# Capabilities of the CMS detector for studies of hard probes in heavy ion
collisions at the LHC

(for the CMS Collaboration)
D.V. Skobeltsyn Institute of Nuclear Physics, M.V. Lomonosov Moscow State
University, Moscow, Russia
E-mail

###### Abstract:

The capabilities of the CMS experiment to study properties of hot and dense
QCD-matter created in heavy ion collisions at the CERN Large Hadron Collider
with the perturbative processes (so-called ”hard probes”) are presented.
Detailed studies from complete simulations of the CMS detectors in Pb+Pb
collisions at $\sqrt{s}=5.5$ TeV per nucleon pair are presented in view of two
hard probes: quarkonium and $\gamma$-jet production.

##  1 Introduction

The study of the fundamental theory of the strong interaction (Quantum
Chromodynamics, QCD) in new, unexplored extreme regimes of super-high
densities and temperatures is one of the primary goals of the modern high
energy physics. The experimental and phenomenological study of multiple
particle production in ultrarelativistic heavy ion collisions is expected to
provide valuable information on the (thermo)dynamical behaviour of QCD matter
in the form of a quark-gluon plasma (QGP), as predicted by lattice QCD
calculation. A detailed description of the potential of CMS to carry out a
series of representative Pb-Pb measurements has been presented in [1].

Heavy ion observables accessible to measurement with CMS include:

  * •

“Soft” probes [2]: global particle and energy rapidity densities, elliptic
flow and spectra of low transverse momentum hadrons. These observables are
mostly sensitive to the space-time evolution of the system once thermalization
has set in and thus carry information about the thermodynamical properties of
the produced QCD matter.

  * •

“Hard” probes: quarkonia, heavy quarks, jets, $\gamma$-jet and high-$p_{T}$
hadrons, which are produced with high transverse momenta $p_{T}$ or large
masses $M$ (much greater than the typical QCD scale of confinement: $p_{T}$,
$M$ $\gg$ $\Lambda_{\rm QCD}=200$ MeV). The hard probes production cross
sections can be described in the framework of perturbative QCD theory. Such
hard particles are produced in the very early stages of the evolution of the
system and thus are potentially affected by final-state interactions as they
traverse the produced medium. Modifications with respect to the “vacuum QCD”
spectra and cross-sections measured in proton-proton collisions, provide
direct information on the dynamical and transport properties of the system:
initial parton densities, transport coefficient of the medium, critical energy
density.

In this contribution, the detailed study for a complete simulation of the CMS
detectors in Pb+Pb collisions at $\sqrt{s}=5.5$ TeV per nucleon pair are
presented in view of two hard probes: quarkonium and $\gamma$-jet production.
The results of similar studies for high-$p_{T}$ hadrons are presented during
this Workshop in another talk [3].

##  2 CMS detector

CMS is a general purpose experiment at the LHC designed to explore the physics
at the TeV energy scale [4]. Since the CMS detector subsystems have been
designed with a resolution and granularity adapted to cope with the extremely
high luminosities expected in the proton-proton running mode, CMS can also
deal with the large particle multiplicities anticipated for heavy-ion
collisions. A detailed description of the detector elements can be found in
the corresponding Technical Design Reports [5, 6, 7, 8]. The central element
of CMS is the magnet, a 13 m long, 6 m diameter, high-field solenoid (a
uniform 4 T field) with an internal radius of $\approx 3$ m.

The tracker covers the pseudorapidity region $|\eta|<2.5$ and is composed of
two different types of detectors: silicon pixels and silicon strips. The pixel
detector consists of three barrel layers located at 4, 7, 11 cm from the beam
axis with granularity $150\times 150$ $\mu$m2 and two forward layers with
granularity $150\times 300$ $\mu$m2 located at the distances of 34 and 43 cm
in z-direction from the center of detector. Silicon strip detectors are
divided into inner and outer sections and fill the tracker area from 20 cm to
110 cm (10 layers) in the transverse direction and up to 260 cm (12 layers) in
longitudinal direction.

The hadronic (HCAL) and electromagnetic (ECAL) calorimeters are located inside
the coil (except the forward calorimeter) and cover (including the forward
calorimeter) from $-$5.2 to 5.2 in pseudorapidity. The HF calorimeter covers
the region $3<|\eta|<5.2$.

The CMS muon stations cover the pseudorapidity region $|\eta|<2.4$ and consist
of drift tube chambers (DT) in the barrel region (MB), $|\eta|<1.2$, cathode
strip chambers (CSCs) in the endcap regions (ME), $0.9<|\eta|<2.4$, and
resistive plate chambers (RPCs) in both barrel and endcaps, for $|\eta|<2.1$.
The RPC detector is dedicated to triggering, while the DT and CSC detectors,
used for precise momentum measurements, also have the capability to self-
trigger up to $|\eta|<2.1$.

Figure 1: CMS coverage for tracking, calorimetry, and muon identification in
pseudo-rapidity ($\eta$) and azimuth ($\phi$). The size of a jet with cone
$R=0.5$ is also depicted for comparison.

Note that CMS is the largest acceptance detector at the LHC (Fig. 1) with
unique detection capabilities also in the very forward hemisphere with the
CASTOR (5.1 $<|\eta|<$ 6.6) and the Zero-Degree (ZDCs, $|\eta_{neut}|>$ 8.3)
calorimeters [9].

Another key aspect of the CMS hard probe capabilities for heavy ion physics is
its unparalleled high-level-trigger (HLT) system running on a filter farm with
an equivalent of ${\cal O}$(104) 1.8 GHz CPU units, yielding few tens of
Tflops [10]. The HLT system is powerful enough to run “offline” algorithms on
every single Pb+Pb event delivered by the level-1 trigger, and select the
interesting events while reducing the data stream from an average 3 kHz L1
input/output event rate down to 10–100 Hz written to permanent storage. The
resulting enhanced statistical reach for hard probes is a factor of $\times$20
to $\times$300 larger, depending on the signal, than for the min-bias (MB)
trigger.

##  3 Simulation and analysis of J/$\psi$ and $\Upsilon$ production

One of the important hard probes at the LHC will be the production of heavy-
quark bound states, which should be suppressed in QGP due to colour screening
[11]. An intriguing phenomenon is the “anomalously” strong suppression of the
J/$\psi$-meson yields, observed in Pb+Pb collisions at SPS [12, 13]. Although
the interpretation of this phenomenon as a result of the formation of a QGP is
quite plausible, alternative explanations have also been put forward, such as
rescattering on co-moving hadrons. The surprisingly similar amount of J/$\psi$
suppression observed at SPS and RHIC energies [14] is not yet fully
understood. Further information on the nature of quarkonia suppression in hot
and dense QCD-matter will come from the results at the LHC at the much higher
temperatures accessible. CMS features the best dimuon mass resolution of any
LHC detector, leading to a clean separation of the various quarkonia states
and an improved signal over background ratio. This fact will open up a unique
opportunity to study the threshold dissociation behaviour of the whole
bottomonium family ($\Upsilon$, $\Upsilon`$, $\Upsilon``$) together with the
charmonium one. Since various quarkonium states are predicted to melt at
different medium temperatures, scan of corresponding suppression factors will
serve as an effective QCD-matter “thermometer”.

Event generator HIJING [15] with full GEANT4-based simulation of the tracking
of secondaries and simulated detector response were used in the analysis
presented. The J/$\psi$ and $\Upsilon$ acceptances on CMS are shown as a
function of $p_{T}$ in Fig. 2 for two $\eta$ ranges: full detector
($|\eta|<2.4$) and central barrel ($|\eta|<0.8$). Because of their relatively
small mass, low momentum J$/\psi$’s ($p<4$ GeV/$c$) are mostly not accepted:
their decay muons do not have enough energy to traverse the calorimeters and
coil, and are absorbed before reaching the muon chambers. The J/$\psi$
acceptance increases with $p_{T}$, flattening out at $\sim$ 15% for $p_{T}>12$
GeV/$c$. The $\Upsilon$ acceptance starts at $\sim\,$40% at $p_{T}=0$ GeV/$c$
and remains constant at $\sim$ 15% (full detector) or 5% (barrel only) for
$p_{T}>4$ GeV/$c$. The $p_{T}$-integrated acceptance is about 1.2% for the
J/$\psi$ and 26% for the $\Upsilon$, assuming the input theoretical
distributions.

In the central barrel of the CMS detector, the dimuon reconstruction
efficiency remains above 80% for all multiplicities whereas the purity
decreases slightly with increasing multiplicities $dN_{ch}/d\eta$ but also
stays above 80% even at as high as $dN_{ch}/d\eta|_{\eta=0}=6500$. If (at
least) one of the muons is detected in the endcaps, the efficiency and purity
drop due to stronger reconstruction cuts. Nevertheless, for the
$dN_{ch}/d\eta|_{\eta=0}\approx 2000$ multiplicity realistically expected in
central Pb+Pb at LHC, the efficiency (purity) remains above 65% (90%) even
including the endcaps.

At the $\Upsilon$ mass, the dimuon mass resolution for muon pairs in the
central barrel, $|\eta|<0.8$, is 54 MeV/$c^{2}$. In the full pseudorapidity
range, the dimuon mass resolution is about 1% of the quarkonium mass: 35
MeV/$c^{2}$ at the J/$\psi$ mass and 86 MeV/$c^{2}$ at the $\Upsilon$ mass.
There is a slight dependence of the mass resolution on the event multiplicity.
Increasing the multiplicity from $dN/d\eta=0$ to 2500 degrades the mass
resolution of the reconstructed $\Upsilon$ from 86 to 90 MeV/$c^{2}$.

Figure 2: J/$\psi$ (top) and $\Upsilon$ (bottom) CMS acceptances (convoluted
with trigger efficiencies) as a function of $p_{T}$, in the full detector
($|\eta|<2.4$, solid line) and only in the central barrel ($|\eta|<0.8$,
dashed line).

Figure 3: Dimuon mass distributions within $|\eta|<2.4$ for Pb+Pb events with
$dN_{ch}/d\eta|_{\eta=0}=5000$ (top) and $dN_{ch}/d\eta|_{\eta=0}=2500$
(bottom) in the J/$\psi$ (left) and $\Upsilon$ (right) mass regions. The main
background contributions are also shown: $h$, $c$ and $b$ stand for $\pi+K$,
charm, and bottom decay muons, respectively.

