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The Rotating Wave Approximation Revisited

The Rotating Wave Approximation was the central approximation made in the semiclassical model. In some situations this approximation was good and in others not so good. As previously discussed, the Jaynes--Cummings model makes this approximation as well. In general, it is a very good approximation, but it will fail under certain circumstances. The question is when will it fail, and what variable determines how well it works.

Minimize JCApp and maximize QEDapp. Turn the RWA on in order for QEDapp to simulate the Jaynes--Cummings model. In the State dialog, make the element one and leave all the other terms zero. Run the application and verify that it provides the same result as observed in the previous exercise with JCApp.

Now place the system in the |a0> state, i.e., set equal to one, and run QEDApp. The probability never changes. An atom in its lowest energy state and a field in its lowest energy state cannot exchange energy. Turn the RWA off and run the program again. Now the probability does change. Notice that a complete transition is not being made. This result is possible because the vacuum state has a non--zero energy and in this configuration the atom's ground state has energy .

  
Figure: The oscillation of the atom when both the atom and field are in the ground state at g=1.00,g=0.50,g=0.25 with RWA off.

Run QEDapp several times and make the atom--field coupling constant, g, closer to zero each time. Notice that in Figure gif, at low g, the atom acts in the same manner that the Jaynes--Cummings model would predict. The fast oscillation that has been observed comes from the terms that were thrown out of the Jaynes--Cummings model by the RWA. Therefore, the RWA is valid for low coupling constants and frequencies close to resonance. This coupling constant takes the place of the field amplitude in BlochApp in limiting the range within which the RWA is valid. The situations which both JCApp and QEDApp are modelling are a single atom interacting with a weak field. In these cases, the coupling constant is in general quite small; so, the RWA is a very good approximation.



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Next: The Atom and Up: Simulations Previous: Spontaneous and Stimulated



Andy Antonelli
Wed May 17 14:34:24 EDT 1995