How trapped particles interact with and sample superfluid vortex excitations
HHow trapped particles interact with and sample superfluid vortex excitations
Umberto Giuriato and Giorgio Krstulovic
Universit´e Cˆote d’Azur, Observatoire de la Cˆote d’Azur, CNRS, Laboratoire Lagrange, Nice, France
Sergey Nazarenko
Universit´e Cˆote d’Azur, Institut de Physique de Nice, CNRS, Nice, France
Particles have been used for more than a decade to visualize and study the dynamics of quantumvortices in superfluid helium. In this work we study how the dynamics of a collection of particles setinside a vortex reflects the motion of the vortex. We use a self-consistent model based on the Gross-Pitaevskii equation coupled with classical particle dynamics. We find that each particle oscillateswith a natural frequency proportional to the number of vortices attached to it. We then studythe dynamics of an array of particles trapped in a quantum vortex and use particle trajectoriesto measure the frequency spectrum of the vortex excitations. Surprisingly, due to the discreetnessof the array, the vortex excitations measured by the particles exhibits bands, gaps and Brillouinzones, analogous to the ones of electrons moving in crystals. We then establish a mathematicalanalogy where vortex excitations play the role of electrons and particles that of the potential barriersconstituting the crystal. We find that the height of the effective potential barriers is proportionalto the particle mass and the frequency of the incoming waves. We conclude that large-scale vortexexcitations could be in principle directly measured by particles and novel physics could emerge fromparticle-vortex interaction.
I. INTRODUCTION
When a fluid composed of bosons is cooled down, aspectacular phase transition takes place. The system be-comes superfluid and exhibits exotic physical properties.Unlike any classical fluid, a superfluid flows with no vis-cosity. This is an intriguing example of the manifestationof pure quantum mechanical effects on a macroscopiclevel. The first discovered superfluid is liquid helium He in its so-called phase II, below the critical temper-ature T λ (cid:39) .
17 K. In one of the first attempts of de-scribing the behavior of superfluid helium, London sug-gested that superfluidity is intimately linked to the phe-nomenon of Bose-Einstein condensation (BEC) [1]. Inthe same years, Landau and Tisza independently put for-ward a phenomenological two-fluid model, wherein super-fluid helium can be regarded as a physically inseparablemixture of two components: a normal viscous componentthat carries the entire entropy and an inviscid componentwith zero entropy [2, 3].Because of its intrinsic long-range order, a superfluidcan be described by a macroscopic complex wave func-tion. A stunning quantum-mechanical constraint is thatvortices appear as topological defects of such order pa-rameter. In three dimensions, such defects are unidimen-sional structures, usually referred to as quantum vor-tices. Indeed, the circulation (contour integral) of theflow around a vortex must be a multiple of the Feynman-Onsager quantum of circulation h/m , where h is thePlanck constant and m is the mass of the Bosons con-stituting the fluid [4]. Such peculiarity is necessary toensure the monodromy of the wave function. In super-fluid helium, quantum vortices have a core size of theorder of an Angstrom. At low temperatures, below 1 K,the normal component is negligible and vortices are sta-ble and do not decay by any diffusion process, unlike their classical counterparts. The understanding of superfluidvortex dynamics has a direct impact on many interesting,complex non-equilibrium multi-scale phenomena, such asturbulence [5–7].Most of the experimental knowledge on superfluid vor-tices is based on indirect measurement of their proper-ties. The early efforts in the observation of quantizedvortices were made in the framework of rotating super-fluid helium, by using electron bubbles (ions) as probes[8]. Since then, impurities have been extensively used tounveil the dynamics of superfluid vortices. An importantbreakthrough occurred in 2006, when micrometer sizedhydrogen ice particles were used to directly visualize su-perfluid helium vortices [9]. Thanks to pressure gradi-ents, particles get trapped inside quantum vortices andare subsequently carried by them. Hence, it has been pos-sible to observe vortex reconnections and Kelvin waves(helicoidal displacements that propagate along the vortexline) by means of standard particle-tracking techniques[10]. Furthermore, the particle dynamics unveiled impor-tant differences between velocity statistics of quantumand classical turbulent states [11, 12]. In experiments,such particles are used as tracers, despite their very largesize compared to the vortex core. Therefore, it is of theutmost importance that the mechanisms driving their dy-namics are fully comprehended. Specifically, how well isvortex dynamics reflected by the motion of the particlestrapped in it? How much do their presence in the coremodify the propagation of Kelvin waves? Would theyaffect the reconnection rates?Describing the interaction of particles with isolatedvortex lines or complex quantum vortex tangles is notan easy task. Depending on the scale of interest, thereare different theoretical and numerical models that canbe adopted. A big effort has been made in adapting thestandard dynamics of particles in classical fluids to the a r X i v : . [ c ond - m a t . o t h e r] M a r case of superfluids described by two-fluid models [13, 14].This is a macroscopic model in which vorticity is a coarse-grained field and therefore there is no notion of quantizedvortices. A medium-scale description is given by the vor-tex filament model, where the superfluid is modeled asa collection of lines that evolve following Biot-Savart in-tegrals. In this approximation, circulation of vortices isby construction quantized but reconnections are absentand have to be implemented via some ad-hoc mechanism.Finite size particles can be studied in the vortex fila-ment framework but the resulting equations are numer-ically costly and limited [15]. A microscopic approachconsists in describing each impurity by a classical fieldin the framework of the Gross-Pitaevskii model [16–18].In principle, such method is valid for weakly interactingBECs, and is numerically and theoretically difficult tohandle if one wants to consider more than just a few par-ticles. In the same context, an alternative possibility isto assume classical degrees of freedom for the particles,while the superfluid is still a complex field obeying theGross-Pitaevskii equation. This idea of modelling par-ticles as simple classical hard spheres has been shownto be both numerically and analytically very powerful[19–22]. In particular, such minimal and self-consistentmodel allows for simulating a relatively large number ofparticles, and describes well the particle-vortex interac-tion [22]. Although formally valid for weakly interactingBECs, it is expected to give a good qualitative descrip-tion of superfluid helium.In this paper we investigate how particles trapped inquantum vortices interact with vortex excitations and inparticular how well they can be used to infer propertiesof superfluid vortices. We use the Gross-Pitaevskii equa-tion coupled with inertial and active particles obeyingclassical dynamics to answer this question. We first ad-dress how the Magnus force acting on trapped particlesinduces oscillations at a certain natural frequency. Thisquantity may be experimentally measured to determinethe number of vortices composing a polarized bundle (seea discussion later in this paper). Secondly, in order tounderstand the effect of particle inertia, we analyze thespectrum of vortex excitations in the case when a contin-uous distribution of mass is contained inside the vortexcore. Then, we study an array of particles trapped in-side a vortex, in a setting similar to the one observedin experiments. Surprisingly, the dispersion relation ofvortex waves measured by the particles is found to con-tain band gaps and the periodicity typically observed inthe energy spectra of solids. We explain the numericalobservation applying the concepts used in the standardKronig-Penney model [23, 24], that describes the motionof electrons in a unidimensional crystal. Finally, basedon our results, we discuss in which regimes particles couldbe reliably used to sample vortex excitations. II. THEORETICAL BACKGROUNDA. Model for superfluid vortices and activeparticles
We consider a superfluid at very low temperature con-taining N p spherical particles of mass M p and radius a p .We describe the system by a self-consistent model basedon the three-dimensional Gross-Pitaevskii equation. Theparticles are modeled by strong localized potentials V p ,that completely deplete the superfluid up to a distance a p from their center position q i . Particles have inertiaand obey a Newtonian dynamics. The Hamiltonian ofthe system is H = (cid:90) (cid:126) m |∇ ψ | + g | ψ | + N p (cid:88) i =1 V p ( r − q i ) | ψ | d r + N p (cid:88) i =1 p i M p + N p (cid:88) i
Excitations are present in quantum vortices because ofthermal, quantum or turbulent fluctuations. They arewaves propagating along the vortex line with a certainfrequency Ω v ( k ), where k is the (one dimensional) wavenumber of the excitation. At scales larger than the vor-tex core size ( kξ (cid:28) v s . i . of the line on itself [8]. .This model involves non-local contributions and a singu-lar integral that needs to be regularized [27]. Note thatthis model has also been derived at large scales also inthe framework of the GP equation [28]. The simplestapproximation that can be done is the well known LocalInduction Approximation (LIA), where only the contri-bution to v s . i . due to the local curvature at each point of the filament is considered. Such approximation is validwhen the curvature is much larger than the vortex coresize. The LIA model reads [29]˙ s ( ζ, t ) = v s . i . ( ζ, t ) , v s . i . ( ζ, t ) = Γ4 π Λ ∂ s ∂ζ × ∂ s ∂ζ , (7)where s ( ζ, t ) is the curve that parametrizes the filament, ζ is the arc-length. The parameter Λ > z -axis, the vortexline can be parametrized as s ( z, t ) = s x ( z, t ) + is y ( z, t ).At the leading order (7) reduces to˙ s ( z, t ) = v s . i . ( z, t ) ,v s . i . ( z, t ) = i Γ4 π Λ ∂ ∂z s ( z, t ) . (8)The LIA equation (8) admits solutions in the form ofhelicoidal waves propagating along the vortex line witha dispersion relationΩ LIA ( k ) = − ΓΛ4 π k . (9)A better description of vortex waves was formally derivedfrom the Euler equations for an ideal incompressible fluidby Sir W. Thomson (Lord Kelvin) [30] in the case ofa hollow vortex, namely if the vorticity is concentratedin a thin tube of radius a . In this case the frequencyof propagation is given by the well known Kelvin wavedispersion relationΩ KW ( k ) = Γ2 πa (cid:34) − (cid:115) a | k | K ( a | k | ) K ( a | k | ) (cid:35) (10)where K n ( x ) is the modified Bessel function of order n and a depends on the model of the vortex core. It hasbeen shown by Roberts [31] that the small wave numberlimit of expression (10) is valid also for large-scale wavespropagating along the superfluid vortex described by theGP equation:Ω v ( k ) −→ kξ (cid:28) Ω KW ( ka →
0) = − Γ4 π k (cid:18) ln 2 a | k | − γ E (cid:19) , (11)where a = 1 . ξ and γ E ∼ . v ( k ) −→ kξ (cid:29) − Ω B ( kξ → ∞ ) = − Γ4 π k , (12)Note that all the frequencies (9-12) have an opposite signwith respect to the circulation Γ, namely KWs rotate op-posite to the vortex flow v v . Since there is not an ana-lytic expression for the full dispersion relation of vortexexcitations of the GP model, in the numerics presentedin this work we use a fit of the dispersion relation thatmatches both asymptotic (10) and (12). It readsΩ fitv ( k ) =Ω KW ( k ) (cid:18) (cid:15) ( a | k | ) + (cid:15) ( a | k | ) + 12 ( a | k | ) (cid:19) . (13)The dimensionless parameters (cid:15) = − .
20 and (cid:15) = 0 . k limit Ω KW ( k ) ∼ Γ2 πa ( a | k | ) can be straightforwardlyadjusted to obtain the free particle dispersion relation(12) (dash-dotted magenta line). − kξ ω τ (a) (b) Ω fitv ( k )Ω KW ( k ) Γ4 π k log( b | k | ) − Γ4 π k .
00 0 . kξ . . . − FIG. 1. (a)
Spatiotemporal spectrum of a GP bare vortexloaded with small amplitude Kelvin waves. Solid green line isthe fit (13). Dashed cyan line is KW dispersion relation (10).Dotted yellow line is the small- k asymptotic (11), with b = a e γ E /
2. Magenta dash-dotted line is tha large k asymptotic(12). The resolution of the simulation is N ⊥ = N (cid:107) = 256 in acomputational domain of size L ⊥ = L (cid:107) = 256 ξ . (b) A zoomclose to small wave numbers.
