Sampsa Vihonen

# Overview of the physics prospects in MOMENT

###### Abstract

MuOn-decay MEdium-baseline NeuTrino beam facility (MOMENT) is a recently
proposed neutrino oscillation experiment that is currently under consideration
in China. Based on a novel accelerator concept, MOMENT is capable of
delivering 15 MW neutrino beam produced with the decay of positively and
negatively charged muons. In this study, the physics performance of MOMENT is
briefly reviewed in its baseline setup, involving a 150 km baseline length and
large gadolinium-doped Water Cherenkov detector. The prospects are discussed
in the case of precision measurements on the Dirac CP phase $\delta_{CP}$,
which MOMENT is shown to be able to measure by about 8∘…18∘ resolution at
1$\,\sigma$ CL. It is examined how MOMENT performs as an independent
experiment and also as a complementary facility to the future superbeam
experiments.

##  1 Introduction

Recent advances in neutrino oscillation physics have led to a significant
improvement in the understanding of three-neutrino mixing [1]. The neutrino
mixing angles $\theta_{12}$, $\theta_{13}$ and $\theta_{23}$ and mass-squared
differences $\Delta m_{21}^{2}$ and $\Delta m_{31}^{2}$ have been measured in
the numerous solar, reactor and accelerator neutrino experiments as well as in
neutrino telescopes. The next-generation neutrino oscillation experiments such
as the Jiangmen Underground Neutrino Observatory (JUNO), Deep Underground
Neutrino Experiment (DUNE) and Tokai-to-HyperKamiokande (T2HK) experiment will
aim to address the remaining questions in the standard neutrino oscillation
parameters111One of the long-term goals in neutrino experiments is to measure
the value of the Dirac CP phase $\delta_{CP}$, which could shed light on the
problem of matter-antimatter asymmetry as well as the origin of neutrino
mixing.. Future experiments are also expected to improve the precision on the
previously measured oscillation parameters and look for signals of new
physics.

MuOn-decay MEdium-baseline NeuTrino beam facility (MOMENT) is a recently
proposed neutrino oscillation experiment currently under consideration in
China [2]. MOMENT features a novel accelerator concept where a high-intensity
low-energy beam of neutrinos and antineutrinos is produced by using a
continuous-wave proton beam and a stack of super-conducting solenoids. The
neutrino beams in MOMENT are created by using positively and negatively
charged muons, which allow to measure neutrino mixing in eight oscillation
channels in low beam-related backgrounds. This approach has many advantages,
including low beam-related backgrounds and greater access to oscillation
channels. In this proceeding, we briefly review the physics prospects of
MOMENT in the precision measurement of the standard neutrino oscillation
parameters. We focus on the sensitivity to Dirac CP phase $\delta_{CP}$.
Potential synergies with the Chinese research planning are also discussed. We
also highlight the complementarity between MOMENT and the proposed next-
generation long-baseline neutrino experiments DUNE, T2HK and T2HKK.

This proceeding is organized as follows. We give a brief overview on the
MOMENT configuration in section 2 and discuss the prospects to measure
$\delta_{CP}$ in section 3. We finally leave the concluding remarks in section
4.

##  2 Overview of MOMENT

MOMENT is a next-generation muon-decay-based neutrino beam facility which
utilizes a novel technology to generate a low-energy high-intensity beam [2].
Following the design and timeline of the China initiative for Accelerator-
Driven Systems (CiADS), the proton beam of MOMENT is expected to reach 15 MW
power at 1.5 GeV energy. The neutrino beam of MOMENT reaches its peak at about
200-300 MeV and it is therefore suitable for a baseline length of 150 km. The
synergy of this configuration can be seen in Figure 1, where the simulated
$\nu_{e}$ fluxes and oscillation probabilities are shown. The expected
neutrino fluxes can easily reach the second oscillation maximum and parts of
the first maximum.

Figure 1: Transition probabilities for the oscillation channel
$\nu_{e}\rightarrow\nu_{\mu}$ for neutrino energies and baseline length
relevant for the MOMENT setup [3]. The probabilities are shown for
$\delta_{CP}=$ 0∘, 90∘, 180∘ and 270∘, as indicated by the solid, dashed,
dotted and dot-dashed curves, respectively. The expected $\nu_{e}$ fluxes in
MOMENT are represented by the red boxes.

As a muon-decay-based neutrino beam facility, MOMENT has many advantages known
from the conventional neutrino factory, such as the access to eight different
oscillation channels and a well-understood neutrino source. MOMENT is
anticipated to have excellent control of beam-related systematic uncertainties
and low beam-induced backgrounds. The average energy of the neutrino beam in
MOMENT places neutrino interactions in the quasi-elastic regime, which
requires a large-scale neutrino detector that is capable of studying neutrinos
at the desired energy range. Based on the recent progress in the gadolinium-
doped Water Cherenkov technology, a neutrino detector of 500 kt fiducial mass
has been considered for the far detector of MOMENT [2, 3]. Alternative designs
for the neutrino detector have also been examined [4].

The prospects of the MOMENT facility have been studied in the precision
measurement of the Dirac CP phase [3] and in various searches for new physics,
see Ref. [5] for more information. MOMENT could be established by constructing
entirely new facilities for source and detector. Alternatively, the
configuration could be realized using the existing research infrastructure in
China. Table 1 presents the geographical locations of five accelerator
laboratories and two underground laboratories which are either currently under
construction or are under active consideration. Placing MOMENT beamline at
CiADS and detector at JUNO for instance would give access to 221 km baseline
length.

