# Correlation harvesting between particle detectors in uniform motion

Lana Bozanic<EMAIL_ADDRESS>Department of Applied Mathematics,
University of Waterloo, Waterloo, Ontario, N2L 3G1, Canada  Manar Naeem
<EMAIL_ADDRESS>Department of Physics and Astronomy, University of
Waterloo, Waterloo, Ontario, N2L 3G1, Canada  Institute for Quantum Computing,
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  Robert B. Mann
<EMAIL_ADDRESS>Department of Physics and Astronomy, University of
Waterloo, Waterloo, Ontario, N2L 3G1, Canada

###### Abstract

We investigate the correlation harvesting protocol using two Unruh-DeWitt
particle detectors moving along four classes of uniformly accelerated
trajectories categorized by Letaw: linear, catenary, cusped, and circular
motions. For each trajectory, two types of configurations are carried out: one
possesses a stationary (time-translation invariant) Wightman function and the
other is nonstationary. We find that detectors undergoing linear, catenary,
and cusped motions gain fewer correlations in the nonstationary configurations
compared to those in stationary configurations. Detectors in circular motion
have similar behavior in both configurations. We discuss the relative
suppression of correlation harvesting due to high acceleration for each case.
Remarkably we find that under certain circumstances detectors in both linear
and circular states of motion can harvest genuine (non-communication assisted)
entanglement even though they are in causal contact.

##  I Introduction

The field of relativistic quantum information (RQI) has undergone rapid
development in recent years, resulting in the emergence of groundbreaking
concepts such as entanglement degradation due to non-inertial motion [1, 2,
3], entanglement harvesting [4, 5, 6, 7, 8], and the long-established Unruh
effect [9, 10, 11]. The Unruh effect states that a linearly accelerating
observer in Minkowski spacetime will experience a thermal bath, and this
experience is indistinguishable from that of an inertial observer sitting in a
thermal bath. More precisely, the temperature detected by a linearly
accelerating two-level quantum system, known as the Unruh-DeWitt (UDW)
particle detector [11, 12], is proportional to its acceleration $a$ and it
reads

$T_{\text{U}}=\dfrac{\hbar a}{2\pi ck_{\text{B}}}\,,$ (1)

where $\hbar$ is the reduced Planck constant, $c$ is the speed of light, and
$k_{\text{B}}$ is Boltzmann’s constant.

Despite its theoretical significance, the Unruh effect has yet to be verified
experimentally. The main challenge hindering its experimental verification
lies in the large acceleration required to produce experimentally measurable
temperatures. For example, an acceleration on the order of magnitude of
$a\approx 10^{20}$ m/s2 is needed to achieve a temperature of
$T_{\text{U}}\sim 1$ Kelvin.

Given this situation, researchers have explored other detector trajectories
that could induce a phenomenon similar to the Unruh effect. In 1981, Letaw
found five classes of stationary trajectories with nonzero constant
acceleration in flat spacetime [13]: linear, circular, cusped, catenary, and
helix trajectories. In $(3+1)$-dimensional Minkowski spacetime, these
trajectories are characterized by three parameters: two torsions and the
magnitude of the proper acceleration. The effective temperatures observed by a
detector undergoing these motions have been studied over subsequent years in
various contexts [14, 15, 16, 17, 18, 19, 20]. Among these, the circular
trajectory has attracted considerable attention for potential experimental
realizations of the Unruh effect [14, 15, 21, 22, 23, 19, 24].

Less studied is the entanglement harvesting protocol for these various classes
of motion, apart from detectors undergoing linearly accelerated motion, which
has been extensively analyzed [8, 25, 26, 27, 28]. In this paper, we consider
this problem and investigate how two UDW detectors can extract entanglement
from the vacuum whilst undergoing these various types of non-inertial motion.

In the entanglement harvesting protocol [4, 5, 6, 7], two initially
uncorrelated detectors interact locally with a quantum field in some state
(typically the vacuum state) to extract preexisting entanglement [29, 30].
More generally, detectors can harvest classical and quantum correlations in
what is called the correlation harvesting protocol. The amount of harvested
correlations is sensitive to the background spacetime [31, 32, 33, 34, 35, 36,
37, 38, 39, 40, 41, 42, 43, 44, 45, 46] and the motion of the detectors [47,
8, 48, 25, 26, 49, 50, 27, 28]. Implementing this protocol is very close to
implementation as recent experiments detecting correlations of the
electromagnetic ground state in a ZnTe crystal have demonstrated [51, 52, 53].

In this paper we investigate the correlation harvesting protocol with
detectors in four classes of uniform acceleration motion: linear, catenary,
cusped, and circular trajectories. Unlike the helical case, these motions can
all be realized in 2 spatial dimensions, and so are more amenable to
experimental testing [21, 22, 23, 20, 19, 24]. We categorize the
configurations for the detectors into two configurations: stationary, in which
the Wightman function is time-translation invariant, and nonstationary, in
which the Wightman function is not time-translation invariant. The Wightman
functions for these two scenarios are similar, except in nonstationary
configurations they possess an additional term that breaks time-translation
invariance.

After introducing the UDW detector model in section II and the four uniform
acceleration trajectories in section III, we then focus in section IV.1 on a
single detector following the four trajectories to examine its transition
probability (or response function). We study its dependence on the magnitudes
of the acceleration and torsions, and numerically evaluate the effective
temperature of the detector.

We then consider the correlation harvesting protocol in section IV.2.
Specifically, concurrence of entanglement and quantum mutual information –
which measures the harvested total correlations – are numerically evaluated.
We find that the stationary and nonstationary configurations behave in a
similar manner since their Wightman functions have terms in common. However,
the amount of correlations extracted by the detectors in the nonstationary
configurations differs from those of the stationary ones due to an additional
term in the Wightman function. We also look into the acceleration dependence
of the harvested correlations and conclude that sufficiently high
accelerations prevent any uniformly accelerating detectors from extracting
correlations. This point is consistent with previous papers that focused on
linear and circular motions [47, 8, 48, 25, 26, 27, 28]. Finally, we show that
constant acceleration makes it challenging to extract ‘genuine entanglement’
(entanglement preexistent in a quantum field that has no possible assistance
from detector communication) in section IV.3. In general, genuine entanglement
can be harvested from causally disconnected spacetime regions due to
microcausality. For inertial detectors with Gaussian switching in Minkowski
spacetime, it is shown that a sufficiently large energy gap allows the
detectors to extract genuine entanglement from such regions [7, 54]. While we
find that this is generally not the case for uniformly accelerated detectors,
remarkably we find small but non-negligible regions of parameter space where
detectors in causal contact can harvest genuine entanglement.

Throughout this manuscript, we use the mostly-plus metric convention,
$(-,+,+,+)$ and the natural units $\hbar=k_{\text{B}}=c=1$. A point in
spacetime is denoted by $\mathsf{x}$.

##  II Unruh-DeWitt detectors

###  II.1 Density matrix of detectors

Let us first review the correlation harvesting protocol. Consider two
pointlike UDW detectors A and B with an energy gap
$\Omega_{j},\,j\in\\{\text{A},\text{B}\\}$ between ground $\ket{g_{j}}$ and
excited states $\ket{e_{j}}$. These detectors interact with the quantum Klein-
Gordon field $\hat{\phi}$ along their trajectories
$\mathsf{x}_{j}(\tau_{j})=(t(\tau_{j}),{\bm{x}}(\tau_{j}))$, where $\tau_{j}$
is the proper time of detector-$j$.

In the interaction picture, the interaction Hamiltonian (as a generator of
time-translation with respect to $\tau_{j}$) describing the coupling between
detector-$j$ and $\hat{\phi}$ is given by

$\displaystyle\hat{H}_{j}^{\tau_{j}}(\tau_{j})$
$\displaystyle=\lambda_{j}\chi_{j}(\tau_{j})\hat{\mu}_{j}(\tau_{j})\otimes\hat{\phi}(\mathsf{x}_{j}(\tau_{j}))\,,~{}j\in\\{\text{A},\text{B}\\}$
(2)

where $\lambda_{j}$ is a coupling constant and $\chi_{j}(\tau_{j})$ is the
switching function that governs the time-dependence of the coupling. Here,
$\hat{\mu}_{j}(\tau_{j})$ is the monopole moment given by

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

which describes each detector’s internal dynamics, and the field operator
$\hat{\phi}(\mathsf{x}_{j}(\tau_{j}))$ is pulled back along the trajectory of
detector-$j$. The superscript on $\hat{H}_{j}^{\tau_{j}}(\tau_{j})$ indicates
the time-translation that the Hamiltonian is generating.

