rel="nofollow" href="#fb3_img_img_76a611fe-4436-593e-8e46-76c4356ffe93.jpg" alt="image"/>
where we have used definition (20) for μ to replace 2gn0 — μ by gn0. If we now look for solutions having a plane wave spatial dependence:
the differential equation can be written as:
The eigenvalues ħw(k) of this matrix satisfy the equation:
(30)
that is:
(31)
The solution of this equation is:
(the opposite value is also a solution, as expected since we calculate at the same time the evolution of
(33)
relation (32) can be written:
The spectrum given by (32) is plotted in Figure 1, where one sees the intermediate regime between the linear region at low energy, and the quadratic region at higher energy. It is called the “Bogolubov spectrum” of the boson system.
β. Discussion
Let us compute the spatial and time evolution of the particle density n(r, t) when δφ(r, t) obeys relation (28). The particle density at each point r of space is the sum of the densities associated with each particle, that is N times the squared modulus of the wave function φ(r, t). To first-order in δφ(r, t), we obtain:
(35)
(where c.c. stands for “complex conjugate”). Using (26) and (28), we can finally write:
(36)
Consequently, the excitation spectrum we have calculated corresponds to density waves propagating in the system with a phase velocity w(k)/k.
In the absence of interactions, (g = k0 = 0), this spectrum becomes:
Figure 1: Bogolubov spectrum: variations of the function ω(k) given by equation (32) as a function of the dimensionless variable κ = k/k0. When κ ≪ 1, we get a linear spectrum (the arrow in the figure shows the tangent to the curve at the origin), whose slope is equal to the sound velocity c; when κ ≫ 1, the spectrum becomes quadratic, as for a free particle.
which simply yields the usual quadratic relation for a free particle. Physically, this means that the boson system can be excited by transferring a particle from the individual ground state, with wave function φ0 (r) and zero kinetic energy, to any state φk(r) having an energy ħ2k2/2m.
In the presence of interactions, it is no longer possible to limit the excitation to a single particle, which immediately transmits it to the others. The system’s excitations become what we call “elementary excitations”, involving a collective motion of all the particles, and hence oscillations in the density of the boson system. If k ≪ k0, we see from (34) that:
(38)
where c is defined as:
For small values of k, the interactions have the effect of replacing the quadratic spectrum (37) by a linear spectrum. The phase velocity of all the excitations in this k value domain is a constant c. It is called the “sound velocity “ in the interacting boson system, by analogy with a classical fluid where the sound wave dispersion relation is linear, as predicted by the Helmholtz equation. We shall see in § 3 that the quantity c plays a fundamental role in the computations related to superfluidity, especially for the critical velocity determination. If, on the other hand, k ≫ k0, the spectrum becomes:
(40)
(the following corrections being in