Table 1: Geographical locations of the existing and proposed accelerator and
underground laboratories [5]. The baseline lengths as well as the energies
needed for the first and second oscillation maximum are shown for each
potential configuration. The neutrino energies are shown for $\Delta
m_{31}^{2}\simeq$ 2.517$\times$10-3 eV2.

Accelerator facility | JUNO (22.12∘, 112.51∘) | CJPL (28.15∘, 101.71∘)
---|---|---
Baseline | 1st maximum | 2nd maximum | Baseline | 1st maximum | 2nd maximum
CAS-IMP (36.05∘, 103.68∘) | 1759 km | 3.6 GeV | 1.2 GeV | 894 km | 1.8 GeV | 600 MeV
CiADS (23.08∘, 114.40∘) | 221 km | 450 MeV | 150 MeV | 1389 km | 2.8 GeV | 940 MeV
CSNS (23.05∘, 113.73∘) | 162 km | 330 MeV | 110 MeV | 1329 km | 2.7 GeV | 900 MeV
Nanjing (32.05∘, 118.78∘) | 1261 km | 2.6 GeV | 850 MeV | 1693 km | 3.4 GeV | 1.1 GeV
SPPC (39.93∘, 116.40∘) | 1871 km | 3.8 GeV | 1.3 GeV | 1736 km | 3.5 GeV | 1.2 GeV

##  3 Physics prospects

MOMENT has been shown to be very capable of performing precision measurements
on the neutrino oscillation parameters. The sensitivity to the Dirac CP phase
$\delta_{CP}$ in the baseline setup of MOMENT is comparable with the expected
sensitivities that have been projected for the future long-baseline
experiments [3]. This is demonstrated in Figure 2, where the sensitivities to
$\delta_{CP}$ are presented for the presently running experiments T2K and
NO$\nu$A, future experiments DUNE and T2HK. Sensitivities are also shown for
the proposed T2HKK configuration, where the second detector of T2HK is
constructed in South Korea. The sensitivities are displayed separately for the
normal ordering (NO) and inverted ordering (IO) of neutrino masses. The true
values of the neutrino oscillation parameters are taken to be identical with
the present fit result from the nu-fit 5.1 database [1]. In this regard,
MOMENT is able to provide sensitivities that are comparable to those obtained
for DUNE, T2HK and T2HKK.

There exists synergy between MOMENT and the proposed long-baseline neutrino
experiments. This complementarity is illustrated in Figure 3, where the
expected resolution to $\delta_{CP}$ is projected in the proposed
configurations of MOMENT, DUNE, T2HK and T2HKK. The figure shows the absolute
precision for $\delta_{CP}$ as function of its theoretically allowed values
222It should be noted that at 3$\,\sigma$ CL, the recent experimental data
disfavours values $\delta_{CP}=$ -10∘…144∘ in normal ordering and -15∘…194∘ in
inverted ordering, respectively [1]. in various experiment combinations. It is
shown that the combined data of MOMENT, DUNE and T2HKK is sufficient to
constrain the $\delta_{CP}$ below 12∘ at 1$\,\sigma$ CL. This resolution is
maintained regardless of the true value of $\delta_{CP}$ or the neutrino mass
ordering. In contrast, MOMENT can be anticipated to measure $\delta_{CP}$ at
approximately 8∘…18∘ precision on its own. MOMENT is therefore expected to
have promising prospects in constraining the value of $\delta_{CP}$.

Figure 2: Sensitivity to Dirac CP phase $\delta_{CP}$ in MOMENT and several
long-baseline neutrino experiments. The true value is taken to be
$\delta_{CP}\simeq$ 230∘ for normal ordering (NO) and 278∘ for inverted
ordering (IO). The figure is adapted updated from the similar one in Ref. [3].
Figure 3: Expected precisions on the CP phase $\delta_{CP}$ in MOMENT [3]. The
combined sensitivities for MOMENT and the long-baseline experiments DUNE, T2HK
and T2HKK are shown in terms of absolute precision at 1$\,\sigma$ CL. Here
LBLs+MOMENT stands for the combined run of DUNE, T2HKK and MOMENT.
Sensitivities are shown for normal ordering (NO) and inverted ordering (IO).

##  4 Summary and outlook

The near-future of neutrino oscillation physics can be anticipated to show
advances in the precision measurements of standard neutrino oscillation
parameters. In this proceeding, we have briefly reviewed the prospects to
determine the value of the Dirac CP in the proposed medium-baseline muon-decay
neutrino experiment MOMENT. Owing to the low beam-related backgrounds, greater
selection of oscillation channels and its well-understood neutrino source, the
baseline configuration of MOMENT has been shown to be able to provide
competitive sensitivities to $\delta_{CP}$. As an independent experiment,
MOMENT is expected to reach about 8∘…18∘ precision on $\delta_{CP}$ at
1$\,\sigma$ CL. In combination with the proposed long-baseline experiments
DUNE and T2HKK, the precision can be pushed below 12∘ at 1$\,\sigma$ CL
regardless of the true value of $\delta_{CP}$.

