11.4: Four-Wave Mixing

A more advanced ultrafast spectroscopy technique than pump-probe is four- wave mixing (FWM). It enables to investigate not only energy relaxation processes, as is the case in pump-probe measurements, but also dephasing processes in homogenous as well as inhomogenously broadened materials. The typical set-up is shown in Figure 11.12

Figure 11.12: Typical Four-Wave-Mixing (FWM) beam geometry.

Lets assume these pulses interact resonantely with a two-level system modelled by the Bloch Equations derived in chapter 2 (2.1592.162).

\[\left (\Delta - \dfrac{1}{c_0^2} \dfrac{\partial^2}{\partial t^2} \right ) \vec{E}^{(+)} (z, t) = \mu_0 \dfrac{\partial^2}{\partial t^2} \vec{P}^{(+)} (z, t), \nonumber \]

\[\vec{P}^{(+)} (z, t) = -2N \vec{M}^* d(z, t) \nonumber \]

\[\dot{d} (z, t) = -(\dfrac{1}{T_2} - j \omega_{eg})d + \dfrac{1}{2j \hbar} \vec{M} \vec{E}^{(+)} w, \nonumber \]

\[\dot{w} (z, t) = -\dfrac{w - w_0}{T_1} + \dfrac{1}{j \hbar} (\vec{M}^* \vec{E}^{(-)} d - \vec{M} \vec{E}^{(+)} d^*) \nonumber \]

The two-level system, located at \(z = 0\), will be in the ground state, i.e. \(d(t = 0) = 0\) and \(w(t = 0) = -1\), before arrival of the first pulses. That is, no polarization is yet present. Lets assume the pulse interacting with the two-level system are weak and we can apply perturbation theory. Then the arrival of the first pulse with the complex field

\[\vec{E}^{(+)} (\vec{x}, t) = \vec{E}_0^{(+)} \delta (t) e^{j(\omega_{eg} t - j \vec{k}_1 \vec{x})} \nonumber \]

will generate a polarization wave according to the Bloch-equations

\[d(\vec{x}, t) = -\dfrac{\vec{M} \vec{E}_0^{(+)}}{2j \hbar} e^{j(\omega_{eg} - 1/T_2)t} e^{-j \vec{k}_1 \vec{x}} \delta (z), \nonumber \]

which will decay with time. Once a polarization is created the second pulse will change the population and induce a weak population grating

\[\Delta w(\vec{x}, t) = \dfrac{|\vec{M} \vec{E}_0^{(+)}|^2}{\hbar^2} e^{-t_{12}/T_2} e^{-j(\vec{k}_1 - \vec{k}_2)\vec{x}} e^{-(t - t_2)/T_1} \delta (z) + c.c., \nonumber \]

When the third pulse comes, it will scatter of from this population grating, i.e. it will induce a polarization, that radiates a wave into the direction \(\vec{k}_3 + \vec{k}_2 - \vec{k}_1\) according to

\[d (\vec{x}, t) = \dfrac{\vec{M} \vec{E}_0^{(+)}}{2j \hbar} \dfrac{|\vec{M} \vec{E}_0^{(+)}|^2}{\hbar^2} e^{-t_{12}/T_2} e^{-t_{32}/T_1} e^{-j(\vec{k}_3 + \vec{k}_2 - \vec{k}_1)\vec{x}} \delta (z) \nonumber \]

Thus the signal detected in this direction, see Figure 11.12, which is proportional to the square of the radiating dipole layer

\[S(t) \sim |d(\vec{x}, t)|^2 \sim e^{-2t_{12}/T_2} e^{-2t_{32}/T_1} \nonumber \]

will decay on two time scales. If the time delay between pulses 1 and 2, \(t_{12}\), is only varied it will decay with the dephasing time \(T_2/2\). If the time delay between pulses 2 and 3 is varied, \(t_{32}\), the signal strength will decay with the energy relaxation time \(T_1/2\)

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