Let us begin our examination of semiclassical atom--field interactions by placing a single atom at some point in space and aiming a laser at it. This laser will be tuned in such a way that only two energy levels of the atom are linked by the laser's frequency which does not have to be exactly equal to the atom's transition frequency but must be quite close. If the laser frequency is not close enough, then we lose the ability to approximate the atom by two energy levels.
Setting up this system in BlochApp is simple. Let us assume that initially the atom is in the lower state, i.e., . Since the atom is in the lower state, the probability of being in that state is one, and the probability of being in the upper state is zero; so, and . The off--diagonal elements are also zero.
Turn off all the switches in the Atomic item under Parameters. These options are necessary only when we consider a statistical ensemble of atoms. The sliders have the two most important variables: Amp and Freq. Freq refers to the laser frequency, , and Amp refers to the scaled amplitude of the electric field, . Remember that
The Amp in BlochApp is a scaled amplitude which absorbs both e and .
For this experiment, set Freq=1.00 and Amp=0.05. The transition frequency of the two level atom has been defined as 1.00 for simplicity. Set the upper analysis window to examine , and ignore the lower analysis window for the time being. Click on the run button, and let it run for a few seconds. Stop the simulation and reset it. Click on the density matrix window and run the program again. Notice how the probability of being in the lower state decreases and how the probability of being in the upper state increases. Also note that off--diagonal terms appear and disappear. When do they disappear? Why?
After one run, click on the upper analysis plot with the right mouse button and go to the menu. Now click on the Inspect Scale option and select the Clone Graph option. You can make as many clones of a graph as you like. Return to BlochApp via the Windows Task Manager (Press Control and Escape simultaneously to obtain the Task Manager.) and run BlochApp again with Amp=0.10. Go to Inspect Scale and archive the plot with Copy to Archive. Move to the graph that you cloned via the Windows Task Manager, and double click on the cloned window. This action will bring up the cloned plot's Inspect Scale option. Click on Paste from Archive. The final result will be a vs Time graph for both Amp=0.05 and Amp=0.10. Continue doing the same thing for a few more values of Amp. What do you notice?
Figure: Rabi oscillation at Freq=1.00; Amp=0.05,0.10,0.20
In Figure , we see that the amplitude of the oscillation is always the same. A complete transition is made between the upper and lower states. As Amp is increased, the frequency of the oscillations increases. Suppose that we now perform the same experiment; however, this time the Amp will be set at 0.05, and Freq will be varied from 1.00 to 1.05 to 1.10.
Figure: Rabi oscillation at Amp=0.05; Freq=1.00,1.05,1.10
A drastic change has occurred in this situation as seen in Figure . The amplitude of the oscillations decreases as the laser frequency increases. A complete transition between the upper and lower state is not made any more. Therefore, 1.00 is the resonant frequency of the system! Hold down the left mouse button in the cloned plot and compare the frequencies of the three plots. They are not the same; indeed, as Freq increases so does the frequency of oscillation. The frequency of oscillation of the probability amplitudes is governed by both Freq and Amp. However, the frequency--shifting effect of Freq is much smaller than that of Amp.
The oscillation that was observed in these experiments is called the Rabi oscillation. is called the Rabi frequency. However, there is a generalized Rabi frequency which has the form
where . The generalized Rabi frequency is what is measured in these plots. These terms again refer to nuclear magnetic resonance and were later appropriated by students of atom--field interactions.
Figure is similar to the plots of the Lorentz Oscillator model at various values of Q. The resonance condition as well as the frequency shifts are predicted by the classical model. However, there is a clear distinction between these two. The classical model is predicting a physical oscillation in space. The semiclassical model's plot is an oscillation in state probability. Each has a very different interpretation.