rel="nofollow" href="#ulink_9d39eda4-0863-5d7f-8f8a-60e093e0b882">(49) to obtain the Laplacian:
(54)
We then insert the time derivative of φ(r, t) given by the Gross-Pitaevskii equation (18) in the left-hand side of relation (53), which becomes:
(55)
This result must be equal to the right-hand side of (53). We therefore get, after dividing both sides by —2n(r, t):
(56)
Using (51), we finally obtain the evolution equation for the velocity v(r, t):
This equation looks like the classical Newton equation. Its right-hand side includes the sum of the forces corresponding to the external potential V1(r, t), and to the mean interaction potential with the other particles gn(r, t); the third term in the gradient is the classical kinetic energy gradient1 (as in Bernoulli’s equation of classical hydrodynamics). The only purely quantum term is the last one, as shown by its explicit dependence on ħ2. It involves spatial derivatives of n(r, t), and is only important if the relative variations δn/n of the density occur over small enough distances (for example, this term is zero for a uniform density). This term is sometimes called “quantum potential”, or “quantum pressure term” or, in other contexts, “Bohm potential”. A frequently used approximation is to consider the spatial variations of n(r, t) to be slow, which amounts to ignoring this quantum potential term: this is the so-called Thomas-Fermi approximation.
We have found for a system of N particles a series of properties usually associated with the wave function of a single particle, and in particular a local velocity directly proportional to its phase gradient2. The only difference is that, for the N-particle case, we must add to the external potential V1(r, t) a local interaction potential gn(r, t), which does not significantly change the form of the equations but introduces some nonlinearity that can lead to completely new physical effects.
Figure 2: A repulsive boson gas is contained in a toroidal box. All the bosons are supposed to be initially in the same quantum state describing a rotation around the Oz axis. As we explain in the text, this rotation can only slow down if the system overcomes a potential energy barrier that comes from the repulsive interactions between the particles. This prevents any observable damping of the rotation over any accessible time scale; the fluid rotates 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
