# Nonlocal coherence harvesting from quantum vacuum

Rui-Di Wang, Shu-Min<EMAIL_ADDRESS>(corresponding author), Xiao-Li
Huang222<EMAIL_ADDRESS>Department of Physics, Liaoning Normal
University, Dalian 116029, China

###### Abstract

It is well known that nonlocal coherence reflects nonclassical correlations
better than quantum entanglement. Here, we analyze nonlocal coherence
harvesting from the quantum vacuum to particle detectors adiabatically
interacting with a quantum scalar field in Minkowski spacetime. We find that
the harvesting-achievable separation range of nonlocal coherence is larger
than that of quantum entanglement. As the energy gap grows sufficiently large,
the detectors harvest less quantum coherence, while the detectors could
extract more quantum entanglement from the vacuum state. Compared with the
linear configuration and the scalene configuration, the equilateral triangle
configuration is the best model to harvest tripartite coherence. Finally, we
find a monogamous relationship, which means that tripartite $l_{1}$-norm of
coherence is essentially bipartite types.

###### pacs:

04.70.Dy, 03.65.Ud,04.62.+v

##  I Introduction

Quantum coherence arises from the superposition of quantum states and is a
basic characteristic of quantum physics L1 . Generating and maintaining
quantum coherence is one of the necessary prerequisites for quantum
information processing tasks L2 . Despite its importance, quantum coherence
received increasing attention until Baumgratz $et$ $al.$ successfully applied
the information-theoretic quantification of quantum coherence and introduced
the measures based on the $l_{1}$-norm and the relative entropy of all the
potential metrics L3 ; L47 . Similar to quantum entanglement, quantum
coherence is a significant quantum resource that plays a critical role in the
fields of quantum biology, quantum metrology, quantum thermodynamic, and
quantum computation L4 ; L5 ; L6 ; L7 ; L8 ; L9 ; L10 ; L11 ; L12 ; L13 .
Particularly, the phenomena of quantum entanglement can be attributed to the
nonlocal superposition of quantum states. It was confirmed that quantum
coherence in a quantum system could be converted to quantum entanglement
without providing further quantum coherence, meaning that quantum entanglement
monotone can induce quantum coherence monotone of quantum state L14 . Although
the relationship between quantum coherence and entanglement has been further
studied, it is still an open question worth investigating.

It has long been recognized that the vacuum states of a free quantum field
theory in Minkowski space are highly entangled across spacelike regions L15 .
Employing algebraic methods, Summers and Werner proved that correlations
between field observables across spacelike regions can maximally violate Bell
inequality L16 ; L17 ; L18 ; L19 . Subsequently, it was realized that this
vacuum entanglement could be “harvested” via particle detectors, which are
linearly coupled to the quantum field L20 ; L21 ; L22 . Two-level particle
detectors, called Unruh-DeWitt (UDW) detectors L23 ; L24 ; L25 , are like a
particle in a “box” interacting with the field when the “box” opens. These
detectors passing through the quantum field will be excited, showing that the
vacuum state can be considered as a quantum resource for quantum entanglement
and has been tested in a wide range of scenarios L26 ; L27 ; L28 ; L29 ; L30 ;
L31 ; L32 ; L33 ; L34 ; L35 . In addition, entanglement harvesting can be used
to detect the influences of nontrivial spacetime structure on vacuum
entanglement, such as spacetime curvature, nontrivial spacetime topology,
Unruh effect, Hawking effect, and cosmological effects L36 ; L37 ; L38 ; L39 ;
L40 ; L41 ; L42 ; L42-3 ; L42-4 ; L42-5 ; L42-7 ; L42-8 ; L42-9 ; L42-10 ;
L42-11 ; L42-12 ; L42-13 ; L42-15 . Because the Minkowski vacuum is an
interesting nonclassical correlation and many of its properties remain
unclear, it is worth further exploration.

