# Charmonium and Charmoniumlike States at the BESIII Experiment

Chang-Zheng Yuan<EMAIL_ADDRESS>Institute of High Energy Physics, Chinese
Academy of Sciences, Beijing 100049, China  University of Chinese Academy of
Sciences, Beijing 100049, China

###### Abstract

Charmonium is a bound state of a charmed quark and a charmed antiquark, and a
charmoniumlike state is a resonant structure that contains a charmed quark and
antiquark pair but has properties that are incompatible with a conventional
charmonium state. While operating at center-of-mass energies from 2 to 4.9
GeV, the BESIII experiment can access a wide mass range of charmonium and
charmoniumlike states, and has contributed significantly in this field. We
review BESIII results involving conventional charmonium states, including the
first observation of M1 transition $\psi(2S)\to\gamma\eta_{c}(2S)$ and the
discovery of the $\psi_{2}(3823)$; and report on studies of charmoniumlike
states, including the discoveries of the $Z_{c}(3900)$ and $Z_{c}(4020)$
tetraquark candidates, the resolution of the fine structure of the $Y(4260)$,
the discovery of the new production process $e^{+}e^{-}\to\gamma X(3872)$, and
the uncovering of strong evidence for the commonality among the $X(3872)$,
$Y(4260)$, and $Z_{c}(3900)$ states. The prospects for further research at
BESIII and proposed future facilities are also presented.

charmonium states, charmoniumlike states, exotic hadrons, $e^{+}e^{-}$
annihilation

##  I Introduction

In the conventional quark model, mesons are comprised of a quark and anti-
quark pair, while baryons are comprised of three quarks. A bound state of a
charmed quark ($c$) and a charmed antiquark ($\bar{c}$) is named charmonium.
The first charmonium state, the $J/\psi$, was discovered at BNL ting  and at
SLAC richter  in 1974, and since then, all the charmonium states below the
open-charm threshold and a few vector charmonium states above the open-charm
threshold have been established pdg ; the measured spectrum of states agrees
well with theoretical calculations based on QCD-inspired potential models
eichten ; godfrey ; barnes .

In addition to the charmonium states, the conventional quark model describes
almost all of the other hadrons that have been observed to date quite well,
including baryons and other mesons pdg . Since the very beginning of the quark
model, exotic hadronic states with configurations not limited to two or three
quarks have been the subject of numerous theoretical proposals and
experimental searches Jaffe:2004ph ; klempt . These proposed exotic hadrons
include hadron-hadron molecules, diquark-diantiquark tetraquark states, hadro-
quarkonia, quark-antiquark-gluon hybrids, multi-gluon glueballs, and
pentaquark baryons.

Many charmonium and charmoniumlike states were discovered at the BaBar babar
and Belle belle  $B$-factories during the first decade of this century PBFB .
Whereas some of these are good candidates for conventional charmonium states,
there are other states that have properties that do not match those of any of
the unassigned $c\bar{c}$ states, which may indicate that exotic states have
already been observed reviews . These candidate exotic meson states are
collectively called the $\rm XYZ$ particles, to indicate their underlying
nature is still unclear. Although this is not fully accepted within the high
energy physics community, practitioners in the field use $Z_{Q}({\rm xxxx})$
to denote a quarkoniumlike state with mass roughly ${\rm xxxx}$ MeV/$c^{2}$
that contains a heavy quark pair $Q\bar{Q}$ and with non-zero isospin; $Y({\rm
xxxx})$ for a vector quarkoniumlike state, and $X({\rm xxxx})$ for states with
other quantum numbers.

Although the BaBar babar  and Belle belle  experiments finished data taking in
2008 and 2010, respectively, the data are still used for various physics
analyses. In 2008, two new experiments: BESIII bes3 , a $\tau$-charm factory
experiment at the BEPCII $e^{+}e^{-}$ collider; and LHCb lhcb_detector , a
$B$-factory experiment at the LHC $pp$ collider, started data taking, and have
been contributing to the study of charmonium and charmoniumlike states ever
since.

The BESIII experiment at the BEPCII double ring $e^{+}e^{-}$ collier observed
its first collisions in the $\tau$-charm energy region in July 2008. The
BESIII detector bes3  is a magnetic spectrometer with an effective geometrical
acceptance of 93% of $4\pi$ and state-of-the-art subdetectors for high
precision charged and neutral particle measurements. After a few years of
running at center-of-mass (c.m.) energies for its well-defined physics
programs BESIII_YB , i.e., at the $J/\psi$ and $\psi(2S)$ peaks in 2009 and
the $\psi(3770)$ peak in 2010 and 2011, the BESIII experiment began to collect
data for the study of the $\rm XYZ$ particles, a program that was not
described in the BESIII Yellow Book BESIII_YB . The first data sample was
collected at the $\psi(4040)$ resonance in May 2011 with an integrated
luminosity of about $0.5~{}{\rm fb}^{-1}$. This sample was used to search for
the production of the $X(3872)$ and the excited $P$-wave charmonium spin-
triplet states via $\psi(4040)$ radiative transitions. The size of the sample
was limited by the brief, one-month running time following the $\psi(3770)$
data taking in the 2010–2011 run.

In summer 2012, the LINAC of the BEPCII was upgraded so that the highest beam
energy was increased from $2.1$ to $2.3~{}\rm GeV$, which made it possible to
collect data at higher c.m. energies (up to $4.6~{}\rm GeV$). A data sample of
$525~{}{\rm pb}^{-1}$ was collected at a c.m. energy of $4.26~{}\rm GeV$ from
December 14, 2012 to January 14, 2013, with which the $Z_{c}(3900)$ charged
charmoniumlike state was discovered zc3900 . This observation changed the data
collection plan for the 2012–2013 run and had considerable impact on the
subsequent running schedule of the experiment; more data points between $4.13$
and $4.60~{}\rm GeV$ dedicated to the $\rm XYZ$ related analyses were recorded
lum_songwm . The highest beam energy was further increased from $2.3$ to
$2.45~{}\rm GeV$ in summer 2019, making it possible to collect data at even
higher c.m. energies (up to $4.9~{}\rm GeV$).

The data samples used for the $\rm XYZ$ study cover the energy range between
4.0 and 4.7 GeV, with a typical integrated luminosity of $500~{}{\rm pb}^{-1}$
at each energy point. These data were also used for charmonium studies
together with a 448 million $\psi(2S)$ event sample. Data samples with a total
$826~{}{\rm pb}^{-1}$ integrated luminosity at 104 energy points between $3.8$
and $4.6~{}\rm GeV$ lum_Rscan  was also used for the $\rm XYZ$ study.

In this article, we review studies of charmonium and charmoniumlike states
from the BESIII bes3  experiment. We first introduce the study of conventional
charmonium states and then the $\rm XYZ$ states. Finally, we discuss prospects
for future studies with the BESIII experiment, and also point out possible
studies at next generation facilities.

##  II Conventional charmonium states

The search for new charmonium states has always been a high priority topic.
With the data taken at c.m. energies above 4 GeV, it is possible to search for
states predicted by the potential models that are still unobseved eichten ;
godfrey ; barnes . These states include the excited $P$-wave spin-triplet
states $\chi_{cJ}(2P)$ ($J=0$, $1$, $2$), the excited $P$-wave spin-singlet
state $h_{c}(2P)$, the $D$-wave spin-triplet states $\psi_{J}(1D)$ ($J=2$,
$3$; $J=1$ state, the $\psi(3770)$, was observed many years ago pdg ), and the
$D$-wave spin-singlet state $\eta_{c2}(1D)$.

The predicted mass of the $D$-wave charmonium states (excluding the
$\psi(3770)$, which is, in fact, a mixture of the $1\,^{3}D_{1}$ and
$2\,^{3}S_{1}$ vector states) is in the $3.81\sim 3.85$ GeV/$c^{2}$ range
predicted by several phenomenological calculations eichten ; godfrey ; barnes
; 3d2-mass . Since the mass of $\psi_{2}(1D)$ is above the $D\bar{D}$
threshold but below the $D\bar{D}^{*}$ threshold, and $\psi_{2}(1D)\to
D\bar{D}$ violates parity, the $\psi_{2}(1D)$ is expected to be narrow and its
dominant decay mode is $\psi_{2}(1D)\to\gamma\chi_{c1}$ 3d2-mass ; ratio . The
$\psi_{2}(1D)$ state, also called the $\psi_{2}(3823)$, was discovered at
BESIII BES3x3823  in this final state, and the $\psi_{3}(1D)$ state was
observed by LHCb in its decay into $D\bar{D}$ final state lhcb-3d3 .

The spin-triplet charmonium states are produced copiously in $e^{+}e^{-}$
annihilation and in $B$ decays and, thus, they are understood much better than
the spin-singlet charmonium states, including the lowest lying $S$-wave state,
the $\eta_{c}$, its radial excited partner the $\eta_{c}(2S)$, and the
$P$-wave spin-singlet state the $h_{c}$. Since these three states are all
produced in $\psi(2S)$ decays, the world largest $\psi(2S)$ data sample at
BESIII made it possible to study their properties with improved precision. In
addition, the unexpected large production cross section for
$e^{+}e^{-}\to\pi^{+}\pi^{-}h_{c}$ in BESIII energy region
bes3_pipihc_lineshape  opened a new mechanism of studying the $h_{c}$ and
$\eta_{c}$ (from $h_{c}\to\gamma\eta_{c}$), and BESIII contributed world best
measurements of the properties of these states bes3_etacbr . We report here
the observation of the M1 transition $\psi(2S)\to\gamma\eta_{c}(2S)$ at BESIII
bes3_etacp , a transition that has been sought for since the first generation
BES experiment in 1980’s.

###  II.1 Discovery of M1 transition $\psi(2S)\to\gamma\eta_{c}(2S)$

The production of the $\eta_{c}(2S)$ through a radiative transition from the
$\psi(2S)$ involves a charmed-quark spin-flip and, thus, proceeds via a
magnetic dipole (M1) transition. The branching fraction has been calculated by
many authors, with predictions in the range ${\cal
B}(\psi(2S)\to\gamma\eta_{c}(2S))=(0.1-6.2)\times 10^{-4}$ gaoky ; zhaoq ;
mabq . Experimentally, this transition has been searched for by Crystal Ball
cbal , BES ycz ; initial_psp , and CLEO etacp_cleo_c . No convincing signal
was observed by any of these experiments.

With a sample of 106 million $\psi(2S)$ events collected at BESIII, the
process $\psi(2S)\to\gamma\eta_{c}(2S)$ was observed for the first time with
$\eta_{c}(2S)\to K_{S}^{0}K^{\pm}\pi^{\mp}$ and $K^{+}K^{-}\pi^{0}$ modes. The
final $K\bar{K}\pi$ mass spectra and the fit results are shown in Fig. 1. The
fit yields for the number of the $\eta_{c}(2S)$ signal events are $81\pm 14$
for the $K_{S}^{0}K^{\pm}\pi^{\mp}$ mode and $46\pm 11$ for the
$K^{+}K^{-}\pi^{0}$ mode; the overall statistical significance of the signal
is larger than $10\sigma$ bes3_etacp .