III. MOTION OF PARTICLES TRAPPED BYQUANTUM VORTEX
We are interested in the behavior of particles capturedby quantum vortices. Since hydrogen and deuterium par-ticles used to visualize vortices in superfluid helium ex-periments are considerably larger than the vortex core(typically a p ∼ ξ ) they could be captured not by anisolated vortex but by bundles of many polarized vor-tices. In such complex system, the large particle size and inertia might affect the vortex dynamics. It is then nat-ural to try to understand how the dynamics of vorticesis modified by the presence of the particles, or in otherterms, how well particles track superfluid vortices.An amazing experimental evidence is that trapped par-ticles distribute themselves at an almost equal spacing(see for instance Ref. [10]). In this work we do notaddress the physical origins of this distribution, but weadopt it as a hypothesis for setting the initial conditionof our simulations.We start our discussion by presenting the settings ofthe GP-P model in our simulations. The GP-P equa-tions are integrated in a 3D periodic domain of dimen-sions L ⊥ × L ⊥ × L (cid:107) . The initial conditions consist ina perturbed straight vortex containing small amplitudevortex excitations. The vortex is loaded with a num-ber of particles and then evolved under GP-P dynamics.The computational domain contains three other imagevortices in order to preserve periodicity. Only one vor-tex contains particles whereas the three other are bare.We have used resolutions up to 256 × × collocation points. We express the particle massas M p = M M , where M is the mass of the displacedsuperfluid. Therefore, light, neutral and heavy particleshave M < M = 1 and M > ξ , times in units of τ = ξ/c andvelocities in units of c . Further details on the numericalimplementation are given in Appendix A.Figure 2 displays the four different configurations stud-ied in this work. Figure 2.a shows one particle moving in (a)
Asingle particle of size a p = 13 . ξ trapped in a vortex filament. (b) An array of particles of size a p = 13 . ξ and relative dis-tance d = 51 . ξ . (c) A wire made of 50 overlapping particlesof size 2 . ξ trapped in a vortex filament. (d) An array ofparticles of size a p = 13 . ξ trapped in a bundle of 4 vor-tex filaments. Movies of the simulations can be found in theSupplemental Material. a quantum vortex which clearly induces KWs on the fil-ament. Figure 2.b displays an array of particles initiallyset at equal distances. We have checked that providedthat particles are distant enough, they remain equallydistributed along the vortex, with very small fluctuationsalong its axis. Figure 2.c displays a snapshot in the casewhere particles strongly overlap creating an almost con-tinuous distribution of mass inside the vortex. Produc-ing this state is possible by properly adjusting the repul-sive potential V ij rep in Eq. (3). The purpose of studyingthis configuration is two-fold. First, from the theoreticalpoint of view it will provide an easier way to describethe role of the particle mass in the vortex dynamics andits effect on vortex excitations. On the other hand, suchsetting is similar to recent experiments that study thenanowire formation by gold nano-fragments coalescenceon quantum vortices [32] or experiments with vibratingwires inside quantum vortices in superfluid He and He[33, 34]. Finally, Fig.2.d displays a bundle of four equallycharged vortices loaded with an array of particles. Inall cases, we clearly see the interaction between parti-cles and vortices producing sound (phonon) and Kelvinwaves. Movies of the simulations can be found in theSupplemental Material.
A. Natural frequency of particles trapped bysuperfluid vortices
We first consider the dynamics of a particle trapped byan almost straight superfluid vortex. At the leading orderthis is the classical hydrodynamical problem of a movingsphere with non-zero circulation in an ideal fluid. Themain force acting on the particle is the Magnus force, thatarises from the pressure distribution generated at theboundary of the particle in such configuration [35, 36].We introduce the complex variable q ( t ) = q x ( t ) + iq y ( t )for the center of the particle in the plane orthogonal tothe vortex filament, and v = v x + iv y for the velocity ofthe ambient superfluid flow. In these variables, the equa-tion of motion for the particle in absence of any externalforce is [36]¨ q ( t ) = i Ω p ( ˙ q ( t ) − v ) , Ω p = 32 ρ Γ a p M effp , (14)where M effp = M p + M = ( M + ) M is the effectivemass of the particle and M = πρa is the displacedmass of the fluid. In equation (14), the fluid is assumedto be incompressible with density ρ ∼ ρ ∞ , which is agood approximation when the particle size is larger thanthe healing length. From (14) we can derive the temporalspectrum of the particle position | ˆ q ( ω ) | = Ω | ˆ v ( ω ) | ω ( ω − Ω p ) (15)where ˆ q ( ω ) = (cid:82) q ( t ) e − iωt d t and ˆ v ( ω ) = (cid:82) v ( t ) e − iωt d t .The vortex line tension, which is responsible for the prop-agation of Kelvin waves [37], is implicitly contained in the superfluid flow v in Eq. (14). It generates particleoscillations in the rotation direction opposite to the flowgenerated by the vortex. However, from Eq. (15) wesee that the particle motion is dominated by a preces-sion with frequency Ω p , which has the same sign of Γand therefore has the same direction of the vortex flow.Such frequency is the natural frequency of the particle:expressing it as a function of M we get:Ω p = 94 π Γ a (2 M + 1) . (16)For current experiments using particles as probes, suchcharacteristic frequency is of order 10–100Hz, which isactually measurable [38].We have performed a series of numerical experimentswith particles trapped in a superfluid vortex excited withsmall amplitude Kelvin waves. Measurements of tempo-ral spectra (15) for particles characterized by differentvalues of Ω p are reported in Fig.3. In the x–axis of theplot we have the angular frequencies with the same sign ofΓ. The different natural frequencies have been obtained !/ ⌦ theoryp | ˆ q ( ! ) | / | ˆ q ( ⌦ p ) | a p = 13 . ⇠ M = 1 † a p = 7 . ⇠ M = 1 a p = 13 . ⇠ M = 1 †⇤ a p = 5 . ⇠ M = 1 † a p = 7 . ⇠ M = 0 . a p = 2 . ⇠ M = 2 a p = 2 . ⇠ M = 1 | ⌦ theoryp | .
