Inelastic scattering of xenon atoms by quantized vortices in superfluids
IInelastic scattering of xenon atoms by quantized vortices in superfluids
I.A. Pshenichnyuk and N.G. Berloff , Skolkovo Institute of Science and Technology Novaya St., 100, Skolkovo 143025, Russian Federation and Department of Applied Mathematics and Theoretical Physics,University of Cambridge, Cambridge CB3 0WA, United Kingdom (Dated: October 3, 2018)We study inelastic interactions of particles with quantized vortices in superfluids by using a semi-classical matter wave theory that is analogous to the Landau two-fluid equations, but allows forthe vortex dynamics. The research is motivated by recent experiments on xenon doped heliumnanodroplets that show clustering of the impurities along the vortex cores. We numerically simulatethe dynamics of trapping and interactions of xenon atoms by quantized vortices in superfluid heliumand the obtained results can be extended to scattering of other impurities by quantized vortices.Different energies and impact parameters of incident particles are considered. We show that inelasticscattering is closely linked to the generation of Kelvin waves along a quantized vortex during theinteraction even if there is no capture. The capture criterion of an impurity is formulated in termsof the binding energy.
I. INTRODUCTION
In-depth understanding of the dynamics of quantumfluids, and in particular understanding of processes oc-curring in quantum turbulence during the formation andevolution of a vortex tangle, requires advanced theoret-ical modelling and precise experimental probing tech-niques. Superfluid helium, being the first quantum fluidavailable for experiments, and probably the most studiedone, generates vorticity at the length scale of angstromswhich makes the direct observation of vortices compli-cated. Indirect measurements usually involve probingvortices with impurities, which are often used as dopingfor subsequent optical detection. Early experiments wereperformed with electrons and ions . Later, manytypes of other impurities including molecules, molecularclusters and excimers were used as doping to visual-ize and study quantized vortices. Modern particle im-age velocimetry techniques allow to use various kinds ofmicron size tracer particles to visualize flow patterns inhelium . This methods allow one to trace both the nor-mal and superfluid components (through the interactionwith vortices) and thus provide a useful tool to studytwo-fluid hydrodynamics . It is shown experimentally,that the coalescence of metal particles trapped on quan-tized vortices may lead to the formation of centimetrelong wires . Such a mechanism provides not only away to visualize the structure of quantized vortices butalso a new approach for producing long metal nanowires.Zmeev et al. have shown that a moving vortex tanglecan transport molecules through superfluid helium, sothe composite particles and molecules can be used toprobe the density and orientation of the vortex tangleand lead to some new and unusual types of matter orga-nization with potentially peculiar properties.Recently, nanodroplets experiments that embed sin-gle atoms and molecules into liquid helium droplets havebecome a new tool to study various aspects of super-fluid behaviour. In these experiments ultracold heliumworks as a homogenous matrix for subsequent spectro- scopic studies . In the experiments of Gomez et al. femtosecond x-ray coherent diffractive imaging techniquewas used to demonstrate the existence of vortex arraysin helium droplets through the observation of Bragg pat-terns. Xenon atoms were used as doping in this experi-ments. The analysis revealed an unusual form of dropletsand line associations of xenon atoms which was explainedby the formation of vortices in a rotating helium dropletwith subsequent trapping of the xenon atoms at the vor-tex cores.Most commonly used theoretical approach to study thestatic behavior of impurities in nanodroplets is based onDFT calculations . It is particularly successful in find-ing the minimal energy configurations and so capable ofdescribing the various aspects of the particle-vortex in-teraction. The dopant of choice