## References

  * [1] I. Esteban et al., JHEP 09 (2020) 178, arXiv:2007.14792.
  * [2] J. Cao et al., Phys. Rev. ST Accel. Beams 17 (2014) 090101, arXiv:1401.8125.
  * [3] J. Tang, S. Vihonen and T.C. Wang, JHEP 12 (2019) 30, arXiv:1909.01548.
  * [4] J. Tang, S. Vihonen and T.C. Wang, Phys. Rev. D 102 (2020) No. 1, 013006, arXiv:2003.02792.
  * [5] J. Tang, S. Vihonen and Y. Xu, arXiv:2108.11107.

## Acknowledgments

This project is partially supported by China Postdoctoral Science Foundation
under Grant No. 2020M672930.


11institutetext: Faculty of Mathematics, Computer Science and Econometrics,
University of Zielona Góra, 65-246 Zielona Góra, Poland,
11email<EMAIL_ADDRESS>
WWW home page: http://staff.uz.zgora.pl/akarczew 22institutetext: Faculty of
Physics and Astronomy, University of Zielona Góra,
65-246 Zielona Góra, Poland,
22email<EMAIL_ADDRESS>
# Signatures of chaotic dynamics in wave motion according to the extended KdV
equation

Anna Karczewska 11 Piotr Rozmej 22

###### Abstract

In this communication we test the hypothesis that for some initial conditions
the time evolution of surface waves according to the _extended KdV equation_
(KdV2) exhibits signatures of the deterministic chaos.

###### keywords:

Extended KdV equation, numerical evolution, deterministic chaos

##  1 Introduction

Korteweg-de Vries equation (KdV) is the most famous nonlinear partial
differential equation modelling long-wave, weekly dispersive gravity waves of
small amplitude on a surface of shallow water. In scaled variables and a fixed
reference frame KdV takes the following form

$\eta_{t}+\eta_{x}+\frac{3}{2}\alpha\eta\eta_{x}+\frac{1}{6}\beta\eta_{xxx}=0.$
(1)

In (1), $\eta(x,t)$ represents the wave profile, $\alpha\\!=\\!A/H$,
$\beta\\!=\\!(H/L)^{2}$, where $A$ is wave’s amplitude, $L$ \- it’s average
wavelength and $H$ is water depth. Indexes indicate partial derivatives. KdV
equation (1) is derived from Euler equations in perturbation approach, under
assumption that parameters $\alpha\\!\approx\\!\beta$ are small. It has
several analytic solutions: single soliton, multi-soliton, and periodic
(cnoidal) ones. KdV is integrable. It also has the unique property of the
infinite number of integral invariants, see, e.g., [1].

In 1990, Marchant and Smyth, extending perturbation approach to second order
in small parameters $\alpha,\beta$, derived the _extended KdV equation_ (KdV2)
[2]

$\displaystyle\eta_{t}+\eta_{x}$
$\displaystyle+\frac{3}{2}\alpha\eta\eta_{x}+\frac{1}{6}\beta\eta_{xxx}-\frac{3}{8}\alpha^{2}\eta^{2}\eta_{x}$
(2)
$\displaystyle+\alpha\beta\left(\frac{23}{24}\eta_{x}\eta_{xx}+\frac{5}{12}\eta\eta_{xxx}\right)+\frac{19}{360}\beta^{2}\eta_{xxxxx}=0.$

Studying this equation, we showed that KdV2 has only one exact invariant,
representing mass (volume) of displaced fluid. The other integral invariants
are only approximate, with deviations of the order of $O(\alpha^{2})$ [3].
Next, we showed that KdV2 equation, despite being non-integrable, has exact
single soliton and periodic solutions in the same form as KdV equation, but
with slightly different coefficients [4, 5, 6].

An exact single soliton solution of KdV2 has the same form as the KdV soliton,
that is,

$\eta(x,t)=A\,\,{\sf sech}[B(x-vt)].$ (3)

However, coefficients $A,B,v$ are uniquely determined by the coefficients of
the KdV2 equation [4]. This property is entirely different from KdV solitons’
properties, for which there is a one-parameter family of possible solutions.
Therefore KdV equation admits multi-soliton solutions, whereas the KdV2
equation does not. Since the equation (2) is second-order in small parameters
$\alpha,\beta$ (assuming that $\alpha\approx\beta$), it should be a good
approximation for much larger values of small parameters than KdV.

Figure 1: Snapshots of time evolution according to the KdV2 equation (2).
Subsequent profiles corresponding to times $t_{n}=n*64$, with
$n=0,1,\ldots,5$, are shifted up by one unit to avoid overlaps. Initial
condition in the form of the Gaussian of the same volume as the soliton, the
same amplitude and velocity.

Figure 2: The same as in Fig. 1 but for initial condition in the form of the
Gaussian which volume is three times greater, with the same amplitude and
velocity but the inverted displacement.

However, in the KdV2 case, for some initial conditions we obtained unexpected
results. In particular, when the initial condition was chosen in the form of
depression (instead of ‘usual’ elevation), the time evolution calculated
according to the KdV2 equation (2) appeared to be entirely different than that
when initial conditions are not much different from the exact soliton. We
first encountered these facts in [7]. We show an example of this behavior in
Figs. 1 and 2.

At the first glance, time evolution presented in Fig. 2 (bottom) looks
_chaotic_. In the next section, we try to verify this observation
quantitatively.

