
G/c (cid:28) for slow impurity In the weak coupling regime the backscattering amplitude varies strongly with the velocity of the depleton. Herewe present results for slow impurity which can be found by evaluating Γ + − for V = 0.There are two distinct physical regimes corresponding to a slow impurity: one, corresponding to the solution withΦ (cid:39) P (cid:39) M V (cid:39) (cid:39) π andmomentum P (cid:39) πn dominated by the background supercurrent. For Φ ≈ N ( V, n ) = 2 κ G c − (cid:115) (cid:18) G c (cid:19) , Φ ( V, n ) = 2 Vc − G c + (cid:115) (cid:18) G c (cid:19) , (77)9while near Φ = π , we find N ( V, n ) = 2 κ , Φ ( V, n ) = π − Vc (cid:18) Gc (cid:19) . (78)These expressions are correct up to terms O ( V ) for the number of particles and O ( V ) for the phase.In the case Φ (cid:39) + − . For weak coupling to the leading order in both g/c = 1 /κ and G/c we have, Γ + − = 1 mc (cid:18) Gc (cid:19) (cid:18) mGM g − (cid:19) , G/c (cid:28) . (79)This result agrees with Eq.(5) in Ref. [14] and indicates that friction vanishes near integrable point M = m and G = g , at least for small velocities near Φ = 0. The results for arbitrary coupling G/c are presented in Figs. 7.In the case Φ (cid:39) π , we use Eqs.(78) and calculate effective mass M ∗ = M − mN + n∂ V Φ = M − κm − κm (1+ G/c ).Using Eq.(74) we find for arbitrary couplingΓ + − = 1 mc ( M/κm ) (1 + 2
G/c ) − G/cM/κm − G/c − , (80)It comes as no surprise that Eq.(80) vanishes in the limit M, G →
0, where one recovers the dark soliton resultfrom Appendix B. Indeed, there it is mentioned that such soliton excitation is transparent for phonons due to itsintegrability and, consequently, Γ + − = 0 for all velocities. In the limit of weak coupling Eq.(80) becomesΓ + − = 1 mc (cid:18) Gc − Mκm (cid:19) . (81)At the Yang-Gaudin integrable point ([15], see also [25]) of the quantum system defined as M = m and G = g ,the corresponding limit of the second equation in (81) is proportional g/c = 1 /κ which is a small parameter in thesemiclassical limit and therefore our calculation goes beyond the accuracy of the approximations adopted above. B. Bloch oscillations in the presence of viscous friction
Imagine the impurity is dragged by the weak external force F . There is a velocity V < V c such that F + F fr ( V ) = 0and the system reaches a steady state with constant drift velocity V D and no oscillations, V B = 0. If the applied forceis very small, the drift velocity can be found from the small velocity limit of Eq. (73) in the linear response form, V D = σ T F , where the finite temperature mobility is given by σ − T = 16 π T c (cid:12)(cid:12) Γ + − ( V = 0) (cid:12)(cid:12) . (82)in accordance with [13, 14]. Increasing the external force, leads to a slightly nonlinear dependence of the drift velocityon F (due to the non-linear velocity dependence of F fr ( V ), cf. Eq. (73)), until the the maximal possible velocity V c isreached for F = F min = − F fr ( V c ). Beyond this critical force no steady state solution can be found and the impurityperforms Bloch oscillations along with the drift. In such a nonlinear regime the amplitude, period of the oscillations,drift velocity, as well as the momentum-dependent viscous backscattering amplitude, Eq. (75), are controlled by theequilibrium dispersion relation.We illustrate the above scenario using the model of a strongly coupled impurity. In this case the critical velocity issmall and backscattering amplitude is velocity-independent. The critical force is then found from the linear responseas F min = V c /σ T . We can use the relation V = V c sin Φ + κ ˙Φ / n (i.e., the second of Eqs. (61)) and the fact that P = n Φ to write down the equation of motion for the superfluid phase,˙Φ = F min n (1 + κ/ σ T n ) (cid:16) λ − sin Φ (cid:17) , (83)where λ = F/F min . Introducing dimensionless time variable s = ( F min /n (1 + κ/ σ T n )) t and assuming, without lossof generality, Φ(0) = 0, the solution of Eq. (83) is found to betan Φ2 = λ tanh (cid:16) √ − λ s (cid:17) √ − λ + tanh (cid:16) √ − λ s (cid:17) (84)0 T (cid:144) mc V D (cid:144) V c F (cid:61) (cid:61) (cid:61) (cid:61) FIG. 8. Drift velocity of a strongly coupled impurity in a weakly interacting bose liquid as a function of temperature T for various forces as predicted by Eq. (86). F is given in units of F max V c /c , while the Luttinger parameter is taken to be K = πκ = 8 π / ≈
52 so that, to a good approximation, F min ( T = mc ) = F max V c /c . For λ < V ∞ = V c sin Φ( ∞ ) = λV c . For λ > τ B = ts π √ λ − πn (1 + κ/ σ T n ) (cid:112) F − F . (85)The corresponding drift velocity for F > F min can be found by averaging the momentum relation n ˙Φ = F − V /σ T over a single period with the result V D = V c (cid:32) FF min − (cid:112) F /F −
11 + κ/ σ T n (cid:33) . (86)For F min (cid:28) F (cid:28) F max , one may expand Eq. (86) to obtain a small temperature correction V D /c ≈ FF max + cV c F min F (cid:18) V c /c − ( F/F max ) (cid:19) , (87)which gives the previous result Eq. (59) in the limit of small T , or more specifically, small F min . Interestingly, Eq. (87)predicts an increase in the drift with increasing T if F/F max (cid:46) V c /c . This suggests that for a small enough force,a significant drop in the drift velocity may occur as the system is cooled below the critical temperature, when theimpurity enters the regime of Bloch oscillations, see Fig. 8.The above analysis demonstrates, inter alia , a non-monotonic dependence of the drift velocity on the parameter F/F min as it is increases past 1 and enters the regime of Bloch oscillations. This may occur either by fixing thetemperature and increasing the force, or by fixing the external force and cooling the system. In contrast to thevanishing amplitude V B near F max , indicated by the second equation in (64), it is rather the divergent period, Eq. (85)for F approaching F min from above which leads to the disappearance of Bloch oscillations. VII. DISCUSSION OF THE RESULTS
The coupling with phonons governed by the universal term, Eq. (29), results in transfer of energy and momentumbetween the depleton and the background. At zero temperature it takes the form of radiation of phonons by acceleratedimpurity similar to the radiative damping in classical electrodynamics [40] and leads to the finite mobility of thedepleton. For sufficiently weak forces the mobility, defined as the ratio of the drift velocity to the applied force, seeEq. (56) can be expressed via equilibrium dispersion, Eq. (57) reminiscent of the Kubo linear response theory. Thedrift, however, is superimposed with the essentially non-linear
Bloch oscillations of the velocity. The amplitude of1 F max Bloch Oscillations + Drift Drift F min T (cid:61) m c T F FIG. 9. Schematic force vs. temperature diagram of impurity motion in a 1D quantum liquid. In the region
