Experimental set-up and definition of the problem

The experimental set-up for studying rotating superfluids is simple: a cylindrical container with radius R, filled with a superfluid, is rotated around its symmetry axis at an angular velocity . Suppose the container is slowly accelerated into rotation from rest at a constant rate . The normal component is viscously coupled to the walls, and for sufficiently slow , it will co-rotate with the container like a solid body. In contrast, the superfluid component remains initially at rest in the laboratory frame. This gives rise to a superfluid counterflow with a relative velocity . At a critical angular velocity the first vortex is nucleated at some rough spot on the cylindrical side wall of the container, where the critical value of the superfluid counterflow is first reached. In contrast to ⁴He-II, for the ³He superfluids in a smooth-walled container the critical velocity is often a well-defined reproducible quantity, which is of the same order of magnitude as the Landau instability of bulk superflow.

After nucleation the Magnus force pulls the vortex line to the center of the container, where it remains stretched between the bottom and top plates. At the cylindrical wall the counterflow now drops by the equivalent of the nucleated circulation quanta, i.e., , where is the number of circulation quanta for one vortex. If is increased further, the second vortex line will be nucleated when the critical velocity vc is again reached. During a slow linear acceleration of , quantized vortex lines are thus accumulated one by one in the center of the container, where they form a vortex cluster, separated by an annular layer of vortex-free counterflow from the cylindrical wall. Within the vortex cluster the average of the superfluid velocity follows the solid-body behavior and .

This pattern of vortices and counterflow is formed by the orbital part of the superfluid order parameter; it represents the minimum of the total hydrodynamic flow energy for a given critical velocity , and it can be probed by various macroscopic measurements with sufficient resolution to discern the nucleation of individual vortex lines. The fine-scale structure of the order parameter texture depends on the details of the quantized vortex line and can be reconstructed from measurements with microscopic resolution, such as the attenuation of transmitted ultrasound or the shape of the NMR spectrum. In the majority of our work we have focused on vortex textures to determine the topology and structure of the different types of vortex lines.