The research so far has mainly focused on entanglement harvesting of UDW
detectors. However, nonlocal coherence is a better proxy for nonclassical
correlation than quantum entanglement. Therefore, in this paper, we study
nonlocal coherence harvesting protocol with two UDW detectors adiabatically
interacting with a quantum scalar field in $(3+1)$-dimensional Minkowski
spacetime. Compared with entanglement harvesting, we will obtain the novel
properties of nonlocal coherence and the differences between them. We also
study nonlocal coherence of three detectors, considering three configurations
of equilateral triangular, linear, and scalene triangular for three detectors,
respectively. We will try to explore the relationship between bipartite
coherence and tripartite coherence of three detectors.

The structure of the paper is as follows. In Sec. II, we briefly introduce the
UDW model. In Sec. III, we study nonlocal coherence of two UDW detectors. In
Sec. IV, we extend the relevant research to three detectors. The last section
is devoted to a brief conclusion.

##  II The Unruh-DeWitt model

Without losing generality, the UDW model of a detector $D$ with the ground
state $|0\rangle_{D}$ and the excited state $|1\rangle_{D}$ is regarded as a
two-level quantum system. We consider two detectors labeled $A$ and $B$ that
interact locally with the massless quantum scalar field $\phi[x_{D}(\tau)]$.
Then the interaction Hamiltonian between the detectors and the field can be
expressed as

$\displaystyle
H_{D}(\tau)=\lambda\chi(\tau)[e^{i\Omega\tau}\sigma^{+}+e^{-i\Omega\tau}\sigma^{-}]\phi[x_{D}(\tau)],\quad
D\in\\{A,B\\}.$ (1)

Here, $\Omega_{D}$ is the energy gap of the detector, the proper time $\tau$
parameterizes the classical spacetime trajectory of the detector, $\lambda$ is
the coupling strength, $\sigma^{+}=|1_{D}\rangle\langle 0_{D}|$ and
$\sigma^{-}=|0_{D}\rangle\langle 1_{D}|$ denote the ladder operators of the
SU(2) algebra, $\chi(\tau)$ is the switching function, and $\phi[x_{D}(\tau)]$
is the pullback of the field operator on detector $D$’s trajectory.

For simplicity, we assume that the coupling constants are all identical, and
$\lambda\ll 1$ means weak coupling. In the process of interaction, the unitary
operator $U$ is a description of the time evolution of the detectors and field
and is generated by the interaction Hamiltonian in Eq.(1) as follows

$\displaystyle U:$ $\displaystyle=$
$\displaystyle\mathcal{T}\exp\left(-i\int_{\mathbb{R}}d\tau\
H_{D}(\tau)\right)\ =1+(-i\lambda)\int_{\mathbb{R}}d\tau\ H_{D}(\tau)$ (2)
$\displaystyle+\frac{(-i\lambda)^{2}}{2}\int_{\mathbb{R}}d\tau\int_{\mathbb{R}}d\tau^{\prime}\mathcal{T}H_{D}(\tau)H_{D}(\tau^{\prime})+\mathcal{O}(\lambda^{3}).$

Initially, the two detectors are prepared (as $\tau\rightarrow-\infty$) in the
ground state $|0_{A}0_{B}\rangle$, and the field is in a vacuum state
$|0_{M}\rangle$. The initial density matrix is thus
$\rho_{0}=|0_{A}0_{B}\rangle\langle 0_{A}0_{B}|\otimes|0_{M}\rangle\langle
0_{M}|$. With the interaction Hamiltonian, the final state of the system is
found to be

$\displaystyle\rho_{AB}={\rm Tr}_{\phi}[U(|0_{A}0_{B}\rangle\langle
0_{A}0_{B}|\otimes|0_{M}\rangle\langle 0_{M}|)U^{{\dagger}}].$ (3)

By performing the standard perturbation theory L35 ; L37 , the final state of
the system can be rewritten as a density matrix in the basis
$\\{|0_{A}\rangle\langle 0_{B}|,|0_{A}\rangle\langle
1_{B}|,|1_{A}\rangle\langle 0_{B}|,|1_{A}\rangle\langle 1_{B}|\\}$,