00 0 .
15 0 . | ⌦ theoryp | ⌧ . . . ⌦ p / ⌦ t h e o r y p FIG. 3. Temporal spectra of the particle positions for differentvalues of the natural frequency Ω p , obtained varying mass andsize of the particles. The expected natural frequency | Ω theoryp | (16) is the dotted vertical line. Inset : Comparison of themeasured natural particle frequency with the theory. † : theparticle considered belongs to a particle array. ∗ : the particleconsidered is trapped in a bundle of 4 vortices. varying the mass and the size of the particles. The ob-served peak at Ω p is well predicted by Eq. (15). Thenatural frequency is also observed for particles in theparticle-array configuration. In particular, if particlesare attached to a bundle of N v quantum vortices insteadof a single filament, the corresponding characteristic fre-quency is N v times larger. The case of a bundle of N v = 4is also reported in Fig.3, in a remarkable agreement withtheory. This has an important experimental implication.Measuring the natural frequency Ω p could give an in-dependent estimate of the circulation (and therefore ofthe number of vortices) in the bundles visualized by theparticles in superfluid helium experiments.Note that in general the vortex line tension could havea non-trivial coupling with the particles and lead to amodification of the precession frequency Ω p . Indeed, inthe idealized derivation of Eq. (14), it is assumed thatthe particle center coincides with the center of a straightvortex line. In principle, one should solve Eq. (14) to-gether with the equation of motion of the vortex, takinginto account the proper boundary conditions between asphere and a vortex filament [27], that will include restor-ing forces maintaining the particle trapped. Accountingfor such phenomena might lead to a more accurate pre-diction of the precession frequency. However, the GPsystem naturally contains all these effects. Therefore,given the agreement between the prediction (16) and GPnumerical simulations, we conclude that the modificationof the particle natural frequency Ω p due to the couplingat the particle-vortex boundary is a negligible effect. Thesimple formula (16) can be thus safely used as a first es-timate in current experiments. B. Dispersion relation of a massive quantum vortex
As already mentioned above, in order to study the dy-namics of an array of particles and their interaction withvortex waves in a setting like Fig.2.b or Fig.2.d, It isinstructive to first analyze the case of a massive quan-tum vortex, as the one in Fig.2.c. Our considerations arenecessary to give a picture of the role of inertia in thepropagation of vortex wave excitations. They are notmeant to model a real wire, for which some results arewell known in literature [39, 40] and has been used tomeasure the quantized circulation in superfluid helium[41, 42]. We consider a wire of length L w , radius a w and mass M w , filling a superfluid vortex. The effectivemass is M effw = M w + M and the displaced mass is now M = ρL w πa . Since such wire possesses a circulation,each mass element is driven by Magnus force as in Eq.(14), but with a different prefactor [35]Ω w = ρ Γ L w M effw , (17)which arises because of the geometrical difference be-tween a spherical particle and a cylinder. We allow thewire to deform, that means that the complex variable q is now a function of the z component too. Such physicalsystem is analogous to a massive quantum vortex with afinite size core, which is already well known in literature[39, 40], and it has been used to measure the quantizedcirculation in superfluid helium [41, 42]. If the curva-ture radius is much greater than the wire radius and thehealing length, the flow velocity v can be approximatedby the self-induced velocity of the vortex filament on it-self. In the LIA approximation, the self-induced velocityis simply given by v s , i in Eq. (8). The dynamics of the wire is therefore driven by the equation¨ q ( z, t ) = i Ω w (cid:18) ˙ q ( z, t ) − i Γ4 π Λ ∂ ∂z q ( z, t ) (cid:19) . (18)In this simplified model, we are neglecting modes propa-gating along the wire due to elastic tension and the wavenumber dependence of the added mass. This choice isdone because we want to focus on the inertial effectsthat will be relevant in the case of a particle array, de-veloped in the following section. Equation (18) allowsas solution linear circularly polarized waves in the form q ( z, t ) = q e i (Ω ± M t − kz ) , where the frequency is given byΩ ± M ( k ) = Ω w ± (cid:114) Ω + Ω w ΓΛ π k . (19)More generally, one can consider a phenomenological ex-trapolation based on a more realistic model for the self-induced velocity of the vortex in the equation (18), sothat the dispersion relation of waves propagating alongthe wire is generalized asΩ ± M ( k ) = 12 (cid:104) Ω w ± (cid:112) Ω − w Ω v ( k ) (cid:105) , (20)where Ω v ( k ) is the bare vortex wave frequency and de-pend on the model chosen for the self-induced velocity.We will refer to (20) as the “massive vortex wave” dis-persion relation. In the LIA approximation we haveΩ v ( k ) = Ω LIA ( k ) (9) and we recover Eq. (19), but amore accurate result is expected if the wave propagationis instead described by Ω KW ( k ) or by the measured dis-persion relation Ω fitv ( k ) (13). Note that the zero-mode ofthe branch Ω +M coincides with Ω w and does not vanisheven if M w = 0 because of the added mass M . Thisis related to the fact that the wire possesses an effectiveinertia because during its motion it has to displace somefluid [39, 43]. In the limit kξ (cid:28)
1, the result (20) canbe obtained from the one derived in Ref. [40] using fluiddynamic equations to study ions in superfluid helium.We build numerically a massive vortex placing a largenumber of small overlapping particles along a vortex fila-ment. We set the repulsion between particles at a radius r = 2 L w / ( N p a p ) (see Appendix A), so that they arekept at constant distance r /
2. Such system mimics acontinuum of matter with total mass given by the sumof all particle masses M w = N p M p = N p M M . Wehave checked that the repulsion among particles leads tomatter sound waves with frequencies that are sub-leadingwith respect to other terms present in (18). We initiallyexcite the system with small amplitude Kelvin waves andwe let it evolve under GP-P dynamics. Figure 2.c showsa typical snapshot of the system but in the case of alarger initial perturbation (in order to enhance visibility).We then use the particle positions to construct the spa-tiotemporal spectrum S q ( k, ω ) ∼ | ˆ q ( k, ω ) | , with ˆ q ( k, ω )the time and space Fourier transform of q ( z, t ) (see Ap-pendix B for further details). Density plots of S q ( k, ω )are displayed in Fig.4 for different values of the particlemass. For a better presentation, we have chosen Γ < . . . k⇠ . . . ! ⌧ . . . k⇠ . . . ! ⌧ k⇠ . . . ! ⌧ . . . k⇠ . . . ! ⌧ M = 1
1, as expected. The corresponding massivevortex wave predictions (20) are also displayed in greendashed and solid lines. For low masses, the effect of in-ertia is negligible, so that massive vortex wave (20) andbare vortex wave (13) predictions are similar. As the mass increases, the wire inertia becomes important andthe measured frequencies of the wire excitations decreaseat small scales, in good agreement with the massive vor-tex wave prediction. The model (20) is not expected togive a good explanation for the negative branches, as itneglects the details of the internal structure of the wire,as well as the dependence on the wave number of the ef-fective mass. Such features, that are out of the scope ofthe present work, are taken into account in Ref. [40] inthe case of an elastic and massive hollow vortex (with nonotion of the free-particle behavior of vortex excitationsat small scales). The predicted natural frequency of thewire Ω W = | Ω +M (0) | is clearly reproduced by the numer-ical measurements and it does not become infinite when M → kξ ∼ .