Figure 1: The fit to the invariant-mass spectra for
$K_{S}^{0}K^{\pm}\pi^{\mp}$ (left panel) and $K^{+}K^{-}\pi^{0}$ (right
panel). Dots with error bars are data, and the curves are total fit and each
component. The lowest peaks correspond to the $\eta_{c}(2S)$ signals.

The mass of the $\eta_{c}(2S)$ is measured to be $(3637.6\pm 2.9\pm 1.6)$
MeV/$c^{2}$, the width $(16.9\pm 6.4\pm 4.8)$ MeV, in good agreement with the
PDG world average values pdg , and the product branching fractions ${\cal
B}(\psi(2S)\to\gamma\eta_{c}(2S))\times{\cal B}(\eta_{c}(2S)\to
K\bar{K}\pi)=(1.30\pm 0.20\pm 0.30)\times 10^{-5}$. Combining this result with
a BaBar measurement of ${\cal B}(\eta_{c}(2S)\to K\bar{K}\pi)$, the M1
transition rate is determined to be ${\cal
B}(\psi(2S)\to\gamma\eta_{c}(2S))=(6.8\pm 1.1_{\mathrm{stat}}\pm
4.5_{\mathrm{sys}})\times 10^{-4}$. This agrees with theoretical calculations
gaoky ; zhaoq ; mabq  and naive estimates based on the
$J/\psi\to\gamma\eta_{c}$ transition etacp_cleo_c .

This study benefited from the BESIII detector’s high resolution
electromagnetic calorimeter, which makes the detection of the radiative photon
with 50 MeV energy possible bes3 . Given the tiny transition rate and the low
photon energy, it is understandable why this transition was not observed in
previous studies cbal ; ycz ; etacp_cleo_c . This is the third M1 transition
observed in a charmonium system (the other two are $J/\psi\to\gamma\eta_{c}$
and $\psi(2S)\to\gamma\eta_{c}$ observed in 1980 cbal_etac ); improved
measurements of these transitions and discovery of more M1 transitions would
improve the understanding of the high order effects involved in these
transitions barnes .

###  II.2 Observation of $\psi_{2}(3823)$

The processes of $e^{+}e^{-}\to\pi^{+}\pi^{-}\gamma\chi_{c1,2}$ are studied at
BESIII experiment using $4.1~{}{\rm fb}^{-1}$ data samples collected at c.m.
energies from $4.23$ to $4.60~{}\rm GeV$ BES3x3823 . The $\chi_{c1,2}$ are
reconstructed via their decays into $\gamma J/\psi$, with $J/\psi$ to
$\ell^{+}\ell^{-}$ ($\ell=e,\mu$). A clear signal is observed as a $19\pm 5$
event peak in the $\gamma\chi_{c1}$ invariant mass distribution that is
evident in Fig. 2(1eft); its mass is determined to be $(3821.7\pm 1.3\pm
0.7)~{}{\rm MeV}/c^{2}$, and its properties are in good agreement with the
$\psi(1^{3}D_{2})$ charmonium state. The statistical significance of the
$\psi_{2}(3823)$ signal is $6.2\sigma$. The upper limit on the intrinsic width
of the $\psi_{2}(3823)$ is determined as $16~{}\rm MeV$ at the 90% confidence
level (C.L.). This observation is in good agreement with the $3.8\sigma$
“evidence” in $B$ decays reported by Belle belle-3d2 . For the
$\gamma\chi_{c2}$ mode, no significant $\psi_{2}(3823)$ signal is observed
(Fig. 2(right)), and an upper limit on its production rate is determined.
BESIII obtains the ratio $\frac{{\cal
B}[\psi_{2}(3823)\to\gamma\chi_{c2}]}{{\cal
B}[\psi_{2}(3823)\to\gamma\chi_{c1}]}<0.42$ at the 90% C.L., which also agrees
with expectations for the $\psi(1\,^{3}D_{2})$ state ratio .

Figure 2: Simultaneous fit to the $M_{\rm recoil}(\pi^{+}\pi^{-})$
distribution of $\gamma\chi_{c1}$ events (left) and $\gamma\chi_{c2}$ events
(right), respectively BES3x3823 . The small peak in the left panel is the
$\psi_{2}(3823)$ signal. Dots with error bars are data, red solid curves are
total fit, dashed blue curves are background, and the green shaded histograms
are $J/\psi$ mass sideband events.

With the observation of three $D$-wave spin-triplet states, their center-of-
gravity, 3822 MeV/$c^{2}$, is a good estimation of the mass of the $D$-wave
spin-singlet state, $\eta_{c2}(1D)$. Since it cannot decay into open charm
final states, the $\eta_{c2}(1D)$ is expected to be very narrow, and the
identification of it should be clear, if it is produced with large enough
rate, in $e^{+}e^{-}\to\gamma\eta_{c2}(1D)$ or
$e^{+}e^{-}\to\pi^{+}\pi^{-}h_{c}(2P)\to\pi^{+}\pi^{-}\gamma\eta_{c2}(1D)$.

##  III Exotic charmoniumlike states

A revival of the study of charmonium spectroscopy occurred in the early 21st
century when the BaBar and Belle $B$-factories started accumulating large data
samples at $\Upsilon(4S)$ peak. The high luminosity at these $B$-factories
enabled studies of charmonium states that are produced in a variety of ways,
including $B$ decays, initial-state-radiation (ISR) processes, double-
charmonium production, two-photon processes, etc. While the discovery of the
conventional charmonium states such as $\eta_{c}(2S)$ and $\chi_{c2}(2P)$ were
more-or-less rountine, the observations of the $X(3872)$ by Belle in 2003
bellex  and the $Y(4260)$ by BaBar in 2005 babar_y4260 , the first of the $\rm
XYZ$ mesons, came as big surprises; although these new states decay to final
states that contain both a $c$\- and a $\bar{c}$-quark, they have properties
that do not match those of any $c\bar{c}$ meson reviews .

All studies of $\rm XYZ$ states at the $B$-factories suffer from low
statistics and limited precision. In contrast, BESIII can tune the c.m. energy
to match the peaks of the $Y$ states, where event rates are high enough to
facilitate precise measurements of their resonance parameters and search for
new states among their decay products.

###  III.1 New insights into the $Y$ states

The $Y$ states, such as the $Y(4260)$ babar_y4260 , the $Y(4360)$ babar_y4360
; belle_y4660  and the $Y(4660)$ belle_y4660 , are produced directly or via
the ISR process in $e^{+}e^{-}$ annihilation and, thus, are vectors with
quantum numbers $J^{PC}=1^{--}$. These states have strong couplings to hidden-
charm final states in contrast to the established vector charmonium states in
the same mass region, such as the $\psi(4040)$, $\psi(4160)$, and
$\psi(4415)$, which dominantly couple to open-charm meson pairs bes2_psis ;
uglov-kmatrix .

In potential models, five vector charmonium states are expected to be in the
mass region between 4.0 and 4.7 GeV/$c^{2}$, namely the $\psi(3S)$,
$\psi(2D)$, $\psi(4S)$, $\psi(3D)$, and $\psi(5S)$ 3d2-mass , with the first
three identified with the well-established $\psi(4040)$, $\psi(4160)$, and
$\psi(4415)$ charmonium mesons; the masses of the as yet undiscovered
$\psi(3D)$ and $\psi(5S)$ are expected to be higher than 4.4 GeV/$c^{2}$.
However, six vector states in this mass region have been identified, as listed
above. These makes the $Y(4260)$, $Y(4360)$ and perhaps the $Y(4660)$ states
good candidates for new types of exotic particles and this stimulated many
theoretical interpretations, including tetraquark states, molecular states,
hybrid states, or hadro-charmonia reviews .

The $Y(4260)$ was first observed at the $B$-factories as a distinct peak in
the $\pi^{+}\pi^{-}J/\psi$ invariant mass distribution for ISR-produced
$e^{+}e^{-}\to\gamma_{\rm ISR}\pi^{+}\pi^{-}J/\psi$ events belley ;
babar_y4260 . Improved measurements from both BaBar babar_y4260_new  and Belle
belley_new  with their full data samples confirmed the existence of both the
$Y(4260)$ and a non-$Y(4260)$ component in $e^{+}e^{-}\to\pi^{+}\pi^{-}J/\psi$
around $4.0~{}\rm GeV$, but the line-shape was parameterized with different
models. The parameters of the $Y(4260)$ determined by fit to the combined data
from the two $B$-factory experiments and the CLEO measurements cleo_isry  are
$M_{Y(4260)}=(4251\pm 9)~{}{\rm MeV}/c^{2}$ and $\Gamma_{Y(4260)}=(120\pm
12)~{}\rm MeV$ pdg2016 . High precision BESIII measurements of the direct
cross section for the $Y(4260)$ production in different final states supply
new insight into its nature. These measurements include:
$e^{+}e^{-}\to\pi^{+}\pi^{-}J/\psi$ bes3_pipijpsi_lineshape ,
$e^{+}e^{-}\to\pi^{+}\pi^{-}h_{c}$ bes3_pipihc_lineshape ,
$e^{+}e^{-}\to\omega\chi_{cJ}$ bes3_omegachic0 ; bes3_omegachic0_new ,
$e^{+}e^{-}\to D^{0}D^{*-}\pi^{+}+c.c.$ bes3_ddstarpi , and so on bes3_white .

Figure 3 shows the measured cross sections for each of these final states. The
$Y(4260)$ structure is evident, but its line shape is, in fact not well
described by a single Breit-Wigner (BW) resonance function. Instead, its line-
shape is peaked at around $4.22~{}\rm GeV$, which is substantially lower than
the average value from previous measurements pdg2016 , and a distinct shoulder
is observed on its high-mass side that is especially pronounced in the
$\pi^{+}\pi^{-}J/\psi$ mode. In order to describe this line shape, two
resonant structures in the $Y(4260)$ peak region are needed. The lower one has
a mass of $(4222.0\pm 3.1\pm 1.4)~{}{\rm MeV}/c^{2}$ and a width of $(44.1\pm
4.3\pm 2.0)~{}\rm MeV$, while the higher one has a mass of $(4320.0\pm 10.4\pm
7.0)~{}{\rm MeV}/c^{2}$ and a width of $(101.4^{+25.3}_{-19.7}\pm 10.2)~{}\rm
MeV$. The mass of the first resonance is $\sim 30~{}{\rm MeV}/c^{2}$ lower
than the world average value at that time pdg2016  for the $Y(4260)$ and its
width is about a factor of three narrower. The second resonance is observed in
the $e^{+}e^{-}\to\pi^{+}\pi^{-}J/\psi$ process for the first time. The
resonance parameters for the $Y(4260)$ structures are also measured in other
decay channels and listed in Table 1.