##  2 Dynamics determined by the KdV2 equation

It is well known [8] that the _deterministic chaos_ occurs when trajectories
of the system’s motion, starting from very close initial conditions, diverge
exponentially with time. How can we define the distance between the
trajectories? Let $\eta_{1}(x,t)$ and $\eta_{2}(x,t)$ denote two different
trajectories. Define the following measures of the distance between them:

$\displaystyle M_{1}(t)$
$\displaystyle=\int_{-\infty}^{\infty}|\eta_{1}(x,t)-\eta_{2}(x,t)|\,dx$ (4)
$\displaystyle M_{2}(t)$
$\displaystyle=\int_{-\infty}^{\infty}\left[\eta_{1}(x,t)-\eta_{2}(x,t)\right]^{2}dx.$
(5)

In numerical simulations, we utilize the finite difference method (FDM)
described in detail in [4, 7] with $N$ of the order 5000-10000. Integrals are
approximated by sums, so
$M_{1}(t)\approx\sum_{i=1}^{N}|\eta_{1}(x_{i},t)-\eta_{2}(x_{i},t)|dx$ and
$M_{2}(t)\approx\sum_{i=1}^{N}[\eta_{1}(x_{i},t)-\eta_{2}(x_{i},t)]^{2}dx$. In
calculations, we use the periodic boundary conditions. Therefore, the interval
$x\in[x_{1},x_{2}]$ has to be much wider than the region where the surface
wave is localized. In numerical calculations, presented in [7], when the
initial conditions were exact KdV2 solitons, the invariant
$I_{1}=\int_{-\infty}^{\infty}\eta(x,t)dx$ was conserved with the precision
$10^{-10}-10^{-12}$. In calculations shown in this note, since initial
conditions are much different, $I_{1}$ is conserved up to $10^{-7}$.

Figure 3: Distances between trajectories, which start from almost identical
initial conditions, as functions of time. Lines marked by S correspond to
initial conditions in the form of exact KdV2 soliton (3), whereas those marked
by G correspond to initical conditions in the form of Gaussians. Open symbols
indicate $M_{1}$ measures, whereas the filled ones indicate $M_{2}$ measures.

First, we check the divergence of trajectories when the initial conditions are
close to the exact KdV2 soliton. This is presented in Fig. 3 with lines marked
by S. The lines marked by G in Fig. 3 represent the divergence of trajectories
when $\eta_{1}(x,t=0)$ was the Gaussian distortion having the same amplitude
$A_{1}$ as the KdV2 soliton, with the width $\sigma_{1}$ providing the same
volume. Then, for $\eta_{2}(x,t=0)$ we chose the similar Gaussian form but
with parameters slightly changed, namely with $A_{2}=A_{1}(1+\varepsilon)$ and
$\sigma_{2}=\sigma_{1}/(1+\varepsilon)$, which ensures the same initial
volume. In both cases, initial velocity was assumed to be equal to soliton’s
velocity. Note, that profiles shown in Fig. 2 represent the evolution of
$\eta_{1}(x,t=0)$. Results displayed in Fig. 3 were obtained for
$\varepsilon=10^{-4}$. For $\varepsilon=10^{-3}$ and $\varepsilon=10^{-5}$ the
relative distances $M(t)/M(0)$ behave almost exactly the same. It is clear
that in all cases presented in Fig. 3 the distances between trajectories which
start from neighbor initial conditions diverge linearly with time. For
dynamical systems, it means that such conditions belong to this phase-space
region in which motion is regular.

Figure 4: Time dependence of measures $M_{1}$ for $\varepsilon=10^{-4}$ and
different volumes of initial conditions.

In the below-presented calculations, we studied the motion according to the
KdV2 equation when initial displacements have the sign opposite from the
soliton. For $\eta_{1}(x,t=0)$ we chose the Gaussian distortion having the
amplitude $-A_{1}$ of the KdV2 soliton, but with the width $\sigma_{1}$
ensuring multiple soliton’s volumes. For $\eta_{2}(x,t=0)$ we chose the
similar Gaussian form but with parameters slightly changed, namely with
$A_{2}=-A_{1}(1+\varepsilon)$ and $\sigma_{2}=\sigma_{1}/(1+\varepsilon)$,
which ensures the same initial volume. In both cases, the initial velocity is
chosen the same as the velocity of the KdV2 soliton.

Since in all cases shown below, the distances between trajectories increased
much faster than linearly with time, the next figures are plotted on a
semilogarithmic scale. In Fig. 4, we show the time dependence of the distance
measures $M_{1}$ for $\varepsilon=10^{-4}$. The notation Vol=$n$, with
$n=1,2,3,4$ denotes the initial volume of the $\eta_{1}(x,t=0)$ in the units
of the KdV2 soliton volume. In Fig. 5 we present the analogous results but for
$M_{2}$ measures. In both figures we observe almost perfect exponential
divergence of trajectories, even for $n=1$. Fits to the plots displayed in
Fig. 4 give the following exponents: 0.00475, 0.00785, 0.0086 and 0.0089 for
Vol=1,2,3,4, respectively. For $M_{2}$ measures the fitted exponents are:
0.00477, 0.0051, 0.00539 and 0.00561, respectively. All these exponents are
obtained by fitting in the interval $t\in[150:300]$.

Figure 5: The same as in Fig. 4 but for $M_{2}$ measures.

The results shown above allow us to conclude that for initial conditions
substantially different from the exact KdV2 soliton, the dynamics of motion
governed by the KdV2 equation exhibits exponential divergence of trajectories.

