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Linear Response#

Here we consider the linear response of a more complicated system described by the Gross-Pitaevskii equation (GPE):

\[\begin{gather*} \I\hbar\dot{\psi}(\vect{x}, t) = \left( -\frac{\hbar^2\vect{\nabla}}{2m} + g\abs{\psi(\vect{x}, t)}^2 + V(\vect{x}, t)\right)\psi(\vect{x}, t). \end{gather*}\]

Although this is intended to represent a quantum many-body system (a Bose-Einstein Condensate or BEC), \(\psi(\vect{x}, t)\) is formally a complex-valued classical field.

In this document, we shall restrict ourselves to 1D. The goal is to perform the equivalent of a normal-modes analysis to see how the system will respond to small perturbations. Specifically, we will drive it with a small periodic potential

\[\begin{gather*} V(x, t) = \epsilon \sin(\omega t)f(x) \end{gather*}\]

and see how it responds.

As usual, we must start with a stationary state \(\psi_0(x, t) = e^{\mu t/\I\hbar}\psi_0(x)\) such that

\[\begin{gather*} -\frac{\hbar^2\psi_0''(x)}{2m} + g\abs{\psi_0(x)}^2\psi_0(x) = \mu \psi_0(x). \end{gather*}\]

We then look for states \(\psi_{\pm n}(x)\) such that the GPE is satisfied to order \(\epsilon^2\) for

\[\begin{gather*} \psi(x, t) = e^{\mu t/\I\hbar}\left( \psi_0(x) + \epsilon e^{\I\omega t}\psi_{+}(x) + \epsilon e^{-\I\omega t}\psi_{-}(x) \right). \end{gather*}\]

Inserting, expanding, collecting the coefficients of \(e^{\I(\mu \pm\omega) t}\) respectively, and conjugating the second equation, we obtain the following equations to linear order in \(\epsilon\):

\[\begin{align*} (\mu - \omega)\psi_+ &= \frac{-\hbar^2}{2m}\psi_+'' + g(\psi_0^*\psi_+ + \psi_-^*\psi_0)\psi_0 + (g\abs{\psi_0}^2 + V)\psi_+,\\ (\mu + \omega)\psi_-^* &= \frac{-\hbar^2}{2m}\psi_-''^* + g(\psi_0\psi_-^* + \psi_0^*\psi_+)\psi_0^* + (g\abs{\psi_0}^2 + V)\psi_-^*,\\ \begin{pmatrix} \omega\\ & -\omega \end{pmatrix} \begin{pmatrix} \psi_+\\ \psi_-^*\\ \end{pmatrix} &= \begin{pmatrix} \frac{-\hbar^2}{2m}\nabla^2 + 2g\abs{\psi_0}^2 - \mu + V & g\psi_0^2\\ g(\psi_0^*)^2 & \frac{-\hbar^2}{2m}\nabla^2 + 2g\abs{\psi_0}^2 - \mu + V \end{pmatrix} \underbrace{ \begin{pmatrix} \psi_+\\ \psi_-^*\\ \end{pmatrix}}_{\ket{\Psi}}. \end{align*}\]