The total interaction Hamiltonian, $\hat{H}_{\text{I}}^{t}(t)$, can be written
as a generator of time-translation with respect to the time $t$ that is common
to both detectors:

$\displaystyle\hat{H}_{\text{I}}^{t}(t)$
$\displaystyle=\dfrac{\text{d}\tau_{\text{A}}}{\text{d}t}\hat{H}_{\text{A}}^{\tau_{\text{A}}}\big{(}\tau_{\text{A}}(t)\big{)}+\dfrac{\text{d}\tau_{\text{B}}}{\text{d}t}\hat{H}_{\text{B}}^{\tau_{\text{B}}}\big{(}\tau_{\text{B}}(t)\big{)}\,,$
(4)

Note that the proper times $\tau_{\text{A}}$ and $\tau_{\text{B}}$ are each
now functions of $t$. From this Hamiltonian, one obtains the time-evolution
operator $\hat{U}_{\text{I}}$ [55, 56]:

$\displaystyle\hat{U}_{\text{I}}$
$\displaystyle=\mathcal{T}_{t}\exp\left(-\mathsf{i}\int_{\mathbb{R}}\text{d}t\,\hat{H}_{\text{I}}^{t}(t)\right)\,,$
(5)

where $\mathcal{T}_{t}$ is a time-ordering symbol with respect to the common
time $t$.

One can then use a perturbative analysis and obtain the final density matrix
of a joint system $\mathcal{H}_{\text{A}}\otimes\mathcal{H}_{\text{B}}$, where
$\mathcal{H}_{j}$ is a Hilbert space for detector-$j$. Assuming a small
coupling strength, $\lambda\ll 1$, the Dyson series expansion of
$\hat{U}_{\text{I}}$ reads

$\displaystyle\hat{U}_{\text{I}}$
$\displaystyle=\mathds{1}+\hat{U}_{\text{I}}^{(1)}+\hat{U}_{\text{I}}^{(2)}+\mathcal{O}(\lambda^{3})\,,$
(6a) $\displaystyle\hat{U}_{\text{I}}^{(1)}$
$\displaystyle=-\mathsf{i}\int_{-\infty}^{\infty}\text{d}t\,\hat{H}_{\text{I}}^{t}(t)\,,$
(6b) $\displaystyle\hat{U}_{\text{I}}^{(2)}$
$\displaystyle=-\int_{-\infty}^{\infty}\text{d}t_{1}\int_{-\infty}^{t_{1}}\text{d}t_{2}\,\hat{H}_{\text{I}}^{t}(t_{1})\hat{H}_{\text{I}}^{t}(t_{2})\,.$
(6c)

By assuming that the initial state, $\rho_{0}$, of the detectors-field system
is

$\displaystyle\rho_{0}$
$\displaystyle=\ket{g_{\text{A}}}\bra{g_{\text{A}}}\otimes\ket{g_{\text{B}}}\bra{g_{\text{B}}}\otimes\ket{0}\bra{0}\,,$
(7)

where $\ket{0}$ is the vacuum state of the field, one finds the final total
density matrix $\rho_{\text{tot}}$ after the interaction to be

$\displaystyle\rho_{\text{tot}}$
$\displaystyle=\hat{U}_{\text{I}}\rho_{0}\hat{U}_{\text{I}}^{\dagger}$
$\displaystyle=\rho_{0}+\rho^{(1,1)}+\rho^{(2,0)}+\rho^{(0,2)}+\mathcal{O}(\lambda^{4})\,,$
(8)

where $\rho^{(i,j)}=\hat{U}^{(i)}\rho_{0}\hat{U}^{(j)\dagger}$ and all the
odd-power terms of $\lambda$ vanish [7]. Then the final density matrix of the
detectors, $\rho_{\text{AB}}$, is obtained by tracing out the field part:
$\rho_{\text{AB}}=\operatorname{Tr}_{\phi}[\rho_{\text{tot}}]$. By employing
the basis
$\ket{g_{\text{A}}g_{\text{B}}}=[1,0,0,0]^{\top},\ket{g_{\text{A}}e_{\text{B}}}=[0,1,0,0]^{\top},\ket{e_{\text{A}}g_{\text{B}}}=[0,0,1,0]^{\top},\ket{e_{\text{A}}e_{\text{B}}}=[0,0,0,1]^{\top}$,
the density matrix $\rho_{\text{AB}}$ reads

$\displaystyle\rho_{\text{AB}}$
$\displaystyle=\left[\begin{array}[]{cccc}1-\mathcal{L}_{\text{AA}}-\mathcal{L}_{\text{BB}}&0&0&\mathcal{M}^{*}\\\
0&\mathcal{L}_{\text{BB}}&\mathcal{L}_{\text{AB}}^{*}&0\\\
0&\mathcal{L}_{\text{AB}}&\mathcal{L}_{\text{AA}}&0\\\
\mathcal{M}&0&0&0\end{array}\right]+\mathcal{O}(\lambda^{4})\,,$ (13)

where

$\displaystyle\mathcal{L}_{ij}$
$\displaystyle=\lambda^{2}\int_{\mathbb{R}}\text{d}\tau_{i}\int_{\mathbb{R}}\text{d}\tau_{j}^{\prime}\,\chi_{i}(\tau_{i})\chi_{j}(\tau_{j}^{\prime})e^{-\mathsf{i}\Omega(\tau_{i}-\tau_{j}^{\prime})}$
$\displaystyle\qquad\qquad\qquad\qquad\times
W\big{(}\mathsf{x}_{i}(\tau_{i}),\mathsf{x}_{j}(\tau_{j}^{\prime})\big{)}\,,$
(14a) $\displaystyle\mathcal{M}$
$\displaystyle=-\lambda^{2}\int_{\mathbb{R}}\text{d}\tau_{\text{A}}\int_{\mathbb{R}}\text{d}\tau_{\text{B}}\,\chi_{\text{A}}(\tau_{\text{A}})\chi_{\text{B}}(\tau_{\text{B}})e^{-\mathsf{i}\Omega(\tau_{\text{A}}+\tau_{\text{B}})}$
$\displaystyle\hskip
14.22636pt\times\big{[}\Theta\big{(}t(\tau_{\text{A}})-t(\tau_{\text{B}})\big{)}W\big{(}\mathsf{x}_{\text{A}}(\tau_{\text{A}}),\mathsf{x}_{\text{B}}(\tau_{\text{B}})\big{)}$
$\displaystyle\hskip
28.45274pt+\Theta\big{(}t(\tau_{\text{B}})-t(\tau_{\text{A}})\big{)}W\big{(}\mathsf{x}_{\text{B}}(\tau_{\text{B}}),\mathsf{x}_{\text{A}}(\tau_{\text{A}})\big{)}\big{]}\,,$
(14b)

where $\Theta(t)$ is the Heaviside step function and
$W(\mathsf{x},\mathsf{x}^{\prime})\coloneqq\bra{0}\hat{\phi}(\mathsf{x})\hat{\phi}(\mathsf{x}^{\prime})\ket{0}$
is the vacuum Wightman function. In $(3+1)$-dimensional Minkowski spacetime,
the Wightman function reads

$\displaystyle W(\mathsf{x},\mathsf{x}^{\prime})$
$\displaystyle=-\dfrac{1}{4\pi^{2}}\dfrac{1}{(t-t^{\prime}-\mathsf{i}\epsilon)^{2}-({\bm{x}}-{\bm{x}}^{\prime})^{2}}\,,$
(15)

where $\epsilon$ is the UV cutoff. The elements
$\mathcal{L}_{jj},\,j\in\\{\text{A},\text{B}\\}$ are the so-called transition
probabilities (or response functions), which describe the probability of a
detector transitioning from the ground to excited states,
$\ket{g_{j}}\to\ket{e_{j}}$. The off-diagonal elements $\mathcal{M}$ and
$\mathcal{L}_{\text{AB}}$ are responsible for harvesting entanglement and
quantum mutual information, respectively, as we shall see in the next
subsection.

Throughout this paper, we use a Gaussian switching function

$\displaystyle\chi_{j}(\tau_{j})$
$\displaystyle=e^{-\tau_{j}^{2}/2\sigma^{2}}\,,$ (16)

where $\sigma>0$ is the characteristic Gaussian width, which has the units of
time. We will use $\sigma$ to make all quantities unitless (such as
$\Omega\sigma$).

###  II.2 Correlation measure

Let us introduce two measures for correlation: concurrence
$\mathcal{C}_{\text{AB}}$ and quantum mutual information $I_{\text{AB}}$.

Concurrence is a measure of entanglement [57, 58]. Let $\rho_{\text{AB}}$ be
the density matrix of a two-qubit system. We first define a matrix
$\tilde{\rho}_{\text{AB}}$ as

$\displaystyle\tilde{\rho}_{\text{AB}}$
$\displaystyle\coloneqq(\hat{\sigma}_{y}\otimes\hat{\sigma}_{y})\rho_{\text{AB}}^{*}(\hat{\sigma}_{y}\otimes\hat{\sigma}_{y})\,,$
(17)

where $\hat{\sigma}_{y}$ is the Pauli-$y$ operator and $\rho_{\text{AB}}^{*}$
is the complex conjugate of $\rho_{\text{AB}}$. Then by denoting
$w_{i}\in\mathbb{R}$, $(i=1,2,3,4)$ as eigenvalues of a Hermitian operator
$\sqrt{\sqrt{\rho_{\text{AB}}}\tilde{\rho}_{\text{AB}}\sqrt{\rho_{\text{AB}}}}$,
the concurrence is defined as follows.

