What does it mean for half of an empty cavity to be full?

It is well known that the vacuum state of a quantum field is spatially entangled. This is true in both free and confined spaces, for example, in an optical cavity. The obvious consequence of this, however, is surprising and intuitively challenging: namely, that in a mathematical sense, half of an empty cavity is not empty. Formally this is clear, but what does this physically mean in terms of, say, measurements that can actually be made? In this paper we utilize a local quantization procedure along with the tools of Gaussian quantum mechanics to characterize the particle content in the reduced state of a subregion within a cavity and expose the spatial profile of its entanglement with the opposite region. We then go on to discuss a thought experiment in which a mirror is very quickly introduced between the regions. In so doing we expose a simple and physically concrete answer to the above question: the real excitations created by slamming down the mirror are mathematically equivalent to those previously attributed to the reduced states of the subregions. Performing such an experiment in the laboratory may be an excellent method of verifying vacuum entanglement, and we conclude by discussing different possibilities of achieving this aim.

Figure 1

Sketch of the one-dimensional cavity setting. We start ($t<0$) considering a cavity in its vacuum state $|0{⟩}_G$. At some instant ($t=0$) we slam a mirror separating the cavity into two regions. As explained in the text, the normal modes in these separated subcavities correspond to the localized modes of the cavity without mirror, which we will show suffice to analize states, correlations and particle production after slamming the mirror. The horizontal line corresponds to $t=0$, the diagonal lines represent the light cone starting at the slamming event.

Figure 2

The cavity for the cases studied in the paper. The figures on the left correspond to the full cavity without mirrors, the light dotted vertical bars indicating the border of the regions chosen in Secs. 3a and 3c to study localization into two (top) or three (bottom) spatial regions. The right figures show the decomposition in terms of local modes at $t=0$ for both settings.

Figure 3

The logarithmic negativity $E_N$ between local modes $u_m$ and ${\overline{u}}_n$ on the left and right sides of the cavity, respectively. The cavity is divided in two equal regions $r=0.5R$. Left: a field mass of $\mu =0$. Right: a field mass of $\mu =15/R$.

Figure 4

The logarithmic negativity $E_N$ between local, diagonalizing modes $v_m$ and ${\overline{v}}_n$ on the left and right sides of the cavity, respectively. The cavity is split into two equal sides, $r=0.5R$, and $N=200$ for both the left and right sides. Left: a field mass of $\mu =0$. Right: a field mass of $\mu =15/R$.

Figure 5

The function $|v_1(x)|$ in the left side of the cavity, representing the spatial distribution of entanglement with the opposite side. The parameters are given by $r=0.5R$, and $N=200$, with different field masses $\mu$ considered: 0 (blue), $10/R$ (light blue) and $50/R$ (green). As can be seen: the larger the mass of the field, the closer the entanglement straddles the boundary between the two sides of the cavity, as expected.

Figure 6

Evolution of the entanglement spatial distribution for the massive case $\mu =50/R$ after an elapsed time $t=r/2$. We can see a peak for the correlations at exactly the position of the particle-burst front as originated from the slamming. The cavity parameters are the same as in Fig. 5.

Figure 7

Two-mirror case: logarithmic negativity $E_N$ between local, diagonalizing modes $v_m^A$ and $v_n^C$ on the left and rightmost sides of the cavity, respectively, in the case that the field is massless $\mu =0$. The cavity is in this case split into three regions, ${\mathrm{\Delta }}_A=[0,0.5R-B/2]$, ${\mathrm{\Delta }}_B=[0.5R-B/2,0.5R+B/2]$, ${\mathrm{\Delta }}_C=[0.5R+B/2,R]$. We have taken $N=200$. Left: Size of the middle section $B=0.1\mathrm{R}$. Right: Size of the middle section $B=0.2R$.

Figure 8

The number expectation value of local modes $u_m$ for the case of a massless field $\mu =0$ and a cavity split in half $r/R=0.5$.