$\displaystyle\rho_{AB}=\left(\\!\\!\begin{array}[]{cccccccc}\rho_{11}&0&0&\rho_{14}\\\
0&\rho_{22}&\rho_{23}&0\\\ 0&\rho^{*}_{23}&\rho_{33}&0\\\
\rho^{*}_{14}&0&0&\rho_{44}\\\
\end{array}\\!\\!\right)=\left(\\!\\!\begin{array}[]{cccccccc}1-P_{A}-P_{B}&0&0&X\\\
0&P_{B}&C&0\\\ 0&C^{*}&P_{A}&0\\\ X^{*}&0&0&0\\\
\end{array}\\!\\!\right)+\mathcal{O}(\lambda^{4}),$ (12)

where

$\displaystyle P_{D}:=\lambda^{2}\int\int{d\tau
d\tau^{\prime}\chi(\tau)\chi(\tau^{\prime})e^{-i\Omega(\tau-\tau^{\prime})}W(x_{D}(t),x_{D}(t^{\prime}))}\
\quad D\in\\{A,B\\},$ (13) $\displaystyle C:=\lambda^{2}\int\int{d\tau
d\tau^{\prime}\chi(\tau)\chi(\tau^{\prime})e^{-i\Omega(\tau-\tau^{\prime})}W(x_{A}(t),x_{B}(t^{\prime}))},$
(14) $\displaystyle X:$ $\displaystyle=$
$\displaystyle-\lambda^{2}\int\int{d\tau
d\tau^{\prime}\chi(\tau)\chi(\tau^{\prime})e^{-i\Omega(\tau+\tau^{\prime})}}$
(15)
$\displaystyle[\theta(t^{\prime}-t)W(x_{A}(t),x_{B}(t^{\prime}))+\theta(t-t^{\prime})W(x_{B}(t^{\prime}),x_{A}(t))].$

Here, $W(x,x^{\prime})=\langle 0_{M}|\phi(x)\phi(x^{\prime})|0_{M}\rangle$ is
the vacuum Wightman function of the field, $\theta(t)$ represents the
Heaviside’s step function, $t$ is a common time, $P_{D}$ is called the
transition probability, and the quantities $C$ and $X$ characterize
nonclassical correlations.

In the following, to make the parameter space less complex, we make all
detectors identical, with the same energy gap $\Omega\sigma$ and switching
function $\chi(\tau)$. Thus, the two detectors will have the same transition
probabilities

$\displaystyle P_{A}=P_{B}=P.$ (16)

Especially, $\chi(\tau)=\exp[-\tau^{2}/(2\sigma^{2})]$ is the Gaussian
switching function, which controls the duration of interaction via parameter
$\sigma$. Therefore, the matrix elements can be specifically expressed as L451
; L45

$\displaystyle
P_{D}=\frac{\lambda^{2}}{4\pi}[e^{-\Omega^{2}\sigma^{2}}-\sqrt{\pi}\Omega\sigma{\rm
erfc}(\Omega\sigma)],$ (17) $\displaystyle
C=\frac{\lambda^{2}}{4\sqrt{\pi}}\frac{\sigma}{L}e^{-L^{2}/4\sigma^{2}}\bigg{[}{\rm
Im}\left(e^{i\Omega{L}}{\rm
erf}\left(i\frac{L}{2\sigma}+\Omega\sigma\right)\right)-\sin(\Omega{L})\bigg{]}\in\mathbb{R},$
(18) $\displaystyle
X=\frac{-i\lambda^{2}}{4\sqrt{\pi}}\frac{\sigma}{L}e^{-\sigma^{2}\Omega^{2}-L^{2}/4\sigma^{2}}{\rm
erfc}\left(i\frac{L}{2\sigma}\right),$ (19)

where

$\displaystyle{\rm erf}(x):=\frac{2}{\sqrt{\pi}}\int^{x}_{0}dte^{-t^{2}},$
$\displaystyle{\rm erfc}(x):=1-{\rm erf}(x).$

