Adiabatic theorem with stochastic variables

Question

Suppose a system which is driven by a stochastic complex variable $\alpha$(t). Looking at the eigensystem, both eigenvectors and eigenvalues are then stochastic variables. In my case, after building a time-dependent Hamiltonian, its eigenvalues are:

$\lambda_{0}=0$, $\lambda_{+}=-i\gamma/2+\sqrt{2\gamma |\alpha(t)|^2}$, and $\lambda_{-}=-i\gamma/2-\sqrt{2\gamma |\alpha(t)|^2}$

and the eigenvectors are

$|\lambda_{0}\rangle=(0,2\sqrt{2\gamma}\alpha(t),1)^T$, $|\lambda_{\pm}\rangle$= not important

where ^T means transpose.

Given an ansatz for my state $|\psi\rangle=c_{0}|\lambda_{0}\rangle$, and following the adiabatic theorem, the coefficient is given by

$\dot{c}_{0}(t)=-(i\lambda_{0}+\langle\lambda_{0}|\dot{\lambda_{0}}\rangle)c_{0}$. While $\lambda_{0}=0$, I still need to calculate the bra-ket with the time-derivative. Here is here the problem stats. How do I take this derivative with the stochastic variable?

All I know about my stochastic process is that $d\alpha=-\kappa/2\alpha+\sqrt{\kappa n_{\rm th}}dW$, where $dW$ is a Wiener process and that the ensemble average is $<<\alpha(t)>>=0$ and $<<|\alpha|^2(t)>>=g(t)$, where $g(t)$ is a known function.

My process so far: Given the eigenvector $|\lambda_{0}\rangle$, the first thing I do is to normalize it, such that $|\lambda_{0}\rangle\rightarrow |\lambda_{0}\rangle_{N}=(0,\frac{2\sqrt{2}\alpha(t)}{\sqrt{(\gamma+8|\alpha(t)|^2)}},\frac{1}{\sqrt{1+8|\alpha(t)|^2/\gamma}})^T$

Now I can symbolically take the time-derivative of this vector, which will give me a vector in function of $\alpha$ and $\dot{\alpha}$ and finally, I can take the bra, such that I finally get $_{N}\langle\lambda_{0}|\dot{\lambda_{0}}\rangle_{N}=f(\alpha,\dot{\alpha})$

My idea is now that I just take the ensemble-average of the stochastic process and evaluate $<<f(\alpha,\dot{\alpha})>>=f(g(t))$. It will be in function of variables I know.

However, this approach means that taking the ensemble-average of the stochastic variable and then taking the time-derivative is the same as taking the time-derivative of the variable and then averaging, which does not sound right to me.

Any help would be appreciated.

Answer 1

I was not able to follow all the details. But in general, if you have a function $$ f(t,X_t)$$ where $X_t$ is a stochastic process defined by the Itô stochastic differential equation $$\dot{x}_t=A(t,X_t)+\sigma(t,X_t)\xi(t), $$ then the time derivative is obtained through the Itô's chain rule (Itô's lemma): $$\dot{f}=\frac{d}{dt}f(t,X_t)=\partial_t f(t,X_t)+\partial_x f(t,x)|_{x=X_t}\dot{x}_t+\frac{1}{2}\partial_x^2f(t,x)|_{x=X_t}\sigma(t,X_t)^2.$$ Then, if you want to do the ensemble average you also have to take care of the rules of the calculus you are using. For example, in Itô calculus the process is not correlated with the noise, $$<g(X_t)dW_t>=<g(X_t)><dW_t>= 0, \forall g.$$ This is not true in Stratonovich calculus.

I don't think you can change the order of the time derivative and the average since $<\dot{\alpha}(t)>$ and $\frac{d}{dt}<\alpha(t)>$ mean different things. Still, I can not think of a counter-example right now.