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A Coherent State

The interaction of an atom with a coherent field state is the closest quantum mechanical analog to the situation modeled in BlochApp. The spontaneous emission would have to be turned on in BlochApp because the quantized model has already included this contingency. Set the field in a coherent state with equal to one.

  
Figure: The oscillation of the probability of an atom in the upper state interacting with a coherent state, =1.

Does the plot in Figure gif appear to be chaotic? The answer might seem to be yes. Indeed, many early theorists were frustrated by this picture. The experimentalists told them that a definite structure existed in this situation, but no theorist could find it.[23] The reason for this problem is that is too small. Set the number of field states to 75 and make a coherent state with equal to fifty.

  
Figure: The oscillation of the probability of the lower state of an atom initially in the upper state interacting with a coherent state, =50

This field is still relatively weak, but note the distinct difference in the structure between Figures gif and gif. The oscillations die out as they do in BlochApp. However, at some time later the oscillations revive. They then die out again only to revive again. This collapse and revival scheme continues until the two effects flow together, setting the atom into a quasi--random motion. The initial collapse was predicted by Cummings and is called the Cummings collapse. This collapse is due merely to the relative dephasing of the various elements of the field mode, or semiclassically the saturation of the atomic transition. The revivals are entirely due to the grainy nature of the field.[24] The observed effect is similar to an inhomogeneous effect in the semiclassical model called a photon echo.[25]

This collapse and revival scheme is one of the most studied areas of quantum optics. At this point in history, it has become possible to experimentally observe these effects in fields even smaller than the one simulated above. This result is a general one for all coherent states. Indeed, the collapse time and revival time have been analytically calculated using various approximations. When

where is the amount of time required for the system to decay by a factor of . This approximation was derived by Eberly[23] and has been shown to correlate quite well with experiments. Calculate the and for the situation above. You will notice that is close but is not. The reason for this difference is simple. The run required to observe this phenomena is in the range of a hundred seconds. The g is one, and the square root of fifty is a little greater than seven. We are not in a range in which the analytical approximation of is valid. In order to observe the proper scaling, the would have to be much higher, and the g would have to be quite small (Note: In this range, the RWA is a very good approximation.). A run of this kind would not necessarily be computationally prohibitive, but it would be temporally prohibitive.

Run JCApp for various other coherent states. Try making higher and higher. What do you observe about the revivals relative to each other? Does the increase as increases? Try to draw the envelope of atomic populations for each run. Do you observe a pattern?



next up previous contents index
Next: A Squeezed State Up: Simulations Previous: A Thermal Field



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