## References

  * [1] Drazin, P.G., Johnson, T.S.: _Solitons: An Introduction._ Cambridge University Press, Cambridge 1989.
  * [2] Marchant, T.R., Smyth, N.F.: _The extended Korteweg-de Vries equation and the resonant flow of a fluid over topography._ J Fluid Mech 1990:221:263-288.
  * [3] Karczewska, A., Rozmej, P., Infeld E., Rowlands G.: _Adiabatic invariants of the extended KdV equation_. Phys Lett A 2017:381:270-275.
  * [4] Karczewska, A., Rozmej, P., Infeld, E.: _Shallow water soliton dynamics beyond Korteweg-de Vries equation_. Phys Rev E 2014:90:012907.
  * [5] Infeld, I., Karczewska, A., Rowlands, G., Rozmej, P.: _Exact cnoidal solutions of the extended KdV equation_. Acta Phys Pol 2018:133:1191-1199.
  * [6] Rozmej, P., Karczewska, A., Infeld E.: _Superposition solutions to the extended KdV equation for water surface waves._ Nonlinear Dynamics 2018:91:1085-1093.
  * [7] Karczewska, A., Rozmej, P.: _Generalized KdV-type equations versus Boussinesq’s equations for uneven bottom – numerical study._ Comput Meth Sci Tech 2020:26(4):121-136.
  * [8] Alligood, K.T., Sauer, T., Yorke, J.A.: _Chaos: an introduction to dynamical systems._ Springer-Verlag, 1997.


# Quantum Coherence of Orbital Angular Momentum Multiplexed Continuous-
Variable Entangled State

Hong Wen  1,2 Li Zeng  1,2 Rong Ma  1,2 Haijun Kang  1,2 Jun Liu  1,2
Zhongzhong Qin  1,2,* and Xiaolong Su1,2  1State Key Laboratory of Quantum
Optics and Quantum Optics Devices, Institute of Opto-Electronics, Shanxi
University, Taiyuan, 030006, People’s Republic of China
2Collaborative Innovation Center of Extreme Optics, Shanxi University,
Taiyuan,Shanxi 030006, People’s Republic of China<EMAIL_ADDRESS>
###### Abstract

Orbital angular momentum (OAM) multiplexed entangled state is an important
quantum resource for high dimensional quantum information processing. In this
paper, we experimentally quantify quantum coherence of OAM multiplexed
continuous-variable (CV) entangled state and characterize its evolution in a
noisy environment. We show that the quantum coherence of the OAM multiplexed
CV entangled state carrying topological charges $l=1$ and $l=2$ are the same
as that of the Gaussian mode with $l=0$ in a noisy channel. Furthermore, we
show that the quantum coherence of OAM multiplexed entangled state is robust
to noise, even though the sudden death of entanglement is observed. Our
results provide reference for applying quantum coherence of OAM multiplexed CV
entangled state in a noisy environment.

††journal: osac††articletype: Research Article

##  1 Introduction

Quantum coherence, which characterizes the quantumness and underpins quantum
correlation in quantum systems, plays a significant role in quantum
information processing [1]. Recently, quantum coherence has been identified as
important quantum resource besides quantum entanglement and steering, and has
attracted rapidly increasing interests [2, 3]. In 2014, Baumgratz et al.
established a framework to quantify quantum coherence by referring to the
method of quantifying entanglement [1]. The quantum coherence of a quantum
state is defined as the minimum distance between the quantum state and an
incoherent state in the Hilbert space, and can be quantified by relative
entropy and $l_{1}$-norm [1]. Besides, it has been shown that quantum
coherence can also be quantified by Fisher information [4], skew information
entropy [5], Tsallis relative $\alpha$ entropy [6], robustness [7], and so on.
Furthermore, quantum coherence with infinite-dimensional systems, i.e.
continuous-variable (CV) quantum states, can be quantified by relative entropy
[8]. Both theoretical investigations and experimental demonstrations of
quantum coherence have achieved significant progresses [9, 10, 11, 12, 13, 14,
15, 16, 17, 18]. Recently, we have experimentally demonstrated the robustness
of Gaussian quantum coherence in quantum channels [19], as well as the
conversion of local and correlated Gaussian quantum coherence [20].

As an important quantum resource, Einstein-Podolsky-Rosen (EPR) entangled
state has been widely applied in quantum communication, quantum computation,
and quantum precision measurement [21, 22, 23, 24, 25, 26, 27]. Besides
optical parametric amplifier, four-wave mixing (FWM) process in warm alkali
vapor cell is another efficient method to generate CV EPR entangled state [28,
29]. The FWM process has been widely used in quantum state engineering [30,
31, 32], quantum beam splitter [33], and quantum precision measurement [34,
35]. Especially, spatial-multi-mode advantage of the FWM process, attributed
to its cavity-free configuration, makes it an ideal method to generate
entangled images [28]. Orbital angular momentum (OAM) multiplexing of light
has been found to be an efficient way to improve data-carrying capacity in
both classical and quantum communications due to its infinite range of
possibly achievable topological charges [36, 37]. Recently, OAM multiplexed
bipartite and multipartite CV entangled states have been generated based on
the FWM process [38, 39, 40], and they have been applied in OAM multiplexed
quantum teleportation [41] and quantum dense coding [42]. It is essential to
distribute OAM multiplexed quantum resources in quantum channels to realize
these quantum information processing protocols. Recently, our group has
experimentally demonstrated that quantum entanglement and quantum steering of
the OAM multiplexed states carrying topological charges $l=1$ and $l=2$ are
the same as that of the Gaussian mode with $l=0$ in lossy and noisy channels
[43]. However, it remains unclear whether the quantum coherence of OAM
multiplexed CV entangled state is also as robust as that of the Gaussian mode
with $l=0$.

