Set equal to one and equal to one. Place the field in a thermal state with a temperature of one. Run FieldApp with the analysis windows examining **<N>** and . The result is shown in Figure .

**Figure:** The evolution of a thermal state with a temperature of one interacting with a reservoir of temperature one,

is non--zero, but the system is completely unaffected. The system is in equilibrium. A thermal state is defined as the equilibrium state for a reservoir at temperature interacting with a single mode of an electromagnetic field. The reservoir in this case has the same temperature as the thermal state; so, the thermal state was the equilibrium state for that configuration.

Reset the simulation and set for the reservoir. Run the simulation again. In Figure the average and variance of the photon number are not constant as in Figure .

**Figure:** The evolution of a thermal state with a temperature of one interacting with a reservoir of temperature one--half,

After a short period of time, the average and variance of the photon number become constant. The general shape of this new curve is that of a thermal state. By plugging into the relations provided for **<N>** and for a thermal state, it should be clear that this new equilibrium state is a thermal state with a temperature of 0.5.

**Figure:** The evolution of a thermal state with a temperature of one interacting with a reservoir at absolute zero,

Reset the simulation again and set for the reservoir and run the simulation. The steady state for this configuration, seen in Figure , has both the average and variance in the photon number equal to zero. The **n=0** state also has a probability of one, while all other states are zero, i.e., there is no probability of a photon existing in the field. This state is called the vacuum state. It can only be reached when the temperature is equal to absolute zero, and it has very special properties. The vacuum state is by definition a Fock state; however, since and , it is also a coherent state. One of most important properties of this state is that it has a non--zero energy of where is the mode's frequency.

The thermal state is special because it is the equilibrium state for a given reservoir temperature. All initial configurations of field states will decay into a thermal state. Try setting up a coherent, squeezed coherent, and Fock state at and equal to one. Notice that they all decay to the same thermal state.

Wed May 17 14:34:24 EDT 1995