F < F min (lightgray) the impurity drifts with mobility σ T ∝ T − for T (cid:28) mc . For T (cid:29) mc , the mobility scales σ T ∝ T − (dashed are thesmall and large temperature asymptotics). For F min < F < F max (dark gray) one has Bloch oscillations + drift. Above F max ,our theory is inapplicable and we expect some kind of incoherent acceleration, possibly corresponding to a supersonic impuritywhich has escaped its self-induced depletion cloud. the oscillations is shown to vanish as the external force attains the upper critical force F max reflecting the limit ofvalidity for the description based on the scale separation between the healing length and the phonon wavelength.At finite temperature the thermal phonons present in the system are scattered by the depleton leading to the viscousfriction force, Eq. (73). This in turn leads to the appearance of the lower critical force F min ( T ) ∼ T which sets thelower limit for the external force driving the Bloch oscillation. In contrast to the situation in the vicinity of F max , theapproach to F min makes the Bloch oscillation disappear through the divergence of the period, Eq. (85). Below F min the system enters the regime of non-oscillatory drift characterized by the velocity V D < V c , where all the momentumprovided by the external force is dissipated into phononic bath.The viscous force contains valuable information about the fine details of the interactions between particles. In thiswork we have confirmed and extended the earlier observation [14], that the backscattering amplitude, given in termsof equilibrium properties by Eqs. (74), (75) vanishes if the depleton is an elementary excitation of an integrable model.This includes the dark soliton excitation of Lieb-Liniger model [45] as well as spinons in bosonic Yang-Gaudin model[15]. The microscopic mechanism responsible for the absence of dissipation is due to destructive quantum interferencebetween various two-phonon processes. It can be traced back to the absence of three-body processes lying at heartof integrability in one dimension. In contrast, the radiative processes due to the presence of external force are alwayspresent in the dynamics of the depleton. This is because the external potential is not, in general, compatible with theintegrability.Various dynamical regimes of depleton are summarized in diagram Fig. 9. At sufficiently low temperatures thereis a wide parametric window F min < F < F max for the external force in which the Bloch oscillations can be observedexperimentally. At temperatures higher than chemical potential the mobility of the particle becomes inversely pro-portional to temperature [43, 44] and moderates the growth of F min (see Fig. 9). The above range of forces can beincreased further by exploring the dependence of F min and F max on the interaction parameters. In particular, atintegrable point F min vanishes for any temperature.Our approach to the dynamics of depleton is essentially classical, the quantum mechanics enters only via parametersof the effective action. The equations of motion, Eqs. (46) neglect therefore the quantum and thermal fluctuationsof the collective variables of the depleton. These fluctuations can be taken into account by either simulating theLangevin equation in the presence of equilibrium noises or by writing appropriate Fokker-Planck equation for thedistribution function. One of the most important consequence of the fluctuations is expected to be the smearing theboundaries between dynamical regimes in Fig. 9. We leave this as well as the question about the role of quantumfluctuation for further investigation. Another important extension of the present work would be the studies of thefinite size effects due to the trap geometry relevant for experiments with ultracold atoms.2 VIII. ACKNOWLEDGMENTS
We are indebted to L. Glazman, A. Lamacraft, M. Zvonarev, I. Lerner and J.M.F. Gunn for illuminating discus-sions. MS and AK were supported by DOE contract DE-FG02-08ER46482. DMG acknowledges support by EPSRCAdvanced Fellowship EP/D072514/1. DMG and AK are thankful to Abdus Salam ICTP in Trieste for hospitality atthe early stages of this work.