Here, we experimentally quantify quantum coherence of OAM multiplexed CV
entangled state and characterize its evolution in a noisy channel. We show
that quantum coherence of CV entangled state carrying topological charges
$l=1$ and $l=2$ are as robust against loss and noise as that of Gaussian mode
entangled state with $l=0$. More interestingly, the quantum coherence of CV
entangled state always exists unless one mode is completely lost, while sudden
death of entanglement is observed in the presence of certain amounts of loss
and noise. Our results pave the way for applying the quantum coherence of OAM
multiplexed entangled state in high data-carrying capacity quantum
communication protocols.

##  2 Theory

The Hamiltonian of the OAM multiplexed FWM process can be expressed as [43]:

$\hat{H}=\sum_{l}i\hbar\gamma_{l}\hat{a}^{\dagger}_{l,P}\hat{a}^{\dagger}_{-l,C}+h.c.$
(1)

where $\gamma_{l}$ is defined as the interaction strength of each OAM pair.
$\hat{a}^{\dagger}_{l,P}$ and $\hat{a}^{\dagger}_{-l,C}$ are the creation
operators related to OAM modes of the Pr and Conj fields, respectively. Since
the pump field does not carry OAM ($l=0$), the topological charges of the Pr
and Conj fields are opposite. The output state of the OAM multiplexed FWM
process is:

$\ket{\Psi}_{out}=\ket{\Psi}_{-l}\otimes\cdots\otimes\ket{\Psi}_{0}\otimes\cdots\otimes\ket{\Psi}_{l}$
(2)

where $\ket{\Psi}_{l}$ presents a series of independent OAM multiplexed CV EPR
entangled states of $\ket{\psi_{l,P}}$ and $\ket{\psi_{-l,C}}$ generated in
the FWM process. $\ket{\psi_{l,P}}$ and $\ket{\psi_{-l,C}}$ represent Pr field
carrying topological charge $l$ and Conj field carrying topological charge
$-l$ , respectively.

A Gaussian state $\hat{\rho}(\bar{\mathbf{x}},\mathbf{V})$ can be completely
represented by the displacement $\bar{\mathbf{x}}$ and the covariance matrix
$\mathbf{V}$ in phase space, which correspond to the first and second
statistical moments of the quadrature operators, respectively [24, 25]. The
displacement $\mathbf{\bar{x}}=\langle\hat{x}\rangle$, where
$\hat{x}\equiv(\hat{X}_{-l,C},\hat{Y}_{-l,C},\hat{X}_{l,P},\hat{Y}_{l,P})^{T}$,
$\hat{X}=\hat{a}+\hat{a}^{{\dagger}}$ and
$\hat{Y}=(\hat{a}-\hat{a}^{{\dagger}})/i$ are the amplitude and phase
quadratures of an optical mode, respectively, and $T$ denotes transpose. The
elements of covariance matrix $\mathbf{V}$ are defined as
$\mathbf{V}_{ij}=\frac{1}{2}\langle\hat{x}_{i}\hat{x}_{j}+\hat{x}_{j}\hat{x}_{i}\rangle-\langle\hat{x}_{i}\rangle\langle\hat{x}_{j}\rangle$.
The Gaussian quantum coherence of EPR entangled state is expressed by [9]

$\mathcal{C}_{rel.~{}ent.}\left[\hat{\rho}(\bar{\mathbf{x}},\mathbf{V})\right]=S\left[\hat{\rho}(\bar{\mathbf{x}}_{th},\mathbf{V}_{th})\right]-S\left[\hat{\rho}(\bar{\mathbf{x}},\mathbf{V})\right],$
(3)

where
$S\left[\hat{\rho}(\bar{\mathbf{x}},\mathbf{V})\right]\\!=-\\!\underset{i=1}{\overset{2}{\sum}}\\!\left[\\!\left(\frac{\nu_{i}-1}{2}\right)\\!\log\\!_{2}\\!\left(\frac{\nu_{i}-1}{2}\right)\\!-\\!\left(\frac{\nu_{i}+1}{2}\right)\\!\log\\!_{2}\\!\left(\frac{\nu_{i}+1}{2}\right)\\!\right]$
and

$S\left[\hat{\rho}(\bar{\mathbf{x}}_{th},\mathbf{V}_{th})\right]\\!\\!=\\!-\\!\underset{i=1}{\overset{2}{\sum}}\\!\left[\\!\left(\frac{\mu_{i}-1}{2}\right)\\!\log\\!_{2}\\!\left(\frac{\mu_{i}-1}{2}\right)\\!-\\!\left(\frac{\mu_{i}+1}{2}\right)\\!\log\\!_{2}\\!\left(\frac{\mu_{i}+1}{2}\right)\\!\right]$
are the von Neumann entropy of $\hat{\rho}(\bar{\mathbf{x}},\mathbf{V})$ and a
thermal state $\hat{\rho}(\bar{\mathbf{x}}_{th},\mathbf{V}_{th})$,
respectively. $\nu_{i}$ and $\mu_{i}$ are the symplectic eigenvalues of
$\mathbf{V}$ and $\mathbf{V}_{th}$, respectively. The elements of the diagonal
covariance matrix $\mathbf{V}_{th}$ are given by $\mathbf{V}$ with $V_{th}\
{}_{2i-1,2i-1}=V_{th}\
{}_{2i,2i}=\frac{1}{2}\left(V_{2i-1,2i-1}+V_{2i,2i}+\left[\mathbf{\bar{x}}_{2i-1}\right]^{2}+\left[\mathbf{\bar{x}}_{2i}\right]^{2}\right)$.

