# Entanglement-enhanced test proposal for local Lorentz-symmetry violation via
spinor atoms

Min Zhuang  College of Physics and Optoelectronic Engineering, Shenzhen
University, Shenzhen 518060, China  Jiahao Huang  Email:
<EMAIL_ADDRESS><EMAIL_ADDRESS>Guangdong Provincial Key
Laboratory of Quantum Metrology and Sensing $\&$ School of Physics and
Astronomy, Sun Yat-Sen University (Zhuhai Campus), Zhuhai 519082, China  State
Key Laboratory of Optoelectronic Materials and Technologies, Sun Yat-Sen
University (Guangzhou Campus), Guangzhou 510275, China  Chaohong Lee  Email:
<EMAIL_ADDRESS><EMAIL_ADDRESS>College of Physics and Optoelectronic
Engineering, Shenzhen University, Shenzhen 518060, China  Guangdong Provincial
Key Laboratory of Quantum Metrology and Sensing $\&$ School of Physics and
Astronomy, Sun Yat-Sen University (Zhuhai Campus), Zhuhai 519082, China  State
Key Laboratory of Optoelectronic Materials and Technologies, Sun Yat-Sen
University (Guangzhou Campus), Guangzhou 510275, China

###### Abstract

Invariance under Lorentz transformations is fundamental to both the standard
model and general relativity. Testing Lorentz-symmetry violation (LSV) via
atomic systems attracts extensive interests in both theory and experiment. In
several test proposals, the LSV violation effects are described as a local
interaction and the corresponding test precision can asymptotically reach the
Heisenberg limit via increasing quantum Fisher information (QFI), but the
limited resolution of collective observables prevents the detection of large
QFI. Here, we propose a multimode many-body quantum interferometry for testing
the LSV parameter $\kappa$ via an ensemble of spinor atoms. By employing an
$N$-atom multimode GHZ state, the test precision can attain the Heisenberg
limit $\Delta\kappa\propto 1/(F^{2}N)$ with the spin length $F$ and the atom
number $N$. We find a realistic observable (i.e. practical measurement
process) to achieve the ultimate precision and analyze the LSV test via an
experimentally accessible three-mode interferometry with Bose condensed
spin-$1$ atoms for example. By selecting suitable input states and unitary
recombination operation, the LSV parameter $\kappa$ can be extracted via
realizable population measurement. Especially, the measurement precision of
the LSV parameter $\kappa$ can beat the standard quantum limit and even
approach the Heisenberg limit via spin mixing dynamics or driving through
quantum phase transitions. Moreover,the scheme is robust against nonadiabatic
effect and detection noise. Our test scheme may open up a feasible way for a
drastic improvement of the LSV tests with atomic systems and provide an
alternative application of multi-particle entangled states.

##  1 Introduction

Local Lorentz-symmetry violation is one of the most fundamental principles of
modern physics and a key feature of the standard model [1, 2, 3, 4, 5]. It
means that the outcome of any local non-gravitational experiment is
independent of the velocity and the orientation of the apparatus. However,
some theories aim at unifying gravity with other fundamental interactions,
including string theory and field theory, imply the possibility or even
necessity for Lorentz-symmetry violation (LSV) [7, 8, 6, 9, 10, 11, 12, 13,
16, 14, 15, 17, 18]. The high-precision tests of LSV with matter, gravity, or
light have the potential to offer insight about new physics and set limits on
various theories. The effects of LSV can be classified in the framework of the
standard model extension (SME) [8, 10, 9, 19]. The SME is an effective field
theory that augments the standard model Lagrangian with all combination of the
standard model fields that are not term-by-term invariant under Lorentz
transformations, but maintain gauge invariance, energy-momentum conservation,
and Lorentz invariance of the total action.

Based on the SME, the LSV tests with atomic systems have attracted vast
attention [20]. The LSV tests with electronic states have conducted via
neutral Dy atoms [21], Ca+ ions [22] and a pair of two entangled trapped Yb+
ions [23]. Recently, it was proposed to test LSV via decoherence free subspace
(DFS) [23] or dynamical decoupling [24] to cancel out the influence of
magnetic field fluctuations. For an atom with fixed total angular momentum
$F$, the LSV in bound electronic states result in a small shift of the energy
levels [25, 21, 22, 23, 24, 27, 26], one energy level shift term is
proportional to the square of the $z$ component of the total angular momentum
[25, 24, 26], i.e., $\Delta E_{LSV}\propto\kappa\hat{s}_{z}^{2}$ with the LSV
parameter $\kappa$, which is analogous to quadratic Zeeman shift.

It is well known that quantum multi-particle entanglement can implement high-
precision measurement beyond the standard quantum limit (SQL) [28, 29, 30, 33,
31, 32, 34]. Recently, utilizing multi-particle entangled state to enhance
test precision of LSV has been proposed [26]. The study indicates that when
ensembles of entangled particles are employed [26], the QFI of LSV parameter
$\kappa$, an equivalent quadratic Zeeman shift can asymptotically approach the
Heisenberg limit [28, 29, 30, 33, 31, 32]. The QFI provides an ultimate bound
for the test precision of LSV via multipartite entanglement, but it is very
important to find a realistic observable or practical measurement process to
achieve the ultimate precision.

Bose-condensed spinor atoms involve multiple spin degrees of freedom [35], can
serve as an excellent candidate for implementing multimode quantum
interferometry to achieve high-precision measurement [35, 36]. In the presence
of magnetic field, the hyperfine states $|F,m_{F}\rangle$ are split separately
and then can act as multiple paths or multiple modes. Naturally, the following
questions arise: i) can one use multimode many-body quantum states to test the
LSV parameter $\kappa$ ? ii) what is the ultimate measurement precision bound?
and iii) is it possible to implement in a state-of-the-art experiment and find
a realistic observable?

In this paper, by considering an ensemble of Bose condensed spinor atoms, we
propose a multimode many-body quantum interferometry for estimating the LSV
parameter $\kappa$. We find that, if the atoms are prepared in a multimode GHZ
state, the measurement precision of $\kappa$ can attain the Heisenberg limit,
and it depends on the single-atom spin length $F$ according to
$\Delta\kappa\propto 1/(F^{2}N)$. Further, we take an experimentally
accessible three-mode interferometry with spin-$1$ Bose-condensed atoms for
example and show how to achieve the sub-SQL measurement precision via
population measurement without request of single-particle resolution [31, 37,
38, 39, 40, 41, 42, 43, 45, 44]. Our test scheme may open up a feasible way to
achieve high-precision estimation of the LSV parameter $\kappa$. Given the
broad interest in testing LSV and quantum-enhanced metrology, our study not
only may pave the way to test some theories, such as string theory [6], field
theory [7, 12, 9, 8], Einstein-aether theories [46] and quantum gravity [47],
but also provides an alternative application of multi-particle entangled
states [48]. Meanwhile, the energy shift resulted from LSV is similar to the
quadratic Zeeman shifts in atomic clocks [49, 50, 51] and atomic magnetometry
[52, 53, 54], thus our test proposal also can be used in practical quantum
sensors.

The paper is organized as follows. In Sec. 2, we introduce the scheme of
multimode interferometry via spin-$F$ Bose-condensed atoms for testing the LSV
parameter $\kappa$ . In Sec. 3, we discuss the ultimate precision bound for
the LSV parameter $\kappa$ via individual atoms as well as entangled atoms. In
particular, using a specific $N$-atom multimode GHZ state as an input state,
the measurement precision of the LSV parameter $\kappa$ can reach the
Heisenberg limit. In Sec. 4, as an example, we propose an experimentally
accessible three-path interferometric scheme via spin-$1$ Bose-condensed atoms
and show how to push the test precision beyond SQL via population measurement.
Finally, we give a brief summary in Sec. 5.

