# Help with Power Spectral Density Function derivation

1. Mar 29, 2007

### logic smogic

1. The problem statement, all variables and given/known data
Given a two-level atom with transition frequency $${ \omega }_{ ji } \equiv { \omega }_{a}$$ and spontaneous decay rate $$\gamma$$, we are asked to find an expression for the "power spectral density function" $$S(\omega)$$, in terms of $$\omega, {\omega}_{a}$$, and $$\gamma$$.

2. The attempt at a solution

Of course, it should be normalized to 1,
$$\int_{- \infty }^{ \infty } d \omega S(\omega) = 1$$

I believe that the PSD function is just a Fourier Transform of the decay rate, right? If so,
$$S(\omega)= \frac{ 1}{ \sqrt[ ]{ 2 \pi } } \int_{ - \infty }^{ \infty } \gamma (t) { e}^{ - \imath \omega t} dt$$

And then I would just need to double-check that it's normalized.

Thoughts on this?