to detect vortices bymeans of spectroscopic experiments is discussed in An-cilotto et al. where the adsorption properties of dif-ferent atomic impurities are compared. This approachwas used to study vortex array equilibrium configura-tions in rotating nanodroplets, the properties of xenonchains trapped by the vortex lines, and to explain shapesand surprising stability of nanodroplets .Despite the large amount of studies, the details ofparticle-vortex scattering and especially processes whichtake place at the vortex core during the interaction arenot well understood firstly, because of the interatomicdistances involved, secondly, because there is no firstprinciples models that allow one to describe such a dy-namics correctly. Minimalistic models of particles mov-ing in superfluids at zero temperature usually assumethat the Bernoulli’s force is a dominant one and thatit adequately describes the motion far from the vortexcores . Close to the vortex, substitution energy basedanalysis is often used to explain the existence of the po-tential energy barrier with certain parameters which de-fine the capture and escape probabilities . At thesame time, 3D simulations based on the Gross-Pitaevskiiequation and the self-trapping model demonstratethat the capture of an electron by a quantized vortex is a r X i v : . [ c ond - m a t . o t h e r] A ug accompanied by the emission of Kelvin waves which prop-agate along the vortex core and carry a certain portion ofenergy with them. It makes the particle-vortex scatteringprocess inelastic and renders more detailed energy redis-tribution analysis. Non-elasticity of the trapping processis similar to inelastic scattering of electrons on molecules,where electrons can be captured by molecules, as a resultof internal energy redistribution through the electron-phonon coupling mechanism, forming long-living nega-tive ions.Xenon particles used as doping in the experiments ofGomez et al. are very different from electrons, con-sidered in Berloff and Roberts . Electron in helium,through its zero-point motion, forms an electron bubbleof a radius of about 16 ˚A, that brings about a large (incomparison with the electron) effective mass and the dis-tortion of the soft bubble boundary. This effect for theelectrically neutral xenon is minimal and we expect itsradius in helium to be of the order of the size of vortexcores. It results in a significant difference of substitutionenergies of electrons and xenon atoms. Moreover, atomsare much heavier than electrons and can potentially pro-duce more disturbance along the vortex lines when usedas doping.In this paper we develop ideas formulated in Berloffand Roberts to study scattering of Xe atoms by quan-tized vortices in different regimes. We shall elucidate therole of the binding energy and attachment/detachmentcriteria. The paper is organized as following. We presentthe model representing the mathematical equivalent ofthe Landau two-fluid theory which is the basis for our nu-merical and analytical study in Section II. We discuss mo-tion of a xenon atom next to a straight line quantized vor-tex and analyze various scenarios of the impurity-vortexinteractions in Section III. We conclude with Section IVsummarizing the main findings. II. MODELLING OF THE VORTEX-IMPURITYINTERACTIONS
A useful approach in modelling the dynamics and in-teractions of particles with quantized vortices was origi-nally formulated by Gross . In this approach the nonlin-ear Schr¨odinger equation (NLSE) also known as Gross-Pitaevskii equation (GPE) which describes the wavefunc-tion of a Bose-Einstein condensate is coupled with thelinear Schr¨odinger equation for the particle’s wavefunc-tion. In reality only about 10% of superfluid heliumis in a condensed phase and the fluid is dominated bymany-body effects, so its approximation by the conden-sate order parameter is at best phenomenological. It waslater demonstrated that the NLSE in the context of thesemi-classical matter field description corresponds to theLandau two-fluids model and, therefore, describes boththe superfluid and the normal fluid as long as the low oc-cupancy modes and their coupling to the highly occupiedmodes are neglected. The framework of the coupled GP-