Figure 1: (a) Experimental setup for generating and distributing quantum
coherence of OAM multiplexed CV quantum entangled state in a noisy channel.
Cs: cesium vapor cell; Pr: probe field; Conj: conjugate field; LO-l,P and
LOl,C: local oscillators of Pr and Conj fields; AM: amplitude modulator; PM:
phase modulator; GL: Glan-laser polarizer; GT: Glan-Thompson polarizer; PBS:
polarization beam splitter; HWP: half-wave plate; VPP: vortex phase plate; M:
mirror; BS: 50:50 beam splitter; BHD1, BHD2: balanced homodyne detectors; SA:
spectrum analyzer. (b) Beam patterns and interference patterns of the Pr and
Conj fields for $l=1$. (c) Beam patterns and interference patterns of the Pr
and Conj fields for $l=2$.

In our experiment, the displacements $\bar{\mathbf{x}}$ of the EPR entangled
state are zero, so the state can be completely represented by its covariance
matrix $\mathbf{V}$. The covariance matrix of the OAM multiplexed entangled
state after distribution in a noisy channel is

$\displaystyle\mathbf{V}$
$\displaystyle=\left(\begin{array}[]{cccc}\mathbf{A}&\mathbf{C}\\\
\mathbf{C}^{T}&\mathbf{B}\end{array}\right),$ (6)

where $\mathbf{A}=\frac{V+V^{\prime}}{2}~{}\mathbf{I}$,
$\mathbf{B}=[\eta\frac{V+V^{\prime}}{2}+(1-\eta)(1+\delta)]~{}\mathbf{I}$,
$\mathbf{C}=\sqrt{\eta}\frac{V^{\prime}-V}{2}~{}\mathbf{Z}$,
$\mathbf{I}=\begin{pmatrix}\begin{smallmatrix}1&0\\\
0&1\end{smallmatrix}\end{pmatrix}$ and
$\mathbf{Z}=\begin{pmatrix}\begin{smallmatrix}1&0\\\
0&-1\end{smallmatrix}\end{pmatrix}$. The submatrices $\mathbf{A}$ and
$\mathbf{B}$ correspond to the states of Alice’s and Bob’s subsystems,
respectively. $V$ and $V^{\prime}$ represent the variances of correlation and
anti-correlation quadratures of the EPR entangled state, respectively. Note
that $VV^{\prime}\geq 1$ is always satisfied according to the uncertainty
principle, and the equality holds only for pure state. $\eta$ and $\delta$ (in
the units of SNL) represent transmission efficiency and excess noise of the
noisy channel, respectively. Therefore, $\delta=0$ represents a lossy but
noiseless channel, while $\delta>0$ represents a noisy channel.

The Peres-Horodecki criterion of positivity under partial transpose (PPT)
criterion is a sufficient and necessary criterion to characterize the
entanglement of CV bipartite entanglement [44]. The PPT value is determined by
$\sqrt{\frac{\Gamma-\sqrt{\Gamma^{2}-4\textup{Det}\mathbf{V}}}{2}}$ for a
bipartite Gaussian state, where
$\Gamma=\textup{Det}\mathbf{A}+\textup{Det}\mathbf{B}-2\textup{Det}\mathbf{C}$.
If the PPT value is smaller than 1, bipartite entanglement exists. Otherwise,
it’s a separable state. Furthermore, smaller PPT values represent stronger
entanglement.

Figure 2: Dependence of PPT values and quantum coherence of the OAM
multiplexed CV entangled states on the transmission efficiency $\eta$ for
$l=0$, $l=1$ and $l=2$. (a,b) Case for a lossy channel with the excess noise
$\delta=0$ ; (c,d) Case for a noisy channel with the excess noise $\delta=1$.
The initial PPT value is 0.46$\pm$0.01 at $\eta=1$. Curves and data points
show theoretical predictions and experimental results, respectively. Error
bars of experimental data represent one standard deviation and are obtained
based on the statistics of the measured data. The light blue plane in (c)
shows the boundary for entanglement where PPT value is equal to 1. The three
vertical dashed lines indicate corresponding transmission efficiencies where
sudden death of entanglement starts to appear.

##  3 Experimental setup

Figure 1(a) shows the schematic of experimental setup. The Gaussian mode pump
field and probe field carrying topological charge $l$ of OAM mode cross each
other in the center of the cesium vapor cell at an angle of 6 mrad [29]. In
this way, a conjugate field carrying topological charge $-l$ of OAM mode is
generated on the other side of the pump field. The topological charge of OAM
mode $l=1$ or $l=2$ is added to the probe field by passing it through a vortex
phase plate. The pump field is filtered out by using a Glan-Thompson polarizer
after the vapor cell. The Conj field is kept by Alice, while the Pr field is
distributed to a remote quantum node owned by Bob through a noisy channel. The
noisy channel is realized by overlapping the Pr field with an auxiliary field
at a polarization beam splitter (PBS) followed by a half-wave plate (HWP) and
a PBS. The auxiliary field, which is modulated by an amplitude modulator and a
phase modulator with white noise, has the same topological charge and
frequency with the Pr field so that they can interfere. The amount of excess
noise is controlled by tuning the optical power of the auxiliary field and the
white noise added on it. Covariance matrix of the OAM multiplexed CV entangled
state is experimentally measured by utilizing two sets of balanced homodyne
detectors. In our experiment, the spatially mode-matched local oscillator
beams used in the balanced homodyne detectors are obtained from a second set
of FWM process in the same vapor cell [43].