Fig. 3 shows the opposite-sign dimuon mass distributions, for the high and low
multiplicity cases and full acceptance ($|\eta|<2.4$). The different quarkonia
resonances appear on top of a continuum due to the various sources of decay
muons: $\pi+K$, charm and bottom decays. Assuming that the CMS trigger and
acceptance conditions treat opposite-sign and like-sign muon pairs equally,
the combinatorial like-sign background can be subtracted from the opposite-
sign dimuon mass distribution, giving us a better access to the quarkonia
decay signals. The statistics of J/$\psi$ and $\Upsilon$,
$\Upsilon^{{}^{\prime}}$ and $\Upsilon^{{}^{\prime\prime}}$ with both muons in
$|\eta|<2.4$ region expected in one month of data taking are 140000, 20000,
5900 and 3500 correspondingly for the multiplicity
$dN_{ch}/d\eta|_{\eta=0}=5000$ and 180000, 25000, 7300, 4400 for the
multiplicity $dN_{ch}/d\eta|_{\eta=0}=2500$. The signal-to-background ratios
are 0.6, 0.07 for J/$\psi$ and $\Upsilon$’s for $dN_{ch}/d\eta|_{\eta=0}=5000$
correspondingly and 1.2, 0.12 for $dN_{ch}/d\eta|_{\eta=0}=2500$. The signal-
to-background ratio (the number of events) collected in one month for the
dimuons in J/$\psi$ and $\Upsilon$ mass regions with both particles in
$|\eta|<0.8$ region are 2.75 (12600) and 0.52 (6000) for
$dN_{ch}/d\eta|_{\eta=0}=5000$ and 4.5 (11600), 0.97 (6400) for
$dN_{ch}/d\eta|_{\eta=0}=2500$. The background and reconstructed resonance
numbers are in a mass interval $\pm\sigma$, where $\sigma$ is the mass
resolution.

These quantities have been calculated for an integrated luminosity of 0.5 nb-1
assuming an average luminosity $\cal L$ = $4\times 10^{26}$ cm-2s-1 and a
machine efficiency of 50%. The expected statistics are large enough to allow
further offline analysis for example in correlation with the centrality of the
collision or the transverse momentum of the resonance.

##  4 Photon-tagged jet production

Another important hard probe very sensitive to the initial conditions of the
produced QCD-matter in heavy-ion collisions, is the QCD jet production. It is
expected that final state in-medium interactions should reduce the energy of
the jet partons (“jet quenching”) and result in medium-modified jet
fragmentation [16]. Recent RHIC data on high-pT hadron production [17] are
consistent with jet quenching predictions. However full event-by-event
reconstruction of jets in heavy ion collisions is rather complicated in the
lower energy RHIC experiments. In CMS, large transverse momentum probes can be
isolated experimentally from the soft particle background of the collision. In
particular, full jet reconstruction and high-pT particle reconstruction in the
high multiplicity environment of a Pb+Pb collision are possible [1]. At the
LHC, the production rates for jet pairs with transverse energy $E_{T}>50$ GeV
are several orders of magnitude larger than at RHIC. Thus, high statistics
systematic studies will be possible in a controlled perturbative regime, far
beyond the limits of RHIC. The one of the important jet-related observables
accessible to study in heavy ion collisions at CMS will be the $\gamma$-jet
(and $Z$-jet) channel provides a very clean means to determine medium-modified
parton fragmentation functions (FFs) [18]. Since the prompt $\gamma$ is not
affected by final-state interactions, its transverse energy ($E_{T}^{\gamma}$)
can be used as a proxy of the away-side parton energy ($E_{T}^{\rm jet}\approx
E_{T}^{\gamma}$) before any jet quenching has taken place in the medium. The
FF, i.e. the distribution of hadron momenta, $1/{N_{\rm jets}}\,dN/dz$,
relative to that of the parent parton $E_{T}^{\rm jet}$, can be constructed
using $z=p_{T}/E_{T}^{\gamma}$ or, similarly, $\xi=-\ln
z=\ln(E_{T}^{\gamma}/p_{T})$, for all particles with momentum $p_{T}$
associated with the jet.

Figure 4: Generated (histogram) and reconstructed (points) fragmentation
functions as a function of $z$ (left) and $\xi$ (right) for quenched partons.
Statistical errors correspond to an integrated luminosity of 0.5 nb-1. The
estimated systematic error is represented as the shaded band. Figure 5:
Generated (histogram) and reconstructed (points) ratio of quenched (PYQUEN)
and unquenched (PYTHIA) fragmentation functions as a function of $\xi$.

Full CMS simulation-reconstruction studies of the $\gamma$-jet channel have
been carried out [19], where the isolated $\gamma$ is identified in ECAL
($R_{isol}$ = 0.5), the away-side jet axis ($\Delta\phi_{\gamma-jet}>$ 3 rad)
is reconstructed in ECAL+HCAL, and the momenta of hadrons around the jet-axis
($R_{jet}<$ 0.5) are measured in the tracker. The event generators PYTHIA [20]
(non-quenched partons) and PYQUEN [21] (quenched partons) were used to
simulate signal $\gamma$+jet events, and HYDJET [21] was used to model the
underlying (background) heavy ion event (also without or with jet quenching).
For this study, the $10$% most central Pb+Pb collisions were selected by the
impact parameter of the lead nuclei, yielding an average mid-pseudorapidity
density of about 2400 (2200) charged particles in the quenched (unquenched)
HYDJET background. In total, 4000 $\gamma$-jet events in the CMS acceptance
for $E_{T}^{\gamma}>70$ GeV and $\mid{\eta^{\gamma}}\mid<2$ and about 40000
(125000) QCD background events for the quenched (unquenched) case are
simulated. This corresponds to the expected yields for one running year of
Pb+Pb data taking with an integrated luminosity of 0.5 nb-1. The working point
for this analysis was set to 60% signal efficiency, leading to a background
rejection of about 96.5%, and to a signal-to-background ratio of $4.5$ for
$0-10$% central quenched Pb+Pb.

The obtained FFs for photon-jet events with $E_{T}^{\gamma}>70$ GeV — after
subtraction of the underlying-event tracks using a R = 0.5 cone transverse to
the jet — are shown in Fig. 4 for central quenched Pb+Pb collisions. Medium
modified FFs are measurable with high significance (the systematic
uncertainties being dominated by the low jet reconstruction efficiency for
$E_{T}^{\rm jet}$ = 30–70 GeV) in the ranges $z<$ 0.7 or 0.2 $<\xi<$ 5\. The
overall capability to measure the medium-induced modification of jet
fragmentation functions in the $\gamma$+jet channel can be illustrated by
comparing the fully reconstructed quenched fragmentation function to the
unquenched MC truth distribution (Fig. 5).

##  5 Summary

With its large acceptance, nearly hermetic fine granularity hadronic and
electromagnetic calorimetry, and good muon and tracking systems, CMS is an
excellent device for the study of hard probes (such as quarkonia, jets,
photons and high-pT hadrons) in heavy ion collisions at the LHC.

##  6 Acknowledgments

The author wishes to express the gratitude to the members of CMS Collaboration
for providing the materials and the organizers of the Workshop “High-pT
Physics at LHC” for the warm welcome and the hospitality. The author also
gratefully acknowledge support from Russian Foundation for Basic Research
(grants No 08-02-91001 and No 08-02-92496) and Grants of President of Russian
Federation (No 1007.2008.2 and No 1456.2008.2).

## References

  * [1] D. d’Enterria (ed.) et al. (CMS Collaboration), _J. Phys._ G 34 (2007) 2307.
  * [2] F. Sikler (CMS Collaboration), in these Proceedings.
  * [3] K. Krajczar (CMS Collaboration), in these Proceedings.
  * [4] A. de Roeck (ed.) et al.(CMS Collaboration), _J. Phys._ G 34 (2007) 995.
  * [5] _CMS HCAL Technical Design Report_ , CERN/LHCC 97-31, 1997.
  * [6] _CMS MUON Technical Design Report_ , CERN/LHCC 97-32, 1997.
  * [7] _CMS ECAL Technical Design Report_ , CERN/LHCC 97-33, 1997.
  * [8] _CMS Tracker Technical Design Report_ , CERN/LHCC 98-6, 1998.
  * [9] M. Albrow et al. (CMS/TOTEM Collaborations), CERN-LHCC-2006-039/G-124, 2006.
  * [10] G. Roland et al. (CMS Collaboration), _J. Phys._ G 34 (2007) S733.
  * [11] T. Matsui and H. Satz, _Phys. Lett._ B 178 (1986) 416\.
  * [12] B. Alessandro et al. (NA50 Collaboration), _Eur. Phys. J._ C 39 (2005) 335.
  * [13] B. Alessandro et al. (NA50 Collaboration), _Eur. Phys. J._ C 49 (2007) 559.
  * [14] A. Adare et al. (PHENIX Collaboration), _Phys. Rev. Lett._ 98 (2007) 232301.
  * [15] M. Gyulassy and X.-N. Wang, _Comput. Phys. Commun._ 83 (1994) 307.
  * [16] R. Baier, D. Schiff and B.G. Zakharov, _Annual Rev. Nucl. Part. Sci._ 50 (2000) 37.
  * [17] _RHIC White Papers_ , _Nucl. Phys._ A 757 (2005) 28.
  * [18] F. Arleo et al. _JHEP_ 0411 (2004) 009.
  * [19] C. Loizides (CMS Collaboration), arXiv:0804.3679.
  * [20] T. Sjostrand, S. Mrenna and P. Skands, _JHEP_ 0605 (2006) 026\.
  * [21] I.P. Lokhtin and A.M. Snigirev, _Eur. Phys. J._ C 45 (2006) 211\.


#  Room-temperature biphoton source with a spectral brightness near the
ultimate limit

Jia-Mou Chen,1 Chia-Yu Hsu,1 Wei-Kai Huang,1 Shih-Si Hsiao,1 Fu-Chen Huang,1
Yi-Hsin Chen,2,5 Chih-Sung Chuu,1,5 Ying-Cheng Chen,3,5 Yong-Fan Chen,4,5 and
Ite A. Yu1,5<EMAIL_ADDRESS>1Department of Physics, National Tsing Hua
University, Hsinchu 30013, Taiwan
2Department of Physics, National Sun Yat-Sen University, Kaohsiung 80424,
Taiwan
3Institute of Atomic and Molecular Sciences, Academia Sinica, Taipei 10617,
Taiwan
4Department of Physics, National Cheng Kung University, Tainan 70101, Taiwan
5Center for Quantum Technology, Hsinchu 30013, Taiwan

###### Abstract

The biphotons, generated from a hot atomic vapor via the process of
spontaneous four-wave mixing (SFWM), have the following merits: stable and
tunable frequencies as well as linewidth. Such merits are very useful in the
applications of long-distance quantum communication. However, the hot-atom
SFWM biphoton sources previously had far lower values of generation rate per
linewidth, i.e., spectral brightness, as compared with the sources of
biphotons generated by the spontaneous parametric down conversion (SPDC)
process. Here, we report a hot-atom SFWM source of biphotons with a linewidth
of 960 kHz and a generation rate of 3.7$\times$$10^{5}$ pairs/s. The high
generation rate, together with the narrow linewidth, results in a spectral
brightness of 3.8$\times$$10^{5}$ pairs/s/MHz, which is 17 times of the
previous best result with atomic vapors and also better than all known results
with all kinds of media. The all-copropagating scheme together with a large
optical depth (OD) of the atomic vapor is the key improvement, enabling the
achieved spectral brightness to be about one quarter of the ultimate limit.
Furthermore, this biphoton source had a signal-to-background ratio (SBR) of
2.7, which violated the Cauchy-Schwartz inequality for classical light by
about 3.6 folds. Although an increasing spectral brightness usually leads to a
decreasing SBR, our systematic study indicates that both of the present
spectral brightness and SBR can be enhanced by further increasing the OD. This
work demonstrates a significant advancement and provides useful knowledge in
the quantum technology using photons.

