We have theoretically investigated Kelvin waves of quantized vortex lines in trapped Bose-Einstein condensates. Counterrotating perturbation induces an elliptical instability to the initially straight vortex line, driven by a parametric resonance between a quadrupole mode and a pair of Kelvin modes of opposite momenta. Subsequently, Kelvin waves rapidly decay to longer wavelengths emitting sound waves in the process. We present a modified Kelvin wave dispersion relation for trapped superfluids and propose a simple method to excite Kelvin waves of specific wave number.
We propose a different type of Landau instability in trapped Bose-Einstein condensates by a helically moving environment. In the presence of quantized vortices, the instability can cause spontaneous radiation and amplification of Kelvin waves. This study gives a microscopic understanding of the Donnelly-Glaberson instability which was known as a hydrodynamic instability in superfluid helium. The Donnelly-Glaberson instability can be a powerful tool for observing the dispersion relation of Kelvin waves, vortex reconnections, and quantum turbulence in atomic Bose-Einstein condensates.
We determine numerically the steady-state spectrum for Kelvin waves on a superfluid 4He vortex that is driven by shaking its end points and damped by mutual friction. The relaxation rate towards a steady state is determined by the mutual friction. We have reached the zero temperature limit where the steady-state spectrum and the vortex length become independent of the temperature. Our drive with pointlike pinning sites produces a characteristic spectrum with $|w_m|\propto m^{-η}$, where $η\approx$ 1.88. The spatially sharply peaked drive and the fact that we are fixing the oscillation amplitud…
In superfluid helium, vorticity is quantized and constrained on line-like phase singularities, called quantum or quantized vortices. By visualizing the motion of sub-micron frozen particles in superfluid $^{4}$He, we directly observe for the first time the helical Kelvin waves excited after quantized vortex reconnections. We compare the data with self-similar solutions of vortex filament models. We report the results in a fluid dynamic video.
Development of experimental techniques to study superfluid dynamics, in particular, application of nanomechanical oscillators to drive vortex lines, enables potential observation of the Kelvin-wave cascade on quantized vortices. One of the first questions which then arises in analysis of the experimental results is the relation between the energy flux in the cascade and the amplitude of the Kelvin waves. We provide such relation based on the L'vov-Nazarenko picture of the cascade. Remarkably, the amplitude of the waves depends on the energy flux extermely weakly, as power one tenth.
Kelvin waves are expected to play an essential role in the energy dissipation for quantized vortices. However, the identification of these helical distortions is not straightforward, especially in case of vortex tangle. Here we review several numerical methods that have been used to identify Kelvin waves within the vortex filament model. We test their validity using several examples and estimate whether these methods are accurate enough to verify the correct Kelvin spectrum. We also illustrate how the correlation dimension is related to different Kelvin spectra and remind that the 3D energy sp…
We consider a trapped Bose--Einstein condensate (BEC) with a highly quantized vortex. For the BEC with a doubly, triply or quadruply quantized vortex, the numerical calculations have shown that the Bogoliubov--de Gennes equations, which describe the fluctuation of the condensate, have complex eigenvalues. In this paper, we obtain the analytic expression of the condition for the existence of complex modes, using the method developed by Rossignoli and Kowalski [R. Rossignoli and A. M. Kowalski, Phys. Rev. A 72, 032101 (2005)] for the small coupling constant. To derive it, we make the two-mode ap…
We have theoretically studied vortex waves of Bose-Einstein condensates in elongated harmonic traps. Our focus is on the axisymmetric varicose waves and helical Kelvin waves of singly quantized vortex lines. Growth and decay dynamics of both types of vortex waves are discussed. We propose a method to experimentally create these vortex waves on demand.
We study the statistical properties of the Kelvin waves propagating along quantized superfluid vortices driven by the Gross-Pitaevskii equation. No artificial forcing or dissipation is added. Vortex positions are accurately tracked. This procedure directly allows us to obtain the Kevin-waves occupation-number spectrum. Numerical data obtained from long time integration and ensemble-average over initial conditions supports the spectrum proposed in [L'vov and Nazarenko, JETP Lett 91, 428 (2010)]. Kelvin wave modes in the inertial range are found to be Gaussian as expected by weak-turbulence pred…
We theoretically study the Kelvin-Helmholtz instability in phase-separated two-component Bose-Einstein condensates using the Gross-Pitaevskii and Bogoliubov-de Gennes models. A flat interface between the two condensates is shown to deform into sawtooth or Stokes-like waves, leading to the formation of singly quantized vortices on the peaks and troughs of the waves. This scenario of interface instability in quantum fluids is quite different from that in classical fluids.