Figure 3: From top to bottom are the measured cross sections of $e^{+}e^{-}\to\pi^{+}\pi^{-}J/\psi$ bes3_pipijpsi_lineshape , $e^{+}e^{-}\to\pi^{+}\pi^{-}h_{c}$ bes3_pipihc_lineshape , $e^{+}e^{-}\to\omega\chi_{c0}$ bes3_omegachic0 ; bes3_omegachic0_new , and $e^{+}e^{-}\to D^{0}D^{*-}\pi^{+}+c.c.$ bes3_ddstarpi . Dots with error bars are the data and the dotted vertical line is the peak of the $Y(4220)$. Table 1: Resonance parameters of the $Y(4220)$ from different modes measured at BESIII. The cross sections measured at c.m. energy of 4.226 GeV are also listed. Mode | Mass (GeV/$c^{2}$) | Width (MeV) |  $\sigma$ at $\sqrt{s}=4.226$ GeV (pb)
---|---|---|---
$e^{+}e^{-}\to\pi^{+}\pi^{-}J/\psi$ | $4222.0\pm 3.1\pm 1.4$ | $44.1\pm 4.3\pm 2.0$ | $85.1\pm 1.5\pm 4.9$
$e^{+}e^{-}\to\pi^{+}\pi^{-}h_{c}$ | $4218.4^{+5.5}_{-4.5}\pm 0.9$ | $66.0^{+12.3}_{-8.3}\pm 0.4$ | $55.2\pm 2.6\pm 8.9$
$e^{+}e^{-}\to\omega\chi_{c0}$ | $4218.5\pm 1.6\pm 4.0$ | $28.2\pm 3.9\pm 1.6$ | $55.4\pm 6.0\pm 5.9$
$e^{+}e^{-}\to\pi^{+}D^{0}D^{*-}+c.c.$ | $4228.6\pm 4.1\pm 6.3$ | $77.0\pm 6.8\pm 6.3$ | $252\pm 5\pm 15$

Since the resonant structure around $4.2~{}{\rm GeV}/c^{2}$ is present in all
of the above channels with similar resonance parameters, the authors of Ref.
gaoxy  applied a combined fit to the measured cross sections to determine the
resonance parameters of the low-mass $Y(4220)$ peak with a resultant mass of
$(4219.6\pm 3.3\pm 5.1)~{}{\rm MeV}/c^{2}$ and width of $(56.0\pm 3.6\pm
6.9)~{}\rm MeV$. These values are very different from those obtained from
previous experiments pdg2016 . The fit also gives the product of the leptonic
decay width and the decay branching fraction to the considered final state.
After accounting for the unmeasured isospin partners of the measured channels,
a lower limit on the leptonic partial width of the $Y(4220)$ is determined to
be $\Gamma_{e^{+}e^{-}}>(29.1\pm 7.4)~{}\rm eV$, where the error is the
combined fit error and those from different fit scenarios. The authors of Ref.
zhenghq  analyzed BESIII, Belle, and BaBar data on charmonium as well as open
charm final states, and a leptonic width of $O(0.1\sim 1)$ keV is obtained.
This partial width is much larger than LQCD predictions for a hybrid vector
charmonium state chenying .

In spite of the limited experimental information that had been available
between the time of its discovery in 2005 and the recent BESIII measurements,
the $Y(4260)$ has attracted considerable attention. The BESIII measurements of
its production, decay, and line shape in a variety of final states enable more
sophisticated theoretical investigations and some analyses have been
performed, such as those in Ref. zhenghq  and those quoted in Ref. reviews .
The presence of the nearby $D_{s}^{*+}D_{s}^{*-}$, $D\bar{D}_{1}(2420)$, and
$\omega\chi_{cJ}$ production thresholds, and its mass overlap with the
$\psi(4160)$ and $\psi(4415)$ conventional charmonium states complicate its
interpretation. Joint experimental and theoretical efforts will likely be
required to gain a full understanding of the nature of this state.

###  III.2 Discovery of the iso-triplet charmoniumlike $Z_{c}(3900)$ and
$Z_{c}(4020)$ states

Searching for charged charmoniumlike states is one of the most promising ways
of establishing the existence of the exotic hadrons, since such a state must
contain at least four quarks and, thus, could not be a conventional meson.
These searches have been concentrated on decay final states that contain one
charged pion and a charmonium state, such as the $J/\psi$, $\psi(2S)$, and
$h_{c}$, since they are narrow and their experimental identification is
relatively unambiguous.

The first reported charged charmoniumlike state, the $Z_{c}(4430)^{-}$, was
found in the $\pi^{-}\psi(2S)$ invariant mass distribution in $B\to
K\pi^{-}\psi(2S)$ decays by the Belle experiment in 2008 Belle_zc4430 ;
Belle_zc4430pwa . It was confirmed by the LHCb experiment seven years later
LHCb_zc4430 . The $Z_{c}(3900)^{-}$ was observed in the $\pi^{-}J/\psi$
invariant mass distribution in the study of
$e^{+}e^{-}\to\pi^{+}\pi^{-}J/\psi$ at BESIII zc3900  and Belle belley_new
experiments, and the $Z_{c}(4020)^{-}$ was observed in the $\pi^{-}h_{c}$
system in $e^{+}e^{-}\to\pi^{+}\pi^{-}h_{c}$ zc4020  only at BESIII.

####  III.2.1 Observation of the $Z_{c}(3900)$

The BESIII experiment studied the $e^{+}e^{-}\to\pi^{+}\pi^{-}J/\psi$ process
using a $525~{}{\rm pb}^{-1}$ data sample at a c.m. energy of $4.26~{}\rm GeV$
zc3900 . About $1500$ signal events were observed and the cross section was
measured to be $(62.9\pm 1.9\pm 3.7)$ pb, which agrees with the previous
existing results from the Belle belley  and BaBar babar_y4260_new
experiments. The intermediate states in this three-body system were studied by
examining the Dalitz plot of the selected candidate events, as shown in Fig.
4.

Figure 4: Dalitz plot for selected $e^{+}e^{-}\to\pi^{+}\pi^{-}J/\psi$ events
in the $J/\psi$ signal region (left, the inset show background events from the
$J/\psi$ mass sidebands) and the $Z_{c}(3900)$ signal in the
$M_{\mathrm{max}}(\pi J/\psi)$ (right). Points with error bars are data, the
curves are the best fit, the dashed histograms are the phase space
distributions and the shaded histograms are the non-$\pi^{+}\pi^{-}J/\psi$
background estimated from the normalized $J/\psi$ sidebands.

In addition to the known $f_{0}(500)$ and $f_{0}(980)$ structures in the
$\pi^{+}\pi^{-}$ system, a structure at around $3.9~{}{\rm GeV}/c^{2}$ was
observed in the $\pi^{\pm}J/\psi$ invariant mass distribution with a
statistical significance larger than $8\sigma$, which is referred to as the
$Z_{c}(3900)$. A fit to the $\pi^{\pm}J/\psi$ invariant mass spectrum (see
Fig. 4) determined its mass to be $(3899.0\pm 3.6\pm 4.9)~{}{\rm MeV}/c^{2}$
and its width to be $(46\pm 10\pm 20)~{}\rm MeV$.

An article from the Belle experiment that was released subsequent to the
BESIII paper reported the observation of the $Z_{c}(3900)$ state (referred to
as $Z(3900)^{+}$ in the Belle paper) produced via the ISR process with a mass
of $(3894.5\pm 6.6\pm 4.5)~{}{\rm MeV}/c^{2}$ and a width of $(63\pm 24\pm
26)~{}\rm MeV$ with a statistical significance larger than $5.2\sigma$
belley_new . These observations were later confirmed by an analysis of CLEO-c
data at a c.m. energy of $4.17~{}\rm GeV$ seth_zc , with a mass and width that
agree with the BESIII and Belle measurements.

BESIII studied the spin-parity of the $Z_{c}(3900)$ with a partial wave
analysis (PWA) of about 6000 $e^{+}e^{-}\to\pi^{+}\pi^{-}J/\psi$ events at
$\sqrt{s}=4.23$ and $4.26~{}\rm GeV$ zc3900_jpc . The fit indicated that the
spin-parity $J^{P}=1^{+}$ assignment for the $Z_{c}(3900)$ is favored over
other quantum numbers ($0^{-}$, $1^{-}$, $2^{-}$, and $2^{+}$) by more than
7$\sigma$.

The $Z_{c}(3900)$ mass determined from its $\pi J/\psi$ invariant mass
distribution is slightly above the $D\bar{D}^{*}$ mass threshold. The open-
charm decay $Z_{c}(3900)^{\pm}\to(D\bar{D}^{*})^{\pm}$ was observed with much
larger rate than that to $\pi J/\psi$ zc3885st ; zc3885dt , and the pole mass
and width were determined with high precision to be $(3882.2\pm 1.1\pm
1.5)~{}{\rm MeV}/c^{2}$ and $(26.5\pm 1.7\pm 2.1)~{}\rm MeV$, respectively.

In both the QCD tetraquark and the molecular pictures, the $Z_{c}(3900)^{\pm}$
states are the $I_{3}=\pm 1$ members of an isospin triplet. BESIII confirmed
this by observing their neutral, isospin $I_{3}=0$ partners: the
$Z_{c}(3900)^{0}$, in both the $\pi^{0}J/\psi$ Ablikim:2015tbp  and
$(D\bar{D}^{*})^{0}$ Ablikim:2015gda  decay modes. These observations
establish the $Z_{c}(3900)$ as isovector states with even $G$-parity. From a
PWA to the $e^{+}e^{-}\to\pi^{0}\pi^{0}J/\psi$ data in the vicinity of the
$Y(4260)$ resonance, it is found that the cross section line shape of
$e^{+}e^{-}\to\pi^{0}Z_{c}(3900)^{0}\to\pi^{0}\pi^{0}J/\psi$ is in agreement
with that of the $Y(4220)$ (see Fig. 5) zc0-lineshape .

Figure 5: The cross sections of
$e^{+}e^{-}\to\pi^{0}Z_{c}(3900)^{0}\to\pi^{0}\pi^{0}J/\psi$ zc0-lineshape .
Points with error bars are data, the red solid curve is the total fit result,
the red-dashed (blue-dotted) curve is the resonant (non-resonant) component,
and the magenta dash-dotted line represents the interference of the two
components.

BESIII also searched for the $Z_{c}(3900)$ isospin violating decay mode $\eta
J/\psi$ Zc_jpsieta_bes3  as well as to the light hadron final states
$\omega\pi$ Zc_omegapi_bes3 , $K\bar{K}\pi$ and $K\bar{K}\eta$ Zc_kkpi_bes3 ,
these were not observed and the upper limits of the decay rates are one order
of magnitude or even smaller than that for $Z_{c}(3900)\to\pi J/\psi$, as
naively expected.

