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arXiv:1803.00147

Non-Hamiltonian Kelvin wave generation on vortices in Bose-Einstein condensates

Strong, Scott A., Lincoln D. Carr

Published Mar 1, 2018• 1 citation
arXivHTMLPDF

Abstract

Ultra-cold quantum turbulence is expected to decay through a cascade of Kelvin waves. These helical excitations couple vorticity to the quantum fluid causing long wavelength phonon fluctuations in a Bose-Einstein condensate. This interaction is hypothesized to be the route to relaxation for turbulent tangles in quantum hydrodynamics. The local induction approximation is the lowest order approximation to the Biot-Savart velocity field induced by a vortex line and, because of its integrability, is thought to prohibit energy transfer by Kelvin waves. Using the Biot-Savart description, we derive a generalization to the local induction approximation which predicts that regions of large curvature can reconfigure themselves as Kelvin wave packets. While this generalization preserves the arclength metric, a quantity conserved under the Eulerian flow of vortex lines, it also introduces a non-Hamiltonian structure on the geometric properties of the vortex line. It is this non-Hamiltonian evolution of curvature and torsion which provides a resolution to the missing Kelvin wave motion. In this work, we derive corrections to the local induction approximation in powers of curvature and state them for utilization in vortex filament methods. Using the Hasimoto transformation, we arrive at a nonlinear integro-differential equation which reduces to a modified nonlinear Schrödinger type evolution of the curvature and torsion on the vortex line. We show that this modification seeks to disperse localized curvature profiles. At the same time, the non-Hamiltonian break in integrability bolsters the deforming curvature profile and simulations show that this dynamic results in Kelvin wave propagation along the dispersive vortex medium.

