It has been stated that the resonant frequency of this system is the transition frequency, which in this case is one. This result was not proven. The proof was by analogy to the classical system. At this point, one begins to think that any analogy to the classical system could be rather dangerous. In order to show that is in fact the resonance condition for this system, one can perform a frequency scan through resonance. This scan requires that we parse a function that will alter the frequency. The best function to use is a linear one such as
from t=0 up to t=2c. This function will scan from up to . An accurate scan should not be done quickly. A good value of c is 1000, which means that the total run time should be 2000. The above function should not be put into the parser since the parser adds to Freq whatever is registered on the slider. Let the frequency slider read 1.00 and enter
into the parser. Initially, we will utilize the RWA; so, make the amplitude small, Amp=0.05, where the RWA is quite good. Place all the atoms in the lower state. Also, turn on both elastic collisions and spontaneous emission with values and . These values will make both the longitudinal and transverse lifetimes equal to 12.50. These lifetimes are extremely short compared to the total run time. Thus, the Rabi oscillation will damp out very quickly. This damping is required so that the effect of the frequency scan will be completely isolated. After performing the frequency scan with the damping, compare it to the result without the damping, and explain the result. Again all the atoms should be in the lower state. Making the steps per calculation equal to 50 is advisable because the general shape, as opposed to the oscillation, is what contains the interesting physics. Run BlochApp and examine the uvw vs time plot.
Figure: A frequency scan through resonance with the RWA on and Freq=1.00, Amp=0.05, =0.04, and A=0.08
Notice that initially the curves in Figure oscillate rapidly and then stop. Ignore these transient pieces of u, v, and w. The w curve is peaked at t=1000 seconds. At 1000 seconds, the frequency function was at . Therefore, is the resonant condition for this system. The v curve has a minimum at the resonant frequency, while the u curve is zero. The u curve is the dispersive element of the dipole moment. At resonance, there is no dispersion. -v is the absorptive or constructive component of the dipole moment; so, the maximum amount of absorption occurs at resonance. Therefore, resonance is a very special frequency at which the population inversion is maximized, dispersion is non--existent, and the atom is absorbing the maximum amount of energy from the field. These curves are exactly analogous to those for a classical oscillator which demonstrates that this system is really not as far removed from its classical cousin as one would think.
The amplitude of the electric field was intentionally made small in order that the RWA could be assumed. Run the previous situation again with the RWA off. Make the steps per calculation equal to 10 and the time step equal to 0.1. Notice that the non--RWA and RWA results for Amp=0.05 are exactly the same. The resonant frequency occurs at the same point.
Figure: A frequency scan through resonance with RWA off and Freq=1.00, Amp=0.50, A=0.08
Let us now make the amplitude large, Amp=0.50, and run BlochApp again. The resonant frequency is significantly different in this case Figure . The frequency appears to have shifted about 250 seconds to the right of 1000 seconds. Examining the parser display of the frequency function, this corresponds to a shift in the resonant frequency of 0.060. This shift in the resonant frequency is called the Bloch--Siegert Shift and has been theoretically predicted and observed in the laboratory. It can be quantified in our notation as 
For Amp= = 0.50 and = 1.00, the frequency shift should be: . The parsed frequency function has this value at about 1250 seconds. The theoretical data corresponds quite well with the observed results. eq:bsShift for the Bloch--Siegert shift is an approximation; thus, it should not be surprising that at very high amplitudes this approximation seems to fail. The Bloch--Siegert shift is another effect which the RWA dismisses.
The RWA is an excellent simplifying approximation under a number of circumstances; however, it is an approximation and one must know when to use and when not to use it. The semiclassical theory is itself an approximation which works quite well at large photon numbers. The rest of this text will construct a fully quantized theory of atom--field interactions which will work where the semiclassical theory fails at small photon numbers.