##  2 LSV test via quantum interferometry of spinor atoms

Invariance under Lorentz transformations is fundamental in modern physics. In
relativistic physics, Lorentz symmetry implies the laws of physics stay the
same for all observers moving at constant velocities with respect to each
other in inertial frames. It is also described as the outcome of any local
experiment is independent of the velocity and the orientation of the (freely
falling) apparatus. However, the Lorentz symmetry might be violated at
experimentally accessible energy scales due to spontaneous symmetry breaking
[14]. In addition, some theories that unify gravitation and the standard model
assert that Lorentz symmetry is valid only at large length scales [16]. For
atomic systems, the effect of various sources of LSV on atoms has been
discussed in Ref. [19]. Within the framework of SME, the QED Lagrangian (for
electron) for an electron sector with LSV can be written as [9, 10, 22]

$L=\frac{1}{2}i\bar{\psi}(\gamma_{\nu}+c_{\mu\nu}\gamma^{\mu})\stackrel{{\scriptstyle\leftrightarrow}}{{D}}\\!^{\nu}{\psi}-\bar{\psi}m_{e}{\psi},$
(1)

where $m_{e}$ is the electron mass, $\psi$ is a Dirac spinor, $\gamma^{\mu}$
are the Dirac matrices and
$\bar{\psi}\\!\stackrel{{\scriptstyle\leftrightarrow}}{{D}}^{\nu}\\!{\psi}\equiv\bar{\psi}{D}^{\nu}{\psi}-{\psi}{D}^{\nu}\bar{\psi}$,
with ${D}^{\nu}$ being the covariant derivative. The tensor $c_{\mu\nu}$ in
the above Eq. (1) quantifies the strength of LSV by the frame dependent
interaction term [7, 8, 9]. According to Eq. (1), the LSV in bound electronic
states results in a small energy level shift described by the Hamiltonian [25,
21, 22]:

$\delta
H_{LSV}=-(C_{0}^{(0)}-\frac{2U}{3c^{2}}c_{00})\frac{\emph{{p}}^{2}}{2}-\frac{1}{6}C_{0}^{(2)}T_{0}^{(2)}.$
(2)

Here, $U$ is the Newtonian gravitational potential, p is the momentum of a
bound electron, and $c$ is the speed of light. The more specific parameters
$C_{0}^{(0)}$, $c_{00}$ and $C_{0}^{(2)}$ contain elements of $c_{\mu\nu}$
tensor, thus they can quantify the strength of LSV [22]. We just consider the
second term of Eq. (2) which describe the dependence of electron’s energy on
the distribution of the total momentum among the three spatial components. The
non-relativistic form of the $T_{0}^{(2)}$ operator is
$T_{0}^{(2)}=(\emph{{p}}^{2}-3p_{z}^{2})/m_{e}$ with the electron mass $m_{e}$
and the component of the electron’s momentum $p_{z}$ along the quantization
axis fixed in the laboratory frame. Thus, for a bound electron system with
momentum p, the non-relativistic form of the second term of $\delta H_{LSV}$
reduces to [25, 21, 22, 23, 24, 27]

${\hat{h}}=-C_{0}^{(2)}\frac{\emph{{p}}^{2}-3p_{z}^{2}}{6m_{e}}.$ (3)

The Hamiltonian Eq. (3) is responsible for breaking the symmetry of
orientation or velocity in a bound electron system. It illustrates that the
energy shift depends on the distribution of total momentum p among the three
spatial components. The $c_{\mu\nu}$ tensor is frame-dependent, thus
$C_{0}^{(2)}$ varies in time as the Earth rotates, resulting in a time
variation of the electron’s energy correlated with the Earth’s motion.

Now, we consider an ensemble of spinor Bose condensed atoms with orbital
angular momentum $L=0$ and spin angular momentum $F$, thus the total angular
momentum of the spinor atom is $F$. In the presence of magnetic field, there
are $2F+1$ Zeeman sublevels denoted by $|{F,m_{F}}\rangle$, with
$m_{F}=-F,-F+1,...,F$ being the magnetic quantum number. According to the
Wigner-Eckart theorem, the matrix element of operator $T_{0}^{(2)}$ can be
expressed through the reduced matrix element $\langle F|T^{(2)}|F\rangle$ and
it is [26]

$\displaystyle\langle
F,m_{F}|\frac{\emph{{p}}^{2}\\!-\\!3p_{z}^{2}}{6m_{e}}|F,m_{F}\rangle$
$\displaystyle=\frac{[-F(F+1)\\!+\\!3m_{F}^{2}]\langle
F|T^{(2)}|F\rangle}{\sqrt{(2F+3)(F+1)(2F\\!+\\!1)(2F-1)}}.$

According to Eq. (2), the LSV signal contains a term proportional to
$m_{F}^{2}$, thus the LSV dynamics can be described by an equivalent
Hamiltonian $\hat{h}^{{L}}=\kappa\hat{s}_{z}^{2}$ with the magnetic angular
momentum operator $\hat{s}_{z}$ [24, 26]. The equivalent Hamiltonian is
analogous to a quadratic Zeeman shift. Thus, the LSV dynamics can be described
by the equivalent Hamiltonian $\hat{h}^{{L}}$, and the goal of placing a bound
on the parameter $C_{0}^{(2)}$ is equivalent to placing a bound on the
parameter $\kappa$.

In the following, we describe the LSV dynamics via the equivalent Hamiltonian
$\hat{h}^{{L}}$. For $N$ spin-$F$ atoms, the LSV Hamiltonian can be expressed
as

$\displaystyle\hat{H}^{{L}}_{F}=\sum\limits_{n=1}^{N}\hat{h}^{{L}[n]}=\kappa\sum\limits_{n=1}^{N}\hat{s}^{2[n]}_{z}=\kappa\hat{G},$
(5)

where $n$ denotes the $n$-th atom with $n=1,2,\ldots,N$. The procedures of
multimode quantum interferometer are as follows. In the initialization stage,
an $N$-particle input state $|{\Psi_{{in}}}\rangle$ involving $2F+1$ Zeeman
sublevels is prepared. Then, the multimode input probe state undergoes the LSV
dynamics and acquires a phase shift. The information of the LSV parameter
$\kappa$ is imprinted in the evolved output state
$|\Psi_{{{out}}}\rangle={e}^{-i\hat{H}^{{L}}_{F}T}|\Psi_{{in}}\rangle$ with
the evolution time $T$. In the readout stage, the unknown parameter $\kappa$
can be extracted by selecting suitable measurement on the output state.

##  3 Measurement precision bounds

In this section, we discuss the ultimate measurement precision bound for the
LSV parameter $\kappa$ via our multimode interferometer with different input
states. Without loss of generality, we assume the evolution time $T=1$ in our
calculation. In Sec. 3.1 and Sec. 3.2, we show the ultimate measurement
precision bound of $\kappa$ via individual and entangled Bose condensed
spin-$F$ atoms, respectively.

According to the parameter quantum estimation theory [60, 55, 56, 57, 58, 59],
the ultimate measurement precision of $\kappa$ is determined by the quantum
Cramér-Rao bound(QCRB) and it is

$\displaystyle\Delta\kappa\geq\Delta\kappa_{QCRB}\equiv\frac{1}{\sqrt{\eta
F_{Q}}}.$ (6)

Here, $\eta$ corresponds to the number of trials, $F_{Q}$ is the quantum
Fisher information(QFI) and it can be written as

$\displaystyle
F_{Q}=4(\langle\Psi^{\prime}|\Psi^{\prime}\rangle-|\langle\Psi^{\prime}|\Psi_{{out}}\rangle|^{2})=4\Delta^{2}\hat{G},$
(7)

where $|{\Psi^{\prime}}\rangle=d|\Psi_{{out}}\rangle/d\kappa$ denotes the
derivative of the output state respective to the parameter $\kappa$, and
$\Delta^{2}\hat{G}=\langle\Psi_{{in}}|\hat{G}^{2}|\Psi_{{in}}\rangle-(\langle\Psi_{{in}}|\hat{G}|\Psi_{{in}}\rangle)^{2}$
is the variance of $\hat{G}$.

###  3.1 Measurement precision bounds offered by individual atoms

We first consider individual atoms without any entanglement. Thus, the system
state can be described by a product state of $N$ atoms, which is written as
[36]

$|{\Psi}\rangle^{{Pro}}=\left(\sum\limits_{m_{F}=-F}^{F}\\!\\!\alpha_{m_{F}}|{F,m_{F}}\rangle\right)^{\otimes
N},$ (8)

where $\alpha_{m_{F}}$ is the complex amplitude and satisfies the
normalization condition:
$\sum\limits_{m_{F}=-F}^{F}\\!\\!|\alpha_{m_{F}}|^{2}=1$. Taking the $N$-atom
product state (8) as the input state, the ultimate bound of $\kappa$ is
${\Delta^{2}\kappa}_{QCRB}=\frac{1}{4N(M_{2}-M_{1}^{2})}$, with
$M_{1}=\sum_{{m_{F}}=-F}^{F}|\alpha_{{m_{F}}}|^{2}m_{F}^{2}$ and
$M_{2}=\sum_{{m_{F}}=-F}^{F}|\alpha_{{m_{F}}}|^{2}m_{F}^{4}$. It is obvious
that the QFI only scales linearly with $N$.

If the input state is in uniform distribution among the Zeeman sublevels [36],
the corresponding ultimate measurement precision bound for parameter $\kappa$
is ${\Delta\kappa}_{{QCRB}}=\sqrt{\frac{45}{4NF(1+F)(4F^{2}+4F-3)}}$. Further,
the optimal state for individual atoms can be obtained via minimizing the
ultimate measurement precision bound [36] and the corresponding ultimate
measurement precision bound is
${\Delta\kappa}_{{QCRB}}=\frac{1}{\sqrt{N}F^{2}}$.