####  III.2.2 Observation of the $Z_{c}(4020)$

The process $e^{+}e^{-}\to\pi^{+}\pi^{-}h_{c}$ was observed at c.m. energies
of $3.90-4.42~{}\rm GeV$ zc4020  with cross section that is similar to that
for $e^{+}e^{-}\to\pi^{+}\pi^{-}J/\psi$ bes3_pipijpsi_lineshape . Intermediate
states of this three-body system were studied by examining the Dalitz plot of
the selected $\pi^{+}\pi^{-}h_{c}$ candidate events, similar to what was done
for $e^{+}e^{-}\to\pi^{+}\pi^{-}J/\psi$ process zc3900 . Although there are no
clear structures in the $\pi^{+}\pi^{-}$ system, there is distinct evidence
for an exotic charmoniumlike structure in the $\pi^{\pm}h_{c}$ system, as
clearly evident in the Dalitz plot shown in Fig. 6. This figure also shows
projections of the $M(\pi^{\pm}h_{c})$ (two entries per event) distribution
for the signal events as well as the background events estimated from
normalized $h_{c}$ mass sidebands. There is a significant peak at around
$4.02~{}{\rm GeV}/c^{2}$ (the $Z_{c}(4020)$), and there are also some events
at around $3.9~{}{\rm GeV}/c^{2}$ that could be due to the $Z_{c}(3900)$. The
mass and width of the $Z_{c}(4020)$ were measured to be $(4022.9\pm 0.8\pm
2.7)~{}{\rm MeV}/c^{2}$ and $(7.9\pm 2.7\pm 2.6)~{}\rm MeV$, respectively. The
statistical significance of the $Z_{c}(4020)$ signal is greater than
$8.9\sigma$.

Figure 6: Dalitz plot ($M^{2}_{\pi^{+}h_{c}}$ vs. $M^{2}_{\pi^{+}\pi^{-}}$)
for selected $e^{+}e^{-}\to\pi^{+}\pi^{-}h_{c}$ events (left) and the
$Z_{c}(4020)$ signal observed in $\pi h_{c}$ invariant mass spectrum (right).
Points with error bars are data, the solid curves are the best fit, the shaded
histograms are the non-$\pi^{+}\pi^{-}h_{c}$ background estimated from the
normalized $h_{c}$ sidebands.

In an analysis of $e^{+}e^{-}\to\pi^{0}\pi^{0}h_{c}$ process, the
$Z_{c}(4020)^{0}$, the neutral isospin partner of the $Z_{c}(4020)^{\pm}$ was
observed in $\pi^{0}h_{c}$ system. This indicates that the $Z_{c}(4020)$ is an
isovector state zc0_4020 . The open charm decay of the $Z_{c}(4020)$ was
observed in $e^{+}e^{-}\to(D^{*}\bar{D}^{*})\pi$, with a rate that is much
larger than that for its decay into $\pi h_{c}$ zc4025 ; zc0_4025 .

####  III.2.3 Nature of the $Z_{c}$ states

Although many measurements have been performed on the $Z_{c}(3900)$ and
$Z_{c}(4020)$, the experimental information is still not very precise. From
the experience of the $Z_{c}(4430)$, we know that the resonance parameters
determined from a simple one-dimensional fit to the invariant mass
distribution Belle_zc4430  may differ from those based on a full amplitude
analysis with the interference effects between different amplitudes considered
properly Belle_zc4430pwa . The same thing may happen with the $Z_{c}(3900)$
and $Z_{c}(4020)$. Amplitude analyses that are applied to the relevant final
states that extract the resonant parameters as well as the couplings to
different modes are essential to obtain more refined information for
understanding the nature of these states. In addition, PWA also can provide
measurements of Argand plots of the $Z_{c}$ amplitudes, which can be used to
discriminate between different models for the $Z_{c}$ states.

The production of $Z_{c}$ states at a variety of c.m. energies can reveal
whether these states are from resonance decays or continuum production. So far
only the $Z_{c}(3900)$ has been observed both in $e^{+}e^{-}$ annihilation
zc3900  and in $B$-hadron decays D0-zc3900-1 ; D0-zc3900-2 . Searches for
these states in different production modes is of great importance.

These states seem to indicate that a new class of hadrons has been observed.
Since there are at least four quarks within each of these $Z_{c}$ states, they
have been alternatively interpreted as compact tetraquark states, molecular
states of two charmed mesons ($D\bar{D}^{*}+D^{*}\bar{D}$, $D^{*}\bar{D}^{*}$,
etc.), hadro-quarkonium states, or other multiquark configurations; in some
phenomenological studies they have been attributed to purely kinematical
effects reviews . Since many of these models suffer from assumptions that are
hard to prove, it is essential that non-perturbative studies such as lattice
QCD (LQCD) provide a way to understand their underlying nature; if the $Z_{c}$
structures are not purely kinematical effects, they should appear on the
lattice since they are strong interaction phenomena.

The currently available LQCD calculations that are relevant to the
$Z_{c}(3900)$ suffer from a number of uncertainties, as has been recently
reviewed in Ref. LiuChuan . These include the lattice spacing, the volume, the
physical $\pi$ mass, and the channels that are considered in the calculation.

An early lattice study performed by Prelovsek et al. investigated the energy
levels of two-meson systems including $\pi J/\psi$, $\pi\psi(2S)$,
$\rho\eta_{c}$, $D\bar{D}^{*}$, $D^{*}\bar{D}^{*}$, etc., as well as
tetraquark operators. However, no convincing signals for extra new energy
levels apart from the almost free scattering states of the two mesons were
identified Prelovsek:2014swa . Taking $D\bar{D}^{*}$ as the main relevant
channel, the CLQCD collaboration performed a calculation that was based on the
single-channel Lüscher finite-size formalism and found a slightly repulsive
interaction between the two charmed mesons Chen:2014afa ; Chen:2015jwa . The
results therefore do not support the possibility of a shallow bound state for
the two mesons for the pion mass values of 300, 420, and 485 MeV/$c^{2}$. A
preliminary study using staggered quarks finds no $J^{PC}=1^{+-}$ state
distinct from the noninteracting scattering states either, but the authors
also pointed out that future calculations with a larger interpolating operator
basis may be able to resolve this state Lee:2014uta .

The HALQCD collaboration studied the problem using an approach where an
effective potential is extracted from the lattice data and then used to solve
the Schrödinger-like equations Ikeda:2016zwx ; Ikeda:2017mee . A fully
coupled-channel potential for $\pi J/\psi$, $\rho\eta_{c}$, and $D\bar{D}^{*}$
interactions is obtained, and a strong off-diagonal transition between $\pi
J/\psi$ and $D\bar{D}^{*}$ indicates that the $Z_{c}(3900)$ can be explained
as a threshold cusp within their current configuration ($m_{\pi}=400-700$
MeV/$c^{2}$). In order to establish a definite conclusion on the structure of
the $Z_{c}(3900)$ in the real world, full QCD simulations near the physical
point are being carried out Ikeda:2016zwx ; Ikeda:2017mee .

Recently, in order to clarify the mismatch between these two approaches, CLQCD
performed a two-channel lattice study using the two-channel Ross-Shaw
effective range expansion Chen:2019iux . They considered the $\pi J/\psi$ and
$D\bar{D}^{*}$ channels that are most strongly coupled to $Z_{c}(3900)$ and
found that the parameters of the Ross-Shaw matrix do not seem to support the
HALQCD scenario. The parameters turn out to be large and the Ross-Shaw $M$
matrix is far from singular, which is required for a resonance close to the
threshold. However, since only two channels are studied, it is still not a
direct comparison with the HALQCD approach, in which three channels were
studied. In Ref. LiuChuan , the same three channels that the HALQCD
collaboration analyzed were considered, namely $\pi J/\psi$, $\rho\eta_{c}$,
and $D\bar{D}^{*}$. However, the final results will not come very soon.

Whatever the nature of the $Z_{c}$ states turn to be, they will teach us a lot
about the hadronic structures. Unless all these structures are purely
kinematical effects (in which case it would have to be an as yet unknown
kinematic effect), they will suggest a new category of hadrons beyond the
conventional meson and baryon picture. Recent discoveries of structures with
two pairs of charm-anticharm quarks lhcb_x6900  and with a minimal 4-quark
configuration $cs\bar{u}\bar{d}$ lhcb_x2900  confirm this expectation.
Additional searches for other conceivable states should be performed and the
theoretical consequences of these new types of hadrons should be investigated.

###  III.3 Comprehensive study of the $X(3872)$ in $e^{+}e^{-}$ collision

The $X(3872)$ was first observed in $B^{\pm}\to K^{\pm}\pi^{+}\pi^{-}J/\psi$
decays 17 years ago by Belle bellex . It was confirmed subsequently by several
other experiments CDFx ; D0x ; babarx ; x3872_JPC_LHCb_2013 . Prior to 2014,
the $X(3872)$ was only observed in $B$ meson decays and hadron collisions.
Since the quantum numbers of $X(3872)$ are $J^{PC}=1^{++}$, it can be produced
via radiative decays of excited vector charmonium or charmoniumlike states
such as the $\psi$s and the $Y$s.

The $X(3872)$ was observed at BESIII in the process $e^{+}e^{-}\to\gamma
X(3872)\to\gamma\pi^{+}\pi^{-}J/\psi$, $J/\psi\to\ell^{+}\ell^{-}$ BES3x  (see
the left panel of Fig. 7) and this first measurement was subsequently improved
with more data BES3x_omegajpsi . The c.m. energy dependence of the product of
the cross section $\sigma[e^{+}e^{-}\to\gamma X(3872)]$ and the branching
fraction ${\cal B}[X(3872)\to\pi^{+}\pi^{-}J/\psi]$ is shown in the right
panel of Fig. 7, where the red curve shows the results of a fit to a BW
resonance line shape with a mass of $(4200.6^{+7.9}_{-13.3}\pm 3.0)~{}{\rm
MeV}/c^{2}$ and a width of $(115^{+38}_{-26}\pm 12)~{}{\rm MeV}$. These
resonance parameters are consistent with those of the $\psi(4160)$ charmonium
states pdg  or the $Y(4220)$ (see Sec. III.1) within errors.

Figure 7: (Left panel) Fit to the $M(\pi^{+}\pi^{-}J/\psi)$ distribution BES3x
and (right panels) fit to $\sigma^{B}[e^{+}e^{-}\to\gamma X(3872)]\times{\cal
B}[X(3872)\to\pi^{+}\pi^{-}J/\psi]$ BES3x_omegajpsi . Dots/triangles with
error bars are data, and the curves are the best fits.

Using all the data samples available at c.m. energies between $4.0$ and
$4.6~{}\rm GeV$, BESIII is able to observe for the first time significant
signals of $X(3872)\to\omega J/\psi$ BES3x_omegajpsi  and
$X(3872)\to\pi^{0}\chi_{c1}$ BES3x_pi0chic1 , and search for other possible
decays.

BESIII confirmed earlier observations of a large $X(3872)\to
D^{*0}\bar{D}^{0}+c.c.$ branching fraction and finds evidence for
$X(3872)\to\gamma J/\psi$ with a significance of $3.5\sigma$ BES3x_anything .
No evidence is found for the decays $X(3872)\to\gamma\psi(2S)$. The upper
limit on the ratio ${\cal B}(X(3872)\to\gamma\psi(2S))/{\cal
B}(X(3872)\to\gamma J/\psi)<0.59$ is obtained at the 90% C.L. BES3x_anything ,
which is inconsistent with LHCb R_in_lhcb  and BaBar measurements R_in_babar
but consistent with a Belle upper limit R_in_belle . No significant
$X(3872)\to\pi^{0}\chi_{c0,2}$ signals are observed.