10 5 0 5 10 15X [ ]0.50.00.51.01.52.0 M a tt e r f i e l d ψ / W a v e f un c t i o n ϕ [ ψ ∞ / ξ − / ] Vortex Particle v p | ψ ( x, , | Im ϕ ( x, , Re ϕ ( x, , | ϕ ( x, , | FIG. 1. A snapshot of the one-dimensional cross-sections ofthe matter field ψ and the wavefunction ϕ along the particle-vortex interaction line. The left drop in the modulus of theamplitude of ψ corresponds to the quantized vortex, the rightdrop corresponds to the position of the xenon atom. The plotsof the real and imaginary part of ϕ are given to indicate thatthe atom is moving towards the vortex. type equation for the superfluid and normal fluid compo-nents and the equation for the particle’s wavefuction cantherefore be used at finite temperature. We can furtherremedy this description and incorporate the equation ofstate correct for the superfluid helium using a higher or-der NLSE . The higher order nonlinearity appears fordense fluids with the equation of state given by a poly-nomial expression . Such an equation is mathematicallyequivalent to the Landau two-fluid model and allows one,in addition, to account for the processes associated withquantized vortices. In this sense it provides a frameworkto describe the behavior of superfluid helium at finitetemperatures. In the Appendix A we show how to re-cover the Landau two-fluid model from our theory.We formulate the Hamiltonian of the system by in-troducing various contributions: the kinetic and internalenergies of superfluid helium E kin and E int , the particle-helium interaction energy E h (which is the most signif-icant in the healing layer between the particle and thefluid), the energy of the xenon particle E p (it includesthe kinetic energy of motion E pk , which will be discussedlater, and the zero-point energy) and explicitly introduc-ing the Lagrange multiplier (the chemical potential) µ inthe view of the constraint on the total number of matter (cid:82) | ψ | dV = N , where N is a number of bosons in thesystem: E = E kin + E int + E h + E p − µN, (1) E kin = (cid:90) (cid:126) m |∇ ψ | dV , (2) FIG. 2. (Color online) Visualization of two scattering processes with (the top row) and without (the bottom row) trapping.Initially the impurity is located 11 ˚A away from the vortex and moving with an initial kinetic energy 0.16 meV (the top row)or 0.38 meV (the bottom row). The panels (a) and (e) show the trajectories of the atom (blue line) and the vortex (grey line).Other panels show two-dimensional cross sections of the modulus of the amplitude of the fluid | ψ ( x, , z ) | at different momentsof time. Small three-dimensional insets show the corresponding isosurfaces | ψ ( x, y, z ) | = 0 . ψ ∞ . E int = (cid:90) ε int ( | ψ | ) dV , (3) E h = (cid:90) U | ψ | | ϕ | dV , (4) E p = (cid:90) (cid:126) M |∇ ϕ | dV . (5)Here m and M are the masses of the helium atom andthe xenon atom, respectively, ψ is classical complex mat-ter field which describes the superfluid and the normalfluid components, ϕ is the wave function of the particle.The parameter U = 2 πl (cid:126) /M ∗ is the local He-Xe inter-action potential strength, where l = 3 . . This value is also close to the sumof the Van der Waals radii of xenon and helium atoms. M ∗ is the reduced mass of the interaction.The internal energy functional is based on the phe-nomenological equation of state of liquid helium andhas the form ε int ( n ) = − V n − V n + V n , (6)where n = | ψ | . Coefficients V = 719 k b K ˚A , V =3 . · k b K ˚A and V = 2 . · k b K ˚A (where k b is the Boltzmann constant) are chosen to reproduce thebinding energy, the density and the sound velocity of liq-uid helium . The Hamiltonian of Eq. (1) was used in Berloff et al. and Pshenichnyuk to study the multi-plication of vortex rings in a superfluid during pressureoscillations.Performing a variation of the full energy E with respectto ψ ∗ and ϕ ∗ we get the system of equations, where thefirst one we will refer to as the NLSE-7: i (cid:126) ∂ψ∂t = − (cid:126) m ∇ ψ + U | ϕ | ψ ++( − V | ψ | − V | ψ | + V | ψ | ) ψ − µψ, (7) i (cid:126) ∂ϕ∂t = − (cid:126) M ∇ ϕ + U | ψ | ϕ. (8)Function ϕ is normalized by (cid:82) | ϕ | dV = 1. Away fromthe impurity the