###  3.2 Measurement precision bounds offered by entangled atoms

As we have known, entanglement is a useful quantum resource to improve the
measurement precision [59, 60, 61, 62, 63]. In the following, we show how the
entangled input states of $N$ atoms can enhance the measurement precision of
the LSV parameter $\kappa$. In particular, we consider a multimode GHZ state
as an input state and the form is [36]

$|{\Psi}\rangle^{{GHZ}}=\sum\limits_{{m_{F}}=-F}^{F}\alpha_{{m_{F}}}|{F,{m_{F}}}\rangle^{\otimes
N}.$ (9)

Substituting the input state $|{\Psi}\rangle^{{GHZ}}$ into Eq. (6), the
corresponding ultimate precision of $\kappa$ is
${\Delta^{2}{\kappa}}_{{QCRB}}=\frac{1}{4N^{2}(M_{2}-M_{1}^{2})}$. Compared
with individual atoms, the scaling of the ultimate measurement precision bound
versus total atom number $N$ changes from $1/\sqrt{N}$ to $1/N$. It is evident
that the GHZ’s form of $N$-atom state can improve the measurement precision of
the LSV parameter $\kappa$ to the Heisenberg limit. For the multimode GHZ
state, the optimal one reads as
$|{\Psi}\rangle_{{opt}}^{{GHZ}}=\frac{1}{2}|{F,F}\rangle^{\otimes
N}+\frac{1}{\sqrt{2}}|{F,0}\rangle^{\otimes
N}\\!+\frac{1}{2}|{F,-F}\rangle^{\otimes N}$ and the corresponding ultimate
measurement precision bound for parameter $\kappa$ is
${\Delta\kappa}_{{QCRB}}=\frac{1}{NF^{2}}$.

The ultimate measurement precision of the LSV parameter $\kappa$ decreases
monotonously as $F$ increases for individual or entangled atoms, thus one can
improve the measurement precision of $\kappa$ by using the atoms with larger
spin $F$ [36].

##  4 LSV test via three-mode interferometry with Bose-condensed spin-1 atoms

In experiments, how to prepare the desired input states for sensing is an
important problem. Despite multimode GHZ states can make measurement precision
attain the Heisenberg limit in theory, they are not easy to be generated in
experiments. Meanwhile, to perform the interferometry with GHZ states in
practice, parity measurement is required [28, 30]. However, parity measurement
requires single-particle resolution technique and limits the experimental
feasibility of quantum metrology with GHZ states. How to prepare an available
entangled state and beat the SQL without single-particle-resolved detection
become essential.

[width=]Fig3.pdf .

Figure 1: (color online). Schematic of three-mode interferometry. Firstly, a
suitable input state is prepared. Secondly, interrogation for accumulating
phase is implemented. Thirdly, an unitary operation $\hat{U}$ is applied for
recombination. Lastly, the population measurement $\hat{N}_{0}$ is applied to
extract the parameter $\kappa$

In this section, we discuss the test of LSV via three-mode interferometry with
spin-$1$ Bose-condensed atoms [64, 65, 66, 67]. For spin-$1$ Bose-condensed
atoms, the LSV dynamics can be written as

$\hat{H}_{F=1}^{L}=\kappa(\hat{N}_{-1}+\hat{N}_{1}).$ (10)

Here $\hat{N}_{m_{F}}=\hat{a}^{\dagger}_{m_{F}}\hat{a}_{m_{F}}$ denotes the
particle number operator of atoms in state $|{1,{m_{F}}}\rangle$, with
$\hat{a}^{\dagger}_{m_{F}}$ and $\hat{a}_{m_{F}}$ being the creation and
annihilation operators for state $|{1,{m_{F}}}\rangle$, respectively. For
convenience, we abbreviate the Hamiltonian of LSV dynamics $\hat{H}_{F=1}^{L}$
to $\hat{H}_{1}^{L}$, i.e.,
$\hat{H}_{1}^{L}=\kappa(\hat{N}_{-1}+\hat{N}_{1})$. Now, we illustrate the
procedures of three-mode interferometry in detail (see Fig. 1). Firstly, the
system is initialized in a desired input state $|{\Psi_{{in}}}\rangle$. Then,
the input state $|{\Psi_{{in}}}\rangle$ undergoes the LSV dynamics and a phase
dependent on $\kappa$ is accumulated . The output state
$|{\Psi_{{out}}}\rangle=e^{-i\hat{H}_{1}^{L}T}|{\Psi_{{in}}}\rangle$ contains
the information of the parameter $\kappa$. Finally, a suitable unitary
operation $\hat{U}$ is performed on the output state $|{\Psi_{{out}}}\rangle$
for recombination and the final state can be written as

$|{\Psi_{final}}\rangle=\hat{U}e^{-i\hat{H}_{1}^{L}T}|{\Psi_{{in}}}\rangle.$
(11)

After applying the population measurement $\hat{N}_{0}$ on the final state,
one can obtain the expectation and standard deviation of $\hat{N}_{0}$,

$\displaystyle\langle\hat{N}_{0}\rangle=\langle{\Psi_{{{final}}}}|\hat{N}_{0}|{\Psi_{{final}}}\rangle$
$\displaystyle\Delta{\hat{N}_{0}}\\!=\\!\sqrt{\langle{\Psi_{{final}}}|\hat{N}_{0}^{2}|{\Psi_{{final}}}\rangle\\!-\\!(\langle{\Psi_{{{final}}}}|\hat{N}_{0}|{\Psi_{{final}}}\rangle)^{2}}.$

According to the error propagation formula, the measurement precision of the
estimated parameter $\kappa$ read as

$\displaystyle\Delta\kappa=\frac{\Delta{\hat{N}_{0}}}{|\partial{\langle\hat{N}_{0}\rangle}/\partial{\kappa}|}.$
(14)

For spin-$1$ Bose condensed atoms, spin exchange collision can be used to
generate different kinds of input states. In the following, we study how to
realize the high-precision measurement for the LSV parameter $\kappa$ via
spin-mixing dynamics (SMD) and driving through quantum phase transitions
(QPTs), and study its measurement precision via population measurement.

###  4.1 LSV test via spin-mixing dynamics

[width=1]Fig4.pdf

Figure 2: (color online). The log-log scaling of $\Delta\kappa_{min}$ versus
the total atom number $N$ under SMD. (a) The circles show results for the
input state $|{\Psi_{{in}}}\rangle=e^{-i\hat{H}_{SMD}t}|{0,N,0}\rangle$; (b)
The circles show results for the input state
$|{\Psi_{{in}}}\rangle=(|{0,N,0}\rangle+|{N/2,0,N/2}\rangle)/\sqrt{2}$. Here,
$\kappa T\in[-0.1\pi,0.1\pi]$, $t\in(0,2\pi]$, $\chi=1$, $T=1$

The input state $|{\Psi_{{in}}}\rangle$ can be generated by time evolution
under spin-mixing dynamics [70, 71, 68, 69]. The governed Hamiltonian is
written as

$\hat{H}_{SMD}=\chi\left(\hat{a}^{\dagger}_{0}\hat{a}^{\dagger}_{0}\hat{a}_{1}\hat{a}_{-1}+\hat{a}_{0}\hat{a}_{0}\hat{a}^{\dagger}_{1}\hat{a}^{\dagger}_{-1}\right),$
(15)

where $\chi$ is the strength of spin exchange collision. Initially with all
$N$ atoms in state $|{1,{m_{F}=0}}\rangle$, i.e.,
$|{\Psi_{0}}\rangle=|{0,N,0}\rangle$. Then, the input state can be generated
via SMD,

$|{\Psi_{{in}}}\rangle=e^{-i\hat{H}_{SMD}t}|{\Psi_{0}}\rangle=e^{-i\hat{H}_{SMD}t}|{0,N,0}\rangle,$
(16)

where $t$ is the evolution time. For a fixed $\chi$, different evolution time
can result in different input state. Especially, the input state satisfies
$N_{1}=N_{-1}$, thus it is immune to stray magnetic fields and forms a DFS
[23, 26]. For a fixed total atom number $N$, one can numerically obtain all
the input state during the time evolution. Here, we choose
$\hat{U}=e^{i\hat{H}_{SMD}t}$ to realize the recombination, which can be
regarded as a nonlinear detection [40, 72, 73, 44, 41, 42, 43, 45, 74, 75].
The final state can be written as

$\displaystyle|{\Psi_{{final}}}\rangle$ $\displaystyle=$ $\displaystyle
e^{i\hat{H}_{SMD}t}e^{-i\hat{H}^{{L}}_{1}T}e^{-i\hat{H}_{SMD}t}|{\Psi_{0}}\rangle$
$\displaystyle=$ $\displaystyle
e^{i\hat{H}_{SMD}t}e^{-i\kappa(\hat{N}_{-1}+\hat{N}_{1})T}e^{-i\hat{H}_{SMD}t}|{\Psi_{0}}\rangle.$

Finally, one can extract the estimated parameter $\kappa$ via measuring the
population $\hat{N}_{0}$ on the final state.