8. Note that the KW disper-sion relation (dashed cyan line) seems to be very similarto the fitted one (solid cyan line). However, the differ-ence between the two is apparent in Fig.4.d. Moreover, itis clear how the massive vortex wave dispersion relationcomputed using Ω v ( k ) = Ω fitv ( k ) (solid green line) fits thedata for all the masses analyzed. In particular, in Fig.4.d,it is shown that it can predict the dispersion relation ofa massive vortex wire with relative mass M = 1 up toa wave number kξ ∼ .
7. This is not the case for themassive vortex wave dispersion relation computed usingΩ v ( k ) = Ω KW ( k ) (dashed green line). We thus concludethat the main effect of the inertia of the particles con-stituting the wire is to modify the frequency spectrumof vortex wave, as follows from simple hydrodynamicalconsiderations. C. Frequency gaps and Brillouin zones for an arrayof trapped particles
Now we shall address the main question of this work.How well do particles, seating in a quantum vortex,track vortex waves? In order to study this problem,we consider an array of particles as the one displayedin Fig.2.b. Particles are placed in a quantum vortex,initially separated by a distance d . The system is ex-cited by superimposing small amplitude KWs. We canbuild a discrete spatio-temporal spectrum S q ( k, ω ) of themeasured vortex excitations by using the displacementof particles in the plane perpendicular to the vortex.In Fig.5.a and Fig.5.c we display the particle spatio-temporal spectra for an array of N p = 20 particles ofsize a p = 2 . ξ with masses M = 5 and M = 1 respec-tively, placed at a distance d = 12 . ξ . The Bogoliubovwaves are still weakly sampled by the particles, as dis-played by yellow dotted lines. Surprisingly, a higher fre-quency branch appears. Such pattern is similar to thoseobserved in the typical energy spectra of crystals [24].Particles are actually able to sample the vortex excita-
10 0 10 kd . . . . ! ⌧
10 0 10 kd . . . . ! ⌧ kd . . . . ! ⌧ (d)
10, giving access toall the small scales solved by the numerical simulations.Several Brillouin zones are clearly appreciated, as well asthe opening of band gaps in the dispersion relation. Atthe same time, Bogoliubov modes can be observed andalso bare vortex waves. The latter belong to the imagevortices in the computational domain, where no particleshave been attached.The presence of particles clearly affects the propaga-tion of waves along the vortex line inducing high fre-quency excitations not only for small but also for large wave lengths. The intuitive idea is that when a vortexwave reaches a particle, it is partially reflected or trans-mitted, depending on the mass and the size of the parti-cles, and eventually on its own frequency. This remindsus of the standard quantum-mechanical problem of anelectron described by the (linear) Schr¨odinger equationhitting a potential barrier. Furthermore, if particles areset at almost equal distances, the system is similar toan electron propagating in a periodic array of potentialbarriers, as in the Kronig-Penney model [23, 24]. In or-der to apply quantitatively this intuition and explain theopening of band gaps in the dispersion relation of vortexwave excitations, we start by considering an artificial sys-tem made of segments of bare quantum vortex of length( d − L w ), alternated with massive vortex wires of length L w . A sketch of the problem is given in Fig.6.a. To re- z
0, keeping the mass of thewires equal to the effective mass of the particles. Theresulting effective theory must be intended as an asymp-totic limit of the actual system for long waves ka p (cid:28)
1, inwhich the nonlinear interactions of the vortex excitationsare neglected and the complexity of the vortex–particleboundary is ignored. The accuracy of such model has tobe checked by comparing its predictions with the resultsof the GP simulations. The motion of the bare vortices isdriven by the self-induced velocity that leads to the prop-agation of vortex waves, while the wires are driven by theMagnus force. For the sake of simplicity, we first considerthe LIA approximations (8) and (18) respectively. Thedynamics is thus given in each zone by˙ q ( z, t ) = i Γ4 π Λ ∂ ∂z q ( z, t ) (I)¨ q ( z, t ) = i Ω w (cid:20) ˙ q ( z, t ) − i Γ4 π Λ ∂ ∂z q ( z, t ) (cid:21) (II) (21)where (I) is the region 0 < z < d − L w and (II) is theregion d − L w < z < d . Note that the use of LIA in thesystem (21) is rather qualitative, given the high level ofcomplexity of the problem. In particular it ignores thenonlocal dynamics of the vortex, does not reproduce thegood dispersion relation of vortex excitations and maynot be able to take into account the exact boundary con-dition between the particles and the vortex. However,it allows us to introduce some general physical conceptsand perform a fully analitically treatment of the problem.The effective model will be then generalized in order totake into account a more realistic description of vortexwaves and provide quantitative predictions. The disper-sion relation can be found borrowing standard techniquesfrom solid state physics, in particular by adapting the so-lution of the Kronig-Penny model [23, 24]. We look for awave solution q ( z, t ) = Φ( z ) e iωt , where the spatial func-tion Φ( z ) can be written in the form Φ( z ) = e ikz u ( z ) ac-cording to Bloch theorem, where u ( z ) is a periodic func-tion of period d [44]. The key point is the imposition ofcontinuity and smoothness of the function Φ( z ) as well asperiodicity of u ( z ) and its derivative. These constrainslead to an implicit equation relating the frequency of theexcitations ω , the wavenumber k and all the physical pa-rameters. The full derivation is explained in AppendixC. The last