$\displaystyle\mathcal{C}_{\text{AB}}$
$\displaystyle\coloneqq\max\\{0,\,w_{1}-w_{2}-w_{3}-w_{4}\\}\,,$ (18)
$\displaystyle\hskip 42.67912pt(w_{1}\geq w_{2}\geq w_{3}\geq w_{4})\,.$

The concurrence is zero if and only if the state $\rho_{\text{AB}}$ is
separable. In the case of our density matrix (13), the concurrence is known to
be

$\displaystyle\mathcal{C}_{\text{AB}}$
$\displaystyle=2\max\\{0,\,|\mathcal{M}|-\sqrt{\mathcal{L}_{\text{AA}}\mathcal{L}_{\text{BB}}}\\}+\mathcal{O}(\lambda^{4})\,.$
(19)

Quantum mutual information [59], on the other hand, quantifies the amount of
total correlation, both classical and quantum. Quantum mutual information
$I_{\text{AB}}$ between two qubits A and B up to second order in $\lambda$ is
[7]

$\displaystyle I_{\text{AB}}$
$\displaystyle=\mathcal{L}_{+}\ln\mathcal{L}_{+}+\mathcal{L}_{-}\ln\mathcal{L}_{-}$
$\displaystyle\hskip
14.22636pt-\mathcal{L}_{\text{AA}}\ln\mathcal{L}_{\text{AA}}-\mathcal{L}_{\text{BB}}\ln\mathcal{L}_{\text{BB}}+\mathcal{O}(\lambda^{4})\,,$
(20)

where

$\displaystyle\mathcal{L}_{\pm}$
$\displaystyle\coloneqq\dfrac{1}{2}\left(\mathcal{L}_{\text{AA}}+\mathcal{L}_{\text{BB}}\pm\sqrt{(\mathcal{L}_{\text{AA}}-\mathcal{L}_{\text{BB}})^{2}+4|\mathcal{L}_{\text{AB}}|^{2}}\right).$
(21)

Note that, while concurrence (19) vanishes when the “noise term”
$\sqrt{\mathcal{L}_{\text{AA}}\mathcal{L}_{\text{BB}}}$ exceeds the nonlocal
element $|\mathcal{M}|$, the mutual information becomes zero when
$|\mathcal{L}_{\text{AB}}|=0$. In addition, if $\mathcal{C}_{\text{AB}}=0$ but
the mutual information is nonvanishing, then the extracted correlation by the
detectors is either classical correlation or nondistillable entanglement.

Figure 1: Four trajectories characterized by $\bar{b}\equiv b/a$.

##  III Uniform acceleration trajectories

###  III.1 Single detector trajectory classification

The most well-known trajectory for a uniformly accelerating (i.e., $a=const.$)
pointlike particle is linear accelerated motion. However, Letaw pointed out
that there are, in fact, five classes of uniformly accelerated trajectories,
excluding the case where $a=0$. Along with the linear case, the other classes
are circular, catenary, cusped, and helix [13]. Consider a trajectory in
$(3+1)$-dimensional Minkowski spacetime. Such a trajectory can be
characterized by three geometric invariants: the curvature $a(\tau)$, which
represents the magnitude of proper acceleration, the first torsion $b(\tau)$,
and the second torsion (also known as hypertorsion) $\nu(\tau)$ of the
worldline. The torsions $b(\tau)$ and $\nu(\tau)$ correspond to the proper
angular velocities in a given tetrad frame [13]. Assuming that these
invariants are constants, the trajectory becomes stationary. In a nutshell,
these motions are characterized by the following:

  1. 1.

linear: $a\neq 0,b=\nu=0$

  2. 2.

catenary: $a>b$, $\nu=0$

  3. 3.

cusped: $a=b,\nu=0$

  4. 4.

circular: $a<b,\nu=0$

  5. 5.

helix: $\nu\neq 0$

In this subsection, we review these trajectories and consider the
corresponding vacuum Wightman functions. We will suppress the UV cutoff
$\epsilon$ for readability.

####  III.1.1 Linear motion

The linear acceleration motion of a detector is defined solely by the constant
acceleration $a$, with all other parameters set to zero. The trajectory reads

$\displaystyle\mathsf{x}(\tau)$
$\displaystyle=\left(\dfrac{1}{a}\sinh(a\tau),\dfrac{1}{a}\cosh(a\tau),0,0\right)\,,$
(22)

and the Wightman function along this trajectory is given by

$\displaystyle W_{\text{lin}}(\Delta\tau)$
$\displaystyle=-\dfrac{1}{4\pi^{2}}\dfrac{1}{\dfrac{4}{a^{2}}\sinh^{2}\left(\frac{a\Delta\tau}{2}\right)}\,,$
(23)

where $\Delta\tau\coloneqq\tau-\tau^{\prime}$.

####  III.1.2 Circular motion

The circular trajectory is defined by $a$ and $b$ satisfying $a<b$. Let us
begin with a commonly used trajectory

$\displaystyle\mathsf{x}(\tau)$
$\displaystyle=\left(\gamma\tau,R\cos(\omega\gamma\tau),R\sin(\omega\gamma\tau),0\right)\,,$
(24)

where $R,\omega$, and $\gamma$ are the radius of the circular motion, angular
velocity, and the Lorentz factor defined as $\gamma\coloneqq
1/\sqrt{1-v^{2}}$. Here, $v\coloneqq R\omega(\leq 1)$ is the speed of the
detector. Introducing the acceleration of the detector
$a=R\omega^{2}\gamma^{2}$, these parameters can be related by

$\displaystyle\omega$ $\displaystyle=\sqrt{\dfrac{a}{(1+aR)R}}\,,$ (25a)
$\displaystyle\gamma$ $\displaystyle=\sqrt{1+aR}\,,$ (25b) $\displaystyle v$
$\displaystyle=\sqrt{\dfrac{aR}{1+aR}}\,.$ (25c)

In terms of the acceleration $a$ and the torsion $b$, we can further express
$\omega$ and $v$ as

$\omega=b(1-a^{2}/b^{2})\qquad v=a/b$

respectively. The Wightman function is then

$\displaystyle
W_{\text{cir}}(\Delta\tau)=-\dfrac{1}{4\pi^{2}}\dfrac{1}{\gamma^{2}\Delta\tau^{2}-4R^{2}\sin^{2}(\omega\gamma\Delta\tau/2)}\,.$
(26)

####  III.1.3 Cusped motion

Cusped motion is described by the acceleration and torsion with $a=b$. The
trajectory reads

$\displaystyle\mathsf{x}(\tau)$
$\displaystyle=\left(\tau+\dfrac{1}{6}a^{2}\tau^{3},\,\dfrac{1}{2}a\tau^{2},\,\dfrac{1}{6}a^{2}\tau^{3},\,0\right)\,,$
(27)

and the corresponding Wightman function is

$\displaystyle W_{\text{cus}}(\Delta\tau)$
$\displaystyle=-\dfrac{1}{4\pi^{2}}\dfrac{1}{\Delta\tau^{2}+\dfrac{a^{2}}{12}\Delta\tau^{4}}\,.$
(28)

####  III.1.4 Catenary motion

Catenary motion can be characterized by $a$ and $b$ with $a>b$. The trajectory
is given by

$\displaystyle\mathsf{x}(\tau)$
$\displaystyle=\bigg{(}\dfrac{a}{a^{2}-b^{2}}\sinh\big{(}\sqrt{a^{2}-b^{2}}\,\tau\big{)},$
$\displaystyle\quad\dfrac{a}{a^{2}-b^{2}}\cosh\big{(}\sqrt{a^{2}-b^{2}}\,\tau\big{)},\dfrac{b\tau}{\sqrt{a^{2}-b^{2}}},0\bigg{)}\,,$
(29)

and the Wightman function reads

$\displaystyle W_{\text{cat}}(\Delta\tau)$
$\displaystyle=-\dfrac{1}{4\pi^{2}}\dfrac{1}{-\dfrac{b^{2}\Delta\tau^{2}}{a^{2}-b^{2}}+\dfrac{4a^{2}}{(a^{2}-b^{2})^{2}}\sinh^{2}\big{(}\frac{\sqrt{a^{2}-b^{2}}\Delta\tau}{2}\big{)}}.$
(30)

We immediately see that catenary motion reduces to the linear motion as $b\to
0$. Catenary motion also reduces to cusped motion as $b\to a$ after a
coordinate transformation consisting of a Lorentz boost a translation [18].