DOI
10.48550/arxiv.1803.00147

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Ultra-cold quantum turbulence is expected to decay through a cascade of Kelvin waves. These helical excitations couple vorticity to the quantum fluid causing long wavelength phonon fluctuations in a Bose-Einstein condensate. This interaction is hypothesized to be the route to relaxation for turbulent tangles in quantum hydrodynamics. The local induction approximation is the lowest order approximation to the Biot-Savart velocity field induced by a vortex line and, because of its integrability, is thought to prohibit energy transfer by Kelvin waves. Using the Biot-Savart description, we derive a generalization to the local induction approximation which predicts that regions of large curvature can reconfigure themselves as Kelvin wave packets. While this generalization preserves the arclength metric, a quantity conserved under the Eulerian flow of vortex lines, it also introduces a non-Hamiltonian structure on the geometric properties of the vortex line. It is this non-Hamiltonian evolution of curvature and torsion which provides a resolution to the missing Kelvin wave motion. In this work, we derive corrections to the local induction approximation in powers of curvature and state them for utilization in vortex filament methods. Using the Hasimoto transformation, we arrive at a nonlinear integro-differential equation which reduces to a modified nonlinear Schrödinger type evolution of the curvature and torsion on the vortex line. We show that this modification seeks to disperse localized curvature profiles. At the same time, the non-Hamiltonian break in integrability bolsters the deforming curvature profile and simulations show that this dynamic results in Kelvin wave propagation along the dispersive vortex medium.
The dynamics of quantized vortices in weakly interacting superfluids are often modeled by a nonlinear Schrödinger equation. In contrast, we show that quantized vortices in fact obey a non-Hamiltonian evolution equation, which enhances dispersion along the vortex line while introducing a gain mechanism. This allows the vortex medium to support a helical shock front propagating ahead of a dissipative soliton. This dynamic relaxes localized curvature events into Kelvin wave packets. Consequently, a beyond local induction model provides a pathway for decay in low-temperature quantum turbulence.
Geometric quantum hydrodynamics merges geometric hydrodynamics with quantum hydrodynamics to study the geometric properties of vortex structures in superfluid states of matter. Here the vortex line acts as the fundamental building block and is a topological defect of the fluid medium about which the otherwise irrotational fluid circulates. In this thesis, we show that except for the simplest fluid flows, a vortex line seeks to decompose localized regions of curvature into helical configurations. The simplest flow, known as the local induction approximation, is also an integrable one. Integrability makes the transfer of energy into helical modes impossible. In the following, we demonstrate that any arclength conserving correction to this approximation defines a non-Hamiltonian evolution of the vortex geometry, which is capable of supporting dissipative solitons and helical wavefronts. Quantum turbulence in ultracold vortex tangles relies on energy transfer between helical or Kelvin modes to decay. Thus, models of vortex lines beyond the lowest order integrable cases are vitally important to our mathematical description of free decay of turbulent tangles. To motivate the results of this thesis we connect our theory of vortex line dynamics to continuum fluid mechanics. The Navier-Stokes equations are a statement of momentum balance for a fluid whose response to shear stress is proportional to the fluid's velocity gradients. Building off of Onsager's rather obscure work in fluid turbulence, others have shown that solutions to Navier-Stokes limit to Euler evolutions of distributional velocity profiles for large Reynolds number. Using this as our context, we show that the Euler equation can be transformed, in an inverse Madelung sense, to the Gross-Pitaevskii equation associated with the mean-field quantum dynamics of a dilute Bose gas. This theory predicts that an irrotational fluid is capable of circulating around regions of density depletion known as vortex lines. Furthermore, if a vortex line is used as the ansatz for the Gross-Pitaevskii equation then it is possible to show that the Biot-Savart integral over the vortex source results. This connection between the vortex line geometry and the induced velocity field provides the basis for the application of geometric quantum hydrodynamics to Bose-Einstein condensates. If the Biot-Savart integral is the basis of geometric quantum hydrodynamics, then Hasimoto's transformation is the structure built on top. The fundamental theorem of space curves states that up to rotations and translations, a curve is defined by its curvature and torsion. Through the Hasimoto transformation, it is possible to map flows defined by the Biot-Savart integral to scalar partial differential equations evolving the curvature and torsion of a vortex line. In this thesis, we conduct an asymptotic expansion of the Biot-Savart integral and apply the Hasimoto transformation to show that vortex lines prefer to relax curved abnormalities through the excitation of helical waves along the vortex. Thus, our…
In a uniform fluid, a quantized vortex line with circulation h∕Mh∕Mh∕M can support long-wavelength helical traveling waves $\ensuremath{\propto}{e}^{i(kz\ensuremath{-}{\ensuremath{\omega}}_{k}t)}$ with the well-known Kelvin dispersion relation ${\ensuremath{\omega}}_{k}\ensuremath{\approx}(\ensuremath{\hbar}{k}^{2}∕2M)\mathrm{ln}(1∕\ensuremath{\mid}k\ensuremath{\mid}\ensuremath{\xi})$, where $\ensuremath{\xi}$ is the vortex-core radius. This result is extended to include the effect of a nonuniform harmonic trap potential, using a quantum generalization of the Biot-Savart law that determines the local velocity V\mathbf{V}V of each element of the vortex line. The normal-mode eigenfunctions form an orthogonal Sturm-Liouville set. Although the line's curvature dominates the dynamics, the transverse and axial trapping potential also affect the normal modes of a straight vortex on the symmetry axis of an axisymmetric Thomas-Fermi condensate. The leading effect of the nonuniform condensate density is to increase the amplitude along the axis away from the trap center. Near the ends, however, a boundary layer forms to satisfy the natural Sturm-Liouville boundary conditions. For a given applied frequency, the next-order correction renormalizes the local wave number k(z)k(z)k(z) upward near the trap center, and k(z)k(z)k(z) then increases still more toward the ends.
In a recent paper, we give a study of the purely rotational motion of general stationary states in the two-dimensional local induction approximation (2D-LIA) governing superfluid turbulence in the low-temperature limit [B. Svistunov, “Superfluid turbulence in the low-temperature limit,” Phys. Rev. B 52, 3647 (1995)]. Such results demonstrated that variety of stationary configurations are possible from vortex filaments exhibiting purely rotational motion in addition to commonly discussed configurations such as helical or planar states. However, the filaments (or, more properly, waves along these filaments) can also exhibit translational motion along the axis of orientation. In contrast to the study on vortex configurations for purely rotational stationary states, the present paper considers non-stationary states which exhibit a combination of rotation and translational motions. These solutions can essentially be described as waves or disturbances which ride along straight vortex filament lines. As expected from our previous work, there are a number of types of structures that can be obtained under the 2D-LIA. We focus on non-stationary states, as stationary states exhibiting translation will essentially take the form of solutions studied in [R. A. Van Gorder, “General rotating quantum vortex filaments in the low-temperature Svistunov model of the local induction approximation,” Phys. Fluids 26, 065105 (2014)], with the difference being translation along the reference axis, so that qualitative appearance of the solution geometry will be the same (even if there are quantitative differences). We discuss a wide variety of general properties of these non-stationary solutions and derive cases in which they reduce to known stationary states. We obtain various routes to Kelvin waves along vortex filaments and demonstrate that if the phase and amplitude of a disturbance both propagate with the same wave speed, then Kelvin waves will result. We also consider the self-similar solutions to the model and demonstrate that these types of solutions can model vortex kinks that gradually smooth and radiate Kelvin waves as time increases. Such solutions qualitatively agree with what one might expect from post-reconnection events.
The dynamics of vortex solitons is studied in a BEC superfluid. A quantum lattice-gas algorithm (measurementbased quantum computation) is employed to examine the dynamical behavior vortex soliton solutions of the Gross-Pitaevskii equation (ø4 interaction nonlinear Schroedinger equation). Quantum turbulence is studied in large grid numerical simulations: Kolmogorov spectrum associated with a Richardson energy cascade occurs on large flow scales. At intermediate scales, a new k-6 power law emerges, due to vortex filamentary reconnections associated with Kelvin wave instabilities (vortex twisting) coupling to sound modes and the exchange of intermediate vortex rings. Finally, at very small spatial scales a k-3power law emerges, characterizing fluid dynamics occurring within the scale size of the vortex cores themselves. Poincaré recurrence is studied: in the free non-interacting system, a fast Poincaré recurrence occurs for regular arrays of line vortices. The recurrence period is used to demarcate dynamics driving a nonlinear quantum fluid towards turbulence, since fast recurrence is an approximate symmetry of the nonlinear quantum fluid at early times. This class of quantum algorithms is useful for studying BEC superfluid dynamics and, without modification, should allow for higher resolution simulations (with many components) on future quantum computers.