Appendix A: Derivation of dissipative action
Gaussian integration of the interaction term Eq. (38) using the following phononic propagators − i (cid:68) χ cl ( x, t ) χ † cl ( x (cid:48) , t (cid:48) ) (cid:69) = D K ( x − x (cid:48) , t − t (cid:48) ) , − i (cid:68) χ q ( x, t ) χ † cl ( x (cid:48) , t (cid:48) ) (cid:69) = D A ( x − x (cid:48) , t − t (cid:48) ) − i (cid:68) χ cl ( x, t ) χ † q ( x (cid:48) , t (cid:48) ) (cid:69) = D R ( x − x (cid:48) , t − t (cid:48) ) , − i (cid:68) χ q ( x, t ) χ † q ( x (cid:48) , t (cid:48) ) (cid:69) = 0 (A1)leads to the quadratic nonlocal action S eff = 12 (cid:90) d t d t (cid:48) ˙Λ † cl ( t ) ∂ t (cid:104) ∆ R ( t − t (cid:48) ) − ∆ A ( t − t (cid:48) ) (cid:105) (cid:104) Λ q ( t (cid:48) ) + V − ˙Λ cl ( t (cid:48) ) X q ( t (cid:48) ) (cid:105) − (cid:90) d t d t (cid:48) (cid:104) Λ † q ( t ) + X q ( t ) ˙Λ † cl ( t ) V − (cid:105) ∂ t ∆ K ( t − t (cid:48) ) (cid:104) Λ q ( t (cid:48) ) + V − ˙Λ cl ( t (cid:48) ) X q ( t (cid:48) ) (cid:105) , (A2)where ∆ R,A,K ( t ) = D R,A,K ( V t, t ) are phonon propagators restricted to the impurity trajectory.Inverting the matrix in Eq. (33) and taking appropriate analytic structure in the complex ω plane we obtain Fouriercomponents of the retarded and advanced propagators, D R ( q, ω ) = (cid:104) D A ( q, ω ) (cid:105) † = (cid:90) dxdt e − iqx + iωt D R ( x, t ) = (cid:32) q ( ω − cq + i − q ( ω + cq + i (cid:33) . (A3)This leads to D R ( q, ω ) − D A ( q, ω ) = 2 πiq (cid:18) δ ( ω − cq ) 00 − δ ( ω + cq ) (cid:19) (A4)and, subsequently, ∂ t (cid:104) ∆ R ( t ) − ∆ A ( t ) (cid:105) = 12 π (cid:90) d q d ω e i ( qV − ω ) t (cid:18) qV − ωq (cid:19) (cid:18) δ ( ω − cq ) 00 − δ ( ω + cq ) (cid:19) = − (cid:18) δ ( t ) 00 δ ( t ) (cid:19) (A5)The noise terms, i.e. the second line of Eq. (A2) are controlled by the Keldysh component D K and its derivatives.Assuming thermal equilibrium of phonons in the laboratory frame and using Fluctuation-Dissipation Theorem wehave D K ( q, ω ) = coth (cid:16) ω T (cid:17) (cid:16) D R ( q, ω ) − D A ( q, ω ) (cid:17) . (A6)Using Eq. (A4) we find the Fourier component of the Keldysh propagator restricted to the classical trajectory∆ K ( ω ) = (cid:90) d q π coth (cid:18) ω + qV T (cid:19) (cid:16) D R ( q, ω + qV ) − D A ( q, ω + qV ) (cid:17) = 1 iω (cid:18) coth ω T cc − V
00 coth ω T cc + V (cid:19) , Substituting its Fourier transform it into Eq. (A2) and taking double time derivative leads to the second term inEq. (40).
