# Extraction of entanglement from quantum fields with entangled particle
detectors

Dyuman Bhattacharya<EMAIL_ADDRESS>Department of Physics and
Astronomy, University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada  Kensuke
Gallock-Yoshimura<EMAIL_ADDRESS>Department of Physics and Astronomy,
University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada  Laura J. Henderson
Centre for Engineered Quantum Systems, School of Mathematics and Physics, The
University of Queensland, St. Lucia, Queensland 4072, Australia
<EMAIL_ADDRESS>Department of Physics and Astronomy, University of
Waterloo, Waterloo, Ontario, N2L 3G1, Canada  Robert B. Mann
<EMAIL_ADDRESS>Department of Physics and Astronomy, University of
Waterloo, Waterloo, Ontario, N2L 3G1, Canada

###### Abstract

We consider two initially entangled Unruh-DeWitt particle detectors and
examine how the initial entanglement changes after interacting with a quantum
scalar field. As initially nonentangled detectors extract entanglement from
the field, entangled detectors also can gain more entanglement so long as they
are weakly correlated at the beginning. For initially sufficiently entangled
detectors, only degradation takes place. We then apply our analysis to a
gravitational shockwave spacetime and show that a shockwave could enhance the
initial entanglement of weakly entangled detectors. Moreover, we find that
this enhancement can occur for greater detector separations than in Minkowski
spacetime.

##  I Introduction

It is well-known by now that a vacuum state of a quantum field is an entangled
state [1, 2]. One can examine such entanglement by making use of a particle
detector, which is a nonrelativistic first-quantized system locally coupled to
the field. A commonly used model is a two-level quantum system known as an
Unruh-DeWitt (UDW) detector [3, 4], which is known to capture the essence of
light-matter interactions when no angular momentum is exchanged [5, 6, 7]. For
example, two initially uncorrelated UDW detectors (e.g., both in their ground
states) can become entangled after interacting with a quantum scalar field
even when they are causally disconnected. In other words, the detectors
extract entanglement from the field without directly exchanging quanta. Such a
protocol is known as entanglement harvesting [8, 9, 10, 11, 12, 13], and the
amount of harvested entanglement is sensitive to the state of motion of the
detectors and the geometry of underlying spacetime [14, 15, 16, 17, 18, 19,
20, 21, 22].

One can also think of initially entangled detectors and ask how much their
initial entanglement changes as a function of their motion and the structure
of spacetime. If their initial entanglement is maximal (e.g., a Bell state)
then their entanglement will necessarily decrease since after the interaction
the detectors will be correlated with the field, and the monogamy of
entanglement tells us that the detectors must lose their initial correlation.
In general, the phenomenon of losing initial entanglement is known as
entanglement degradation. This phenomenon has been known for quite some time
for accelerating observers [23, 24, 25], with a number of subsequent studies
[26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36] carried out for entangled UDW-
type detectors.

Entanglement harvesting and degradation suggest that two initially entangled
detectors can either gain or lose entanglement after interacting with the
field. One then can ask what the conditions are for enhancement and
degradation of entanglement. Such a question was considered for two inertial
harmonic-oscillator type UDW detectors in Minkowski spacetime [27], in which
the time dependence of the initial entanglement was analyzed by using the
quantum Langevin equation with an assumption that the detectors interact with
the field for an infinitely long time after being suddenly switched on. The
initial entanglement was found to decrease with time, vanishing at the end.

Here we instead consider how much the initial entanglement of a pair of UDW
detectors changes after interacting with a field for a finite duration of
time, working perturbatively in the field coupling. It is known that such a
method has a technical difficulty: in a perturbative analysis, the density
matrix of entangled pointlike UDW detectors contains divergent elements [32].
One way to circumvent this issue is to use finite-sized detectors. Although
one must face tedious calculations in a generic curved spacetime, inertial
detectors in Minkowski spacetime are tractable and yet the fate of
entanglement in such a simple setting has not been examined. Another route is
to apply an approximation to a measure of entanglement [35]: the divergent
elements in the density matrix only appear in the higher order terms in the
coupling constant. Quantifying the amount of entanglement between the
detectors via negativity, we can ignore such a divergent term for sufficiently
small coupling. However, this approximation is valid only for initially
sufficiently entangled detectors, and cannot be applied to initially weakly
entangled ones.

We investigate in this paper the extraction of vacuum entanglement by two
entangled inertial detectors in Minkowski spacetime, considering situations in
which the detectors are initially nearly maximally entangled and initially
weakly entangled, complementing previous work in this subject [12] for
initially separable detectors. Introducing finite-sized UDW detectors to make
their density matrix well-defined, we employ concurrence as a measure of
entanglement. We examine the effects of the divergent terms on this quantity,
and provide an approximation that consistently allows us to neglect such
terms. We show that these approximations are valid even for pointlike
detectors, and utilize them to show that weakly entangled detectors can
extract more entanglement from the field even when they are causally
disconnected.

We then extend our analysis to consider a gravitational shockwave spacetime
[20] and explore the effect of a shockwave on the initial entanglement between
pointlike detectors. We find that a shockwave can enhance entanglement
compared to Minkowski spacetime case: the detectors are able to gain
entanglement with smaller energy gaps and larger amounts of initial
entanglement.

Our paper is organized as follows. We begin with providing a density matrix of
initially entangled detectors after interacting with a quantum scalar field in
Sec. II.1. The divergence in the density matrix does not show up in the
concurrence after performing approximations in Sec. II.2. We then examine the
fate of initial entanglement in $(3+1)$-dimensional Minkowski (Sec. III) and
shockwave spacetimes (Sec. IV), followed by Conclusion in Sec. V. We will use
natural units, $\hbar=c=1$, and denote spacetime points by
$\mathsf{x}=(t,{\bm{x}})$.

##  II Initially entangled detectors

###  II.1 Density matrix

We shall consider two UDW detectors A and B, linearly coupled to a quantum
scalar field. Let $\tau_{j}$ ($j\in\\{\text{A},\text{B}\\}$) be a proper time
of detector-$j$ and $t$ a common time (i.e., a time parameter that describes
both detectors). The interaction Hamiltonian in the interaction picture is
given by

$\displaystyle\hat{H}_{\text{I}}(t)$
$\displaystyle=\dfrac{\text{d}\tau_{\text{A}}}{\text{d}t}\hat{H}_{\text{A}}\big{(}\tau_{\text{A}}(t)\big{)}+\dfrac{\text{d}\tau_{\text{B}}}{\text{d}t}\hat{H}_{\text{B}}\big{(}\tau_{\text{B}}(t)\big{)}\,,$
(1)

where $\hat{H}_{j}(\tau_{j})$ are

$\displaystyle\hat{H}_{j}(\tau_{j})$
$\displaystyle=\lambda_{j}\chi_{j}(\tau_{j})\hat{\mu}_{j}(\tau_{j})\otimes\int\text{d}^{3}x\,F_{j}({\bm{x}}-{\bm{x}}_{j})\hat{\phi}(\mathsf{x}_{j}(\tau_{j}))\,.$
(2)

Here, $\lambda_{j}$ is a coupling constant between detector-$j$ and the field
$\hat{\phi}$, $\chi_{j}(\tau_{j})$ is a switching function that specifies how
a detector interacts, and $\hat{\mu}_{j}(\tau_{j})$ is a monopole moment given
by

$\displaystyle\hat{\mu}_{j}(\tau_{j})$ $\displaystyle\coloneqq
e^{\mathsf{i}\Omega_{j}\tau_{j}}\ket{e_{j}}\bra{g_{j}}+e^{-\mathsf{i}\Omega_{j}\tau_{j}}\ket{g_{j}}\bra{e_{j}}$
(3)

with $\Omega_{j}$ being an energy gap between the ground $\ket{g_{j}}$ and
excited $\ket{e_{j}}$ states. $F_{j}({\bm{x}}-{\bm{x}}_{j})$ is the so-called
smearing function, which specifies the spatial profile and the center of mass
position ${\bm{x}}_{j}$ of detector-$j$. As an example, one can choose the
Dirac delta distribution to make the detector pointlike.

