Unruh-DeWitt detectors and entanglement: The anti–de Sitter space

We investigate entanglement harvesting in ${\mathrm{AdS}}_4$. Applying the general results of [K. K. Ng, R. B. Mann, and E. Martín-Martínez, Phys. Rev. D 97, 125011 (2018)], we consider two scenarios: one where two particle detectors are geodesic, with equal redshift; and one where both are static, at unequal redshift. As expected, at large AdS length $L$, our results approximate flat space. However at smaller $L$ we observe nontrivial effects for various field boundary conditions. Furthermore, in the static case we observe a novel feature of the entanglement as a function of switching time delay, which we attribute to different (coordinate) frequencies of the detectors. We also find an island of separability in parameter space, analogous to that observed in ${\mathrm{AdS}}_3$, to which we compare and contrast our other results. The variety of features observed in both cases suggest further study in other spacetimes.

Figure 1

The one-detector transition rate for the static detector, $L=\sigma$, varying $\mathrm{\Omega }$ and $r$ separately. Note the resonances visible in Fig. 1.

Figure 2

The one-detector transition rate at the center, with fixed gap $\mathrm{\Omega }=2/\sigma$, varying curvature length $L$.

Figure 3

Negativity values for varying separation and gap. $ϵ=-1$. Region where ${\mathcal{N}}^{(2)}>{\mathcal{C}}_{\mathrm{AB}}$ appears to be limited in size, even as $L\to \infty$. Zero contour marked in red.

Figure 4

The negativity of two geodesic detectors for varying boundary conditions, one at the origin, one at proper separation $\mathrm{\Delta }x$. Zero contours in red.

Figure 5

Negativity ${\mathcal{N}}^{(2)}$ and mutual information $I({\rho }_{AB})$ for geodesic detectors, one at the origin, as a function of separation in space and time. $L=5\sigma ,\mathrm{\Omega }=3/\sigma$. Zero contour in red.

Figure 6

Negativity for geodesic detectors, one at the origin, as a function of separation in space and time. $L=\sigma ,\mathrm{\Omega }=2/\sigma$. Zero contours in red.

Figure 7

The negativity for a pair of static detectors separated by $\mathrm{\Delta }x$ and having proper gap $\mathrm{\Omega }$, with AdS length $L=5\sigma$. Note that coarseness of ${\mathcal{N}}^{(2)}$ appears to be an artifact of the plot grid. The zero contour is in red; we see it bounds an island of spacelike entanglement.

Figure 8

The negativity ${\mathcal{N}}^{(2)}$ for static detectors, one at the origin, as a function of curvature length $L$ and proper separation $\mathrm{\Delta }x$. Zero contours in red. Note the island of separability, in which there is no entanglement.

Figure 9

The negativity and mutual information for static detectors, one at the origin, as a function of separation in space and time. $L=5\sigma ,\mathrm{\Omega }=3/\sigma$. Zero contours in red.

Figure 10

The negativity and mutual information for static detectors, one at the origin, as a function of separation in space and time. $L=\sigma ,\mathrm{\Omega }=2/\sigma$. Zero contours in red.