Rotation and Formation of Vortices

One of the most astonishing consequences of superfluidity is the formation of vortices in a rotating liquid. This section gives a brief description of this phenomenon, for a

comprehensive overview see the text book by Donnelly [92]. As a measure of rotation, the circulation of a fluid is given by the curl of its velocity field𝑣_l, also known as the vorticity 𝜔_v [92]:

𝜔v= curl𝑣l. (2.30)

Since a superfluid flows with zero viscosity, no shearing forces and therefore no turbulence occur. Hence, Landau assumed in his two-fluid model that the flow of the superfluid component is irrotational, i.e.,

curl𝑣sf = 0. (2.31)

In 1955, Feynman chose a microscopic approach (i.e., solving the Schrödinger equation) to describe superfluid helium, resulting in the prediction of quantized vortices as possible excitations below the Landau velocity [93]. The following considerations are based on Ref. [66]. Coming back to London’s original idea of interpreting the superfluid transition in liquid helium as a form of Bose-Einstein condensation, the condensate’s macroscopic wavefunction describing the movement of the atoms condensed in the ground state is given as

𝜓(𝑟) =𝜓0e^{𝑖𝜙(𝑟)}. (2.32)

Here,𝜓0 is the amplitude that is assumed constant and 𝜙(𝑟) is the position-dependent phase that is related to the velocity of the atoms in the superfluid component. The momentum𝑝 of a condensed atom can be described by [66]

−𝑖ℏ∇𝜓=𝑝𝜓. (2.33)

Since the momentum of an atom in the superfluid component can be identified as𝑚4𝑣sf

(where𝑚₄ is the effective mass of a⁴He atom), inserting Eq. (2.32) into Eq. (2.33) gives 𝑝=ℏ∇𝜙(𝑟) =𝑚₄𝑣_sf and therefore

𝑣sf = ℏ

𝑚4∇𝜙(𝑟), (2.34)

meaning that the superfluid velocity depends on the gradient of the phase. For the possible motion of the condensate, it immediately follows

curl𝑣_sf =∇ ×𝑣_sf = ℏ

𝑚₄∇ × ∇𝜙(𝑟) = 0, (2.35) which is the curl-free constraint from Eq. (2.31). In consequence, the local motion of the superfluid component has to be free of rotation, meaning that a superfluid in a rotating cylinder should stay at rest. In contrast to a normal fluid liquid that would form a parabola-shaped meniscus in a rotating container, a flattened surface is expected for helium II, since the superfluid component does not participate in the rotation and hence the centrifugal force is reduced. In Fig.2.10 (a), the layout of such a rotating bucket experiment carried out by Osborne [94] is shown. As is already indicated in the sketch, it turned out that the meniscus of the superfluid exhibits the same parabola-shape as one would expect for a normal liquid. This unanticipated result can be understood by considering the hydrodynamic circulation𝜅 of the superfluid along a contour 𝐶 that is given for a single connected region by the line integral

𝜅=^∮︁

𝐶

𝑣sfd𝑙. (2.36)

In the superfluid, the contour can be continuously contracted to a point, yielding

2.2. Aspects of Superfluidity 21

FIG. 2.10: Formation of vortices in a rotating bucket filled with superfluid helium.

(a) Rotating bucket experiment. The helium II inside the inner cylinder exhibits a parabola-shaped meniscus upon rotation. Reprinted from Ref. [94]. (b) In a rotating container filled with superfluid helium, vortices form carrying the rotational energy.

The liquid helium close to the vortex cores is normal fluid, i.e., the vortices impose the parabola shape on the liquid’s surface. Reprinted from Ref. [66].

zero circulation, unless there is a singularity or hole within the contour (e.g., helium II circulating in a torus). In the latter case, using Eq. (2.34), the circulation becomes

𝜅=^∮︁

𝐶

ℏ

𝑚₄∇𝜙(𝑟) d𝑙= ℏ

𝑚₄Δ𝜙_𝑐, (2.37)

with Δ𝜙𝑐 the phase difference along the contour 𝐶. Since the wavefunction must be single-valued, the phase change for a complete cycle can only be an integer multiple of 2𝜋, hence Δ𝜙_𝑐 = 2𝜋𝑛_𝜙 and therefore

𝜅= ℎ 𝑚4

𝑛𝜙 (𝑛𝜙 = 0,1,2, . . .), (2.38) which means that the circulation is quantized in multiples of ℎ/𝑚4 (with ℎ the Planck constant). This quantization has been experimentally demonstrated by Vinen [95] in a cylinder filled with liquid helium II and a thin wire placed in the center. For a singly connected region in helium II, as in Osborne’s rotating bucket experiment, the superfluid component seems to rotate because of vortex lines that form in the liquid. The superfluid velocity increases with decreasing distance to the vortex center and as soon as it exceeds the Landau velocity𝑣_L, a normal fluid is formed in vicinity of the vortex core. Therefore, the vortices act as holes in the superfluid component. In Fig.2.10 (b), the situation is shown for a rotating bucket filled with helium II. The forming vortices impose the classical parabola shape on the superfluid meniscus. The energy of a vortex increases quadratically with the circulation, it is therefore energetically favorable to have multiple vortices with 𝑛_𝜙 = 1 instead of fewer vortices with larger circulation. Further, the vortices repel each other, thus forming vortex arrays as shown in Fig. 2.11. It is the top view of a rotating cylinder filled with helium II, where the number of vortices increases with the cylinder’s angular velocity. The diameter of the dark circles corresponds to the cylinder diameter which is about 2 mm, hence the quantized vortex lines indeed

FIG. 2.11: Vortex arrays in bulk superfluid helium. Top view of a rotating cylinder filled with helium II. As the camera co-rotates with the cylinder, static vortex arrays are observed. The number of vortices increases with increasing angular velocity of the cylinder. Adapted from Ref. [98].

demonstrate a quantum phenomenon on a macroscopic scale. In the context of helium nanodroplets, after a long-lasting discussion whether vortices do exist in the droplets or not, similar vortex lattices have been observed [12,96,97].