fluid wavefunction acquires its groundstate value ψ = ψ ∞ fixing the chemical potential to µ = − V ψ ∞ − V ψ ∞ + V ψ ∞ . For superfluid helium at atmo-spheric pressure ρ ∞ = mψ ∞ = 145 . . The heal-ing length, ξ , is given by the characteristic length-scaleon which fluid heals itself to the unperturbed value fromzero value and is determined by matching the kinetic andthe potential energy of interactions ξ = (cid:126) / √ mµ = 0 . x → ξx , t → ξ m (cid:126) t , ψ → ψ ∞ ψ , ϕ → ξ − / ϕ and numeri-cally integrate it using the 4-th order space discretizationand the 4-th order Runge-Kutta time propagation. Scat-tering processes are modelled in a computational box ofthe size (37 . ξ ) with the resolution of 4 points per heal-ing length ξ . Before the beginning of the dynamical com-putation, the initial guess for ψ and ϕ is optimized usingthe imaginary time evolution for a few time steps . Theinitial kinetic energy is given to the particle by multi-plying its wavefunction ϕ by the factor e i k · r . A typicalone-dimensional cross-section of the fields prepared bythis procedure is shown on Fig. 1, where the initial ve-locity of the particle points towards the vortex along theplotted axis. The figure shows the fluid and the parti-cle amplitudes and oscillating real and imaginary partsof the particle’s wavefunction. Two minima in the fluid’samplitude correspond to the vortex and the depletion dueto the repulsive interactions with the impurity.For comparison, we have also performed computationsusing a simple classical model for the interaction of theparticle with a vortex. It is based on the theory developedin Poole et al. and Sergeev et al. to study the motionof tracer particles in superfluid helium in the presence ofquantized vortex lines. This approach takes into accounta number of forces which are associated with both super-fluid and normal components of superfluid helium. Atsufficiently low temperatures (below 1 K) where the su-perfluid component dominates, this approach reduces tothe Newton equation of motion for the xenon atom withthe dominating effect coming from the Bernoulli’s forcethat appears as the result of the existence of the pressuregradients, produced by the inhomogenious velocity fieldof the vortex. The equation of motion reads M d v p dt = (cid:90) S P ( r )ˆ n dS, (9)where P ( r ) is the superfluid pressure field, ˆ n is the unitvector normal to the surface of the particle S and theintegral is taken over the impurity’s surface which is as-sumed to be spherical with the radius 2 . as it can describe only elastic scatteringand the particle’s motion away from any vortex cores. III. INTERACTIONS OF THE VORTEX WITHA MOVING IMPURITY
It is energetically favourable for a particle to be cap-tured by a vortex since the particle-vortex bindingenergy, ∆ E , defined as the difference between the en-ergy of the system when the vortex and the particle arefar away from each other and the energy of the parti-cle located on the vortex core, is positive. Both energieshave the same logarithmic divergencies linked to the di-vergence of the energies of the vortex velocity field whichfalls as (cid:126) /mr with the distance r away from the vortex.The standard approach to deal with such integrals is to K i n e t i c e n e r g y E p k [ m e V ] ∆ E pk
15 10 5 0 5 10 15X [ ]15105051015 Z [] (b)(a) FIG. 3. (Color online) Time evolution of kinetic energies ofparticles during the scattering events for different initial ve-locities. The insets show (a) the corresponding trajectoriesof the impurity and (b) the corresponding time evolutions ofthe kinetic energy based on the classical Bernoulli’s force cal-culations. Curves in the inset (b) are plotted in the sametime/energy window as the main figure. introduce a finite radius, R , of integration, which givesthe energy of the vortex line in the NLSE-7 to be (seeAppendix B for derivation) E vort = Lπψ ∞ (cid:126) m ln (cid:18) . Rξ (cid:19) , (10)where L is the length of the straight line vortex. Thelogarithmic divergencies of the vortex-particle complexand the vortex line cancel out to give proper integralsthat are evaluated numerically to give ∆ E = 0 .
19 meVfor the xenon atom and ∆ E = 6 .
