Next: The Jaynes--Cummings Model Up: Simulations Previous: The Decay of

## The Decay of Squeezed States

The decay of a squeezed coherent state is similar to that of a coherent state in many ways. The interesting point to examine in this decay is how the variances change. For any temperature a squeezed coherent state must unsqueeze because both the vacuum state and the thermal state have variances of momentum and position which are equal.

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

The end result is a thermal state, as in Figure . Unlike the devolution of a coherent state, there may be an intermediate state which has well--defined statistics. The plot has a value of one at some time in the devolution. If the temperature is zero, then the state will remain there. If the temperature is non--zero, then the plot will continue towards some value dictated by the reservoir temperature. When and , doe the squeezed coherent state become a coherent state?

Figure: The squeezed variance() increases, while the stretched variance() decreases with and .

No. An examination of the variances of momentum and position in Figure where will show that neither of these quantities is equal to one half. The equality of the variance and average photon number is a necessary but not a sufficient condition for a field being coherent. Similar points can be found in other decays like that of a Fock state. Does a squeezed coherent state become coherent if ? (Hint: Remember what happens to a coherent state at .)

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

The Wigner function shown in Figure shows how a coherent state unsqueezes. The system is initially squeezed with respect to position. The ellipse begins its journey and slowly attempts to transfer the squeeze from position to momentum; however, the momentum variance never squeezes. Indeed, the ellipse becomes a circle and decays to the origin. The path and the end result are basically identical to that of a coherent state.

FieldApp is capable of generating two types of squeezed states: the squeezed coherent state and the squeezed vacuum state. A squeezed vacuum state has equal to zero and a non--zero squeezing parameter. If r=0, then what would this state be? Set to zero and plot the result in the Wigner function. How are the momentum and position versus time plots related? Run the simulation with a equal to zero and a equal to both zero and one. What are the differences in these two simulations? similarities? Is a squeezed vacuum state very stable?

Next: The Jaynes--Cummings Model Up: Simulations Previous: The Decay of

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