Introduction to the vortex sheet of superfluid He-A

It is well known that superfluids respond to rotation by forming vortex lines. It has been recently discovered that a different type of state consisting of a vortex sheet, instead of lines, can be created in the A phase of superfluid ³He. This paper presents an introduction to the vortex sheet. We first discuss ⁴He, where a vortex sheet is unstable. The way to realize a stable sheet in ³He-A is called a vortex soliton. It consists of a topologically stable domain wall to which nonsingular vorticity is bound. The vortex soliton has been observed by nuclear magnetic resonance, and its most prominent experimental properties are explained. The macroscopic shape of the sheet and the superfluid flow in a rotating container are discussed.     

Vortex sheet in superfluid<Superscript>3</Superscript>He-A

The vortex sheet (VS) is the most unexpected discovery in rotating superfluids during the last ten years. Usually superfluids respond to rotation by creating an array of vortex lines, which are parallel to the rotation axis, and the circulation around them is quantized. In the VS the vorticity is located on a 2 dimensional sheet that folds to equidistant layers in a rotating container. The VS is one out of five stable vortex structures in³He-A. The stability of the VS in³He-A arises from a special structure, which consists of a nonsingular vorticity bound to a topologically stable domain wall. The vortex sheet forms the equilibrium state of³He-A at rotation velocities larger than ～3 rad/s, but it is also created as metastable state at lower velocities. Experimentally the vortex sheet is distinguished from its NMR response.     

Superfluid vortex front at T→0: decoupling from the reference frame

Steady-state turbulent motion is created in superfluid (3)He-B at low temperatures in the form of a turbulent vortex front, which moves axially along a rotating cylindrical container of (3)He-B and replaces vortex-free flow with vortex lines at constant density. We present the first measurements on the thermal signal from dissipation as a function of time, recorded at 0.2T(c) during the front motion, which is monitored using NMR techniques. Both the measurements and the numerical calculations of the vortex dynamics show that at low temperatures the density of the propagating vortices falls well below the equilibrium value, i.e., the superfluid rotates at a smaller angular velocity than the container. This is the first evidence for the decoupling of the superfluid from the container reference frame in the zero-temperature limit.     

Vortex-lattice dynamics in rotating spinor Bose-Einstein condensates

We observe interlaced square vortex lattices in rotating dilute-gas spinor Bose-Einstein condensates (BEC). After preparing a hexagonal vortex lattice in a one-component BEC in an internal atomic state |1, we coherently transfer a fraction of the superfluid to a different state |2. The subsequent evolution of this pseudo-spin-1/2 superfluid towards a state of offset square lattices involves an intriguing interplay of phase-separation and -mixing dynamics, both macroscopically and on the length scale of the vortex cores, and a stage of vortex turbulence. The stability of the square structure is proved by its response to applied shear perturbations. An interference technique shows the spatial offset between the two vortex lattices. Vortex cores in either component are filled by fluid of the other component, such that the spin-1/2 order parameter forms a Skyrmion lattice.     

Critical velocity of continuous vortex nucleation in a slab of superfluid 3He-A

A flow-induced Fréedericksz transition is observed in a 0.26 mm thick disk-shaped slab of superfluid 3He-A using a rotating cryostat and a torsional oscillator, and it is used to detect vortices in zero magnetic field. The phenomena are studied as a function of magnetic field normal to the slab. In defect-free l texture the critical velocity for vortex nucleation is 0.5 mm/s, but in the presence of a domain wall it is reduced to approximately Planck's over 2pi /2ma(c), where a(c)(H) is the field-dependent radius of the vortex soft core. The vortices nucleate at a distance at least 0.3 mm from the outer edge of the disk.     

Composite defect extends analogy between cosmology and 3He

Spin-mass vortices have been observed to form in rotating superfluid 3He-B, following the absorption of a thermal neutron and a rapid transition from the normal to the superfluid state. The spin-mass vortex is a composite defect which consists of a planar soliton (wall) which terminates on a linear core (string). This observation fits well within the framework of a cosmological scenario for defect formation, known as the Kibble-Zurek mechanism. It suggests that in the early Universe analogous cosmological defects might have formed.     

Vortex molecules in coherently coupled two-component Bose-Einstein condensates

A vortex molecule is predicted in rotating two-component Bose-Einstein condensates whose internal hyperfine states are coupled coherently by an external field. A vortex in one component and one in the other are connected by a domain wall of the relative phase, constituting a "vortex molecule," which features a nonaxisymmetric (pseudo)spin texture with a pair of merons. The binding mechanism of the vortex molecule is discussed based on a generalized nonlinear sigma model and a variational ansatz. The anisotropy of vortex molecules is caused by the difference in the scattering lengths, yielding a distorted vortex-molecule lattice in fast rotating condensates.     

Vortex arrays of coexisting singly and doubly quantized vortex lines in<Superscript>3</Superscript>He-A

In superfluid³He-A singly and doubly quantized vortex lines can coexist in a rotating container. We measure with NMR techniques the radial distribution of the two vortex types in an array of vortex lines. The radial composition is found to depend on the procedure by which the array has been formed. The result shows that in superfluid³He the energy barriers separating different configurations of the vortex array are inpenetrably high for a metastable state to relax.     

Structure of the surface vortex sheet between two rotating 3He superfluids

We study a two-phase sample of superfluid 3He where vorticity exists in one phase (3He-A) but cannot penetrate across the interfacial boundary to a second coherent phase (3He-B). We calculate the bending of the vorticity into a surface vortex sheet on the interface and solve the internal structure of this new type of vortex sheet. The compression of the vorticity from three to two dimensions enforces a structure which is made up of 1 / 2-quantum units, independently of the structure of the source vorticity in the bulk. These results are consistent with our NMR measurements.     

