Linear theory without a condensate

Linearizing the GP equation, to investigate the behavior of wavepackets $\psi$ without the presence of a condensate, we obtain the usual linear Schrödinger equation:

$$i \partial_{t}\psi + \triangle \psi -U\psi = 0,$$ (3)

where $U$ is a slowly varying potential. Let us apply the Gabor transformation to (). Note that the Gabor transformation commutes with the Laplacian, so that $\widehat{\Delta \Psi}=\Delta \hat{\Psi}$. Also note that

$$\widehat{U\Psi}\simeq U \hat{\Psi} + i (\nabla_x U)\nabla_k\hat{\Psi},$$

where we have neglected the quadratic and higher order terms in $\epsilon$ because $\Psi$ changes on a much shorter scale than the large scale function $U$. Combining the Gabor transformed equation with its complex conjugate we find the following WKB transport equation,

$$D_{t}\vert\hat{\psi}\vert^{2} = 0,$$ (4)

where

$$D_{t}\equiv \partial_{t} + \dot{\vec{x}}\cdot\nabla +\dot{\vec{k}}\cdot\partial_{k},$$

represents the total time derivative along the wavepacket trajectories in phase-space. The ray equations are used to describe wavepacket trajectories in $(\vec{k},\vec{x})$ phase-space,

$$\dot{\vec{x}} = \partial_{k}\omega, \dot{\vec{k}} = -\nabla\omega.$$ (5)

The frequency $\omega$, in this case, is given by $\omega = k^{2}+U$, (again we use the notation $k=\vert\vec{k}\vert$). Equations () and () are nothing more than the famous Ehrenfest theorem from quantum mechanics. According to (), the wavepackets will get reflected by the potential at points $r_{R}$ where $U(r_{R}) = k^{2}_{max}$. We will now move on to consider linear wavepackets in the presence of a background condensate.