####  III.1.5 Helix motion

Finally, helix motion is a combination of circular and linear acceleration
motions characterized by three parameters, $a,b$, and $\nu$:

$\displaystyle\mathsf{x}(\tau)$
$\displaystyle=\bigg{(}\dfrac{\mathcal{P}}{\Gamma_{+}}\sinh(\Gamma_{+}\tau),\dfrac{\mathcal{P}}{\Gamma_{+}}\cosh(\Gamma_{+}\tau),$
$\displaystyle\quad\dfrac{\mathcal{Q}}{\Gamma_{-}}\cos(\Gamma_{-}\tau),\dfrac{\mathcal{Q}}{\Gamma_{-}}\sin(\Gamma_{-}\tau)\bigg{)}\,,$
(31)

where $\mathcal{P}\coloneqq\Xi/\Gamma$, $\mathcal{Q}\coloneqq ab/\Xi\Gamma$,
and

$\displaystyle\Xi^{2}$
$\displaystyle\coloneqq\dfrac{1}{2}(\Gamma^{2}+a^{2}+b^{2}+\nu^{2})\,,$ (32a)
$\displaystyle\Gamma^{2}$
$\displaystyle\coloneqq\Gamma_{+}^{2}+\Gamma_{-}^{2}\,,\quad\Gamma_{\pm}^{2}\coloneqq\sqrt{A^{2}+B^{2}}\pm
A\,,$ (32b) $\displaystyle A$
$\displaystyle\coloneqq\dfrac{1}{2}(a^{2}-b^{2}-\nu^{2})\,,\quad B\coloneqq
a\nu\,.$ (32c)

The Wightman function reads

$\displaystyle W_{\text{hel}}(\Delta\tau)=$
$\displaystyle-\dfrac{1}{4\pi^{2}}\dfrac{1}{\dfrac{4\mathcal{P}^{2}}{\Gamma_{+}^{2}}\sinh^{2}\left(\frac{\Gamma_{+}\Delta\tau}{2}\right)-\dfrac{4\mathcal{Q}^{2}}{\Gamma_{-}^{2}}\sin^{2}\left(\frac{\Gamma_{-}\Delta\tau}{2}\right)}\,.$
(33)

Note that the trajectory and the corresponding Wightman function reduce to the
aforementioned trajectories when $\nu\to 0$. In this sense, the helix is the
general motion that contains other motions.

####  III.1.6 Wightman function at $\nu=0$

We now turn our attention to the special case where $\nu=0$. Although the
Wightman functions for linear, circular, catenary, and cusped motions may
initially appear to take different forms, they can actually be expressed in a
unified manner. Let $\bar{b}\equiv b/a$ with the condition that $a\neq 0$. The
Wightman functions for all trajectories with $\nu=0$ can be written in the
following compact form:

$\displaystyle W_{\nu=0}(\Delta\tau)$
$\displaystyle=-\dfrac{1}{4\pi^{2}}\dfrac{1}{-\dfrac{\bar{b}^{2}}{1-\bar{b}^{2}}\Delta\tau^{2}+\dfrac{4}{(1-\bar{b}^{2})^{2}a^{2}}\sinh^{2}\left(\frac{\sqrt{1-\bar{b}^{2}}a\Delta\tau}{2}\right)}\,.$
(34)

The parameter $\bar{b}$ serves to specify the particular trajectory, as
illustrated in figure 1: linear ($\bar{b}=0$), catenary ($0<\bar{b}<1$),
cusped ($\bar{b}=1$), and circular ($\bar{b}>1$). For circular motion, we
employ the identity $\sin(\mathsf{i}x)=\mathsf{i}\sinh(x)$. Note that one
obtains the Wightman function for the cusped motion, as given in (28), by
performing a series expansion around $\bar{b}=1$.

The corresponding transition probability $\mathcal{L}_{jj}$,
$j\in\\{\text{A},\text{B}\\}$, in (13) reads

$\displaystyle\mathcal{L}_{jj}$
$\displaystyle=\lambda^{2}\sigma\sqrt{\pi}\int_{\mathbb{R}}\text{d}u\,e^{-u^{2}/4\sigma^{2}}e^{-\mathsf{i}\Omega
u}W_{\nu=0}(u)\,.$ (35)

Figure 2: Stationary and nonstationary configurations for four classes of
uniformly accelerating detectors. Red and blue strips represent detectors A
and B, respectively.

###  III.2 Two detectors in uniform acceleration

We now consider two UDW detectors A and B, both undergoing uniform
acceleration motion. In particular, we categorize the detector configurations
into two classes: stationary (time-translation invariant) and nonstationary
scenarios.

####  III.2.1 Stationary scenario

Consider two detectors undergoing the same uniform acceleration (e.g., both
linearly accelerated). The Wightman function can be made time-translation
invariant, meaning it depends only on the time difference
$\Delta\tau\coloneqq\tau_{\text{A}}-\tau_{\text{B}}$, by imposing that the
angle between the velocity vector of a detector and the spatial displacement
vector from one detector to the other is time-independent. For example, two
linearly accelerating detectors along the trajectories

$\displaystyle\mathsf{x}_{\text{A}}(\tau_{\text{A}})$
$\displaystyle=\left(\dfrac{1}{a}\sinh(a\tau_{\text{A}}),\dfrac{1}{a}\cosh(a\tau_{\text{A}}),0,0\right)\,,$
(36a) $\displaystyle\mathsf{x}_{\text{B}}(\tau_{\text{B}})$
$\displaystyle=\left(\dfrac{1}{a}\sinh(a\tau_{\text{B}}),\dfrac{1}{a}\cosh(a\tau_{\text{B}}),L,0\right)$
(36b)

give the following stationary Wightman function:

$\displaystyle
W_{\text{lin}}(\tau_{\text{A}},\tau_{\text{B}})=-\dfrac{1}{4\pi^{2}}\dfrac{1}{\dfrac{4}{a^{2}}\sinh^{2}\left(\frac{a\Delta\tau}{2}\right)-L^{2}}\,$
(37)

where $L\coloneqq|{\bm{x}}_{\text{A}}-{\bm{x}}_{\text{B}}|$ is the spatial
separation between the two detectors. As depicted in figure 2 top-left, each
velocity vector of the detector is always perpendicular to the displacement
vector ${\bm{x}}_{\text{AB}}\coloneqq{\bm{x}}_{\text{A}}-{\bm{x}}_{\text{B}}$
throughout the interaction.

We can also construct stationary Wightman functions for other motions:

circular:

$\displaystyle\mathsf{x}_{\text{A}}=\left(\gamma\tau_{\text{A}},R\cos(\omega\gamma\tau_{\text{A}}),R\sin(\omega\gamma\tau_{\text{A}}),0\right)\,,$
(38a)
$\displaystyle\mathsf{x}_{\text{B}}=\left(\gamma\tau_{\text{B}},R\cos(\omega\gamma\tau_{\text{B}}),R\sin(\omega\gamma\tau_{\text{B}}),L\right)\,.$
(38b)

cusped:

$\displaystyle\mathsf{x}_{\text{A}}=\left(\tau_{\text{A}}+\dfrac{1}{6}a^{2}\tau_{\text{A}}^{3},\,\dfrac{1}{2}a\tau_{\text{A}}^{2},\,\dfrac{1}{6}a^{2}\tau_{\text{A}}^{3},\,0\right)\,,$
(39a)
$\displaystyle\mathsf{x}_{\text{B}}=\left(\tau_{\text{B}}+\dfrac{1}{6}a^{2}\tau_{\text{B}}^{3},\,\dfrac{1}{2}a\tau_{\text{B}}^{2},\,\dfrac{1}{6}a^{2}\tau_{\text{B}}^{3},\,L\right)\,,$
(39b)

catenary:

$\displaystyle\mathsf{x}_{\text{A}}=\bigg{(}\dfrac{a}{a^{2}-b^{2}}\sinh\big{(}\sqrt{a^{2}-b^{2}}\,\tau_{\text{A}}\big{)},$
$\displaystyle\hskip
28.45274pt\dfrac{a}{a^{2}-b^{2}}\cosh\big{(}\sqrt{a^{2}-b^{2}}\,\tau_{\text{A}}\big{)},\dfrac{b\tau_{\text{A}}}{\sqrt{a^{2}-b^{2}}},0\bigg{)}\,,$
(40a)
$\displaystyle\mathsf{x}_{\text{B}}=\bigg{(}\dfrac{a}{a^{2}-b^{2}}\sinh\big{(}\sqrt{a^{2}-b^{2}}\,\tau_{\text{B}}\big{)},$
$\displaystyle\hskip
28.45274pt\dfrac{a}{a^{2}-b^{2}}\cosh\big{(}\sqrt{a^{2}-b^{2}}\,\tau_{\text{B}}\big{)},\dfrac{b\tau_{\text{B}}}{\sqrt{a^{2}-b^{2}}},L\bigg{)}\,,$
(40b)

As for a single detector, the Wightman functions along the trajectories given
above take the following compact form:

$\displaystyle W_{\text{s}}(\tau_{\text{A}},\tau_{\text{B}})\equiv
W_{\text{s}}(\Delta\tau)=-\dfrac{1}{4\pi^{2}}\dfrac{1}{-\dfrac{\bar{b}^{2}}{1-\bar{b}^{2}}\Delta\tau^{2}+\dfrac{4}{(1-\bar{b}^{2})^{2}a^{2}}\sinh^{2}\left(\frac{\sqrt{1-\bar{b}^{2}}a\Delta\tau}{2}\right)-L^{2}}\,,$
(41)

where $\Delta\tau\coloneqq\tau_{\text{A}}-\tau_{\text{B}},\bar{b}\equiv b/a$,
and the subscript ‘s’ stands for stationary. Since the Wightman function
depends only on $\Delta\tau$, the elements in the density matrix (13),
$\mathcal{M}$ and $\mathcal{L}_{\text{AB}}$, can be simplified to single
integrals when the Gaussian switching function (16) is used:

$\displaystyle\mathcal{M}$
$\displaystyle=-2\lambda^{2}\sigma\sqrt{\pi}e^{-\Omega^{2}\sigma^{2}}\int_{0}^{\infty}\text{d}u\,e^{-u^{2}/4\sigma^{2}}W_{\text{s}}(u)\,,$
(42a) $\displaystyle\mathcal{L}_{\text{AB}}$
$\displaystyle=\lambda^{2}\sigma\sqrt{\pi}\int_{\mathbb{R}}\text{d}u\,e^{-u^{2}/4\sigma^{2}}e^{-\mathsf{i}\Omega
u}W_{\text{s}}(u)\,.$ (42b)

Here, we used the fact that the Heaviside step function in (14b) can be
written as
$\Theta(t(\tau_{\text{A}})-t(\tau_{\text{B}}))=\Theta(\tau_{\text{A}}-\tau_{\text{B}})$
for any of the uniform acceleration scenarios mentioned earlier.