Appendix B: Grey Solitons
Dynamic properties of one-dimensional bosons of mass m weakly interacting via repulsive short range potentialproportional to coupling constant g can be studied within the Gross-Pitaevskii description using the following La-grangian L = (cid:90) d x (cid:18) i ¯ ψ∂ t ψ − m | ∂ x ψ | − g | ψ | + µ | ψ | (cid:19) . (B1)3Here ψ is the quasi-condensate wavefunction corresponding to the asymptotic density | ψ ( ±∞ , t ) | = n and vanishingsupercurrent at infinity. The condition of weak interactions is mg/n (cid:28)
1. Minimizing the action defined by theLagrangian Eq. (B1) leads to the Gross-Pitaevskii equation, i∂ t ψ = (cid:18) − ∂ x m + g | ψ | − µ (cid:19) ψ. (B2)Substituting a constant solution ψ = √ n one obtains the chemical potential µ = gn = mc related to the soundvelocity c = (cid:112) gn/m . In addition to uniform solution, Gross-Pitaevskii equation (C1) admits a one-parameter familyof solutions ψ s ( x − V t ; Φ s ) = √ n (cid:18) cos (cid:18) Φ s (cid:19) − i sin (cid:18) Φ s (cid:19) tanh (cid:18) x − V tl (cid:19)(cid:19) , (B3)known as grey solitons [28, 31]. They can be visualized as a dip moving with velocity V and having a core size l = 1 /m (cid:112) ( c − V ). The solution, Eq. (B3) is characterized by the total phase drop and number of expelled particlesrelated to the velocity asΦ s ( V, n ) = 2 arccos
Vc , N s ( V, n ) = (cid:90) d x (cid:0) n − | ψ s ( x ) | (cid:1) = 2 g (cid:112) c − V . (B4)Momentum and energy of the soliton are given by [28, 48, 49] P s = n Φ s − mN s V , E s = 43 cn sin (Φ s /
2) = mg N (B5)which allows to define the Lagrangian L s ( V, n ) = ( n Φ s − mN s V ) V − mg N , (B6)Using soliton Lagrangian (B6) and putting M = 0 in Eqs. (17) and (18) one sees immediately that the variables in(B4) coincide with the collective variables Φ, N . Therefore the grey soliton can be viewed as a model for a massless impurity consisting of depletion cloud only.We invert Eqs. (B4) which yields V = gN Φ2 , n = mgN . (B7)Using Eq. (20) together with Eqs. (B4) and (B6) yields the internal energy, Eq. (23) of the soliton, H d ( N, Φ) = mg N + N (cid:18) mV − gn (cid:19) = mg N (cid:18) − Φ / (cid:19) . (B8)The matrix of second derivatives at the equilibrium solution reads H = (cid:18) H ΦΦ H Φ N H N Φ H NN (cid:19) = (cid:32) − cn Φ / / mc / / mc / / − m c n Φ / / (cid:33) (B9)The backscattering amplitude Γ + − , calculated with the help of Eq. (E1), vanishes identically due to the integrability[50] of the Gross-Pitaevskii equation (B2).As it was shown in [44, 46] a weak cubic nonlinearity − ( α/ | ψ | in the Lagrangian (B1) breaks the integrabilityof the model. Cubic terms describe three-body interactions which arise from virtual transitions to higher transversestates of tightly confined one-dimensional liquid [47]. Here we extent the results in Ref.[43] and calculate the amplitudeof the corresponding dissipation processes for the whole range of soliton velocities.The corresponding correction to the Lagrangian, Eq. (B6) of the soliton can be calculated to the leading order byevaluating it with the unperturbed solution, δL d = − α (cid:90) (cid:0) | ψ d ( x ) | − n (cid:1) = 845 αn mc (cid:18) − V c (cid:19) / = αm g N = − δH d . (B10)4 (cid:45) (cid:45) (cid:45) mc (cid:72) x (cid:45) Vt (cid:76) (cid:200) Ψ (cid:144) n FIG. 10. Superfluid density in the presence of an impurity moving with velocity
