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Physical Review

Theory of Liquid Helium II in a Rotating Annulus: Ginzburg-Pitaevskii Approach

Alan J. Bennett, L. M. Falicov

Published Apr 8, 1966• 1 citation
Publisher

Abstract

The Ginzburg-Pitaevskii two-fluid formulation of the superfluidity problem is used to describe the behavior of He II in a rotating annular cylinder. The equations are solved with the assumption that the thickness of the annulus is (a) much smaller than the mean radius and (b) much larger than the superfluid healing length. Two cases are considered; (i) constant angular velocity, (ii) constant angular momentum. Curves are given for the free energy and the superfluid-component density as a function of angular velocity or angular momentum.

DOI
10.1103/physrev.144.162

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The Ginzburg-Pitaevskii two-fluid formulation of the superfluidity problem is used to describe the behavior of He II in a rotating annular cylinder. The equations are solved with the assumption that the thickness of the annulus is (a) much smaller than the mean radius and (b) much larger than the superfluid healing length. Two cases are considered; (i) constant angular velocity, (ii) constant angular momentum. Curves are given for the free energy and the superfluid-component density as a function of angular velocity or angular momentum.
We report on the first mathematically rigorous proofs of a transition to a giant vortex state of a superfluid in rotating anharmonic traps. The analysis is carried out within two-dimensional Gross-Pitaevskii theory at large coupling constant and large rotational velocity and is based on precise asymptotic estimates on the ground state energy. An interesting aspect is a significant difference between 'soft' anharmonic traps (like a quartic plus quadratic trapping potential) and traps with a fixed boundary. In the former case vortices persist in the bulk until the width of the annulus becomes comparable to the size of the vortex cores. In the second case the transition already takes place in a parameter regime where the size of vortices is very small relative to the width of the annulus. Moreover, the density profiles in the annulus are different in the two cases. In both cases rotational symmetry of the density in a true ground state is broken, even though a symmetric variational ansatz gives an excellent approximation to the energy.
We report on the first mathematically rigorous proofs of a transition to a giant vortex state of a superfluid in rotating anharmonic traps. The analysis is carried out within two-dimensional Gross-Pitaevskii theory at large coupling constant and large rotational velocity and is based on precise asymptotic estimates on the ground state energy. An interesting aspect is a significant difference between 'soft' anharmonic traps (like a quartic plus quadratic trapping potential) and traps with a fixed boundary. In the former case vortices persist in the bulk until the width of the annulus becomes comparable to the size of the vortex cores. In the second case the transition already takes place in a parameter regime where the size of vortices is very small relative to the width of the annulus. Moreover, the density profiles in the annulus are different in the two cases. In both cases rotational symmetry of the density in a true ground state is broken, even though a symmetric variational ansatz gives an excellent approximation to the energy.
The Bose-Einstein condensation mechanism is shown to be capable of accounting for the existence of separate hydrodynamic velocity fields for the normal and the superfluid, provided suitable assumptions are made with regard to the single-particle energy spectrum. We consider the effect on the microcanonical gas distribution of imposing a nonzero value of the total momentum P. For an ordinary gas the effect is trivial, the whole distribution being merely shifted in momentum space. However, in the case of a degenerate Bose gas whose single-particle energy spectrum has a sharp minimum (gap or cusp), only the excited part of the gas (normal fluid) participates in the imposed motion, the condensate (superfluid) remaining "frozen" in momentum space. This rigidity of the condensate in momentum space plays the same role as the rigidity of the superelectrons on imposition of a magnetic field in the London theory of superconductivity. P acts as an additional thermodynamic variable, states with P \ensuremath{\ne} 0 being macroscopically metastable and corresponding to the existence of a relative velocity between normal and superfluid. The basic hydrodynamic assumption of the two-fluid model is thus reduced to an assumption concerning the form of an effective single-particle energy spectrum, and the parallelism between the theories of superconductivity and superfluidity is clearly exhibited. The present theory permits, in particular, the introduction of the mathematical form of Landau's phonon and "roton" spectra within the framework of the Bose-Einstein condensation picture. The statistical-thermodynamic formulas are derived and are shown to lead to characteristic two-fluid equations derived previously from a variational principle.
The famous two-fluid model of finite-temperature superfluids has been recently extended to describe the mixed classical-superfluid dynamics of the newly discovered supersolid phase of matter. We show that for rigidly rotating supersolids one can derive a more appropriate single-fluid model, in which the seemingly classical and superfluid contributions to the motion emerge from a spatially varying phase of the global wave function. That allows one to design experimental protocols to excite and detect the peculiar rotation dynamics of annular supersolids, including partially quantized supercurrents, in which each atom brings less than ℏ units of angular momentum. Our results are valid for a more general class of density-modulated superfluids.
A quantum liquid of an almost ideal Bose gas brought into rotation is investigated from a physical and mechanical point of view on the basis of the Gross–Pitaevskii (GP) equation by applying a quantum-mechanical scenario, i.e. the London scenario. This scenario allows a superfluid to have rotational states. Considering that the equation governs an interacting Bose gas, it is proposed that the GP equation admits rotational flows of a superfluid. This is carried out without incurring essential change of the equation. By this reformulation, a superfluid placed in a rotating vessel is able to have a solid body rotation with the same angular velocity as its container and also to have a meniscus approximated by a parabolic profile. The solid body rotation is accompanied by a density increase proportional to the square of its angular velocity. These are consistent with experimental observations. In addition, this formulation allows a vortex of quantized circulation with coaxial rotational core whose density does not necessarily vanish at its center.