Recent experiments on quantum turbulence in superfluid helium made use of small tracer particles to track the motion of quantized vortices, determine velocity statistics and visualize vortex reconnections. A problem with this visualization technique is that it may change the turbulent flow which it visualizes. To address this problem, we derive and solve the equations of motion of a quantized vortex ring which contains a given number of particles trapped on the vortex core, hence derive a quantitative criterion to determine in which measure small particles trapped in quantized vortices affect …
Kelvin waves (kelvons)--helical waves on quantized vortex lines--are the normal modes of vortices in a superfluid. At zero temperature, the only dissipative channel of vortex dynamics is phonon emission. Starting with the hydrodynamic action, we derive the Hamiltonian of vortex-phonon interaction, thereby reducing the problem of the interaction of Kelvin waves with sound to inelastic elementary excitation scattering. On the basis of this formalism, we calculate the rate of sound radiation by superfluid turbulence at zero temperature and estimate the value of short-wavelength cutoff of the turb…
Peculiar dynamics of a free surface of the superfluid 4He has been observed experimentally with a newly established technique utilizing a number of electrically charged fine metal particles trapped electrically at the surface by Moroshkin et al. They have reported that some portion of the particles exhibit some irregular motions and suggested the existence of quantized vortices interacting with the metal particles. We have conducted calculations with the vortex filament model, which turns out to support the idea of the vortex-particle interactions. The observed anomalous metal particle motions…
We study the nonlinear dynamics of the splitting of a doubly quantized vortex in a trapped condensate. The dynamics is studied in detail by solving the Gross-Pitaevskii equation. The main dynamical features are explained in terms of a nonlinear three-level system. We find an analytical solution for the characteristics of the dynamics. It is concluded that the time scale for the splitting is mainly determined by the instability of the linearized system, and nonlinear effects contribute logarithmically.
The Bogoliubov equations are solved for a three-dimensional Bose-Einstein condensate containing a doubly quantized vortex, trapped in a harmonic potential. Complex frequencies, signifying dynamical instability, are found for certain ranges of parameter values. The existence of alternating windows of stability and instability, respectively, is explained qualitatively and quantitatively using variational calculus and direct numerical solution. It is seen that the windows of stability are much smaller for a cigar shaped condensate than for a pancake shaped one, which is consistent with the findin…
We report on the analysis of the root mean square curvature as a function of the numerical resolution for a single reconnection of two quantized vortex rings in superfluid helium. We find a similar scaling relation as reported in the case of decaying thermal counterflow simulations by L. Kondaurova et al. There the scaling was related to the existence of a Kelvin-wave cascade which was suggested to support the L'vov-Nazarenko spectrum. Here we provide an alternative explanation that does not involve the Kelvin-wave cascade but is due to the sharp cusp generated by a reconnection event in a sit…
We study Kelvin modes and translational zero modes excited along a quantized vortex and relativistic global string in superfluids and a relativistic field theory, respectively, by constructing the low-energy effective theory of these modes. We find that they become exact gapless Nambu-Goldstone modes only in a system with infinite volume limit. On the other hand, in a system with the finite volume, we find an imaginary massive gap causing the tachyonic instability above some critical wavelength in the relativistic theory. We also find in the non-relativistic theory that Kelvin modes with wavel…
We study the statistical and dynamical behavior of turbulent Kelvin waves propagating on quantized vortices in superfluids, and address the controversy concerning the energy spectrum that is associated with these excitations. Finding the correct energy spectrum is important because Kelvin waves play a major role in the dissipation of energy in superfluid turbulence at near-zero temperatures. In this paper, we show analytically that the solution proposed in Ref. \cite{10LN} enjoys existence, uniqueness and regularity of the pre-factor. Furthermore, we present numerical results of the dynamical …
We comment on the claim by Lebedev and L'vov [arXiv:1005.4575] that the symmetry with respect to a tilt of a quantized vortex line does not yet prohibit coupling between Kelvin waves and the large-scale slope of the line. Ironically, the counterexample of an effective scattering vertex in the local induction approximation (LIA) attempted by Lebedev and L'vov invalidates their logic all by itself being a notoriously known example of how symmetries impose stringent constraints on kelvon kinetics---not only the coupling in question but the kinetics in general are absent within LIA. We further exp…
