Negative energy density in superposition and entangled states

We examine the maximum negative energy density which can be attained in various quantum states of a massless scalar field. We consider states in which either one or two modes are excited, and show that the energy density can be given in terms of a small number of parameters. We calculate these parameters for several examples of superposition states for one mode, and entangled states for two modes, and find the maximum magnitude of the negative energy density in these states. We consider several states which have been, or potentially will be, generated in quantum optics experiments.

Figure 1

The quantity $R-n$ for two superposed squeezed vacua, Eq. (31), is plotted for the case $\theta =0$ for various values of $\eta$. The case $\eta =0$ is the single squeezed vacuum, and gives more negative energy density than do any of the superpositions. For nonzero $\eta$, there is a maximum value for $R-n$ at a finite value of $r$.

Figure 2

Here $R-n$ is plotted for the superposition of a coherent and a squeezed vacuum state, Eq. (39) with $\eta =1$, as a function of $r$ for various values of $\alpha$. Note that for $\alpha \ne 0$, $R-n$ can be positive, corresponding to negative energy, for both smaller and larger values of $r$ in the range illustrated, but has an intermediate region where there is no negative energy.

Figure 3

Here $R-n$ is plotted as a function of $r$ for $\alpha =0$ and $\eta =-1$, a superposition of the vacuum and a squeezed vacuum. The maximum negative energy occurs when $r\approx 2$, where $R-n\approx 0.3$.

Figure 4

The quantities $R_1-n_1$ (solid lines) and $n_1$ (dashed lines) are plotted for two values of $\theta$ as functions of $r$ for the entangled squeezed state defined in Eqs. (55, 56). In the limit that $\theta$ is close to $\pi$, one can have appreciable negative energy at small values of $r$. We see that the peak negative energy, $R_1-n_1\approx 0.25$ occurs at $r\approx 0.007$ for $\theta =0.99\pi$ and at $r\approx 0.035$ for $\theta =0.95\pi$, whereas the mean particle number is about the same for both cases, $n_1\approx 0.2$.

Figure 5

The function $f(\sigma )$, given by Eq. (72), is plotted. Its maximum, at $\sigma \approx 0.7$, describes the case of maximal negative energy density for the entangled coherent state defined in Eqs. (64, 65, 66).