The hadronic transitions of the $X(3872)$ to low mass charmonum states via a
single pion or a rho meson violate isospin, and the large decay rates of
$X(3872)\to\pi^{0}\chi_{c1}$ and $\rho J/\psi\to\pi^{+}\pi^{-}J/\psi$ relative
to the isospin-conserved mode $X(3872)\to\omega J/\psi$ indicate that
$X(3872)$ is very unlikely to be a pure charmonium state, such as the
$\chi_{c1}(2P)$. The order of magnitude larger decay rate to
$D^{*0}\bar{D}^{0}+c.c.$ than to charmonium final state favors the
$D^{*}\bar{D}+c.c.$ molecule interpretation of the $X(3872)$, as is the
relatively smaller production rate of $X(3872)\to\gamma\psi(2S)$ compared with
$X(3872)\to\gamma J/\psi$, or at least that there is a large fraction of
molecular component in its wave function in addition to a charmonium
component.

BESIII measured the ratios of branching fractions for $X(3872)\to\gamma
J/\psi$, $\gamma\psi(2S)$, $\omega J/\psi$, $\pi^{0}\chi_{c1}$,
$D^{*0}\bar{D}^{0}+c.c.$, $\pi^{0}D\bar{D}$, and $\gamma D\bar{D}$ to that for
$X(3872)\to\pi^{+}\pi^{-}J/\psi$. By combining these with the measurements of
the $X(3872)$ properties from the $B$-factories, the authors of Ref.
X_br_lich_ycz  obtained the absolute branching fractions of the $X(3872)$
decays into six modes by globally fitting the measurements provided by Belle,
BaBar, BESIII, and LHCb experiments (see Table 2). The branching fraction for
$X(3872)\to\pi^{+}\pi^{-}J/\psi$ is determined to be $(4.1^{+1.9}_{-1.1})\%$,
which is in good agreement with earlier estimates in Refs. ycz_pic2009  and
Esposito:2014rxa . By combining the branching fractions of all of the observed
modes, the fraction of the unknown decays of the $X(3872)$ is found to be
$(31.9_{-31.5}^{+18.1})\%$, which is an important challenge for future
experimental studies of $X(3872)$ decays.

Table 2:  The fitting results of the absolute branching fractions of the $X(3872)$ decays X_br_lich_ycz . The branching fraction of $X(3872)$ decays into unknown modes is calculated from the fit results. Parameter index | Decay mode | Branching fraction
---|---|---
1 | $X(3872)\to\pi^{+}\pi^{-}J/\psi$ | $(4.1^{+1.9}_{-1.1})\%$
2 | $X(3872)\to D^{*0}\bar{D}^{0}+c.c.$ | $(52.4^{+25.3}_{-14.3})\%$
3 | $X(3872)\to\gamma J/\psi$ | $(1.1^{+0.6}_{-0.3})\%$
4 | $X(3872)\to\gamma\psi(2S)$ | $(2.4^{+1.3}_{-0.8})\%$
5 | $X(3872)\to\pi^{0}\chi_{c1}$ | $(3.6^{+2.2}_{-1.6})\%$
6 | $X(3872)\to\omega J/\psi$ | $(4.4^{+2.3}_{-1.3})\%$
| $X(3872)\to{\rm unknown}$ | $(31.9_{-31.5}^{+18.1})\%$

With a very large sample of $X(3872)\to\pi^{+}\pi^{-}J/\psi$ events, the LHCb
experiment reported an improved measurement of its mass and a first
measurement of its width lhcb_x_width . Limited by its capability of $D^{*0}$
reconstruction and mass resolution, it is still not possible for LHCb to
measure the line shape of the resonance.

###  III.4 Commonality between the $X(3872)$, $Y(4260)$, and $Z_{c}(3900)$

With data taken with c.m. energy at and near the $Y(4260)$ resonance peak,
BESIII discovered a clear signal for $X(3872)$ production in association with
a $\gamma$-ray BES3x , as shown in Fig. 7, and a clear signal for
$Z_{c}(3900)$ production in association with a $\pi$ meson zc0-lineshape , as
shown in Fig. 5. The c.m.-energy-dependence of the $e^{+}e^{-}\to\gamma
X(3872)$ cross section is suggestive of a $Y(4260)\to\gamma X(3872)$ decay
process, and that of $e^{+}e^{-}\to\pi^{0}Z_{c}(3900)^{0}$ cross section is
suggestive of a $Y(4260)\to\pi Z_{c}(3900)$ decay process, these indicate that
there might be some common features to the internal structures of the
$Z_{c}(3900)$, $Y(4260)$, and $X(3872)$.

Many of the models developed to interpret the nature of one of these three
states do not consider the possibility of a connection between them. With data
supplied by the BESIII experiments, some of these models may be ruled out and
others may need to be revisited in the light of these new observations.

##  IV Summary and Perspectives

With the capability of adjusting the $e^{+}e^{-}$ c.m. energy to the peaks of
resonances, combined with the clean experimental environments due to near-
threshold operation, BESIII is uniquely able to perform a broad range of
critical measurements of charmonium physics, and the production and decays of
many of the nonstandard $\rm XYZ$ states, as discussed above in the context of
the studies of the $X(3872)$, $Y(4220)$, $Z_{c}(3900)$ and $Z_{c}(4020)$.
Table 3 shows the operating times associated the discoveries of the $\rm XYZ$
states at BESIII and other experiments, including the previous generation
$B$-factories BaBar and Belle, and the new generation super-$B$-factories LHCb
and Belle II. BESIII’s special advantages for studying the $\rm XYZ$ states
are evident.

Table 3: The numbers of observed events of discovery modes of the $\rm XYZ$ states at the BESIII and other experiments. Here the states are detected with $X(3872)\to\pi^{+}\pi^{-}J/\psi$, $Y(4260)\to\pi^{+}\pi^{-}J/\psi$, $Z_{c}(3900)^{\pm}\to\pi^{\pm}J/\psi$, $Z_{c}(4020)\to\pi^{\pm}h_{c}$, and $Y(4660)\to\pi^{+}\pi^{-}\psi(2S)$. The numbers for Belle II experiments are a simple scale according to those of Belle experiment. “–” means no measurement available. BESIII can detect other decay modes of these states while other experiments can barely do.  Experiment | Data taking time | $X(3872)$ | $Y(4260)$ | $Z_{c}(3900)$ | $Z_{c}(4020)$ | $Y(4660)$
---|---|---|---|---|---|---
BESIII | 3 months | 20 | 6,000 | 1,300 | 180 | 250
BaBar | 1999-2008 | 90 | 270 | 80 | – | 45
Belle | 1999-2010 | 170 | 550 | 160 | – | 90
LHCb | 2011-12 ($B$ decays) | 4,000 | – | – | – | –
| 2011-18 ($pp$ collision) | 16,000 | – | – | – | –
Belle II | 2019-30 | 8,000 | 28,000 | 8,000 | – | 5,000

We emphasize here that BESIII measured all the known decay modes of the
$X(3872)$ and discovered its new decay modes even though the numbers of
produced $X(3872)$ events are much smaller than those of other experiments.
This is because the very clean experimental environment of $e^{+}e^{-}$
collisions in the $\tau$-charm threshold energy region uniquely facilitates
the isolation of signals for $X(3872)$ decays into final states with one or
more photons with high efficiency. This is especially true for final states
that contain an $h_{c}$ charmonium state like the BESIII discovery of the
$Z_{c}(4020)$ state and measurements of $Y(4220)$ and
$Y(4390)\to\pi^{+}\pi^{-}h_{c}$ decays. Neither the BaBar and Belle
$B$-factory nor the LHCb experiment has ever seen an $h_{c}$ signal.

BESIII has produced a considerable amount of information about the $\rm XYZ$
and the conventional charmonium states. In addition, there are still data that
are still being analyzed and more data that will be accumulated at other c.m.
energies fop ; bes3_white . Analyses with these additional data samples will
provide an improved understanding of the $\rm XYZ$ states, especially the
$X(3872)$, $Y(4260)$, $Z_{c}(3900)$, and $Z_{c}(4020)$. The maximum c.m.
energy accessible at BEPCII was upgraded from 4.6 to 4.9 GeV in 2019, and a
3.5 fb-1 sample of data between 4.6 and 4.7 GeV was accumulated in the 2019-20
running period, with more data planned for the future. This will enable a full
coverage of the $Y(4660)$ belle_y4660  resonance and a search for possible
higher mass vector mesons and states with other quantum numbers, as well as
improved measurements of their properties.

At the same time, other experiments will also supply information on these
states. At the LHCb, in addition to the 3 fb-1 data at 7 and 8 TeV that have
been used for most of their published analyses, there is a 6 fb-1 data sample
at 13 TeV that is being used for improved analyses of many of the topics
discussed above such as the $X(3872)$ decay properties and the searches for
the $Y$ and $Z_{c}(3900)$ states in $B$ decays.

Belle II belle2  has collected about 70 fb-1 data by mid-2020, and will
accumulate 50 ab-1 data at the $\Upsilon(4S)$ peak by the end of 2030. These
data samples can be used to study the $\rm XYZ$ and charmonium states in many
different ways PBFB , among which ISR can produce events in the same energy
range covered by BESIII. A 50 ab-1 Belle II data sample will correspond to
2.0–2.8 fb-1 data for every 10 MeV from 4–5 GeV. Similar statistics will be
available for modes like $e^{+}e^{-}\to\pi^{+}\pi^{-}J/\psi$ at Belle II and
BESIII (after considering the fact that Belle II has lower efficiency). Belle
II has the advantage that data at different energies will be accumulated at
the same time, making the analysis much simpler than at BESIII.

There are two super $\tau$-charm factories proposed, the STC in China HIEPA
and the SCT in Russia SCT_charm2018 . Both machines would run at c.m. energies
of up to 5 GeV or higher with a peak luminosity of $10^{35}$ cm-2s-1 which is
a factor of 100 improvement over the BEPCII. This would enable systematic
studies of the $\rm XYZ$ and charmonium states with unprecedented precision.

## Acknowledgments

I thank my BESIII collaborators for producing these fantastic results
presented in this review, I thank Steve Olsen for comments and suggestions on
the manuscript. This work is supported in part by National Key Research and
Development Program of China under Contract No. 2020YFA0406300, National
Natural Science Foundation of China (NSFC) under contract Nos. 11961141012,
11835012, and 11521505; and the CAS Center for Excellence in Particle Physics
(CCEPP).

Note added: After submission of this manuscript, BESIII announced observation
of a near-threshold structure in the $K^{+}$ recoil-mass spectra in
$e^{+}e^{-}\to K^{+}(D^{-}_{s}D^{*0}+D^{*-}_{s}D^{0})$ which could be the
strange partner of the $Z_{c}(3900)$ with the $d$ quark replaced with an $s$
quark zcs .