The time-evolution operator, $\hat{U}_{\text{I}}$, can be written by using a
time-ordering symbol, $\mathcal{T}_{t}$, with respect to the common time $t$:

$\displaystyle\hat{U}_{\text{I}}$
$\displaystyle=\mathcal{T}_{t}\exp\left(-\mathsf{i}\int_{\mathbb{R}}\text{d}t\,\hat{H}_{\text{I}}(t)\right)$
(4)
$\displaystyle=\hat{U}_{\text{I}}^{(0)}+\hat{U}_{\text{I}}^{(1)}+\hat{U}_{\text{I}}^{(2)}+\mathcal{O}(\lambda^{3})\,,$
(5)

where $\hat{U}_{\text{I}}^{(j)}$ is the $j$-th power of the coupling constant
$\lambda$:

$\displaystyle\hat{U}_{\text{I}}^{(0)}$ $\displaystyle\coloneqq\mathds{1}\,,$
(6) $\displaystyle\hat{U}_{\text{I}}^{(1)}$
$\displaystyle\coloneqq-\mathsf{i}\int_{\mathbb{R}}\text{d}t\,\hat{H}_{\text{I}}(t)\,,$
(7) $\displaystyle\hat{U}_{\text{I}}^{(2)}$
$\displaystyle\coloneqq-\int_{\mathbb{R}}\text{d}t_{1}\int_{-\infty}^{t_{1}}\text{d}t_{2}\,\mathcal{T}_{t}[\hat{H}_{\text{I}}(t_{1})\hat{H}_{\text{I}}(t_{2})]\,.$
(8)

Let us assume that the field is in the vacuum state, $\ket{0}$, and the
detectors A and B are initially entangled

$\displaystyle\ket{\Psi_{0}}$
$\displaystyle=\big{(}\alpha\ket{g_{\text{A}}}\ket{g_{\text{B}}}+\beta
e^{\mathsf{i}\theta}\ket{e_{\text{A}}}\ket{e_{\text{B}}}\big{)}\ket{0}\,,$ (9)

where $\alpha,\beta\in[0,1]$ with $\alpha^{2}+\beta^{2}=1$ and
$\theta\in[0,2\pi)$ is the relative phase. For example, $\alpha=0\,,1$
correspond to separable states while $\alpha=1/\sqrt{2}$ gives a Bell state.
We will refer to $\alpha=1$ as the entanglement harvesting scenario. Unless
otherwise stated, we let $\beta=\sqrt{1-\alpha^{2}}$ throughout this paper.

One can obtain a state of the detectors, $\rho_{\text{AB}}$, after the
interaction by tracing out the field degree of freedom. Writing
$\rho_{0}\coloneqq\ket{\Psi_{0}}\bra{\Psi_{0}}$, we obtain

$\displaystyle\rho_{\text{AB}}$
$\displaystyle=\operatorname{Tr}_{\phi}[\hat{U}_{\text{I}}\rho_{0}\hat{U}_{\text{I}}^{\dagger}]$
(10)
$\displaystyle=\operatorname{Tr}_{\phi}[\rho_{0}]+\operatorname{Tr}_{\phi}[\hat{U}_{\text{I}}^{(1)}\rho_{0}\hat{U}_{\text{I}}^{(1)\dagger}]$
$\displaystyle\quad+\operatorname{Tr}_{\phi}[\hat{U}_{\text{I}}^{(2)}\rho_{0}]+\operatorname{Tr}_{\phi}[\rho_{0}\hat{U}^{(2)\dagger}]+\mathcal{O}(\lambda^{4})\,.$
(11)

In the basis
$\\{\ket{g_{\text{A}}g_{\text{B}}},\ket{g_{\text{A}}e_{\text{B}}},\ket{e_{\text{A}}g_{\text{B}}},\ket{e_{\text{A}}e_{\text{B}}}\\}$,
the density matrix $\rho_{\text{AB}}$ can be obtained as follows [32].

$\displaystyle\rho_{\text{AB}}$
$\displaystyle=\left[\begin{array}[]{cccc}r_{11}&0&0&r_{14}\\\
0&r_{22}&r_{23}&0\\\ 0&r_{23}^{*}&r_{33}&0\\\
r_{14}^{*}&0&0&r_{44}\end{array}\right]+\mathcal{O}(\lambda^{4})\,,$ (12e)
$\displaystyle r_{11}$
$\displaystyle=\alpha^{2}+2\alpha^{2}\text{Re}[J_{\text{AA}}^{(-+)}+J_{\text{BB}}^{(-+)}]+2\alpha\sqrt{1-\alpha^{2}}\text{Re}[e^{\mathsf{i}\theta}(J_{\text{AB}}^{(--)}+J_{\text{BA}}^{(--)})]$
(12f) $\displaystyle r_{14}$
$\displaystyle=\alpha\sqrt{1-\alpha^{2}}e^{-\mathsf{i}\theta}(1+J_{\text{AA}}^{(-+)}+J_{\text{AA}}^{(+-)*}+J_{\text{BB}}^{(-+)}+J_{\text{BB}}^{(+-)*})+\alpha^{2}(J_{\text{AB}}^{(++)*}+J_{\text{BA}}^{(++)*})+(1-\alpha^{2})(J_{\text{AB}}^{(--)}+J_{\text{BA}}^{(--)})$
(12g) $\displaystyle r_{22}$
$\displaystyle=(1-\alpha^{2})I_{\text{AA}}^{(+-)}+2\alpha\sqrt{1-\alpha^{2}}\text{Re}[e^{-\mathsf{i}\theta}I_{\text{AB}}^{(++)}]+\alpha^{2}I_{\text{BB}}^{(-+)}$
(12h) $\displaystyle r_{23}$
$\displaystyle=\alpha\sqrt{1-\alpha^{2}}e^{\mathsf{i}\theta}I_{\text{AA}}^{(--)}+(1-\alpha^{2})I_{\text{BA}}^{(+-)}+\alpha^{2}I_{\text{AB}}^{(-+)}+\alpha\sqrt{1-\alpha^{2}}e^{-\mathsf{i}\theta}I_{\text{BB}}^{(++)}$
(12i) $\displaystyle r_{33}$
$\displaystyle=\alpha^{2}I_{\text{AA}}^{(-+)}+2\alpha\sqrt{1-\alpha^{2}}\text{Re}[e^{\mathsf{i}\theta}I_{\text{AB}}^{(--)}]+(1-\alpha^{2})I_{\text{BB}}^{(+-)}$
(12j) $\displaystyle r_{44}$
$\displaystyle=(1-\alpha^{2})+2(1-\alpha^{2})\text{Re}[J_{\text{AA}}^{(+-)}+J_{\text{BB}}^{(+-)}]+2\alpha\sqrt{1-\alpha^{2}}\text{Re}[e^{-\mathsf{i}\theta}(J_{\text{AB}}^{(++)}+J_{\text{BA}}^{(++)})]$
(12k)

where

$\displaystyle I_{jk}^{(pq)}$
$\displaystyle\coloneqq\lambda_{j}\lambda_{k}\int_{\mathbb{R}}\text{d}\tau_{j}\int_{\mathbb{R}}\text{d}\tau_{k}\,\chi_{j}(\tau_{j})\chi_{k}(\tau_{k})e^{\mathsf{i}(p\Omega_{j}\tau_{j}+q\Omega_{k}\tau_{k})}\mathcal{W}\big{(}\mathsf{x}_{j}(\tau_{j}),\mathsf{y}_{k}(\tau_{k})\big{)}\,,$
(13) $\displaystyle J_{jk}^{(pq)}$
$\displaystyle\coloneqq-\lambda_{j}\lambda_{k}\int_{\mathbb{R}}\text{d}\tau_{j}\int_{-\infty}^{t_{1}(\tau_{j})}\text{d}\tau_{k}\,\chi_{j}(\tau_{j})\chi_{k}(\tau_{k})e^{\mathsf{i}(p\Omega_{j}\tau_{j}+q\Omega_{k}\tau_{k})}\mathcal{W}\big{(}\mathsf{x}_{j}(\tau_{j}),\mathsf{y}_{k}(\tau_{k})\big{)}\,,$
(14)

and we have used the fact that
$I_{\text{AB}}^{(\pm\pm)}=I_{\text{BA}}^{(\mp\mp)*}\in\mathbb{C}$ in $r_{22}$
and $r_{33}$ respectively. The elements $I_{\text{AB}}^{(\pm\pm)}$ and
$J_{kk}^{(\pm\mp)}$ appear in the density matrix (12e) only for initially
entangled detectors. The quantity

$\displaystyle\mathcal{W}(\mathsf{x}_{j},\mathsf{y}_{k})$
$\displaystyle\coloneqq\int\text{d}^{3}x\int\text{d}^{3}y\,F_{j}({\bm{x}}-{\bm{x}}_{j})F_{k}(\bm{y}-\bm{y}_{j})W(\mathsf{x},\mathsf{y}),$
(15)

where
$W(\mathsf{x},\mathsf{y})\coloneqq\bra{0}\hat{\phi}(\mathsf{x})\hat{\phi}(\mathsf{y})\ket{0}$
is the Wightman function (two-point vacuum correlation function). In the above
equations, the Wightman functions,
$W(\mathsf{x}_{j}(\tau_{j}),\mathsf{y}_{k}(\tau_{k}))$, are pulled back along
the trajectories of detectors-$j$ and $k$.