step in order to describe the excitations ofthe particle array, is to take the limit L w → M eff . The dispersion relation is finally determined by theimplicit equationcos( kd ) = cos( α ω d ) − sin( α ω d ) α ω d P ω , (22)where P = 3 πda p / ΛΓΩ p and α ω satisfies the equationΩ LIA ( α ω ) = ω : α ω = (cid:114) − πω ΓΛ . (23)In Figures 6.b-c the r.h.s. of Eq. (22) is plotted as afunction of ωτ for heavy and light small particles (thatis low and high Ω p ). The curve must be equal to cos( kd )and this selects the only allowed frequencies (displayedin gold). It is exactly the same mechanism that leads tothe formation of energy bands in crystals [24].The previous calculations can be directly generalizedfor more realistic wave propagators (see Appendix C). Inparticular, if we consider a dispersion relation Ω v ( k ) forthe vortex excitations, the only change in the result (22)is the functional dependence of α ω (23), that must satisfyΩ v ( α ω ) = ω . Furthermore, the constant P becomes inde-pendent of any adjustable parameter: P = 3 πda p / ΓΩ p .We consider the dispersion relation Ω fitv ( ω ) (13) thatmatches large and and short scales excitations and weinvert it numerically to find α ω .In Fig.5 the contour-plot of the theoretical predic-tion (22) obtained this way is compared with the nu-merical data (solid green lines), exhibiting a remarkableagreement with the observed excited frequencies. FromFig.6.b, we remark that the only allowed negative fre-quencies lay a in a thin band around Ω p . This is also in qualitative agreement with the data. Note that the bareKelvin wave dispersion relation (10) (dashed cyan line),and the fitted bare vortex wave dispersion relation (13)(solid cyan line) are very similar in Fig.5. The reason isthat the smallest scale that can be solved by the consid-ered array of particles is kξ = 0 .
25 (i.e. kd = π ) andfor wave numbers smaller than this value Ω fitv ( k ) tends toΩ KW ( k ) by construction.In order to make a closer connection with experi-ments, we now describe an array of larger particles of size a p = 13 . ξ and relative mass M = 1 set in a single quan-tum vortex and in a bundle of composed of four vortices.The corresponding spatio-temporal spectra S p ( k, ω ) aredisplayed in Fig.7. In principle such setting should not be kd . . . . . ! ⌧ kd . . . . . ! ⌧ (d)
In this work we have presented a theoretical and nu-merical study of the interaction between quantum vor-tices and a number of particles trapped in it. We havefirst pointed out that a trapped particle oscillates with awell defined natural frequency that depends on its massand the circulation of the flow surrounding it. Because ofthe typical values of particle parameters used in currentsuperfluid helium experiments, such frequency should bemeasurable. This measurement can thus provide an in-dependent way of estimating the number of vortices con-stituting the bundles at which particles are attached.Based on the experimental evidence that particlesspread along quantum vortices keeping a relatively con-stant inter-particle distance, we have studied how theparticles modifies the vortex excitations. The most ex-citing result of this work is the strong analogy with solidstate physics. Here, particles play the role of ions in theperiodic structure of a crystal and vortex excitations thatof the electrons. When an electron propagates, it feels theions as the presence of a periodic array of potential barri-ers. One of the simplest and idealized descriptions of thisphysical phenomenon is the Kronig-Penney model, wherethe barriers have a constant height U . Similarly, vortexwaves propagate and interact with particles and we haveshown that a similar theoretical approach can be used.The main difference is that the constant height of thebarriers in the standard Kronig-Penney model inducesconstant shift of the energy (frequency here). As a con-sequence, the lowest energy level in a crystal is differentfrom zero (unlike the case of free electrons). Instead, inthe vortex case, the interaction potential is due to Mag-nus force and depends on the frequency. Comparing themodels, we can then establish a mathematical analogy(see Eq. (22) and Ref.[23, 24]) by noticing that the ef-fective potential in the case of vortex excitations is givenby U ∼ ω / Ω p ∝ ω M effp (24)The height of the potential is thus proportional to thesquared frequency of the incoming wave and to the par-ticle mass. In particular, for very low frequencies thepresence of particles does not perturb much the vorticesand large scale Kelvin waves could be tracked by directlymeasuring the particle dynamics. Moreover, we observethat for particles with a higher natural frequency Ω p (namely lighter and smaller particles), the value of U and of P in Eq. (22) decrease. As a consequence, thebands of allowed frequencies are broadened. Ideally, inthe limiting case of particles with zero mass, the natu-ral frequency is infinite and P and U vanish. Thereforeequation (22) gets simplified dramatically and becomescos( kd ) = cos( α ω d ). This implies ω ( k ) = Ω v (cid:18) k + 2 nπd (cid:19) , n ∈ Z (25) that is just vortex wave dispersion relation, but repeatedwith period k d = 2 π/d . In other words, light and smallparticles can follow the filament without modifying thevortex waves. On the contrary, particle inertia reducesthe excited frequencies (in absolute value) of vortex ex-citations. This fact (actually coming from simple linearphysics) should be taken into account, when the Kelvinwaves are tried to be measured experimentally.In this work we did not take into account the rele-vance of buoyancy effects for light and heavy particles.We can estimate it by comparing the buoyancy force F b = ( M p − M )˜ g , where ˜ g ∼ . m/s is the gravita-tional acceleration, with the Magnus force that drivesthe particles F M = ρ Γ a p u , where u is the typical parti-cle velocity estimated as u ∼ Ω p a p . It turns out that F b /F M = C ( M −