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# A superradiant two-level laser with intrinsic light force generated gain

Anna Bychek1 Helmut Ritsch1 1Institut für Theoretische Physik, Universität
Innsbruck, Technikerstr. 21a, A-6020 Innsbruck, Austria

###### Abstract

The implementation of a superradiant laser as an active frequency standard is
predicted to provide better short-term stability and robustness to thermal and
mechanical fluctuations when compared to standard passive optical clocks.
However, despite significant recent progress, the experimental realization of
continuous wave superradiant lasing still remains an open challenge as it
requires continuous loading, cooling, and pumping of active atoms within an
optical resonator. Here we propose a new scenario for creating continuous gain
by using optical forces acting on the states of a two-level atom via
bichromatic coherent pumping of a cold atomic gas trapped inside a single-mode
cavity. Analogous to atomic maser setups, tailored state-dependent forces are
used to gather and concentrate excited state atoms in regions of strong atom-
cavity coupling while ground-state atoms are repelled. To facilitate numerical
simulations of a sufficiently large atomic ensemble, we rely on a second-order
cumulant expansion and describe the atomic motion in a semi-classical point-
particle approximation subject to position-dependent light shifts which induce
optical gradient forces along the cavity axis. We study minimal conditions on
pump laser intensities and detunings required for collective superradiant
emission. Balancing Doppler cooling and gain-induced heating we identify a
parameter regime of a continuous narrow-band laser operation close to the bare
atomic frequency.

##  I Introduction

In view of establishing a new outstanding and robust optical time and
frequency standard the quest to build a continuous superradiant laser
operating on a very narrow atomic transition has been the subject of intense
theoretical and experimental research in the past decade Meiser et al. (2009);
Bohnet et al. (2012); Maier et al. (2014); Bychek et al. (2021); Norcia et al.
(2018); Debnath et al. (2018); Liu et al. (2020); Wu et al. (2022); Zhang et
al. (2022); Kazakov et al. (2022). These studies are also fueled by the
remarkable operating characteristics and relative simplicity of its microwave
analogue, the hydrogen maser Weiss and Walls (1996); Bauch et al. (2015).
Recently pulsed superradiance has been experimentally observed using laser-
cooled atomic ensembles Norcia et al. (2016); Laske et al. (2019); Tang et al.
(2021). Some proof of principle setups based on magneto-optical trapping
demonstrated quasi-continuous operation on kHz transitions Norcia and Thompson
(2016); Kristensen et al. (2023). The major remaining challenge is to achieve
sufficient gain via continuous inversion on the relevant clock transition
without significantly perturbing the atomic levels.

One straightforward approach which is currently pursued is based on a
continuous ultracold beam of excited atoms passing through an optical
resonator Bohnet et al. (2012); Liu et al. (2020); Chen et al. (2019);
Escudero et al. (2021); Chen et al. (2022); Tang et al. (2022); Cline et al.
(2022); Takeuchi et al. (2023). In the past years, considerable progress has
been made in this direction, yet the main challenge is to create a sufficient
inverted atomic flux needed for collective superradiant emission. As an
alternative, some calculations suggest optimized multilevel pumping schemes,
where a careful choice of laser powers and detunings minimizes the transition
level shifts at reasonable pumping rates Hotter et al. (2022). Still this
needs many lasers, which have to be combined to pump and cool the atomic
ensemble simultaneously in order to keep the density of the active gas
constant.

As in a standard micromaser, where population inversion is created by coherent
pumping of atoms followed by magneto-mechanical separation of ground- and
excited-state atoms in the mode volume, one could look for an analogous scheme
for an optical setup. As the length scales are several orders of magnitude
smaller at optical frequencies, the magnetic gradients for a sufficient state
separation are very difficult to achieve and also potentially shift the clock
transition in a detrimental way. However, state-dependent optical forces could
do the trick, and thus excitation and separation could be done by suitably
designed laser fields. At the same time, it has already been shown in previous
experiments on BEC formation in Strontium that dressing lasers can create
large enough optical level shifts in a dimple configuration to manipulate only
a chosen sub-ensemble of the atoms with an extra laser without affecting the
majority of the unperturbed atoms outside the dimple Chen et al. (2022).

Here, we suggest a new scenario for creating an intrinsic light force
generated inversion mechanism. The idea is to combine the internal degrees of
freedom of atoms with the motional ones to create the necessary inversion. We
will study configurations, where pumping and gain occur in different spatial
regions of the cavity by taking into account atomic motion and state-dependent
forces resulting from the spatially dependent periodic drive of the
transition. After all, an inverted ensemble of only very weakly perturbed
atoms can be created in regions of strong atom-cavity coupling. For
sufficiently many atoms we show that this should lead to collective narrow-
band lasing close to the bare atomic transition frequency.

A detailed quantum description including the necessary number of atoms to
achieve sufficient gain at low excitation powers goes beyond the available
computational power to numerically simulate the coupled atom-field dynamics.
Therefore, we have to resort to approximations and only limited atom numbers
from which we are able to extract predictions for scaling towards larger
ensembles. Hence we treat the atomic motion semi-classically and use a quantum
description only for the internal atomic dynamics and the cavity field. As it
has been observed for instance in Ref. Hechenblaikner et al. (1998), a semi-
classical description of motion shows a good agreement with the full quantum
description, where the external degrees of freedom are quantized. Still, we
have to make use of a cumulant expansion approach Kubo (1962) for studying
larger atom numbers.

The paper is organized as follows. First in Sec. II.1 we introduce the spatial
light shifts and optical forces present in the system. In Sec. II.2, we
present the system overview and calculate the coupled atom-field dynamics
under the coherent laser drive. It is then shown in Sec. II.3, that a
bichromatic coherent drive can lead to a continuous narrow-band laser
operation. We extend the model for an ensemble of atoms with light force
induced inversion in Sec. III. We start with the full quantum approach in Sec.
III.1 and proceed with the second-order cumulant expansion in Sec. III.2 in
order to describe the collective atomic dynamics and spectrum of the light
field in the cavity.

##  II Model definition

###  II.1 Light shifts and forces

A two-level atom coherently driven by a laser detuned from the atomic
resonance frequency experiences energy light shifts Dalibard and Cohen-
Tannoudji (1985). Under this drive the ground and excited states of the atom
are no longer eigenstates of the system. The Hamiltonian of the atom
($\omega_{a}$) under the coherent laser drive ($\omega_{\Omega}$) in a
rotating frame of the laser field can be written as ($\hbar=1$)

$H_{a}=-\Delta_{a}\sigma^{+}\sigma^{-}+\Omega(\sigma^{+}+\sigma^{-})=-\frac{\Delta_{a}}{2}-\frac{\Delta_{a}}{2}\sigma^{z}+\Omega\sigma^{x},$
(1)

where $\Delta_{a}=\omega_{\Omega}-\omega_{a}$, $\Omega$ is the transition Rabi
frequency, and $\sigma^{x}=\sigma^{+}+\sigma^{-}$,
$\sigma^{z}=\sigma^{+}\sigma^{-}-\sigma^{-}\sigma^{+}$ are the Pauli matrices.
It is known that any 2x2 Hermitian matrix can be expressed in a unique way as
a linear combination of the Pauli matrices

$H=h_{0}\mathbb{1}+\vec{h}\vec{\sigma},$ (2)

with all coefficients being real numbers $h_{0}=const$,
$h_{1}=h\sin{\Theta}\cos{\phi}$, $h_{2}=h\sin{\Theta}\sin{\phi}$,
$h_{3}=h\cos{\Theta}$, where
$h=|\vec{h}|=\sqrt{h_{1}^{2}+h_{2}^{2}+h_{3}^{2}}$. It is easy to show that
the eigenvalues of this matrix are

$E_{\pm}=h_{0}\pm h,$ (3)

and the corresponding eigenvectors can be expressed as an effective rotation
of the uncoupled states,

$\displaystyle|+\rangle$
$\displaystyle=\sin{\frac{\Theta}{2}}\exp{(\frac{i\phi}{2})}|g\rangle+\cos{\frac{\Theta}{2}}\exp{(-\frac{i\phi}{2})}|e\rangle$
(4) $\displaystyle|-\rangle$
$\displaystyle=\cos{\frac{\Theta}{2}}\exp{(\frac{i\phi}{2})}|g\rangle-\sin{\frac{\Theta}{2}}\exp{(-\frac{i\phi}{2})}|e\rangle.$

Therefore, the eigenvalues of the Hamiltonian (1) can be written as

$E_{\pm}=-\frac{\Delta_{a}}{2}\pm\sqrt{\Omega^{2}+\Delta_{a}^{2}/4},$ (5)

with the corresponding eigenstates known as the dressed states,

$\displaystyle|+\rangle$
$\displaystyle=\sin{\frac{\Theta}{2}}|g\rangle+\cos{\frac{\Theta}{2}}|e\rangle$
(6) $\displaystyle|-\rangle$
$\displaystyle=\cos{\frac{\Theta}{2}}|g\rangle-\sin{\frac{\Theta}{2}}|e\rangle,$

where $tg\>\Theta=-\frac{2\Omega}{\Delta_{a}}$.

When the atom is illuminated by a laser whose Rabi frequency has a spatial
periodic distribution ${\Omega(x)=\Omega\cos(kx)}$ formed by a standing wave
with the wavelength $\lambda=2\pi/k$ the energy shifts can be plotted as shown
in figure 1(a). This creates a mean dipole force ${\langle
F\rangle=-\langle\nabla H\rangle}$ acting on the atom moving along the cavity
axis, which has the opposite sign for $|+\rangle$ and $|-\rangle$ states. In
this regard one could think of a population inversion scheme shown in figure
1(b). An atom located at position (1) with some non-zero initial velocity is
pumped by a laser into state (2) and experiences the dipole force as it
continues to move. This force pushes the atom to position (3) where there is
no force acting on the atom. If this process occurs at a faster rate than the
lifetime of the excited state, then the atom emits a photon with the frequency
close to the bare atomic transition frequency and undergoes the transition
into state (4) where it is dragged by the light force left or right to
position (1) and the process repeats itself.

Figure 1: Eigenvalues of the Hamiltonian (1) under a spatially dependent
coherent laser drive $\Omega(x)=\Omega\cos(kx)$
with negative detuning $\Delta_{a}=\omega_{\Omega}-\omega_{a}$.

Thus such a scheme could be used to spatially separate the region of pumping
from the lasing in the system. In other words, we create the population
inversion in the specific regions of the cavity, those regions where we would
like to have a maximal coupling to the cavity.

