4. Post-Processing the acquired data

Theory

The Rabi frequency is proportional to the square root of the Intensity. Let’s write this relation as \(\Omega = A \sqrt{I}\). Let’s assume we know \(\Omega\) for a RF amplitude \(V_0\), and write this Rabi frequency \(\Omega_0\). Using our setup, we measure \(I_0 = I(V_0)\), and deduce \(A = \frac{\Omega_0}{I_0}\). Therefore we now have \(\Omega^2(V) = \frac{\Omega^2_0}{I^2_0} I(V)\). \(V\) is obtained from \(\Omega^2\) as \(V(\Omega) = I^{-1}(\frac{I^2_0}{\Omega^2_0} \Omega^2)\).

Implementation

The core of the problem is about building the inverse function of the intensity \(I^{-1}\), that takes an intensity and outputs a voltage. From the output of the previous simulation, we have two arrays of data associating intensities i to voltage a. The function looks like a bijection, and it also makes sense to plot the voltage a function of the intensity i. This disrete V(I) set of points can be interpolated using scipy.interpolate.interp1d(), building the wanted \(V = I^{-1}(I)\) curve. Note that lots of points are necessary in the zones where the I(V) is almost constant (for small amplitudes for example).

To find the RF amplitude associated to a Rabi frequency, you then just have to call the interpolated function just built with parameter \(\frac{I^2_0}{\Omega^2_0} \Omega^2(V)\)