indefinitely, and is said to be superfluid.
3. Metastable currents, superfluidity
Consider now a system of repulsive bosons contained in a toroidal box with a rotational axis Oz (Figure 2); the shape of the torus cross-section (circular, rectangular or other) is irrelevant for our argument and we shall use cylindrical coordinates r, φ and z. We first introduce solutions of the Gross-Pitaevskii equation that correspond to the system rotating inside the toroidal box, around the Oz axis. We will then show that these rotational states are metastable, as they can only relax towards lower energy rotational states by overcoming a macroscopic energy barrier: this is the physical origin of superfluidity.
3-a. Toroidal geometry, quantization of the circulation, vortex
To prevent any confusion with the azimuthal angle φ we now call χ the Gross-Pitaevskii wave function. The time-independent Gross-Pitaevskii equation then becomes (in the absence of any potential except the wall potentials of the box):
We look for solutions of the form:
where l is necessarily an integer (otherwise the wave function would be multi-valued). Such a solution has an angular momentum with a well defined component along Oz, equal to lħ per atom. Inserting this expression in (58), we obtain the equation for ul(r, z):
which must be solved with the boundary conditions imposed by the torus shape to obtain the ground state (associated with the lowest value of μ). The term in l2ħ2/2mr2 is simply the rotational kinetic energy around Oz. If the tore radius R is very large compared to the size of its cross-section, the term l2/r2 may, to a good approximation, be replaced by the constant l2/R2. It follows that the same solution of (60) is valid for any value of l as long as the chemical potential is increased accordingly. Each value of the angular momentum thus yields a ground state and the larger l, the higher the corresponding chemical potential. All the coefficients of the equation being real, we shall assume, from now on, the functions ul(r, z) to be real.
As the wave function is of the form (59), its phase only depends on φ and expression (51) for the fluid velocity is written as:
where eφ is the tangential unit vector (perpendicular both to r and the Oz axis). Consequently, the fluid rotates along the toroidal tube, with a velocity proportional to l. As v is a gradient, its circulation along a closed loop “equivalent to zero” (i.e. which can be contracted continuously to a point) is zero. If the closed loop goes around the tore, the path is no longer equivalent to 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