Model¶
Exponentical decay¶
In pump-probe time resolved spectroscopy, we usually model the dynamics as sum of the exponential decay. For simplicity, I will consider one exponential decay model.
\[\begin{equation*}
{Model}(t) = \begin{cases}
0& \text{if $t<0$}, \\
\exp(-kt)& \text{if $t \geq 0$}.
\end{cases}
\end{equation*}\]
where \(k\) is rate constant, inverse of the life time.
Damped Oscillation¶
One can observe vibrational feature in pump-probe time resolved spectroscopy experiment, such vibrational feature can be modeled to damped oscillation.
\[\begin{equation*}
{Model}(t) = \begin{cases}
0 & \text{if $t<0$}, \\
\exp(-kt)\cos(2\pi t/T+\phi) & \text{if $t \geq 0$}.
\end{cases}
\end{equation*}\]
where \(k\) is damping constant, inverse of the lifetime of vibration, \(T\) is the period of vibration and \(\phi\) is phase factor. One can view damped oscillation as generalized form of exponential decay.