We present a comprehensive statistical study of free decay of the quantized vortex tangle in superfluid $^4$He at low and ultra-low temperatures, $0\leqslant T \leqslant 1.1\,$K. Using high resolution vortex filament simulations with full Biot-Savart vortex dynamics, we show that for ultra-low temperatures $T\lesssim 0.5 \,$K, when the mutual friction parameters $α\simeq α' < 10^{-5}$, the vortex reconnections excite Kelvin waves with wave lengths $λ$ of the order of the inter-vortex distance $\ell$. These excitations cascade down to the resolution scale $Δξ$ which in our simulations is of the…
On the basis of recently proposed scenario of the transformation of the Kolmogorov cascade into the Kelvin-wave cascade, we develop a theory of low-temperature cutoff. The theory predicts a specific behavior of the quantized vortex line density, $L$, controlled by the frictional coefficient, $α(T) \ll 1$, responsible for the cutoff. The curve $\ln L(\ln α)$ is found to directly reflect the structure of the cascade, revealing four qualitatively distinct wavenumber regions. Excellent agreement with recent experiment by Walmsley {\it et al.} [arXiv:0710.1033]--in which $L(T)$ has been measured do…
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 for the vortex dynamics. The research is motivated by recent experiments on xenon doped helium nanodroplets that show clustering of the impurities along the vortex cores. We numerically simulate the dynamics of trapping and interactions of xenon atoms by quantized vortices in superfluid helium and the obtained results can be extended to scattering of other impurities by quantized vortices. Different ener…
We explore the possibility that the fast and exotic negative ions in superfluid helium are electrons bound to quantized vortex structures, the simplest being a ring. In the states we consider, the electron energy is only slightly below the conduction band minimum of bulk helium. To support our proposal we present two calculations. In the first, we show that the electron pressure on the vortex core is insufficient to cavitate the helium and form an electron bubble. In the second, we estimate the equilibrium radius of the vortex ring that would bind an electron and find it is much smaller than t…
We show that a multiply-quantized vortex line core can be energetically stable in a harmonically trapped two-component atomic Bose-Einstein condensate. The structure, in which the condensate component with the vortex line is surrounded by the second component forming a vortex ring, can be identified as a particlelike Skyrmion with a multiply-quantized winding number, and closely resembles cosmic vortons.
An apparatus for producing atomic-gas Bose-Einstein condensates (BECs) of 87-Rb atoms is described. The apparatus produces 87-Rb BECs in a dual-chamber vacuum system that incorporates magnetic transport of trapped atoms from the magneto-optical trapping cell to the BEC production cell via the operation of a series of overlapping magnet coils. The design, construction, and operation of the apparatus are described in detail. The apparatus is used to study the creation of quantized vortices in BECs by the merging and interference of multiple trapped condensates. In this experiment, a single harmo…
Dynamics of quantized vortex lines in a superfluid feature symmetries associated with the geometric character of the complex-valued field, $w(z)=x(z)+iy(z)$, describing the instant shape of the line. Along with a natural set of Noether's constants of motion, which---apart from their rather specific expressions in terms of $w(z)$---are nothing but components of the total linear and angular momenta of the fluid, the geometric symmetry brings about crucial consequences for kinetics of distortion waves on the vortex lines---the Kelvin waves. It is the geometric symmetry that renders Kelvin-wave ca…
We study the stability of singly- and doubly-quantized vortex states of harmonically trapped dipolar Bose-Einstein Condensates (BECs) by calculating the low-lying excitations of these condensates. We map the dynamical stability of these vortices as functions of the dipole-dipole interaction strength and trap geometry by finding where their excitations have purely real energy eigenvalues. In contrast to BECs with purely contact interactions, we find that dipolar BECs in singly-quantized vortex states go unstable to modes with an increasing number of angular and radial nodes for more oblate trap…
We derive general approximate formulas that provide with remarkable accuracy the ground-state properties of any mean-field scalar Bose-Einstein condensate with short-range repulsive interatomic interactions, confined in arbitrary cylindrically symmetric harmonic traps. Our formulation is even applicable for condensates containing a multiply quantized axisymmetric vortex. We have checked the validity of our formulas by numerically solving the 3D Gross-Pitaevskii equation.