V /c = 0 . G/c = 0 . Here we have used the expression Eq. (B4) for the number of expelled particles N to calculate the correction to theenergy relying on the theorem of small increments. The corresponding change in the matrix (B9) of second derivatives δ H = − αm g N × (cid:18) (cid:19) (B11)can be taken perturbatively in the calculation of the off-diagonal matrix element δ Γ + − . We have δ Γ = − Γ (cid:16) T − δ H (cid:0) T − (cid:1) † (cid:17) Γ . (B12)Substituting Eqs. (B11), (31), yields δ Γ + − = ακm g N Γ ++ Γ −− = ακm g N det H (B13)Here we have used the fact that the matrix Γ is diagonal in the leading order in α and det( T † T ) = 1. The determinantof the matrix (B9) is det H = (cid:0) mc sin Φ / (cid:1) . Substituting it into Eq. (B13) and using (B4) leads toΓ + − = − αn m c (cid:18) − V c (cid:19) / (B14)in agreement with the results of Ref.[43]. Appendix C: Impurity in a weakly interacting liquid
To model the impurity coupled to a weakly interacting superfluid at T = 0, the Gross-Pitaevskii equation, Eq. (B2)is modified in the presence of a delta function potential moving with constant velocity Vi∂ t ψ = (cid:18) − ∂ x m + g | ψ | − µ + Gδ ( x − V t ) (cid:19) ψ. (C1)For G >
0, the soliton solution, Eq. (B3) still satisfies Eq.(C1) except at the location of the impurity. Thus, one mayconstruct a solution to Eq.(C1) by matching two soliton solutions, Eq. (B3) at the location of the impurity, as shownin Fig. 10. The proper solutions of Eq, (C1) can thus be written ψ ( y ) = (cid:40) ψ s ( y + x ; Φ s ) e i Φ / , y > ψ s ( y − x ; Φ s ) e − i Φ / , y < y = x − V t and velocity is always related as
V /c = cos(Φ s /
2) to the phase Φ s parameterizing the solitonconfiguration, Eq. (B3). The solutions ψ ± to the right (left) of the impurity satisfy the boundary conditions: ψ + (0) = ψ − (0) , ψ (cid:48) + (0) − ψ (cid:48)− (0) = 2 mGψ (0). Using (C2), the boundary conditions give rise to the following two equations forΦ and z = tanh( x /l ) tan (cid:18) Φ (cid:19) = z tan (cid:18) Φ s (cid:19) (C3)sin (cid:18) Φ s (cid:19) (cid:0) − z (cid:1) z = Gc (cid:20) cos (cid:18) Φ s (cid:19) + z sin (cid:18) Φ s (cid:19)(cid:21) . (C4)Equations (C3), (C4) permit a solution only for V < V c ( G/c ) where V c is some critical velocity that depends onlyupon the parameter G/c [17, 41]. This can be seen by considering the right and left hand sides of the second equation(C4). While the left hand side is bounded by the maximum at z max = 1 / √ z and therefore solution exist only for a limited range of Φ s , which leads to the above-mentioned limitation onvelocity.For this reason we choose to parameterize the solution, Eq.(C2), by the total phase drop across the impurity,Φ = Φ s − Φ , which happens to permit a solution for any Φ. Thus, upon solving Eqs. (C3), (C4) one finds therelations z = z (Φ , G/c ) and Φ s = Φ s (Φ , G/c ). It can easily be seen from Eqs. (C3), (C4) that these functions areperiodic in Φ. The number of expelled particles and momentum can be calculated and expressed through the phaseΦ as N = (cid:90) dx ( n − | ψ | ) = 2 κ sin Φ s − z ); P = n Φ + ( M − mN ) V, (C5)The energy may also be calculated and expressed in terms of Φ as E = 12 M V + (cid:90) dx (cid:20) | ψ x | m + g n − | ψ | ) (cid:21) + G | ψ (0) | = 12 M V + 43 nc sin Φ s (cid:20) − z − z (cid:21) . (C6)Alternatively, we may solve for Φ = Φ( P ) by inverting the second of Eqs. (C5). Substituting it into the energyfunction, Eq. (C6) one obtains the dispersion relation E ( P ) plotted in Fig. 1 for the impurity in a weakly interactingbose liquid.In the weak coupling regime G/c (cid:28) z -dependence of ther.h.s. Eq. (C4). This is justified a posteriori as at the solution Φ s (cid:28)