We note that all stationary configurations can only be realized in $(3+1)$
dimensions, with the exception of the linear configuration.

####  III.2.2 Nonstationary scenario

One can also consider configurations similar to those in section III.2.1,
where the Wightman function depends not only on $\Delta\tau$ but also on
$\Delta_{+}\tau\coloneqq\tau_{\text{A}}+\tau_{\text{B}}$. In this case, the
Wightman function is no longer time-translation invariant (hence,
nonstationary).

In particular, consider two linearly accelerating UDW detectors whose
trajectories are given by

$\displaystyle\mathsf{x}_{\text{A}}(\tau_{\text{A}})$
$\displaystyle=\left(\dfrac{1}{a}\sinh(a\tau_{\text{A}}),\dfrac{1}{a}\cosh(a\tau_{\text{A}})+L,0,0\right)\,,$
(43a) $\displaystyle\mathsf{x}_{\text{B}}(\tau_{\text{B}})$
$\displaystyle=\left(\dfrac{1}{a}\sinh(a\tau_{\text{B}}),\dfrac{1}{a}\cosh(a\tau_{\text{B}}),0,0\right)\,.$
(43b)

The correlation harvesting protocol along these trajectories was examined in
[26, 28]. The corresponding Wightman function reads

$\displaystyle W_{\text{lin}}(\tau_{\text{A}},\tau_{\text{B}})$
$\displaystyle=-\dfrac{1}{4\pi^{2}}\dfrac{1}{\dfrac{4}{a^{2}}\sinh^{2}\left(\frac{a\Delta\tau}{2}\right)-L^{2}-\dfrac{4L}{a}\sinh\left(\frac{a\Delta\tau}{2}\right)\sinh\left(\frac{a\Delta_{+}\tau}{2}\right)}\,.$
(44)

The term $\Delta_{+}\tau$ comes from the fact that the angle between the
velocity vector and the displacement vector is time-dependent ($0^{\circ}$ or
$180^{\circ}$).

Similarly, other uniformly accelerating trajectories that yield a
nonstationary Wightman function are

circular:

$\displaystyle\mathsf{x}_{\text{A}}=\left(\gamma\tau_{\text{A}},R\cos(\omega\gamma\tau_{\text{A}})+L,R\sin(\omega\gamma\tau_{\text{A}}),0\right)\,,$
(45a)
$\displaystyle\mathsf{x}_{\text{B}}=\left(\gamma\tau_{\text{B}},R\cos(\omega\gamma\tau_{\text{B}}),R\sin(\omega\gamma\tau_{\text{B}}),0\right)\,.$
(45b)

cusped:

$\displaystyle\mathsf{x}_{\text{A}}=\left(\tau_{\text{A}}+\dfrac{1}{6}a^{2}\tau_{\text{A}}^{3},\,\dfrac{1}{2}a\tau_{\text{A}}^{2}+L,\,\dfrac{1}{6}a^{2}\tau_{\text{A}}^{3},\,0\right)\,,$
(46a)
$\displaystyle\mathsf{x}_{\text{B}}=\left(\tau_{\text{B}}+\dfrac{1}{6}a^{2}\tau_{\text{B}}^{3},\,\dfrac{1}{2}a\tau_{\text{B}}^{2},\,\dfrac{1}{6}a^{2}\tau_{\text{B}}^{3},\,0\right)\,,$
(46b)

catenary:

$\displaystyle\mathsf{x}_{\text{A}}=\bigg{(}\dfrac{a}{a^{2}-b^{2}}\sinh\big{(}\sqrt{a^{2}-b^{2}}\,\tau_{\text{A}}\big{)},$
$\displaystyle\hskip
28.45274pt\dfrac{a}{a^{2}-b^{2}}\cosh\big{(}\sqrt{a^{2}-b^{2}}\,\tau_{\text{A}}\big{)}+L,\dfrac{b\tau_{\text{A}}}{\sqrt{a^{2}-b^{2}}},0\bigg{)}\,,$
(47a)
$\displaystyle\mathsf{x}_{\text{B}}=\bigg{(}\dfrac{a}{a^{2}-b^{2}}\sinh\big{(}\sqrt{a^{2}-b^{2}}\,\tau_{\text{B}}\big{)},$
$\displaystyle\hskip
28.45274pt\dfrac{a}{a^{2}-b^{2}}\cosh\big{(}\sqrt{a^{2}-b^{2}}\,\tau_{\text{B}}\big{)},\dfrac{b\tau_{\text{B}}}{\sqrt{a^{2}-b^{2}}},0\bigg{)}\,,$
(47b)

The Wightman function for these nonstationary motions can be compactly
expressed as

$\displaystyle W_{\text{ns}}(\tau_{\text{A}},\tau_{\text{B}})=$
$\displaystyle-\dfrac{1}{4\pi^{2}}\dfrac{1}{-\dfrac{\bar{b}^{2}}{1-\bar{b}^{2}}\Delta\tau^{2}+\dfrac{4}{(1-\bar{b}^{2})^{2}a^{2}}\sinh^{2}\left(\frac{\sqrt{1-\bar{b}^{2}}a\Delta\tau}{2}\right)-L^{2}-\dfrac{4L}{(1-\bar{b}^{2})a}\sinh\left(\frac{\sqrt{1-\bar{b}^{2}}a\Delta\tau}{2}\right)\sinh\left(\frac{\sqrt{1-\bar{b}^{2}}a\Delta_{+}\tau}{2}\right)}\,,$
(48)

and possesses an additional term in the denominator compared to the stationary
Wightman function (41). Here, the subscript ‘ns’ designates nonstationary. Due
to this additional term, the correlations harvested by nonstationary detectors
will exhibit behavior similar to those of stationary detectors.

Note that the presence of $\Delta_{+}\tau$ prevents us from reducing the
double integrals in (14) into single integrals. Furthermore, all nonstationary
configurations can be realized in $(2+1)$ dimensions, except for the helix
case, which we are not considering.

Figure 3:  (a) Transition probabilities $\mathcal{L}_{jj}/\lambda^{2}$ as a
function of the magnitude of acceleration $a\sigma$ with $\Omega\sigma=2$. (b)
$\mathcal{L}_{jj}/\lambda^{2}$ as a function of $\log_{10}\bar{b}$ with
$a\sigma=1$ and (c) with $a\sigma=6$. (d) The effective temperature
$T_{\text{eff}}$ as a function of $\log_{10}\bar{b}$ with $a\sigma=1$.

##  IV Numerical results

Here, we numerically compute the concurrence (19) and quantum mutual
information (20) harvested by two uniformly accelerating detectors by
inserting the Wightman functions (34), (41) and (48) into $\mathcal{L}_{ij}$
and $\mathcal{M}$ given in (14). For stationary detectors in III.2.1, we
utilize the expressions given by (42).

###  IV.1 Transition probability of uniformly accelerating detectors

Let us begin by considering the transition probability $\mathcal{L}_{jj}$ for
a uniformly accelerating detector. We are particularly interested in the cases
of linear ($\bar{b}=0$), catenary ($0<\bar{b}<1$), cusped ($\bar{b}=1$), and
circular ($\bar{b}>1$) motions, and their respective transition probabilities
are given by (35). We consider $\mathcal{L}_{jj}/\lambda^{2}$ and write the
parameters in units of $\sigma$, which makes the transition probability a
function of three variables: $a\sigma,\bar{b}$, and $\Omega\sigma$. It is
important to note that $\mathcal{L}_{\text{AA}}=\mathcal{L}_{\text{BB}}$, as
we are assuming both detectors are identical.

Figure 3 depicts the transition probability $\mathcal{L}_{jj}/\lambda^{2}$ as
a function of the magnitude of acceleration $a\sigma$ for fixed $\Omega$
(panel (a)) and $\log_{10}\bar{b}$ for different values of the acceleration
(panels (b), (c)). In figure 3(a), the transition probabilities for a detector
with $\Omega\sigma=2$ in linear ($\bar{b}=0$), catenary ($\bar{b}=0.5$),
cusped ($\bar{b}=1$), and circular ($\bar{b}=2$) motions are shown. We find
that in all these cases, $\mathcal{L}_{jj}/\lambda^{2}$ increases with the
acceleration $a\sigma$.111$\mathcal{L}_{jj}/\lambda^{2}$ is not guaranteed to
monotonically increase with $a\sigma$ for a finite interaction duration. For a
detector in the linearly accelerated motion, such phenomenon is known as the
(weak) anti-Unruh effect [60, 61].