##  III Nonlocal coherence harvesting of two Unruh-DeWitt detectors

A quantum state is considered coherent for a complete set of states if it can
be represented as a nontrivial linear superposition of these states L2 . The
concept of quantum coherence comes directly from the superposition principle,
which is one of the conceptual pillars of quantum theory L46 . In this paper,
we examine nonlocal coherence harvesting for UDW detectors. Note that nonlocal
coherence does not exist in a single subsystem. We briefly introduce two
measures of quantum coherence: the $l_{1}$-norm of coherence and the relative
entropy of coherence (REC) L47 . In a reference basis
$\\{|i\rangle\\}_{i=1,...,n}$ of a $n$-dimensional system, the $l_{1}$-norm of
quantum coherence can be defined as the sum of the absolute values of all the
off-diagonal elements of the system density matrix $\rho$,

$\displaystyle C_{l_{1}}(\rho)=\sum_{i\neq j}|{\rho}_{i,j}|,$ (20)

and the measure of the REC can be written as

$\displaystyle C_{REC}(\rho)=S(\rho_{diag})-S(\rho),$ (21)

where $S(\rho)$ indicates the von Neumann entropy of quantum state $\rho$, and
$\rho_{diag}$ is the state obtained from $\rho$ by removing all off-diagonal
elements L47 .

Employing Eqs.(12) and (20), we can obtain the $l_{1}$-norm of quantum
coherence as

$\displaystyle C_{l_{1}}(\rho_{AB})=2|C|+2|X|+\mathcal{O}(\lambda^{4}).$ (22)

Next, we study the REC from the quantum vacuum. For this purpose, we need to
calculate the eigenvalues of the density matrix $\rho_{AB}$. The density
matrix of Eq.(12) has four nonzero eigenvalues

$\displaystyle\lambda_{1}=-C+P,$ $\displaystyle\lambda_{2}=C+P,$
$\displaystyle\lambda_{3}=\frac{1}{2}(1-2P-\sqrt{1-4P+4P^{2}+4|X|^{2}}),$
$\displaystyle\lambda_{4}=\frac{1}{2}(1-2P+\sqrt{1-4P+4P^{2}+4|X|^{2}}).$

Thus, the REC of state $\rho_{AB}$ becomes

$\displaystyle
C_{REC}(\rho_{AB})=-\sum_{i}\alpha_{i}\log_{2}\alpha_{i}+\sum_{j=1}^{4}\lambda_{j}\log_{2}\lambda_{j}+\mathcal{O}(\lambda^{4}),$
(23)

where $\alpha_{i}$ are the diagonal elements of $\rho_{AB}$ and $\lambda_{j}$
are nonzero eigenvalues of quantum state $\rho_{AB}$. Here, we emphasize that
quantum coherence between two UDW detectors is nonlocal and does not exist in
any subsystem.

In order to compare nonlocal coherence and quantum entanglement, we use the
negativity to calculate quantum entanglement. Quantum entanglement between $A$
and $B$ in $\rho_{AB}$ is measured by the negativity $N(\rho_{AB})$, which can
be defined as L48

$\displaystyle N(\rho_{AB})=\frac{\parallel\rho_{AB}^{T_{A}}\parallel-1}{2},$
(24)

where $T_{A}$ denotes the partial transpose of $\rho_{AB}$ concerning
subsystem $A$. Here, $\parallel\cdot\parallel$ is the trace norm of a matrix,
and $\parallel\rho_{AB}^{T_{A}}\parallel-1$ equals two times of the sum of
absolute values of negative eigenvalues of the matrix $\rho_{AB}^{T_{A}}$. By
using Eqs.(12) and (24), we can obtain the negativity $N(\rho_{AB})$ as

$\displaystyle N(\rho_{AB})=\max[0,|X|-P]+\mathcal{O}(\lambda^{4}).$ (25)

So, the negativity $N(\rho_{AB})$ is a result of the competition between off-
diagonal matrix element $X$ and transition probabilities $P$, but it does not
depend on the correlation term $C$. From Eqs.(22) and (23), we can see that
$l_{1}$-norm of quantum coherence just depends on elements $C$ and $X$, while
the REC depends on not only $C$ and $X$, but also $P$. Through the analysis of
their analytic expressions, the nonlocal coherence can reflect the
nonclassical correlation better than quantum entanglement.

