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In the book of $\textit{The Quantum World of Ultra-Cold Atoms and Light: Book 1 Foundations of Quantum Optics}$ by Peter Zoller and Crispin Gardiner on page 75, they derive the phase-amplitude stochastic differential equation for a thermalized oscillator.

From a complex Ornstein-Uhlenbeck process of the form \begin{equation} d\alpha=-(i\omega+\gamma/2)\alpha dt +\sqrt{\gamma n_{th}} dW_{t} \end{equation} where $dW_t$ is a complex Wiener increment, they define two new variables such that $\mu+i\phi=log \alpha$. Then, by defining $a=e^\mu$, they derive two real stochastic differential equations

\begin{equation} d a= \left(-\gamma/2 a + \frac{\gamma n_{th}}{4a}\right)dt + \sqrt{\frac{\gamma n_{th}}{2}} dW_{a}(t) \end{equation} \begin{equation} d \phi= -i\omega dt +\sqrt{\frac{\gamma n_{th}}{2}} \frac{dW_{\phi}(t)}{a} \end{equation}

If we are on resonance, we can set $\omega=0$ and forget about the phase differential equation. The original complex-valued stochastic equation can be formally integrated to

\begin{equation} \alpha(t)=\alpha(0)e^{-\gamma/2 t} + \sqrt{\gamma n_{t}} \int_0^t e^{-\gamma/2(t-s)} dW(s) \end{equation}

and from it, one can compute its mean and its covariance, $\overline{\alpha(t_1)\alpha(t_2)}$.

My question is how to do it with the amplitude stochastic differential equation. I have been struggling with how to formally integrate the equation (mostly because if 1/a term) and then find its covariance. Is there some way to relate the complex-valued covariance $\overline{\alpha(t_1)\alpha(t_2)}$ to the new variable $\overline{a(t_1)a(t_2)}$?

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Using the fact that: $$ a=|\alpha| $$ You can use it to calculate: $$ \langle a(t_1)a(t_2)\rangle =\langle |\alpha(t_1)\alpha(t_2)|\rangle $$ You can calculate in very special cases, but there is no simple general formula. Perhaps what is more relevant is to look at large separation times to extract the correlation time.

Btw, correlations are most relevant when the process is Gaussian, but here, the amplitudes are not a gaussian process so perhaps it’s not so the best quantity to compute.

Furthermore, physically, it is rather the square of the amplitude that is of interest $a^2=|\alpha|^2$ which is related to the number operator/energy. In this case the calculation of the square amplitude correlations is easier.

Hope this helps.

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