###  II.2 Entanglement measure

To quantify the entanglement between the detectors we use the concurrence,
$\mathcal{C}_{\text{AB}}$. For density matrices of the form (12e), it is known
that the qubits are entangled if and only if one of the following is satisfied
[14].

$\displaystyle|r_{14}|^{2}>r_{22}r_{33}\,,\qquad|r_{23}|^{2}>r_{11}r_{44}\,.$
(16)

We find that $|r_{23}|^{2}>r_{11}r_{44}$ can never be realized; thereby the
concurrence is defined as

$\displaystyle\mathcal{C}_{\text{AB}}\coloneqq
2\max\\{0,\,|r_{14}|-\sqrt{r_{22}r_{33}}\\}\,.$ (17)

Note that $0\leq\mathcal{C}_{\text{AB}}\leq 1$ and $\mathcal{C}_{\text{AB}}=0$
if and only if two detectors are not entangled. One can easily verify that the
initial entanglement in (9) is

$\displaystyle\mathcal{C}_{\text{AB,0}}\coloneqq
2\alpha\sqrt{1-\alpha^{2}}~{}(=2\beta\sqrt{1-\beta^{2}})\,.$ (18)

Although evaluating the concurrence (17) may seem straightforward, inspection
of (12g), (12h), and (12j) indicates that care must be taken in perturbatively
approximating $\mathcal{C}_{\text{AB}}$. This is because each of $r_{22}$ and
$r_{33}$ are of order $\lambda^{2}$ (and so their geometric mean
$\sqrt{r_{22}r_{33}}$ is of order $\lambda^{2}$) whereas a perturbative
expansion of $r_{14}$

$\displaystyle r_{14}$
$\displaystyle=r_{14}^{(0)}+\lambda^{2}r_{14}^{(2)}+\lambda^{4}r_{14}^{(4)}+\mathcal{O}(\lambda^{6})\,,$
(19)

yields

$\displaystyle|r_{14}|^{2}$
$\displaystyle=|r_{14}^{(0)}|^{2}+\lambda^{2}2\text{Re}[r_{14}^{(0)}r_{14}^{(2)*}]$
$\displaystyle\quad+\lambda^{4}(|r_{14}^{(2)}|^{2}+2\text{Re}[r_{14}^{(0)}r_{14}^{(4)*}])+\mathcal{O}(\lambda^{6})\,.$
(20)

For initially separable states, (the standard harvesting protocol)
$r_{14}^{(0)}=0$, and so
$|r_{14}|^{2}=\lambda^{4}|r_{14}^{(2)}|^{2}+\mathcal{O}(\lambda^{6})$,
indicating that there values of $\alpha$, close to 0 and 1, for which
$\lambda^{4}$ terms must make a significant contribution to $|r_{14}|$. In
order to keep the perturbative expansion of $|r_{14}|$ consistent, we will
consider the cases where the detectors are nearly separable and the case where
the detectors are sufficiently entangled separately.

Another issue has to do with the structure of $r_{14}$, which from (12g) can
be written as

$\displaystyle r_{14}^{(0)}$
$\displaystyle=\alpha\sqrt{1-\alpha^{2}}e^{-\mathsf{i}\theta}\,,$ (21)
$\displaystyle\lambda^{2}r_{14}^{(2)}$
$\displaystyle=\lambda^{2}\Big{[}\alpha\sqrt{1-\alpha^{2}}e^{-\mathsf{i}\theta}(Y_{\text{A}}+Y_{\text{B}})$
$\displaystyle\qquad+\alpha^{2}X_{\text{AB}}^{*}(-\Omega)+(1-\alpha^{2})X_{\text{AB}}(\Omega)\Big{]}\,,$
(22)

where

$\displaystyle\lambda^{2}Y_{k}\equiv J_{kk}^{(-+)}+J_{kk}^{(+-)*}\,,$ (23a)
$\displaystyle\lambda^{2}X_{\text{AB}}(\Omega)\equiv
J_{\text{AB}}^{(--)}+J_{\text{BA}}^{(--)}\,,$ (23b)
$\displaystyle\lambda^{2}X_{\text{AB}}^{*}(-\Omega)\equiv
J_{\text{AB}}^{(++)*}+J_{\text{BA}}^{(++)*}\,.$ (23c)

The imaginary part of $Y_{k}$ is divergent and causes problems in the
pointlike limit [32]. Although it can be regularized by introducing a smearing
function, we shall demonstrate that it does not appear in the concurrence
under the approximations we consider.

###  II.3 Approximating the Concurrence

We consider three scenarios:

(i) Initial weak entanglement with $\alpha\approx 0$;
(ii) Initial weak entanglement with $\alpha\approx 1$;
(iii) Initial sufficient entanglement

$\bullet$ Initial weak entanglement: $\alpha\approx 0$

Let us assume that $\alpha\ll 1$ and perform a series expansion of Eq. (20) in
terms of $\lambda^{n}\alpha^{m}$:

$\displaystyle|r_{14}|^{2}$
$\displaystyle=\underbrace{\alpha^{2}(1-\alpha^{2})}_{\text{initial}}+\underbrace{2\lambda^{2}\alpha\text{Re}[X_{\text{AB}}(\Omega)e^{\mathsf{i}\theta}]}_{\text{neutral}}$
$\displaystyle\quad+\underbrace{\lambda^{4}|X_{\text{AB}}(\Omega)|^{2}}_{\text{harvesting}}+\underbrace{2\lambda^{2}\alpha^{2}\text{Re}[Y_{\text{A}}+Y_{\text{B}}]}_{\text{degradation}}$
$\displaystyle\quad+\mathcal{O}(\lambda^{n}\alpha^{m})\,.\qquad(n+m>4)$ (24)

As one can see, $r_{14}^{(4)}$ and the divergent element $\text{Im}[Y_{k}]$ do
not exist in the lower-order terms.

$\bullet$ Initial weak entanglement: $\alpha\approx 1$

In the same manner, one can obtain a similar expression for $\alpha\approx 1$.
For simplicity, let us use $\beta=\sqrt{1-\alpha^{2}}$ and expand Eq. (20)
around $\beta=0$:

$\displaystyle|r_{14}|^{2}$
$\displaystyle=\underbrace{\beta^{2}(1-\beta^{2})}_{\text{initial}}+\underbrace{2\lambda^{2}\beta\text{Re}[X^{*}_{\text{AB}}(-\Omega)e^{\mathsf{i}\theta}]}_{\text{neutral}}$
$\displaystyle\quad+\underbrace{\lambda^{4}|X_{\text{AB}}(-\Omega)|^{2}}_{\text{harvesting}}+\underbrace{2\lambda^{2}\beta^{2}\text{Re}[Y_{\text{A}}+Y_{\text{B}}]}_{\text{degradation}}$
$\displaystyle\quad+\mathcal{O}(\lambda^{n}\beta^{m})\,.\qquad(n+m>4)$ (25)

As before, the elements $r_{14}^{(4)}$ and $\text{Im}[Y_{k}]$ are absent. Note
that $\beta=0$ reduces to the entanglement harvesting scenario:
$|r_{14}|=\lambda^{2}|X_{\text{AB}}(-\Omega)|$.

In both cases $\alpha\approx 0$ and 1, the approximated $|r_{14}|^{2}$
consists of four parts. The initial entanglement contribution is
$\alpha^{2}(1-\alpha^{2})$ in (24) [equivalent to $\beta^{2}(1-\beta^{2})$ in
(25)]. The third term containing $X_{\text{AB}}(\Omega)$ is the part
contributing to entanglement harvesting, and so will enhance entanglement. The
last term, containing $\text{Re}[Y_{\text{A}}+Y_{\text{B}}]$, depends only on
each detector and not on their correlations. This term corresponds to an
outcome of local operations and so never enhances entanglement. Indeed, one
can straightforwardly show that
$\lambda^{2}\text{Re}[Y_{j}]=-(I_{jj}^{(-+)}+I_{jj}^{(+-)})/2\leq 0$ and so
degrades entanglement. The second term, which we denote as the neutral
contribution, can be either positive or negative. By choosing the relative
phase $\theta$ appropriately, we can ensure that
$\text{Re}[e^{-\mathsf{i}\theta}X_{\text{AB}}]>0$, which enhances
entanglement.