Figure 1: Quantum coherence and entanglement of two UDW detectors as a
function of the detector separation $L/\sigma$ for various values of the
energy gap $\Omega\sigma$ with fixed $\lambda=0.1$.

In Fig.1, we plot quantum coherence (nonlocal coherence) and entanglement as a
function of the detector separation $L/\sigma$ for different values of the
energy gap $\Omega\sigma$. We find that quantum coherence first decreases and
then tends to zero with the increase of the detector separation $L/\sigma$,
meaning that the smaller detector separation is more conducive to nonlocal
coherence harvesting of two UDW detectors. We also find that, for a fixed
$L/\sigma$, the smaller energy gap $\Omega\sigma$ can harvest the larger
quantum coherence, and nonlocal coherence harvesting for any energy gap
$\Omega\sigma$ has a similar trend. However, from Fig.1 (c) we can see that
quantum entanglement first monotonically decreases and then suffers “sudden
death” with the increase of the detector separation $L/\sigma$. This means
that the conditions for quantum entanglement harvesting are more demanding
than nonlocal coherence harvesting. In other words, the harvesting-achievable
separation range of nonlocal coherence is larger than that of quantum
entanglement. We can also see that the variation of nonlocal coherence with
the energy gap $\Omega\sigma$ is monotonous. Unlike nonlocal coherence,
quantum entanglement is not monotonous with the energy gap $\Omega\sigma$,
meaning that the smaller energy gap $\Omega\sigma$ is conducive to nonlocal
coherence harvesting and may be detrimental to quantum entanglement
harvesting.

##  IV Nonlocal coherence harvesting of three Unruh-DeWitt detectors

In the previous section, we studied nonlocal coherence of two UDW detectors
from the quantum vacuum. To investigate the relationship between tripartite
coherence and bipartite coherence, we will consider the case of three UDW
detectors L49 . Thus, using a similar method of two detectors, we can get the
final state of the system of three detectors as

$\displaystyle\rho_{ABC}=\left(\\!\\!\begin{array}[]{cccccccc}1-(P_{A}+P_{B}+P_{C})&0&0&0&X_{BC}&X_{AC}&X_{AB}&0\\\
0&P_{C}&C_{BC}&C_{AC}&0&0&0&0\\\ 0&C_{BC}^{*}&P_{B}&C_{AB}&0&0&0&0\\\
0&C_{AC}^{*}&C_{AB}^{*}&P_{A}&0&0&0&0\\\ X_{BC}^{*}&0&0&0&0&0&0&0\\\
X_{AC}^{*}&0&0&0&0&0&0&0\\\ X_{AB}^{*}&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\
\end{array}\\!\\!\right)+\mathcal{O}(\lambda^{4}).\ $ (34)

Here, the basis vectors of the density matrix $\rho_{ABC}$ are
$|0_{A}0_{B}0_{C}\rangle$, $|0_{A}0_{B}1_{C}\rangle$,
$|0_{A}1_{B}0_{C}\rangle$, $|1_{A}0_{B}0_{C}\rangle$,
$|0_{A}1_{B}1_{C}\rangle$, $|1_{A}0_{B}1_{C}\rangle$,
$|1_{A}1_{B}0_{C}\rangle$, $|1_{A}1_{B}1_{C}\rangle$. We will take into
account three types of detector configurations, which are shown in Fig.2. In
the first configuration, we place the three detectors on each of the three
vertices of an equilateral triangle and investigate how quantum entanglement
varies with the detector separation $L/\sigma$. The second configuration is
that the detectors are placed in a line of total length 2$L$, and the center
detector is the same distance from the other two detectors, which changes the
separation of the end detectors from the center one. Lastly, we consider a
scalene triangular configuration. Detector $B$ is moved along the line
parallel to the line connecting $A$ and $C$. The distance moved is denoted by
$D$, where $A$ and $C$ remain at a fixed distance L26 .