1. Expanding the trigonometric functions andusing z max = 1 / √ s / = (3 √ / G/c , which justifies our approximation. We thus havethe critical velocity V c c = 1 − (cid:18) Φ s (cid:19) = 1 − (cid:32) √ Gc (cid:33) / . (C7)Solving Eqs. (C3), (C4) for arbitrary velocity or momentum is cumbersome and we resort to numerical methods.In the strong coupling limit G (cid:29) c we determine the dependence of z and Φ s on Φ to order c/G . This is done bywriting Φ s (cid:39) π + cG Φ (1)s and z (cid:39) cG z (1) and finding coefficients Φ (1)s and z (1) from Eqs. (C3), (C4). We have z (Φ) (cid:39) cG cos Φ2 ; Φ s (Φ) (cid:39) π − cG sin Φ . (C8)Using Eqs. (C8), (C5) one has V (cid:39) c G sin Φ; N (cid:39) κ (cid:18) − cG cos Φ2 (cid:19) . (C9)The first equation has the Josephson form with the critical velocity V c = c / G in the strong coupling limit. It isalso clear from the second equation that in the leading approximation N is a constant N (cid:39) κ . This results in theJosephson form for the energy, Eq. (24).At V = 0 the equations (C3), (C4) simplify considerably by putting Φ s = π . There are two solutions, z = (cid:115) (cid:18) G c (cid:19) − G c , z π = 0 , (C10)The z root corresponds to Φ = 0 and describes a background only slightly perturbed by the stationary impurity. The z π root corresponds to the dark soliton solution with Φ = π , which persists for G > i.e. , there is no additional energy cost to put the impurity in the center of a dark soliton.6
Appendix D: Solution of equation of motion for a strongly coupled impurity
We write the sinusoidal forms for the velocity and the phase of depleton V ( t ) = V D + V B sin( ω B t + δ V ); ˙Φ( t ) = ω B (1 + A cos( ω B t + δ Φ )) , (D1)depending on a priori unknown parameters V D , V B , ω B , δ V , δ Φ , A and substitute them into Eq. (61). One first arrivesat the following relations between the time independent components. nω B = F − κV D ω c − V ; κω B / n = V D = ⇒ ω B / mc = V D /c (D2)Solving these equations one obtains V D and ω B in Eq. (64). In the limit F (cid:28) F max = 2 nmc we recover the lineardependence V D /c = F/F max . The leading order deviation from linearity is given by V D /c ≈ FF max (cid:18) − F F (cid:19) . (D3)Substituting the ansatz (D1) into the first of Eqs. (61) gives a relation between the time dependent components, A (cid:20) nω B + κω V D c − V (cid:21) cos( ω B t + δ Φ ) + V B ( M − mN ) ω B (cid:20) cos( ω B t + δ V ) + 12 κω B M − mN c + V ( c − V ) sin( ω B t + δ V ) (cid:21) = 0 , (D4)where we neglected terms O ( A ), O ( V ) and O ( AV B ) to keep the calculation to first order in V c /c , as both V B and A will be seen to scale with V c /c . The second term in brackets in Eq. (D4) is simplified employing the formulacos x + α sin x = − (cid:112) α cos( x + π/ /α )) . (D5)In order to cancel the term ∝ cos( ω B t + δ Φ ), we require a definite relation between the phases and amplitudes. Theconstraint for the phase is δ V + π/ /α ) = δ Φ . From Eq. (D4) we have α = 12 κω B M − mN c + V ( c − V ) = mκM − mN V /c (1 − V /c ) V D c . (D6)The equality of amplitude implies a relation between A and V B , namely A = V B c M − mNmκ (cid:112) α (cid:18) − V /c V /c (cid:19) . (D7)Finally, we substitute the ansatz into the second of Eqs. (61) to obtain (cid:20) AV D + αV B √ α (cid:21) cos( ω B t + δ Φ ) + V B √ α sin( ω B t + δ Φ ) = − V c sin( ω B t + Φ ) , (D8)where we used sin( ω B t + δ V ) = − √ α (sin( ω B t + δ Φ ) + α cos( ω B t + δ Φ )) and Φ is some initial phase of Φ whichcomes from integrating the second of Eqs. (D1). Using Eq. (D5) again finally gives V B = V c − V /c (cid:112) V /α c = V c − V /c (cid:113) (1 + V /c ) + (cid:0) M − mNκm (cid:1) (1 − V /c ) V /c , (D9)where V D is given by Eq. (64). As expected, V B is an even function of F since V D is odd. For small F (cid:28) F max Eq. (D9) gives Eq. (64).
