# Exctitation spectrum of Bose-Einstein condensate

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 the linearized Gross-Pitaevskii equation, which is aimed to describe coherent motion of the condensate, at the same time describes 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?

• Why do you say that they are incoherent? – Abhijit Mar 31 '17 at 11:12
• @Abhijit I meant these excitations are in incoherent mixture with the condensate state in a thermal ensemble. Besides, they form the normal component in the two-fluid model and thus do not belong to the condensate. – Alexey Sokolik Mar 31 '17 at 12:28
• An example of when this is applied to incoherent thermal excitations would be helpful. – Rococo Mar 31 '17 at 14:43
• @Rococo See, e.g. [C.J. Pethick, H. Smith, Bose-Einstein Condensation in Dilute Gases (2002)], where the Gross-Pitaevskii equation is used to derive the spectrum of Bogolyubov excitations (pp. 174-176) and then the same spectrum of excitations is used to calculate the density of normal fluid, which consists of thermal excitations (pp. 267-269). Or this coincidence of the spectra is just accidental? – Alexey Sokolik Mar 31 '17 at 18:14

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.

• I agree that it is perhaps more correct to treat $\delta\hat{\psi}$ as operators. But, on the other hand, we have a full right to consider slow and smooth perturbations $\delta\psi$ of the condensate (now $\delta\psi$ is a $c$-number!), so my 2-nd question remains: what is a difference between slow smooth motion of the condensate and non-condensed fluctuations? – Alexey Sokolik Apr 2 '17 at 22:32
• Yes, I think there is a subtlety here that I don't have a full grasp on either. Pethick and Smith consider first perturbations of the form $n+\delta n$, which they say is variations of the ground state density, and then perturbations of the above form, which accounts for depletion of the ground state. – Rococo Apr 3 '17 at 0:04
• Okay, after some more reading (mainly Leggett, Quantum Liquids), I think the difference can be thought of as adiabatic vs diabatic evolution. See the edit above. – Rococo Apr 3 '17 at 4:07
• Thank you for the remark about adiabaticity, I need to think about it. Still, I will leave the question open to retain an opportunity of getting any other answers from, maybe, different points of view. – Alexey Sokolik Apr 3 '17 at 20:56