Analog model for light propagation in semiclassical gravity

We treat a model based upon nonlinear optics for the semiclassical gravitational effects of quantum fields upon light propagation. Our model uses a nonlinear material with a nonzero third order polarizability. Here a probe light pulse satisfies a wave equation containing the expectation value of the squared electric field. This expectation value depends upon the presence of lower frequency quanta, the background field, and modifies the effective index of refraction, and hence the speed of the probe pulse. If the mean squared electric field is positive, then the pulse is slowed, which is analogous to the gravitational effects of ordinary matter. Such matter satisfies the null energy condition and produces gravitational lensing and time delay. If the mean squared field is negative, then the pulse has a higher speed than in the absence of the background field. This is analogous to the gravitational effects of exotic matter, such as stress tensor expectation values with locally negative energy densities, which lead to repulsive gravitational effects, such as defocusing and time advance. We give some estimates of the magnitude of the effects in our model and find that they may be large enough to be observable. We also briefly discuss the possibility that the mean squared electric field could be produced by the Casimir vacuum near a reflecting boundary.

Figure 1

Behavior of the quantity $\mathrm{\Delta }$ as a function of time. Notice that subvacuum effects ($\mathrm{\Delta }<0$) occur periodically in time intervals of duration ${\tau }^{-}$. In this plot we have set $q=0.5$.

Figure 2

The refraction of the probe ray in the nonlinear medium is illustrated. In the absence of the background field, the angle of refraction is ${\theta }_0$. When the index of refraction attains its maximum value, the angle becomes ${\theta }^{+}$, and at its minimum value, the angle becomes ${\theta }^{-}$. The former case models the gravitational effect of positive energy, and the latter that of negative energy.