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Assuming as before that the densities associated with the states l and l′ are practically the same, we obtain, after integration over r and z:
(71)
Adding (69), we finally obtain:
(72)
We have shown that the density modulation associated with the superposition of states always increases the internal repulsion energy: this modulation does lower the energy in the low density region, but the increase in the high energy region outweighs the decrease (since the repulsive energy is a quadratic function of the density). The internal energy therefore varies between
α. Other geometries, different relaxation channels
There are many other ways for the Gross-Pitaevskii wave function to go from one rotational state to another. We have limited ourselves to the simplest geometry to introduce the concept of energy barriers with minimal mathematics. The fluid could transit, however, through more complex geometries, such as the frequently observed creation of a vortex on the wall, the little swirl we briefly talked about at the end of § 3-a. A vortex introduces a 2π phase shift around a singular line along which the wave function is zero. Once the vortex is created, and contrary to what was the case in (62), the velocity circulation along a loop going around the torus is no longer independent of its path: it will change by 2πħ/m depending on whether the vortex is included in the loop or not. Furthermore, as the vortex moves in the fluid from one wall to another, it can be shown that the proportion of fluid conserving the initial circulation decreases while the proportion having a circulation where the quantum number l differs by one unit increases. Consequently, this vortex motion changes progressively the rotational angular momentum. Once the vortex has vanished on the other wall, the final result is a decrease by one unit of the quantum number l associated with the fluid rotation.
The continuous passage of vortices from one wall to another therefore yields another mechanism that allows the angular moment of the fluid to decrease. The creation of a vortex, however, is necessarily accompanied by a non-uniform fluid density, described by the Gross-Pitaevskii equation (this density must be zero along the vortex core). As we have seen above, this leads to an increase in the average repulsive energy between the particles (the fluid elastic energy). This process thus also encounters an energy barrier (discussed in more detail in the conclusion). In other words, the creation and motion of vortices provide another “relaxation channel” for the fluid velocity, with its own energy barrier, and associated relaxation time.
Many other geometries can be imagined for changing the fluid flow. Each of them is associated with a potential barrier, and therefore a certain lifetime. The relaxation channel with the shortest lifetime will mainly determine the damping of the fluid velocity, which may take, in certain cases, an extraordinarily long time (dozens of years or more), hence the name of “superfluid”.
3-c. Critical velocity, metastable flow
For the sake of simplicity, we will use in our discussion the simple geometry of § 3-a. The transposition to other geometries involving, for example, the creation of vortices in the fluid would be straightforward. The main change would concern the height of the energy barrier4.
With this simple geometry, the potential to be used in (60) is the sum of a repulsive potential g |ul(r, z)|2 and a kinetic energy of rotation around Oz, equal to l2ħ2/2mr2. We now show that, in a given l state, these two contributions can be expressed as a function of two velocities. First, relation (61) yields the rotation velocity vl associated with state l:
(73)
and the rotational energy is simply written as:
As for the interaction term (term in g on the left-hand side), we can express it in a more convenient way, defining as before the numerical density n0:
(75)
and using the definition (39) for the sound velocity c. It can then be written in a form similar to (74):
(76)
The two velocities vl and c allow an easy comparison of the respective importance of the kinetic and potential energies in a state l.
We now compare the contributions of these two terms either for states with a given l, or for a superposition of states (63). To clarify the discussion and be able to draw a figure, we will use a continuous variable defined as the average 〈Jz〉 of the component along Oz of the angular momentum:
This expression varies continuously between lħ and l′ħ when the relative weights of |cl (t)|2 and |cl' (t)|2 are changed while imposing relation (64); the continuous variable:
(78)
allows making interpolations between the discrete integer values of l.
Using the normalization relation (64) of the wave function (63), we can express x as a function of |cl′ (t)|2:
(79)