46 meV for the elec-tron bubble, see Table I. We have also considered vari-ous energy contributions to the binding energy to showthat the main contribution comes from the kinetic en-ergy: when trapped the impurity replaces a significantvolume of circulating fluid . The second contributionto ∆ E comes from the zero-point energy of the par-ticle E p (the particle doesn’t move and there is no ki-netic energy component in E p ). It is connected with theconfinement radius of its wave function and the uncer-tainty principle. Since the vortex core is ”hollow” insideit provides a weaker confinement than the bulk helium,decreasing the uncertainty in momentum and the zero-point energy. The density of the xenon atom captured bythe vortex has an ellipsoidal shape, in contrast with thespherical shape in the bulk. Changing of the form andstaying inside the vortex core rearranges the healing layerbetween the particle and the fluid, which decreases thehealing energy E h as well. The internal energy change∆ E int is negligible. TABLE I. Binding energies ∆ E of the xenon atom and the electron attached to the quantized vortex. Values are obtainedusing stationary numerical computations. Corresponding energy terms and their contributions to the total binding energy areshown. ∆ E , meV ∆ E kin , meV ∆ E int , meV ∆ E h , meV ∆ E p , meVXenon 0.19 0.10 (53%) 0.01 (5%) 0.03 (16%) 0.05 (26%)Electron 6.46 5.48 (85%) 0.53 (8%) 0.03 ( < E n e r g y [ m e V ] d = 0 d = 0.92 d = 1.84 d = 2.76 d = 3.68 d = 5.52
15 10 5 0 5 10 15X [ ] 15105051015 Z [] B i n d i n g E n e r g y [ m e V ] (b)(a) displaced liquidzero point energy FIG. 4. (Color online) Kinetic energies of particles scatteredwith different values of the impact parameter d . The insetsshow (a) the corresponding trajectories of the impurity andof the vortex core and (b) the inelastic energy loss ∆ E fordifferent d . Numerical estimations of the two main contribu-tions to ∆ E are shown with dashed and dotted lines in (b). When the xenon atom approaches the vortex core andgets trapped it releases a portion of energy ∆ E . In com-parison with a weakly interacting condensate modelledby the GPE a system described by the NLSE-7 is notas compressible, so only a negligible amount of energyis converted into sound waves . The dominant effectis the generation of the Kelvin waves along the vortexline, carrying the excess energy away from the interac-tion site . The emission of the Kelvin waves plays animportant role during the scattering of xenon atoms onvortices, when particles possess some initial kinetic en-ergy. If the particle’s kinetic energy is large enough theparticle may pass through the vortex. Since some por-tion of the full energy stay locked in the Kelvin waves,the impurity should sacrifice the same amount of its ki-netic energy, and slow down or get trapped. This makesthe particle-vortex scattering a purely inelastic process.It has a certain resemblance to the well-studied inelasticscattering of electrons on molecules, where vibrationalmodes of the molecule may accept a certain portion of en-ergy, keeping the electron trapped for a long time . The difference between our case and the scattering ofelectrons on molecules is that the spectrum of the Kelvinwaves is continuous (while molecular electronic andvibrational spectra are discrete) and the particle-vortexinteraction is likely to be non-resonant. A discrete spec-trum can be introduced in our system by considering anarrow channel where the vortex line is pinned by thecontainer walls and therefore only certain wavelengths ofthe Kelvin waves can be exited.First we consider a head-on collision of the impuritywith the vortex line. On Fig. 2 we present the visualiza-tion of two scattering processes with (the top row) andwithout (the bottom row) trapping. The vortex line isinitially located along the vertical axis. The particle isplaced 11 ˚A away from the vortex, with the initial velocitydirected towards the vortex. The left panels in each rowshow the trajectory of the particle (blue line), recordedat the position of particle’s density maximum. Motionof a selected point of the vortex core (slightly above theparticle) is shown by the grey line. The maximum am-plitude of the Kelvin waves generated is approximately1 ˚A in this case. The Kelvin waves appear in both caseswhether or not the trapping took