In summary, we see from (17) that the detectors gain entanglement after the
interaction provided the harvesting part is greater than other two parts in
$|r_{14}|$ and $\sqrt{r_{22}r_{33}}$. Indeed, if the harvesting part does not
exceed both the initial and degradation contributions to $|r_{14}|$, then
entanglement degradation is inevitable, regardless of the value of
$\sqrt{r_{22}r_{33}}$. This is not the case for harvesting: even if the
harvesting contribution is greater than the other two in $|r_{14}|$,
entanglement enhancement is not ensured because this depends on
$\sqrt{r_{22}r_{33}}$.

$\bullet$ Initial sufficient entanglement

In this approximation we assume
$|r_{14}^{(0)}|^{2}\gg\mathcal{O}(\lambda^{4})$ and write Eq. (20) explicitly
as [35]

$\displaystyle|r_{14}^{(0)}|^{2}=\alpha^{2}(1-\alpha^{2})\,,$ (26a)
$\displaystyle\text{Re}[r_{14}^{(0)}r_{14}^{(2)*}]=\alpha\sqrt{1-\alpha^{2}}\Big{[}\alpha\sqrt{1-\alpha^{2}}\text{Re}[Y_{\text{A}}+Y_{\text{B}}]+\Big{(}\alpha^{2}\text{Re}[X_{\text{AB}}^{*}(-\Omega)e^{\mathsf{i}\theta}]+(1-\alpha^{2})\text{Re}[X_{\text{AB}}(\Omega)e^{\mathsf{i}\theta}]\Big{)}\Big{]}\,,$
(26b)
$\displaystyle|r_{14}^{(2)}|^{2}=\left|\alpha^{2}X_{\text{AB}}^{*}(-\Omega)+(1-\alpha^{2})X_{\text{AB}}(\Omega)\right|^{2}+\alpha^{2}(1-\alpha^{2})|Y_{\text{A}}+Y_{\text{B}}|^{2}$
$\displaystyle\qquad+2\alpha\sqrt{1-\alpha^{2}}\Big{(}\alpha^{2}\text{Re}[X_{\text{AB}}^{*}(-\Omega)(Y_{\text{A}}+Y_{\text{B}})^{*}e^{\mathsf{i}\theta}]+(1-\alpha^{2})\text{Re}[X_{\text{AB}}(\Omega)(Y_{\text{A}}+Y_{\text{B}})^{*}e^{\mathsf{i}\theta}]\Big{)}\,.$
(26c)

Omitting the $\lambda^{4}$ terms, we obtain

$\displaystyle|r_{14}|^{2}$
$\displaystyle\approx|r_{14}^{(0)}|^{2}+\lambda^{2}2\text{Re}[r_{14}^{(0)}r_{14}^{(2)*}]$
(27)

which is also independent of $r_{14}^{(4)}$ and $\text{Im}[Y_{k}]$. Obviously
this approximation does not work for $\alpha=0$ and 1 since it cannot recover
the entanglement harvesting results. In fact, as in the previous
approximations, Eq. (27) consists of three components: the initial
entanglement, the harvesting contribution, and the degradation contribution.
However now the harvesting contribution does not contain
$|X_{\text{AB}}(\Omega)|^{2}$ or $|X_{\text{AB}}^{*}(-\Omega)|^{2}$. Note that
for $\alpha\ll 1$ the leading term in (26c) yields the harvesting contribution
in (24) (and likewise for $\beta\ll 1$ the harvesting contribution in (25) is
recovered).

Henceforth we shall consider these approximations (which are applicable even
to pointlike detectors) in evaluating the concurrence in the scenarios we
consider.

###  II.4 Smeared detectors in a flat spacetime

We now compute the elements in the density matrix (12e) in $(3+1)$-dimensional
Minkowski spacetime. We assume that the detectors are identical (have the same
shape and internal structure) and are both at rest in a single frame of
reference. We choose the switching function, $\chi_{j}(\tau_{j})$, and
smearing function, $F_{j}({\bm{x}}-{\bm{x}}_{j})$, to be Gaussian functions:

$\displaystyle\chi_{j}(\tau_{j})=e^{-(\tau_{j}-\tau_{j,0})^{2}/T^{2}}\,,$ (28)
$\displaystyle
F_{j}({\bm{x}}-{\bm{x}}_{j})=\dfrac{1}{(\sqrt{\pi}\sigma)^{3}}e^{-({\bm{x}}-{\bm{x}}_{j})^{2}/\sigma^{2}}\,,$
(29)

where $\tau_{j,0}$ and ${\bm{x}}_{j}$ are their respective centers and $T$ and
$\sigma$ are their respective temporal and spatial widths.

As we show in the Appendix, the quantities
$J_{kk}^{(-+)}+J_{kk}^{(+-)*},J_{\text{AB}}^{(++)*}$, and
$J_{\text{AB}}^{(--)}$ in $r_{14}$, and $I_{jk}^{(\pm\pm)}$ in $r_{22}$ and
$r_{33}$ are then given by

$\displaystyle\lambda_{k}^{2}Y_{k}\equiv J_{kk}^{(-+)}+J_{kk}^{(+-)*}$
$\displaystyle\qquad=-\dfrac{\lambda_{k}^{2}e^{-T^{2}\Omega^{2}/2}}{8\pi(1+\sigma^{2}/T^{2})^{3/2}}\left[2\sqrt{1+\sigma^{2}/T^{2}}+e^{T^{2}\Omega^{2}/2(1+\sigma^{2}/T^{2})}\sqrt{2\pi}T\Omega\text{erf}\left(\dfrac{T\Omega}{\sqrt{2(1+\sigma^{2}/T^{2})}}\right)\right]$
$\displaystyle\qquad-\mathsf{i}\dfrac{\lambda_{k}^{2}T^{2}}{8\pi}\int_{0}^{\infty}\text{d}|\bm{k}|\,|\bm{k}|e^{-|\bm{k}|^{2}\sigma^{2}/2}\left[e^{-T^{2}(|\bm{k}|-\Omega)^{2}/2}\text{erfi}\left(\dfrac{T(|\bm{k}|-\Omega)}{\sqrt{2}}\right)-e^{-T^{2}(|\bm{k}|+\Omega)^{2}/2}\text{erfi}\left(\dfrac{T(|\bm{k}|+\Omega)}{\sqrt{2}}\right)\right]\,,$
(30a) $\displaystyle
J_{\text{AB}}^{(--)}=-\dfrac{\lambda_{\text{A}}\lambda_{\text{B}}T^{2}}{8\pi
L}e^{-\mathsf{i}\Omega(t_{\text{A,0}}+t_{\text{B,0}})}e^{-T^{2}\Omega^{2}/2}\int_{0}^{\infty}\text{d}|\bm{k}|\,e^{-|\bm{k}|^{2}(\sigma^{2}+T^{2})/2}e^{\mathsf{i}\Delta
t|\bm{k}|}\text{erfc}\left(\dfrac{\Delta
t+\mathsf{i}T^{2}|\bm{k}|}{\sqrt{2}T}\right)\sin(|\bm{k}|L)\,,$ (30b)
$\displaystyle
I_{kk}^{(-+)}=\dfrac{\lambda_{k}^{2}e^{-T^{2}\Omega^{2}/2}}{8\pi(1+\sigma^{2}/T^{2})^{3/2}}\left[2\sqrt{1+\sigma^{2}/T^{2}}-\sqrt{2\pi}T\Omega
e^{T^{2}\Omega^{2}/2(1+\sigma^{2}/T^{2})}\text{erfc}\left(\dfrac{T\Omega}{\sqrt{2(1+\sigma^{2}/T^{2})}}\right)\right]\,,$
(30c) $\displaystyle
I_{\text{AB}}^{(++)}=\dfrac{\mathsf{i}\lambda_{\text{A}}\lambda_{\text{B}}Te^{\mathsf{i}\Omega(t_{\text{A,0}}+t_{\text{B,0}})}e^{-T^{2}\Omega^{2}/2}}{8\pi
L\sqrt{1+\sigma^{2}/T^{2}}}\sqrt{\dfrac{\pi}{2}}\left[e^{-\Gamma_{-}^{2}}\text{erfc}(\mathsf{i}\Gamma_{-})-e^{-\Gamma_{+}^{2}}\text{erfc}(-\mathsf{i}\Gamma_{+})\right]$
(30d)

where $L\coloneqq|\bm{x}_{\text{B}}-\bm{x}_{\text{A}}|$ and $\Delta t\coloneqq
t_{\text{B,0}}-t_{\text{A,0}}$ are spatial and temporal separation of the
detectors’ Gaussian peaks [see Fig. 1(a)], and

$\displaystyle\Gamma_{\pm}$ $\displaystyle\coloneqq\dfrac{L\pm\Delta
t}{T\sqrt{2(1+\sigma^{2}/T^{2})}}\,.$ (31)

We remark that $J_{\text{AB}}^{(++)*}$ can be obtained from
$J_{\text{AB}}^{(--)}$ by $\Omega\to-\Omega$ and then taking a complex
conjugate. In the same manner, one finds
$I_{\text{BA}}^{(--)}=I_{\text{AB}}^{(++)*}$. Note that the real part of (30a)
is always nonpositive, which means that $\text{Re}[Y_{\text{A}}+Y_{\text{B}}]$
in $|r_{14}|$ acts as noise that inhibits the detectors from gaining
entanglement. In fact, in units of $\Omega$, one can verify from (30a) that
$\text{Re}[Y_{k}]\sim-T\Omega$ when $T\Omega\gg 1$ for $\sigma\geq 0$. This
suggests that the longer the interaction is, the more leakage of entanglement
occurs, and thereby entanglement extraction becomes more difficult as the
interaction duration increases.

