zero and its circulation may be computed along a circle where r and z remain constant, and φ varies from 0 to 2π; as the path length equals 2πr, we get:
(with a + sign if the rotation is counterclockwise and a — sign in the opposite case). As l is an integer, the velocity circulation around the center of the tore is quantized in units of h/m. This is obviously a pure quantum property (for a classical fluid, this circulation can take on a continuous set of values).
To simplify the calculations, we have assumed until now that the fluid rotates as a whole inside the toroidal ring. More complex fluid motions, with different geometries, are obviously possible. An important case, which we will return to later, concerns the rotation around an axis still parallel to Oz, but located inside the fluid. The Gross-Pitaevskii wave function must then be zero along a line inside the fluid itself, which thus contains a singular line. This means that the phase may change by 2π as one rotates around this line. This situation corresponds to what is called a “vortex”, a little swirl of fluid rotating around the singular line, called the “vortex core line”. As the circulation of the velocity only depends on the phase change along the path going around the vortex core, the quantization relation (62) remains valid. Actually, from a historical point of view, the Gross-Pitaevskii equation was first introduced for the study of superfluidity and the quantization of the vortices circulation.
3-b. Repulsive potential barrier between states of different l
A classical rotating fluid will always come to rest after a certain time, due to the viscous dissipation at the walls. In such a process, the macroscopic rotational kinetic energy of the whole fluid is progressively degraded into numerous smaller scale excitations, which end up simply heating the fluid. Will a rotating quantum fluid of repulsive bosons, described by a wave function χl(r), behave in the same way? Will it successively evolve towards the state χl – 1(r), then χl-2(r), etc., until it comes to rest in the state χ0(r)?
We have seen in § 4-c of Complement CXV that, to avoid the energy cost of fragmentation, the system always remains in a state where all the particles occupy the same quantum state. This is why we can use the Gross-Pitaevskii equation (18).
α. A simple geometry
Let us first assume that the wave function χ(r, t) changes smoothly from χl(r) to χl′(r) according to:
where the modulus of cl (t) decreases with time from 1 to 0, whereas cl′ (t) does the opposite. Normalization imposes that at all times t:
In such a state, let us show that the numerical density n(r, φ, z;t) now depends on φ (this was not the case for either states l or l′ separately). The transverse dependence of the density as a function of the variables r and z, is barely affected3. The variations of n(r, φ, z; t) are given by:
where c.c. stands for the complex conjugate of the preceding factor. The first two terms are independent of φ, and are just a weighted average of the densities associated with each of the states l and l′. The last term oscillates as a function of φ with an amplitude |cl (t)| × |cl′ (t)|, which is only zero if one of the two coefficients cl (t) or cl′ (t) is zero. Calling φl the phase of the coefficient cl (t) this last term is proportional to:
Whatever the phases of the two coefficients cl (t) and cl′ (t), the cosine will always oscillate between — 1 and 1 as a function of φ. Adjusting those phases, one can deliberately change the value of φ for which the density is maximum (or minimum), but this will always occur somewhere on the circle. Superposing two states necessarily modulates the density.
Let us evaluate the consequences of this density modulation on the internal repulsive interaction energy of the fluid. As we did in relation (15), we use for the interaction energy the zero range potential approximation, and insert it in expression (15) of Complement CXV. Taking into account the normalization (17) of the wave function, we get:
(67)
We must now include the square of (65) in this expression, which will yield several terms. The first one, in |cl (t)|4, leads to the contribution:
(68)
where
Up to now, the superposition has had no effect on the repulsive internal interaction energy. As for the cross terms between the terms independent of φ in (65) and the terms in e±i(l – l′)φ, they will cancel out when integrated over φ. We are then left with the cross terms in e±i(l – l′)φ × e∓;i(l – l′)φ, whose integral over φ yields:
(70)