##  I Introduction

Biphotons are pairs of time-correlated single photons, and the sum frequency
and delay time of the two photons are entangled TEentanglement . One photon of
a pair can trigger or start an operation of a quantum process or device, and
then the other photon in the same pair will be employed in the operation. The
two photons are regarded as the heralding and heralded single photons, which
are convenient and versatile in quantum information manipulation QI1 ; QD1 ;
QD2 ; QD3 ; QD4 . Therefore, biphotons have been widely utilized in
applications, such as quantum memory, quantum communication, quantum
interference, and entanglement generation QM1 ; QM2 ; QM3 ; QM4 ; QC1 ; QC2 ;
QC3 ; interference1 ; interference2 ; interference3 ; Entangle1 ; Entangle2 .

Two major physical mechanisms are used to produce biphotons: the spontaneous
parametric down conversion (SPDC) SPDCReview1 ; SPDCReview2 ; SPDC.Theory ;
brightSPDC5 ; brightSPDC6 ; narrowSPDC1 ; narrowSPDC2 ; narrowSPDC3 ;
narrowSPDC4 ; brightSPDC1 ; brightSPDC2 ; brightSPDC3 ; brightSPDC4  and
spontaneous four-wave mixing (SFWM) SFWMReview1 ; SFWMReview2 ; SFWM.Theory ;
narrowSFWM1 ; narrowSFWM2 ; narrowSFWM3 ; ColdAtoms1 ; ColdAtoms2 ; ColdAtoms3
; ColdAtoms4 ; SFWM.dLambda.HotAtom1 ; SFWM.dLambda.HotAtom2 ;
SFWM.dLambda.HotAtom3 ; SFWM.dLambda.HotAtom4 ; SFWM.dLambda.HotAtom5 ;
SFWM.dLambda.HotAtom6 ; OurOPEX2021 . The SPDC is often employed for nonlinear
crystals. An SPDC biphoton source can have a generation rate larger than 106
pairs/s. With the assistance of an optical cavity, the linewidth of SPDC
biphotons can be narrowed down to a few MHz brightSPDC5 ; brightSPDC6 ;
narrowSPDC1 ; narrowSPDC2 ; narrowSPDC3 ; narrowSPDC4 . The SFWM is often used
for cold or hot atomic vapors. A cold-atom SFWM source can produce biphotons
with a sub-MHz linewidth, because of the low decoherence rate in the cold-atom
system narrowSFWM1 ; narrowSFWM2 ; narrowSFWM3 . However, the cold-atom SFWM
source requires a timing sequence to switch the magneto-optical trap off and
on, before and after the biphoton generation. Considering the duty cycle of
the timing sequence, the SFWM biphoton source of cold atoms typically has a
generation rate in the order of $10^{2}$ pairs/s ColdAtoms1 ; ColdAtoms2 ;
ColdAtoms3 ; ColdAtoms4 .

The SFWM biphoton source of hot atoms has the following merits. The frequency
of its biphotons, being locked to lasers’ frequencies due to the SFWM
transition scheme, are more stable than the frequency of the SPDC biphotons,
being locked to an optical cavity’s temperature. The frequency of the hot-atom
SFWM photons is continuously tunable in a range of 0.6 GHz or larger, based on
the Doppler width of the atoms. This tuning range is comparable to that of the
SPDC biphoton source, and it is significantly larger than that of the cold-
atom SFWM biphoton source. The linewidth of the hot-atom SFWM biphotons can be
varied for more than an order of magnitude and narrowed down to 290 kHz
OurOPEX2021 , while the linewidth of SPDC biphotons is fixed by the optical
cavity. Thus, the SFWM biphoton source of hot atoms is versatile in the
application of long-distance quantum communication.

A biphoton source of a larger generation rate can produce information carriers
faster to enable a higher bandwidth in the information transmission. A quantum
operation utilizing biphotons of a narrower linewidth as information carriers
can usually achieve a better efficiency. Thus, the generation rate per
linewidth, i.e., the spectral brightness, is the combined measure of the
generation rate and linewidth. The spectral brightness is an important figure
of merit of a biphoton source, and it strongly influences the success rate of
quantum communication. Among the single-mode SPDC biphoton sources, Luo et al.
reported a 66-MHz biphoton source with the spectral brightness of 3$\times$
$10^{5}$ pairs/s/MHz brightSPDC2 , while Tsai et al. reported a 6-MHz biphoton
source with the spectral brightness of 2.2$\times$ $10^{5}$ pairs/s/MHz
brightSPDC3 . As for the hot-atom SFWM biphoton sources, Zhu et al. reported a
2.9-MHz biphoton source with the spectral brightness of 1.4$\times$ $10^{4}$
pairs/s/MHz SFWM.dLambda.HotAtom2 , and Hsu et al. reported a 630-kHz biphoton
source with the spectral brightness of 2.3$\times$ $10^{4}$ pairs/s/MHz
OurOPEX2021 . Prior to this work, the hot-atom SFWM biphoton source had a far
lower spectral brightness, compared with the SPDC biphoton source.

The biphoton sources made of integrated photonics devices with micro-
resonators or waveguides have been developed recently IPD1 ; IPD2 ; IPD3 ;
IPD4 ; IPD5 ; IPD6 ; IPD7 ; IPD8 ; IPD9 ; IPD10 ; IPD11 ; IPD12 ; IPDexample ;
IPD.Review1 ; IPD.Review2 ; IPD.Review3 . Either the SFWM or SPDC process can
be employed in the generation scheme. Such a biphoton source has a large
generation rate due to the high $Q$-factor of a resonator or the strong mode
confinement of a waveguide, and also a high signal-to-background ratio (SBR)
or coincidence-to-accidental ratio (CAR) IPD.Review1 ; IPD.Review2 ;
IPD.Review3 ; background . The linewidth of the biphotons emitted from the
device can be as narrow as about 100 MHz. In Ref. IPDexample, , a very large
generation rate of 1.3$\times$107 pairs/s at a high SBR or CAR of about 400
was achieved with a microring-resonator biphoton source, and the biphotons had
a linewidth of 92 MHz. Considering the generation rate and the linewidth, the
spectral brightness of this biphoton source was 1.4$\times$105 pairs/s/MHz.

We introduced the all-copropagating scheme to develop a hot-atom SFWM source.
In the all-copropagating scheme as shown in Fig. 1, the pump and coupling
fields and the anti-Stokes and Stokes photons all propagate in the same
direction and completely overlap. The scheme is nearly free of phase mismatch
and ensures a low decoherence rate. The phase match makes the generation
process efficient, and the low decoherence rate results in a narrow biphoton
linewidth. Thus, a high-rate source of narrow-linewidth biphotons was achieved
in this work. At a vapor cell temperature of 65 ∘C, the source of the 960-kHz
biphotons in this work had a generation rate of 3.7$\times$ $10^{5}$ pairs/s,
which is an order of magnitude better than the previous best result of the
hot-atom SFWM source. The high generation rate, together with the narrow
linewidth, gave a spectral brightness of 3.8$\times$105 pairs/s/MHz, which is
17 times of the previous best result with atomic vapors and also better than
all known results with all kinds of media.

Figure 1: (a) Relevant energy levels and transition scheme in the experiment.
$|1\rangle$, $|2\rangle$, $|3\rangle$, and $|4\rangle$ represent the ground
states of $|5S_{1/2},F=2\rangle$ and $|5S_{1/2},F=1\rangle$ and the excited
states of $|5P_{1/2},F=2\rangle$ and $|5P_{3/2},F=1,2\rangle$ of 87Rb atoms,
respectively. The wavelengths of the pump field and anti-Stokes photons are
about 780 nm, and those of the coupling and Stokes photons are about 795 nm.
Throughout this study, the pump detuning, $\Delta_{p}$, was set to $-2.0$ GHz
with respect to the transition of $|1\rangle\rightarrow|5P_{3/2},F=2\rangle$,
and the coupling field frequency was tuned to the resonance. (b) Schematic
experimental setup. PBS: polarizing beam splitter, DBS: dichroic beam
splitter, GTP: Glan-Thompson polarizer, PMF: polarization-maintained optical
fiber, ETF: etalon filter, and SPCM: single-photon counting module.

The spectral brightness has an ultimate limit, which is explained as follows:
The generation rate is equal to $1/\tau_{d}$, where $\tau_{d}$ is the average
separation time between pairs. While $\tau_{p}$ is the biphoton’s temporal
width, the linewidth is equal to $1/(2\pi\tau_{p})$. Hence, the spectral
brightness or the generation rate divided by the linewidth is given by
$2\pi\tau_{p}/\tau_{d}$. If the value of $2\pi\tau_{p}/\tau_{d}$ gets too
large, biphotons often overlap their wave packets and they are no longer
nonclassical. Consequently, the spectral brightness has an upper limit to
ensure that biphotons are reasonably separated in time and preserve their
nonclassicality. We estimate the limit by using the condition of
$\tau_{p}/\tau_{d}=$ 0.25, i.e., once a pair is generated, the probability of
another pair generated within a time interval of $\tau_{p}$ is 0.25. This
limit is given by $(\pi/2)$$\times$106 pairs/s/MHz. The highest spectral
brightness in this work is 3.8$\times$105 counts/s/MHz, which is about one
quarter of the ultimate limit.

