In superconductors, as in Helium II (He II), the holes round which quantisation occurs can be either physical boundaries, or the cores of quantised vortex lines. For convenience we summarise here the resemblances and differences between vortex lines and vortex states in He II and in superconductors. In contrast, states in which a whole vortex is attached to the wire should be comparatively stable. The rotation of the superfluid can be satisfactorily explained by assuming that it is threaded by a series of parallel straight vortex lines. Consider a single straight vortex line in the superfluid. The energy associated with a single vortex line is mainly the kinetic energy of the circulating superfluid. The number and separation of vortex lines present for a given angular velocity clearly depends on their individual circulations. Two rival models for the ion-vortex transition have been proposed, generally known as the peeling model and the quantum transition model.
In superconductors, as in Helium II (He II), the holes round which quantisation occurs can be either physical boundaries, or the cores of quantised vortex lines. For convenience we summarise here the resemblances and differences between vortex lines and vortex states in He II and in superconductors. In contrast, states in which a whole vortex is attached to the wire should be comparatively stable. The rotation of the superfluid can be satisfactorily explained by assuming that it is threaded by a series of parallel straight vortex lines. Consider a single straight vortex line in the superfluid. The energy associated with a single vortex line is mainly the kinetic energy of the circulating superfluid. The number and separation of vortex lines present for a given angular velocity clearly depends on their individual circulations. Two rival models for the ion-vortex transition have been proposed, generally known as the peeling model and the quantum transition model.
In the previous chapter we described quantized vortex lines, which are one of the characteristic features of superfluids. In a classical fluid, the circulation of vortex lines is not quantized and, in addition, vortex lines decay because of viscous processes. Another feature of a superfluid, the lack of response to rotation for a small enough angular velocity, was also demonstrated. This is analogous to the Meissner effect for superconductors. One characteristic common to superfluids and superconductors is the ability to carry currents without dissipation. Such current-carrying states are not the lowest-energy state of the system. They are metastable states, the existence of which is intimately connected to the nature of the low-lying elementary excitations. The word ‘superfluidity’ does not refer to a single property of the system, but it is used to describe a variety of different phenomena (see Ref. [1]). Historically, the connection between superfluidity and the existence of a condensate, a macroscopically occupied quantum state, dates back to Fritz London's suggestion in 1938, as we have described in Chapter 1. However, the connection between Bose–Einstein condensation and superfluidity is a subtle one. A Bose–Einstein condensed system does not necessarily exhibit superfluidity, an example being the ideal Bose gas for which the critical velocity vanishes, as demonstrated in Sec. 10.1 below. Also lower-dimensional systems may exhibit superfluid behaviour in the absence of a true condensate, as we shall see in Chapter 15.
This chapter focuses on "steady" superfluid flow, in which the superfluid velocity, vs, satisfies the equation div vs=0. Such flow can, in principle, occur only in multiply-connected systems. However, it is possible to achieve an equivalent situation in the bulk of the liquid by setting up vortex lines. The major new feature brought in by superfluidity, and a most important one indeed, is the quantization of vortex lines, arising as a consequence of the quantization of circulation. The square terms in correspond to the energy of the single vortex lines, while the cross term describes an interaction energy between two vortex lines. In the absence of vortex lines, the superfluid and normal components of a Bose liquid are completely decoupled. The velocity pattern of the vortex gives rise to a position-dependent energy, which may be used to calculate the scattering. The dynamics of superfluid flow is thus appreciably modified by the presence of vortex lines.
The authors consider the core structures of quantised vortex lines in Bose and Fermi superfluid systems. They emphasise the result that, within the core of a singly quantised vortex line in a Bose superfluid, there spontaneously appears a condensate of boson pairs in a relative d-wave state, instead of the normal fluid. Also, in the case of the singly quantised Abrikosov vortex line in a superconductor, instead of normal electrons there emerges a novel state of superconductivity, in which a condensate of bosons made of four fermions appears. In particular, they discuss these core structures as further examples for the flaring-out of the vortex singularity into higher dimensions, in analogy with the extended-space topology of the superfluid 3 He-B vortex core; they also relate this generic behaviour to the Laughlin state within the quantum Hall effect.
Evidence is presented that below a rather large angular velocity, ${\ensuremath{\omega}}_{c1}$, there is a vortex-free state in rotating locked helium, which we call the Landau state. This is similar to the Meissner state in superconductors. We further show that all the properties of the rotating helium have counterparts in irreversible type-II superconductors and find that ${\ensuremath{\omega}}_{c2}$ (analogous to Hc2) greatly exceeds our high speeds of rotation.
This doctoral thesis consists of experimental studies on rotating superfluid helium-3 at millikelvin temperatures. Superfluid 3He provides an example of ordered systems where quantum phenomena are visible on a macroscopic scale. The order parameter is described by a 3x3 complex matrix leading to different superfluid phases with different properties. The A and B phases support topologically different kinds of defects, most notably vortex lines. Vortex lines are most easily created and studied in uniform rotation. Because of quantum nature, the circulation around the vortices is quantized in units of κ0 = h/2m3.\n\nIn superfluid 3He nuclear magnetic resonance (NMR) provides a means to distinguish between different types of vortices and count them one by one. During the course of this work the sensitivity of the NMR spectrometer was improved by an order of magnitude, which enabled the direct measurement of the circulation of a vortex in the A phase that carries two circulation quanta (2κ0). Up to now the circulation of the known quantized vortex types in other quantum fluids, which include superfluid 4He, superconductors, and Bose-Einstein -condensates, has been singly quantized.\n\nOther NMR experiments reported in the thesis are: (1) The first qualitative measurements of vortex annihilation in the A and B phases of 3He. These measurements tell the lowest rotation velocity that stabilizes an array of vortex lines with a given number of vortices, and how it depends on the type of the defect. (2) Studies of vortex arrays with both singly and doubly quantized A phase vortex lines coexisting in the experimental container. 3He-A provides a unique system in which this kind of arrangement can be done. (3) In order to improve the vortex detection sensitivity in the B phase even further, the response of different nonlinear spin precession modes to superfluid-normal fluid counterflow was studied. A new mode, where the orbital angular momentum is oriented transverse to the external magnetic field, was found.
A comprehensive overview of the basic principles of vortex dynamics in superfluids, this book addresses the problems of vortex dynamics in all three superfluids available in laboratories (4He, 3He, and BEC of cold atoms) alongside discussions of the elasticity of vortices, forces on vortices, and vortex mass. Beginning with a summary of classical hydrodynamics, the book guides the reader through examinations of vortex dynamics from large scales to the microscopic scale. Topics such as vortex arrays in rotating superfluids, bound states in vortex cores and interaction of vortices with quasiparticles are discussed. The final chapter of the book considers implications of vortex dynamics to superfluid turbulence using simple scaling and symmetry arguments. Written from a unified point of view that avoids complicated mathematical approaches, this text is ideal for students and researchers working with vortex dynamics in superfluids, superconductors, magnetically ordered materials, neutron stars and cosmological models.