place. If the particle de-taches from the vortex core we also detect its vibrationalmotion . Other panels on Fig. 2 illustrate the dynamicsof the vortex interactions with the impurity via the timesnapshots of the absolute value of the matter field | ψ | .On Fig.3 the kinetic energy of the particle E pk = (cid:126) M (cid:20) Im (cid:90) ϕ ∗ ∇ ϕdV (cid:21) (11)is shown as the function of time for the different ini-tial velocities of the impurity. On Fig. 3(a) we presentcorresponding trajectories of the impurity. The initialposition of the vortex is shown with a black dot. Inthree cases out of five, which correspond to lower ini-tial energies, the xenon atom gets trapped. Figure 3(b)shows the results obtained for the same initial configura-tions using the Bernoulli force based classical approachas described in the previous section. Such a minimalisticmodel draws a purely elastic scattering picture in a cen-trally symmetric potential. The Bernoulli’s force causesparticle to accelerate when it approaches the vortex andto decelerate when it moves away from it. The widthand position of the resulting peak depends on the ini-tial velocity. There are obvious similarities with NLSE-7results with respect to the positions of peaks which in-dicates that the Bernoulli force accurately describes thedynamics of particles outside of the interaction region(where the separation between the impurity and the vor-tex is larger than 5˚A, according to our simulations). Theheight of the peaks is higher in classical computations,as the energy in our model is being continuously redis-tributed between various terms. The trapped particlesoscillate around the vortex core along elliptic trajectorieswith an amplitude of about 5 ˚A. Their kinetic energy timedependence contains multiple maxima as it is shown onFig.3. During such motion the particle’s energy contin-uously dissipates and the amplitude of peaks goes downin time. It is accompanied by the increase in the heal-ing energy while no further increase in the Kelvin wavesamplitude is detected. The time evolution for trappedparticles is computed for 1 ns to ensure that the particledoes not detach.When the particle does not become trapped there isan energy drop ∆ E pk ≈ . E pk withinthe accuracy of the simulation coincides with the bindingenergy ∆ E which shows that the portion of energy equalto ∆ E is being transferred to the Kelvin waves duringthe interaction causing the drop in the kinetic energy ofthe particle. If the xenon initial energy is lower than ∆ E it can not escape and gets trapped by the vortex line.This value defines the capture criteria for xenon atomsby vortices in superfluid helium at low temperatures.In some regimes we observe the splitting of the particlewave function, ϕ , between two spacial locations. Duringthe detachment, small part of the particle wave functionmay remain attached to the vortex. This reflects theprobabilistic nature of the process, and is interpreted asthe existence of some finite probability of the particle toget captured even at high energies. In cases which arecharacterized as scattering regimes with no trapping thisprobability is usually less than 5 percent (defined as theportion of the trapped mass of the particle). We stressthat despite the fact that the superfluid is modelled interms of classical fields, for the particle we have usual lin-ear Shr¨odinger equation, which describes quantum effectstypical for a particle in a potential well. Nevertheless, theinteraction picture in our models is more classical thanquantum, there exists a sharp border between attach-ment and detachment regimes.Next we consider the scattering and trapping of thexenon atom which is off-set from a vortex line in the di-rection of its motion. As was shown for the head-on colli-sion, the main contribution to the binding energy comesfrom ∆ E kin (see Table I), which represents the kineticenergy of superfluid displaced from the vortex velocityfield by the particle. The value of ∆ E kin is expected tobe smaller than the one in Table I if the particle is placedat a certain distance from the vortex core, since the su-perfluid velocity decreases with this distance. It shouldbe reflected in scattering events when the particle passesat a certain distance from the core. On Fig. 4 we presentresults for different values of an impact parameter d (the