###  II.2 Semi-classical master equation for the coupled atom-field dynamics

A stable laser operation requires a continuous inversion mechanism to keep the
population inversion on a lasing transition. In the previous section, we
introduced the scenario for creating effective inversion on a lasing
transition using light forces. While in principle many possible geometries to
achieve this purpose can be investigated, we will restrict ourselves here to a
simple generic case, where the underlying mechanisms can be studied in detail.
Hence we consider a 1D motion in a linear Fabry-Perot cavity with a standing
cosine wave pump field and sine wave laser mode. Ground-state atoms will be
trapped and cooled close to the antinodes of the cosine mode. Atoms excited to
the upper level at this points are pushed towards the nodes of the cosine
mode, where they maximally couple to the cavity sine mode. Let us identify a
parameter regime, which leads to the desired local inversion and gain. In
other words, one needs to find a regime of stable narrow-band lasing at the
bare atomic frequency, with a linewidth that is much smaller than the cavity
linewidth and that is well distinguished from the other light sources present
in the cavity. This requires several conditions to be ensured:

$\bullet$ $\kappa>\Gamma$ \- the system is in the bad-cavity regime, which
provides the low intracavity photon number operation and thus reduced
sensitivity to cavity noise;

$\bullet$ $\Delta_{a}<0$, and $|\Delta_{a}|>\Omega$ \- the driving laser is
far red-detuned from the atomic transition frequency to minimize the amount of
coherently scattered photons from the drive into the cavity;

$\bullet$ $2\sqrt{\Omega^{2}+\Delta_{a}^{2}/4}>\kappa$ \- the maximal light
shift in equation (5) is bigger than the cavity linewidth to spatially
separate the region of pumping (1-2) from the lasing (3-4) in figure 1.

Figure 2: Atomic dynamics and lasing under the coherent drive with
$|\Delta_{a}|<\Omega$.
Parameters: $\kappa=20\Gamma$, $g=4\Gamma$, $\Omega=30\Gamma$,
$\Delta_{a}=-10\Gamma$, $\omega_{r}=2.5\Gamma$. Figure 3: Cooling and trapping
of an atom in the cavity under the coherent drive with $|\Delta_{a}|>\Omega$.
Parameters: $\kappa=20\Gamma$, $g=4\Gamma$, $\Omega=10\Gamma$,
$\Delta_{a}=-20\Gamma$, $\omega_{r}=6\Gamma$. Figure 4: (a) The schematics of
an atom under coherent drive from two lasers.
(b) Dipole force acting on the dressed states of an atom.

The interaction of the atom with the cavity in the Jaynes–Cummings model is
given by ${H_{int}=g(a^{\dagger}\sigma^{-}+a\sigma^{+})}$, where $g$ is the
atom-cavity interaction strength. The quantum dynamics of the composite atom-
cavity system can be described by the master equation for the system density
matrix $\rho$ in the Lindblad form

$\dot{\rho}=-\frac{i}{\hbar}[H_{a}+H_{c}+H_{int},\rho]+\mathcal{L}_{\kappa}[\rho]+\mathcal{L}_{\Gamma}[\rho],$
(7)

where $H_{c}=-\Delta_{c}a^{\dagger}a$, with
$\Delta_{c}=\omega_{\Omega}-\omega_{c}$. The loss of photons through the
cavity mirrors and individual atomic decay are given by

$\displaystyle\mathcal{L}_{\kappa}[\rho]$
$\displaystyle=\frac{\kappa}{2}(2a\rho a^{\dagger}-a^{\dagger}a\rho-\rho
a^{\dagger}a)$ (8) $\displaystyle\mathcal{L}_{\Gamma}[\rho]$
$\displaystyle=\frac{\Gamma}{2}(2\sigma^{-}\rho\sigma^{+}-\sigma^{+}\sigma^{-}\rho-\rho\sigma^{+}\sigma^{-}),$

with the cavity loss rate $\kappa$ and single-atom spontaneous emission rate
$\Gamma$, respectively. In order to approximate the atomic motion and light
forces acting on the atom, we include the semi-classical equations of motion
in the system description,

$\displaystyle\dot{\langle x\rangle}$ $\displaystyle=2\omega_{r}\langle
p\rangle$ (9) $\displaystyle\dot{\langle p\rangle}$
$\displaystyle=-\langle\nabla H\rangle,$

where we use the dimensionless variables ${t\rightarrow\Gamma t^{\prime}}$,
${\langle x\rangle\rightarrow k_{a}\langle x^{\prime}\rangle}$, ${\langle
p\rangle\rightarrow\langle p^{\prime}\rangle/(\hbar k_{a})}$, and
${\omega_{r}=\hbar k_{a}^{2}/(2m\Gamma)}$ is the atomic recoil frequency given
by the atomic mass and wave number of the atomic transition. Here we neglect
the effects of momentum diffusion arising from spontaneous emission, because
for the ultimate case of a superradiant laser the rate $\Gamma$ is usually
very small compared to other parameters.

First of all, we would like to calculate the coupled atom-photon dynamics in
the cavity. For a non-moving atom under a coherent drive the solution is known
as the damped Rabi oscillations eventually leading to population of the
excited state by no more than fifty percent Fox (2006), i.e. no inversion.
However, due to atomic motion and forces acting differently on the states of
the atom, the local population inversion can become positive in certain
positions.

For the remainder of this work we consider a linear cavity with a cosine wave
pump and a sine wave cavity mode with the coupling strength $g(x)=g\sin(kx)$,
where $k=k_{a}$ is the cavity mode wave number. The cavity mode is tuned on
resonance with the bare atomic transition frequency. Since we are in the
parameter regime where $\kappa>\Gamma$ and $g<\kappa$, the Hilbert space of
the photon field can be truncated at low photon numbers. Figures 2-3 show the
atomic dynamics and lasing under the coherent drive $\Omega(x)=\Omega\cos(kx)$
for two different cases of $|\Delta_{a}|<\Omega$ and $|\Delta_{a}|>\Omega$.
The position and momentum of the atom are given in units of $\pi/k$ and $\hbar
k$, respectively, in figures 2-3(a-b). The mean photon number in the cavity
and atomic inversion are shown as a function of time in figures 2-3(c-d), and
as a function of position in figures 2(e-f).

In the case of $|\Delta_{a}|<\Omega$, we have found the parameter regime,
where atomic cooling is balanced by heating from the driving laser, see
figures 2(a-b). Starting from a given initial momentum, the atom experiences
laser cooling until it reaches a quasi-stationary state. As seen in figure
2(a), the atom oscillates between a particular node and the neighboring
antinodes of the driving field while the atomic inversion
$\langle\sigma_{z}\rangle=\langle\sigma^{+}\sigma^{-}\rangle-\langle\sigma^{-}\sigma^{+}\rangle$
in figure 2(f) becomes positive towards the points of the maximal coupling
strength. Followed by photon emission and a maximum in average photon number
in figure 2(e) this dynamics is close to the ideal scenario depicted in figure
1(b). The results demonstrate that in principle the above scenario may take
place, unfortunately this can not be used as a good pumping scheme due to the
fact that mostly scattered photons from the drive will dominate the cavity
output spectrum.

Figure 5: Atomic dynamics and lasing under the bichromatic coherent drive.
Parameters: $\kappa=20\Gamma$, $g=4\Gamma$, $\Omega=10\Gamma$,
$\Delta_{a}=-20\Gamma$, $\eta=8\Gamma$, $\Delta_{\eta}=-25\Gamma$,
$\omega_{r}=6\Gamma$.

On the other side, after studying the case of $|\Delta_{a}|>\Omega$, we have
observed that the drive from a single laser is never sufficient enough to
create the desired population inversion. As presented in figure 3, in this
case the applied far-detuned drive results in strong cooling for the atom and
there is no inversion on the lasing transition at any point. Therefore, one
may think of the idea of adding an extra laser drive to populate the excited
state, with the parameters particularly chosen and optimized to ensure the
above conditions in the cavity.

###  II.3 Two-level dynamics with a bichromatic coherent drive

In the previous section, we have indicated the parameter regime of our
interests in the context of the superradiant laser. We observed that the
driving from a single laser is not sufficient to create the desired population
inversion on the lasing transition. In this section, the idea is to use this
laser exclusively to create the spatial light shifts, as depicted in figure
4(a). In order to populate the excited state we introduce the second laser
drive $\eta(x)=\eta\cos(kx)$ into the system. The frequency of the second
laser drive is now tuned to the resonance with the dressed states given by
equations (5-6), however not at their maximal light shifts, but at the points
where the dipole force acting on the atom is close to its maxima, see figure
4(b). This allows the excited atom to reach the lasing position more
efficiently since there is strong acceleration from the force. We expect that
a combination of these laser drives acting together can lead to collective
narrow-band emission, provided that optimal driving intensities and detunings
are found. However, the master equation becomes significantly more difficult
to solve as the second laser drive does not allow to eliminate the time
dependence in the Hamiltonian. Thus the Hamiltonian of the system in the
rotating frame of the first laser can be written as

$\displaystyle
H=-\Delta_{a}\sigma^{+}\sigma^{-}-\Delta_{c}a^{\dagger}a+g\sin(kx)(a^{\dagger}\sigma^{-}+a\sigma^{+})+...$
(10)
$\displaystyle...+\Omega\cos(kx)(\sigma^{+}+\sigma^{-})+\eta\cos(kx)(\sigma^{+}e^{i\Delta_{\eta}t}+\sigma^{-}e^{-i\Delta_{\eta}t}),$

where $\Delta_{\eta}=\omega_{\Omega}-\omega_{\eta}$. Note, that if the laser
drive is not strong enough it will not be able to create population inversion.
On the other hand, a strong laser drive will produce both a lot of coherently
scattered photons and strong heating in the system. In order to find the
dynamics of the system we solve the master equation (7) with the Hamiltonian
(LABEL:Hamiltonian_2lasers). It is then the Rabi frequency and detuning of the
second laser which have to be scanned and mutually adjusted.

Figure 5 shows the atomic dynamics and lasing under the bichromatic coherent
drive in the regime, where the light from the second drive pushes the atom to
move along the cavity axis such that the atomic momentum reaches its quasi-
stationary state, see figures 5(a-b). This results in a linear motion of the
atom, where it is being pumped at the maximal driving intensity and continues
to undergo the inversion scheme depicted in figure 1(b). As the atom moves,
the atomic inversion becomes positive in the vicinity of the unperturbed bare
atomic transition frequency, which is followed by the photon emission, as can
be seen in figures 5(d-f). Figure 5(c) shows the mean photon number in the
cavity (blue line) and can be split into the laser part (orange line) coming
from the atoms (given by $\langle a^{\dagger}a\rangle-\langle a\rangle^{2}$)
and the scattered part (green line) coming from the laser drive, which should
have only a minor contribution. One can see a stable laser operation with the
oscillations of the photon number around the mean value. This example
demonstrates how the specially tuned bichromatic coherent drive can lead to
continuous lasing with intrinsic light force generated inversion.

Figure 6: (a) Dynamics of $N=8$ atoms under the bichromatic coherent drive.
(b) Mean photon number in the cavity depending on time (in units of $\Gamma$).
(c) The position dependence of the atomic inversion during the stable final
stage of the time evolution. The initial distribution of momenta
$|p^{0}_{m}|\in[2,2.5]\hbar k$ and positions $x^{0}_{m}=m\pi/k$ for
$m=1,...,N$. The parameters are the same as in figure 5.