1) (2 M + 1), where C = π ga Γ .This expression strongly depends on the particle size. Forinstance, given that the quantum of circulation in super-fluid helium is Γ ∼ − m /s , we get that C ∼ · − for a particle of size a p = 1 µm and therefore the buoy-ancy is negligible. However C becomes of order 1 for aparticle of size a p = 7 µm . We conclude that small andlight particles would be the most suitable for tracking thevortex excitations.Several questions can be immediately raised. If parti-cles are not actually equally distributed along the vortexbut instead they present some randomness, vortex waveswill then propagate in a disordered media. It will be nat-ural then to study the possibility of Anderson localizationin such a system [45, 46]. Such situation could perhapsappear if the vortex lines are excited by external means,for instance close the onset of the Donnelly-Glabersoninstability [47, 48].The physical system studied in this work is a first ide-alized picture of what happens in real superfluid heliumexperiments. The most evident difference is that the sizeof particles is typically orders of magnitude larger thanthe vortex core size ( a p ∼ ξ ). However, the prediction(22) comes from an asymptotic theory in which ka p (cid:28) v ( k ). Therefore, we expectthat our result should still apply for wave lengths largerthan the particle size. Such long waves are indeed ob-served in experiments [10]. In particular, the fact thatparticle inertia does not affect the (low) frequency Kelvinwaves should be still valid. A more quantitative predic-tion for vortices in He II would be always Eq. (22), butwith α ω such that ω = Ω He ( α ω ), where Ω He ( k ) is thetrue vortex excitations dispersion relation in superfluidhelium. In any case all the main conclusions remain valid,since the analogy with a crystal is independent of Ω v ( k ).Moreover, the behaviour at large scales is expected towork quantitatively also for superfluid helium vorticesbecause Ω He ( ka → ∼ Ω KW ( ka ).Furthermore, we have used arrays of particles with allidentical masses. Instead, in actual experiments there isnot a perfect control on the mass and size of particles.In particular, the mass distribution of particles could be1poly-dispersed. In this case, new gaps in the dispersionrelation are opened revealing much more complex con-figurations. A preliminary numerical study confirms thisbehavior and it will be reported in a future work. Inany case, the basic interaction between one particle andvortex waves remains the same regardless the presenceof some disorder. Therefore, large-scale Kelvin waves arenot disturbed by the particles. Studying in detail the ef-fects of different species of particles trapped in a vortexcan be done systematically in the same spirit of the effec-tive theory developed in the present work, for exampleadapting tight-binding models [24] to the vortex-particlessystem. We think that this is a worthy research directionthat could establish new and deeper connections withconcepts already known in solid state physics, introduc-ing a plethora of novel phenomena in the framework ofquantum fluids.Last but not least, note that the basic equations con-sidered in this work to build up the effective model arebased on classical hydrodynamics. Therefore, one couldexpect that most of the phenomenology remains valid ina classical fluid provided that a mechanism to sustain avortex exists. Such mechanism could be for instance pro-vided by two co-rotating propellers at moderate speeds.Since these systems are achievable in much less extremeconditions than in cold superfluid helium and because themanipulation of particle parameters is much simpler, itcould be possible to build analogs of solid state physicsphenomena by using classical fluid experiments. Appendix A: Numerical scheme and parameters
Equations (2-3) are solved with a standard pseudo-spectral code and a 4 th order Runge-Kutta scheme forthe time stepping in a 3D periodic domain of dimensions L ⊥ × L ⊥ × L (cid:107) with N ⊥ × N ⊥ × N (cid:107) collocation points.We set c = ρ ∞ = 1.The ground states with particles and straight vorticesare prepared separately by performing imaginary timeevolution of the GP equation. In order to have an ini-tial state with zero global circulation (and therefore en-sure periodic boundary conditions) we need to add inthe computational box three image vortices with alter-nating charges. The state with bundles of N v = 4 vor-tices (Fig.2.4) is prepared imposing a phase jump of 2 N v π around a vortex (including its images). Then, imaginarytime evolution of GP equation is performed for a time ∼ τ , so that the vortex filaments separate and thebundles form. KWs are generated from the state withstraight vortices slightly shifting each xy plane of thecomputational domain. Then the states with KWs andparticles are multiplied to obtain the wished initial con-dition. Just one vortex filament is loaded with particles,while the three other images remain bare. The initialcondition is evolved for a short time ( ∼ τ ) using GPwithout the particle dynamics in order to adapt the sys-tem. The particle potential is a smoothed hat-function V p ( r ) = V (1 − tanh (cid:104) r − η l (cid:105) ) and the mass displacedby the particle is measured as M = ρ ∞ L ⊥ L (cid:107) (1 − (cid:82) | ψ p | d x / (cid:82) | ψ ∞ | d x ), where ψ p is the steady statewith just one particle. Since the particle boundariesare not sharp, we measure the particle radius as a p =(3 M / πρ ∞ ) for given values of the numerical parame-ters η and ∆ l . For all the particles V = 20. The param-eters used are the following. For a p = 2 . ξ : η = ξ and∆ l = 0 . ξ . For a p = 7 . ξ : η = 2 ξ and ∆ l = 2 . ξ . For a p = 13 . ξ : η = 10 ξ and ∆ l = 2 . ξ .The parameter r of the potential V ij rep = ε ( r / | q i − q j | ) is the radius of the repulsion between particles.The parameter ε is fixed numerically in order to imposean exact balance between the repulsive force and the GPforce − (cid:82) V p ( | x − q i | ) ∇| ψ | d x in the ground state withtwo particles placed at distance 2 a p when r = 2 a p . Theparameters used for the repulsion are the following. Forthe wires in Fig.4: r = 2 L w / ( N p a p ) and ε = 4 . · − .For the array of particles in Fig.5: r = 4 a p and ε =4 . · − . For the array of particles in Fig.7: r = 2 a p and ε = 1 . · − . Appendix B: Spatio-temporal spectra
