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Physical Review

Vortices in an Imperfect Bose Gas. I. The Condensate

Alexander L. Fetter

Published Apr 19, 1965• 146 citations
Publisher

Abstract

A theoretical study of rectilinear vortices in an imperfect Bose gas shows a close correspondence with classical hydrodynamics. The energy and momentum of a vortex pair in an unbounded fluid are calculated. The similarity between a vortex pair and a vortex ring leads to an estimate of the critical velocity ${v}_{c}$ of liquid He II in a tube of radius $R$ that includes the effect of the walls ${v}_{c}=\frac{C\ensuremath{\hbar}}{2mR}$, where $C$ is a constant of order unity. A variational treatment of a system of many identical vortices in a container shows that the energy is lowest for a uniform distribution, and that the number of vortices per unit area $\ensuremath{\nu}$ agrees with Feyman's result $\ensuremath{\nu}=\frac{2m\ensuremath{\omega}}{h}$. In the classical limit ($h\ensuremath{\rightarrow}0$), the angular momentum and energy approach the values for solid-body rotation.

DOI
10.1103/physrev.138.a429

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Journal of Physics A Mathematical and General•Jul 20, 1999•56 citations
A theoretical study of rectilinear vortices in an imperfect Bose gas shows a close correspondence with classical hydrodynamics. The energy and momentum of a vortex pair in an unbounded fluid are calculated. The similarity between a vortex pair and a vortex ring leads to an estimate of the critical velocity vc{v}_{c}vc​ of liquid He II in a tube of radius RRR that includes the effect of the walls vc=C\ensuremathℏ2mR{v}_{c}=\frac{C\ensuremath{\hbar}}{2mR}vc​=2mRC\ensuremathℏ​, where CCC is a constant of order unity. A variational treatment of a system of many identical vortices in a container shows that the energy is lowest for a uniform distribution, and that the number of vortices per unit area \ensuremathν\ensuremath{\nu}\ensuremathν agrees with Feyman's result \ensuremathν=2m\ensuremathωh\ensuremath{\nu}=\frac{2m\ensuremath{\omega}}{h}\ensuremathν=h2m\ensuremathω​. In the classical limit (h\ensuremath→0h\ensuremath{\rightarrow}0h\ensuremath→0), the angular momentum and energy approach the values for solid-body rotation.
The translational velocity of a vortex with circulation hm\frac{h}{m}mh​ in an imperfect Bose gas is calculated by considering the time dependence of the condensate wave function corresponding to a given initial configuration of vortices. Each vortex in a system of rectilinear vortices is shown to move with the local fluid velocity at its core; a vortex ring of radius RRR is shown to move with velocity $(\frac{h}{4\ensuremath{\pi}\mathrm{mR}})\mathrm{ln}(\frac{8R}{\ensuremath{\alpha}})$, where $\ensuremath{\alpha}$ is a length the order of the core size. Both results agree with the predictions of classical hydrodynamics.
The energy and momentum associated with vortex formation in an imperfect Bose gas have been computed quantum mechanically. In the computation, explicit account is taken of the boundary condition to be satisfied by the wave function at the walls surrounding the fluid. Numerical estimates of the critical velocity obtained thereby throw considerable light on the question of the most favorable location (relative to the walls) for the formation of the vortex. It is concluded that the optimum location for this formation could be well within the fluid---significantly away from the walls.
The single-particle excitations of an imperfect Bose gas with a vortex in the condensate are studied using the Bogoliubov approximation. The problem is reduced to an eigenvalue equation, whose solution yields both the single-particle energies and the eigenfunction expansion of the single-particle Green's function. The spectrum of the eigenvalues contains a discrete portion (bound states) and a continuous portion (scattering states). The lowest eigenvalue corresponding to angular momentum l=0l=0l=0 and $l=\ifmmode\pm\else\textpm\fi{}1$ is determined variationally. In the long-wavelength limit, the eigenvalue for $l=\ifmmode\pm\else\textpm\fi{}1$ agrees with the frequency of normal modes calculated by Kelvin and Pitaevskii. The density of noncondensed particles is expressed in terms of the single-particle Green's function and is shown to be finite at the center of the vortex.
The theory of the condensate of a weakly interacting Bose gas is developed. The condensate is described by a wavefunction ψ(x, t) normalized to the number of particles. It obeys a nonlinear self-consistent field equation. The solution in the presence of a rigid wall with the boundary condition of vanishing wavefunction involves a de Broglie length. This length depends on the mean potential energy per particle. The self-consistent field term keeps the density uniform except in localized spatial regions. In the hydrodynamical version, a key role is played by the quantum potential. A theory of quantized vortices and of general potential flows follows immediately. In contrast to classical hydrodynamics, the cores of vortices are completely determined by the de Broglie length and all energies are finite. Nonstationary disturbances of the condensate correspond to phonons, rotons, vortex waves etc. They can exchange momentum with rigid boundaries. This is compatible with the vanishing of the wavefunction at a boundary. This condition fully determines the dynamics of the system. These points are illustrated by considering the motion of a foreign ion in a Bose gas, a rotating container of fluid, and the Landau criterion for superfluidity.
Vortex rings are considered as flow patterns of a phenomenological Hartree-type fluid with parameters fitted to the observed phonon-roton spectrum. The basis of the calculations is variational, and density patterns which are constant except in vortex cores are employed. The energy of vortex lines and rings is expressed in terms of a core radius ccc, defined as the distance from the singular line of the vortex to the point of half the asymptotic density. The core size is determined by minimizing the energy and is found to be independent of the ring radius when the latter is larger than ten times the core radius. Comparing the results with the data of Rayfield and Reif, one finds c=1.7c=1.7c=1.7 \AA{}, while the phenomenological model yields half this value. The critical velocity associated with ring formation in a channel is considered following the lines of Fetter's earlier calculation. In the present calculation it is possible to incorporate the Fineman-Chase analysis of channel wall effects. Agreement with experiment is considerably improved.
Nonlocal nonlinear Schrödinger equations are considered as models of liquid helium II. The models contain a nonlocal interaction potential that leads to a phonon-roton-like dispersion relation. Also, a higher-order term in the local density approximation for the correlation energy is introduced into the model to overcome nonphysical mass concentrations. These equations are solved for straight-line vortices. It is demonstrated that the parameters of the equation can be chosen to bring into agreement the vortex core parameter and the healing length. The structure of vortex rings of large radius is studied. The family of the vortex rings of different radii propagating with different velocities is found numerically. As the velocity of the vortex ring reaches the Landau critical velocity the sequence of rings terminates.