minimal distance between the straight line trajectory ofthe particle in the absence of the vortex and position ofthe vortex core). The particle trajectories are plottedon Fig. 4(a). It is clearly seen how the inelastic energydrop, ∆ E pk , decreases with d . This dependence is plot-ted on Fig. 4(b) by the solid line. We have shown abovethat for the head-on collision ∆ E pk coincides with thebinding energy ∆ E . It is impossible to use the samemethod of evaluation for ∆ E when d (cid:54) =0, since such con-figurations are not steady. In Fig. 4 (b) we show roughnumerical estimations how ∆ E kin (dashed line) and E p (dotted line) depend on the distance from the core. E p is associated with the zero-point energy variation dur-ing the interaction, and not with the kinetic energy ofthe particle. Their sum constitute almost 80% of ∆ E .This analysis again points out that the effective radiusof interaction for xenon atoms and quantized vortices inhelium is about 5 ˚A (see Fig. 4(b)).The theory described in this manuscript can be easilyextrapolated to other types of particles. To illustrate,in Table I we compare binding energies ∆ E with corre-sponding components for the xenon and an electron. Thefraction of ∆ E kin in the binding energy is much lager forthe electron than for the xenon in the view of the largeradius of the electron bubble as compared to the xenonradius and therefore larger volume of displaced fluid. Thevalue of ∆ E kin obtained here for the electron is close tothe one obtained using the GPE . For the basic analy-sis of the electron capture we may assume ∆ E ≈ ∆ E kin and compute it using the model suggested by Parks andDonnelly . CONCLUSION
In this manuscript we studied the inelastic scatteringof xenon atoms on quantized vortices in liquid helium.The theoretical framework based on the modified versionof the self-trapping wave function approach is used tomodel the dynamics of the vortex-particle interactions.It is argued that NLSE-7 as a model of superfluid he-lium is mathematically analogous to the Landau two-fluid model and in this sense can be used to model thedynamical effects in superfluid helium. It is shown thatKelvin waves are excited along the vortex filament duringthe interaction with a particle whether or not the particleis trapped at the vortex core, keeping a certain portionof energy and providing a mechanism for the inelastictrapping or scattering of particles. The simple capturecriteria for xenon atoms is formulated. It states that inhead-on collisions the particle is captured if its kineticenergy is less than the binding energy, which is equal to0.2 meV for xenon. For the nonzero impact parameter d the capture criteria becomes weaker and starting from d ≈ Appendix A: Derivation of the Landau two-fluidmodel from classical field equations
The idea to use classical fields approximation to modelsuperfluid helium can be traced back to the works ofPutterman and Roberts . Using the scale separationin GPE they derived an equivalent set of kinetic equa-tions which describe both the condensate and the ther-mal cloud, as well as their interaction, so the classicalfield ψ is no longer directly associated with the conden-sate. Instead, the separation of scales leads to associa-tion of the slowly varying, large-scale, background fieldwith the superfluid component, and the short, rapidlyevolving excitations with the normal component. There-fore, ψ in this context gives rise to both components.This result allows one to generalize the classical fieldapproach and perform finite temperature GPE basedcomputations . Another important step in this direc-tion was made by demonstrating the equivalence of GPEand the Landau two-fluid model using the local gaugetransformation . Gauge field in this case is relatedto additional macroscopic degrees of freedom and allowsone to switch from one-fluid to two-fluid system. In thissection we use the similar procedure to demonstrate theequivalence of NLSE-7 and Landau two-fluid model.The Lagrangian density for NLSE-7 reads L = i (cid:126) (cid:104) ψ ˙ ψ ∗ − ψ ∗ ˙ ψ (cid:105) + (cid:126) m |∇ ψ | − V | ψ | − V | ψ | + V | ψ | . (A1)We apply the local gauge transformation ψ → ψe iα ( r ,t ) m/ (cid:126) , which provides 4 additional independentvariables for the nonzero temperature two-fluid modeldescription. Newly introduced scalar and vector