Entanglement harvesting from the electromagnetic vacuum with hydrogenlike atoms

We study how two fully featured hydrogenlike atoms harvest entanglement from the electromagnetic field vacuum, even when the atoms are spacelike separated. We compare the electromagnetic case—qualitatively and quantitatively—with previous results that used scalar fields and featureless, idealized atomic models. Our study reveals the new traits that emerge when we relax these idealizations, such as anisotropies in entanglement harvesting and the effect of exchange of angular momentum. We show that, under certain circumstances, relaxing previous idealizations makes vacuum entanglement harvesting more efficient.

Figure 1

Euler angles characterizing the relative orientation of atom B (blue) with respect to atom A’s reference frame (red).

Figure 2

Negativity (leading order) when the two atoms are in full light-contact $d=t_{\mathrm{BA}}$ for two hydrogenoid atoms with energy gap $\mathrm{\Omega }T=1$, atomic radius $a_0\mathrm{\Omega }=0.001$ and spatial separations $d/T=1$ (solid blue line), $d/T=1.15$ (dashed red line) and $d/T=1.25$ (dashed-dotted green line), as a function of the relative orientation between the atoms.

Figure 3

Examples of relative orientation of the atoms that (a) minimize and (b) maximize the entanglement harvested from the field when all the $1s\to 2p$ transitions are considered. The Euler angles for the configuration of maximal harvesting shown are $(\pi /4,1.231,\pi /4)$.

Figure 4

Binary plot showing in (dark) red the values of $d/T$ and $\mathrm{\Omega }T$ for which entanglement harvesting is possible, and in (light) grey the values for which there is zero entanglement harvesting. We consider $a_0\mathrm{\Omega }=0.001$ (hydrogen proportions) and that the atomic interactions with the field are delayed by a time delay $t_{\mathrm{BA}}/T=10$. The dashed black lines represent the boundaries of the regions of the plot where atom B is in the light cone of atom A. We have located these boundaries at 8 standard deviations of the switching function (38), that is $d=t_{\mathrm{BA}}\pm 8T/\sqrt{2}$. Notice that we can harvest entanglement from arbitrarily far away distances by increasing the transition energy gap.

Figure 5

(a) Negativity to leading order as a function of spatial distance $d$ and time delay between the atoms’ interactions $t_{\mathrm{BA}}$ of two hydrogenoid atoms satisfying $a_0\mathrm{\Omega }=0.001$ for a fixed energy gap of $\mathrm{\Omega }T=12$. The white, dashed lines represent the boundaries of the light cone, located at $d=t_{\mathrm{AB}}\pm 8T/\sqrt{2}$. (b) Zoom-in on the area marked with a white rectangle in (a), to demonstrate spacelike entanglement harvesting. The white dashed lines represent the distance to the point of light contact $d=t_{\mathrm{AB}}$ in the number of standard deviations $\sigma =T/\sqrt{2}$. Both for the Gaussian switching and the compactly supported cropped version of the Gaussian switching at $8\sigma$ we observe spacelike entanglement harvesting.

Figure 6

Negativity as a function of the spatial separation between the detectors in the different models studied: hydrogen atom dipolarly coupled to the electromagnetic field (blue, solid line); monopole spherically symmetric detector coupled to a scalar field (red, dashed line); and scalar derivative coupling (purple, dotted line). In the electromagnetic coupling case, the atomic $2p$ orbitals are parallel. For all models, the energy gap is chosen to be $\mathrm{\Omega }T=13$, the detectors’ size is given by $a_0\mathrm{\Omega }=0.001$ and the time delay between the switchings is $t_{\mathrm{BA}}/T=10$. The black dashed lines represent the boundaries of the light cone, located at $d=t_{\mathrm{BA}}\pm 8T/\sqrt{2}$. Electromagnetic harvesting is stronger, but has a shorter reach, than the scalar cases. Note that the sharp drop in all cases is due to the logarithmic scale.