Thanks to the smearing function, all the elements in the density matrix are
finite and well-defined. One can then perform the approximations in Sec. II.2
and take a limit $\sigma\to 0$ if pointlike detectors are chosen. In the
following sections, we will adopt pointlike detectors. The qualitative
behavior for smeared detectors is not so different from the pointlike ones.

Figure 1:  (a) Finite-sized detectors in Minkowski spacetime. The center of
mass of detector-B is located at $(t_{\text{B,0}},{\bm{x}}_{\text{B}})$; each
detector has the respective spatial and temporal Gaussian widths $\sigma$ and
$T$. The diagonal orange-shaded strip represents the null trajectories from
detector-A; detector-A can directly signal to B by exchanging field quanta
once the support of detector-B crosses this region. (b) A plot of the positive
part of difference $\mathcal{C}_{\text{AB}}-\mathcal{C}_{\text{AB,0}}$ between
the concurrence before and after the interaction, with
$\beta=10^{-15},\lambda=1/10,\Omega T=5,\sigma/T=0$, and $\theta=0$. The green
dots represent $\mathcal{C}_{\text{AB}}-\mathcal{C}_{\text{AB,0}}\leq 0$,
which means the initial entanglement is degraded, and the dashed lines
indicate the null trajectory corresponding to the one in (a).
Figure 2:  (a) The positive part of
$\mathcal{C}_{\text{AB}}-\mathcal{C}_{\text{AB,0}}$ as a function of $\Omega
T$ and $\log_{10}\beta$ when $\lambda=1/10,L/T=7,\Delta t/T=0,\sigma/T=0$, and
$\theta=0$. The green dots represent degradation. The detectors can gain
entanglement when $\beta$ is small enough. The range of $\Omega T$ that
enables the detectors to gain entanglement enlarges as $\beta\to 0$. (b)
$\max\\{0,\,\mathcal{C}_{\text{AB}}-\mathcal{C}_{\text{AB,0}}\\}$ as a
function of $\log_{10}\beta$ for $\lambda=1/10,L/T=7,\Delta t/T=0,\sigma/T=0$,
and $\Omega T=7$. The blue curve shows a slice in (a) at $\Omega T=7$, whereas
the orange dashed curve represents the case when $\theta=\pi$. (c) Each
component $\mathcal{C}_{\text{AB}}$ and $\mathcal{C}_{\text{AB,0}}$ in (b)
with the same parameters.

##  III The fate of entanglement between inertial detectors

In this section, we consider pointlike ($\sigma\to 0$) detectors in
$(3+1)$-dimensional Minkowski spacetime and see how the initial entanglement
changes after the interaction.

As shown in Fig. 1(a), suppose detector-A’s center of mass is located at the
origin and the peak of Gaussian switching is at $t=0$, with detector-B’s
position in a spacetime, $(t_{\text{B,0}},{\bm{x}}_{\text{B}})$ fixed but
variable. In $(3+1)$-dimensional Minkowski spacetime, the detectors can
potentially communicate when they are lightlike separated. This is depicted as
an orange strip in the figure; detector-A can send quanta to detector-B once
the support of detector-B has overlap with this region.

We first ask if initially entangled detectors can gain more entanglement after
interacting with their local fields? The answer is yes, though we find that
the condition is very limited; only initially weakly entangled detectors can
gain entanglement.

Consider weakly entangled detectors with $\beta\approx 0$ (i.e.,
$\alpha\approx 1$). Using (25) to calculate the concurrence
$\mathcal{C}_{\text{AB}}$ after the interaction, we illustrate in Figure 1(b)
the difference between the initial and final concurrences of the detectors for
$\beta=10^{-15},\lambda=1/10,\Omega T=5,\sigma/T=0$, and $\theta=0$. In
particular, we take
$\max\\{0,\,\mathcal{C}_{\text{AB}}-\mathcal{C}_{\text{AB,0}}\\}$ so that we
can see the region of entanglement enhancement. We depict with green dots
values of the concurrence that are less than its initial value
$\mathcal{C}_{\text{AB,0}}$ after the interaction. We first note that
communication between the detectors greatly assists entanglement extraction as
one can see from Fig. 1(b) that two detectors gain entanglement when they are
in causal contact (within the dashed lines). This phenomenon can be explained
from the entanglement harvesting viewpoint; communication enhances the value
of $|X_{\text{AB}}(-\Omega)|$ in the entanglement harvesting scenario
($\beta=0$) [37], and this is also happening in our $|r_{14}|$ in (25).
Nevertheless, as described in the previous section, the degradation term
$\text{Re}[Y_{\text{A}}+Y_{\text{B}}]$ will suppress this communication
assistance if the interaction duration is too long.

One can also ask if noncommunicating detectors can gain entanglement from the
field. The answer is also yes. To see this, we plot concurrence for pointlike
detectors ($\sigma/T=0$) when they are effectively111Since we are using a
Gaussian switching, the interaction is not compactly supported in the
spacetime. In other words, the detectors are always communicating. However,
due to a Gaussian function’s exponential suppression, it can be thought of as
an effectively compactly supported interaction in
$\tau\in[-3.5T+\tau_{j,0},3.5T+\tau_{j,0}]$, where $\tau$ is a proper time and
$\tau_{j,0}$ is the center of Gaussian of detector-$j$ [37]. causally
disconnected ($L/T=7,\Delta t/T=0$) in Fig. 2. Figure 2(a) shows the values of
$\Omega T$ and $\log_{10}\beta$ that enhance entanglement without
communication. The parameters are set to be $\lambda=1/10,L/T=7,\Delta
t/T=0,\sigma/T=0$, and $\theta=0$. Again, the detectors experience degradation
when $(\Omega T,\log_{10}\beta)$ is chosen to be a point in the green dots in
Figure 2(a). We find that a pair of causally disconnected detectors can
enhance their initial entanglement as long as they are weakly entangled at the
beginning. In addition, one must choose $\Omega T$ from a suitable range: if
the initial entanglement is, for example, $\log_{10}\beta=-30$ with the given
parameters, then the energy gap must be $\Omega T\in[6.8,11]$ to gain more
correlation. This energy gap range increases as $\beta\to 0$, namely, the
upper bound of $\Omega$ to extract entanglement gets pushed to infinity in the
limit $\beta\to 0$. We also note that there is a maximum value of $\beta$ for
fixed $L/T$ beyond which entanglement cannot be extracted for any $\Omega T$.

The blue curve in Fig. 2(b) represents a slice at $\Omega T=7$ in (a), and
each factor $\mathcal{C}_{\text{AB}}$ and $\mathcal{C}_{\text{AB,0}}$ in
$\max\\{0,\,\mathcal{C}_{\text{AB}}-\mathcal{C}_{\text{AB,0}}\\}$ is shown in
(c) as blue and green curves, respectively. It is interesting to note from (c)
that for sufficiently large $\beta$, entanglement is completely extinguished
after the interaction. There is also a dependence on the relative initial
phase $\theta$; for $\theta=\pi$ [the orange curve in (b) and (c)], the
positive difference $\mathcal{C}_{\text{AB}}-\mathcal{C}_{\text{AB,0}}$
persists for larger values of $\beta$, allowing for a wider range of
entanglement extraction. This dependence on the relative phase $\theta$ arises
from the term $\text{Re}[X_{\text{AB}}^{*}(-\Omega)e^{-\mathsf{i}\theta}]$ in
(25) and elements in $r_{22}r_{33}$.

