A coherent state is the most wavelike state. If the temperature is non--zero, then the field will decay towards a thermal state which is non--wavelike; indeed, thermal light is often called chaotic because it has no periodic order due to its purely random phase. As a coherent state decays it moves from a highly ordered form into a highly disordered form. When in the decay does the state cease having its wavelike properties? FieldApp allows the user to understand this problem in a number of ways.

**Figure:** As a Coherent state with decays it loses its Poissonian character for

One of the defining characters of a coherent state is the equality of the variance and the average photon number. Figure shows the end result of the evolution of a coherent state towards equilibrium. The field state is a thermal state as expected. The plot clearly shows that the equality of these two variables is lost at the very beginning of the simulation. Thus, the statistics of the field cannot be classified in a simple way after the decay process has begun.

**Figure:** The Wigner function for a Coherent state with as time increases.

The Wigner function in Figure offers a compact way of viewing the decay. The state begins as it would if . The uncertainty circle decays towards the origin as a satellite might fall to earth; however, as time increases the radius increases towards a maximum value obtained at the origin. The final position of the Winger function shows that the field is in a state with no average momentum and position, i.e, a thermal state. The **q**, and **p**, versus time plots in Figure show the same effect from a different perspective.

**Figure:** **q**, vs **t** and **p**, vs **t** show that the wave--like character of the coherent state disappears as the state decays.

**Figure:** and increase as the Coherent state decays.

The system initially oscillates in a wavelike manner but as time increases the uncertainties in the momentum and position increase and the amplitude decreases. The increase in uncertainty coupled with the exponential decay of the density matrix's off--diagonal terms destroys the coherence of the state. Figure shows the variance of the momentum and position plotted versus time. Notice that these plots show that the variances increase from their initial values of a half and never return to them. Again this shows that the coherent state is destroyed the moment that the simulation is begun because a coherent state is classified as a state whose variances in position and momentum are one half.

Each of these different visualizations illustrates the same result. However, placing all these perspectives together allows the user to gain a great deal of understanding about the system. Try this experiment again with different values of , , and . What are the differences? similarities? How do the variances in momentum and position vary in the decay? Do they decay at the same rate? Are they always equal? Examine both B and E as the coherent state decays.

The condition of non--zero temperature has been cautiously placed in each of the above statements. What if this restriction is lifted? Set to zero and run FieldApp in a coherent state with the analysis types set to **<N>** and .

**Figure:** As a Coherent state with decays it does not lose its Poissonian character at

The ratio of the variance to the average of the photon number is always one in Figure . This result is indicative of a coherent state. Plot the variance of momentum and position and verify that the uncertainties are always equal to one half. The basic shape of the state nonetheless is changing. Each state is different, but all are coherent with different values of . In all the other decays that have been examined, the statistics were destroyed by the decay. In this case, the statistics do not change---only does. A reference to this effect was not found in the literature, but it is reasonable. The vacuum state, which all field states decay to at absolute zero, is coherent. Thus, a coherent state with not equal to zero evolves towards a coherent state with equal to zero. In other words at absolute zero a coherent state decays only in the sense that the number of photons that it can contain decays. This constancy in statistics highlights the coherent state as a fundamentally important and special field state.

Wed May 17 14:34:24 EDT 1995