20
$\begingroup$

I am trying to linearize the following GP eq: \begin{equation} i\partial_{t}\psi(r,t)=\left[-\frac{\nabla^{2}}{2m}+g\left|\psi(r,t)\right|^{2}+V_{d}(r)\right]\psi(r,t) \end{equation}

The ansatz for the mean-field wavefunction is: \begin{equation} \psi_{0}(r,t)=\psi_{0}\, e^{i(k_{0}r-\omega_{0}t)} \end{equation}

One then has to add the fluctuations on top: \begin{equation} \psi(r,t)=\big[\psi_{0}(r)+\delta\psi(r,t)\big]\, e^{-i\omega_{0}t} \end{equation}

Pluggin this into the original equation we get to 0 order that \begin{equation} \omega_{0}-\frac{k_{0}^{2}}{2m}=g|\psi_0|^2. \end{equation}

Expanding to first order (linear response) we get \begin{equation} i\partial_{t}\delta\vec{\psi}=\mathcal{L}\cdot\delta\vec{\psi}+\vec{F}_{d} \end{equation}

with \begin{equation} \delta\vec{\psi}(r,t)=\left(\begin{array}{c} \delta\psi(r,t)\\ \delta\psi^{\star}(r,t) \end{array}\right) \end{equation}

\begin{equation} \vec{F}_{d}(r)=V_{d}(r)\,\left(\begin{array}{c} \psi_{0}(r)\\ -\psi_{0}^{*}(r) \end{array}\right) \end{equation}

\begin{equation} \mathcal{L}=\left(\begin{array}{cc} -\frac{k_{0}^{2}}{2m}-\frac{\nabla^{2}}{2m}+g|\psi_{0}|^2 & g\psi_{0}^{2}\, e^{2ik_{0}r}\\ -g\psi_{0}^{2\star}\, e^{-2ik_{0}r} & -\left(-\frac{k_{0}^{2}}{2m}-\frac{\nabla^{2}}{2m}+g|\psi_{0}|^2\right) \end{array}\right) \end{equation}

The goal here is to determine $\delta\vec{\psi}(r,t)$, by diagonalizing $\mathcal{L}$ and expanding on the corresponding eigenmodes. I am trying to follow these notes http://arxiv.org/abs/cond-mat/0105058v1, which give the general formalism starting on page 66.

However, the author says $\mathcal{L}$ is not diagonalizable in general therefore one has to do the trick of splitting $\delta\psi$ into a part along $\psi_0$ and a part orthogonal to it (see eq. 229). This leads to a new operator $\mathcal{L}$, given by (235) which is diagonalizable.

How can I apply this formalism to my problem? How do I construct the new operator $\mathcal{L}$? How do I deal with the projection operators (233) and (234), etc

$\endgroup$
  • $\begingroup$ What is $V_d(r)$ in your Lagrangian? Is it just the harmonic trap? What is your end goal - is it to obtain the eigenmodes of the system, or is something else? $\endgroup$ – ffc Apr 18 '15 at 12:36
  • $\begingroup$ $V_d(r)$ is a defect potential, which in the simplest case is a delta function $\delta(r-r_0)$. The end goal is to calculate the effect of this perturbation caused by the defect, i.e. calculate $\delta \psi$. $\endgroup$ – Andrei Apr 18 '15 at 14:53
  • $\begingroup$ But then I do not understand the question, as it seems to me that Y. Castin does just what you want in his review. His perturbation $\delta U$ can depend on the coordinate and be time independent, such that $\delta U = V_d(\boldsymbol r)$. It seems to me Eq. 255 is exactly what you need. Could you please clarify if I am wrong? $\endgroup$ – ffc Apr 22 '15 at 8:58
  • $\begingroup$ I agree that his formalism should be applicable, that is why I mention that reference :) I just don't know how to apply it in practice to my problem. $\endgroup$ – Andrei Apr 24 '15 at 10:01
  • $\begingroup$ I think that your problem was actually explicitly treated in the reference! Otherwise, it could be that I do not understand your problem. Could you explain to me what is the difference between your problem and the one treated in the reference? :) $\endgroup$ – ffc Apr 24 '15 at 15:44

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.