Figure 2: Detector configurations we consider for the system of three
detectors: (a) equilateral triangular, (b) linear, and (c) scalene triangular.
(c) indicates the space disposition corresponding to (a) when $t=0$.

###  IV.1 Equilateral triangle

For the equilateral triangular arrangement, the system is viewed as symmetric.
All distances between the identical detectors are the same. So we obtain the
conditions as

$\displaystyle C_{BC}=C_{AC}=C_{AB}\equiv C,\quad X_{BC}=X_{AC}=X_{AB}\equiv
X,$

and then the density matrix of Eq.(34) can be rewritten as

$\displaystyle\rho_{ABC}=\left(\\!\\!\begin{array}[]{cccccccc}1-3P&0&0&0&X&X&X&0\\\
0&P&C&C&0&0&0&0\\\ 0&C&P&C&0&0&0&0\\\ 0&C&C&P&0&0&0&0\\\
X^{*}&0&0&0&0&0&0&0\\\ X^{*}&0&0&0&0&0&0&0\\\ X^{*}&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0\\\ \end{array}\\!\\!\right)+\mathcal{O}(\lambda^{4}).\ $ (43)

Using Eqs.(20) and (43), we can get the $l_{1}$-norm of quantum coherence of
three detectors as

$\displaystyle C_{l_{1}}(\rho_{ABC})=6|C|+6|X|+\mathcal{O}(\lambda^{4}),$ (44)

and according to Eqs.(21) and (43), the REC becomes

$\displaystyle
C_{REC}(\rho_{ABC})=-\sum_{i}\alpha_{i}\log_{2}\alpha_{i}+\sum_{j=5}^{9}\lambda_{j}\log_{2}\lambda_{j}+\mathcal{O}(\lambda^{4}),$
(45)

where $\alpha_{i}$ are the diagonal elements of $\rho_{ABC}$ of Eq.(43), and
$\lambda_{j}$ are nonzero eigenvalues of quantum state $\rho_{ABC}$,

$\displaystyle\lambda_{5}=\lambda_{6}=-C+P,$ $\displaystyle\lambda_{7}=2C+P,$
$\displaystyle\lambda_{8}=\frac{1}{2}(1-3P-\sqrt{1-6P+9P^{2}+12|X|^{2}}),$
$\displaystyle\lambda_{9}=\frac{1}{2}(1-3P-\sqrt{1-6P+9P^{2}+12|X|^{2}}).$

Note that quantum coherence of three UDW detectors is nonlocal and does not
exist in any subsystem. Similar to the approach in the case of two detectors,
we calculate the negativity of three detectors. The tripartite negativity of
the three-qubit state $\rho_{ABC}$ is expressed as L50

$\displaystyle N(\rho_{ABC})$
$\displaystyle=(N_{A-BC}N_{B-AC}N_{C-AB})^{\frac{1}{3}}$ (46)
$\displaystyle=\max[0,\frac{1}{2}\sqrt{C^{2}+|X|^{2}}-\frac{1}{2}C-P]+\mathcal{O}(\lambda^{4}).$

Figure 3: Tripartite coherence and entanglement of three detectors in the
equilateral triangular configuration as a function of the detector separation
$L/\sigma$ for various values of the energy gap $\Omega\sigma$ with fixed
$\lambda=0.1$.

In Fig.3, we plot tripartite coherence and entanglement as a function of the
detector separation $L/\sigma$ for different $\Omega\sigma$ in the case of the
equilateral triangular arrangement. From Fig.3 (a) and (b), we see that
tripartite coherence harvesting first monotonically reduces and then
approaches zero with the growth of the $L/\sigma$. Tripartite coherence
harvesting is larger for the smaller $\Omega\sigma$ with fixed $L/\sigma$.
From Fig.3 (c), we also see that tripartite entanglement first monotonically
reduces and then suffers “sudden death” with increasing the $L/\sigma$.
Tripartite coherence decreases with growing the $\Omega\sigma$, while
tripartite entanglement may increase with increasing the $\Omega\sigma$.
Therefore, with the increase of the $L/\sigma$, tripartite entanglement may
suffer “sudden death” more prematurely for the smaller $\Omega\sigma$.

