Maximal subvacuum effects: A single mode example

We discuss an example of a subvacuum effect, where a quantum expectation value is below the vacuum level, and is hence negative. The example is the time average of the mean squared electric field in a nonclassical state where one mode is excited. We give some specific examples of such states, and discuss the lower bound on the squared field or its time average. We show when a lower bound can be obtained by diagonalization of the squared electric field operator, and calculate this bound. We also discuss the case of an instant time mean squared electric field, when the operator cannot be diagonalized. In this case, a lower bound still exists but is attained only by the limit of a sequence of quantum states. In general, the optimum lower bound on the mean squared electric field is minus one-half of the mean squared electric field in a one-photon state. This provides a convenient estimate of the subvacuum effect, and may be useful for attempts to experimentally measure this effect.

Figure 1

The mean number of particles in the lowest eigenstate of the Lorentzian averaged squared electric field is plotted as a ratio of the sampling time $\tau$ and the period $T$ of the excited mode. This is the state which maximizes the subvacuum effect in this case, and we see that unless $\tau$ is very small, this mean number is relatively small.

Figure 2

The finite duration function, $g_F(t)$ is plotted.

Figure 3

Here the mean number of particles in the lowest eigenstate of the squared electric field averaged over a finite time is plotted as a function of $\tau /T$. In this case, this mean number is larger than for the case of Lorentzian averaging.