In the last section, we observed the phenomenon of Rabi oscillations from the point of view of the density matrix. We can perform the same analysis in the Bloch vector picture. Select the analysis type in the the Bloch window to be 3D under the Projections submenu. Set Amp=0.05 and the Freq=1.00 and run the program.
The Bloch vector draws a circle which is in the v--w plane. To see that these points are coplanar, click on the right mouse button to change the analysis type to a projection onto the u--v, u--w, and v--w planes. The image appears as a line in both the u--v and u--w planes and a circle in the v--w plane. u=0 and both v and w vary sinusoidally when on resonance. Remember that u is defined as twice the real part of the off--diagonal terms. What are the real and imaginary parts of the off--diagonal terms in the density matrix doing? On resonance, the off--diagonal terms are totally imaginary; so, u must be zero.
Try placing different values in Amp leaving Freq=1.00. The rate with which the Bloch vector spins is directly related to the Amp that is set. Select uvw vs Time from the Time Dependent submenu. Compare the frequency changes in the components to what you observe in the diagonal elements of the density matrix.
All the plots that we have examined are in the rotating frame. This frame can best be understood in the Bloch vector picture. Set Amp=0.05 and Freq=1.00 and in the Numerical item in Parameters set the number of calculations between drawing to one. Click on the Rotating Frame item in the Bloch window. This toggle moves one out of the rotating frame and into the original frame which will be called the Schrödinger frame. The frame change only affects the Bloch window, i.e., the density matrix has not been altered. Run the program again.
Figure: Rabi oscillation in the rotating frame with Freq=1.00, Amp=0.05
Figure shows the Bloch vector moving between the upper and lower state; however, there is now a rotation about the w axis as well. The u component is no longer non--existent. Select the uvw vs t analysis and notice that w is exactly the same; however, both u and v are beating as in Figure .
Figure: The components of the Bloch vector in the rotating frame with Freq=1.00, Amp=0.05
This effect is derived from the fact that two distinct sets of oscillation exist in this system. The is changing so quickly that is not able to complete a precession about before it has moved. When we move into the rotating frame, we remove the motion of so that only the precession remains. The physics of this system is the same in both frames, but it is easier to interpret in the rotating frame because one layer of complexity has been stripped off. This result is similar to a top which is spinning and slowly precessing. This slow precession is in many ways distinct from the actual spinning of the top and is termed nutation. Sometimes physicists speak of Optical Nutation instead of Rabi Oscillations. Both refer to the same phenomenon; however, the former relies on the density matrix while the latter relies on the Bloch vector. Again this terminology was born in the realm of nuclear magnetic resonance. The Rotating Wave Approximation and rotating frame may sound the same, but they are completely different. RWA is a numerical approximation that makes simplifying assumptions about the physics of the system, while the rotating frame is merely a change in the point of view of the system.
Perform the second part of the Rabi oscillation experiment. Superimpose the plots that you obtain for Amp=0.05 and Freq=1.00, 1.01, 1.05, and 1.10. What do you observe? What is the position of in each case? Try using various projections. The result should look like Figure .
Figure: Rabi oscillation in the Bloch vector picture with Amp=0.05; Freq=1.00,1.01,1.05,1.10
As varies from zero, the position of changes. Try making Freq smaller than 1.00. What happens to ?