Entanglement harvesting with coherently delocalized matter

We study entanglement harvesting for matter systems such as atoms, ions or molecules whose center of mass degrees of freedom are quantum delocalized and which couple to a relativistic quantum field. We employ a generalized Unruh-deWitt detector model for the light-matter interaction, and we investigate how the coherent spreading of the quantum center of mass wave function of two delocalized detector systems impacts their ability to become entangled with one another, via their respective interaction with a quantum field. For very massive detectors with initially highly localized centers of mass, we recover the results of entanglement harvesting for pointlike Unruh-deWitt detectors with classical center of mass degrees of freedom. We find that entanglement harvesting is Gaussian suppressed in the initial center of mass delocalization of the detectors. We further find that spatial smearing profiles, which are commonly employed to model the finite size of atoms due to their atomic orbitals, are not suited to model center of mass delocalization. Finally, for coherently delocalized detectors, we compare entanglement harvesting in the vacuum to entanglement harvesting in media. We find that entanglement harvesting is significantly suppressed in media in which the wave propagation speed is much smaller than the vacuum speed of light.

Figure 1

Emission and absorption processes which contribute to the final state of the two UdW detectors (6). The arrows denote the state of the detector ($\downarrow$ represents $|g⟩$ and $\uparrow$ represents $|e⟩$) [10, 31].

Figure 2

The negativity $\mathcal{N}$ after the end of the interaction, as a function of the energy gap $\mathrm{\Omega }$ and the separation $S$ of the detectors, plotted (top) for pointlike UdW detectors and (bottom) for spatially smeared UdW detectors with $L=\sigma$. The regions of zero negativity are marked in gray.

Figure 3

The transition probability for a detector as a function of its mass, after the end of the interaction, for different energy gaps and with $L=1000\sigma$. The dotted lines represent the excitation probabilities of pointlike UdW detectors with the same energy gaps as the respective massive detectors.

Figure 4

The entangling term $\mathcal{M}$ for pointlike UdW detectors as well as for massive detectors, after the end of the interaction, as a function of the detector’s separation $S$ (with $\mathrm{\Omega }\sigma =0.1$). For the massive detectors, we chose different values for $M$ and $L$ such as to keep their product constant ($ML=500$), which fixes the virtual velocities at which the detectors dynamically delocalize.

Figure 5

The negativity $\mathcal{N}$ for two coherently delocalizing detectors, after the end of the interaction, plotted as a function of the energy gap $\mathrm{\Omega }$ and the separation $S$ of the two detectors. Regions of zero negativity are marked in gray. We chose the detector masses $M$ and the initial center of mass localization widths $L$ so that $\gamma$ decreases from left to right and from top to bottom. In the first five plots we fixed $1/(lmc)=2.5\times {10}^{-3}$, while in the sixth plot we chose parameters satisfying $1/(lmc)=5\times {10}^{-4}$, such as to see what happens to the negativity as we further decrease $1/(lmc)$. As expected, as we approach the limit $\gamma \to 0$ and $1/(lmc)\to 0$, we find that the negativity resembles more and more the negativity displayed in Fig. 2 for two pointlike UdW detectors.

Figure 6

We consider two detectors (with detector masses $M\sigma =900$ and initial localization widths $L\sigma =4/9$) in a medium with wave propagation speed $c_s=0.26c$. We plot (top) the transition probability $P$, the entangling term $\mathcal{M}$ and the negativity $\mathcal{N}$, after the end of the interaction, as function of the energy gap $\mathrm{\Omega }$ and for a detector separation $S=\sigma /10$, and (bottom) the negativity $\mathcal{N}$, after the end of the interaction, as a function of the energy gap and the detector separation. The region of zero negativity is marked in gray.

Figure 7

We consider two detectors (with detector masses $M\sigma =900$, initial localization widths $L\sigma =4/9$ and detector separation $S=\sigma /10$) in a medium with wave propagation speed $c_s=0.01c$. We plot the transition probability $P$ and the entangling term $\mathcal{M}$, after the end of the interaction and as function of the energy gap, and we find that the negativity $\mathcal{N}$ vanishes for this choice of parameters.