For the case of initially sufficiently entangled detectors, no entanglement
extraction can occur — only degradation takes place. One of the reasons comes
from the fact that the degradation part $\text{Re}[Y_{\text{A}}+Y_{\text{B}}]$
is no longer suppressed in (25) as $\beta$ grows. Extracting entanglement with
$\alpha\approx 0$ is also difficult since $I_{kk}^{(+-)}$ in
$\sqrt{r_{22}r_{33}}$ dominates with large $\Omega$.

One notable difference from entanglement harvesting ($\beta=0$) is the energy
gap dependence. In the harvesting scenario, detectors with a Gaussian
switching222This is not the case for sudden switchings [12]. can extract
entanglement even when they are far apart ($L/T\gg 1$) if $\Omega$ is large
enough [12]. That is, once the detector separation $L$ is fixed, there exists
a minimum energy gap $\Omega_{\text{min}}$ such that the detectors can harvest
entanglement for all $\Omega\geq\Omega_{\text{min}}$. In our case, however,
this is no longer true. Figure 3 shows
$\max\\{0,\,\mathcal{C}_{\text{AB}}-\mathcal{C}_{\text{AB,0}}\\}$ as a
function of $L/T$ and $\Omega T$ for pointlike detectors with $\Delta
t=0,\beta=10^{-15}$, and $\theta=0$. That is, we are exploring the region of
entanglement extraction in $(L/T,\Omega T)$-plane when the centers of Gaussian
switching in Fig. 1(a) are on the same time-slice. We see that there exist
minimum and maximum values of $\Omega$ for some $L$ that allow detectors to
extract entanglement; once $\Omega$ gets large enough, only degradation takes
place. The maximum energy gap $\Omega_{\text{max}}$ allowing harvesting is
pushed to infinity as $\beta\to 0$, recovering the harvesting scenario [12].

##  IV Entanglement in the presence of a shockwave

We now extend our results in Minkowski spacetime to a nontrivial spacetime. We
consider in particular a gravitational shockwave spacetime, since it has been
shown that a weak gravitational field can enhance entanglement harvesting from
the vacuum [38]. Furthermore, for the type of gravitational shockwave that we
consider in this paper, the Wightman function for a massless scalar field has
a non-trivial, but closed form expression [20].

This latter study considered initially separable detectors in their ground
states; we consider here initially entangled detectors with the aim of
understanding if the shockwave mitigates or enhances entanglement degradation.
We find that a shockwave admits for greater entanglement harvesting relative
to the flat space setting, and extends the region in Fig. 2(a) dramatically.

Figure 3:  The positive part of difference
$\mathcal{C}_{\text{AB}}-\mathcal{C}_{\text{AB,0}}$ as a function of $L/T$ and
$\Omega T$. Here, $\Delta t/T=0,\beta=10^{-15}$, and $\theta=0$. Entanglement
degradation occurs in the region with green dots.

###  IV.1 Shockwave spacetime

As in [20] we shall be working with the Dray and ’t Hooft generalization [39]
of the Aichelburg-Sexl shockwave spacetime [40]. Consider a planar shockwave
propagating in the $z$-direction in $D$-dimensional Minkowski spacetime. The
metric describing such a spacetime in the so-called Brinkmann coordinates [41]
is:

$\text{d}s^{2}=-\text{d}u\text{d}v+f(\vec{x})\delta(u-u_{0})\text{d}u^{2}+\delta_{ij}\text{d}x^{i}\text{d}x^{j}\,,$
(32)

where $u=t-z$ and $v=t+z$ are null coordinates, and $\vec{x}$ or $x^{i}$,
$i\in\\{1,\cdots,D-2\\}$ are the transverse coordinates (i.e., the remaining
spatial coordinates). The wavefront of the shockwave is located at $u=u_{0}$.
We consider an idealized scenario in which the spacetime is exactly Minkowski
on either side of $u_{0}$.

Figure 4:  Diagram for two pointlike inertial detectors in a shockwave
spacetime. Detectors A and B are static at $z=z_{\text{A}}$ and
$z_{\text{B}}$, respectively, and the shockwave travels along $u=0$ (the green
wiggling line). The red and blue bars indicate the effective interaction
duration of the detectors.

The term $f(\vec{x})$ is the shockwave profile, which is directly related to
the energy density $\varrho(\vec{x})$ through the Einstein field equations:

$\displaystyle\Delta f(\vec{x})$ $\displaystyle=-16\pi
G_{\text{N}}\varrho(\vec{x})\,,$ (33)

where $\Delta=\delta^{ij}\partial_{i}\partial_{j}$ is the flat Laplacian in
the transverse direction. We use a quadratic profile given by

$f(\vec{x})=-\vec{x}\cdot A\cdot\vec{x}=-\sum_{i=1}^{D-2}a_{i}(x^{i})^{2},$
(34)

where $A$ is a symmetric constant matrix, whose eigenvalues are $a_{i}$. Such
a profile represents an ultrarelativistic domain wall in the transverse plane
[42] as one can see from (33) that the energy density is constant everywhere:
$\varrho(\vec{x})=\operatorname{Tr}[A]/8\pi G_{\text{N}}$.

Figure 5:  (a-i) Initial and final concurrences when $\Omega
T=2,\beta=10^{-5},\lambda=1/10$, and $\theta=0$. The location of detector-B is
fixed at $z_{\text{B}}/T=7$. Detector-A encounters the shockwave near
$z_{\text{A}}/T=0$. (a-ii) The concurrences when $\Omega
T=2,z_{\text{A}}/T=-0.5$ and $z_{\text{B}}/T=7$ with varying initial amount of
entanglement. Two curves will cross and degradation takes place after
$\log_{10}\beta>-3$. (b) Final concurrence $\mathcal{C}_{\text{AB}}$ for
initially maximally entangled detectors ($\mathcal{C}_{\text{AB,0}}=1$) with
$\Omega T=2$ and $\theta=0$. The green dotted curve corresponds to detectors
in Minkowski spacetime. In this case, the shockwave can either reduce or
amplify the degradation effect.

Using the notation $\mathsf{x}=(u,v,\vec{x})$ and $\mathsf{y}=(U,V,\vec{X})$
the exact form of the Wightman function is [20]

$\begin{split}W(\mathsf{x},\mathsf{y})&=\frac{(-\mathsf{i})^{D-2}\Gamma(D/2-1)}{4\pi^{D/2}}\prod_{i=1}^{D-2}\bigg{(}1+a_{i}\Delta\Theta\frac{(u-u_{0}-\mathsf{i}\epsilon)(U-u_{0}+\mathsf{i}\epsilon)}{\Delta
u-\mathsf{i}\epsilon}\bigg{)}^{-1/2}\\\ &\quad\times\bigg{(}(\Delta
v-\mathsf{i}\epsilon)(\Delta
u-\mathsf{i}\epsilon)-\Delta\vec{x}^{2}+\sum_{i=1}^{D-2}\frac{a_{i}\Delta\Theta\big{(}[U-u_{0}+\mathsf{i}\epsilon]x^{i}-[u-u_{0}-\mathsf{i}\epsilon]X^{i}\big{)}^{2}}{\Delta
u-\mathsf{i}\epsilon+a_{i}\Delta\Theta(u-u_{0}-\mathsf{i}\epsilon)(U-u_{0}+\mathsf{i}\epsilon)}\bigg{)}^{(2-D)/2}\,,\end{split}$
(35)

where $\Delta v\equiv v-V$, $\Delta u\equiv u-U$,
$\Theta_{u}\equiv\Theta(u-u_{0})$, and $\Delta\Theta=\Theta_{u}-\Theta_{U}$
with the Heaviside step function $\Theta(u)$. $\Gamma(x)$ is the gamma
function and by choosing $D=4$ it gives $\Gamma(1)=1$. The Wightman function
reduces to the standard Minkowski space form if $f(\vec{x})=0$ (in which case
there is no shockwave at all) or $\Delta\Theta=0$ (so the events $u$ and $U$
are localized at the same side of the shockwave).