According to Eq. (14), we obtain the corresponding $\Delta\kappa$ via
population measurement. Minimizing $\Delta\kappa$, we can obtain the optimal
input state generated via SMD. Although we cannot give the analytical form of
the optimal input state for a given $N$, we numerically confirm that SMD could
be an effective way for preparing the suitable states for estimation. To
confirm the dependence of measurement precision $\Delta\kappa$ on the total
particle number $N$, we numerically calculate the optimal measurement
precision versus particle number in the range of $t\in[0,2\pi]$ and $\kappa
T\in[-0.1\pi,0.1\pi]$, as shown in Fig. 2(a). According to the fitting
results, the optimal measurement precision is given by
$\textrm{{In}}(\Delta\kappa_{{min}}T)=-\textrm{In}(N)+0.033$.

Further, if the atoms are prepared in state
$|{\Psi_{{in}}}\rangle=\frac{1}{\sqrt{2}}\left(|{0,N,0}\rangle+|{N/2,0,N/2}\rangle\right)$,
the corresponding final state is
$|{\Psi_{{final}}}\rangle=e^{i\hat{H}_{SMD}t}e^{-i\hat{H}^{{L}}_{1}T}|{\Psi_{{in}}}\rangle$.
Similarly, we numerically calculate the optimal measurement precision versus
particle number in the range of $t\in[0,2\pi]$ and $\kappa
T\in[-0.1\pi,0.1\pi]$ , as shown in Fig. 2(b). According to the fitting
result, the optimal measurement precision is given by
$\textrm{{In}}(\Delta\kappa_{{min}}T)=-\textrm{{In}}(N)+0.048$. Our results
indicate that SMD could be an effective way for states preparation and beam
recombination for measuring the LSV parameter $\kappa$, the measurement
precision $\Delta\kappa$ can beyond SQL and exhibit the Heisenberg limit via
population measurement.

###  4.2 LSV test via driving through quantum phase transitions

[width=1]Fig5.pdf

Figure 3: (color online). The measurement precision via population
measurement. (a)The log-log scaling of $\Delta\kappa_{min}T$ versus the total
atom number $N$ under driving through QPTs. (b) Variation of measurement
precision versus $\kappa$ under $N=20$. Here, $\kappa T\in[-0.1\pi,0.1\pi]$,
$q_{0}=q_{f}=3$, $\beta=0.01$, $T=1$

For Bose-condensed spin-$1$ atoms, an alternative way for preparing entangled
input state is to drive the system through QPTs [64, 65, 66, 67]. Despite SMD
can generate the desired input state, it sensitively depends on the control of
evolution time and these states are always not steady. In contrast, driving
system through QPTs can deterministically generate the entangled states [30,
31, 32, 64, 65, 66, 67, 77, 76]. Recently, entanglement generation by driving
through QPTs has been realized in ${}^{87}Rb$ spinor condensate [65, 66, 67].
The evolution of the initial state $|{\Psi_{0}}\rangle=|{0,N,0}\rangle$ is
governed by the following Hamiltonian,

$\displaystyle\hat{H}_{{{QPT}}}=$
$\displaystyle\frac{c_{2}}{2N}[2(\hat{a}^{\dagger}_{0}\hat{a}^{\dagger}_{0}\hat{a}_{1}\hat{a}_{-1}+\hat{a}_{0}\hat{a}_{0}\hat{a}^{\dagger}_{1}\hat{a}^{\dagger}_{-1})$
$\displaystyle+(2\hat{N_{0}}-1)(N-\hat{N_{0}})]-q(t)\hat{N_{0}}.$

Here, $|c_{2}|$ describes the rate of spin mixing process,
$q=(\varepsilon_{+1}+\varepsilon_{-1})/2-\varepsilon_{0}$, with
$\varepsilon_{m_{F}}$ being the energy of state $|{1,m_{F}}\rangle$, and
$q(t)$ can be tuned linearly with time in experiment. The system possesses
three distinct phases through the competition between $|c_{2}|$ and $q$ [65,
66, 67]. For $q\gg 2|c_{2}|$, the ground state is a polar state with all atoms
in state $|{1,0}\rangle$. For $q\ll-2|c_{2}|$, the ground state becomes a twin
Fock state with atoms equally populated in $|{1,-1}\rangle$ and
$|{1,1}\rangle$. When $-2|c_{2}|<q<2|c_{2}|$, the ground state corresponds to
a superposition of all three components. If we ramp $q(t)$ from $q\gg
2|c_{2}|$ towards $q\ll-2|c_{2}|$ with very slow ramping rate, any
instantaneous ground states can be adiabatically prepared. Especially, if $q$
is adiabatically ramped from $q\gg 2|c_{2}|$ to zero, an initial polar state
will evolve into the balanced spin-1 Dicke state $|{D}\rangle$, which can
realize the measurement precision bound of the LSV parameter $\kappa$ over the
SQL and asymptotically reach the Heisenberg limit [26].

Our protocol for testing LSV parameter $\kappa$ via driving through QPTs is
presented below. Firstly, we ramp $q(t)=q_{0}-\beta t$ from $q\gg 2|c_{2}|$
towards $q=0$ adiabatically to generate the balanced spin-$1$ Dicke state
$|{D}\rangle=e^{-i\int_{0}^{\tau_{1}}H_{{QPT}}(t)dt}|{0,N,0}\rangle$. Here,
$\beta$ denotes sweeping rate. Then, a rotation operation $\hat{{R}}_{\pi/2}$
is applied on the balanced spin-1 Dicke state $|{D}\rangle$ to generate the
input state $|{\Psi_{{in}}}\rangle=\hat{{R}}_{\pi/2}|{D}\rangle$ with
$\hat{{R}}_{\pi/2}=e^{-i\frac{\pi}{2}\hat{L}_{x}}$ and
$\hat{L}_{x}=(\hat{a}_{1}^{{\dagger}}\hat{a}_{0}+\hat{a}_{1}\hat{a}_{0}^{{\dagger}}+\hat{a}_{0}^{{\dagger}}\hat{a}_{-1}+\hat{a}_{0}\hat{a}_{-1}^{{\dagger}})/\sqrt{2}$.
In the interrogation step, each atom undergoes the LSV dynamics of duration
$T$ and the system goes through a phase accumulation process. In the
recombination step, a second rotation operation
$\hat{{R}}_{\pi/2}^{{\dagger}}$ is applied at first and then we ramp $q(t)$
from $q=0$ towards $q\gg 2|c_{2}|$ adiabatically. The corresponding unitary
operation $\hat{U}$ for recombination is
$\hat{U}=e^{-i\int_{\tau_{1}+T}^{\tau_{2}}H_{{QPT}}(t)dt}\hat{{R}}_{\pi/2}^{{\dagger}}$.
The final state can be written as,

$\displaystyle|{\Psi_{{final}}}\rangle=$ $\displaystyle
e^{-i\int_{\tau_{1}+T}^{\tau_{2}}H_{{QPT}}(t)dt}\hat{{R}}_{\pi/2}^{{\dagger}}e^{-i\kappa(\hat{N}_{-1}+\hat{N}_{1})T}$
(19)
$\displaystyle\hat{{R}}_{\pi/2}e^{-i\int_{0}^{\tau_{1}}H_{{QPT}}(t)dt}|{0,N,0}\rangle.$

Similarly, the LSV parameter $\kappa$ can be estimated by measuring the final
population in component $m=0$. Ideally, if $\kappa T=\frac{M\pi}{2}$ with $M$
an integer, all atoms would populate into $m=0$ again, leading to
$\langle{\hat{N}_{0}}\rangle=N$. For other values of $\kappa T$, the
population $\langle{\hat{N}_{0}}\rangle$ would be less than $N$ and dependent
on $\kappa T$. In Fig. 3 (b), the variation of measurement precisions
$\Delta\kappa$ versus $\kappa$ for $N=20$ is shown. According to our result,
we find that the optimal measurement precision $\Delta\kappa_{{{min}}}$ occurs
near $\kappa T=0$.

Further, to confirm the dependence of measurement precision $\Delta\kappa$ on
particle number $N$, we numerically calculate the optimal measurement
precision versus particle number $N$ in the range of $\kappa
T\in[-0.1\pi,0.1\pi]$, as shown in Fig. 3 (a). According to the fitting
result, we find the optimal measurement precision of LSV parameter $\kappa$ is
${{\textrm{In}}}(\Delta\kappa_{{min}}T)=-0.8\textrm{In}(N)+0.43$. Our study
indicates that the beam recombination via reversed sweeping through QPTs also
can make the measurement precision of the LSV parameter $\kappa$ beyond SQL.