However, the relationship between the transition probabilities of detectors in
different uniform motions is highly nontrivial. For instance, when a detector
has $\Omega\sigma=2$ and $a\sigma\lesssim 5$, as depicted in figure 3(a), a
detector in circular motion with $\bar{b}=2$ shows the largest value of
$\mathcal{L}_{jj}/\lambda^{2}$, whereas a detector in linear motion
($\bar{b}=0$) shows the smallest. This relation, however, flips for
$a\sigma\gtrsim 5$. We will numerically demonstrate that such relationships
depend on the interplay between $a\sigma$ and $\Omega\sigma$.

Figure 4:  Concurrence (a) and quantum mutual information (b) harvested by
stationary and nonstationary detectors as a function of $\bar{b}$. For each
case, $\Omega\sigma=0.1$ and $L/\sigma=1$. (a-i) and (b-i) are respectively
$\mathcal{C}_{\text{AB}}/\lambda^{2}$ and $I_{\text{AB}}/\lambda^{2}$ when
$a\sigma=1$, while (a-ii) and (b-ii) are respectively
$\mathcal{C}_{\text{AB}}/\lambda^{2}$ and $I_{\text{AB}}/\lambda^{2}$ when
$a\sigma=2$.

In figure 3(b), the magnitude of the acceleration is fixed at $a\sigma=1$, and
$\mathcal{L}_{jj}/\lambda^{2}$ is plotted as a function of $\log_{10}\bar{b}$.
Each curve in this figure corresponds to a different value of $\Omega\sigma$,
with the curve for $\Omega\sigma=2$ corresponding to figure 3(a) at
$a\sigma=1$. For each value of $\Omega\sigma$ in 3(b), the transition
probability has a peak for $\log_{10}\bar{b}>0$ (i.e., $\bar{b}>1$), and then
decreases with increasing $\log_{10}\bar{b}$, becoming smaller than the value
for the linear case ($\log_{10}\bar{b}\to-\infty$). This means that
$\mathcal{L}_{jj}/\lambda^{2}$ at $a\sigma=1$ in figure 3(a) increases with
$\bar{b}$ until it reaches a maximum and then decreases. We note that the
presence of the peak is contingent on larger values of $\Omega\sigma$ relative
to $a\sigma$; In fact, the peak does not appear for smaller energy gaps, in
which case the transition probability monotonically decreases with $\bar{b}$,
as shown in figure 3(b). This trend is further illustrated in figure 3(c),
where $a\sigma=6$ is chosen. In this scenario, the peak is nonexistent for
$\Omega\sigma=1$ and 1.5 (as well as for $\Omega\sigma<1$), but becomes
manifest when $\Omega\sigma\gtrsim 2$. Thus we infer that detectors with
smaller energy gaps $\Omega\sigma$ compared to $a\sigma$ do not have a peak in
$\mathcal{L}_{jj}(\bar{b})/\lambda^{2}$.

The behavior of $\mathcal{L}_{jj}/\lambda^{2}$ is related to the concept of
the “effective temperature” perceived by a detector. For now, let us denote
the transition probability as $\mathcal{L}_{jj}(\Omega,\sigma)$. The effective
temperature, $T_{\text{eff}}$, is defined as

$\displaystyle T_{\text{eff}}^{-1}$
$\displaystyle\coloneqq\dfrac{1}{\Omega}\ln\dfrac{\mathcal{L}_{jj}(-\Omega,\sigma)/\lambda^{2}\sigma}{\mathcal{L}_{jj}(\Omega,\sigma)/\lambda^{2}\sigma}\,$
(49)

where this formula is derived in the Appendix. We divide
$\mathcal{L}_{jj}(\Omega,\sigma)$ by $\sigma$ so that it is well defined in
the long interaction limit, $\sigma\to\infty$ [62, 61]. Note that if the
Wightman function obeys the Kubo-Martin-Schwinger (KMS) condition [63, 64],
then the effective temperature converges to the KMS temperature (which is the
temperature of the field formally defined in quantum field theory) in the
limit $\sigma\to\infty$. However, in the case of finite interaction duration,
the effective temperature is an estimator for the actual field temperature.
For a detector in a uniform acceleration motion, the effective temperature for
each scenario has been examined in, e.g., [14, 15, 16, 17, 18, 19, 20].

We plot the effective temperature $T_{\text{eff}}$ as a function of
$\log_{10}\bar{b}$ when $\sigma=1$ and $a\sigma=1$ in figure 3(d), which
corresponds to figure 3(b). We see that the locations of the peaks in
$T_{\text{eff}}$ align with those of $\mathcal{L}_{jj}(\Omega)$ in 3(b). This
suggests that, for a given acceleration and energy gap, a detector in circular
motion within a certain range of $\log_{10}\bar{b}$ can register higher
effective temperatures than those in other types of motion. However, as
$\bar{b}\to\infty$, which corresponds to the speed of a detector in circular
motion with $v_{\text{circ}}(=\bar{b}^{-1})\to 0$, the temperature becomes
colder.

###  IV.2 Concurrence and quantum mutual information between uniformly
accelerating detectors

We now move on to the correlation harvesting protocol using two uniformly
accelerating detectors, exploring both stationary and nonstationary
configurations as described in section III.2.

We first examine the difference between the stationary and nonstationary
configurations by plotting concurrence $\mathcal{C}_{\text{AB}}/\lambda^{2}$
and quantum mutual information $I_{\text{AB}}/\lambda^{2}$ as a function of
$\bar{b}$ in figure 4. In these plots, we fix $\Omega\sigma=0.1$ and
$L/\sigma=1$, and consider $a\sigma=1$ and $a\sigma=2$. We notice two
characteristics: (i) In the vicinity of $\bar{b}\approx 0$, stationary
detectors consistently harvest greater correlations than nonstationary
detectors, for both concurrence and mutual information. (ii) As $\bar{b}$
becomes larger, both plots begin to oscillate with $\bar{b}$, and the curve
representing correlations harvested by nonstationary detectors oscillates
around the curve for the stationary case. The frequency of the oscillation
increases as $a\sigma$ grows.

These observations can be traced back to the form of the Wightman functions
(41) and (48). Let us recall that the denominators of these expressions
contain $\sinh(x)$ when $\bar{b}\in[0,1)$ and transform into $\sin(x)$ when
$\bar{b}>1$. Therefore, within the range $\bar{b}\in[0,1)$, the correlations
are characterized by an exponential pattern, while for $\bar{b}>1$, an
oscillatory behavior emerges. These traits explain the observation above. In
particular, the suppression of correlations near $\bar{b}\approx 0$ for
nonstationary detectors can be attributed to an additional term in the
denominator of (48), which is absent in the stationary Wightman function (41).
This extra term diminishes the amount of harvested correlations relative to
the stationary scenario, and simultaneously gives rise to the oscillations
noticed in the nonstationary case around the stationary one.

Figure 5: Concurrence $\mathcal{C}_{\text{AB}}/\lambda^{2}$ with
$\Omega\sigma=0.1,L/\sigma=0.5$ (a) and quantum mutual information
$I_{\text{AB}}/\lambda^{2}$ with $\Omega\sigma=0.1,L/\sigma=3$ (b) as a
function of the magnitude of acceleration $a\sigma$. (a-i) and (b-i)
correspond to the stationary configuration while (a-ii) and (b-ii) show the
nonstationary one, and each figure has four curves indicating four different
motions.

We next examine the acceleration dependence of concurrence
$\mathcal{C}_{\text{AB}}$ and quantum mutual information $I_{\text{AB}}$ as
illustrated in figures 5(a) and (b), respectively. The stationary (figure
5(a-i) and (b-i)) and nonstationary (figure 5(a-ii) and (b-ii)) configurations
are depicted, and all four uniformly accelerated motions, linear
($\bar{b}=0$), catenary ($\bar{b}=0.5$), cusped ($\bar{b}=1$), and circular
($\bar{b}=2$) are shown in each figure.

As we pointed out earlier, the correlations harvested by nonstationary
detectors for $\bar{b}\in[0,1)$ (figure 5(a-ii) and (b-ii)) decay with
increasing $a\sigma$ faster than those extracted by the stationary detectors
(figure 5(a-i) and (b-i)). Meanwhile, the correlations extracted by
nonstationary detectors in circular motion ($\bar{b}>1$) (figure 5(a-ii) and
(b-ii)) exhibit oscillatory behavior around the corresponding stationary
curves (figure 5(a-i) and (b-i)).