###  IV.2 Results

We restrict to $D=4$ and consider two inertial UDW detectors

$\begin{split}&\mathsf{x}_{\text{A}}(\tau_{\text{A}})=(t(\tau_{\text{A}}),x(\tau_{\text{A}}),y(\tau_{\text{A}}),z(\tau_{\text{A}}))=(\tau_{\text{A}},0,0,z_{\text{A}})\,,\\\
&\mathsf{x}_{\text{B}}(\tau_{\text{B}})=(t(\tau_{\text{B}}),x(\tau_{\text{B}}),y(\tau_{\text{B}}),z(\tau_{\text{B}}))=(\tau_{\text{B}},0,0,z_{\text{B}})\,.\end{split}$
(36)

We further assume that $a_{x}=a_{y}\equiv a(>0)$ and $u_{0}=0$ as in Fig. 4
and $\vec{x}=\vec{X}=\vec{0}$. The Wightman function then reduces to

$\displaystyle W(\mathsf{x},\mathsf{y})=$
$\displaystyle-\dfrac{1}{4\pi^{2}}\dfrac{1}{1+a\Delta\Theta\dfrac{(u-\mathsf{i}\epsilon)(U+\mathsf{i}\epsilon)}{\Delta
u-\mathsf{i}\epsilon}}\dfrac{1}{(\Delta v-\mathsf{i}\epsilon)(\Delta
u-\mathsf{i}\epsilon)}\,.$ (37)

It is worth noting that in this setup, the proper spatial distance between the
two detectors, $L=|z_{\text{A}}-z_{\text{B}}|$, is the same as in Minkowski
space, and that the two detectors are placed in the longitudinal direction of
the shockwave.

In Fig. 5(a), we plot the initial ($\mathcal{C}_{\text{AB,0}}$) and final
($\mathcal{C}_{\text{AB}}$) concurrences in the shockwave spacetime with
$aT=1$. In all figures, detector-B is fixed at the position $z_{\text{B}}/T=7$
which is located on the $u<u_{0}$ part of shockwave spacetime (see Fig. 4) and
the energy gap is $\Omega T=2$. Note that in Minkowski space, the
noncommunicating detectors cannot extract entanglement with $\Omega T=2$, as
shown in Figs. 2(a) and 3.

Figure 5(a-i) depicts the concurrences as functions of detector-A’s static
positions $z_{\text{A}}/T$ when $\beta=10^{-5}$. As shown in Fig. 4,
detector-A encounters the shockwave around $z_{\text{A}}/T=0$. For
$z_{\text{A}}/T\lesssim-1$ the plot indicates that full entanglement
degradation from the initial value of $\mathcal{C}_{\text{AB,0}}$ occurs, due
to the fact that the geometry is Minkowski except along the shockwave
trajectory. In other words, the behavior in $z_{\text{A}}/T\lesssim-1$ follows
from the results in Minkowski spacetime given in Figs. 2 and 3. This is also
true for $z_{\text{A}}/T\gtrsim 0.5$. The growth of $\mathcal{C}_{\text{AB}}$
in $z_{\text{A}}/T>3$ comes from the communication effect in
$|X_{\text{AB}}(-\Omega)|$ between the two detectors, which corresponds to the
bottom left corner in Fig. 1(b). The effect of the shockwave can be seen in
the region $z_{\text{A}}/T\in[-1,0.5]$; not only is the degradation effect
reduced, but we find $\mathcal{C}_{\text{AB}}>\mathcal{C}_{\text{AB,0}}$ — the
initial entanglement is enhanced! Somewhat surprisingly, the shockwave allows
causally disconnected detectors to gain more entanglement even when it is
impossible to do this in Minkowski spacetime.

Figure 5(a-ii) shows the $\beta$-dependence of the concurrences when
$z_{\text{A}}/T=-0.5$ (i.e., $L/T=7.5$), where maximal entanglement
enhancement occurs, with $\Omega T=2,\lambda=1/10$, and $\theta=0$. We find
that entanglement enhancement by the shockwave is possible so long as
$\beta\lesssim 10^{-3}$. Recall that for any $\beta$, only degradation occurs
for these parameters in Minkowski spacetime. Furthermore, even when $\Omega
T=7$ as in Figs. 2(b) and (c), the initial amount of entanglement needs to be
$\beta\lesssim 10^{-17}$ in order to extract entanglement. In this sense, a
shockwave drastically assists extraction of entanglement.

However, the shockwave does not always enhance entanglement. In Fig. 5(b), we
plot the final concurrence when the detectors are initially maximally
entangled, $\mathcal{C}_{\text{AB,0}}=1$. In both Minkowski (dotted curve) and
shockwave (solid) spacetimes, degradation takes place as expected. Notice that
the shockwave could either reduce or assist degradation. This is due to the
fact that the concurrence is now determined by (27); the degradation part
$\text{Re}[Y_{\text{A}}+Y_{\text{B}}]$ is no longer suppressed by small
$\beta^{2}$ and the harvesting part $|X_{\text{AB}}(-\Omega)|$ is negligible.

##  V Conclusion

We considered two initially entangled UDW detectors interacting with a quantum
scalar field and examined how their initial entanglement behaves after the
interaction. The entanglement harvesting protocol allows initially
uncorrelated detectors to extract entanglement from the field in its vacuum
state without signalling, whereas entanglement degradation is the process in
which initial entanglement decreases due to the detectors becoming entangled
with the field. We have investigated which phenomenon is dominant when the
detectors are neither uncorrelated nor maximally entangled.

Such an analysis is known to be difficult when a perturbative method is
employed since there is a UV-divergent element in the density matrix for the
pointlike detectors (and hence in entanglement measures). One can circumvent
this issue by introducing a smearing function and performing an approximation
to an entanglement measure such as concurrence and negativity. We have dealt
with this by introducing an approximation for weakly entangled detectors,
supplementing the method used for initially sufficiently entangled detectors
[35]. Under any of these approximations, the divergent element is absent in
the entanglement measure.

With these approximations, we examined the fate of entanglement in detectors
at rest in $(3+1)$-dimensional Minkowski spacetime. We found that initially
weakly entangled detectors can gain entanglement even when they are causally
disconnected. This can be understood by looking at the distinct contributions
to the concurrence: one part is from the initial entanglement, another yields
entanglement harvesting, and a third causes entanglement degradation. The
detectors extract entanglement if the entanglement harvesting contribution to
the concurrence exceeds the other two. Otherwise, entanglement degradation
takes place and the initial amount of entanglement will be reduced after the
interaction.

The fact that only weakly entangled detectors can gain entanglement is
consistent with the results in Ref. [34]. It was nonperturbatively shown that
quantum mutual information (i.e., the total correlation including classical
ones) between entangled detectors can be enhanced only when their initial
entanglement is weak333The nonperturbative method employed in [34] uses a
delta-switching function, which cannot harvest entanglement at all if each
detector switches only once [43, 44]. It is important to note, however, that
the detectors can gain mutual information even when they are causally
disconnected.

We have also analyzed the energy gap $\Omega$ dependence for entanglement
extraction. In the entanglement harvesting scenario [12], detectors with
Gaussian switching can extract entanglement so long as their energy gap is
above the minimum value: $\Omega\geq\Omega_{\text{min}}$. This feature allows
one to harvest entanglement with detectors arbitrarily far away from each
other once a sufficiently large energy gap is chosen. This is not the case for
entangled detectors. Instead there is a range
$\Omega\in[\Omega_{\text{min}},\Omega_{\text{max}}]$ within which the
detectors can extract entanglement. This range tends to zero as the distance
between the detectors increases. This suggests that entanglement extraction is
not allowed for detectors arbitrarily far away from each other. Nevertheless,
the maximum value $\Omega_{\text{max}}$ will increase as the initial state
approaches a separable state, recovering the results of the standard
entanglement harvesting protocol.

We found that the presence of a gravitational shockwave, in which the metric
is identical to that of Minkowski spacetime except along a single null
trajectory, significantly modifies these effects. Detectors far from the
shockwave exhibit the same behavior as in Minkowski spacetime. However as they
come closer to the shockwave their behavior markedly changes. The shockwave
enhances entanglement as compared to Minkowski spacetime. The range
$\Omega\in[\Omega_{\text{min}},\Omega_{\text{max}}]$ of energy gap that
enables entanglement extraction becomes wider, which in turn allows the
detectors to gain entanglement for smaller values of $\Omega$.

Although the shockwave enhances entanglement, its extraction is still limited
to weakly entangled detectors. If the initial state of the detectors is
sufficiently entangled, the shockwave weakens the effect of degradation rather
than enhancing extraction of correlations. Nevertheless, our results show that
spacetime geometry can enhance initial entanglement harvesting, indicating
that the quantum vacuum is indeed a resource (at least in principle) for
carrying out quantum information tasks.

We close by commenting on a recent similar investigation of the negativity
[45] of two initially entangled detectors in Minkowski spacetime with constant
switching. Using an approximation corresponding to our sufficiently entangled
case, the detectors were found to only experience entanglement degradation,
consistent with our results for sufficiently entangled detectors with an
infinitely long interaction.

## Acknowledgments

This work was supported in part by the Natural Sciences and Engineering
Research Council of Canada.