###  IV.2 Linear

For the simple linear configuration, the system is seen as a straight line and
detectors are placed at equal intervals. In this situation, $P_{D}$ will stay
the same. But the matrix elements $X$ and $C$ will change. They are

$\displaystyle X_{BC}=X_{AB}\equiv X_{L},\quad C_{BC}=C_{AB}\equiv C_{L},$

and

$\displaystyle X_{AC}\equiv X_{2L},\quad C_{AC}\equiv C_{2L}.$

The density matrix of Eq. (34) has the form

$\displaystyle\rho_{ABC}=\left(\\!\\!\begin{array}[]{cccccccc}1-3P&0&0&0&X_{L}&X_{2L}&X_{L}&0\\\
0&P&C_{L}&C_{2L}&0&0&0&0\\\ 0&C_{L}&P&C_{L}&0&0&0&0\\\
0&C_{2L}&C_{L}&P&0&0&0&0\\\ X_{L}^{*}&0&0&0&0&0&0&0\\\
X_{2L}^{*}&0&0&0&0&0&0&0\\\ X_{L}^{*}&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\
\end{array}\\!\\!\right)+\mathcal{O}(\lambda^{4}).$ (55)

Employing Eqs.(20), (21) and (55), the $l_{1}$-norm of quantum coherence and
the REC are found to be

$\displaystyle
C_{l_{1}}(\rho_{ABC})=4|C_{L}|+2|C_{2L}|+4|X_{L}|+2|X_{2L}|+\mathcal{O}(\lambda^{4}),$
(56) $\displaystyle
C_{REC}(\rho_{ABC})=-\sum_{i}\alpha_{i}\log_{2}\alpha_{i}+\sum_{j=10}^{14}\lambda_{j}\log_{2}\lambda_{j}+\mathcal{O}(\lambda^{4}),$
(57)

where $\alpha_{i}$ are the diagonal elements of $\rho_{ABC}$ of Eq.(55),
$\lambda_{j}$ are nonzero eigenvalues of quantum state $\rho_{ABC}$,

$\displaystyle\lambda_{10}=P-C_{2L},$
$\displaystyle\lambda_{11}=\frac{1}{2}(2P+C_{2L}-\sqrt{8C_{L}^{2}+C_{2L}^{2}}),$
$\displaystyle\lambda_{12}=\frac{1}{2}(2P+C_{2L}+\sqrt{8C_{L}^{2}+C_{2L}^{2}}),$
$\displaystyle\lambda_{13}=\frac{1}{2}(1-3P-\sqrt{1-6P+9P^{2}+8|X_{L}|^{2}+4|X_{2L}|^{2}}),$
$\displaystyle\lambda_{14}=\frac{1}{2}(1-3P+\sqrt{1-6P+9P^{2}+8|X_{L}|^{2}+4|X_{2L}|^{2}}).$

Fig.4 shows how the detector separation $L/\sigma$ influences tripartite
coherence for different energy gaps $\Omega\sigma$ under the linear
arrangement. Comparing Fig.3 and Fig.4, we find that nonlocal coherence
obtained in the equilateral triangle configuration is larger than in the
linear configuration under the same conditions.

Figure 4: Quantum coherence of three detectors in the linear configuration as
a function of the detector separation $L/\sigma$ for different energy gaps
$\Omega\sigma$ with fixed $\lambda=0.1$.