##  III Collective dynamics with light force induced inversion

###  III.1 Full quantum approach

An extension of the full quantum model to an ensemble of atoms remains
feasible only for a small atom number due to the exponential growth of the
Hilbert space. Here we present the results for the case of $N=8$, where we
truncate the Hilbert space of the field in the cavity at low photon numbers.
We choose the initial atomic momenta to be randomly distributed around a
selected velocity as $|p^{0}_{m}|=p_{0}\pm\epsilon$, which is of the order of
several $\hbar k$, see figure 6. As the atoms move one can see the
stabilization of photon emission and increase in the mean photon number.
Figure 6(c) shows how the atomic inversion changes with the position similar
to the case of a single atom in figure 5.

###  III.2 Second-order cumulant expansion

Figure 7: Atomic dynamics and lasing in the ensemble of $N=100$ atoms under
the bichromatic coherent drive.
The parameters are the same as in figure 5. Figure 8: Spectrum of the cavity
light for the ensemble of $N=40$ (left panel) and $N=100$ atoms (right panel)
presented in figure 7. The emission intensity is normalized and the inset
shows a zoom-in of the central peak profile.

Next, we would like to extend our model to an ensemble of $N\gg 1$ atoms
subjected to the bichromatic coherent drive, as described in the previous
section. As each atom in the ensemble behaves differently depending on its
initial position and momentum, the numerical solution of the master equation
(7) for the full density matrix becomes challenging. In order to describe a
large ensemble of atoms we make use of the second-order cumulant expansion
Kubo (1962) to write down the closed set of the Heisenberg equations for the
system operators Bychek et al. (2021):

$\displaystyle\frac{d}{dt}\langle a\rangle=-(\kappa/2-i\Delta_{c})\langle
a\rangle-ig\sum_{j}N_{j}\sin(x_{j})\langle\sigma^{-}_{j}\rangle$ (11)
$\displaystyle\frac{d}{dt}\langle\sigma^{-}_{m}\rangle=-(\Gamma/2-i\Delta_{am})\langle\sigma^{-}_{m}\rangle+ig\sin(x_{m})\langle
a\rangle(2\langle\sigma^{+}_{m}\sigma^{-}_{m}\rangle-1)+i(\Omega+\eta
e^{i\Delta_{\eta}t})\cos(x_{m})(2\langle\sigma^{+}_{m}\sigma^{-}_{m}\rangle-1)$
$\displaystyle\frac{d}{dt}\langle a^{\dagger}a\rangle=-\kappa\langle
a^{\dagger}a\rangle+ig\sum_{j}N_{j}\sin(x_{j})(\langle
a\sigma^{+}_{j}\rangle-\langle a^{\dagger}\sigma^{-}_{j}\rangle)$
$\displaystyle\frac{d}{dt}\langle
a\sigma^{+}_{m}\rangle=-(\kappa/2+\Gamma/2+i\Delta_{am}-i\Delta_{c})\langle
a\sigma^{+}_{m}\rangle+ig\sin(x_{m})(\langle a^{\dagger}a\rangle-2\langle
a^{\dagger}a\rangle\langle\sigma^{+}_{m}\sigma^{-}_{m}\rangle-\langle\sigma^{+}_{m}\sigma^{-}_{m}\rangle)+...$
$\displaystyle\quad...-ig\sum_{j;m\neq
j}\sin(x_{j})\langle\sigma^{+}_{m}\sigma^{-}_{j}\rangle-i(\Omega+\eta
e^{-i\Delta_{\eta}t})\cos(x_{m})\langle
a\rangle(2\langle\sigma^{+}_{m}\sigma^{-}_{m}\rangle-1)$
$\displaystyle\frac{d}{dt}\langle\sigma^{+}_{m}\sigma^{-}_{m}\rangle=-\Gamma\langle\sigma^{+}_{m}\sigma^{-}_{m}\rangle-
ig\sin(x_{m})(\langle a\sigma^{+}_{m}\rangle-\langle
a^{\dagger}\sigma^{-}_{m}\rangle)+...$ $\displaystyle\quad...-i(\Omega+\eta
e^{i\Delta_{\eta}t}))\cos(x_{m})\langle\sigma^{+}_{m}\rangle+i(\Omega+\eta
e^{-i\Delta_{\eta}t}))\cos(x_{m})\langle\sigma^{-}_{m}\rangle$
$\displaystyle\frac{d}{dt}\langle\sigma^{+}_{m}\sigma^{-}_{j}\rangle=-\Gamma\langle\sigma^{+}_{m}\sigma^{-}_{j}\rangle-
ig_{m}\sin(x_{m})\langle
a^{\dagger}\sigma^{-}_{j}\rangle(2\langle\sigma^{+}_{m}\sigma^{-}_{m}\rangle-1)+ig_{j}\sin(x_{j})\langle
a\sigma^{+}_{m}\rangle(2\langle\sigma^{+}_{j}\sigma^{-}_{j}\rangle-1)+...$
$\displaystyle\quad...+i(\Omega+\eta
e^{i\Delta_{\eta}t}))\cos(x_{j})\langle\sigma^{+}_{m}\rangle(2\langle\sigma^{+}_{j}\sigma^{-}_{j}\rangle-1)-i(\Omega+\eta
e^{-i\Delta_{\eta}t}))\cos(x_{m})\langle\sigma^{-}_{j}\rangle(2\langle\sigma^{+}_{m}\sigma^{-}_{m}\rangle-1)$
$\displaystyle\frac{d}{dt}\langle x_{m}\rangle=2\omega_{r}\langle
p_{m}\rangle$ $\displaystyle\frac{d}{dt}\langle
p_{m}\rangle=-2g\cos(x_{m})\Re\\{\langle
a\sigma^{+}_{m}\rangle\\}+2\Omega\sin(x_{m})\Re\\{\langle\sigma^{+}_{m}\rangle\\}+\eta\sin(x_{m})(\langle\sigma^{+}_{m}\rangle
e^{i\Delta_{\eta}t}+\langle\sigma^{-}_{m}\rangle e^{-i\Delta_{\eta}t}).$

Figure 7 shows the solution of equations (LABEL:Heisenberg_eqs) for $N=100$
atoms with the distribution of initial positions and momenta used in the full
model. The resulting dynamics becomes much more complicated to describe, but
one can see a similar behavior with the case of a single atom. Although a
small part of atoms gets cooled down and does not contribute to the lasing
process, the majority of atoms display the lasing we are interested in. In
figure 7(b) one can see continuous lasing from the atoms reaching the order of
one photon on average in the cavity. We expect fluctuations in the photon
number to be mitigated in the case of significantly larger atomic ensembles,
where each atom only weakly contributes to the emission process. However, due
to the number of equations growing as $\mathcal{O}(N^{2})$ we are limited to a
system of a few hundred atoms.

Let us calculate the spectrum of the light in the cavity. According to the
Wiener–Khinchin theorem Puri et al. (2001) the spectrum can be found as a
Fourier transform of the first-order correlation function
$g^{(1)}(\tau)=\langle a^{\dagger}(t_{0}+\tau)a(t_{0})\rangle$,

$S(\omega)=2\Re\left\\{\int_{0}^{\infty}d\tau
e^{-i\omega\tau}g^{(1)}(\tau)\right\\}.$ (12)

Here, we use the quantum regression theorem Carmichael and Scully (2000) to
write down a set of differential equations for the first-order correlation
function, where $t_{0}$ is normally given by the time the system reaches its
steady state. However, since in our case the dynamics does not have a steady
state, we have to include these equations in the full system of equations
(LABEL:Heisenberg_eqs) and average it over a set of equidistant initial
conditions ${g^{1}(0)=\langle a^{\dagger}a\rangle(t_{0})|_{t_{0}=t_{end}}}$
chosen from the final stage of the dynamics.

After averaging process one can see the resulting spectrum, as presented in
figure 8 for $N=40$ and $N=100$ atoms. The spectrum averages quite well
already after several averaging steps and reveals the main spectral peak
coming from the atoms at the frequency close to the bare atomic transition.
The inset shows a zoom-in of the central peak profile which broadening can be
attributed to the emission from different atoms at slightly different
frequencies.

Additionally to the central peak there are numerous sidebands located left and
right from the atomic transition frequency. They appear to be independent of
the number of atoms, coupling constant, and Rabi frequencies of the lasers. We
associate them with the motion of atoms with a constant velocity, which is
observed in figure 7(a). One can calculate the frequency of these motional
sidebands as

$\omega_{SB}=\omega_{a}\pm\omega_{\pm}=\omega_{a}\pm 2\pi\frac{v}{\lambda},$
(13)

where $v/\lambda=\omega_{r}p/(\pi\hbar k)$. Since the atomic motion is linear
one can write

$\omega_{\pm}=\pm\frac{2\omega_{r}\langle p\rangle^{st}}{\hbar k}.$ (14)

Substituting the real parameters used in figure 8 into equations (14) we
calculate the frequencies $w_{\pm}$, which agree well with the central
frequencies of the sidebands observed in the spectrum.

##  IV Conclusions and outlook

We have studied population inversion and gain within an optical cavity in a
cold ensemble of coherently driven clock atoms with an intrinsic light force
generated inversion mechanism. Using numerical simulations of the coupled
atom-field dynamics we have found the operating conditions producing
continuous narrow-band emission close to the unperturbed atomic line. In the
limit of low photon number operation, the central frequency is predicted to be
largely insensitive to cavity fluctuations. Note, that the driving field has
to be far-detuned from the atomic resonance such that there are no pump laser
photons coherently scattered into the cavity. As we have shown, adding an
extra specially tuned driving field significantly improves the performance of
the system.

At this point our simulations are limited to a few hundred atoms since we use
a second-order cumulant expansion as a minimum model to reliably predict the
laser spectrum. Much higher output laser power with a cleaner spectrum can be
expected here for realistic atom numbers of up to a million, where each atom
only needs to contribute very weakly to the gain and thus less pump power is
required for lasing. Unfortunately such system sizes are beyond our present
numerical capabilities. Similarly, a reliable evaluation of the photon
statistics, as for instance the calculation of the second-order correlation
function $g^{(2)}(\tau)$, requires even higher order expansions thereby
limiting the tractable atom number even further.

Conceptually, the chosen example setup constitutes a minimalist implementation
of a superradiant laser requiring only a single standing wave pump field and a
single mode within the cavity to facilitate trapping, pumping, and lasing
simultaneously. Via state-dependent light forces atoms excited at the
antinodes of the pump standing wave are drawn towards its nodes, where the
coupling to the cavity mode can be made maximal by a suitable mode choice. The
operating principle here is reminiscent of maser implementations, where in
order to implement gain one uses coherent excitation and magnetic separation
of the excited state fraction. In our model, the excited state separation is
facilitated by differential optical gradient forces, which typically are much
stronger than magnetic gradient forces for neutral atoms.

Eventually more complex geometries involving higher-order transverse modes and
special state-dependent optical guiding fields can be envisaged for better
performance to increase gain and pump efficiency. As there is a large number
of options here, we have restricted ourselves to only one generic
implementation to exhibit the basic principle more clearly. Future more
refined models need to be developed in collaboration with a specific
experimental implementation.

##  V Acknowledgements

We acknowledge funding from the European Union’s Horizon 2020 research and
innovation program under the Marie Sklodowska-Curie Grant Agreement No. 860579
(project MoSaiQC).

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