Fluid analog model for boundary effects in field theory

Quantum fluctuations in the density of a fluid with a linear phonon dispersion relation are studied. In particular, we treat the changes in these fluctuations due to nonclassical states of phonons and to the presence of boundaries. These effects are analogous to similar effects in relativistic quantum field theory, and we argue that the case of the fluid is a useful analog model for effects in field theory. We further argue that the changes in the mean squared density are, in principle, observable by light scattering experiments.

Figure 1

The geometry of rays reflecting from a parabolic mirror is illustrated. An incoming ray at an angle of $\theta$ reflects at an angle of ${\theta }^{\prime }$ to reach the point $P$, which is a distance $a$ from the focus $F$, and at an angle of $\gamma$. The distance from the focus to the mirror is $b/2$, as illustrated, where $b\gg a$.

Figure 2

The function $f({\theta }^{\prime })$ is plotted for various values of $\gamma$. This function relates the angle of the incident ray, $\theta$, to the angle of the reflected ray, ${\theta }^{\prime }$, through the relation $\theta =(a/b)f({\theta }^{\prime })$.

Figure 3

Two rays reflecting from a parabolic mirror to the point $P$ are illustrated. The first arrives at an angle of ${\theta }^{\prime }=\alpha$, and the second at ${\theta }^{\prime }=\beta$. The points of intersection with the mirror are $(x_1,y_1)$ and $(x_2,y_2)$, respectively. The difference in path lengths (lower path minus upper path) is $\Delta \ell =\Delta {\ell }_1-\Delta {\ell }_2$, where $\Delta {\ell }_2$ is illustrated.

Figure 4

The function $g({\theta }_0)$, defined in Eq. (58), is plotted.

Figure 5

The scattering of photon wave packets by a region in the vicinity of a boundary is illustrated. The incident and scattered wave packets overlap in a scattering region (shaded area). Here the characteristic size of this region, $\ell$, is assumed to be smaller than the characteristic distance to the boundary, $a$.