Wavepacket dynamics on a condensate background

One of the common assumptions in the BEC theory is that the presence of a condensate acts on the higher levels by just modifying the confining potential $U$, see for example [25]. If this was the case, the linear dynamics would still be described by the Ehrenfest theorem with some new effective potential. We will show below that this is not the case.

Let us define the condensate $\psi_{0}$ as a nonlinear coordinate dependent solution of equation (), with a lengthscale of the order of the ground state size (although it does not need to be exactly the same as the ground state). In what follows, we will use Madelung's amplitude-phase representation for $\psi_{0}$, namely

$$\psi_{0} = \sqrt{\rho({\bf r})} \, e^{ i \theta},$$ (6)

where ${\bf v} = 2 \nabla\theta$ is the macroscopic speed of the condensate. It is well known that in this representation $\rho$ obeys a continuity equation,

$$\rho_t+ \hbox{div}({\rho {\bf v}}) = 0.$$ (7)

For future reference, one should note that the second term in this expression is $O(\epsilon^2)$. Thus, $\rho_t$ is $O(\epsilon^2)$ too and it must be neglected in the WKB theory which takes into account only linear in $\epsilon$ terms. We start by considering a small perturbation $\phi \ll 1$, such that

$$\psi = \psi_{0}(1+\phi).$$ (8)

Substituting () into () we find

$$i \partial_{t}\phi + \triangle\phi + 2\frac{\nabla\psi_{0}}{\ps... ...rho\Big(\phi+\phi^{*} +2 \vert\phi\vert^2 +\phi^2 +\vert\phi\vert^2\phi\Big)=0.$$ (9)

where $\varrho=\varrho(\vec{x})=\vert\psi_{0}\vert^{2}$ is a slowly varying condensate density.

In a similar manner to the previous subsection, the rest of this derivation consists of Gabor transforming (), combining the result with its complex conjugate and finding a suitable waveaction variable such that the transport equation represents a conservation equation along the rays. Such a derivation is given in Appendix A. It yields to the following expression for the waveaction,

$$n({\bf k},x,t) = \frac{1}{2} \frac{\omega\rho}{k^{2}} \left\ver... ...{ \Re \phi} - \frac{ i k^{2} }{\omega} \widehat{ \Im \phi} \right\vert^{2},$$ (10)

where $\Re$ and $\Im$ mean the real and imaginary parts respectively. As usual, the transport equation takes the form of a conservation equation for waveaction along the rays,

$$D_{t}n(\vec{x},\vec{k},t) = 0,$$ (11)

where

$$D_{t}\equiv \partial_{t} + \dot{\vec{x}}\cdot\nabla +\dot{\vec{k}}\cdot\partial_{k},$$ (12)

is the time derivative along trajectories

$$\dot{\vec{x}} = \partial_{k} \omega, \hspace{1cm} \dot{\vec{k}} = -\nabla \omega.$$ (13)

The frequency is given by the following expression,

$$\omega = k \sqrt{k^{2}+2\varrho}.$$ (14)

One can immediately recognize in () the Bogolubov's formula [21] which was derived before for systems with a coordinate independent condensate and without a trapping potential. It is remarkable that presence of the potential $U$ does not affect the frequency so that expression () remains the same. Obviously, the dynamics in this case cannot be reduced to the Ehrenfest theorem with any shape of potential $U$. Therefore, an approach that models a condensate's effect by introducing a renormalized potential would be misleading in this case.