The counter-propagating scheme was commonly used in the previous hot-atom SFWM
sources SFWM.dLambda.HotAtom1 ; SFWM.dLambda.HotAtom2 ; SFWM.dLambda.HotAtom3
; SFWM.dLambda.HotAtom4 ; SFWM.dLambda.HotAtom5 ; SFWM.dLambda.HotAtom6 . In
the counter-propagating scheme, the pump field and anti-Stokes photons
propagate in one direction and the coupling field and Stokes photons propagate
in the opposite direction. Furthermore, the propagation directions of the
fields and single photons can have a small angle separation. The phase
mismatch, $\Delta\phi$, is given by

$\Delta\phi=L\left(\vec{k}_{p}-\vec{k}_{as}+\vec{k}_{c}-\vec{k}_{s}\right)\cdot\hat{z},$
(1)

where $L$ is the length of the medium, and $\vec{k}_{p}$, $\vec{k}_{c}$,
$\vec{k}_{as}$, and $\vec{k}_{s}$ are the wave vectors of the pump and
coupling fields and the anti-Stokes and Stokes photons. Suppose the pointing
stabilities or uncertainties and beam divergence of the fields and single
photons are all $\pm$0.1∘. With $L=75$ mm in this study, it is estimated that
the average value of $\Delta\phi$ within $\pm$0.1∘ in the all-copropagating
scheme is 0.91 rad, and that in the counter-propagating scheme is 3.7 rad.
Based on the theoretical predictions, the generation rate (or the biphoton
temporal width) of $\Delta\phi=0.91$ rad is about 14 times greater (or 2.5
times longer) than that of $\Delta\phi=3.7$ rad. Therefore, the all-
copropagating scheme employed in our biphoton source is the key of a high
generation rate.

Although the all-copropagating scheme was utilized in our previous work of
Ref. OurOPEX2021,  and in this work, the emphases of the previous work and
this work are completely different. In Ref. OurOPEX2021, , we focused on the
biphoton linewidth. Here, we focused on the generation rate of the biphoton
source. The all-copropagating scheme is the key ingredient for the study of
the generation rate in this work, but it is just a good experimental
arrangement for the study of the linewidth in the previous work. Furthermore,
the enhancement of the generation rate by the pump power usually decreases
SBR. In this work, we systematically studied the effect of the atomic vapor
temperature, i.e., the optical depth (OD), on the biphoton source, which was
not done in Ref. OurOPEX2021, . Our study revealed that the OD played an
important role to increase both generation rate and SBR together.

At a spectral brightness of 3.8$\times$ $10^{5}$ pairs/s/MHz, our biphoton
source had a signal-to-background ratio (SBR) of 2.7, which violated the
Cauchy-Schwartz inequality for classical light by about 3.6 folds. The
spectral brightness can be further enhanced by the pump power. However, such
an enhancement by increasing the pump power will make biphotons no longer be
nonclassical. Nevertheless, our systematic study showed that both of the
present spectral brightness and SBR can be further enhanced by optimizing the
vapor cell temperature and pump power. A hot-atom SFWM biphoton source, with a
spectral brightness close to the ultimate limit and a sufficiently large SBR,
is feasible.

##  II Experimental Setup

We carried out the experiment with a glass cell containing only 87Rb atomic
vapor. The inner wall of the cell is coated with paraffin. Two strong fields,
pump and coupling, of the $p$ and $s$ polarizations were applied to the cell.
Biphotons, i.e., pairs of anti-Stokes and Stokes photons, of the $s$ and $p$
polarizations were generated via the process of spontaneous four-wave mixing
(SFWM) as depicted in Fig. 1(a). In the SFWM process, the pump field and anti-
Stokes photon form one Raman transition and the coupling field and Stokes
photon form another. The spontaneous decay rates, denoted as $\Gamma$, of
excited states $|3\rangle$ and $|4\rangle$ differ by 5.5%. For simplicity, we
used the average value of $\Gamma=2\pi\times$5.9 MHz in this study.

The pump (or coupling) field comes from an external-cavity diode laser of
Toptica DL DLC pro 780 (or 795). The $e^{-2}$ full width of the pump field is
1.4 mm and that of the coupling field is 1.5 mm. A hyperfine optical pumping
(HOP) field was employed in this experiment, although its transition is not
shown in Fig. 1(a). The frequency of the HOP field is red-detuned from the
transition of $|5S_{1/2},F=1\rangle$ $\rightarrow$ $|5P_{3/2},F=2\rangle$ by
80 MHz. We set the power of the HOP field to 9 mW, which is strong enough to
empty the population in the state of $|5S_{1/2},F=1\rangle$ driven by the
coupling field. The HOP field has the hollow-shaped beam profile, while the
biphoton is generated in its hollow region. Hence, the HOP-induced decoherence
rate in the SFWM process is negligible.

The pump and coupling fields and the Stokes and anti-Stokes photons all
propagate in the same direction as shown in Fig. 1(b). After the vapor cell,
we first install a dichroic mirror to separate 780 nm light and 795 nm light.
Then, the pump and coupling fields are attenuated by polarization filters and
etalons. The polarization filter consists of a quarter-wave plate, a half-wave
plate, and a polarizer. We put two etalons in series after the polarization
filter. According to the measurements in which the vapor cell was moved away,
the leakages of the pump and coupling fields merely contribute 18 and 1.4
counts/s/mW to the anti-Stokes and Stokes single-photon counting modules
(SPCMs), respectively. The two SPCMs have the dark-count rates of 140$\pm$5
and 220$\pm$20 counts/s.

We estimated the collection efficiencies of anti-Stokes and Stokes photons by
using two laser beams, signal and probe, respectively. The signal (or probe)
beam has the same frequency and polarization as the anti-Stokes (or Stokes)
photons. The $e^{-2}$ full widths of these two beams are 0.6 mm. The signal
(or probe) beam attenuates to 13% (or 20%) after passing through all optics
including the polarization filter and two etalons. Considering the quantum
efficiency of the anti-Stokes (or Stokes) SPCM, the overall collection
efficiency of anti-Stokes (or Stokes) photons is 8.4% (or 13%).

The net linewidth of the two etalons in series for the anti-Stokes (or Stokes)
photons was about 46 (or 43) MHz, and the photons’ transmission through the
etalons was kept to 56$\pm$2% (or 62$\pm$2%) throughout this work. The
precision of all the etalons’ temperature controllers is 1 mK, corresponding
to the change of etalon’s resonance frequency of about 3 MHz. Due to the drift
of the ambient temperature, we typically needed to tune the etalon’s resonance
frequency once per 4 hours.

The important elements, key issues, and new changes of the experimental setup
have been described in this section. Other details can be found in our
previous work of Ref. OurOPEX2021, .

##  III Results and Discussion

We systematically studied the generation rate and linewidth of biphotons and
the background counts as functions of the pump power, $P_{\rm pump}$, and the
vapor cell temperature, $T_{\rm cell}$, [i.e., equivalently the optical depth
(OD) of the atomic vapor]. $P_{\rm pump}$ was varied from 2 to 16 mW, and
$T_{\rm cell}$ ranged between 38 and 65 ∘C. We kept the coupling power to 4.0
mW throughout the study, which corresponds to the Rabi frequency of
5.4$\Gamma$ as determined by the EIT spectrum HotREIT ; HOTEIT ; OurOPEX2021 .
This coupling Rabi frequency enables the biphoton linewidths measured here to
be always around 1 MHz, which is well below the natural linewidths of the
atomic transitions.

Figure 2: Coincidence count of the anti-Stokes and Stokes photons as a
function of the delay time between them is shown in each main plot. Black dots
represent numbers of coincidence counts per time bin accumulated in 120 s,
where the width of time bin is 0.8 ns. The pump and coupling powers are 16 and
4 mW. At the vapor cell temperatures of 65 and 38 ∘C, the detection rates in
(a) and (b) are 4.0$\times$103 and 560 pairs/s, respectively, corresponding to
the generation rates of 3.7$\times$105 and 5.1$\times$104 pairs/s. Green and
red lines are the best fits and theoretical predictions. In the theoretical
calculation, $\alpha=$ 370 in (a) or 93 in (b), $\Omega_{c}$ = 5.4$\Gamma$,
and $\gamma=$ 0.030$\Gamma$ in (a) or 0.020$\Gamma$ in (b). The full width at
the half maximum (FWHM) of the best fit in (a) is 180 ns and that in (b) is
100 ns. In the insets, blue lines are the Fourier transforms of the best fits,
and their FWHMs are 960 kHz and 1.6 MHz.

Representative data of the biphoton wave packet in Figs. 2(a) and 2(b) were
taken at the highest and lowest vapor cell temperatures in this study. The
biphoton wave packet is the measurement of two-photon correlation function,
i.e., coincidence count of anti-Stokes and Stokes photons as a function of the
delay time between the two photons. In Fig. 2(a), the highest generation rate
of 3.7$\times$105 pairs/s of our biphoton source was achieved at the maximum
pump power. To calculate the generation rate, we divided the detection rate by
the total collection efficiency (1.1%) of the photon pair. The linewidth of
the biphoton wave packet is 960 kHz, and its determination method will be
illustrated in the paragraph consisting of Eq. (6). The high generation rate
plus narrow linewidth results in a spectral brightness or generation rate per
linewidth of 3.8$\times$105 pairs/s/MHz, which, to our knowledge, is the
highest among all biphoton sources, including both of the SFWM and SPDC
generation processes.

As compared with Fig. 2(a), Fig. 2(b) shows that with the same pump power the
generation rate is greatly reduced and the linewidth of biphoton wave packet
significantly increases at the lowest temperature. Furthermore, the success
probability of detecting a Stokes photon (excluding the background) upon an
anti-Stokes trigger in (a) is 3.4% and that in (b) is 2.0%. A higher vapor
cell temperature, i.e., a larger OD, helps the success probability.
Nevertheless, the signal-to-background ratio (SBR) of 12 in (b) is much better
than that of 3.1 in (a), where the SBR is the ratio of amplitude (equal to
subtract baseline from peak) count to baseline (i.e., background) count in the
data of biphoton wave packet. A higher biphoton generation rate is usually
accompanied with a smaller SBR, which has been systematically studied and will
be presented later.

Figure 3: (a) The auto-correlation function of anti-Stokes photons,
$g^{(2)}_{as,as}$, and (b) that of Stokes photons, $g^{(2)}_{s,s}$, as
functions of the delay time. Circles are the experimental data, measured at
the condition of the highest generation rate whose representative data are
shown in Fig. 2(a). The accumulation time was 120 s and the width of time bin
was 12.8 ns. Red lines are the best fits, which determine (a)
$g^{(2)}_{as,as}(0)=$ 1.95$\pm$0.03 and the FWHM of 150 ns, and (b)
$g^{(2)}_{s,s}(0)=$ 1.97$\pm$0.07 and the FWHM of 140 ns.