##  5 Robustness against imperfections

In this section, we study the robustness of the multimode many-body quantum
interferometry. In practical experiments, many imperfections can limit the
final measurement precision. Here, we discuss two imperfections that mainly
exist in the scheme of driving through QPTs: (i) the non-adiabatic effect in
the initialization and recombination steps, and (ii) the detection noise in
the measurement step.

###  5.1 Influence of nonadiabatic effect

[width=1]Fig6.pdf

Figure 4: (color online). The measurement precision $\Delta\kappa$ versus
$\kappa$ for different sweeping rate $\beta$ under $N=10$. Here, $c_{2}=-1$,
$q_{0}=-q_{f}=3$, $T=1$ and $N=10$.

Firstly, we consider the non-adiabatic effect. To realize the proposal of
driving through QPTs perfectly, the sweeping process should be adiabatic.
However, non-adiabatic effect always exists in practical experiments. In
general, the adiabaticity of the driving process can be characterized by the
sweeping rate $\beta$ [76]. If the sweeping rate $\beta$ is sufficiently
small, the adiabatic evolution can be achieved. To confirm the influence of
non-adiabatic effect on the measurement precision, the variation of
$\Delta\kappa$ with $\kappa$ for different sweeping rate $\beta$ is shown in
Fig. 4. It can be observed that the measurement precision of $\Delta\kappa$
becomes worse as $\beta$ increases, but the measurement precision
$\Delta\kappa$ can still beat the SQL when the sweeping rate $\beta$ is
moderate.

###  5.2 Robustness against detection noise

[width=1]Fig7.pdf

Figure 5: (color online). The optimal measurement precision
$\Delta{\kappa}_{\textrm{min}}T$ versus detection noise $\sigma$ for
$N=10,20,30,40$. The dotted lines denote the corresponding SQL, i.e.,
$1/\sqrt{N}$. Here, $c_{2}=-1$, $q_{0}=-q_{f}=3$, $T=1$ and $\beta=0.05$.

Here, we study the influence of detection noise by considering the additional
classical noise in the measurement process. In an ideal situation, the
population measurement on the final state can be rewritten as
$\langle\hat{N}_{0}\rangle=\sum_{N_{0}}{P(N_{0}|\kappa)}N_{0}$, where
${P(N_{0}|\kappa)}$ is the ideal conditional probability which obtain
measurement result $N_{0}$ with a given $\kappa$. However, in realistic
experiments, the detection noise can limit the measurement precision of
frequency. For an imperfect detector with Gaussian detection noise [76, 75,
68, 74], the population measurement becomes

$\langle\hat{N}_{0}\rangle=\sum_{N_{0}}\tilde{P}(N_{0}|{\sigma})N_{0},$ (20)

with

${\tilde{P}(N_{0}|{\sigma})}=\sum_{\tilde{N}_{0}}A_{\tilde{N}_{0}}{e^{-(N_{0}-\tilde{N}_{0})^{2}/2{\sigma}^{2}}}{P({N}_{0}|{\kappa})}.$
(21)

the conditional probability depends on the detection noise $\sigma$. Here,
$A_{\tilde{N}_{0}}$ is a normalization factor. To study the influence of the
detection noise, the optimal measurement precision
$\Delta{\omega}_{\textrm{min}}$ versus the detection noise $\sigma$ for
different particle number $N$ is shown in Fig. 5. For $N=10,20,30,40$, our
numerical results indicate that the measurement precision can still beat the
SQL when $\sigma\leq 0.35\sqrt{N}$. There is no need to use single-particle
resolved detector for measurement when $N$ is modestly large. Thus our
proposal is robust against to the detection noise.

##  6 Summary and discussion

In summary, we have presented an entanglement-enhanced multimode quantum
interferometry for testing the LSV effect. The LSV effect for a bound electron
system results in an energy shift, which is the directional dependent
interaction (from the SME). Based upon the proposed scheme, we study the
ultimate measurement precision bound for the LSV parameter $\kappa$ via
individual atoms and entangled atoms. We find that the ultimate measurement
precision bound depends on both the spin length $F$ and the total atom number
$N$. Larger spin length $F$ and total atomic number $N$ may offer better
measurement precision. Especially, if the input state is the $N$-atom
multimode GHZ state, the measurement precision of $\kappa$ can attain the
Heisenberg limit, i.e., $\Delta\kappa\propto 1/(F^{2}N)$.

Moreover, we study the LSV test via the experimentally accessible three-mode
interferometry with Bose condensed spin-$1$ atoms [36]. The three-mode
interferometer is available in experiments with spin-1 atoms [65, 66, 67]. By
selecting suitable input states and recombination operations, the LSV
parameter $\kappa$ can be extracted via population measurement. Especially,
the measurement precision of the LSV parameter $\kappa$ can beat the SQL and
even approach the Heisenberg limit via SMD [38] or driving through QPTs [64,
65, 66, 67, 66]. To realize the entanglement-enhanced test for local LSV with
the three-mode interferometer in experiments, the precise control of system
parameters and the implementation of adiabatic sweeping process are both
necessary. Owing to the well-developed techniques in quantum control, the
entangled states can be generated in 87Rb condensate via SMD or adiabatically
driving through QPTs with state-of-the-art techniques.

According to the relationship between the parameters $\kappa$ and
$C^{(2)}_{0}$ [26], the measurement precision of parameter $C^{(2)}_{0}$ can
be improved the same orders of magnitude. According to Eq. (2), the
relationship between the parameters $\kappa$ and $C^{(2)}_{0}$ reads as

$\displaystyle\frac{\kappa}{2\pi}=\frac{\Delta
E/(hC^{(2)}_{0})}{\Delta(j_{z}^{2})}C^{(2)}_{0},$ (22)

here $h$ is the Planck constant, $\Delta E$ and $\Delta(j_{z}^{2})$ denote the
energy deviation and angular momentum fluctuation for the experimentally
selected states, respectively. One can obtain $\Delta E$ via calculating the
matrix element of operator $T_{0}^{(2)}$, which is proportional to $m_{F}^{2}$
and the reduced matrix element $\langle F|T^{(2)}|F\rangle$ simultaneously.
For the ${}^{87}\textrm{Rb}$ atoms, a rough estimate using the method of Refs.
[24, 26] gives $\Delta E/(hC^{(2)}_{0})=8.6\times 10^{15}$ Hz and
$\Delta(j_{z}^{2})=1$ for the $m=1$ and $m=0$ states. Compared with the
measurement precision reported in Ref. [26] which just asymptotically
approaches the SQL via observable measurement, our scheme can be expected near
the Heisenberg limit via population measurement. Thus, also assuming a year-
long measurement with $10^{4}$ atoms, the measurement precision of parameter
$C^{(2)}_{0}$ is about $2$ orders magnitude higher than the result reported in
Ref. [26]. Compared with the LSV tests via two fully entangled Yb+ ions [23]
with the parameter $C^{(2)}_{0}$ will be bounded at the level of $10^{-23}$
for year-long measurement, our scheme can also be used for improving the
corresponding measurement precision by utilizing multi-particle entangled
states.

Our proposal may open up a feasible way for a drastic improvement of the LSV
tests with atomic systems and also provide an experimentally accessible scheme
to realize entanglement-enhanced test for LSV. Besides, it may be used for
testing some fundamental theories, such as field theory, string theories,
Einstein-aether theories, and quantum gravity.

## Acknowledgements

This work is supported by the National Key Research and Development Program of
China (Grant No. 2022YFA1404104), the National Natural Science Foundation of
China (Grant No. 12025509, 11874434), and the Key-Area Research and
Development Program of GuangDong Province (Grant No. 2019B030330001). J. H. is
partially supported by the Guangzhou Science and Technology Projects
(202002030459).

