# Linear response theory for Gross Pitaevskii equation

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

• 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?
– jarm
Apr 18, 2015 at 12:36
• $V_d(r)$ 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 $\delta \psi$. Apr 18, 2015 at 14:53
• 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?
– jarm
Apr 22, 2015 at 8:58
• 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. Apr 24, 2015 at 10:01
• 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? :)
– jarm
Apr 24, 2015 at 15:44

The decay of a density of particles governed by stochastic differential equation (SDE) dynamics to the equilibrium density is a well known topic in statistical mechanics, for which you may see the book by Risken. For the stochastic process $$x_s \sim p(s,x)$$, for all $$0 \leq s$$, with the SDE dynamics $$d x_s = b(x_s) ds + \sigma dw_s$$ and the corresponding linear Fokker-Planck (FP) partial differential equation (PDE) dynamics $$\partial_t p = \mathcal{L}^\dagger p$$, the question is whether any initial density will decay to the invariant density $$p^\infty(x)$$ which solves the stationary FP equation and at what rate? This problem has been answered for certain systems, namely, Ornstein-Uhlenbeck and Langevin systems. The key idea is to consider the initial density as a perturbation from the equilibrium given as $$p(0,x) = p^\infty(x) + p^\infty(x) \tilde{p}(0,x)$$ and prove that $$\tilde{p}(s,x)$$ vanishes under the limit $$s \rightarrow +\infty$$, by using the eigen properties of the generator $$\mathcal{L}$$ of the SDE which is indeed the adjoint of the forward FP operator $$\mathcal{L}$$. Thus, properties of the SDE dynamics determine whether or not we can show that the equilibrium density is indeed attained by particles with any arbitrary initial density.