region at low energies, to the quadratic region at high energies.
Comment:
As we assumed the interactions to be repulsive in (21), the square roots in (32) and (39) are well defined. If the coupling constant g becomes negative, the sound velocity c will become imaginary, and, as seen from (32), so will the frequencies ω(k) (at least for small values of k). This will lead, for the evolution equation (29), to solutions that are exponentially increasing or decreasing in time, instead of oscillating. An exponentially increasing solution corresponds to an instability of the system. As already encountered in § 4-c of Complement CXV, we see that a boson system becomes unstable in the presence of attractive interactions, however small they might be. In § 4-b of Complement HXV, we shall see that this instability persists even for non-zero temperature. In a general way, an attractive condensate occupying a large region in space tends to collapse onto itself, concentrating into an ever smaller region. However, when it is confined in a finite region (as is the case for experiments where cold atoms are placed in a magneto-optical trap), any change in the wave function that brings the system closer to the instability also increases the gas energy; this results in an energy barrier, which allows the system of condensed attractive bosons to remain in a metastable state.
2. Hydrodynamic analogy
Let us return to the study of the time evolution of the Gross-Pitaevskii wave function and of the density variations n(r, t), without assuming as in § 1-c that the boson system stays very close to uniform equilibrium. We will show that the Gross-Pitaevskii equation can take a form similar to the hydrodynamic equation describing a fluid’s evolution. In this discussion, it is useful to normalize the Gross-Pitaevskii wave function to the particle number, as in equation (17). Equation (16) can then be written as:
where the local particle density n(r, t) is given by:
(42)
2-a. Probability current
Since:
(43)
the time variation of the density may be obtained by first multiplying (41) by φ*(r, t), then its complex conjugate by φ(r, t), and then adding the two results; the potential terms in V1(r, t) and g n(r, t) cancel out, and we get:
Let us now define a vector J(r, t) by:
If we compute the divergence of this vector, the terms in ▽φ* · ▽φ cancel out and we are left with terms identical to the right-hand side of (44), with the opposite sign. This leads to the conservation equation:
J(r, t) is thus the probability current associated with our boson system. Integrating over all space, using the divergence theorem, and assuming φ(r, t) (hence the current) goes to zero at infinity, we obtain:
(47)
This shows, as announced earlier, that the Gross-Pitaevskii equation conserves the norm of the wave function describing the particle system.
We now set:
The gradient of this function is written as:
Inserting this result in (45), we get:
(50)
or, defining the particle local velocity v(r, t) as the ratio of the current to the density:
We have defined a velocity field, similar to the velocity field of a fluid in motion in a certain region of space; this field velocity is irrotational (zero curl everywhere).
2-b. Velocity evolution
We now compute the time derivative of this velocity. Taking the derivative of (48), we get:
(52)
so that we can isolate the time derivative of α(r, t) by the following combination:
The left-hand side of this relation can be computed with the Gross-Pitaevskii equation (18) and its complex conjugate, as we now show. We first take the divergence