The top rows of Fig. 1(b) and 1(c) show the beam patterns of the Pr and Conj
fields for $l=1$ and $l=2$, respectively, while the bottom rows show their
interference patterns with plane waves at the same frequencies, from which the
topological charges of these fields are inferred. It is obvious that the
topological charges of the Pr and Conj fields are opposite, which confirms the
OAM conservation in the FWM process.

Figure 3: Dependence of PPT values (a) and quantum coherence (b) of the OAM
multiplexed CV entanglement on transmission efficiency $\eta$ and excess noise
$\delta$ for $l=2$ in a noisy channel. Curve planes and data points show
theoretical predictions and experimental results, respectively. The light blue
plane in (a) shows the boundary for entanglement where PPT value is equal to
1, and the red curve shows the boundary where sudden death of entanglement
starts to appear.

##  4 Experimental results and discussion

In our experiment, the correlation and anti-correlation levels of the initial
CV entangled states carrying topological charges $l=0$, $l=1$, and $l=2$ are
all around $-3.3$ dB and 6.1 dB, which correspond to $V=0.47$ and
$V^{\prime}=4.11$, respectively. The evolution of PPT values and relative
entropy of CV bipartite entangled states carrying different topological
charges in lossy channels ($\delta=0$) are shown in Fig. 2(a) and 2(b),
respectively. The entanglement and quantum coherence between the Pr and Conj
fields both degrade as the transmission efficiency decreases. However, the
entanglement and quantum coherence are both robust against loss, i.e., they
always exist until the transmission efficiency reaches 0. It is obvious that
the entanglement and quantum coherence of the CV bipartite entangled states
carrying topological charges $l=1$, $l=2$ are as robust to loss as their
Gaussian counterpart $l=0$.

The evolution of PPT values and relative entropy of CV bipartite entangled
states carrying different topological charges in a noisy channel with
$\delta=1$ (in the units of SNL) are shown in Fig. 2(c) and 2(d),
respectively. Compared with the results shown in Fig. 2(a) and 2(b) for the
case of lossy channel, both entanglement and quantum coherence degrade faster
as the transmission efficiency decreases. Furthermore, entanglement disappears
at a certain transmission efficiency of the Pr field $\eta=0.44$ in the
presence of excess noise, which demonstrates the sudden death of CV quantum
entanglement. In contrast, the quantum coherence always exists until the
transmission efficiency decreases to 0. We note that OAM multiplexed CV
entangled states carrying high order topological charges $l=1$, $l=2$ exhibit
the same quantum entanglement and quantum coherence evolution tendency as
their Gaussian counterpart $l=0$ in a noisy channel.

Figure 3(a) and 3(b) show the dependence of PPT values and relative entropy on
transmission efficiency $\eta$ and excess noise $\delta$ for CV bipartite
entangled state carrying $l=2$ in noisy channels. Experimental results at four
different amounts of noise $\delta=0,0.15,0.5$ and 1 (in the units of SNL) are
taken. It is obvious that both entanglement and quantum coherence degrade with
the increase of loss and excess noise. The degradations of entanglement and
quantum coherence can be attributed to the incoherent operations of the lossy
and noisy channels [1, 9]. The red curve in Fig. 3(a) shows the boundary where
the sudden death of entanglement appears. It is clear that the sudden death of
entanglement appears at higher transmission efficiency as the excess noise
increases. In contrast, the quantum coherence of the OAM multiplexed CV
entangled state still exists even though the entanglement disappears, i.e.
quantum coherence of the state is robust against noise. The physical reason
for the robustness of quantum coherences in noisy channels is that the
proportion of quantum coherence is decreased when it is mixed with thermal
noise, but the quantum coherence disappears completely only when infinite
thermal noise is involved.

##  5 Conclusions

Here, we experimentally quantify quantum coherence of OAM multiplexed CV
entangled state in a noisy channel. Our results demonstrate that quantum
coherence of OAM multiplexed CV entangled state is robust against loss and
noise, although the sudden death of entanglement is observed at a certain
noise level. Recently, it has been shown that entanglement can be transferred
in a single-mode cavity [45], which is a promising application of robustness
of quantum coherence. Our results lay the foundation of applying quantum
coherence of OAM multiplexed entangled state in noisy environments.

Funding National Natural Science Foundation of China (NSFC) (No. 11974227, No.
11834010, and No. 61905135); Fundamental Research Program of Shanxi Province
(No. 20210302122002); Research Project Supported by Shanxi Scholarship Council
of China (2021-003); Fund for Shanxi “1331 Project" Key Subjects Construction.

Disclosures The authors declare no conflicts of interest.

Data availability Data underlying the results presented in this paper are not
publicly available at this time but may be obtained from the authors upon
reasonable request.

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