Topological Vortices in Superfluid Films

We study the topological structure of the vortex system in a superfluid film. Explicit expressions for the vortex density and velocity field as functions of the superfluid order parameter are derived. The evolution of vortices is also studied from the topological properties of the superfluid order parameter field.     

Topological Vortices in Superfluid Films

We study the topological structure of the vortex system in a superfiuid film. Explicit expressions for the vortex density and velocity field as functions of the superfluid order parameter are derived. The evolution of vortices is also studied from the topological properties of the superfluid order parameter field.     

Intrinsic pinning of vorticity by domain walls of l texture in superfluid 3He-A

We present the first observation of substantial persistent flow in superfluid 3He-A in thick simply connected slabs in a zero magnetic field, but only in l textures with domain walls. The flow is induced in a rotating cryostat using a torsional oscillator as a probe. The hysteretic dependences of the trapped vorticity on the maximal angular velocity of rotation are fairly universal for different densities of domain walls and slab thicknesses. A model of a critical state set by either the critical velocity for vortex nucleation or pinning strength explains all observations.     

Twisted vortex state

We study a twisted vortex bundle where quantized vortices form helices circling around the axis of the bundle in a "force-free" configuration. Such a state is created by injecting vortices into a rotating vortex-free superfluid. Using continuum theory we determine the structure and the relaxation of the twisted state. This is confirmed by numerical calculations. We also present experimental evidence of the twisted vortex state in superfluid 3He-B.     

Vortex lattices in a rotating Fermi superfluid in the BCS–BEC crossover with many Landau levels

We present an explicit analytical analysis of the ground state of vortex lattice structure, based on a minimization of the generalized Gross–Pitaevskii energy functional in a trapped rotating Fermi superfluid gas. By a Bogoliubov-like transformation we find that the coarse-grained average of the atomic density varies as inverted parabola in three dimensional cases; the Fermi superfluid in the BEC regime enters into the lowest Landau level at fast rotation, in which the vortices form an almost regular triangular lattice over a central region and the vortex lattice is expanded along the radial direction in the outer region; the fluid in the unitarity and BCS regimes occupies many low-lying Landau levels, in which a trapped gas with a triangular vortex lattice has a superfluid core surrounded by a normal gas. The calculation is qualitatively consistent with recent numerical and experimental data both in the vortex lattice structure and vortex numbers and in the density profiles versus the stirring frequency in the whole BCS–BEC crossover.     

Spin wave and vortex excitations of superfluid 3He-A in parallel-plate geometry

Quantized vortices with half-integer circulation, which are forbidden from existing in a conventional superfluid because of the single valueness of the wave function, are theoretically predicted to exist in superfluid 3He-A if the order parameters l over and d over form l over perpendicular d over texture. To form the l over perpendicular d over texture, we confined the superfluid between parallel plates with a 12.5 microm gap and applied a magnetic field of H=26.7 mT perpendicular to the plates to take NMR and orient d over perpendicular to l over. NMR spectra exhibit a negative-shift peak which probes that the uniform l over perpendicular d over texture is realized in our cell and show a new satellite signal under rotation. The rotation dependence of the satellite signal is interpreted that a Fréedericksz transition of l over texture is induced by rotation above 1.0 rad/s and vortices start to appear above 1.8 rad/s.     

Annihilation of quantized vortex lines in rotating<Superscript>3</Superscript>He-A

The maximum number of vortex lines at a given angular velocity Ω is obtained during deceleration when vortices annihilate at the wall of the container. The width of the vortex-free region in the decelerated state has been measured by cw NMR for vortex lines with singular and continuous core structures. It corresponds to the limit of stability of peripheral vortices for single and double quantization respectively.     

Majorana bound state in rotating superfluid 3He-A between parallel plates

A concrete and experimentally feasible example for testing the putative Majorana zero-energy state bound in a vortex is theoretically proposed for a parallel plate geometry of superfluid 3He-A phase. We examine the experimental setup in connection with ongoing rotating cryostat experiments. The theoretical analysis is based on the well-established Ginzburg-Landau functional, supplemented by microscopic calculations of the Bogoliubov-de Gennes equation, both of which allow the precise location of the parameter regions of the Majorana state to be found in realistic situations.     

Equilibrium number of quantized vortex lines in rotating<Superscript>3</Superscript>He-B

A slow cool-down at constant rotation velocity Ω from the normal to the superfluid state is expected to give the equilibrium state of vortex lines. We have checked this by counting the number of vortex lines in³He-B using cw NMR. Our measurements with 0≤Ω≤0.28 rad/s confirm the expectation.     

Surface Andreev bound states of superfluid (3)He and Majorana fermions

Superfluid (3)He is an intensively investigated and well characterized p-wave superfluid. In the bulk Balian-Werthamer state, which is commonly called the (3)He B phase, the superfluid gap is opened isotropically but near a flat boundary such as a wall of a container it can harbor interesting quasi-particle states inside the gap. These states are called surface Andreev bound states, and have not been experimentally explored in detail. Transverse acoustic impedance measurement has revealed their existence and provided spectroscopic details of the dispersion of the bound states. Recent theoretical arguments claim that the surface Andreev bound states of the superfluid (3)He B phase can be recognized as the edge states of the topological superfluid and be regarded as a Majorana fermion, a fancy particle which has not been confirmed in elementary particle physics. In this review, we present up-to-date knowledge on the surface Andreev bound states of the (3)He B phase revealed by acoustic spectroscopy and the possible realization of a Majorana fermion, along with related studies on this topic.