##  Appendix A Smeared detectors in Minkowski spacetime

The imaginary part of $\lambda^{2}Y_{k}\equiv J_{kk}^{(-+)}+J_{kk}^{(+-)*}$ in
$r_{14}$ for pointlike detectors is known to be divergent. We will show that
this term can be regularized by introducing finite-sized UDW detectors as
suggested in [32]. In particular, we consider initially entangled smeared
detectors at rest in $(3+1)$-dimensional Minkowski spacetime.

Assuming a massless scalar field, the mode decomposition of
$\hat{\phi}(\mathsf{x}(t))$ is known to be

$\displaystyle\hat{\phi}(t,\bm{x})$
$\displaystyle=\int\dfrac{\text{d}^{3}k}{\sqrt{(2\pi)^{3}2|\bm{k}|}}\left(\hat{a}_{\bm{k}}e^{-\mathsf{i}|\bm{k}|t+\mathsf{i}\bm{k}\cdot\bm{x}}+\hat{a}_{\bm{k}}^{\dagger}e^{\mathsf{i}|\bm{k}|t-\mathsf{i}\bm{k}\cdot\bm{x}}\right),$
(38)

and so the Wightman function becomes

$\displaystyle W(\mathsf{x}_{j}(t_{1}),\mathsf{y}_{k}(t_{2}))$
$\displaystyle=\int\dfrac{\text{d}^{3}k}{(2\pi)^{3}2|\bm{k}|}e^{-\mathsf{i}|\bm{k}|(t_{1}-t_{2})+\mathsf{i}\bm{k}\cdot(\bm{x}-\bm{y})}.$
(39)

Figure 6:  Real and imaginary parts of $J_{kk}^{(-+)}+J_{kk}^{(+-)*}$ with
$\sigma/T=1$ and $\lambda=1/10$. The real part converges quickly to 0 with
large $|\Omega T|$ while the imaginary part is nonzero for a wide range of
$\Omega T$.

Let us first consider $\lambda^{2}Y_{\text{A}}\equiv
J_{\text{AA}}^{(-+)}+J_{\text{AA}}^{(+-)*}$ for detector-A in $r_{14}$. We
find

$\displaystyle J_{\text{AA}}^{(-+)}+J_{\text{AA}}^{(+-)*}$
$\displaystyle=-2\lambda_{\text{A}}^{2}\int_{\mathbb{R}}\text{d}t_{1}\int_{-\infty}^{t_{1}}\text{d}t_{2}\,\chi_{\text{A}}(t_{1})\chi_{\text{A}}(t_{2})e^{-\mathsf{i}\Omega_{\text{A}}(t_{1}-t_{2})}\int\text{d}^{3}x\,F_{\text{A}}(\bm{x}-\bm{x}_{\text{A}})\int\text{d}^{3}y\,F_{\text{A}}(\bm{y}-\bm{x}_{\text{A}})\text{Re}[W(\mathsf{x}_{\text{A}}(t_{1}),\mathsf{y}_{\text{A}}(t_{2}))]$
(40)
$\displaystyle=-2\lambda_{\text{A}}^{2}\int\dfrac{\text{d}^{3}k}{2|\bm{k}|}\tilde{F}_{\text{A}}(\bm{k})\tilde{F}_{\text{A}}(-\bm{k})\int_{\mathbb{R}}\text{d}t_{1}\int_{-\infty}^{t_{1}}\text{d}t_{2}\,\chi_{\text{A}}(t_{1})\chi_{\text{A}}(t_{2})e^{-\mathsf{i}\Omega_{\text{A}}(t_{1}-t_{2})}\cos[|\bm{k}|(t_{1}-t_{2})]\,,$
(41)

where we have performed the Fourier transform:

$\displaystyle\tilde{F}_{j}(\bm{k})$
$\displaystyle=\int\dfrac{\text{d}^{3}x}{\sqrt{(2\pi)^{3}}}F_{j}(\bm{x})e^{\mathsf{i}\bm{k}\cdot\bm{x}}\,.$
(42)

Note that $\tilde{F}_{j}(-\bm{k})=\tilde{F}_{j}(\bm{k})$ if
$F_{j}(\bm{x})\in\mathbb{R}$. With a Gaussian switching centered at
$t=t_{\text{A,0}}$, $\chi_{\text{A}}(t)=e^{-(t-t_{\text{A,0}})^{2}/T^{2}}$,

$\displaystyle\int_{\mathbb{R}}\text{d}t_{1}\int_{-\infty}^{t_{1}}\text{d}t_{2}\,e^{-(t_{1}-t_{\text{A,0}})^{2}/T^{2}}e^{-(t_{2}-t_{\text{A,0}})^{2}/T^{2}}e^{-\mathsf{i}\Omega_{\text{A}}(t_{1}-t_{2})}\cos[|\bm{k}|(t_{1}-t_{2})]$
$\displaystyle=\dfrac{T\sqrt{2\pi}}{2}\int_{0}^{\infty}\text{d}u\,e^{-u^{2}/2T^{2}}e^{-\mathsf{i}\Omega
u}\cos(|\bm{k}|u)$ (43) $\displaystyle=\dfrac{\pi
T^{2}}{4}e^{-T^{2}(|\bm{k}|+\Omega)^{2}/2}\left[1+e^{2|\bm{k}|T^{2}\Omega}\left(1+\mathsf{i}\text{erfi}\dfrac{T(|\bm{k}|-\Omega)}{\sqrt{2}}\right)-\mathsf{i}\text{erfi}\dfrac{T(|\bm{k}|+\Omega)}{\sqrt{2}}\right],$
(44)

and therefore, using the smearing function (29) and its Fourier transform,
$\tilde{F}_{j}({\bm{k}})=e^{-|\bm{k}|^{2}\sigma^{2}/4}/\sqrt{(2\pi)^{3}}$,

$\displaystyle J_{\text{AA}}^{(-+)}+J_{\text{AA}}^{(+-)*}$
$\displaystyle=-\dfrac{\lambda_{\text{A}}^{2}\pi
T^{2}}{2}\int\dfrac{\text{d}^{3}k}{2|\bm{k}|}\tilde{F}_{\text{A}}^{2}(\bm{k})e^{-T^{2}(|\bm{k}|+\Omega)^{2}/2}\left[1+e^{2|\bm{k}|T^{2}\Omega}\left(1+\mathsf{i}\text{erfi}\dfrac{T(|\bm{k}|-\Omega)}{\sqrt{2}}\right)-\mathsf{i}\text{erfi}\dfrac{T(|\bm{k}|+\Omega)}{\sqrt{2}}\right]$
(45)
$\displaystyle=-\dfrac{\lambda_{\text{A}}^{2}e^{-T^{2}\Omega^{2}/2}}{8\pi(1+\sigma^{2}/T^{2})^{3/2}}\left[2\sqrt{1+\sigma^{2}/T^{2}}+e^{T^{2}\Omega^{2}/2(1+\sigma^{2}/T^{2})}\sqrt{2\pi}T\Omega\text{erf}\dfrac{T\Omega}{\sqrt{2(1+\sigma^{2}/T^{2})}}\right]$
$\displaystyle\quad-\mathsf{i}\dfrac{\lambda_{\text{A}}^{2}T^{2}}{8\pi}\int_{0}^{\infty}\text{d}|\bm{k}|\,|\bm{k}|e^{-|\bm{k}|^{2}\sigma^{2}/2}\left[e^{-T^{2}(|\bm{k}|-\Omega)^{2}/2}\text{erfi}\dfrac{T(|\bm{k}|-\Omega)}{\sqrt{2}}-e^{-T^{2}(|\bm{k}|+\Omega)^{2}/2}\text{erfi}\dfrac{T(|\bm{k}|+\Omega)}{\sqrt{2}}\right]\,,$
(46)

where we have used
$\text{d}^{3}k=|\bm{k}|^{2}\sin\theta\text{d}|\bm{k}|\text{d}\theta\text{d}\varphi$
with $|\bm{k}|\in[0,\infty),\theta\in[0,\pi]$, and $\varphi\in[0,2\pi)$. The
real and imaginary parts of $J_{kk}^{(-+)}+J_{kk}^{(+-)*}$ for smeared
detectors are shown in Fig. 6.

Similar calculations yield the remaining terms in $r_{14},r_{22}$, and
$r_{33}$. In particular, the expressions for $I_{kk}^{(-+)}$ in $r_{22}$ and
$r_{33}$ in (30c) and for $J_{\text{AB}}^{(++)}$ (and its counterparts) in
(30b) that contribute to $\lambda^{2}X_{\text{AB}}^{*}(-\Omega)\equiv
J_{\text{AB}}^{(++)*}+J_{\text{BA}}^{(++)*}$ in $r_{14}$ can be found in Ref.
[12].

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