Another observation we make is that, for both stationary and nonstationary
configurations and for any value of $\bar{b}$,
$\mathcal{C}_{\text{AB}}/\lambda^{2}$ becomes 0 at sufficiently high
$a\sigma$. This can be attributed to the high transition probability at large
$a\sigma$ as shown in figure 3(a), leading to
$|\mathcal{M}|<\sqrt{\mathcal{L}_{\text{AA}}\mathcal{L}_{\text{BB}}}$ in (19).
Furthermore, the high accelerations prevent the detectors from extracting
quantum mutual information, as $I_{\text{AB}}/\lambda^{2}\to 0$ at
$a\sigma\to\infty$ in figure 5(b). This indicates that any correlations cannot
be harvested as $a\sigma\to\infty$ if the detectors are uniformly accelerated.
These findings are consistent with previous results [47, 8, 48, 25, 26, 27,
28], where linearly and circularly accelerated detectors are considered. Our
paper extends these insights, providing a more general understanding that
encompasses arbitrary uniformly accelerated motion.

###  IV.3 Genuine entanglement

Figure 6: The boundaries between $\mathcal{C}_{\text{AB}}>0$ and
$\mathcal{C}_{\text{AB}}=0$ for the four stationary trajectories as a function
of the proper separation $L/\sigma$ and the energy gap $\Omega\sigma$. Here,
linear ($\bar{b}=0$), catenary ($\bar{b}=0.5$), cusped ($\bar{b}=1$), and
circular ($\bar{b}=2$) with $a\sigma=1$ are depicted. Concurrence is nonzero
in the left region of each curve.

We finally consider how much of entanglement is coming from the quantum field.
It is known that the Wightman function can be decomposed into two parts: the
anticommutator and the commutator of the field operator. The anticommutator
part
$\braket{\\{\hat{\phi}(\mathsf{x}),\hat{\phi}(\mathsf{x}^{\prime})\\}}_{\rho_{\phi}}$
(also known as the Hadamard function), where $\braket{\cdot}_{\rho_{\phi}}$ is
the expectation value with respect to the field state $\rho_{\phi}$, depends
on the state of the field $\rho_{\phi}$. Conversely, the commutator part
$\braket{[\hat{\phi}(\mathsf{x}),\hat{\phi}(\mathsf{x}^{\prime})]}_{\rho_{\phi}}=[\hat{\phi}(\mathsf{x}),\hat{\phi}(\mathsf{x}^{\prime})]\in\mathbb{C}$
(also known as the Pauli-Jordan function) is state-independent. This means
that even if the field state is not entangled, the commutator part in the
Wightman function allows detectors to be entangled with each other. Such
entanglement does not come from preexisting entanglement in the field; rather
it is associated with communication between the detectors, and thus we cannot
say (for an unentangled field state) that entanglement is ‘extracted’ from the
field if the commutator part is the only contribution [54]. We say that
entanglement is harvested if the anticommutator contribution in the element
$\mathcal{M}$ is nonzero, and in particular we qualify the harvested
entanglement as being genuine if the commutator part in $\mathcal{M}$ is zero.
Microcausality tells us that the two detectors can harvest genuine
entanglement if they are causally disconnected. Here, we explore the
circumstances under which two uniformly accelerating detectors can extract
genuine entanglement from the field. Remarkably we find that this can be
possible even if the detectors are in causal contact.

We begin by plotting the concurrence $\mathcal{C}_{\text{AB}}/\lambda^{2}$ as
a function of the proper separation $L/\sigma$ between the detectors and the
energy gap $\Omega\sigma$ in figure 6. The respective curves correspond to
linear ($\bar{b}=0$), catenary ($\bar{b}=0.5$), cusped ($\bar{b}=1$), and
circular motions ($\bar{b}=2$) in the stationary configurations depicted in
figure 2. The left region of each curve represents the parameters
$(L/\sigma,\Omega\sigma)$ that enable the detectors to become entangled,
manifest as $\mathcal{C}_{\text{AB}}/\lambda^{2}>0$. Conversely, the right
region corresponds to $\mathcal{C}_{\text{AB}}/\lambda^{2}=0$. Therefore, the
stationary linear configuration ($\bar{b}=0$) has the broadest parameter space
that leads to $\mathcal{C}_{\text{AB}}/\lambda^{2}>0$ compared to any other
stationary configurations.

It has been shown [7] that two detectors at rest in Minkowski spacetime with a
Gaussian switching function can be entangled with an arbitrary detector
separation $L/\sigma$ if the energy gap $\Omega\sigma$ is large enough.
However, we see that this is not the case for uniformly accelerating detectors
– they can be entangled only when they are close to each other, no matter how
large $\Omega\sigma$ is.

We further ask how much entanglement stems from the anticommutator and
commutator parts in the Wightman function. To see this, let us decompose the
Wightman function as

$\displaystyle W(\mathsf{x},\mathsf{x}^{\prime})$
$\displaystyle=\text{Re}[W(\mathsf{x},\mathsf{x}^{\prime})]+\mathsf{i}\,\text{Im}[W(\mathsf{x},\mathsf{x}^{\prime})]\,,$
(50)

where

$\displaystyle 2\text{Re}[W(\mathsf{x},\mathsf{x}^{\prime})]$
$\displaystyle=\braket{0}{\\{\hat{\phi}(\mathsf{x}),\hat{\phi}(\mathsf{x}^{\prime})\\}}{0}\,,$
(51a) $\displaystyle 2\text{Im}[W(\mathsf{x},\mathsf{x}^{\prime})]$
$\displaystyle=-\mathsf{i}\,[\hat{\phi}(\mathsf{x}),\hat{\phi}(\mathsf{x}^{\prime})]\,.$
(51b)

Then the matrix element $\mathcal{M}$ can be decomposed into

$\displaystyle\mathcal{M}$
$\displaystyle=\mathcal{M}_{+}+\mathsf{i}\mathcal{M}_{-}\,,$ (52)

where $\mathcal{M}_{+}$ and $\mathcal{M}_{-}$ are (14b) with the Wightman
function being replaced by $\text{Re}[W(\mathsf{x},\mathsf{x}^{\prime})]$ and
$\text{Im}[W(\mathsf{x},\mathsf{x}^{\prime})]$, respectively.
$\mathcal{M}_{+}$ contains the information about the genuine entanglement
whereas $\mathcal{M}_{-}$ is state-independent and does not necessarily
exhibit the preexisted entanglement in the field. For the stationary
detectors, these expressions can be simplified to single integral forms:

$\displaystyle\mathcal{M}_{+}$
$\displaystyle=-\lambda^{2}\sigma\sqrt{\pi}e^{-\Omega^{2}\sigma^{2}}\int_{0}^{\infty}\text{d}u\,e^{-u^{2}/4\sigma^{2}}W_{\text{s}}(u)$
$\displaystyle\quad-\lambda^{2}\sigma\sqrt{\pi}e^{-\Omega^{2}\sigma^{2}}\int_{0}^{\infty}\text{d}u\,e^{-u^{2}/4\sigma^{2}}W_{\text{s}}^{*}(u)\,,$
(53a) $\displaystyle\mathcal{M}_{-}$
$\displaystyle=\mathsf{i}\lambda^{2}\sigma\sqrt{\pi}e^{-\Omega^{2}\sigma^{2}}\int_{0}^{\infty}\text{d}u\,e^{-u^{2}/4\sigma^{2}}W_{\text{s}}(u)$
$\displaystyle\quad-\mathsf{i}\lambda^{2}\sigma\sqrt{\pi}e^{-\Omega^{2}\sigma^{2}}\int_{0}^{\infty}\text{d}u\,e^{-u^{2}/4\sigma^{2}}W_{\text{s}}^{*}(u)\,.$
(53b)

We then define harvested concurrence $\mathcal{C}_{\text{AB}}^{+}$ and
communication-assisted concurrence $\mathcal{C}_{\text{AB}}^{-}$ as [54]

$\displaystyle\mathcal{C}_{\text{AB}}^{\pm}$ $\displaystyle\coloneqq
2\max\\{0,\,|\mathcal{M}_{\pm}|-\sqrt{\mathcal{L}_{\text{AA}}\mathcal{L}_{\text{BB}}}\\}+\mathcal{O}(\lambda^{4})\,.$
(54)

Figure 7: Harvested and communication-assisted concurrence
$\mathcal{C}_{\text{AB}}^{\pm}/\lambda^{2}$ for the stationary linear and
circular configurations as a function of the detector separation $L/\sigma$.
(a) When $\Omega\sigma=1$. The linear and circular cases are very similar and
the detectors can harvest genuine entanglement near $L/\sigma=2$. (b) When
$\Omega\sigma=2$. Although the linear case does not change much compared to
the $\Omega\sigma=1$ case, the circular case cannot harvest genuine
entanglement anymore. It is the mixture of the anticommutator and commutator
contributions, or in the worst case
$\mathcal{C}_{\text{AB}}^{+}/\lambda^{2}=0$ around $L/\sigma=1.3$.