At the highest generation rate achieved in this work, we also measured the
auto-correlation function of the anti-Stokes photons, $g^{(2)}_{as,as}(\tau)$,
and that of the Stokes photons, $g^{(2)}_{s,s}(\tau)$ as shown in Figs. 3(a)
and 3(b), where $\tau$ is the delay time between two single photons that
contributed a coincidence count in the auto-correlation measurement. The solid
lines are the best fits, revealing that the anti-Stokes photons had
$g^{(2)}_{as,as}(0)=$ 1.95$\pm$0.03 with a full width at the half maximum
(FWHM) of 150 ns, and the Stokes photons had $g^{(2)}_{s,s}(0)=$ 1.97$\pm$0.07
with a FWHM of 140 ns. On the other hand, the biphoton wave packet of the
highest generation rate exhibited a SBR (denoted as $R_{\rm SB}$) of
2.7$\pm$0.3 with a FWHM of 180 ns, while the representative data in Fig. 2(a)
had a little higher SBR of 3.1. The cross-correlation function between the
anti-Stokes and Stokes photons, $g^{(2)}_{as,s}(0)$, is given by SBRvsCrossg2

$g^{(2)}_{as,s}(0)=1+R_{\rm SB}.$ (2)

Therefore, the measured values of $g^{(2)}_{as,s}(0)$, $g^{(2)}_{as,as}(0)$,
and $g^{(2)}_{s,s}(0)$ reveal that the Cauchy-Schwartz inequality for
classical light, i.e., $[g^{(2)}_{as,s}(0)]^{2}/[g^{(2)}_{as,as}(0)\cdot
g^{(2)}_{s,s}(0)]\leq 1$, is violated by 3.6 folds. The nonclassicality of
these high-rate and narrow-linewidth biphotons has been clearly demonstrated.

Theoretical predictions of the two-photon correlation function,
$G^{(2)}(\tau)$, are calculated with the formulas shown below SFWM.Theory :

$\displaystyle G^{(2)}(\tau)\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\left|\int_{-\infty}^{\infty}d\delta\frac{e^{-i\delta\tau}}{2\pi}\bar{\kappa}(\delta)\,{\rm
sinc}\\!\left[\bar{\rho}(\delta)\right]\exp\\!\left[i\bar{\rho}(\delta)\right]\right|^{2},$
(3) $\displaystyle\bar{\kappa}(\delta)\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\int_{-\infty}^{\infty}d\omega_{D}\frac{e^{-\omega_{D}^{2}/\Gamma_{D}^{2}}}{\sqrt{\pi}\Gamma_{D}}\left[\frac{\alpha\Gamma}{4}\frac{\Omega_{p}}{\Delta_{p}-\omega_{D}+i\Gamma/2}\right.$
(4) $\displaystyle\times$
$\displaystyle\\!\\!\left.\frac{\Omega_{c}}{\Omega_{c}^{2}-4(\delta+i\gamma)(\delta+\omega_{D}+i\Gamma/2)}\right],$
$\displaystyle\bar{\rho}(\delta)\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\int_{-\infty}^{\infty}d\omega_{D}\frac{e^{-\omega_{D}^{2}/\Gamma_{D}^{2}}}{\sqrt{\pi}\Gamma_{D}}$
(5) $\displaystyle\times$
$\displaystyle\\!\\!\left[\frac{\alpha\Gamma}{2}\frac{\delta+i\gamma}{\Omega_{c}^{2}-4(\delta+i\gamma)(\delta+\omega_{D}+i\Gamma/2)}\right],$

where $\tau$ is the delay time between the anti-Stokes and Stokes photons,
$\alpha$ represents the OD of the entire atoms, $\Omega_{p}$ and $\Omega_{c}$
are the Rabi frequencies of the pump and coupling fields, $\Gamma_{D}$ is the
Doppler width, $\Delta_{p}$ is the pump detuning, and $\gamma$ is the
decoherence rate. The other parameters in the above equations are irrelevant
to the experimental condition, and their physical meanings can be found in
Ref. OurOPEX2021, . In all the theoretical calculations, we set $\Gamma_{D}$
to a fixed value of 55$\Gamma$, corresponding to the mediate temperature in
the study. A 5% change in $\Gamma_{D}$ makes a negligible difference in the
calculation result.

The red lines in Figs. 2(a) and 2(b) are the theoretical predictions. In the
theoretical calculation, we set the experimentally-determined values of
$\alpha$ and $\Omega_{c}$, and varied $\gamma$ in a range which is reasonable
according to the EIT spectrum HOTEIT ; OurOPEX2021 . The values of
$\Delta_{p}$ and $\Omega_{p}$ do not affect the temporal profile calculated
from Eq. (3). The agreement between the experimental data and theoretical
predictions is satisfactory.

The green lines in Figs. 2(a) and 2(b) are the best fits. We employ the
following phenomenological function in the fitting:

$\displaystyle\\!\\!\\!\\!\\!\left\\{A\left[1+\tanh\left(\frac{t-t_{0}}{\tau_{1}}\right)\right]^{p}+\epsilon\right\\}\left[1-{\rm
erf}\left(\frac{t-t_{0}-t_{d}}{\tau_{2}}\right)\right]$ $\displaystyle+B,$ (6)

where ${\rm erf}(\,)$ is the error function, and $A$, $B$, $\epsilon$,
$t_{0}$, $p$, $\tau_{1}$, $\tau_{2}$, and $t_{d}$ are the fitting parameters.
The fittings are rather successful as shown by the two figures. We then
perform the numerical Fourier transform of the square root of the best fit,
and square the transform result to obtain the frequency spectrum as shown in
the inset. The spectral profile of the biphoton wave packet is nearly a
Lorentzian function, and its full width at the half maximum (FWHM) gives the
linewidth.

We systematically measured the generation rate of biphotons as a function of
$P_{\rm pump}$ at different $T_{\rm cell}$’s as shown in Fig. 4(a). The
experimental data are fitted with a straight line of zero interception. All of
the best fits are in good agreement with the data. The good agreement
demonstrates that the generation rate is linearly proportional to the pump
power. This is expected from the formula of two-photon correlation function,
$G^{(2)}(\tau)$, in Eq. (3), since the generation rate is given by $\int
G^{(2)}(\tau)d\tau$ and $G^{(2)}(\tau)\propto\Omega_{p}^{2}$. In Fig. 4(a),
one can also see that at a given pump power a higher cell temperatures, i.e.,
equivalent a larger vapor pressure or optical depth of the atoms, results in a
higher generation rate. During the measurement of Fig. 4(a), the detection (or
generation) rate of our biphoton source fluctuated $\pm$4% in 4 hours without
tuning the etalons. As we set the experiment to the same condition, the day-
to-day fluctuation of the detection rate was about $\pm$6%.

To study how the generate rate quantitatively depends on the OD, we plot the
best fit at $P_{\rm pump}$ of 16 mW determined in Fig. 4(a) against the
measured OD, $\alpha^{\prime}$. The data points are shown by the circles in
Fig. 4(b). The value of $\alpha^{\prime}$ was equal to the minus logarithm of
the transmission at the resonance frequency of $|1\rangle$ $\rightarrow$
$|3\rangle$, and it was determined by the fitting of one-photon absorption
spectrum of a probe field. Hence, $\alpha^{\prime}$ relates to $\alpha$ used
in Eqs. (3)-(5) as the following:

$\alpha^{\prime}=\alpha\frac{\sqrt{\pi}}{2}\frac{\Gamma}{\Gamma_{D}}.$ (7)

We calculate the generation rate, i.e., $\int G^{(2)}(\tau)d\tau$, as a
function of $\alpha$, and then convert $\alpha$ to $\alpha^{\prime}$ by using
Eq. (7). In the calculation, $\gamma$ is set to the mean value of the
decoherence rates determined in Figs. 2(a) and 2(b). The black line in Fig.
4(b) is the theoretical prediction of generation rate against
$\alpha^{\prime}$. The consistency between the theoretical prediction and
experimental data is acceptable.

Figure 4: (a) Generation rate as a function of the pump power. In all the
measurements, we set the coupling powers to 4 mW. Green, cyan, blue, magenta,
and red circles are the experimental data measured at the vapor cell
temperatures of 38, 44, 53, 60, and 65 ∘C. Lines are the best fits. (b) Black
circles are the best fit in (a) at the pump power of 16 mW versus the measured
optical depth, $\alpha^{\prime}$, of each temperature. We only consider the
fitting uncertainty in error bars. Black line is the theoretical prediction
which is proportional to $\int G^{(2)}(\tau)d\tau$, where $G^{(2)}(\tau)$ is
given by Eq. (3). In the calculation of the prediction, $\Omega_{c}$ =
5.4$\Gamma$, $\gamma=$ 0.025$\Gamma$, and $\alpha$ used in Eqs. (3)-(5)
relates to $\alpha^{\prime}$ according to Eq. (7).

In the measurement of Fig. 4(a), we also recorded the number of Stokes-photon
triggers per second, i.e., the trigger rate $R_{t}$. The data show $R_{t}$ is
linearly proportional to $P_{\rm pump}$, i.e., $R_{t}=k_{t}P_{\rm pump}$. We
fit the data, and obtain $k_{t}\approx$ (1.8, 2.6, 4.1, 6.2, and 7.6)$\times$
$10^{3}$ counts/s/mW corresponding to the five $T_{\rm cell}$’s from low to
high, respectively. Furthermore, $R_{t}$ is linearly proportional to the OD,
i.e., $k_{t}\approx$ (1.25$\times$ $10^{3}$)$\alpha^{\prime}$ counts/s/mW. The
linear dependences of the trigger rate on the pump power and optical depth are
the expected results of the two-photon Raman transition.

The background count or the baseline in a biphoton wave packet, e.g., 36
counts/bin during the accumulation time of 120 s in Fig. 2(b), can be
contributed from the four major sources: the SPCM’s dark counts, leakage of
the coupling field, fluorescence photons induced by the coupling field, and
untriggered Stokes photons. The untriggered Stokes photon, whose corresponding
anti-Stokes photon of the same pair is unable to make a trigger, contributes a
count due to being detected by the SPCM, after a trigger is made by the anti-
Stokes photon of a different pair. We use the trigger rate of Fig. 2(b) to
estimate the background counts contributed from the first three sources. Since
the SPCM’s dark counts and the coupling leakage produce the backgrounds of
0.60 and 0.015 counts/bin per 120 s, respectively, they are completely
negligible. Besides, the HOP field optically pumps the population out of the
ground state driven by the coupling field, and it was employed in the
experiment. The fluorescence photons, induced by the coupling field and
transmitting through the polarization and etalon filters, contributed 1.4
counts/bin per 120 s, which is also negligible. Thus, the background of a
biphoton wave packet is mostly contributed by the untriggered Stokes photons.

To verify that the counts of untriggered Stokes photons dominate the
background, we plot the background count rate per time bin as a function of
the pump power square ($P_{\rm pump}^{2}$) in Fig. 5(a). Biphotons or pairs of
anti-Stokes and Stokes photons are produced randomly in time. The rate of two
pairs produced within a given time window, $\Delta t$, is proportional to
$[\int G^{(2)}(\tau)d\tau]^{2}\Delta t$, where $\Delta t$ for example is 1.6
$\mu$s in Fig. 2 and $G^{(2)}(\tau)$ is given by Eq. (3). As one of these two
pairs makes a trigger and the other contributes a coincidence count, this
coincidence count of the untriggered Stokes photon becomes the background
Since $G^{(2)}(\tau)\propto P_{\rm pump}$, the background count rate is
linearly proportional to $P_{\rm pump}^{2}$. We fit each data set of a given
$T_{\rm cell}$ with a straight line with a nonzero interception. The nonzero
interception is resulted from the leakage and fluorescence produced by the
coupling field and the SPCM’s dark counts. The consistency between the
straight line and data demonstrates that the background is predominately
contributed from the counts of untriggered Stokes photons.