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## Details of the numerical simulation

Here, we introduce the details of our numerical simulation, we start from all
$N$ atoms in state $|{1,{m_{F}=0}}\rangle$, i.e.,
$|{\Psi_{0}}\rangle=|{0,N,0}\rangle$. Due to the conservation of magnetization
$N_{1}=N_{-1}$ under spin-exchange collisions, the dynamical evolution will be
constrained in the zero magnetization subspace of $\\{|k,N-2k,k\rangle\\}$,
with $k,N-2k,k$ representing number of atoms in the $m_{F}=+1,0,-1$
components, respectively. For the scheme of LSV test via spin-mixing dynamics,
the matrix element of Hamiltonian $\hat{H}_{SMD}$ in this basis is given by

$\displaystyle[\hat{H}_{SMD}]_{k^{\prime},k}$ $\displaystyle\equiv$
$\displaystyle\langle
k^{\prime},N-2k^{\prime},k^{\prime}|\hat{H}_{SMD}|k,N-2k,k\rangle$
$\displaystyle=$ $\displaystyle\chi
k\sqrt{(N-2k+1)(N-2k+2)}\delta_{k^{\prime},k-1}$ $\displaystyle+$
$\displaystyle\chi(k+1)\sqrt{(N-2k)(N-2k-1)}\delta_{k^{\prime},k+1}$

For the scheme of LSV test via driving through quantum phase transitions, the
matrix element of Hamiltonian $\hat{H}_{QPT}$ in this basis is given by

$\displaystyle[\hat{H}_{QPT}]_{k^{\prime},k}$ $\displaystyle\equiv$
$\displaystyle\langle
k^{\prime},N-2k^{\prime},k^{\prime}|\hat{H}_{QPT}|k,N-2k,k\rangle$
$\displaystyle=$
$\displaystyle[\frac{c_{2}}{N}k(2N-4k-1)-q(N-2k)]\delta_{k^{\prime},k}$
$\displaystyle+$
$\displaystyle\frac{c_{2}}{N}k\sqrt{(N-2k+1)(N-2k+2)}\delta_{k^{\prime},k-1}$
$\displaystyle+$
$\displaystyle\frac{c_{2}}{N}(k+1)\sqrt{(N-2k)(N-2k-1)}\delta_{k^{\prime},k+1}$

To simulate dynamical process, we expand the wave function at time $t$ in the
basis $\\{|k,N-2k,k\rangle\\}$ as

$\displaystyle|{\Psi(t)}\rangle=\sum_{k=0}^{N/2}C_{k}(t)|k,N-2k,k\rangle.$
(A3)

According to the Schrodinger equation
$i\partial_{t}|{\Psi(t)}\rangle=H|{\Psi(t)}\rangle$, one can obtain the
coupled equations for the superposition coefficients $C_{k}(t)$. For the
Hamiltonian $\hat{H}_{SMD}$, the coupled equations are

$\displaystyle i\partial_{t}C_{k}^{SMD}(t)$ $\displaystyle=$
$\displaystyle\chi k\sqrt{(N-2k+1)(N-2k+2)}C_{k-1}^{SMD}(t)$ $\displaystyle+$
$\displaystyle\chi(k+1)\sqrt{(N-2k)(N-2k-1)}C_{k+1}^{SMD}(t)$

For the Hamiltonian $\hat{H}_{QPT}$, the coupled equations are

$\displaystyle i\partial_{t}C_{k}^{QPT}(t)$ $\displaystyle=$
$\displaystyle[\frac{c_{2}}{N}k(2N-4k-1)-q(N-2k)]C_{k}^{QPT}(t)$
$\displaystyle=$
$\displaystyle\frac{c_{2}}{N}k\sqrt{(N-2k+1)(N-2k+2)}C_{k-1}^{QPT}(t)$
$\displaystyle+$
$\displaystyle\frac{c_{2}}{N}(k+1)\sqrt{(N-2k)(N-2k-1)}C_{k+1}^{QPT}(t)$

This series of coupled equations can be solved numerically and one can obtain
any expectations for the interested observables.


# Implications of PREX-II on the equation of state of neutron-rich matter

Brendan T. Reed<EMAIL_ADDRESS>Department of Astronomy, Indiana University,
Bloomington, Indiana 47405, USA  Center for Exploration of Energy and Matter
and Department of Physics, Indiana University, Bloomington, IN 47405, USA  F.
J. Fattoyev<EMAIL_ADDRESS>Department of Physics, Manhattan
College, Riverdale, NY 10471, USA  C. J. Horowitz<EMAIL_ADDRESS>Center
for Exploration of Energy and Matter and Department of Physics, Indiana
University, Bloomington, IN 47405, USA  J. Piekarewicz<EMAIL_ADDRESS>Department of Physics, Florida State University, Tallahassee, FL 32306, USA

###### Abstract

Laboratory experiments sensitive to the equation of state of neutron rich
matter in the vicinity of nuclear saturation density provide the first rung in
a “density ladder” that connects terrestrial experiments to astronomical
observations. In this context, the neutron skin thickness of 208Pb ($R_{\rm
skin}^{208}$) provides a stringent laboratory constraint on the density
dependence of the symmetry energy. In turn, an improved value of $R_{\rm
skin}^{208}$ has been reported recently by the PREX collaboration. Exploiting
the strong correlation between $R_{\rm skin}^{208}$ and the slope of the
symmetry energy $L$ within a specific class of relativistic energy density
functionals, we report a value of $L\\!=\\!(106\pm 37)\,{\rm MeV}$—that
systematically overestimates current limits based on both theoretical
approaches and experimental measurements. The impact of such a stiff symmetry
energy on some critical neutron-star observables is also examined.

###### pacs:

21.60.Jz, 24.10.Jv, 26.60.Kp, 97.60.Jd

The updated Lead Radius EXperiment (PREX-II) has delivered on the promise to
determine the neutron radius of 208Pb with a precision of nearly 1%. By
combining the original PREX result Abrahamyan et al. (2012); Horowitz et al.
(2012) with the newly announced PREX-II measurement, the following value for
the neutron skin thickness of 208Pb was reported Adhikari et al. (2021):

$R_{\rm skin}=R_{n}-R_{p}=(0.283\pm 0.071)\,{\rm fm},$ (1)

where the quoted uncertainty represents a $1\sigma$ error, and $R_{n}$ and
$R_{p}$ are the root-mean-square radii of the neutron and proton density
distributions, respectively. Such a purely electroweak measurement is of
critical importance in constraining both models of nuclear structure as well
as the equation of state (EOS) of neutron-rich matter in the vicinity of
nuclear saturation density
($\rho_{\raisebox{-2.0pt}{\tiny\\!0}}\\!\approx\\!0.15\,{\rm fm}^{-3}$). In
turn, the EOS around saturation density provides the first rung in a “density
ladder” that connects laboratory experiment to astronomical observations that
probe the EOS at higher densities. It is the aim of this letter to explore the
impact of PREX-II on certain parameters of the EOS that, in turn, dictates the
behavior of several neutron-star observables.

For two decades the neutron skin thickness of 208Pb has been identified as an
ideal laboratory observable to constrain the EOS of neutron rich matter,
particularly the poorly determined density dependence of the symmetry energy
Brown (2000); Furnstahl (2002); Centelles et al. (2009); Roca-Maza et al.
(2011). The EOS of infinite nuclear matter at zero temperature is enshrined in
the energy per particle which depends on both the conserved neutron
($\rho_{n}$) and proton ($\rho_{p}$) densities; here we assume that the
electroweak sector has been “turned off ”. Moreover, it is customary to
separate the EOS into two contributions, one that represents the energy of
symmetric ($\rho_{n}\\!=\\!\rho_{p}$) nuclear matter and another one that
accounts for the breaking of the symmetry. That is,

$\frac{E}{A}(\rho,\alpha)-\\!M\equiv{\cal E}(\rho,\alpha)={\cal E}_{\rm
SNM}(\rho)+\alpha^{2}{\cal S}(\rho)+{\cal O}(\alpha^{4})\,,$ (2)

where $\rho\\!=\\!(\rho_{n}\\!+\\!\rho_{p})$ is the total baryon density given
by the sum of neutron and proton densities, and
$\alpha\\!=(\rho_{n}\\!-\\!\rho_{p})/\rho$ is the neutron-proton asymmetry.
The first-order correction to the energy of symmetric nuclear matter ${\cal
E}_{\rm SNM}(\rho)$ is encoded in the symmetry energy ${\cal S}(\rho)$. The
symmetry energy quantifies the increase in the energy per particle of infinite
nuclear matter for systems with an isospin imbalance (e.g., more neutrons than
protons). Further, given the preeminent role of nuclear saturation, the energy
of symmetric nuclear matter and the symmetry energy may be described in terms
of a few bulk parameters that characterize their behavior around saturation
density. In this letter we focus on the density dependence of the symmetry
energy Piekarewicz and Centelles (2009):

${\cal
S}(\rho)=J+L\frac{(\rho-\rho_{\raisebox{-2.0pt}{\tiny\\!0}})}{3\rho_{\raisebox{-2.0pt}{\tiny\\!0}}}+\ldots$
(3)

The first term ($J$) represents the correction to the binding energy of
symmetric nuclear matter, whereas the second term ($L$) dictates how rapidly
the symmetry energy increases with density. It is the slope of the symmetry
energy $L$ that displays a strong correlation to the neutron skin thickness of
208Pb. Given that symmetric nuclear matter saturates, namely, its pressure
vanishes at saturation, the slope of the symmetry energy $L$ is closely
related to the pressure of pure neutron matter at saturation density. That is,

$P_{\,\rm
PNM}(\rho_{\raisebox{-2.0pt}{\tiny\\!0}})\\!\approx\\!\frac{1}{3}L\rho_{\raisebox{-2.0pt}{\tiny\\!0}}.$
(4)

Figure 1: (Color online) Left: Slope of the symmetry energy at nuclear
saturation density $\rho_{\raisebox{-2.0pt}{\tiny\\!0}}$ (blue upper line) and
at $(2/3)\rho_{\raisebox{-2.0pt}{\tiny\\!0}}$ (green lower line) as a function
of $R_{\rm skin}^{208}$. The numbers next to the lines denote values for the
correlation coefficients. Right: Gaussian probability distribution for the
slope of the symmetry energy $L\\!=\\!L(\rho_{\raisebox{-2.0pt}{\tiny\\!0}})$
inferred by combining the linear correlation in the left figure with the
recently reported PREX-II limit. The six error bars are constraints on $L$
obtained by using different theoretical approaches Zhang and Chen (2013);
Hebeler et al. (2013); Drischler et al. (2020); Hagen et al. (2015); Chen et
al. (2010); Steiner and Gandolfi (2012); Gandolfi et al. (2014); Roca-Maza et
al. (2015).