We plot $\mathcal{C}_{\text{AB}}^{\pm}/\lambda^{2}$ as a function of
$L/\sigma$ in figure 7. Here, we specifically choose the stationary linear
($\bar{b}=0$) and circular ($\bar{b}=2$) cases as a demonstration. We find
that for $\Omega\sigma=1$ (figure 7(a)), the detectors can harvest
entanglement since $\mathcal{C}_{\text{AB}}^{+}/\lambda^{2}>0$. Most
strikingly, it is possible to extract genuine entanglement for
$L/\sigma\in(1.5,2.2)$ since $\mathcal{C}_{\text{AB}}^{+}>0$ while
$\mathcal{C}_{\text{AB}}^{-}=0$ in this region. However, this is not always
true as one can see from figure 7(b) when $\Omega\sigma=2$. Here, detectors in
circular motion can encounter the case where $\mathcal{C}_{\text{AB}}^{+}=0$
while $\mathcal{C}_{\text{AB}}^{-}>0$, which indicates that the generated
entanglement after the interaction is purely coming from the communication and
not from the field.

However genuine entanglement can still be extracted in the linear
configuration.

##  V Conclusion

We carried out the correlation harvesting protocol using two uniformly
accelerating Unruh-DeWitt (UDW) detectors in $(3+1)$-dimensional Minkowski
spacetime. According to Letaw [13], trajectories with constant (nonzero)
acceleration can be characterized by the magnitude of the acceleration and two
torsion parameters, resulting in five classes: linear, catenary, cusped,
circular, and helix motions. The first four of these classes of motion are
confined to a two-dimensional spatial surface and can be regarded as specific
cases of the helix motion. Since two-dimensional configurations are more
amenable to experimental setups, we employed these four simpler motions for
our analysis.

We first examined the transition probability of a single detector following
the four trajectories in section IV.1. Utilizing a unified expression for the
Wightman functions along these trajectories, we were able to explore the
general characteristics that are common to all these motions. We found that
the transition probabilities of these motions monotonically increase with the
magnitude of acceleration. Moreover, we also evaluated the effective
temperature—an estimator for the temperature as observed by a detector.

We then introduced another UDW detector to consider the correlation harvesting
protocol in Sec. IV.2. Two configurations were explored: stationary and
nonstationary configurations. In the stationary configuration, detectors are
separated in the direction perpendicular to their two-dimensional spatial
planes of motion. Specifically, the displacement vector pointing from one
detector to the other remains orthogonal to the velocity vectors of the
detectors. In such a case, the Wightman function along the stationary
configuration is time-translation invariant. On the other hand, in the
nonstationary configuration, the displacement vector aligns parallel to the
planes of motion. This makes the Wightman function nonstationary (i.e., not
time-translation invariant). Moreover, while this Wightman function shares a
common term with the stationary configuration, an additional term appears that
specifically characterizes the nonstationary nature of this configuration.

We found that the harvested correlations—entanglement and total
correlations—behave in a distinct manner depending on the motion of the
detectors. Specifically, detectors in linear, catenary, and cusped motions
within the nonstationary configuration gain fewer correlations compared to
those in the stationary configuration. On the other hand, in the circular
motion case, both configurations exhibit similar behavior. This difference can
be attributed to the Wightman functions. For linear, catenary, and cusped
motions, the Wightman function contain hyperbolic functions, leading to an
exponential alteration of the results. In contrast, the Wightman function for
circular motion is governed by trigonometric functions.

We also looked into the acceleration dependence of the harvested correlations
and concluded (not surprisingly) that high accelerations prevent the detectors
from acquiring correlations from the field. This point is consistent with
previous papers [47, 8, 48, 25, 26, 27, 28], in which linearly and circularly
accelerated detectors are considered. Our paper generalized these results to
any uniformly accelerating detectors on two-dimensional spatial surfaces.

Finally, we focused on entanglement harvested by the detectors in the
stationary configuration and asked how much of entanglement is coming from the
correlations preexisted in the field. To be precise, the entanglement coming
from the commutator part of the Wightman function is state-independent, which
suggests that the detectors can still be correlated even if the field is not
entangled [54]. Thus, it is important to examine how the anticommutator part
of the Wightman function (which is state-dependent) contributes to the
extracted correlations. One way to eliminate the commutator contribution is to
use causally disconnected detectors. However, we found that the existence of
acceleration prohibits us to extract correlations with detectors separated far
away, no matter what the energy gap is. However we also found the striking
result that detectors in causal contact can harvest genuine entanglement in
certain parameter regimes.

Our results have important implications for experiment. Attempts to realize
the Unruh effect and correlation harvesting generally rely on using laser
pulses to probe what are effectively two dimensional surfaces. To probe the
effects of non-inertial motion on mutual information and entanglement will
therefore involve two detectors (two pulses) in nonstationary configurations,
since only these can be realized in a two-dimensional setting. Experimental
verification of the harvesting of genuine entanglement would be an exciting
confirmation of our understanding of relativistic quantum information.

## Acknowledgments

This work was supported in part by the Natural Sciences and Engineering
Research Council of Canada. KGY is thankful to Dr. Jorma Louko for elaborating
on the relationship among the uniformly accelerating trajectories.

##  Appendix A Effective temperature

Here, we review the concept of effective temperature $T_{\text{eff}}$ and
clarify its relation to the KMS temperature.

Let us first review the KMS temperature. In quantum theory with separable
Hilbert spaces, a trace of an operator, $\operatorname{Tr}[\cdot]$ is well
defined. This enables us to consider the Gibbs state at the inverse
temperature $\beta$, $\rho=e^{-\beta\hat{H}}/Z$, where
$Z\coloneqq\operatorname{Tr}[e^{-\beta\hat{H}}]$ is the partition function.
This is what we consider a thermal state of a system.

However in QFT, a trace is generally not well defined. Instead, we identify
the Kubo-Martin-Schwinger (KMS) state [63, 64] as a thermal state in QFT.
Specifically, if the field is in the KMS thermal state with respect to time
$\tau$ at the inverse KMS temperature $\beta_{\text{KMS}}$, the Wightman
function satisfies

$\displaystyle W(\Delta\tau-\mathsf{i}\beta_{\text{KMS}})$
$\displaystyle=W(-\Delta\tau)\,,$ (55)

where $\Delta\tau\coloneqq\tau-\tau^{\prime}$. The Fourier transform of this
equality with respect to $\Delta\tau$ reads

$\displaystyle\tilde{W}(-\omega)$
$\displaystyle=e^{\beta_{\text{KMS}}\omega}\tilde{W}(\omega)\,.$ (56)

This equality in the Fourier domain is known as the detailed balance
condition. Thus, the thermality of a quantum field is imprinted in these
equalities.

The thermality can also be implemented in the transition probability of a UDW
detector. Recall that the transition probability is written as

$\displaystyle\mathcal{L}$
$\displaystyle=\lambda^{2}\int_{\mathbb{R}}\text{d}\tau\int_{\mathbb{R}}\text{d}\tau^{\prime}\,\chi(\tau)\chi(\tau^{\prime})e^{-\mathsf{i}\Omega(\tau-\tau^{\prime})}W\big{(}\mathsf{x}(\tau),\mathsf{x}(\tau^{\prime})\big{)}\,,$
(57)

where the subscript in $\mathcal{L}_{jj},\,j\in\\{\text{A},\text{B}\\}$ is
omitted for simplicity. Let us assume that the switching function,
$\chi(\tau)$, has a characteristic time length $\sigma$. In our paper, this is
the Gaussian width in $\chi(\tau)=e^{-\tau^{2}/2\sigma^{2}}$. It is convenient
to introduce a quantity related to the transition probability known as the
response function (divided by the characteristic time length)
$\mathcal{F}(\Omega,\sigma)$:

$\displaystyle\mathcal{L}=\lambda^{2}\sigma\mathcal{F}(\Omega,\sigma)\,,$
$\displaystyle\mathcal{F}(\Omega,\sigma)\coloneqq$ $\displaystyle\hskip
14.22636pt\dfrac{1}{\sigma}\int_{\mathbb{R}}\text{d}\tau\int_{\mathbb{R}}\text{d}\tau^{\prime}\,\chi(\tau)\chi(\tau^{\prime})e^{-\mathsf{i}\Omega(\tau-\tau^{\prime})}W\big{(}\mathsf{x}(\tau),\mathsf{x}(\tau^{\prime})\big{)}\,.$
(58)

If the field is in the KMS state and the switching function is a rapidly
decreasing function such as a Gaussian function, then the response function in
the long interaction limit obeys the detailed balance relation [62]:

$\displaystyle\lim_{\sigma\to\infty}\dfrac{\mathcal{F}(-\Omega,\sigma)}{\mathcal{F}(\Omega,\sigma)}=e^{\beta_{\text{KMS}}\Omega}\,.$
(59)

Note that this relation holds when the long interaction limit is taken. On the
other hand, if $\sigma$ is not sufficiently long, the ratio of the response
function (sometimes known as the excited-to-deexcited ratio) does not satisfy
the detailed balance condition.

From this relation, one can define the effective temperature as

$\displaystyle
T_{\text{eff}}^{-1}\coloneqq\dfrac{1}{\Omega}\ln\dfrac{\mathcal{F}(-\Omega,\sigma)}{\mathcal{F}(\Omega,\sigma)}\,.$
(60)

Note that the effective temperature is not necessarily the KMS temperature. If
the field is in the KMS state and the long interaction limit is taken, then
the effective temperature becomes the KMS temperature. In this sense, the
effective temperature is an estimator for the field’s temperature.

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