Figure 5: (a) Rate of the background count per time bin as a function of the
square of the pump power. The biphoton data and legends here are the same as
those in Fig. 4. Lines are the best fits of straight lines with nonzero
interceptions. (b)Black circles are the best fit in (a) at the pump power of
16 mW versus the measured optical depth, $\alpha^{\prime}$, of each
temperature. We only consider the fitting uncertainty in the error bars. Black
line is the theoretical prediction which is proportional to $[\int
G^{(2)}(\tau)d\tau]^{2}$, where $G^{(2)}(\tau)$ is given by Eq. (3). The
calculation parameters of the prediction are the same as those in Fig. 4.

There is another evidence showing that the rate of two biphoton pairs
generated within the triggered time window of 1.6 $\mu$s was significant.
Similar to the trigger rate ($R_{t}$) mentioned previously, the anti-Stokes
photon count rate ($R_{s}$) behaves like $R_{s}=k_{s}P_{\rm pump}$, where
$k_{s}\approx$ (1.9, 2.9, 4.6, 7.2, and 9.2)$\times$ $10^{3}$ counts/s/mW
corresponding to the five $T_{\rm cell}$’s from low to high, respectively.
Thus, $R_{s}$ is significantly larger than $R_{t}$, indicating it happened
many times that the anti-Stokes photon in the first pair contributed a count
to make a trigger, and that in the second pair contributed another count
during the triggered time window. As expected, the difference between $R_{s}$
and $R_{t}$ is also linearly proportional to $P_{\rm pump}^{2}$.

In Fig. 5(b), we show the background count rate per time bin as a function of
the measured OD. The black circles are the best fits at $P_{\rm pump}$ of 16
mW determined in Fig. 5(a). The black line is the theoretical prediction,
which is proportional to $[\int G^{(2)}(\tau)d\tau]^{2}$ as mentioned earlier.
We use the same values of $\Omega_{c}$, $\gamma$, and $\alpha$ (or
equivalently $\alpha^{\prime}$) to calculate the prediction as those used in
Fig. 4(b). The agreement between the prediction and data demonstrates again
that the background is governed by the counts of untriggered Stokes photons.

The singal-to-background ratio (SBR) of the biphoton wave packet relates to
the ratio of the generation rate to the background count rate. Figure 4(a)
experimentally verified that the biphoton generation rate is proportional to
the pump power ($P_{\rm pump}$). Figure 5(a) experimentally verified that the
background count rate is proportional to $P_{\rm pump}^{2}$ with some small
interception, where the interception was the contribution from the sources
other than the untriggered Stokes photons. Thus, one can predict that the SBR
as a function of $P_{\rm pump}$ should behave like

${\rm SBR}=A_{1}\frac{P_{\rm pump}}{P_{\rm pump}^{2}+B_{1}},$ (8)

where $A_{1}$ and $B_{1}$ are constants independent of $P_{\rm pump}$. In Fig.
6(a), the circles represent the experimental data of SBR as a function of
$P_{\rm pump}$ taken at the five temperatures, and the lines are the best fits
of the above function of $P_{\rm pump}$. The well fittings verify the
prediction. While Fig. 4(a) shows that either a larger pump power or a higher
temperature, i.e. a larger OD, makes a larger generation rate, Fig. 6(a)
reveals that it causes a smaller SBR.

Figure 6: Characteristics of the biphoton source as functions of the pump
power: (a) signal-to-background ratio (SBR) in the logarithmic scale, (b)
spectral linewidth, (c) spectral brightness, and (d) product of spectral
brightness and SBR (denoted as $S$). In all the measurements, we set the
coupling powers to 4 mW. Green, cyan, blue, magenta, and red circles are the
experimental data taken at 38, 44, 53, 60, and 65 ∘C. Lines are the best fits,
and Eqs. (8), (10), (11), and (12) are the fitting functions for the data in
(a), (b), (c), and (d), respectively.

For each biphoton data as the examples shown in the main plots of Fig. 2, we
determine the spectral linewidth by the procedure illustrated in the
discussion of Fig. 2. The circles in Fig. 6(b) are the experimental data of
spectral linewidth as a function of $P_{\rm pump}$. According to Eq. (3), the
frequency spectrum, $F(\delta)$, of the biphoton wave packet is given by

$F(\delta)=\left|\bar{\kappa}(\delta)\,{\rm
sinc}\\!\left[\bar{\rho}(\delta)\right]\exp\\!\left[i\bar{\rho}(\delta)\right]\right|^{2},$
(9)

where $\delta$ is the two-photon detuning of the Raman transition between the
Stokes photon and coupling field, or equivalently the frequency deviation of
the Stokes (or anti-Stokes) photon with respect to the resonance. In
principle, the linewidth of $F(\delta)$ is independent of the pump Rabi
frequency $\Omega_{p}$ or $P_{\rm pump}$ based on $\bar{\kappa}(\delta)$ and
$\bar{\rho}(\delta)$ in Eqs. (4) and (5). However, the presence of the pump
field can slightly increase the decoherence rate, $\gamma$, in the system QM2
, which is not considered in Eq. (3). A larger $P_{\rm pump}$ results in a
larger $\gamma$, and a larger $\gamma$ in $F(\delta)$ produces a larger
linewidth HotAtomBiphotonLinewidth . Consequently, the spectral linewidth of
the biphotons slightly increases with the pump power, and the data in Fig.
6(b) behave like

${\rm Spectral~{}Linewidth}=A_{2}P_{\rm pump}+B_{2},$ (10)

where $A_{2}$ and $B_{2}$ are constants independent of $P_{\rm pump}$.

The biphoton linewidth decreases with the OD or the vapor cell temperature as
shown by Fig. 6(b). This is expected from Eqs. (5) and (9). A larger OD
results in a narrower EIT bandwidth of $|\\!\exp[i\bar{\rho}(\delta)]|^{2}$,
and plays a negligible role in the bandwidth of $|\bar{\kappa}(\delta)\;{\rm
sinc}[\bar{\rho}(\delta)]|^{2}$. Thus, one can increase the vapor cell
temperature, i.e., the OD, to enhance the generation rate, and also get the
bonus of narrowing the linewidth.

The generation rate per linewidth, i.e., the spectral brightness, is an
important figure of merit of a biphoton source. The circles in Fig. 6(c) are
the experimental data of the spectral brightness as a function of $P_{\rm
pump}$. Considering the generation rate and the linewidth as functions of
$P_{\rm pump}$ discussed earlier, the spectral brightness should behave like

${\rm Spectral~{}Brightness}=A_{3}\frac{P_{\rm pump}}{P_{\rm pump}+B_{3}},$
(11)

where $A_{3}$ and $B_{3}$ are constants independent of $P_{\rm pump}$. The
lines in Fig. 6(c) are the best fits with the above function. The well
fittings clearly demonstrate the behavior of spectral brightness. As mentioned
in the introduction section, the spectral brightness should have an upper
limit to make biphotons well separated, and the limit is about
$(\pi/2)$$\times$106 pairs/s/MHz. In Fig. 6(c), the highest spectral
brightness is 3.8$\times$105 counts/s/MHz, which is about one quarter of this
limit.

Both a larger spectral brightness and a larger signal-to-background ratio
(SBR) make a higher success rate in quantum communication. However, a larger
spectral brightness always accompanies a smaller SBR as shown by Figs. 6(a)
and 6(c), no matter whether the pump power or the vapor cell temperature,
i.e., the OD, is varied. To study the spectral brightness and SBR together, we
define the quantity $S$ which is the product of spectral brightness and SBR.
The circles in Fig. 6(d) represent the experimental data of $S$ as a function
of $P_{\rm pump}$. The data show that $S$ initially increases with $P_{\rm
pump}$ and approaches an asymptotic limit at $P_{\rm pump}>$ 4 mW. A higher
temperature or a larger OD results in a larger asymptotic limit of $S$.

We illustrate the behavior of $S$ versus $P_{\rm pump}$ as follows. The
“signal” of SBR is proportional to the generation rate divided by the
biphoton’s temporal width, or equivalently the generation rate multiplied by
the biphoton’s linewidth. On the other hand, the “background” of SBR is
proportional to the background count rate per time bin. Thus, $S$ is
proportional to the generation rate square divided by the background count
rate per time bin. The generation rate is linearly proportional to $P_{\rm
pump}$ as demonstrated in Fig. 4(a), while the background count rate per time
bin is linearly proportional to $P_{\rm pump}^{2}$ with a small interception
as demonstrated in Fig. 5(a). Thus, we can expect that $S$ as a function of
$P_{\rm pump}$ behaves like

$S=A_{4}\frac{P_{\rm pump}^{2}}{P_{\rm pump}^{2}+B_{4}},$ (12)

where $A_{4}$ and $B_{4}$ are constants independent of $P_{\rm pump}$. The
lines in Fig. 6(d) are the best fits with the above function. The well
fittings verify the behavior. Furthermore, we plot the SBR against the
spectral brightness in Fig. 7. It can be clearly seen that at a given vapor
cell temperature a larger spectral brightness results in a smaller SBR, and
the product of the two is approximately a constant as the pump power becomes
large.

The biphoton wave packet at the spectral brightness of 3.8$\times$105
pairs/s/MHz had the SBR of 2.7. A larger value of SBR is desirable, e.g., a
SBR of 10 indicates that the probability of the single-photon event is 90%.
However, as shown by Fig. 5(d), we operated the biphoton source in the regime
that $S$ is nearly constant. Further increasing the SBR will result in
decreasing the spectral brightness. Nevertheless, it is also indicated by Fig.
5(d) that a larger optical depth, i.e., a higher atomic vapor temperature, can
further enhance the product of spectral brightness and SBR. Suppose the
present vapor cell could be heated to above 65 ∘C, which was the highest
temperature in this work due to the paraffin coating of the cell. Then, a
spectral brightness around the present best value with a SBR better than 10
can be achievable.

An interesting regime below the threshold of an optical parametric oscillator
was studied in Ref. OPO . In the regime that the biphoton model is no longer
valid, the nonlinear dependence of pump power and the linewidth narrowing of
biphotons were observed. To see whether Eq. (3) based on the biphoton model
can still be used in the present study, we examined the experimental data and
condition. The data in Fig. 4(a) reveal that the generation rate of the
biphotons was linearly proportional to the pump power, and there was no
observable sign of nonlinearity even at the highest pump power and the largest
optical depth. As for the largest generation rate achieved in this study, the
average separation between two biphotons was about 16 times longer than their
temporal width, indicating almost all of the biphotons were away from each
other. The data in Fig. 5(b) show that the linewidth of biphotons was slightly
broadened as the pump power increased, and there was no hint of linewidth
narrowing. Besides, no optical cavity was employed in the experiment. In view
of the above examinations, it is certain that Eq. (3) is valid for the
experimental data in this study.