To assess the impact of the combined PREX–PREX-II measurements (henceforth
referred simply as “PREX-II”)—we provide predictions for several observables
using a set of 16 covariant energy density functionals. These are FSUGold2
Chen and Piekarewicz (2014) together with a set of eight systematically varied
interactions—FSUGold2–L047, L050, L054, L058, L069, L076, L090, L100—with
identical isoscalar properties as FSUGold2, but isovector properties defined
by the associated value of the slope of the symmetry energy $L$. For example,
FSUGold2=FSUGold2–L113 predicts a slope of the symmetry energy of
$L\\!=\\!113\,{\rm MeV}$. Another set of accurately calibrated density
functionals is given by RMF012, RMF016, RMF022, and RMF032 Chen and
Piekarewicz (2015), where now the labels are associated to the predicted value
of $R_{\rm skin}^{208}$. For example, RMF032 predicts a neutron skin thickness
of $R_{\rm skin}^{208}$$\\!=\\!0.32\,{\rm fm}$. Finally, TFa, TFb, and TFc,
with $R_{\rm skin}^{208}$$\\!=\\!0.25,0.30,{\rm and}\,0.33\,{\rm fm}$,
respectively, were created to test whether the large central value of $R_{\rm
skin}^{208}$$\\!=\\!0.33\,{\rm fm}$ originally reported by the PREX
collaboration Abrahamyan et al. (2012) was incompatible with other laboratory
experiments and/or astrophysical observations Fattoyev and Piekarewicz (2013).
We found then, that there was no compelling reason to rule out models with
large neutron skins.

From the compilation of all these 16 models one obtains for the binding energy
per nucleon and the charge radius of 208Pb the following values:
$B/A\\!=\\!7.88\pm 0.01\,{\rm MeV}$ and $R_{\rm ch}\\!=\\!5.51\pm 0.01\,{\rm
fm}$, which compare well against the experimental values of
$B/A\\!=\\!7.87\,{\rm MeV}$ and $R_{\rm ch}\\!=\\!5.50\,{\rm fm}$,
respectively. A detailed description of the fitting protocol—including an
explanation of the model and the observables used in the calibration
procedure—may be found in Refs. Chen and Piekarewicz (2014, 2015); Fattoyev
and Piekarewicz (2013). Moreover we underscore that the models considered here
span a wide range of values for both the neutron skin thickness of 208Pb and
the associated slope of the symmetry energy. Indeed, the range of adopted
values is almost as wide as the one used in the multi-model analysis of the
sensitivity of the symmetry energy to the electric dipole polarizability and
weak-charge form factor of both 48Ca and 208Pb Piekarewicz et al. (2012);
Reinhard et al. (2013).

The strong correlation between $L$ and $R_{\rm skin}^{208}$in the context of
the new PREX-II measurement is illustrated in Fig. 1. The left-hand panel
displays the well-known correlation between the slope of the symmetry energy
at saturation density and the neutron-skin thickness of 208Pb. Also shown in
Fig. 1(a) is the even stronger correlation between $R_{\rm skin}$ and the
slope of the symmetry energy at the slightly lower density of
$\widetilde{\rho}_{\raisebox{-2.0pt}{\tiny\\!0}}\\!=\\!(2/3)\rho_{\raisebox{-2.0pt}{\tiny\\!0}}\\!\approx\\!0.1\,{\rm
fm}^{-3}$ Horowitz and Piekarewicz (2001); Furnstahl (2002); Ducoin et al.
(2010, 2011); Horowitz et al. (2014); Zhang and Chen (2013). At such a lower
density, which represents an average value between the central and surface
densities, the symmetry energy is well constrained by the binding energy of
heavy nuclei with a significant neutron excess. Relying on the strong $R_{\rm
skin}$-$L$ correlation together with the improved PREX-II limit, one obtains
the gaussian probability distribution for $L$ displayed in Fig. 1(b). Using
the same analysis on both $J$ and $\widetilde{L}$—the latter representing the
slope of the symmetry energy at
$\widetilde{\rho}_{\raisebox{-2.0pt}{\tiny\\!0}}$—we derive the following
limits:

$\displaystyle J=(38.1\pm 4.7){\rm MeV},$ (5a) $\displaystyle L=(106\pm
37){\rm MeV},$ (5b) $\displaystyle\widetilde{L}=(71.5\pm 22.6){\rm MeV}.$ (5c)

As indicated in Fig. 1(b), these limits are systematically larger than those
obtained using either purely theoretical approaches or extracted from a
theoretical interpretation of experimental data Zhang and Chen (2013); Hebeler
et al. (2013); Drischler et al. (2020); Hagen et al. (2015); Chen et al.
(2010); Steiner and Gandolfi (2012); Gandolfi et al. (2014); Roca-Maza et al.
(2015). We underscore that the models used in this letter represent a
particular class of relativistic EDFs.

We note that theoretical interpretations of elastic nucleon-nucleus scattering
cross sections together with quasielastic reactions to isobaric analog states
obtained limits on $L$ that are consistent with our findings Danielewicz et
al. (2017). The PREX-II result is also considerably larger—and in many cases
incompatible—with experimental determinations of $R_{\rm skin}^{208}$by
methods that are highly model dependent Trzcińska et al. (2001); Zenihiro et
al. (2010); Tarbert et al. (2014); Thiel et al. (2019). A notable exception is
the dispersive optical model analysis of the Washington University group that
reported a neutron skin thickness of $R_{\rm skin}^{208}$$\\!=\\!(0.25\pm
0.05)\,{\rm fm}$ Atkinson et al. (2020); a revised lower value of $R_{\rm
skin}^{208}$$\\!=\\!(0.18\pm 0.07)\,{\rm fm}$—still consistent with Atkinson
et al. (2020)—was reported shortly thereafter in Ref. Pruitt et al. (2020).

To further underscore the tension between PREX-II and our current
understanding of the EOS, we display in Fig. 2 a summary of simultaneous
constraints on both $J$ and $L$ as reported in Refs. Lattimer (2012);
Drischler et al. (2020). We have adapted Figure 2 from Ref. Drischler et al.
(2020) by including the PREX-II limits on both $J$ and $L$ derived in Eq.(5).
Note that with the exception of the analysis of Ref. Chen et al. (2010), all
other approaches suggest a positive correlation between $L$ and $J$. In the
context of density functional theory, such a positive correlation is easy to
understand. Using Eq.(3) at $\widetilde{\rho}_{\raisebox{-2.0pt}{\tiny\\!0}}$
yields

$S(\widetilde{\rho}_{\raisebox{-2.0pt}{\tiny\\!0}})=J-\frac{L}{9}\rightarrow
J\approx\left(26\,{\rm MeV}+\frac{L}{9}\right).$ (6)

The value of
$S(\widetilde{\rho}_{\raisebox{-2.0pt}{\tiny\\!0}})\\!\approx\\!26\,{\rm MeV}$
Horowitz and Piekarewicz (2001) follows because the symmetry energy at
$\widetilde{\rho}_{\raisebox{-2.0pt}{\tiny\\!0}}$ is tightly constrained by
the binding energy of heavy nuclei. The PREX-II inferred value for $L$ yields
a corresponding value of $J\\!=\\!(37.7\pm 4.1)\,{\rm MeV}$, that is entirely
consistent with the limit obtained in Eq.(5). Although consistent at the
$2\sigma$ level, the “Intersection” region in Fig. 2 obtained from a variety
of experimental and theoretical approaches lies outside the $1\sigma$ PREX-II
limits.