###  IV.3 Scalene triangle

The scalene triangle can be regarded as a promotion of the equilateral
triangle and the linear configuration. $P_{D}$ will remain identical. The
matrix elements $X$ and $C$ will change when the distance between each pair of
detectors is arbitrary. As a consequence, the density matrix of Eq.(34) can be
expressed as

$\displaystyle\rho_{ABC}=\left(\\!\\!\begin{array}[]{cccccccc}1-3P&0&0&0&X_{BC}&X_{AC}&X_{AB}&0\\\
0&P&C_{BC}&C_{AC}&0&0&0&0\\\ 0&C_{BC}^{*}&P&C_{AB}&0&0&0&0\\\
0&C_{AC}^{*}&C_{AB}^{*}&P&0&0&0&0\\\ X_{BC}^{*}&0&0&0&0&0&0&0\\\
X_{AC}^{*}&0&0&0&0&0&0&0\\\ X_{AB}^{*}&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0\\\
\end{array}\\!\\!\right)+\mathcal{O}(\lambda^{4}).$ (66)

Employing Eqs.(20) and (66), we obtain the $l_{1}$-norm of quantum coherence
as

$\displaystyle
C_{l_{1}}(\rho_{ABC})=2|X_{BC}|+2|X_{AC}|+2|X_{AB}|+2|C_{BC}|+2|C_{AC}|+2|C_{AB}|+\mathcal{O}(\lambda^{4}).$
(67)

Since the analytic expression of the REC is complex, we cannot write it down
specifically.

Figure 5: Quantum coherence of three detectors in the scalene configuration as
a function of the detector separation $D/\sigma$ for different energy gaps
$\Omega\sigma$ with $L_{AC}=7\sigma$. The coupling constant is set to
$\lambda=0.1$.

In Fig.5, we plot nonlocal coherence of three UDW detectors of the scalene
triangle arrangement as a function of the detector separation $D/\sigma$ for
different energy gaps $\Omega\sigma$. It is shown that quantum coherence is a
monotonically decreasing function of the separation $D/\sigma$. Combined with
Fig.3-5, we find that the equilateral triangle configuration is the best model
to obtain tripartite coherence from the quantum vacuum.

Through calculation, for any type of detector configuration, we obtain the
monogamy of the $l_{1}$-norm of quantum coherence as

$\displaystyle
C_{l_{1}}(\rho_{AB})+C_{l_{1}}(\rho_{BC})+C_{l_{1}}(\rho_{AC})=C_{l_{1}}(\rho_{ABC}).$
(68)

From Eq.(68), we find that the $l_{1}$-norm of coherence of three detectors is
equal to the sum of the coherence of all two detectors. This indicates that
the global tripartite $l_{1}$-norm of coherence cannot be harvested from the
quantum vacuum.

##  V Conclusion

We have studied nonlocal coherence harvesting of two or three Unruh-DeWitt
detectors from the quantum vacuum. It is shown that bipartite coherence
harvesting first decreases and then tends to zero as the detector separation
increases. However, bipartite entanglement harvesting first decreases and then
suffers “sudden death” with the growth of the detector separation. This
suggests that the harvesting-achievable separation range of nonlocal coherence
is infinite, while the harvesting-achievable separation range of quantum
entanglement is limited. We find that, with the decrease of the energy gap,
bipartite coherence harvesting increases monotonically, while quantum
entanglement harvesting is non-monotonically increasing. Therefore, the
results show that the smaller energy gap is beneficial for nonlocal coherence
harvesting but not necessarily helpful for quantum entanglement harvesting.

We have extended the investigation to three types of detector configurations:
equilateral triangular, linear, and scalene triangle. Besides the similar
properties observed in two Unruh-DeWitt detectors, some new characteristics
have been found in three Unruh-DeWitt detectors. Comparing nonlocal coherence
of three configurations, we find that the equilateral triangle configuration
is the best model for tripartite coherence harvesting from the quantum vacuum.
We obtain a monogamous relationship of tripartite $l_{1}$-norm of coherence,
which means that tripartite coherence equals the sum of all bipartite
coherence. In other words, there is no global tripartite $l_{1}$-norm of
coherence harvesting from the quantum vacuum.

###### Acknowledgements.

This work is supported by the National Natural Science Foundation of China
(Grant Nos. 12205133), LJKQZ20222315 and 2021BSL013.

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