Figure 7: The SBR is plotted against the spectral brightness. Green, cyan,
blue, magenta, and red circles are the experimental data taken at 38, 44, 53,
60, and 65 ∘C. Each line represents the function of SRB $\times$ (spectral
brightness) = $C$, where $C$ is the asymptotic limit, i.e., $A_{4}$ in Eq.
(12), of the best fit in Fig. 6(d).

##  IV Conclusion

Utilizing the all-copropagating scheme in the SFWM process, biphotons were
generated from a paraffin-coated glass cell containing a hot vapor of nearly
all 87Rb atoms. The highest generation rate of our biphoton source was
3.7$\times$ $10^{5}$ pairs/s, which is an order of magnitude better than the
previous best result of all SFWM sources. At the same time, the biphoton wave
packet had a temporal width of 180 ns and a spectral linewidth of 960 kHz. The
high generation rate together with the narrow linewidth resulted in a spectral
brightness of 3.8$\times$ $10^{5}$ pairs/s/MHz, which is the best record to
date among all biphoton sources, including both of the SFWM and SPDC
generation processes. The spectral brightness is an important figure of merit
of a biphoton source, and strongly influences the success rate of quantum
communication. It has an upper limit of ($\pi$/2)$\times$106 counts/s/MHz. The
achieved spectral brightness in this experiment is about one quarter of the
upper limit.

In the experiment, the generation rate, the background count rate, the signal-
to-background ratio (SBR), and the linewidth as functions of the pump power
and the optical depth (OD) have been systematically studied. The functions or
behaviors formed by the experimental data are in good agreement with those
predicted by the theory. Furthermore, both a high spectral brightness and a
large SBR are desirable, but a higher spectral brightness always accompanies a
smaller SBR. The results of this study showed that by increasing the pump
power the product of spectral brightness and SBR approached an asymptotic
limit, and this asymptotic limit increased with the OD. As the spectral
brightness of the biphoton source reached 3.8$\times$ $10^{5}$ pairs/s/MHz,
the SBR decreased to 2.7 due to the asymptotic limit at the highest allowed
temperature of the vapor cell. As long as a vapor cell can be heated to a
higher temperature, both of the spectral brightness and SBR can be enhanced
simultaneously. A biphoton source, with a spectral brightness close to the
ultimate limit and a sufficiently large SBR, is feasible.

In conclusion, the result reported in this work is an important milestone,
showing that the spectral brightness of the SFWM biphotons can approach to the
ultimate limit, and be better than that of the SPDC biphotons. The systematic
study as well as the data shown in Figs. 5(a)-(d) provides useful knowledge of
quantum technology utilizing heralded single photons.

## ACKNOWLEDGMENTS

We thank the Referee who pointed out the relation between the signal-to-
background ratio in biphoton wave packets, and the cross-correlation function
between anti-Stokes and Stokes photons. This work was supported by the
Ministry of Science and Technology of Taiwan, under Grants No.
109-2639-M-007-002-ASP and No. 110-2639-M-007-001-ASP.

