PHYSICAL REVIEW RESEARCH 2, 023149 (2020) How trapped particles interact with and sample superfluid vortex excitations Umberto Giuriato * and Giorgio Krstulovic Université Côte d’Azur, CNRS, OCA, Laboratoire Lagrange, Boulevard de l’Observatoire CS 34229 - F 06304 NICE Cedex 4, France Sergey Nazarenko Université Côte d’Azur, Institut de Physique de Nice (INPHYNI), CNRS UMR 7010, Parc Valrose, 06108 Nice Cedex 2, France (Received 11 July 2019; revised manuscript received 23 March 2020; accepted 25 March 2020; published 11 May 2020) Particles have been used for more than a decade to visualize and study the dynamics of quantum vortices in superfluid helium. In this work we study how the dynamics of a collection of particles set inside 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 oscillates with a natural frequency proportional to the number of vortices attached to it. We then study the dynamics of an array of particles trapped in a quantum vortex and use particle trajectories to measure the frequency spectrum of the vortex excitations. Surprisingly, due to the discreetness of the array, the vortex excitations measured by the particles exhibit bands, gaps, and Brillouin zones, analogous to the ones of electrons moving in crystals. We then establish a mathematical analogy where vortex excitations play the role of electrons and particles that of the potential barriers constituting the crystal. We find that the height of the effective potential barriers is proportional to the particle mass and the frequency of the incoming waves. We conclude that large-scale vortex excitations could be in principle directly measured by particles and novel physics could emerge from particle-vortex interaction. DOI: 10.1103/PhysRevResearch.2.023149 I. INTRODUCTION appear as topological defects of such order parameter. In three dimensions, such defects are unidimensional structures, When a fluid composed of bosons is cooled down, a usually referred to as quantum vortices. Indeed, the circula- spectacular phase transition takes place. The system becomes tion (contour integral) of the flow around a vortex must be superfluid and exhibits exotic physical properties. Unlike any a multiple of the Feynman-Onsager quantum of circulation classical fluid, a superfluid flows with no viscosity. This is h/m, where h is the Planck constant and m is the mass of the an intriguing example of the manifestation of pure quantum- Bosons constituting the fluid [4]. Such peculiarity is necessary mechanical effects on a macroscopic level. The first discov- to ensure the monodromy of the wave function. In superfluid ered superfluid is liquid helium 4He in its so-called phase helium, quantum vortices have a core size on the order of II, below the critical temperature Tλ 2.17 K. In one of the an angstrom. At low temperatures, below 1 K, the normal first attempts of describing the behavior of superfluid helium, component is negligible and vortices are stable and do not London suggested that superfluidity is intimately linked to decay by any diffusion process, unlike their classical coun- the phenomenon of Bose-Einstein condensation (BEC) [1]. In terparts. The understanding of superfluid vortex dynamics has the same years, Landau and Tisza independently put forward a direct impact on many interesting, complex nonequilibrium a phenomenological two-fluid model, wherein superfluid he- multiscale phenomena, such as turbulence [5–7]. lium can be regarded as a physically inseparable mixture of Most of the experimental knowledge on superfluid vortices two components: a normal viscous component that carries the is based on indirect measurement of their properties. The early entire entropy and an inviscid component with zero entropy efforts in the observation of quantized vortices were made in [2,3]. the framework of rotating superfluid helium, by using electron Because of its intrinsic long-range order, a superfluid bubbles (ions) as probes [8]. Since then, impurities have can be described by a macroscopic complex wave function. been extensively used to unveil the dynamics of superfluid A stunning quantum-mechanical constraint is that vortices vortices. An important breakthrough occurred in 2006, when micrometer-sized hydrogen ice particles were used to directly visualize superfluid helium vortices [9]. Thanks to pressure *Corresponding author: [email protected] gradients, particles get trapped inside quantum vortices and are subsequently carried by them. Hence, it has been possible Published by the American Physical Society under the terms of the to observe vortex reconnections and Kelvin waves (helicoidal Creative Commons Attribution 4.0 International license. Further displacements that propagate along the vortex line) by means distribution of this work must maintain attribution to the author(s) of standard particle-tracking techniques [10]. Furthermore, and the published