Entanglement harvesting and divergences in quadratic Unruh-DeWitt detector pairs

Allison Sachs, Robert B. Mann, and Eduardo Martín-Martínez

Phys. Rev. D 96, 085012 – Published 30 October 2017

Abstract

We analyze correlations between pairs of particle detectors quadratically coupled to a real scalar field. We find that, while a single quadratically coupled detector presents no divergences, when one considers pairs of detectors, there emerge unanticipated persistent divergences (not regularizable via smooth switching or smearing) in the entanglement they acquire from the field. We have characterized such divergences, discussed whether a suitable regularization can allow for fair comparison of the entanglement harvesting ability of the quadratic and the linear couplings, and finally found a UV-safe quantifier of harvested correlations. Our results are relevant to future studies of the entanglement structure of the fermionic vacuum.

Received 25 April 2017

DOI:https://doi.org/10.1103/PhysRevD.96.085012

Physics Subject Headings (PhySH)

Entanglement in field theory Entanglement production Quantum field theory

Quantum Information, Science & TechnologyParticles & Fields

Authors & Affiliations

Allison Sachs^1,2,*, Robert B. Mann^1,2,3,†, and Eduardo Martín-Martínez^1,4,3,‡

¹Institute for Quantum Computing, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
²Department Physics and Astronomy, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada
³Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
⁴Department Applied Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

^*asachs@uwaterloo.ca
^†rbmann@uwaterloo.ca
^‡emartinmartinez@uwaterloo.ca

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Issue

Vol. 96, Iss. 8 — 15 October 2017

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Images

Figure 1
All plots illustrate the behavior of relevant quantities as $η$ decreases on a log scale. Plots on the top row are for the usual (linearly coupled) UDW detector. Plots on the bottom row are for the quadratically coupled UDW detector. All plots use parameters $α = 1$ , $δ = 1$ , $γ_{B} - γ_{A} = 0$ , and $β = 4$ , where relevant. Note how all plots (a)–(e) indicate convergence, except (f), which [in contrast to (c)] shows linear growth of $M^{{\hat{ϕ}}^{2}}$ on a logarithmic scale for $η$ , and thus a logarithmic divergence as $η \to 0$ .
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Figure 2
Entanglement harvesting is possible (darker regions colored red or blue) for any detector separation $d$ given a sufficiently large detector gap $Ω$ . These plots show various detector cutoffs, $ε = η / T$ , ranging from $η = 10^{- 2}$ to $η = 10^{- 12}$ . Note the cutoff does not make a significant difference for the linear model (no visible red regions), while there is a marked increase in the harvesting region for smaller gaps and the quadratic detector model [see inset (b) and (c)]. Both plots use parameters $δ = 1$ and $γ_{B} - γ_{A} = 0$ . The vertical white line shows the light cone. The dark lines in the insets are an extrapolation indicating the location where harvesting is no longer possible for a cutoff $η = 10^{- 29}$ (corresponding to setting the cutoff scale to the Planck frequency, as described at the end of Sec. 5a).
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Figure 3
The magnitude of harvested entanglement is dependent on the cutoff chosen. For the same cutoff, either the linear or quadratic models may harvest more entanglement, although the linear model quickly converges to a fixed value, while the quadratic model grows logarithmically with decreasing cutoff. Thus, a cutoff can always be chosen sufficiently high such that the quadratic model harvests more entanglement for a given set of parameters. Here, the plots show the leading order contribution to the negativity with increasing spatial distance. (a) shows these results for the linear model while (b) shows these results for the quadratic model. The inset in (a) shows the convergent behavior of the negativity, while the inset in (b) shows how the distance at which the negativity goes zero increases as the UV cutoff is lifted. All plots use parameters $δ = 1$ and $γ_{B} - γ_{A} = 0$ and $Ω = 1$ .
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Figure 4
Linear mutual information harvesting (left) is greater in magnitude than quadratic mutual entanglement harvesting (right). Legend indicates Log base ten of the mutual information, $I$ . Both plots use parameters $δ = 1$ and $γ_{B} - γ_{A} = 0$ . The thick white line shows the light cone.
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