In this note we provide an analytical proof of the zero helicity condition for systems governed by the Gross–Pitaevskii equation (GPE). The proof is based on the hydrodynamic interpretation of the GPE, and the direct use of Noether's theorem by applying Kleinert's multi-valued gauge theory. As a by-product we also demonstrate the conservation and quantization of the circulation for the GPE.
In this note we provide an analytical proof of the zero helicity condition for systems governed by the Gross–Pitaevskii equation (GPE). The proof is based on the hydrodynamic interpretation of the GPE, and the direct use of Noether's theorem by applying Kleinert's multi-valued gauge theory. As a by-product we also demonstrate the conservation and quantization of the circulation for the GPE.
Abstract Here we show that the Gross–Pitaevskii equation (GPE) for Bose–Einstein condensates (BECs) admits hydrodynamic interpretation in a general Riemannian metric, and show that in this metric the momentum equation has a new term that is associated with local curvature and density distribution profile. In particular conditions of steady state a new Einstein’s field equation is determined in presence of negative curvature. Since GPE governs BECs defects that are useful, analogue models in cosmology, a relativistic form of GPE is also considered to show connection with models of analogue gravity, thus providing further grounds for future investigations of black hole dynamics in cosmology.
Abstract In this paper we show that twist, defined in terms of rotation of the phase associated with quantum vortices and other physical defects effectively deprived of internal structure, is a property that has observable effects in terms of induced axial flow. For this we consider quantum vortices governed by the Gross–Pitaevskii equation (GPE) and perform a number of test cases to investigate and compare the effects of twist in two different contexts: (i) when this is artificially superimposed on an initially untwisted vortex ring; (ii) when it is naturally produced on the ring by the simultaneous presence of a central straight vortex. In the first case large amplitude perturbations quickly develop, generated by the unnatural setting of the initial condition that is not an analytical solution of the GPE. In the second case much milder perturbations emerge, signature of a genuine physical process. This scenario is confirmed by other test cases performed at higher twist values. Since the second setting corresponds to essential linking, these results provide new evidence of the influence of topology on physics.
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…
This work rectifies the hydrodynamic equations commonly used to describe the superfluid velocity field in such a way that vortex dynamics are also taken into account. In the field of quantum turbulence, it is of fundamental importance to know the correct form of the equations which play similar roles to the Navier-Stokes equation in classical turbulence. Here, such equations are obtained by carefully taking into account the frequently overlooked multivalued nature of the U(1) phase field. Such an approach provides exact analytical explanations to some numerically observed features involving the dynamics of quantum vortices in Bose-Einstein condensates, such as the universal t1/2 behavior of reconnecting vortex lines. It also expands these results beyond the Gross-Pitaevskii theory so that some features can be generalized to other systems such as superfluid 4He, dipolar condensates, and mixtures of different superfluid systems.
Using the phi-mapping topological theory, we study the topological structure of vortex lines in a two-dimensional generalized Gross-Pitaevskii theory in (3+1)-dimensional space-time. We obtain the reduced dynamic equation in the framework of the two-dimensional Gross-Pitaevskii theory, from which a conserved dynamic quantity is derived on the stable vortex lines. Such equations can also be used to discuss Bose-Einstein condensates in heterogeneous and highly nonlinear systems. We obtain an exact dynamic equation with a topological term, which is ignored in traditional hydrodynamic equations. The explicit expression of vorticity as a function of the order parameter is derived, where the delta function indicates that the vortices can only be generated from the zero points of Phi and are quantized in terms of the Hopf indices and Brouwer degrees. The phi-mapping topological current theory also provides a reasonable way to study the bifurcation theory of vortex lines in the two-dimensional Gross-Pitaevskii theory.
The aim of this paper is to derive the equations of motion of quantum vortices in a weakly non-ideal Bose–Einstein condensate (BEC) at finite temperatures and in the presence of macroscopic superfluid and normal velocities. The motivation for this work is that the dynamics of quantum vortices in superfluid helium has been studied at the phenomenological level, and a number of issues require clarification, which can be addressed by studying the corresponding problem in BEC. In BEC, quantum vortices are located at points where the macroscopic wave function (MWF) ψ(r,T), which describes the BEC dynamics, vanishes ψ(r,T)=0. The intersection of the surfaces Reψ(r,T)=0 and Imψ(r,T)=0 forms a one-dimensional line, which is just the quantum vortex line. During the evolution of the MWF, the position of the zero surfaces changes, that is, the vortex lines move. Conversely, the motion of these vortex filaments determines the evolution of the MWF. Thus, there is a mutual correspondence, and the quantity ψ(r,T) can be considered as a functional of the entire vortex configuration, i.e., {s(ξ,t)}. This viewpoint allows us to derive the equations of motion for vortices from the Gross–Pitaevskii equation (GPE), which describes the dynamics of MWF. The key finding is the confirmation that the equation derived from the fundamental quantum description (the GPE) for a BEC has the same mathematical structure as the equation derived from macroscopic, phenomenological reasoning for superfluid helium.