fieldsare denoted as ξ ≡ ˙ α ( r , t ), A ≡ −∇ α ( r , t ). They appearas additional terms in the Lagrangian L = L + mξ | ψ | + m A | ψ | − (cid:126) i A · [ ψ ∗ ∇ ψ − ψ ∇ ψ ∗ ] (A2)Switching to hydrodynamic variables ρ and φ such that ψ = (cid:114) ρ ( r , t ) m e iφ ( r ,t ) m/ (cid:126) , (A3)we get L = ρ ˙ φ + (cid:126) m ρ ( ∇ ρ ) + ρ ∇ φ ) + (cid:26) − V ρ m − V ρ m + V ρ m (cid:27) , (A4) L = L + A ρ ρξ − ρ A · ∇ φ. (A5) According to Coste we link scalar and vector fields withphysical variables in a following way ξ = η ( ρ, s ) + v n · A , (A6) A = χ ( ρ, s )( ∇ φ − v n ) , (A7)where χ and η are Galilean invariant scalars which arefunctions of density and entropy only. Thus, the newvariables which we add to the model are the normal fluidvelocity v n and the entropy s . The Lagrangian reads(curly brackets are used to highlight the nonlinear partof NLSE-7) L = ρ ˙ φ + ρ ∇ φ ) + (cid:126) m ρ ( ∇ ρ ) + (cid:26) − V ρ m − V ρ m + V ρ m (cid:27) + ρη + ρχ v n · ( ∇ φ − v n )+ ρχ χ − ∇ φ ) + ρχ (1 − χ ) ∇ φ · v n + ρ χ v n (A8)The Euler-Lagrange equation for φ is ∂ L ∂φ − ∇ ∂ L ∂ ( ∇ φ ) − ∂∂t ∂ L ∂ ˙ φ = 0 . (A9)Substituting L and computing derivatives we get ∂ρ∂t + ∇ · (cid:2) v n ρχ (2 − χ ) + ∇ φρ (1 − χ ) (cid:3) = 0 . (A10)Recalling that v s = ∇ φ and introducing notations ρ (1 − χ ) = ρ s and ρχ (2 − χ ) = ρ n we obtain the first equationof Landau’s model (the equation for mass conservation).The second Landau equation (the equation for thesuperfluid velocity) is derived from the Euler-Lagrangeequation for ρ (one should recall that both χ and ξ arefunctions of ρ ) ∂φ∂t + 12 ( ∇ φ ) + ˜ µ = (cid:126) m (cid:20) ( ∇ ρ ) ρ + ∇ ρ ρ (cid:21) , (A11)where˜ µ ≡ η + ρ ∂η∂ρ + (cid:26) − V m ρ − V m ρ + V m ρ (cid:27) − (cid:20) ρ (1 − χ ) ∂χ∂ρ + χ (2 − χ ) (cid:21) ( v n − v s ) . (A12)The difference of this result with the one obtained inSalman et al. is contained in ˜ µ . Polynomial functionof ρ in curly brackets appears instead of single linearterm in GPE. This doesn’t change the main logic of theoriginal derivation.Remaining two equations of the two-fluid model shouldbe derived from additional constraints which appear asLagrange multipliers in L and correspond to the conser-vation of entropy and relative fluid velocity. This part ofthe derivation is the same for GPE and NLSE-7 . η f ( η ) GPENLSE-7
FIG. 5. Dimensionless radial part f of vortical solutions inNLSE-7 and GPE models as a function of dimensionless co-ordinate η . The superfluid density is given by n = f ψ ∞ Appendix B: Energy of the vortex in NLSE-7
Stationary NLSE-7 reads − (cid:126) m ∇ ψ + ( − V | ψ | − V | ψ | + V | ψ | ) ψ − µψ = 0 . (B1)We switch to cylindrical coordinates ( r ,Φ, z ) and searchthe vortical solution in the form ψ = e i Φ | ψ ( r ) | . (B2)Using the dimensionless units such that | ψ | = f ( η ) ψ ∞ and r = ηξ along with definitions of chemical potential µ and healing length ξ given in the section II we can derivethe following equation for the radial part f of the vortical solution of Eq. (B2)1 η ddη (cid:18) η dfdη (cid:19) − (cid:18) η + 1 (cid:19) f + c f + c f − c f = 0 , (B3)where c = 2 . c = 2 . c = 3 . f (0) = 0, df (0) dη = k . Parameter k is chosen to fulfil another known physical boundarycondition f ( ∞ ) = 1. The resulting function is plotted onFig. 5 along with the vortex amplitude of the GPE forcomparison.The energy of the vortex is given by the full Hamilto-nian of the system, where ψ represents the vortex solutioncomputed above E v = (cid:90) (cid:18) (cid:126) m |∇ ψ | − V | ψ | − V | ψ | ++ V | ψ | − µ | ψ | (cid:19) dV − E gs . (B4)The ground state energy E gs is given by the Eq. (1)with ψ = ψ ∞ . Substituting the solution of Eq. (B2) andusing dimensionless variables as above we can expressthis integral in terms of fE v = πLψ ∞ (cid:126) m R/ξ (cid:90) (cid:40)(cid:18) dfdη (cid:19) + (cid:18) η + 1 (cid:19) f − c f − c f + c f − c (cid:111) ηdη, (B5)where c = 0 . L andradius R .If we consider large R (cid:29) ξ this formula can be signif-icantly simplified, since f → R . We simplyconsider f = 1 when R > a , where a is some constant.The integral splits into two parts and the second one canbe taken analytically. The resulting formula reads E v = πLψ ∞ (cid:126) m ln (cid:18) . Rξ (cid:19) . (B6)The numerical coefficient 1 .
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