article’s title, journal citation, and DOI. the particle dynamics unveiled important differences between 2643-1564/2020/2(2)/023149(12) 023149-1 Published by the American Physical Society GIURIATO, KRSTULOVIC, AND NAZARENKO PHYSICAL REVIEW RESEARCH 2, 023149 (2020) velocity statistics of quantum and classical turbulent states standard Kronig-Penney model [23,24], which describes the [11,12]. In experiments, such particles are used as tracers, motion of electrons in a unidimensional crystal. Finally, based despite their very large size compared to the vortex core. on our results, we discuss in which regimes particles could be Therefore, it is of the utmost importance that the mechanisms reliably used to sample vortex excitations. driving their dynamics are fully comprehended. Specifically, how well is vortex dynamics reflected by the motion of the II. THEORETICAL BACKGROUND particles trapped in it? How much does their presence in the core modify the propagation of Kelvin waves? Would they A. Model for superfluid vortices and active particles affect the reconnection rates? We consider a superfluid at very low temperature con- Describing the interaction of particles with isolated vortex taining Np spherical particles of mass Mp and radius ap.We lines or complex quantum vortex tangles is not an easy describe the system by a self-consistent model based on the task. Depending on the scale of interest, there are different three-dimensional Gross-Pitaevskii equation. The particles theoretical and numerical models that can be adopted. A big are modeled by strong localized potentials Vp, which com- effort has been made in adapting the standard dynamics of pletely deplete the superfluid up to a distance ap from their particles in classical fluids to the case of superfluids described center position qi. Particles have inertia and obey a Newtonian by two-fluid models [13,14]. This is a macroscopic model in dynamics. The Hamiltonian of the system is which vorticity is a coarse-grained field and therefore there is ⎛ ⎞ N no notion of quantized vortices. A medium-scale description h¯2 g p H = ⎝ |∇ψ|2 + |ψ|4 + V (r − q )|ψ|2⎠dr is given by the vortex filament model, where the superfluid is 2m 2 p i modeled as a collection of lines that evolve following Biot- i=1 Savart integrals. In this approximation, circulation of vortices Np Np p2 is by construction quantized but reconnections are absent and + i + V ij , (1) 2M rep have to be implemented via some ad hoc mechanism. Finite- i=1 p i< j size particles can be studied in the vortex filament frame- ψ work but the resulting equations are numerically costly and where is the wave function that describes the superfluid limited [15]. A microscopic approach consists in describing and m is the mass of the condensed bosons interacting with each impurity by a classical field in the framework of the an s-wave scattering length as, so that the coupling constant = π 2/ ij = ε /| − | 12 Gross-Pitaevskii model [16–18]. In principle, such method is g 4 ash¯ m. The potential Vrep (r0 qi q j ) is a is valid for weakly interacting BECs, and is numerically and repulsive potential of radius r0 between particles. See Refs. theoretically difficult to handle if one wants to consider more [20,22] and the next section for further details about the ψ than just a few particles. In the same context, an alternative model. The equations of motion for the superfluid field and = , , possibility is to assume classical degrees of freedom for the the particle positions qi (qi,x qi,y qi,z )are particles, while the superfluid is still a complex field obeying Np ∂ψ h¯2 the Gross-Pitaevskii equation. This idea of modeling parti- ih¯ =− ∇2ψ + (g|ψ|2 − μ)ψ + V (|x − q |)ψ, ∂t 2m p i cles as simple classical hard spheres has been shown to be i=1 both numerically and analytically very powerful [19–22]. In (2) particular, such minimal and self-consistent model allows for simulating a relatively large number of particles, and describes Np ∂ well the particle-vortex interaction [22]. Although formally M q¨ =− V (|x − q |)∇|ψ|2 dx + V ij . (3) p i p i ∂q rep valid for weakly interacting BECs, it is expected to give a j=i i good qualitative description of superfluid helium. In this paper we investigate how particles trapped in quan- This model has been successfully used to study vortex nucle- tum vortices interact with vortex excitations and in particular ation [19] and trapping of particles by quantum vortices [22]. how well they can be used to infer properties of superfluid We denote by GP the Gross-Pitaevskii model without particles and by GP-P the full coupled system (2) and (3). vortices. We use the Gross-Pitaevskii equation coupled with μ inertial and active particles obeying classical dynamics to In the absence of particles, the chemical potential√ fixes √the value of the condensate ground state ψ∞ = ρ∞/m = answer this question.