Appendix E: Calculation of the backscattering amplitude Γ + − Using the transformation Eq. (35) we have Γ = T † H − T which leads toΓ + − == 1det H (cid:18) κH NN − κ H ΦΦ (cid:19) . (E1)7The matrix of second derivatives H = (cid:18) H ΦΦ H Φ N H N Φ H NN (cid:19) (E2)is calculated at equilibrium values of Φ and N . Differentiating Eq. (11) and taking into account Eq. (10) for thedependence of the velocity V on Φ and N at constant momentum P , one can rewrite Eq. (E2) as H = 1 M − mN (cid:18) n − nmV − nmV m V (cid:19) − Ω (cid:18) N µ − Φ µ − N j Φ j (cid:19) (E3)The last term in this equation represents the matrix of second derivatives of H d (Φ , N ). Owing to properties ofLegendre transformation we have expressed it as the inverse of the Hessian matrix, Ω = (cid:18) Ω jj Ω jµ Ω µj Ω µµ (cid:19) = (cid:18) Φ j Φ µ N j N µ (cid:19) (E4)of the thermodynamical potential Ω (cid:48) d ( j (cid:48) , µ (cid:48) ). Hereafter we drop the primes over Ω, j and µ for clarity. In writingEq. (E4) we used Eq. (5) to express double derivatives as derivatives of equilibrium values Φ and N with respect tounderlying values of the supercurrent and chemical potential.Using relations mV = ∂µ/∂V , n = − ∂j/∂V in the first term of Eq. (E3) simplifies considerably the determinantdet H = 1det Ω (cid:18) − M − mN (cid:2) m V N µ + n Φ j − mnV ( N j + Φ µ ) (cid:3)(cid:19) = 1det Ω M ∗ M − mN . (E5)by using the effective mass Eq. (19) of the equilibrated impurity. We use this fact and invert the matrix in Eq. (E3)by a standard procedure and find for the off-diagonal matrix elementΓ + − = 1 κM ∗ (cid:104) ( M − mN ) (cid:0) κ Φ j − N µ (cid:1) + n (cid:0) − V /c (cid:1) (Φ j N µ − Φ µ N j ) (cid:105) . (E6)In the heavy particle limit M (cid:39) M ∗ (cid:29) m only the first term in the square bracket survives. In this limit Γ + − canbe identified with backscattering amplitude of a static impurity Γ + − = Γ ∞ = κ Φ j − κ − N µ = − ( ∂ V Φ + m∂ n N ) /mc .To obtain this result we inverted the relations in Eq. (16), (cid:18) ∂ j ∂ µ (cid:19) = 1 m ( c − V ) (cid:18) − mc /n mV − V n (cid:19) (cid:18) ∂ V ∂ n (cid:19) , (E7)and used the identity n∂ n Φ − mV ∂ n N = V ∂ V Φ − ( mc /n ) ∂ V N . The latter is obtained from equality of the mixedderivatives of the Lagrangian L ( V, n ), obtained by differentiating Eqs. (17), (18) by n and V respectively. Thesecond term in the square brackets in Eq. (E6) is transformed by applying (E7). Combining it with the term,Γ ∞ ( M − mN ) /M ∗ yields Eq. (74).To deal with equilibrium functions Φ( P, n ), N ( P, n ) obtained for a given momentum P rather than velocity V weuse the fact that ∂ V = M ∗ ∂ P and the following relation for the derivatives with respect to the density (cid:0) ∂/∂n (cid:1) V = (cid:0) ∂/∂n (cid:1) P − M ∗ ( ∂V /∂n ) P ∂ P . 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