Measuring the entanglement of analogue Hawking radiation by the density-density correlation function

We theoretically study the entanglement of Hawking radiation pairs emitted by an analogue black hole. We find that this entanglement can be measured by the experimentally accessible density-density correlation function, vastly simplifying the measurement. We find that while the Hawking radiation exiting the black hole might be Planck-distributed, the correlations between the Hawking radiation and the partner particles has a distribution which is weaker but broader than Planckian. Thus, the high-energy tail of the distribution of Hawking radiation should be entangled, whereas the low-energy part should not be. This confirms previous studies. The full Peres-Horodecki criterion is considered, as well as a simpler criterion in the stationary, homogeneous case. Our method applies to systems which are sufficiently cold that the thermal phonons can be neglected.

Figure 1

The quadrants in position space. The ($d$, $u$) and ($u$, $d$) quadrants show the analytic result of Ref. [10]. The profile along the dotted line in the ($u$, $u$) or ($d$, $d$) quadrants is the Fourier transform of the usual static structure factor. Comparing the Fourier transforms along the dotted and dashed lines gives a measure of the entanglement. For simplicity, the delta function along the $x_1=x_2$ diagonal is not shown. This delta function merely adds a constant when the Fourier transform along the dotted line is computed. $x_1$ and $x_2$ are normalized by the upstream healing length ${\xi }_u$.

Figure 2

The correlations between the Hawking and partner particles seen in the Fourier transform of the $(u,d)$ quadrant.

Figure 3

The two terms in the entanglement parameter. The dashed curve indicates the correlations between the Hawking and partner particles. It is proportional to the Fourier transform along the dashed line of Fig. 1. The dotted curve shows the population of the Hawking particles (the Planck distribution). It is a linear function of the Fourier transform along the dotted line of Fig. 1 (the static structure factor). The dashed curve exceeding the dotted curve corresponds to entanglement.

Figure 4

The entanglement parameter. Positive values of $-\mathrm{\Delta }$ correspond to entanglement.

Figure 5

The profile of the density-density correlation function. The solid curve includes the Hawking radiation, while the dashed curve does not. The dashed curve agrees with the curve in Ref. [16] obtained by other means.

Figure 6

The static structure factor. The solid curve includes the Hawking radiation, while the dashed curve does not.