We use the particle positions to define the spatio-temporal spectra of vortex excitations by computing S q ( k, ω ) = C q (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) N p (cid:88) j =1 q ( z j , t ) e − i ( kz j + ωt ) d t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , (B1)where z j the z component of the particle j . Similarly, thespatio-temporal spectrum of the superfluid wave functionis defined as S ψ ( k, ω ) = C ψ (cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ψ ( x, y, z, t ) e − i ( kz + ωt ) d x d y d z d t (cid:12)(cid:12)(cid:12)(cid:12) . (B2)Note that in Eq. (B2) an average of ψ in the x and y directions is implicit. The normalization constants C q and C ψ are set such that the full ( k, ω ) integrals of thespatio-temporal spectra is one. In order to enhance thesmall scale excitations, in the density plots shown in thepresent work, both the spectra (B1) and (B2) are fur-ther normalized with the frequency-averaged spectra, re-spectively (cid:82) S q ( k, ω ) d ω and (cid:82) S ψ ( k, ω ) d ω . All the colormaps shown in the present work are in log-scale. Appendix C: Derivation of the “Kronig-Penney”dispersion relation for vortex waves
We look for a linear wave solution q ( z, t ) = Φ( z ) e iωt of the system (21) and in particular we want to know2which frequencies ω are excited. The function Φ( z ) mustsatisfy the system ∂ ∂z Φ( z ) + α ω Φ( z ) = 0 (I) ∂ ∂z Φ( z ) + β ω Φ( z ) = 0 (II) (C1)where α ω and β ω are such thatΩ LIA ( α ω ) = ω, Ω LIA ( β ω ) = ω − ω Ω w , (C2)that means α ω = (cid:114) − πω ΓΛ , β ω = (cid:115) π ΓΛ (cid:18) ω Ω w − ω (cid:19) (C3)Since the system (C1) is a linear and homogeneousdifferential equation with periodic coefficients of period d , it admits a solution in the form Φ( z ) = e ikz u ( z ), where u ( z ) is a periodic function of period d . The solutions of(C1) in the two regions (I) and (II) areΦ I ( z ) = e ikz u I ( z ) = e ikz (cid:104) Ae i ( α ω − k ) z + Be − i ( α ω + k ) z (cid:105) Φ II ( z ) = e ikz u II ( z ) = e ikz (cid:104) Ce i ( β ω − k ) z + De − i ( β ω + k ) z (cid:105) (C4)The coefficients A , B , C , D are fixed by imposing continu-ity and smoothness of the function Φ( z ) and periodicityof u ( z ) and its derivative: Φ I (0) = Φ II (0)Φ (cid:48) I (0) = Φ (cid:48) II (0) u I ( d − L w ) = u II ( − L w ) u (cid:48) I ( d − L w ) = u (cid:48) II ( − L w ) (C5)The system (C5) is a homogeneous linear system for thevariables A , B , C , D . It admits non-trivial solutions only ifthe determinant of the coefficients is equal to zero. Thisimply the following conditioncos( kd ) = cos( β ω L w ) cos( α ω ( d − L w )) − α ω + β ω α ω β ω sin( β ω L w ) sin( α ω ( d − L w ))(C6)which determines implicitly the dispersion relation ω ( k ).Such expression is structurally identical to the standardKronig-Penney condition but the function α ω and β ω aredifferent. The limit L w → M effw with the mass of the particle M effp .In this way the system becomes a vortex filament loadedwith massive point-particles (see Fig.6). The limit im-plies β ω → ∞ , β ω L w →
0, sin( β ω L w ) ∼ β ω L w , α ω (cid:28) β ω and β ω L w ∼ πa p ω / ΛΓΩ p , so that Eq. (C6) becomesEq. (22). The previous result can be extended to the case of morerealistic vortex waves with some caveat. We can formallyrewrite the model (21) as˙ q ( z, t ) = i ˆ L v [ q ( z, t )] (I)¨ q ( z, t ) = i Ω w (cid:104) ˙ q ( z, t ) − i ˆ L v [ q ( z, t )] (cid:105) (II) (C7)where ˆ L v is the linear non-local differential operatorthat generates the vortex wave dispersion relation Ω v ( k ).Namely, calling s ( z, t ) = (cid:80) k s k ( t ) e ikz the wave operatorsimply reads ˆ L v [ s ( z, t )] = (cid:88) k Ω v ( k ) s k ( t ) e ikz . (C8)The system (C1) thus becomesˆ L V [Φ( z )] − ω Φ( z ) = 0 (I)ˆ L V [Φ( z )] − (cid:18) ω − ω Ω v (cid:19) Φ( z ) = 0 (II) (C9)The functions (C4) are still a solution of (C9), but now α ω and β ω are defined asΩ v ( α ω ) = ω, Ω v ( β ω ) = (cid:18) ω − ω Ω w (cid:19) . (C10)In general such equations cannot be inverted explicitly,but α ω and β ω can be found numerically. In particu-lar the inversion is intended with respect to Ω v ( k > α ω and β ω are well defined (at least for ω/ Γ >
0) because any model for the self-induced ve-locity of a vortex generates a dispersion relation Ω v ( k )that is monotonically increasing for positive k . For eval-uating the limit L w → M effw → M effp , we note thatlim L w → Ω v ( β ω ) = ∞ . Therefore, we can explicitly usethe asymptotics of Ω v ( k ) for large k , that is just freeparticle dispersion relation (12) and can be inverted ex-plicitly: β ω −→ L w → (cid:115) πω ΓΩ w , (C11)so that β ω L w ∼ πa p / ΓΩ p . In this way we recover Eq.(22), with α ω defined as in (C10) and the amplificationfactor P is now independent of any free parameter: P = 3 πda p ΓΩ p . (C12) ACKNOWLEDGMENTS
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