In each of the discussions up to this point the Rotating Wave Approximation has been turned on. One of BlochApp's unique features is that the RWA can be turned off, and the most exact semiclassical formulation of atom--field interactions can be examined. An analytical solution of these equations requires one to use the RWA. If the RWA is not made, then a numerical analysis of the equations is required.
Place all the atoms in the lower state and set the Freq=1.00 and the Amp=0.05. Turn RWA off. Run BlochApp for a few oscillations and clone the vs time plot. Rerun the situation with the RWA turned on. Superimpose both of the vs time plots.
Figure: Graphs of Rabi oscillation with the RWA on and off with Freq=1.00, Amp=0.05
Both the non--RWA and the RWA curves are very close to one another in Figure . However, if you look closely at the non--RWA curve, you will notice a very high frequency oscillation riding on top of the slower Rabi oscillation. This picture shows that the RWA does in fact remove a high frequency oscillation from this system as the theory predicts.
Throughout these exercises the electric field has always had an amplitude of about 0.05. Set Amp=0.50 and run this experiment again.
Figure: Graphs of Rabi oscillation with the RWA on and off with Freq=1.00, Amp=0.50
The RWA result in Figure does not agree very well at all with its non--RWA counterpart; however, the approximation is still in the same general range. As the amplitude is increased still more the RWA failure becomes even greater. The reason for this failure is simple. As the amplitude increases, the Rabi frequency increases. At some point, the Rabi oscillations will be on the order of the high frequency terms that the RWA neglects. At that point, the Rabi oscillation and the high frequency terms will interfere. This interference will alter the simple model that has thus far been examined.
The generalized Rabi frequency includes the detuning as well; so, if the detuning is large the same result can occur. However, in order for our two--state approximation to be valid the detuning must be kept close to zero.
Figure: The path of the Bloch vector for Freq=1.00, Amp=0.50 when the RWA is off
The Bloch vector in Figure behaves in a most interesting manner at high electric field amplitudes when the RWA is off. The beating of the entries in the density matrix can be clearly seen in a uvw versus time plot. This beating generates some fascinating and quite beautiful curves in three dimensions. Examine the three dimensional plot in all the possible projections and note the symmetry. Use the Export option to save a particular three dimensional plot and import into a package like MathCad or Mathematica in order to animate the rotation of the figure (see appendix). When the RWA is on, and the amplitude of the electric field is constant, a circle with some angle of orientation is always observed. With the RWA off and the field at high amplitudes, it is questionable whether or not the curve ever closes; however, it is always bounded by the Bloch sphere. This possible lack of repetition is a signal that a non--RWA semiclassical atom--field interactions theory could contain a quasiperiodic character.
Cusps also appear on the Bloch vector's path. The Bloch vector spends a significant amount of time in these cusps, especially as the amplitude increases. When the RWA is on, and there is no damping in system, all states in a given oscillation exist for the same amount of time. No state is preferred temporally to any other---all have the same lifetime. As the amplitude of the electric field increases and the RWA is necessarily removed, a preference for certain states occurs. This preference corresponds to particular dipole moments and populations. Notice that at Amp=0.05, there are a great number of these cusps, and they are equally distributed around the curve; however, at large amplitudes, like Amp=0.50, the cusps are still equally distributed, but the number has significantly decreased. These cusps seem to represent short--lived, steady--state configurations of the system. This phenomenon is not observed when RWA is turned on. One could test this observation in the laboratory by setting up an experiment with a large amplitude laser and attempt to measure the collective dipole moment of the system. If these cusps do exist in nature, then the statistics obtained from that experiment should be skewed toward a few particular dipole moment configurations.
Try higher amplitudes. What do you observe? Describe the results in terms of u, v, and w. When is the dipole moment most constructive and destructive in general> (Hint: Examine the projections of the Bloch vector carefully.)