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The time dynamics of an atom interacting with a reservoir of spectral density $J(\omega)$ are obtained by solving the following integro-differential equation:

$$ \frac{\mathrm{d}c(t)}{\mathrm{d}t} = - \int\limits_0^t \mathrm{d}\tau~ c(\tau)f(t-\tau),$$

where

$$f(t-\tau) = \int\limits_0^{\infty}\mathrm{d}\omega ~ J(\omega) \exp(i(\omega_E-\omega)(t-\tau)) $$

I have an analytical solution when $J(\omega)$ is a Lorentzian distribution, however, is there a way to solve the above numerically for $c(t)$, given an arbitrary spectral density?

In particular I don't have an analytical expression for $J$, only the numerical data.

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  • $\begingroup$ Both LHS and RHS are linear operators applied to function c(t), so if c is discretized on a grid in time, then the equation boils down to a linear algebra problem. $\endgroup$ Jul 11 at 15:15

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