## References

  * (1) Y. Mei, Y. Zhou, S. Zhang, J. Li, K. Liao, H. Yan, S.-L. Zhu, and S. Du, Einstein-Podolsky-Rosen Energy-Time Entanglement of Narrowband Biphotons, Phys. Rev. Lett. 124, 010509 (2020).
  * (2) D. Flamini, N. Spagnolo, and F. Sciarrino, Photonic quantum information processing: a review, Rep. Prog. Phys. 82, 016001 (2019).
  * (3) M. D. Lukin, Colloquium: Trapping and manipulating photon states in atomic ensembles, Rev. Mod. Phys. 75, 457 (2003).
  * (4) A. Holleczek, O. Barter, A. Rubenok, J. Dilley, P. B. R. Nisbet-Jones, G. Langfahl-Klabes, G. D. Marshall, C. Sparrow, J. L. O’Brien, K. Poulios, A. Kuhn, and J. C. F. Matthews, Quantum Logic with Cavity Photons from Single Atoms, Phys. Rev. Lett. 117, 023602 (2016).
  * (5) F. Jendrzejewski, S. Eckel, T. G. Tiecke, G. Juzeliūnas, G. K. Campbell, Liang Jiang, and A. V. Gorshkov, Subwavelength-width optical tunnel junctions for ultracold atoms, Phys. Rev. A. 94, 063422 (2016).
  * (6) L. Caspani, C. Xiong, B. J. Eggleton, D. Bajoni, M. Liscidini, M. Galli, R. Morandotti, and D. J. Moss, Integrated sources of photon quantum states based on nonlinear optics, Light Sci. Appl. 6, e17100 (2017).
  * (7) M. P. Hedges, J. J. Longdell, Y. Li, and M. J. Sellars, Efficient quantum memory for light, Nature 465, 1052-1056 (2010).
  * (8) Y.-H. Chen, M.-J. Lee, I.-C. Wang, S. Du, Y.-F. Chen, Y.-C. Chen, and I. A. Yu, Coherent Optical Memory with High Storage Efficiency and Large Fractional Delay, Phys. Rev. Lett. 110, 083601 (2013).
  * (9) Y.-F. Hsiao, P.-J. Tsai, H.-S. Chen, S.-X. Lin, C.-C. Hung, C.-H. Lee, Y.-H. Chen, Y.-F. Chen, I. A. Yu, and Y.-C. Chen, Highly Efficient Coherent Optical Memory Based on Electromagnetically Induced Transparency, Phys. Rev. Lett. 120, 183602 (2018).
  * (10) P.-J. Tsai, Y.-F. Hsiao, and Y.-C. Chen, Quantum storage and manipulation of heralded single photons in atomic memories based on electromagnetically induced transparency, Phys. Rev. Research 2, 033155 (2020).
  * (11) S. Ritter, C. Nölleke, C. Hahn, A. Reiserer, A. Neuzner, M. Uphoff, M. Mücke, E. Figueroa, J. Bochmann, and G. Rempe, An elementary quantum network of single atoms in optical cavities, Nature 484, 195-200 (2012).
  * (12) N. Sangouard, C. Simon, H. de Riedmatten, and N. Gisin, Quantum repeaters based on atomic ensembles and linear optics, Rev. Mod. Phys. 83, 33-80 (2011).
  * (13) K.-F. Chang, T.-P. Wang, C.-Y. Chen, Y.-H. Chen, Y.-S. Wang, Y.-F. Chen, Y.-C. Chen, and I. A. Yu, Low-loss high-fidelity frequency beam splitter with tunable split ratio based on electromagnetically induced transparency, Phys. Rev. Research 3, 013096 (2021).
  * (14) S. Zhang, C. Liu, S. Zhou, C.-S. Chuu, M. M. T. Loy, and S. Du, Coherent Control of Single-Photon Absorption and Reemission in a Two-Level Atomic Ensemble, Phys. Rev. Lett. 109, 263601 (2012).
  * (15) M. Tillmann, S.-H. Tan, S. E. Stoeckl, B. C. Sanders, H. de Guise, R. Heilmann, S. Nolte, A. Szameit, and P. Walther, Generalized Multiphoton Quantum Interference, Phys. Rev. X 5, 041015 (2015).
  * (16) C.-H. Wu, C.-K. Liu, Y.-C. Chen, and C.-S. Chuu, Revival of Quantum Interference by Modulating the Biphotons, Phys. Rev. Lett. 123, 143601 (2019).
  * (17) P. Sekatski, M. Aspelmeyer, and N. Sangouard, Macroscopic Optomechanics from Displaced Single-Photon Entanglement, Phys. Rev. Lett. 112, 080502 (2014).
  * (18) R. Ghobadi, S. Kumar, B. Pepper, D. Bouwmeester, A. I. Lvovsky, and C. Simon, Optomechanical Micro-Macro Entanglement, Phys. Rev. Lett. 112, 080503 (2014).
  * (19) R. Ghosh and L. Mandel, Observation of Nonclassical Effects in the Interference of Two Photons, Phys. Rev. Lett. 59, 1903-1905 (1987).
  * (20) O. Slattery, L. Ma, K. Zong, and X. Tang, Background and review of cavity-enhanced spontaneous parametric down-conversion, J. Resv. Natl. Inst. Stan. 124, 124019 (2019).
  * (21) L. Mandel, Quantum effects in one-photon and two-photon interference, Rev. Mod. Phys. 71, S274-S282 (1999).
  * (22) C.-H. Wu, T.-Y. Wu, Y.-C. Yeh, P.-H. Liu, C.-H. Chang, C.-K. Liu, T. Cheng, and C.-S. Chuu, Bright single photons for light-matter interaction, Phys. Rev. A 96, 023811 (2017).
  * (23) K. Niizeki, K. Ikeda, M. Zheng, X. Xie, K. Okamura, N. Takei, N. Namekata, S. Inoue, H. Kosaka, and T. Horikiri, Ultrabright narrow-band telecom two-photon source for long-distance quantum communication, Appl. Phys. Express 11, 042801 (2018).
  * (24) M. Rambach, A. Nikolova, T. J. Weinhold, and A. G. White, Sub-megahertz linewidth single photon source, APL Photonics 1, 096101 (2016).
  * (25) J. Liu, J. Liu, P. Yu, and G. Zhang, Sub-megahertz narrow-band photon pairs at 606 nm for solid-state quantum memories, APL Photonics 5, 066105 (2020).
  * (26) D. Rieländer, A. Lenhard, M. Mazzera, and H. de Riedmatten, Cavity enhanced telecom heralded single photons for spin-wave solid state quantum memories, New J. Phys. 18, 123013 (2016).
  * (27) J. Fekete, D. Rieländer, M. Cristiani, and H. de Riedmatten, Ultranarrow-Band Photon-Pair Source Compatible with Solid State Quantum Memories and Telecommunication Networks, Phys. Rev. Lett. 110, 220502 (2013).
  * (28) A. Lenhard, M. Bock, C. Becher, S. Kucera, J. Brito, P. Eich, P. Müller, and J. Eschner, Telecom-heralded single-photon absorption by a single atom, Phys. Rev. A 92, 063827 (2015).
  * (29) K.-H. Luo, H. Herrmann, S. Krapick, B. Brecht, R. Ricken, V. Quiring, H. Suche, W. Sohler, and C. Silberhorn, Direct generation of genuine single-longitudinal-mode narrowband photon pairs, New J. Phys. 17, 073039 (2015).
  * (30) P.-J. Tsai and Y.-C. Chen, Ultrabright, narrow-band photon-pair source for atomic quantum memories, Quantum Sci. Technol. 3, 034005 (2018).
  * (31) A. Moqanaki, F. Massa, and P. Walther, Novel single-mode narrow-band photon source of high brightness tuned to cesium D2 line, APL Photonics 4, 090804 (2019).
  * (32) V. Balić, D. A. Braje, P. Kolchin, G. Y. Yin, and S. E. Harris, Generation of Paired Photons with Controllable Waveforms, Phys. Rev. Lett. 94, 183601 (2005).
  * (33) J. K. Thompson, J. Simon, H. Loh, and V. Vuletić, A High-Brightness Source of Narrowband, Identical-Photon Pairs, Science 313, 74-77 (2006).
  * (34) S. Du, J. Wen, and M. H. Rubin, Narrowband biphoton generation near atomic resonance, J. Opt. Soc. Am. B 25, C98-C108 (2008).
  * (35) P. Farrera, G. Heinze, B. Albrecht, M. Ho, M. Chávez, C. Teo, N. Sangouard, and H. de Riedmatten, Generation of single photons with highly tunable wave shape from a cold atomic ensemble, Nat. Commun. 7, 13556 (2016).
  * (36) L. Zhao, X. Guo, Y. Sun, Y. Su, M. M. T. Loy, and S. Du, Shaping the Biphoton Temporal Waveform with Spatial Light Modulation, Phys. Rev. Lett. 115, 193601 (2015).
  * (37) L. Zhao, X. Guo, C. Liu, Y. Sun, M. M. T. Loy, and S. Du, Photon pairs with coherence time exceeding 1 $\mu$s, Optica 1, 84-88 (2014).
  * (38) C. Yang, Z. Gu, P. Chen, Z. Qin, J. F. Chen, and W. Zhang, Tomography of the Temporal-Spectral State of Subnatural-Linewidth Single Photons from Atomic Ensembles, Phys. Rev. Applied 10, 054011 (2018).
  * (39) R. Chinnarasu, C.-Y. Liu, Y.-F. Ding, C.-Y. Lee, T.-H. Hsieh, I. A. Yu, and C.-S. Chuu, Efficient generation of subnatural-linewidth biphotons by controlled quantum interference, Phys. Rev. A 101, 063837 (2020).
  * (40) B. Srivathsan, G. K. Gulati, B. Chng, G. Maslennikov, D. Matsukevich, and C. Kurtsiefer, Narrow Band Source of Transform-Limited Photon Pairs via Four-Wave Mixing in a Cold Atomic Ensemble, Phys. Rev. Lett. 111, 123602 (2013).
  * (41) M. Steiner, V. Leong, M. A. Seidler, A. Cerè, and C. Kurtsiefer, Photon bandwidth dependence of light-matter interaction, Opt. Express 25, 6294-6301 (2017).
  * (42) C. Shu, P. Chen, T. K. A. Chow, L. Zhu, Y. Xiao, M. M. T. Loy, and S. Du, Subnatural-linewidth biphotons from a Doppler-broadened hot atomic vapour cell, Nat. Commun. 7, 12783 (2016).
  * (43) L. Zhu, X. Guo, C. Shu, H. Jeong, and S. Du, Bright narrowband biphoton generation from a hot rubidium atomic vapor cell, Appl. Phys. Lett. 110, 161101 (2017).
  * (44) X. Li, D. Zhang, D. Zhang, L. Hao, H. Chen, Z. Wang, and Y. Zhang, Dressing control of biphoton waveform transitions, Phys. Rev. A 97, 053830 (2018).
  * (45) Y. Liu, K. Li, S. Zhang, H. Fan, W. Li, and Y. Zhang, Generation of correlated biphoton via four wave mixing coexisting with multi-order fluorescence processes, Sci. Rep. 9, 20065 (2019).
  * (46) J. Mika and L. Slodika, High nonclassical correlations of large-bandwidth photon pairs generated in warm atomic vapor, J. Phys. B: At. Mol. Opt. Phys. 53, 145501 (2020).
  * (47) T. Jeong and H. S. Moon, Temporal- and spectral-property measurements of narrowband photon pairs from warm double-$\Lambda$-type atomic ensemble, Opt. Express 28, 3985-3994 (2020).
  * (48) C.-Y. Hsu, Y.-S. Wang, J.-M. Chen, F.-C. Huang, Y.-T. Ke, E. K. Huang, W. Hung, K.-L. Chao, S.-S. Hsiao, Y.-H. Chen, C.-S. Chuu, Y.-C. Chen, Y.-F. Chen, and I. A. Yu, Generation of sub-MHz and spectrally-bright biphotons from hot atomic vapors with a phase mismatch-free scheme, Opt. Express 29, 4632-4644 (2021).
  * (49) E. Engin, D. Bonneau, C. M. Natarajan, A. S. Clark, M. G. Tanner, R. H. Hadfield, S. N. Dorenbos, V. Zwiller, K. Ohira, N. Suzuki, H. Yoshida, N. Iizuka, M. Ezaki, J. L. O’Brien, and M. G. Thompson, Photon pair generation in a silicon micro-ring resonator with reverse bias enhancement, Opt. Express 21, 27826-27834 (2013).
  * (50) P. Sarrafi, E. Y. Zhu, K. Dolgaleva, B. M. Holmes, D. C. Hutchings, J. S. Aitchison, and L. Qian, Continuous-wave quasi-phase-matched waveguide correlated photon pair source on a III-V chip, Appl. Phys. Lett. 103, 251115 (2013).
  * (51) C. Xiong, X. Zhang, A. Mahendra, J. He, D.-Y. Choi, C. J. Chae, D. Marpaung, A. Leinse, R. G. Heideman, M. Hoekman, C. G. H. Roeloffzen, R. M. Oldenbeuving, P. W. L. van Dijk, C. Taddei, P. H. W. Leong, and B. J. Eggleton, Compact and reconfigurable silicon nitride time-bin entanglement circuit, Optica 2, 724-727 (2015).
  * (52) R. Kruse, L. Sansoni, S. Brauner, R. Ricken, C. S. Hamilton, I. Jex, and C. Silberhorn, Dual-path source engineering in integrated quantum optics, Phys. Rev. A 92, 053841 (2015).
  * (53) S. Ramelow, A. Farsi, S. Clemmen, D. Orquiza, K. Luke, M. Lipson, and A. L. Gaeta, Silicon-Nitride Platform for Narrowband Entangled Photon Generation, arXiv:1508.04358 (2015).
  * (54) X. Zhang, Y. Zhang, C. Xiong, and B. J. Eggleton, Correlated photon pair generation in low-loss double-stripe silicon nitride waveguides, J. Opt. 18, 074016 (2016).
  * (55) X. Guo, C. L. Zou, C. Schuck, H. Jung, R. Cheng, and H. X. Tang, Parametric down-conversion photon-pair source on a nanophotonic chip, Light: Sci. Appl. 6, e16249 (2017).
  * (56) C. Ma, X. Wang, V. Anant, A. D. Beyer, M. D. Shaw, and S. Mookherjea, Silicon photonic entangled photon-pair and heralded single photon generation with CAR $>$ 12,000 and $g^{(2)}(0)<$ 0006, Opt. Express 25, 32995-33006 (2017).
  * (57) L. Sansoni, K. H. Luo, C. Eigner, R. Ricken, V. Quiring, H. Herrmann, and C. Silberhorn, A two-channel, spectrally degenerate polarization entangled source on chip, NPJ Quantum Inf. 3, 5 (2017).
  * (58) R. R. Kumar, M. Raevskaia, V. Pogoretskii, Y. Jiao, and H. K. Tsang, Entangled photon pair generation from an InP membrane micro-ring resonator, Appl. Phys. Lett. 114, 021104 (2019).
  * (59) F. Samara, A. Martin, C. Autebert, M. Karpov, T. J. Kippenberg, H. Zbinden, and R. Thew, High-rate photon pairs and sequential Time-Bin entanglement with Si3N4 microring resonators, Opt. Express 27, 19309-19318 (2019).
  * (60) J. Zhao, C. Ma, M. Rüsing, and S. Mookherjea, High Quality Entangled Photon Pair Generation in Periodically Poled Thin-Film Lithium Niobate Waveguides Phys. Rev. Lett. 124, 163603 (2020).
  * (61) T. J. Steiner, J. E. Castro, L. Chang, Q. Dang,W. Xie, J. Norman, J. E. Bowers, and G. Moody, Ultrabright Entangled-Photon-Pair Generation from an AlGaAs-On-Insulator Microring Resonator, Phys. Rev. X Quantum 2, 010337 (2021).
  * (62) L. Caspani, C. Xiong, B. J. Eggleton, D. Bajoni, M. Liscidini, M. Galli, R. Morandotti, and D. J. Moss, Integrated sources of photon quantum states based on nonlinear optics, Light: Sci. Appl. 6, e17100 (2017).
  * (63) G. Moody, L. Chang, T. J. Steiner, and J. E. Bowers, Chip-scale nonlinear photonics for quantum light generation, AVS Quantum Sci. 2, 041702 (2020).
  * (64) Y. Wang and K. D. Jöns and Z. Sun, Integrated photon-pair sources with nonlinear optics, Appl. Phys. Rev. 8, 011314 (2021).
  * (65) A. Christ and C. Silberhorn, Limits on the deterministic creation of pure single-photon states using parametric down-conversion, Phys. Rev. A 85, 023829 (2012).
  * (66) B.-H. Wu, Y.-W. Chuang, Y.-H. Chen, J.-C. Yu, M.-S. Chang, and I. A. Yu, Enhanced spectral profile in the study of Doppler-broadened Rydberg ensembles, Sci. Rep. 7, 9726 (2017).
  * (67) G. Wang, Y.-S. Wang, E. K. Huang, W. Hung, K.-L. Chao, P.-Y. Wu, Y.-H. Chen, and I. A. Yu, Ultranarrow-bandwidth filter based on a thermal EIT medium, Sci. Rep. 8, 7959 (2018).
  * (68) L. Zhao, Y. Su, and S. Du, Narrowband biphoton generation in the group delay regime, Phys. Rev. A 93, 033815 (2016).
  * (69) S.-S. Hsiao, W.-K. Huang, Y.-M. Lin, J.-M. Chen, C.-Y. Hsu, and I. A. Yu, Temporal profile of biphotons generated from a hot atomic vapor and spectrum of electromagnetically induced transparency, arXiv:2203.15242.
  * (70) G. Shafiee, D. V. Strekalov, A. Otterpohl, F. Sedlmeir, G. Schunk, U. Vogl, H. G. L. Schwefel, G. Leuchs, and C. Marquardt, Nonlinear power dependence of the spectral properties of an optical parametric oscillator below threshold in the quantum regime, New J. Phys. 22, 073045 (2020).