Excitation spectrum of Bose-Einstein condensate

Question

The Gross-Pitaevskii equation $$ i\hbar\frac{\partial\psi}{\partial t}=-\frac{\hbar^2\nabla^2}{2m}\psi+V\psi+U_0|\psi|^2\psi $$ ($V$ is the external potential, $U_0$ is the interaction constant) in conventionally used to describe dynamics of a Bose-Einstein condensate.

Also it is often used to find excitation spectrum. For example, in the uniform case $V=0$, assuming small perturbations [following C.J. Pethick, H. Smith, Bose-Einstein Condensation in Dilute Gases] $$ \psi=\left\{\sqrt{n}+[ue^{i(\mathbf{qr}-\omega t)}-v^*e^{-i(\mathbf{qr}-\omega t)}]\right\}e^{-i\mu t/\hbar} $$ and taking the equilibrium concentration $n$ and chemical potential $\mu=U_0n$, we get $$ \hbar\omega=\sqrt{\left(\frac{\hbar^2q^2}{2m}+U_0n\right)^2-(U_0n)^2.} $$ This is usual expression for the spectrum of Bogolyubov excitations which can also be found from microscopic mean-field theory. Excitation energies in more complicated nonuniform cases are also often found using the linearized Gross-Pitaevskii equation.

Now the question:

1. Why does the linearized Gross-Pitaevskii equation, which is aimed to describe coherent motion of the condensate, at the same time describe the spectrum of incoherent thermal excitations, which do not belong to the condensate?

2. Does any qualitative difference between coherent oscillations of the condensate and thermal excitations exist, while they are described by the same Gross-Pitaevskii equation?

Answer 1

Thanks for the reference. A tentative answer:

I believe the resolution can be found at the beginning of Ch. 8 of the same book (Pethick and Smith). The authors expand the state around the ground state with fluctuations:

$$\hat{\psi}=\psi_0+\delta \hat{\psi}$$

If the fluctuations are not included, then one gets a classical field theory for the ground state, which follows the Gross-Pitaevskii equations and indeed cannot account for non-condensed atoms. However, the fluctuation term, since it is describing deviations from the ground state, describes non-condensed bosons. Indeed, later on (in eq. 8.35 and following) the authors show how much the condensate fraction is reduced for a given amount of excitations.

In general, both time-dependent fluctations of the condensate and time-dependent scattering out of the condensate are possible. The difference is essentially adiabatic versus diabatic evolution. If you imagine a time-varying external potential that changes sufficiently slowly that the condensate can evolve adiabatically, then the resulting evolution of the condensate may be described using the G-P equation. However, an external potential that causes diabatic transitions that scatter particles out of the condensate must be analyzed with the quantum treatment described above.