Figure 2: (Color online). Constraints on the $J$–$L$ correlation obtained from
a variety of experimental and theoretical approaches. The figure was adapted
from Refs. Lattimer (2012); Drischler et al. (2020) and noticeably displays
the tension with the recent PREX-II result. Figure 3: (Color online). Direct
Urca thresholds for the onset of enhanced cooling in neutron stars. The
threshold density is depicted by the lower blue line and the corresponding
stellar mass for such a central density with the upper green line. The shaded
area represents PREX-II $1\sigma$ confidence region. For each of these two
quantities, the best-fit line is displayed together with their associated
correlation coefficients.

Next, we explore the impact of PREX-II on a few neutron-star observables. We
start by displaying in Fig. 3 the minimum central density and associated
neutron star mass required for the onset of the direct Urca process. Neutron
stars are born very hot ($T\\!\simeq\\!10^{11}K\\!\simeq 10\,{\rm MeV}$) and
then cool rapidly via neutrino emission through the direct Urca process that
involves neutron beta decay followed by electron capture:

$\displaystyle n\rightarrow p+e^{-}+\bar{\nu}_{e},$ (7a) $\displaystyle
p+e^{-}\rightarrow n+\nu_{e}.$ (7b)

After this rapid cooling phase is completed, neutrino emission proceeds in the
standard cooling scenario through the modified Urca process—a process that may
be millions of times slower as it requires the presence of a bystander nucleon
to conserve momentum at the Fermi surfacePage et al. (2004). The transition
into the much slower modified Urca process is solely based on the expectation
that the proton fraction in the stellar core is too low to conserve momentum
at the Fermi surface. However, given that the proton fraction is controlled by
the poorly known density dependence of the symmetry energy Horowitz and
Piekarewicz (2002), the minimal cooling scenario may need to be revisited. In
particular, a stiff symmetry energy—as suggested by PREX-II—favors large
proton fractions that may trigger the onset of the direct Urca process at
lower central densities. This analysis is particularly timely given that x-ray
observations suggest that some neutron stars may require some form of enhanced
cooling. Indeed, the detected x-ray spectrum of the neutron star in the low-
mass x-ray binary MXB 1659-29 strongly suggests the need for a fast neutrino-
cooling process Brown et al. (2018). For a comprehensive report that explores
the interplay between the direct Urca process and nucleon superfluidity in
transiently accreting neutron stars, see Ref. Potekhin et al. (2019). The
shaded area in Fig. 3 displays the region constrained by PREX-II. In
particular, the $1\sigma$ lower limit of $R_{\rm skin}\\!=\\!0.212\,{\rm fm}$
suggests a threshold mass for the onset of direct Urca cooling of
$M_{\star}\\!\approx\\!1.45\,M_{\odot}$ and a corresponding central density of
$\rho_{\star}\\!\approx\\!0.42\,{\rm fm}^{-3}$. However, if instead one adopts
the larger PREX-II central value of $R_{\rm skin}\\!=\\!0.283\,{\rm fm}$, then
one obtains the considerably lower threshold values of
$M_{\star}\\!\approx\\!0.85\,M_{\odot}$ and
$\rho_{\star}\\!\approx\\!0.24\,{\rm fm}^{-3}$, or a threshold density just
slightly higher than saturation density. Although some stars are likely to
require enhanced cooling, observations of many isolated neutron stars are
consistent with the much slower modified URCA process Page et al. (2009). This
may be because the direct URCA neutrino emissivity is reduced by nucleon
pairing.

Figure 4: (Color online). Showcase of neutron star observables as a function
of $R_{\rm skin}^{208}$ as predicted by the set of energy density functionals
considered in this work. The tidal deformability $\Lambda_{\star}^{1.4}$of a
1.4 M⊙ neutron star is computed for each model, and displayed with blue dots
and connected by a best fit power law that scales as the $4.8\\!\approx\\!5$
power of $R_{\star}^{1.4}$. The combined PREX-II result together with NICER
constraints on the stellar radius is depicted by the small (blue) window of
models allowed.

We close the section by displaying in Fig. 4 the dimensionless tidal
deformability of a 1.4 $M_{\odot}$ neutron star as a function of both the
stellar radius $R_{\star}^{1.4}$ and $R_{\rm skin}$. Although not shown, for
the set of density functionals used in this work a very strong correlation (of
about 0.98) is obtained between $R_{\star}^{1.4}$ and $R_{\rm skin}$. However,
because the central density of a 1.4 $M_{\odot}$ neutron star may reach
densities as high as 2-to-3 times saturation density, the robustness of such a
correlation should be examined in the context of alternative theoretical
descriptions. Moreover, a precise knowledge of the EOS of the crust is needed
to minimize possible systematic uncertainties Piekarewicz et al. (2014). As in
Fig. 3, the $1\sigma$ confidence region is indicated by the shaded area in the
figure. Also shown are NICER constrains on the radius of PSR J0030+0451 Riley
et al. (2019); Miller et al. (2019), that are depicted by the two horizontal
error bars and which suggest an upper limit of
$R_{\star}^{1.4}\\!\leq\\!14.26\,{\rm km}$. Invoking the strong
$R_{\star}^{1.4}$ – $R_{\rm skin}$ correlation observed in our models, one
obtains an upper limit on the neutron skin thickness of $R_{\rm
skin}\\!\lesssim\\!0.31\,{\rm fm}$ and a lower limit on the stellar radius of
$R_{\star}^{1.4}\\!\gtrsim\\!13.25\,{\rm km}$. The region that satisfies both
PREX-II and NICER constraints is indicated by the narrow (blue) rectangle in
Fig. 4, which excludes a significant number of models. In turn, given that the
tidal deformability approximately scales with the fifth power of the stellar
radius Fattoyev et al. (2018), one can also set limits on the tidal
deformability of a $1.4\,M_{\odot}$ neutron star. Combining the constraints
from NICER on $R_{\star}^{1.4}$and PREX-II on $R_{\rm skin}^{208}$one obtains:

$\displaystyle 0.21\lesssim$ $\displaystyle\;R_{\rm skin}({\rm fm})\\!\lesssim
0.31$ (8a) $\displaystyle 13.25\lesssim$ $\displaystyle\;R_{\star}^{1.4}({\rm
km})\\!\lesssim 14.26$ (8b) $\displaystyle 642\lesssim$
$\displaystyle\;\Lambda_{\star}^{1.4}\\!\lesssim 955.$ (8c)

The allowed region for the tidal deformability falls comfortably within the
$\Lambda_{\star}^{1.4}\\!\lesssim\\!800$ limit reported in the GW170817
discovery paper Abbott et al. (2017). Yet, the revised limit of
$\Lambda_{1.4}\\!=\\!190^{+390}_{-120}\\!\lesssim\\!580$ Abbott et al. (2018)
presents a more serious challenge. To confirm whether this tension is real, it
will require a multi-prong approach involving a more precise determination of
$R_{\rm skin}^{208}$, additional NICER observations, and more multi-messenger
detections of neutron star mergers. The prospect of a more precise electroweak
determination of $R_{\rm skin}^{208}$ is challenging as it may require the
full operation of the future Mainz Energy-recovery Superconducting Accelerator
(MESA) which is foreseen to start until 2023 Becker et al. (2018). Future
determinations of stellar radii by NICER for neutron stars with known masses,
such as J0437-4715 Reardon et al. (2016), could be made at a $\pm\,3\%$ level,
or to better than $\pm\,0.5\,{\rm km}$. NICER is also collecting pulse profile
modeling data for the highest mass pulsar (PSR J0740+6620) ever measured
Cromartie et al. (2019). Finally, the LIGO-Virgo-KAGRA collaborations are
preparing for the fourth observing run at a higher detector sensitivity.
Although KAGRA will join LIGO and Virgo promising much better sky
localization, COVID-related delays have pushed the fourth observing run until
June 2022.

In summary, PREX-II has confirmed with improved precision the original PREX
suggestion that the EOS at the typical densities found in atomic nuclei is
stiff. This result challenges our present understanding of the density
dependence of symmetry energy extracted from various experimental and
theoretical analyses Thiel et al. (2019). By assessing the impact of PREX-II
at higher densities, we were able to provide limits on both the radius and
deformability of a 1.4 $M_{\odot}$ neutron star. Given that our analysis of
the tidal deformability reveals some tension with the revised limit of
$\Lambda_{1.4}\\!\lesssim\\!580$ Abbott et al. (2018), we eagerly await the
next generation of terrestrial experiments and astronomical observations to
verify whether the tension remains. If so, the softening of the EOS at
intermediate densities, together with the subsequent stiffening at high
densities required to support massive neutron stars, may be indicative of a
phase transition in the stellar core Fattoyev et al. (2018).

###### Acknowledgements.

This material is based upon work supported by the U.S. Department of Energy
Office of Science, Office of Nuclear Physics under Awards DE-FG02-87ER40365
(Indiana University), Number DE-FG02-92ER40750 (Florida State University), and
Number DE-SC0008808 (NUCLEI SciDAC Collaboration).

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