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Fourier Transform

  1. Jan 21, 2009 #1
    1. The problem statement, all variables and given/known data
    [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.
    2. Relevant equations


    When looking at this expression, the integrals on the right are evaluated first then proceed to the left.
    3. 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]
     
  2. jcsd
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