- #1

Nusc

- 760

- 2

## Homework Statement

[tex]

\begin{subequations}

\begin{eqnarray}

\dot{\hat{{\cal E}}}(t) &=& -\kappa \hat{{\cal E}}(t) + i g\int_{-\infty}^{\infty} d \Delta\; {\cal \rho}(\Delta)\, \Bigg\{

e^{-(\gamma + i\Delta)(t-t_{0})}\hat{\sigma}_{ge}(t_{0},\Delta)+ ig\int_{t_{0}}^{t} d t' \hat{{\cal E}}(t')e^{-(\gamma + i\Delta)(t-t')} \Bigg\}

\nonumber\\

& & + \sqrt{2\kappa}\, \hat{{\cal E}}_{in},

\\

&=& -\kappa \hat{{\cal E}}(t) + i g\int_{-\infty}^{\infty} d \Delta\; {\cal \rho}(\Delta)\, \hat{\sigma}_{ge,0}(t_{0},\Delta)e^{-(\gamma +i\Delta)(t-t_{0})}

\nonumber\\

& & -g^{2} \int_{-\infty}^{\infty} d \Delta\; {\cal \rho}(\Delta)\, \int_{t_{0}}^{t} d t' \hat{{\cal E}}(t')e^{-(\gamma + i\Delta)(t-t')} + \sqrt{2\kappa}\, \hat{{\cal E}}_{in},

\\

\nonumber

\end{eqnarray}

\end{subequations}

[/tex]

I need to find the Fourier Transform of these integrals.

## Homework Equations

When looking at this expression, the integrals on the right are evaluated first then proceed to the left.## The Attempt at a Solution

[tex]

After applying the Fourier transform to the integral, we obtain:

\begin{subequations}

\begin{eqnarray}

i\omega \tilde{\hat{{\cal E}}}(\omega)

&=& -\kappa \tilde{\hat{{\cal E}}}(\omega) + i g\int_{-\infty}^{\infty} d \Delta\; {\cal \rho}(\Delta) \int_{-\infty}^{\infty} dt e^{-i\omega t} \hat{\sigma}_{ge,0}(t_{0},\Delta)e^{-(\gamma +i\Delta)(t-t_{0})}

\nonumber\\

& & -\frac{g^{2}}{\sqrt{2\pi}} \int_{-\infty}^{\infty} d \Delta\; {\cal \rho}(\Delta) \int_{-\infty}^{\infty} dt e^{-i\omega t} \int_{t_{0}}^{t} d t' \hat{{\cal E}}(t')e^{-(\gamma + i\Delta)(t-t')}

\nonumber\\

& & + \sqrt{2\kappa}\, \tilde{\hat{{\cal E}}}_{in}(\omega),

\\

\nonumber

&=& -\kappa \tilde{\hat{{\cal E}}}(\omega) + i g\int_{-\infty}^{\infty} d \Delta\; {\cal \rho}(\Delta) \int_{-\infty}^{\infty} dt e^{-i\omega t} \hat{\sigma}_{ge,0}(t_{0},\Delta)e^{-(\gamma +i\Delta)(t-t_{0})}

\nonumber\\

& & -\frac{g^{2}}{\sqrt{2\pi}} \int_{-\infty}^{\infty} d \Delta\; {\cal \rho}(\Delta) \int_{-\infty}^{\infty} dt e^{-i\omega t} \int_{t_{0}}^{t} d t' \hat{{\cal E}}(t')e^{-(\gamma + i\Delta)(t-t')}

\nonumber\\

& & + \sqrt{2\kappa}\, \tilde{\hat{{\cal E}}}_{in}(\omega),

\\

\nonumber

\end{eqnarray}

\end{subequations}

[/tex]