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Physical review. B, Condensed matter

Unwinding of a single quantized vortex from a wire

Klaus W. Schwarz

Published May 1, 1993• 59 citations
Publisher

Abstract

The dynamical behavior of a quantized vortex partially attached to a wire is studied theoretically, with the aim of interpreting recent experiments on quantized circulation in superfluid $^{3}\mathrm{B}$. The geometry considered consists of a thin wire running parallel to the axis of a circular cylinder enclosing the wire. The circulation is assumed to run part way up the wire, and then to enter the fluid as a free vortex which eventually terminates on the outer wall. It is found that such a vortex achieves a state of steady precession around the wire, accompanied by a steady unwinding motion down the wire due to frictional effects. For an off-center wire, both the precession rate and the unwinding rate develop oscillatory components. Various particulars, such as the effects of friction, of moving the wire off center, and of pinning, are investigated. Excellent agreement is obtained between experiment, analytical theory, and numerical calculations.

DOI
10.1103/physrevb.47.12030

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The dynamical behavior of a quantized vortex partially attached to a wire is studied theoretically, with the aim of interpreting recent experiments on quantized circulation in superfluid 3B^{3}\mathrm{B}3B. The geometry considered consists of a thin wire running parallel to the axis of a circular cylinder enclosing the wire. The circulation is assumed to run part way up the wire, and then to enter the fluid as a free vortex which eventually terminates on the outer wall. It is found that such a vortex achieves a state of steady precession around the wire, accompanied by a steady unwinding motion down the wire due to frictional effects. For an off-center wire, both the precession rate and the unwinding rate develop oscillatory components. Various particulars, such as the effects of friction, of moving the wire off center, and of pinning, are investigated. Excellent agreement is obtained between experiment, analytical theory, and numerical calculations.
This paper reports the discovery of a new vortex phenomenon in superfluid 3B^{3}\mathit{B}3B. A single filament of quantized vortex line precesses as a solid body around a wire extending along the axis of a cylinder. The precession frequency equals the angular velocity of the apparatus at which the presence of a single quantum of circulation minimizes the system's free energy. The period of precession is related to the circulation quantum and the dimensions of the apparatus. Thus a measurement of the precession period is an accurate determination of the quantum unit. We find \ensuremath{\kappa}=(1.020\ifmmode\pm\else\textpm\fi{}0.03)h/2m3{\mathit{m}}_{3}m3​, where m3{\mathit{m}}_{3}m3​ is the mass of the 3He^{3}\mathrm{He}3He atom.
In two dimensions a microscopic theory providing a basis for the naive analogy between a quantized vortex in a superfluid and an electron in a uniform magnetic field is presented. Following the variational approach developed by Peierls, Yoccoz, and Thouless, the cyclotron motion of a vortex is described by the many-body wave function, which is a linear combination of Feynman wave functions centered at different positions. An integral equation for the weighting functions of the superposition is derived by minimizing the energy functional. The matrix elements of the kernel are the overlaps between any two displaced Feynman wave functions. A numerical study is conducted for a bosonic superfluid based on a Hartree ground state. A one-to-one correspondence between the rotational states of a vortex in a cylinder and the cyclotron states of an electron in the central gauge is found. Like the Landau levels of an electron, the energy levels of a vortex are highly degenerate. However, the gap between two adjacent energy levels does not only depend on the quantized circulation, but also increases with energy, and scales with the size of the vortex. The fluid density is finite at the vortex axis and the vorticity is distributed in the core region. The effective mass of a quantized vortex defined by the inverse of the energy-level spacing is shown to be logarithmically divergent with the size of the vortex.
We report the generation of a quantized vortex in superfluid 3He-B^{3}\text{H}\text{e-B}3He-B. Oscillating objects can generate vortices due to breaking of Cooper pairs in superfluid 3He^{3}\text{H}\text{e}3He, resulting that the threshold velocity for vortex generation is higher than the pair-breaking velocity. Using very-low-frequency vibrating wires with smooth surfaces, however, we find that the threshold velocity of vortex generation is lower than the pair-breaking velocity. The threshold velocity remains constant with decreasing 3He^{3}\text{H}\text{e}3He pressure until the pair-breaking velocity equals the vortex-generation threshold velocity at a low pressure. These results indicate that another mechanism that is independent of pressure generates vortices for the low-frequency vibrating wire in superfluid 3He^{3}\text{H}\text{e}3He. The threshold velocity observed in experiments turns out to be nearly equal to the velocity of the instability of the vortex lines attached to the wire, which is characterized by only the circulation quantum.
Insight in the dynamics of quantized vortex lines has been obtained by performing NMR measurements in uniformly rotating superfluid 3B^{3}\mathit{B}3B. Compared to superfluid 4He^{4}\mathrm{He}4He, in 3B^{3}\mathit{B}3B the viscosity of the normal component is three orders of magnitude larger and the pinning of vortices is weaker. The collective modes governing the hydrodynamic response of an array of vortex lines are highly overdamped. Two distinct modes are identified: (1) a relatively fast mutual-friction-resisted mode, which controls the redistribution of the vortex density on a time scale of a few seconds, and (2) an exponentially relaxing slow mode with a time constant of a few minutes, which governs asymptotically the approach to equilibrium at nearly constant vortex density. Measurement of the fast motion allows us to extract the dissipative mutual friction. The slow mode is dominated by the elastic tension along the vortex line and by weak pinning at the top and bottom surfaces of the rotating container. With increasing vortex density the surface pinning becomes a collective process, in which coherently moving groups of vortices are pinned in unison. Collective pinning is a result of the shear elasticity related to the crystalline order of the vortex lattice. Our measurements on the slow mode indicate that the crystalline correlation extends over many lattice spacings.