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Geometric quantum hydrodynamics and Bose-Einstein condensates: non-Hamiltonian evolution of vortex lines
Strong, Scott A.
Strong, Scott A.
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2018
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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 model predicts a geometric mechanism for the generation of Kelvin waves and corrects a nearly 50 year old result to include the dynamics expected in our most primitive states of fluid matter. Our derivation begins with the Biot-Savart integral representation of the velocity field induced by a vortex line. We derive an exact representation of the velocity field in terms of the incomplete elliptic integrals by approximating local regions of vorticity with plane circular arcs. The velocity field is shown to be a combination of three fields defining axial flow, circulation and a binormal flow that appears in the presence of non-trivial curvature. The latter flow explains why vortex rings of smaller diameter travel faster than larger rings. Using known asymptotic formulae for the elliptic integrals allows us to move past the lowest order local induction approximation. The asymptotic representations are valid for the regime where the local curvature is small relative to the inverse of the vortex core size and are applicable to vortex lines in Bose-Einstein condensates. To understand the predictions stemming from our asymptotic representation of the local velocity field, we compute two key quantities. First, we consider the expansion of the local field in powers of curvature to define corrections to the flow when curvature becomes large. Second, we consider the Hasimoto transformation of the general induced binormal flow. The result is a scalar evolution of the curvature and torsion for configurations of vortex lines with significant bending. Moving past the local induction approximation causes Hasimoto's transformation to map the local flow onto a nonlinear integro-differential equation. Through the use of our asymptotic expansion, this evolution reduces to a nonlinear partial differential equation amenable to both symbolic and numerical analysis. Our symbolic analysis shows that a specific nonlinear term in the partial differential equation prohibits a Hamiltonian formulation of the problem. Using conservation laws associated with the local induction approximation, we consider how the non-Hamiltonian term gives rise to a dissipative mechanism allowing gain and loss of curvature in the vortex medium. Additionally, we derive a nonlinear dispersion relation predicting that low wavenumber helical modes jettison from locally curved regions. Consequently, curved regions seek to relax their bending through the production of helical structures. If the curvature is created through a vortex reconnection, then helical Kelvin waves couple the vortex to the Bose-Einstein condensate for the purposes of phonon generation and are expected to be the route to decay in free quantum turbulence. We corroborate these predictions with simulations of corrections to initial states predicted by the local induction approximation. Specifically, we simulate out of plane perturbations of a vortex ring, along with soliton and breathing states. A detailed analysis of the soliton state shows a transition to a log normal distribution where curvature disperses ahead of the traveling wave. On the vortex, this appears as a helical wavefront propagating into an otherwise straight line. The non-Hamiltonian gain mechanism acts to support Kelvin modes as they travel and keeps the initial peak profile from completely eroding, a feature indicative of a dissipative soliton. These dynamics were also seen in both the breathing and ring states. Thus, geometric quantum hydrodynamics predicts that the simplest flow of curved regions on a vortex line generates Kelvin waves providing a route to the anomalous dissipation first predicted by Onsager.
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